{"text":"\n\n\n\\section{A Planning Model for Dialogue}\n\\label{sec:approach}\n\n\n\n\\towrite{\n  Mention somewhere the fact that \n  sometimes the dialogue has to be abandoned, if the user repeatedly\n  misses the point (e.g., they repeatedly provide gibberish input, \n  or reject the recommendations for a predefined number of times).\n  Describe a simple mechanism to implement this.\n}\n\nAI planning problems are often expressed in a domain-independent language from the PDDL family.\nOur PDDL models for different dialogue domains\nshare some commonalities in terms of the high-level design strategy.\nWe present lessons learned when designing dialogue planning models.\n\n\\begin{comment}\n\\subsection{External Context}\n\\label{sec:context}\n\n\\ab{Move context description to preliminaries? \nIt would normally belong there. However, the danger is that\nthat section can become too long, exhausting the reader with preliminary stuff\nbefore they get to the new, interesting stuff.\nWhen more parts of the paper come into place, it will\nbe clearer what to do. I think the context is important\nfor the mapping of plan actions into real actions in \nthe system as well.\n}\n\n\\cm{I'm a bit confused at where this section came from. I viewed the following\nsection (on the level of abstraction) as being the appropriate place for this\nsort of stuff, but at a far reduced size. The paper's meant to cover mainly the\nmodelling aspect of the work, and not the system\/execution monitoring aspect.}\n\n\nThe dialogue orchestrator maintains a so-called dialogue context,\nwhich contains knowledge available in the dialogue.\nThe context offers an important advantage in terms of PDDL modeling.\nSince detailed information is readily available in the context,\nwe can design simplified PDDL models, \nwith no need to redundantly include all details captured in the context.\n\nThe context is implemented as a set of instantiated variables.\nAt the beginning of a dialogue,\nthe context can optionally be initialized with knowledge\nsaved, for example, at the end of the previous conversation with the user at hand.\nOtherwise, a dialogue starts with an empty context.\n\nDuring the dialogue, the context is updated accordingly.\nRecall that user utterances can automatically be annotated with\ninformation such as the intent, and instantiations of variables.\nThese become part of the context.\nFor example, if the system recognizes a place, such as {\\sc New York},\na variable instantiation {\\sc {@place = New York}}\nis automatically added to the annotation.\nThis is further added to the context of the dialogue.\nIf an instantiation of the {\\sc @place} variable was previously present\nin the context, it is replaced with the most recent instantiation.\n\nAs utterances in the dialogue have a timestamp, the history of a dialogue \nallows to reconstruct the history of context snapshots.\nFor simplicity, unless otherwise mentioned, the term context will refer to\nthe most recent snapshot of the context information.\n\\end{comment}\n\n\\subsubsection{Choosing the Level of Abstraction in PDDL}\n\n\nThe availability of the context,\nseparately from a dialogue plan,\neliminates the need to explicitly model in the PDDL problem description\nall objects (i.e., possible values of variables) that could occur in a dialogue.\nThis abstraction avoids an artificial blow up in the problem size,\nand in the solving effort.\n\nAssume, for instance,\nthat at some point in a trip planning dialogue,\nthe destination has been set.\nIn the PDDL modeling, it is sufficient to encode that the destination\nis known, with no need to explicitly name the destination.\nThat is, we use a predicate such as {\\sc have-location-dest}, as opposed to \n{\\sc have-location-dest ?loc}.\nThe latter would have to be instantiated into many grounded fluents,\none for each possible destination.\nIn contrast, the former is grounded into exactly one instantiated fluent,\nwith corresponding savings in the problem size and difficulty.\n\nA PDDL model abstracts away some, but not necessarily all information included in the context.\nAs mentioned in Section~\\ref{sec:architecture},\npart of the predicates used in the PDDL model \nare mirrored with corresponding variables in the context,\nto be able to decide on what branches to continue with the \nexecution of a dialogue plan.\n\n\\begin{comment}\n\\subsection{Basic Actions and Fluents}\n\\label{sec:base-encoding}\n\nHere, we introduce the basic components to the planning model for dialogue which\nare shared across the various domains. \nFurther extensions\nare presented later in Section \\ref{sec:extensions}.\n\\end{comment}\n\n\\subsubsection{Basic Fluents} Following the previous discussion on using the right level\nof abstraction, we introduce the following types of fluents for the PDDL model:\n\n\\begin{tabular}{rl}\n{\\sc ok-*} & To indicate if a Boolean flag holds true. \\\\\n{\\sc have-*} & To indicate we have a context value. \\\\\n{\\sc maybe-*} & To indicate uncertainty of a context value. \\\\\n{\\sc goal} & Specially designated fluent for the goal.\n\\end{tabular}\n\n\\cm{I reduced {\\sc maybe-have-*} to {\\sc maybe-*} for space reasons, but perhaps we could\nuse {\\sc know-*} and {\\sc poss-*} instead of {\\sc have-*} and {\\sc maybe-*} (respectively)\nthroughout the paper.}\n\nFor a context variable, such as {\\sc location-dest}, the fluents {\\sc have-location-dest}\nand {\\sc maybe-location-dest} make a 3-valued logic (at most one can be true). We found\nthe latter mode to be essential for tailored dialogue that responds appropriately to\nuncertain data (e.g., asking ``\\textit{You'll be traveling to Berlin, right?}'' instead\nof ``\\textit{Where will you be traveling to?}'' when {\\sc maybe-location-dest} holds and\nwe have some idea what the location should be).\n\nThe {\\sc goal} fluent captures the fact that we typically achieve the dialogue goal by\nmeans of executing a particular action (e.g., booking a trip or making a successful\nrecommendation). We elaborate on this further in Section \\ref{sec:multi-intent}.\n\n\\subsubsection{Basic Actions}\nWe have identified two key action types that\nare shared across the dialogue domains: dialogue actions and service actions.\nThe model could optionally include other actions, such as an auxiliary action at\nthe end of every plan, to indicate the termination of the dialogue, but the two\npervasive types are what we discuss.\n\n\n\\textit{Dialogue actions} correspond to sending messages to the end-user\nin a conversation.\nWe assume that the executor of a plan has a way to map a given\ndialogue action (along with the current context) to an utterance that should be\nsent to the end user. \nIf the dialogue action has more than one outcome, it is\npresumed to be a message that warrants a response from the user, and the user's\nresponse will correspond to the various action outcomes.\n\n\\ab{The next paragraph might belong to a different section.}\n\nIn the deployed dialogue plans we have created, the execution of a dialogue action\nsends the message to the end-user using a common messaging protocol, and the response\nin situations with more than one outcome is assessed using off-the-shelf NLU technology\n(e.g., services for natural language disambiguation and entity extraction). The effects\nof an outcome for a dialogue action can encode \\emph{whether} various types of information are\navailable.\nSeparately from the dialogue plan,\nthe context will store the actual values of those types of information.\n\nEliciting information from the user is an important feature in multi-turn, goal-oriented\ndialogues. Dedicated fluents encode whether a given type of information has successfully\nbeen elicited. Fluents such as {\\sc have-employee-name} (in a HR \ndialogue where a manager can ask about the performance of various team members)\nand {\\sc have-location-dest}, mentioned\nearlier, are prime examples of this.\nOnce again, during execution, the context is updated to reflect\nthe actual values that have been acquired or modified.\n\nThe preconditions of a dialogue action dictate \\textit{when} such an utterance\nor question would be posed to the user. For example, querying a user's destination location\nonly makes sense if we do not already have it. More subtly, an action such\nas {\\sc ask-user-dest} would be predicated on \\textit{not} having {\\sc maybe-location-dest}\nhold in the state, as we would instead prefer the action {\\sc confirm-user-dest}.\n\n\n\n\n\\textit{Service actions} refer to the actions in the model that do not directly correspond\nto messages that are sent to the end-user. These include system checks that have multiple\noutcomes associated (essentially embedding key components of logic into the process of\ncompiling the dialogue agent) or even web API calls that may be required as part of the\nconversation. An example would be looking up the weather using an online RESTful service.\nThe outcomes of a service action correspond to the possible responses we might expect and\nwish to handle as part of the conversation.\n\nThe specific implementation details are beyond the scope of this paper, but essential to\na service action being used as part of a plan, we assume that the executor is capable of\nmaking the RESTful API calls (or similar such service actions), and resolving the outcome.\nPart of this resolution process is to update the context with new information as appropriate,\nand maintain the corresponding state of the world from the view of the planner's abstraction.\n\n\n\n\\ab{The following two paragraaphs might belong \nto the section on abstraction levels.}\n\nIt is worth emphasizing the role of the outcomes from the dialogue designer's perspective.\nThere may be countless ways that a user could respond to a question, and similarly countless\nerror codes that a RESTful endpoint might return. However, the task of the dialogue designer\nis to \\textit{only specify the outcomes that contribute to changes in state and\/or conversation}.\nThis means that a large variety of possible outcomes are categorized together.\n\nAn example for the service action might be mapping all error codes of the weather service\ninto one {\\sc no-weather-service} outcome.\nAn example for a dialogue question might\nbe all of the ways the user could respond in the affirmative. It was a prevailing design\nphilosophy of the dialogue agent modelling that we should only consider the outcomes that\nare \\textit{required for conversation}, and including a catch-all outcome as needed when\nthe response is unclear (e.g., when the NLU cannot understand the end-user response).\n\n\n\n\n\n\n\\hide{\n\\subsection{Reusing PDDL Actions Across Domains}\n\\cm{I think this is an important point to make, but perhaps not warranting its own (sub-)section.}\n\nReusing PDDL code across domains can further\nboost the benefits constructing dialogue plans\nautomatically, from a library of small building  \nblocks, such as atomic actions.\n\nConsider a dialogue that calls a recommender system,\nand presents the a recommendation to the user.\nIf the recommendation is rejected, \nelicit additional information from the user,\nrefine the recommendation and continue recursively.\n\nSuch a scenario is relevant in multiple domains.\nIn career coaching, recommendations can refer to\na long-term career goal, a career pathway towards the career goal,\nor simply a next job to apply for.\nIn an entertainment domain, recommendations could refer to movies to watch.\n\nPDDL allows to define the building blocks needed for the dialogue\nflow illustrated earlier in this section in a domain-independent manner.\nOnce available, they can automatically be included in any dialogue domain\nusing recommender systems.\nIn each case, they can be combined with other actions, \nresulting in significantly different dialogue plans.\n\n\\ab{The discussion here could be stronger. Leaving it as is for now, to call it a day.}\n}\n\n\n\n\n\\section{Advanced PDDL Features}\n\\label{sec:extensions}\n\n\\ab{I suggest to merge this with the previous section.\nIt's hard to draw a line between what is advanced and what is not.\nOr, if the section would be too long, restructure on another\ncriterion than advanced\/basic.}\n\n\\cm{The split is done assuming the standard form of a theoretical correspondence\nbetween two systems. You don't want to show the entire model (including the syntactic\nsugar that this section represents) and have to draw the correspondence with it all.\nInstead, you present the core ideas, establish the connection, and then show the\nextensions while claiming that it doesn't change the correspondence in any way\n(i.e., the syntactic sugar argument). If we forgo any rigor on the model itself,\nthen I think it makes perfect sense to combine (as long as it still can read well\nwithout reader fatigue).}\n\nHaving the base encoding in hand, we now describe some of the advanced encoding\nfeatures that we have identified and deployed for the dialogue agents we have created.\nThese stem from common patterns observed in addressing the pain points of dialogue\ndesigners.\n\n\\subsubsection{Forced Followup}\n\nGenerally speaking, the declarative nature of planning can offer massive savings\nto the process of dialogue design (and we demonstrate as such later in Section \\ref{sec:eval}).\nThat said, there are some limited forms of imperative-style specification that\nwe found to be common in the domains we have investigated. Almost exclusively,\nthese took the form of immediate followup functionality: examples include responding\nquickly with an affirmation, running a complex service action with many outcomes\nafter a particular response was received, etc. Here, we detail the modelling\nstrategy used for such situations.\n\n\\textit{Forced followup} is a modelling feature that introduces a new set of\nfluents that are incorporated into the actions in a particular way. We use two\nnew fluents: (1) {\\sc forced-followup-$t$} indicates if there is a forced followup\nthat must occur of type $t$; and (2) {\\sc force-reason-$r$} indicates what the\nreason is for the forced followup.\n\nExamples we have considered for type $t$ include {\\sc dialogue} (immediately\nrespond with a message), {\\sc check} (run a system check), and {\\sc abort} (to\nabort the conversation and hand off to a human operator). The type of forced\nfollowup allows us to predicate some subset of the actions with the ability\nto handle the forced followup. If only one action can handle a particular type,\nthen the model should ensure that it is the \\textit{only} applicable action.\nWe achieve this by having the negation of all types that an action $a$ cannot handle\nas a precondition for $a$. For many of the actions, there will be no\ntype of forced followup that they can handle, which means they have a negated\nprecondition for every type (easily specified using quantified preconditions in PDDL).\n\nWe assume that actions which handle a particular type of forced followup\nalways remove the appropriate fluent as part of their effects (i.e., the\n{\\sc forced-followup-$t$} is deleted in every outcome of the action). This\nensures that the remaining actions in the domain are subsequently re-enabled.\n\nThe {\\sc force-reason-$r$} fluents in some sense mirror the {\\sc forced-followup-$t$}\nfluents, as they are both added and deleted at the same time, but they additionally provide a higher\nfidelity to the followup mechanism. As a grounded example, one of the domains\n(discussed later in \nSection~\\ref{sec:eval})\nuses {\\sc dialogue}\nas a type for forced followup with reasons spanning a range of errors (such as\n{\\sc bad-weather}, {\\sc bad-dates}, etc), warnings (e.g., {\\sc no-weather-service}),\nand affirmations (e.g., {\\sc affirm-ok}). The action description in the model\nmakes use of the lifted representation for PDDL, and thus only one action is needed\nto handle the range of forced responses corresponding to each reason. The\nexample action schema for a forced followup of type $dialogue$ would be\n{\\sc (handle-forced-dialogue ?r - reason)}\n\nWe found that the task of declaratively specifying a dialogue agent was greatly\nsimplified by allowing for this single-step imperative pattern to be used directly.\nIt essentially empowers the dialogue designer to specify the immediate followup\nfor key outcomes on certain actions, and additionally had the benefit of simplifying\nthe execution of the dialogue plans (as utterances from the agent to the end-user\nneed not be placed on outcomes).\n\n\n\\subsubsection{Handling Multiple Intents}\n\\label{sec:multi-intent}\n\nIf desired, one can build a dialogue plan with a \ndisjunctive goal. Such a plan would satisfy any\none of a collection of goals (top-level intents)\nconsidered in the disjunctive goal.\nIn such a case,\nwe introduce additional auxiliary fluents and actions to\naddress them: for each intent $i$, we have a fluent {\\sc intent-$i$} that indicates if\nthe user has that intent, and a corresponding action {\\sc assert-intent-$i$} is introduced\nwith the following properties:\n\n\\begin{enumerate}\n  \\item Only applicable when both {\\sc intent-$i$} and the necessary condition for\n        intent $i$ to be satisfied holds.\n  \\item Adds the goal fluent {\\sc goal} as its only effect.\n\\end{enumerate}\n\nIn the domains we have experienced, it is enough to assume that only\na single intent needs to be confirmed.\n\n\\hide{\nA common pattern in creating multi-turn dialogue agents is to model multiple possible\n\\textit{intents} that the user might have.\\footnote{An important distinction\nis that we refer to the top-level intents of a user, and not the common use in dialogue\nliterature i.e., the classification of every end-user utterance as an intent.}\nWhen multiple intents exist, we introduce additional auxiliary fluents and actions to\naddress them: for each intent $i$, we have a fluent {\\sc intent-$i$} that indicates if\nthe user has that intent, and a corresponding action {\\sc assert-intent-$i$} is introduced\nwith the following properties:\n\n\\begin{enumerate}\n  \\item Only applicable when both {\\sc intent-$i$} and the necessary condition for\n        intent $i$ to be satisfied holds.\n  \\item Adds the goal fluent {\\sc goal} as its only effect.\n\\end{enumerate}\n\nThe astute reader will recognize the disjunctive nature of this encoding method: if\nthe user has more than one intent that is recognized, satisfying only one of them would\nsuffice. However, in the domains we have experienced, it is enough to assume that only\na single intent need be confirmed. If multiple intents were of interest simultaneously,\nthen a new intent that represents their combination could be used.\n}\n\n\\section{Architecture Overview}\n\\label{sec:architecture}\n\nHere, we present an architecture that integrates\ndialogue plans into an overall dialogue system,\nstarting with the key definition required for\nmaintaining the dialogue status.\n\n\\begin{definition}[Context]\n    Given a set of variables $\\cal W$,\n    a \\emph{context} is a partial instantiation of $\\cal W$.\n    In other words, a context contains instantiations to a subset \n    of the variables in $\\cal W$.\n\\end{definition}\n\nAs mentioned, the context contains information available in the dialogue system at a given time.\nVariables in $\\cal W$ can include intents, entities, and variables to store\nthe outputs of calls to external services \n(i.e., variables instantiated as described in Section~\\ref{sec:preliminaries}).\nThe context can contain additional variables, with rules about how and when\nto instantiate them.\nFor instance,\na trip planning domain can define variables such as \n{\\sc location-dest} and {\\sc location-orig}.\nDuring the dialogue, the context\ncan assign the automatically recognized value of the {\\sc @place} entity\nto either {\\sc location-dest} or {\\sc location-orig},\ndepending on the state of the dialogue.\n\nEach goal that can be considered in some dialogue plan\nhas a corresponding intent in the dialogue system, called a \\emph{top-level intent}.\nUser utterances classified into a top-level intent\ntrigger the execution of a dialogue plan with the corresponding goal.\n\nRecall\nthat a dialogue plan obtained from a planning system is\na directed graph.\nA unique edge originating from a node represents\na deterministic transition, and multiple edges from a given node\nrepresent the branches of a non-deterministic action.\nEach action in the plan (represented as a string) needs to be mapped into an actual code to execute.\nWe call the code corresponding to an action a \\emph{transformer}.\nAs such, each dialogue trace modelled in the dialogue plan\nhas a corresponding sequence of transformers to call.\nAs a dialogue progresses along a given trace,\nthe context can change after each step.\nFor instance, analysing a user utterance can lead to \nnew entity instantiations.\nLikewise, calling an external service leads to new output.\nA transformer can \nconsume context information (i.e., use context instantiations as input)\nand produce context information (i.e., populate the context with new instantiations).\nWhen the action at hand involves calling an external service\n(e.g., call a career-pathway recommender system),\nthe corresponding transformer takes as an argument the \nlink to the API of the external service.\nThe transformer calls the external service with the input at hand\n(e.g., user profile stored as a context variable)\nand places the results in dedicated context variables.\n\n\\towrite{Modify this accordingly, in case that the definition of a dialog plan changes.}\nConsider a node $n$ in the plan, with multiple outgoing edges,\nto a set of children nodes $c_1, \\dots, c_l$.\nWhen the dialogue continues from a node with \nmultiple outgoing edges (branches), such as $n$,\nthe execution needs to decide what branch to choose.\nThat is, we need a mechanism to observe part of the dialogue state (context) and\nmake a decision based on that observation.\nAt the end of applying the action corresponding to node $n$,\nthe context allows to infer the current planning state.\\footnote{To\n    achieve this, each fluent from the planning problem is also defined as a context variable,\nwith a rule about how to instantiate (evaluate) it to {\\sc true} or {\\sc false}.\n}\nWe use the current planning state, the formulas defined for\nbranches leading to the children nodes $c_1, \\dots, c_l$,\nand the previous planning state (when the execution was at node $n$),\nto infer which branch should be followed.\nWe assume that the effects of exactly one branch are consistent\nwith the transition from the previous planning state to the current one.\n\nConsider the example presented in Figure~\\ref{fig-fsm}.\nThe node corresponding to the action {\\sc ask-how-many} has two outgoing branches, \ncorresponding to two non-deterministic effects of \nwhether the user provides a number or not.\nOne branch is a self loop with no effects (i.e., no number provided)\nand the other progresses to a different state, with the\nnumber of luggage pieces given.\n\n\\hide{\nConditions defined in such a way are evaluated when the execution trace\nreaches the parent state of the corresponding branches.\nThe evaluation is based on context information.\nMore specifically, each PDDL predicate used in a condition has associated an evaluation\nformula based on variables from the context.\nIn our running example, the predicates {\\sc user-accept} and\n{\\sc not-accepted} are evaluated based on \nthe classification (intent) of the most recent utterance\n(acceptance or rejection).\n}\n\nA deeper discussion on monitoring the execution of a plan is beyond the scope of this paper.\n\n\\begin{comment}\n\\subsection{Compiling Plans into Dialogues}\n\\label{sec:pc}\n\n\nDialogues can be conceived as a combination of a small set of functions \\emph{transforming} the dialogue context described in Sect.~\\ref{sec:abstraction}. Such dialogue transformers perform basic tasks such as saying something to the user, or asking a question. In addition, the transformer library provides composite transformers that allow to model complex dialogues in a declarative, compositional way. \n\nOne crucial composite transformer is the finite state machine, allowing to model arbitrary dialogue plans. For example, Figure \\ref{fig:transformers} depicts the translation to dialogue transformers of the plan shown in Figure \\ref{fig-fsm}.\n\n\\newcommand\\jcomm[1]{\\textcolor[rgb]{0.18,0.49,0.19}{\\textit{#1}}}\n\\newcommand\\jstr[1]{\\textcolor[rgb]{0.05,0.27,0.63}{#1}}\n\\begin{figure}\n      \\begin{footnotesize}\n    \\begin{Verbatim}[commandchars=\\\\\\{\\}]\n\\jcomm{\/\/ Initial state}\ninitial(\\jstr{\"get-recomms\"}).call(seq(\n    this::getRecomms,\n    say(this::stringifyRecomms)\n)).jumpTo(\\jstr{\"query-acceptance\"}).always()\n\n\\jcomm{\/\/ Query user acceptance}\non(\\jstr{\"query-acceptance\"}).call(\n    clarify(\\jstr{\"acceptance\"})\n        .asking(\\jstr{\"Accept recommendations?\"})\n        .withExtractor(yesNo())\n).finish().when(getBool(\\jstr{\"acceptance\"}))\n .jumpTo(\\jstr{\"elicit-reject-reason\"}).otherwise()\n\n\\jcomm{\/\/ Elicit reject reason}\non(\\jstr{\"elicit-reject-reason\"}).call(\n    clarify(\\jstr{\"reason\"})\n        .asking(\\jstr{\"Why not?\"})\n        .withExtractor(fromInput())\n).jumpTo(\\jstr{\"get-recomms\"}).always()\n    \\end{Verbatim}\n\\end{footnotesize}\n\\caption{Plan in Figure \\ref{fig-fsm}, as dialogue transformers.}\n\\label{fig:transformers}\n\\end{figure}\n\nFrom the initial state we call the high-level transformer \\texttt{seq} to assemble the two-step sequence [\\texttt{getRecomms}, \\texttt{say(stringifyRecomms)}], thus retrieving recommendations and showing them to the user (using the \\textit{say} transformer). This sequence represents the mapping from the \\texttt{get-recomms} action in the plan to actual dialogue code. The rest of the translation is self-explanatory: the dialogue jumps unconditionally to the \\texttt{query-acceptance} state, thus querying the user. Depending on the intent associated to the answer's utterance (extracted by means of a \\texttt{yesNo} NLU extractor) the dialogue will finalize or go the \\texttt{elicit-reject-reason} state, asking about the rejection reason. From there the dialogue will jump unconditionally to the initial state.\n\nSince plans can be arbitrarily complex, the translation process must be automated. To this end we implemented a compiler that traverses a plan in a depth first fashion, generating the corresponding transformer ensemble along the way. The compiler requires a mapping from plan actions to transformers (e.g., \\texttt{get-recomms} to the sequence [\\texttt{getRecomms}, \\texttt{say(stringifyRecomms)}]), and another from plan predicates to code (e.g., \\texttt{user-accepted} to \\texttt{getBool(\"acceptance\")}). The ability to compile plans leverages dynamic scenarios where a planning service is invoked on the fly, and the resulting plan compiled into a dialogue and executed at runtime.\n\n\\end{comment}\n\n\\section{Dialogue Plans}\n\\label{sec:dial-plans}\n\n\\cm{I think this should be the target output that we shoot for. Keep the connection to planning slim, as that is the main bulk of the approach (i.e., how do we encode things to generate something that maps to our target dialogue plan. Phrase it as, ``a dialogue agent is represented by what we refer to as a dialogue plan...''}\n\n\n\\hide{\n\\begin{definition}\n  A \\emph{dialogue plan} is a structure $\\la \\dialnodes, \\dialedges, \\dialinitnode, \\dialgoals \\ra$.\n  $(\\mathcal{V}, \\mathcal{T})$ is a directed graph,\n  with $\\mathcal{V}$ being the nodes and $\\mathcal{T}$ being the edges.\n  One unique node, labeled $v_0$, represents\n  the \\emph{initial node} of the dialogue.\n  $G \\neq \\emptyset$ is the set of nodes with no outgoing edges,\n  called the goal nodes.\n  Nodes have two labels each: a \\emph{condition label} and an \\emph{action label}.\n  A condition label is a Boolean formula where atoms are fluents from a set $\\mathcal{F}$.\n  An action label is a string.\n\\end{definition}\n}\n\n\\begin{definition}\n  A \\emph{dialogue plan} is a structure $\\la \\dialnodes, \\dialedges, \\dialinitnode, \\dialgoals \\ra$.\n  $(\\mathcal{V}, \\mathcal{T})$ is a directed graph,\n  with $\\mathcal{V}$ being the nodes and $\\mathcal{T}$ being the edges.\n  One unique node, labeled $v_0$, represents\n  the \\emph{initial node} of the dialogue.\n  $G \\neq \\emptyset$ is the set of nodes with no outgoing edges,\n  called the goal nodes.\n  Each node has an \\emph{action label} represented as a string.\n  When a node has multiple outgoing edges, each such an\n  edge has a Boolean formula where atoms are \n  fluents from a set $\\mathcal{F}$.\n\\end{definition}\n\n\n\\hide{\nWe use FOND planning\nto generate dialogue plans.\nComputing dialogue plans with AI planning\nprovides a mechanism\nto construct the conditions associated to nodes.\nA solution to a planning problem (plan) typically contains in each node information\nthat characterizes the problem state reached at that node.\nFor simplicity, assume the common case when\nthe state information in a node is the full\ndescription of a unique state (i.e., a list of fluents from $\\mathcal{F}$\nthat hold in the state at hand).\nConsider a dialogue-plan node $n$ with multiple outgoing branches,\nand its children nodes $c_1, \\dots, c_l$.\nThe formula of a child $c_i$ can be\nas straightforward as the description of the corresponding state.\nOptionally, the formulas can further be simplified to \ncapture differences between the state descriptions of the siblings.\n\n\\begin{figure}\n  \\centering\n\n  \\definecolor{yellowfill}{RGB}{255,242,204}\n  \\definecolor{yellowborder}{RGB}{214,182,86}\n  \\definecolor{redfill}{RGB}{248,206,204}\n  \\definecolor{redborder}{RGB}{184,84,80}\n  \n  \\tikzset{rectangular\/.style={draw, rectangle, minimum height=1cm}}\n  \\tikzset{circular\/.style={draw, circle}}\n  \\tikzset{yellow\/.style={draw=yellowborder,fill=yellowfill}}\n  \\tikzset{red\/.style={draw=redborder,fill=redfill}}\n\n  \\resizebox{.25\\textwidth}{!}{\n\n    \\begin{tikzpicture}[\n    shorten >= 1pt, \n    node distance=1.8cm, scale=0.5, semithick, ->]\n  \\tikzstyle{every node} = [font=\\footnotesize]\n  \\node[\n    state, \n    rectangular, \n    yellow, \n    align=center, \n    text width=2.7cm] (query-checkin) {\\textsf{C: [true] \\\\ A: ask-checkin-luggage}};\n  \\node[\n    state, \n    rectangular, \n    yellow, \n    below of=query-checkin, \n    align=center, \n    xshift=1.85cm, \n    text width=2.5cm] \n    (checkin-nonumber) {\\textsf{C: [ok-checkin] \\\\ A: ask-how-many}};\n  \\node[\n    accepting, \n    rectangular,\n    red,\n    align=center, \n    text width=2.5cm, \n    below of=query-checkin, \n    xshift=-1.85cm, \n    double distance=1pt] (noalarm) {\\textsf{C: [no-checkin] \\\\ A: Done}};\n  \\node[\n    state, \n    rectangular,\n    yellow, \n    align=center, \n    text width=3.8cm, \n    xshift=1.3cm, \n    below of=checkin-nonumber] (checkin-number) {\\textsf{C: [ok-checkin, have-number] \\\\ A: set-luggage-checkin}};\n\n  \\node[\n      accepting, \n      rectangular,\n      red, \n      align=center, \n      text width=2.5cm, \n      below of=checkin-number] (alarm-set) {\\textsf{C: [luggage-checkin-set] \\\\ A: Done}};\n  \n  \\draw[<-] (query-checkin) -- ++(-4cm,0);\n  \\path \n    (query-checkin) edge node[anchor=east, yshift=1mm]{\\textsf{}} (checkin-nonumber)\n    \n    (query-checkin) edge[bend right=20] \n      node[anchor=east, xshift=1mm, yshift=2mm]{\\textsf{}} (noalarm)\n    \n    (query-checkin) edge[bend left=60] \n      node[anchor=east, xshift=1mm, yshift=2mm]{\\textsf{}} (checkin-number)\n\n    (checkin-nonumber) edge[bend left=20] \n      node[anchor=west, xshift=-1mm, yshift=2mm]{\\textsf{}} (checkin-number)\n\n    (checkin-nonumber) edge[loop above] node {} (checkin-nonumber)\n\n    (query-checkin) edge[loop above] node {} (query-checkin)\n\n    (checkin-number) edge[bend left=20] \n      node[anchor=west, xshift=-1mm, yshift=2mm]{\\textsf{}} (alarm-set);\n\n    \\end{tikzpicture}\n  }\n  \\caption{Toy dialogue plan. The plan has two goal states,\n  shown as double-bordered boxes.}\n  \\label{fig-fsm}\n  \\end{figure}\n}\n\nWe use FOND planning\nto generate dialogue plans.\nComputing dialogue plans with AI planning\nprovides a mechanism\nto construct the formulas associated to edges.\nAs the multiple edges originating from a node\nare non-deterministic effects of a given action,\nwe use the outcome of each effect\nas the formula of the corresponding edge.\n\n\\begin{figure}\n  \\centering\n\n  \\definecolor{yellowfill}{RGB}{255,242,204}\n  \\definecolor{yellowborder}{RGB}{214,182,86}\n  \\definecolor{redfill}{RGB}{248,206,204}\n  \\definecolor{redborder}{RGB}{184,84,80}\n  \n  \\tikzset{rectangular\/.style={draw, rectangle, minimum height=1cm}}\n  \\tikzset{circular\/.style={draw, circle}}\n  \\tikzset{yellow\/.style={draw=yellowborder,fill=yellowfill}}\n  \\tikzset{red\/.style={draw=redborder,fill=redfill}}\n\n  \\resizebox{.4\\textwidth}{!}{\n\n    \\begin{tikzpicture}[\n    shorten >= 1pt, \n    node distance=1.8cm, scale=0.5, semithick, ->]\n  \\tikzstyle{every node} = [font=\\footnotesize]\n  \\node[\n    state, \n    rectangular, \n    yellow, \n    align=center, \n    text width=2.7cm] (query-checkin) {\\textsf{A: ask-checkin-luggage}};\n  \\node[\n    state, \n    rectangular, \n    yellow, \n    below of=query-checkin, \n    align=center, \n    xshift=1.85cm, \n    text width=2.5cm] \n    (checkin-nonumber) {\\textsf{A: ask-how-many}};\n  \\node[\n    accepting, \n    rectangular,\n    red,\n    align=center, \n    text width=2.5cm, \n    below of=query-checkin, \n    xshift=-1.85cm, \n    double distance=1pt] (noalarm) {\\textsf{A: Done}};\n  \\node[\n    state, \n    rectangular,\n    yellow, \n    align=center, \n    text width=3.8cm, \n    xshift=1.3cm, \n    below of=checkin-nonumber] (checkin-number) {\\textsf{A: set-luggage-checkin}};\n\n  \\node[\n      accepting, \n      rectangular,\n      red, \n      align=center, \n      text width=2.5cm, \n      below of=checkin-number] (alarm-set) {\\textsf{A: Done}};\n  \n  \\draw[<-] (query-checkin) -- ++(-4cm,0);\n  \\path \n    (query-checkin) edge node[anchor=east, yshift=1mm]{\\textsf{}} (checkin-nonumber)\n    \n    (query-checkin) edge[bend right=20] \n      node[anchor=east, xshift=1mm, yshift=2mm]{\\textsf{}} (noalarm)\n    \n    (query-checkin) edge[bend left=60] \n      node[anchor=east, xshift=1mm, yshift=2mm]{\\textsf{}} (checkin-number)\n\n    (checkin-nonumber) edge[bend left=20] \n      node[anchor=west, xshift=-1mm, left, align=center]{\\textsf{[have-number]}} (checkin-number)\n\n    (checkin-nonumber) edge[loop above] node {[ ]} (checkin-nonumber)\n\n    (query-checkin) edge[loop above] node {} (query-checkin)\n\n    (checkin-number) edge[bend left=20] \n      node[anchor=west, xshift=-1mm, yshift=2mm]{\\textsf{}} (alarm-set);\n\n    \\end{tikzpicture}\n  }\n  \\caption{Toy dialogue plan. The plan has two goal states,\n  shown as double-bordered boxes.\n  Edge formulas shown for only one set of non-deterministic edges,\n  to avoid clutter.}\n  \\label{fig-fsm}\n  \\end{figure}\n\nFigure~\\ref{fig-fsm} shows a toy example of a dialogue plan\ninspired from a trip planning application.\nIn this example, \nthe agent asks the user whether the agent should \ncheck in any luggage in an upcoming flight.\nThe plan captures four possible options for the user's answer:\n1) no luggage should be checked in, in which case the dialogue\ncan progress to the goal state at the left;\n2) the user gives a positive reply, and provides the number of \nsuitcases (e.g., \\emph{``Yes, 2 pieces''});\n3) the user gives a positive reply, without specifying the number;\nand 4) the user gives an irrelevant response, in which case the agent\nasks again.\nIn case 2, the dialogue can progress to calling an external service\nthat marks the corresponding field in a flight booking form,\nafter which the dialogue progresses to a goal state.\nIn case 3 the agent asks about the number. If a number is given,\nthe dialogue progresses to case 2.\nOtherwise, the agent would ask again for a number.\nFor simplicity, we skip details such as interrupting the dialogue\nafter a finite number of iterations, in case that the user\nkeeps providing meaningless answers.\n\n\\section{Use Cases}\n\\label{sec:eval}\n\nWe evaluate our generic approach to computing dialogue plans with AI planning\nin four domains:\nhuman resources, career coaching, trip planning,\nand a synthetic domain.\nThe first two correspond to two products.\nThe synthetic domain allows to \nperform a more focused scalability evaluation.\n\n\n\\ab{Differences in size\/effort compared to more manual approaches. Adapting\/maintainig\/extending the system with ease. Any user study available? That would help too.}\n\n\\cm{Another is to measure the shift a small change in model can have to the final solution. Minor things in the declarative setting can lead to \\textit{massive} changes in the contingent plan (which would be manual edits the way dialogue agents are currently designed).}\n\n\\cm{There is already a todo item to fix up the figure for the first two use cases --\nwould it be possible to switch the order of action \/ state? So nodes correspond to actions\nwhile the state is implicit (then the outgoing edges correspond to outcomes of the action.}\n\n\n\\towrite{\n    Introduce use case: relevant intents, building blocks, how building blocks can \n    pop up in multiple intents. Present the limitations of Watson Assistant,\n    which leads to the need for specialized apps, such as Java apps. Then present the\n    limitations of apps with hardcoded plans. More work to develop, more work \n    to maintain. Which leads to our approach, with one app that performs\n    planning, planning monitoring and planning execution. Give some stats about\n    the size of the apps.\n}\n\n\\subsubsection{Human Resources} We present a HR application with\ndialogues related to employee professional performance.\nRegular employees could have a dialogue about their own performance.\nIn addition, a manager could have a chat about \nthe performance of any team member,\nand the team as a whole.\n\nThe performance is defined along a number of criteria called capabilities, such as\ntechnical skills and knowledge.\n\nIn this domain, dialogue snippets can be partitioned into three main categories:\nquestions with static answers related to the domain\n(e.g., \\emph{``How can I use this tool?\")};\ngeneral-purpose chitchat\n(e.g., \\emph{``Hello\"});\nand goal-oriented, multi-turn conversations.\nThe first two categories are simple snippets of one question and one answer.\nThe third one is handled with the approach presented in this paper.\n\nGoals considered in multi-turn dialogues include:\n(1) presenting the performance rating of a given individual for a given capability;\n(2) giving an explaination about the value of a performance rating for an individual and a capability;\n(3) recommending learning resources to improve the performance of a given individual in a given capability;\n(4) presenting the performance rating of a team for a given capability;\n(5) giving an explaination about the value of a performance rating for a team and a capability;\nand\n(6) identifying the strongest and the weakest performer in a team, for a given capability.\n\nSuch goals often require a multi-turn conversation,\nfor person disambiguation and capability disambiguation.\nFor instance, a person name given in the user utterance\nis used to identify a person record in the team member database.\nA first name provided in a user utterance might correspond to zero, one or several person records in the database. In the first and the third case, the dialogue needs to continue with disambiguating the\nperson.\nLikewise, when no capability is specified, or the capability is ambiguous, the dialogue needs to \ndisambiguate it.\n\nAn early implementation of the goal-oriented conversations used one\nJava program for goals 1 and 4; one Java program for goals 2 and 5;\none Java program for goal 3; and one Java program for goal 6 (using a\ntotal of 3,702 lines of code).  These Java programs are essentially\nimplementing hard-coded dialogue plans.  This is hard to maintain and\nto extend to new goals, and the portability to a different domain is\nvery limited.\n\nWe have replaced such multiple individual Java apps, based on\nhardcoded specific dialogue plans, with one single application (using\nonly 1,166 lines of code).  The application analyzes the goal at hand,\nidentified from the intent of the user utterance,\nand constructs a planning instance accordingly.\nThe instance can be fed into an off-the-shelf planning system to obtain a dialogue plan on demand.\nAlternatively, dialogue plans can be precomputed and stored into a library,\nindexed on the goals they address.\nThe dialogue plan is executed and monitored in the system.\n\n\n\n\n\\subsubsection{Career Coaching} We consider two dialogue goals in career coaching:\nreaching a point where the user has eventually chosen a long-term career goal;\nand eventually chosing a career pathway towards that goal.\nThus, the system implements calls to two APIs: one for a career goal recommender, and one for a pathway recommender.\nEach API implements calls that provide: a list of recommendations; an explanation associated with a given item (goal or pathway) included in a recommendation; additional details about a given item included in a recommendation.\n\nFor long-term career goals,\na service call to a recommender system provides a number of recommendations.\nThese are ordered, and the top three are presented to the user.\nThe user can choose a career goal,\nor request additional information\n(e.g., an explanation of why a given career goal is included),\nor reject the career goals currently provided.\nIn the first case, the goal is achieved.\nIn the second case, additional information is provided and the dialogue continues\nrecursively, from the state where the user is required to choose between the three options again.\nIn the third case, additional information is elicited from the user, regarding the reason\nof not liking any career goal. Based on the newly elicited information, the short list of\nthree recommendations is re-computed, and the dialogue continues recursively.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.27\\textwidth]{figs\/figure3.pdf}\n    \\caption{Dialogue plan for career goals and pathways.}\n    \\label{fig::myca-plan}\n\\end{figure}\n\n\\hide{\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.3\\textwidth]{sections\/myca-cg-plan.pdf}\n    \\caption{Dialogue plan focused on successfully recommending a long-term career goal to a user.}\n    \\label{fig::myca-cg-plan}\n\\end{figure}\n}\n\nAnother dialogue plan focuses on helping the end user choose a career pathway.\nCareer pathways are computed with a call to an external service.\nPathways are presented to the user, which can accept or reject the career pathway at hand.\nIn the former case, the goal is reached and the dialogue concludes.\nIn the latter case, information about the reason of rejecting the pathway is elicited from the user.\nPotential reasons can be related to a job role along the pathway, or constraints\nassociated with roles (e.g., the user might dislike management roles).\nThe context is updated, a career pathway is recomputed, and the dialogue\ncontinues recursively.\n\n\\hide{\nIn practice, dialogue systems should implement the option to conclude the conversation\nif no progress can be made after a given number of repetitions of a cycle in the flow.\nAn example is the case when the user keeps rejecting the recommendations.\nSuch an exit mechanism can be implemented, for instance, at the level of the execution monitoring,\nby counting the executions of a cycle.\n}\n\nComputing a career pathways towards a long-term career goal requires the career goal as an input.\nAs such, the two dialogues could be chained in a sequence. They can also be generated\nas independent dialogue plans. Such variations can easily be obtained with very small\nmodifications in the PDDL problem instance definition.\nFor instance, if the goal is to choose a career pathway, and a career goal is already\navailable in the initial state, there is no need to \nhave the dialogue focused on choosing a career goal.\nOtherwise, the two plans will automatically be chained in one larger dialogue plan.\nFigure~\\ref{fig::myca-plan} illustrates this combined dialogue plan.\n\n\\hide{\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.3\\textwidth]{sections\/myca-cp-plan.pdf}\n\t\\caption{Dialogue plan focused on successfully recommending a career pathway to a user.\n\tDialogue actions and service actions are visualised with different graphical symbols.}\n    \\label{fig::myca-cp-plan}\n\\end{figure}\n}\n\n\\ab{TODO: Add an argument about the value of the work in this domain.}\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{figs\/trip-plan-example}\n    \\caption{Generated dialogue for the Trip Planning use case.}\n    \\label{fig:trip-plan-example}\n\\end{figure}\n\n\\subsubsection{Trip Planning} The goal of our trip planning system is\nto provide booking services while considering weather situation\nrelevant for the trip.  The system needs to collect departure and\narrival destinations and date range of the trip. We provide in the\nsupplementary material the actual PDDL domain and problem instance\nused to drive the dialogue. For the previous two use cases, the PDDL\nspecification is closed.\n\nThe system is requesting the trip parameters through the natural dialogue and verifies their correctness (e.g. validity of provided locations).\nAfter enough information is collected, the system uses an external call to check the weather situation for the specified destination and dates.\nIf the weather is evaluated as inferior, the system informs the user and suggests changing the trip parameters.\n\nFrom the automated planning perspective, the interesting aspect of this use case is handling the uncertainty of collected system parameters.\nThe uncertainty comes from two sources: (1) the location recognition can be ambiguous due to NLU errors; and (2)\nthe system can hypothesize about arrival and destination locations based on historical data and the actual user location.\nThis leads to the planner to confirming information with the user that it has some certainty about, and soliciting information\nfrom scratch when it does not have a sense as to what the true value is.\n\nFigure \\ref{fig:trip-plan-example} shows a high-level view of the generated dialogue plan that comes from a model with only\nnine actions. The node symbols indicates the type of action that corresponds to that part of the plan: either dialogue, API\ncall, or system action (the latter two being specific examples of non-dialogue actions discussed in \nSection~\\ref{sec:approach}).\nEven in this limited setting, we can observe how complex behaviour can be captured in the generated dialogue agent from a simple\ndeclarative specification.\n\n\\subsubsection{Scalability Analysis}\n\n\\hide{\n\\ab{Looks nice. A couple of comments: \n\\begin{itemize}\n\t\\item Is it possible, by any chance, to measure the solution differences in terms of the Graph Edit Distance (GED)?\n\t\\url{https:\/\/en.wikipedia.org\/wiki\/Graph_edit_distance}\n\t\\item Maybe the reviewers wonder why we have a sudden jump from 20 to 40 in Figure~\\ref{fig:SRCbyIGC}. I.e., dot isolated at the bottom-left corner. Any insights?\n\\end{itemize}\n}\n\n\\cm{\\begin{itemize}\n    \\item I don't think so. We'd need to build a search algorithm from scratch to go from one graph to another. I cane envision how to roll this out from scratch (but don't have the time), and can't see a clear path to modeling it as planning (so we could just use a planner as the A* search). It's a very good measure (nice find!), but don't think it's feasible at the moment. Do you think we should cite it?\n    \\item Good point. It was from a single outlier that has only a small change in the solution for a fairly different init\/goal. 7 things changed in the init\/goal and only 134 lines changed in the solution. I've added text to this effect.\n\\end{itemize}}\n}\n\n\nScalability \nis a major advantage\nof using a declarative representation for goal-oriented dialogue agents. To demonstrate this empirically, we created synthetic domains and problems mirroring the properties we observed in the existing dialogue encodings of our three use cases above. We measure the model size of the generated problems and solution size of the computed dialogue agents as the number of unique actions used in the solution and the total number of actions in the dialogue plan respectively.\n\nWe setup our experiments by populating a domain with random non-deterministic actions and a problem with random initial and goal states. Action precondition and effects are generated through random sampling from a set of fluents. We mirror the characteristics of actions inherent to a real dialogue system by: (1) keeping size of action preconditions within the range of 1-5 inclusive; (2) randomly sampling the effect type as either \\textit{select} (exactly one of 2-5 fluents will become true) or \\textit{assign} (1-4 fluents are randomly flipped); (3) randomly sampling 1-5 fluents for the initial state; and (4) randomly sampling 1-2 fluents for the goal state. As parameters to the random problem generator, we provide the number of actions and fluents.\n\nThe two action types correspond to typical dialogue actions that we have encountered. The \\textit{select} action mirrors the determination of one type of response the user could provide (typically with a range of 2 to 5 possibilities). The \\textit{assign} action mirrors dialogue actions where many aspects of the context can be assessed simultaneously (and thus multiple non-deterministic aspects are considered simultaneously). We have additionally confirmed qualitatively that the generated plans appear to contain a similar structure as the known dialogue plans (e.g., with the same expected plan length).\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.36\\textwidth]{images\/SSbyMSHist.png}\n\t\\caption{Plot of ratio of Solution size and Model size}\n\t\\vspace{-1em}\n\t\\label{fig::SSbyMS}\n\\end{figure}\n\nIn total we generated 100 instances.\nFigure \\ref{fig::SSbyMS} shows a histogram of the ratio of solution size divided by model size.\nIn most of the instances, solution size is at least 4 times the size of the model, and in extreme cases it can grow to 16 times the model size. This confirms our assertion that complex dialogue systems can be efficiently designed with very compact declarative representations.\n\n\\hide{\nOur second experiment aims at demonstrating the effect of rapid prototyping of the declarative model for a dialogue system: small changes to the declarative specification can have massive ramifications to the generated dialogue agent. Generating the dialogue agents from a declarative specification means that we can avoid massive amounts of effort to maintain the full dialogue plans, as is currently the case.\n\nTo measure the amount of change that is required, we look at the model difference and solution difference when compared to a candidate instance (one chosen for every group of 10 instances -- the other 9 differ only in their initial and goal states). The model difference for problems $\\langle \\mathcal{F}, \\mathcal{I}, \\mathcal{A}, \\mathcal{G} \\rangle$ and $\\langle \\mathcal{F}, \\mathcal{I}', \\mathcal{A}, \\mathcal{G}' \\rangle$ is measured as:\n\\[ |(\\mathcal{I} \\cup \\mathcal{I}') \\setminus (\\mathcal{I} \\cap \\mathcal{I}')| + |(\\mathcal{G} \\cup \\mathcal{G}') \\setminus (\\mathcal{G} \\cap \\mathcal{G}')| \\]\n\nThe solution difference is measured as the number of lines that differ in the files that represent a solution (using a standard command-line diff utility). While this is a crude approximation of true difference between solutions, it nevertheless serves as a general proxy for the difference in plans. The authors are unaware of a pre-existing measure for determining the difference between two contingent solutions.\n\nFigure \\ref{fig:SRCbyIGC} shows the ratio of solution difference by model difference (instances sorted based on ratio). In most of the instances, the solution difference is at least 35 times the model difference. This confirms our assertion that by generating dialogue agents from a declarative specification, we can avoid massive amounts of effort to maintain the full dialogue plans. The one outlier at the far left of the graph comes from a sizable change to the initial\/goal state (7 according to the measure above), and relatively few lines changed in the produced solution (134 instead of the average of 573 lines changed among the rest of the instances in that group).\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.35\\textwidth]{images\/SRCbyIGC.png}\n\t\\caption{Ratio of Solution Difference and Model Difference}\n\t\\label{fig:SRCbyIGC}\n\\end{figure}\n}\n\n\n\n\\hide{\n\\noindent \\textbf{Full Data}\n\n\\towrite{Remove this from the final version -- just including it for completeness in case there is something else we want to discuss.}\n\nTable \\ref{tab:syntheticData} shows the simulated data. Each row corresponds to a unique set of generated actions and fluents as part of a domain. For any particular row (or domain), when moving across the column, actions and fluents of that domain remain constant while initial and goal state of problem change with respect to the first generated initial and goal state for that domain. Note that for any P*, the initial and goal state is not the same across column. Each cell of the table contains a tuple with four values: size of the plan\/solution (number of plan nodes), size of the model(number of unique actions used in the solution), the difference in initial state \/ goal fluents (set union minus set intersection for each, then summed), and the size of the ``diff'' between solution files of the generated model. The final two entries are done in comparison with the first instance of the row.\n\n\n\\begin{table*}\n\t\\resizebox{\\textwidth}{!}{%\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& P1 & P2 & P3 & P4 & P5 & P6 & P7 & P8 & P9 & P10\\\\\n\t\t\t\\hline\n\t\t\tD1 & (56, 8, 0, 0) & (45, 5, 5, 451) & (51, 8, 4, 410) & (44, 7, 7, 475) & (50, 6, 7, 412) & (75, 8, 7, 483) & (85, 7, 6, 493) & (45, 8, 3, 441) & (42, 9, 8, 448) & (45, 5, 8, 451)\\\\\n\t\t\t\\hline\n\t\t\tD2 & (92, 12, 0, 0) & (79, 8, 7, 637) & (83, 9, 9, 689) & (90, 8, 6, 714) & (92, 9, 10, 680) & (120, 10, 5, 804) & (96, 12, 7, 734) & (71, 5, 10, 665) & (126, 12, 5, 496) & (214, 8, 8, 1008)\\\\\n\t\t\t\\hline\n\t\t\tD3 & (49, 11, 0, 0) & (44, 13, 6, 325) & (81, 16, 10, 580) & (55, 15, 10, 356) & (57, 16, 8, 512) & (59, 13, 7, 504) & (46, 12, 7, 329) & (83, 16, 8, 580) & (42, 11, 10, 375) & (73, 19, 9, 408)\\\\\n\t\t\t\\hline\n\t\t\tD4 & (52, 11, 0, 0) & (54, 12, 8, 464) & (69, 14, 6, 509) & (44, 12, 11, 434) & (49, 12, 6, 447) & (173, 8, 7, 869) & (56, 13, 11, 444) & (48, 11, 7, 454) & (169, 7, 8, 805) & (41, 8, 10, 427)\\\\\n\t\t\t\\hline\n\t\t\tD5 & (41, 8, 0, 0) & (64, 12, 8, 491) & (58, 8, 7, 431) & (46, 10, 9, 564) & (44, 9, 9, 383) & (67, 11, 6, 500) & (45, 8, 9, 448) & (68, 15, 8, 298) & (41, 8, 9, 346) & (49, 9, 10, 410)\\\\\n\t\t\t\\hline\n\t\t\tD6 & (288, 7, 0, 0) & (51, 12, 10, 1135) & (60, 14, 10, 1144) & (45, 9, 11, 1099) & (57, 11, 10, 1135) & (63, 14, 12, 1227) & (49, 10, 11, 1129) & (65, 14, 12, 1151) & (64, 11, 9, 1156) & (64, 15, 8, 1156)\\\\\n\t\t\t\\hline\n\t\t\tD7 & (49, 8, 0, 0) & (75, 9, 7, 526) & (92, 12, 5, 531) & (92, 7, 8, 415) & (47, 10, 8, 440) & (51, 11, 7, 498) & (72, 8, 10, 487) & (100, 13, 9, 619) & (50, 9, 8, 483) & (52, 10, 6, 461)\\\\\n\t\t\t\\hline\n\t\t\tD8 & (45, 13, 0, 0) & (42, 12, 9, 433) & (51, 13, 9, 466) & (41, 8, 6, 344) & (41, 11, 7, 393) & (61, 16, 7, 482) & (49, 12, 7, 456) & (42, 10, 9, 371) & (47, 12, 7, 452) & (51, 11, 6, 460)\\\\\n\t\t\t\\hline\n\t\t\tD9 & (43, 9, 0, 0) & (52, 9, 9, 465) & (51, 11, 9, 386) & (43, 10, 11, 442) & (51, 9, 10, 450) & (43, 9, 7, 252) & (42, 8, 10, 405) & (44, 10, 6, 255) & (135, 9, 10, 718) & (41, 8, 10, 372)\\\\\n\t\t\t\\hline\n\t\t\tD10 & (51, 12, 0, 0) & (74, 6, 9, 555) & (44, 10, 7, 134) & (105, 11, 9, 547) & (207, 9, 9, 916) & (127, 10, 8, 692) & (43, 9, 10, 440) & (41, 7, 8, 434) & (46, 8, 7, 417) & (88, 6, 10, 585)\\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\\caption{Synthetic Data}\n\t\\label{tab:syntheticData}\n\\end{table*}\n\n}\n\n\\section{Integrating Plans into a Dialogue System}\n\\label{plan-integration}\n\n\\towrite{Describe how the compiled plan corresponds to the dialogue plans described in the previous section.}\n\nRecall that a dialogue plan obtained from a planning system is\na directed graph, with nodes representing states.\nA unique transition originating from a state represents\na deterministic action, and multiple transitions from a given state\nrepresent the branches of a non-deterministic action.\nTransitions are labeled with one string each,\nrepresenting the name of the corresponding action and its list of zero or more arguments.\nStates are labeled with the predicates that are true in the state at hand.\nIn this section we discuss how such a plan can be integrated and executed\ninto a dialogue system.\n\nEach action in the plan (string) needs to be mapped into an actual code to execute.\nWe call the code corresponding to an action a \\emph{transformer}.\nAs such, each trajectory (dialogue trace) modelled in the dialogue plan\nhas a corresponding sequence of transformers to call.\nAs mentioned in Section~\\ref{sec:abstraction}, not all information contained\nin the dialogue context is necessarily part of the PDDL planning problem.\nTransformers can use information from the dialogue context\n(i.e., use part of the context as input, and modify part of the context).\nThis provides a mechanism through which transformers can consume (use as input)\nthe effects of transformers previously called in an execution trace.\n\n\\ab{\n    Is ``transformer'' an established term? If not, how about renaming these?\n}\n\nWhen multiple branches originate in a given state of a plan,\nthe execution needs to decide what branch to choose.\nA simple but effective approach is to associate a condition with each branch,\nbased on the differences between the labels of the resulting states of each branch.\nConsider the example presented in Figure~\\ref{fig-fsm}.\nThe state labeled {\\sc has-recomms} has two outgoing branches, \ncorresponding to two non-deterministic effects of an action corresponding to a user utterance.\nBased on the differences between the labels of the two resulting states,\nthe condition associated with the leftmost branch is {\\sc user-accept},\nand the rightmost branch has the condition {\\sc not-accepted}.\nThese are two complementary conditions, for the cases when the user has accepted\nthe recommendation, or the user has rejected it.\n\nConditions defined in such a way are evaluated when the execution trace\nreaches the parent state of the corresponding branches.\nThe evaluation is based on context information.\nMore specifically, each PDDL predicate used in a condition has associated an evaluation\nformula based on variables from the context.\nIn our running example, the predicates {\\sc user-accept} and\n{\\sc not-accepted} are evaluated based on \nthe classification (intent) of the most recent utterance\n(acceptance or rejection).\n\nA deeper discussion on monitoring the execution of a plan is beyond the scope of this paper.\n\n\\begin{comment}\n\\subsection{Compiling Plans into Dialogues}\n\\label{sec:pc}\n\n\nDialogues can be conceived as a combination of a small set of functions \\emph{transforming} the dialogue context described in Sect.~\\ref{sec:abstraction}. Such dialogue transformers perform basic tasks such as saying something to the user, or asking a question. In addition, the transformer library provides composite transformers that allow to model complex dialogues in a declarative, compositional way. \n\nOne crucial composite transformer is the finite state machine, allowing to model arbitrary dialogue plans. For example, Figure \\ref{fig:transformers} depicts the translation to dialogue transformers of the plan shown in Figure \\ref{fig-fsm}.\n\n\\newcommand\\jcomm[1]{\\textcolor[rgb]{0.18,0.49,0.19}{\\textit{#1}}}\n\\newcommand\\jstr[1]{\\textcolor[rgb]{0.05,0.27,0.63}{#1}}\n\\begin{figure}\n      \\begin{footnotesize}\n    \\begin{Verbatim}[commandchars=\\\\\\{\\}]\n\\jcomm{\/\/ Initial state}\ninitial(\\jstr{\"get-recomms\"}).call(seq(\n    this::getRecomms,\n    say(this::stringifyRecomms)\n)).jumpTo(\\jstr{\"query-acceptance\"}).always()\n\n\\jcomm{\/\/ Query user acceptance}\non(\\jstr{\"query-acceptance\"}).call(\n    clarify(\\jstr{\"acceptance\"})\n        .asking(\\jstr{\"Accept recommendations?\"})\n        .withExtractor(yesNo())\n).finish().when(getBool(\\jstr{\"acceptance\"}))\n .jumpTo(\\jstr{\"elicit-reject-reason\"}).otherwise()\n\n\\jcomm{\/\/ Elicit reject reason}\non(\\jstr{\"elicit-reject-reason\"}).call(\n    clarify(\\jstr{\"reason\"})\n        .asking(\\jstr{\"Why not?\"})\n        .withExtractor(fromInput())\n).jumpTo(\\jstr{\"get-recomms\"}).always()\n    \\end{Verbatim}\n\\end{footnotesize}\n\\caption{Plan in Figure \\ref{fig-fsm}, as dialogue transformers.}\n\\label{fig:transformers}\n\\end{figure}\n\nFrom the initial state we call the high-level transformer \\texttt{seq} to assemble the two-step sequence [\\texttt{getRecomms}, \\texttt{say(stringifyRecomms)}], thus retrieving recommendations and showing them to the user (using the \\textit{say} transformer). This sequence represents the mapping from the \\texttt{get-recomms} action in the plan to actual dialogue code. The rest of the translation is self-explanatory: the dialogue jumps unconditionally to the \\texttt{query-acceptance} state, thus querying the user. Depending on the intent associated to the answer's utterance (extracted by means of a \\texttt{yesNo} NLU extractor) the dialogue will finalize or go the \\texttt{elicit-reject-reason} state, asking about the rejection reason. From there the dialogue will jump unconditionally to the initial state.\n\nSince plans can be arbitrarily complex, the translation process must be automated. To this end we implemented a compiler that traverses a plan in a depth first fashion, generating the corresponding transformer ensemble along the way. The compiler requires a mapping from plan actions to transformers (e.g., \\texttt{get-recomms} to the sequence [\\texttt{getRecomms}, \\texttt{say(stringifyRecomms)}]), and another from plan predicates to code (e.g., \\texttt{user-accepted} to \\texttt{getBool(\"acceptance\")}). The ability to compile plans leverages dynamic scenarios where a planning service is invoked on the fly, and the resulting plan compiled into a dialogue and executed at runtime.\n\n\\end{comment}\n\n\\section{Introduction}\n\\label{sec:intro}\n\nDialogue agents are becoming increasingly pervasive across many industries.\n\\towrite{$\\langle \\langle$ list of industry-specific agents $\\rangle \\rangle$.}\nThere is also a recognized demand for goal-oriented agents that are capable of bridging several services to assist human users in multi-turn dialogue \\cite{DBLP:journals\/aim\/Ortiz18}. It is natural, then, to view the generation of dialogue agents through the lens of a technology well-suited for multi-step and goal-oriented settings: automated planning.\n\nState-of-the-art dialogue systems \ntypically can fall into two categories, dialogue trees and conversation learners. ELIZA is a first dialogue system that used dialogue trees~\\cite{Weizenbaum:1966:ECP:365153.365168}.\nIn ELIZA and more recent tree-based dialogue systems,\nthe conversation is structured by a complex series of branches with options depending on the user responses, and the dialogue at each branching node is hard coded. Dialogue trees can rapidly become unwieldy as the choices become more complex. Additionally, a great deal of repetition may exist where similar dialogue interactions may be required at different points in the conversation.\n\nConversation learners use machine learning to compute the next response from examples or historical interactions \\cite{DBLP:conf\/ijcai\/IlievskiMHB18,DBLP:conf\/acl\/GaoWPLL18}. They allow the\nquick development of a complex dialogue without the laborious task of constructing a dialogue tree. Drawbacks include the lack of control from the application owner and the potentially unpredictable results. In certain domains, uncertainty and the risk of erroneous responses may be acceptable, but in domains such as health, finance and human resources (HR)\nwe require guarantees and predictability of results.\n\nA modern dialogue agent should integrate important properties such as:\nmulti-turn, goal-oriented dialogues;\ncalls to external services whose output is needed in the dialogue;\nand handling contingencies.\nInformally, the latter covers the fact that, \nat certain points in a dialogue, the conversation\ncould steer in more than one direction.\nFor instance, the user could accept or reject\na suggestion from the agent,\nand the follow-up dialogue\ncould be very different in each case.\n\n\nWe introduce an approach to constructing dialogue agents with AI planning.\nOur dialogue plans feature the properties mentioned above.\nThese properties are not necessarily new,\nbut our novelty stems  from\ncomputing such dialogue plans \\textit{automatically},\nusing domain-independent planning.\n\nThis allows to create plans\ntailored to specific scenarios\n(e.g., focused on achieving a given goal).\nThe input includes a library of actions representing individual steps in a dialogue.\nThese are the building blocks used to construct dialogue plans.\nAs typical in AI planning,\nthe input further contains a problem instance,\nwhich states the initial state and the goal of the dialogue plan to compute.\nThe system computes a dialogue plan that \nstarts from the initial state and continues\nuntil the goal is achieved.\n\nA library of actions can be used to generate a range of dialogue plans,\nand a given subset of actions can be used across several domains.\nFor instance, the user can ask about the weather in a range of domains,\nsuch as trip planning and family event planning.\nThe ability to automatically compute dialogue plans\nallows us to easily maintain deployed dialogue systems.\nFixing a bug, or slightly modifying the behaviour\nof an action further implies the need to recreate\nall dialogue plans impacted by the changes.\nManually fixing a collection of plans could be tedious and error prone.\nAvoiding this is a key advantage to our approach.\n\nUse cases show the feasability and the scalability of the approach.\nIn addition, our work has been integrated into a human resources product,\nand it is being added to another product, for career coaching.\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\n\n\n\n\nWe introduce basic concepts needed in a dialogue system\ncapable to integrate our dialogue plans.\nIn the second part of this section we overview\nfully observable non-deterministic planning (FOND).\nThis is the main planning approach we use,\ntogether with elements of contingent planning.\n\n\\subsubsection{Dialogue} \nIn a dialogue, user utterances are classified\ninto so-called \\emph{intents}.\nFor example, a statement such as \\emph{``I wonder how the weather is in Paris''}\ncould be classified into an intent such as {\\sc \\#asked-about-weather}.\nWe assume that intents are predefined (when the system is designed),\nand the system is pretrained for intent classification.\n\n\\emph{Entities} are variables defined in a dialogue system.\nBesides general-purpose variables,\nsuch as places (e.g., \\emph{Paris}), people names, dates and numbers,\ndesigners can define entities specific to the domain at hand,\nwith a range of values that an entity can take.\nA given value could possibly be defined in multiple ways, as a list of synonyms.\nValue assignments to entities can automatically be recognized\nin a user utterance, using \nNatural Language Understanding (NLU)~\\cite{Manning:1999:FSN:311445,Jurafsky:2000:SLP:555733,Florian:2010:IMD:1870658.1870691}.\nUser utterances are annotated with recognized intents\nand entity instantiations.\nThe sample utterance presented earlier\nwould be annotated with both the intent mentioned,\nand with a variable assignment such as {\\sc @place = Paris}.\nAs mentioned earlier, we allow calls to external services\nin the middle of a dialogue.\nCalls to external services are annotated with their inputs and outputs.\n\nSuch annotations will allow us to build more advanced \\emph{context} information.\nInformally, the context contains \ninformation that is relevant at a given point in a dialogue.\nSee details in Section~\\ref{sec:architecture}.\n\n\n\\subsubsection{Fully Observable Non-deterministic Planning}\n\nA \\textit{FOND problem} is defined by a set of \\textit{fluents} that\ncan be true or false in the environment, an \\textit{initial state}\nrepresented as a set of fluents, a set of \\textit{non-deterministic\n  actions} and a \\textit{goal} condition that must hold at the end of\nthe plan. \n\nThe FOND representation is similar to classical planning, except that\nin FOND actions can have multiple effects (see also\n\\cite{DBLP:series\/synthesis\/2013Geffner} for additional details).\n\nA \\textit{non-deterministic action} consists of a\n\\textit{precondition} that must be satisfied for the action to be\napplicable and a set of \\textit{outcomes}, one of which will be used\nduring execution. Each outcome is a set of fluents or their negation,\nindicating if the fluent should be added or deleted from the state of\nthe world.\n\nExactly one outcome \noccurs when an action is executed, and thus a solution to a FOND\nproblem must adequately handle every possible outcome.  Among several\nequivalent ways to define a solution, we will use one resembles more\nclosely our dialogue plans.\n\nConsider a new action ${\\mathbf{Done}}$ whose precondition satisfies the\ngoal. A \\emph{solution} to a FOND problem is a directed graph\n$(\\mathcal{N}, \\mathcal{E})$ where $\\mathcal{N}$ is a set of nodes and $\\mathcal{E}$ is a\nset of directed edges. Each node $n \\in \\mathcal{N}$ is associated\nwith an action (including ${\\mathbf{Done}})$ and has one outgoing directed\nedge for each outcome of its corresponding action. $\\mathcal{E}$ is the\nunion of all edges across all nodes. Nodes labeled by the\n${\\mathbf{Done}}$ action have no outgoing edges. We call these leaf\nnodes. A solution contains at least one leaf node and exactly one root\nnode without any incoming edges.\n\nThe action associated with the root node must be applicable in the\ninitial state. \\textit{Reachable states} are recursively defined by\napplying the action in a state reached at node $n$ (starting with the\ninitial state and root node, respectively). The outcome that occurs\nduring execution dictates both the following state of the world, and\nthe following node for the solution to proceed to. A solution must\nadditionally have the following properties: for every reachable state\nand node pair $(s,n)$, the action corresponding to $n$ must be\napplicable in $s$.  For every reachable node $n$, some leaf node $n'$\ncan be reached through some selection of outcomes.\n\n\n\n\n\n\\hide{\n\\begin{definition}[FOND Solution]\n  A \\textit{solution} to a given FOND problem $\\la \\fluents, \\init, \\actions, \\goal \\ra$ is a tuple $\\la \\nodes, \\edges, \\initnode \\ra$ where $\\mathcal{N}$ is a set of nodes with associated function $\\actmap{}: \\mathcal{N} \\rightarrow \\mathcal{A}$ that dictates what action corresponds to the node; $\\mathcal{E}$ is a set of edges that correspond directly to the outcomes of $\\actmap{n}$ (edges originating at node $n$); and initial node $n_0$. We further assume that every leaf node $n \\in \\mathcal{N}$ (i.e., those without an outgoing edge) has a specially designated $\\actmap{n} = {\\mathbf{Done}}$ action included, where $\\pre{{\\mathbf{Done}}} = \\mathcal{G}$. \\textit{Reachable states} are recursively defined by applying $\\actmap{n}$ in a state reached at node $n$ (starting with the initial state and node, $\\mathcal{I}$ and $n_0$ respectively). The outcome that occurs during execution dictates both the following state of the world, and the following node for the solution to proceed to. A solution must additionally have the following properties:\n  \\begin{enumerate}\n    \\item $\\actmap{n_0}$ must be applicable in $\\mathcal{I}$.\n    \\item For every reachable state and node pair $(s,n)$, the action $\\actmap{n}$ must be applicable in $s$.\n    \\item For every reachable node $n$, some leaf node $n'$ can be reached through some selection of outcomes.\n  \\end{enumerate}\n  \\end{definition}\n}\n\nIn summary, a solution is a directed graph where the nodes correspond to actions the agent takes and edges correspond to how the uncertain environment responds. There must always be some path to arrive at a goal node.\n\n\\cm{Do we want to get into fairness at all? This would be the place...}\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\n\nIn the past, \\citeauthor{Kuijpers1998}~\\shortcite{Kuijpers1998}\npresented an approach where a pre-existing library of plans can be\nused by an agent in a\ndialogue. \\citeauthor{Steedman07planningdialog}~\\shortcite{Steedman07planningdialog}\nadvocated the use of AI planning to facilitate mixed-initiative\ncollaborative discourse. More recently,\n\\cite{DBLP:conf\/sigdial\/NothdurftBBBM15} described a system capable of\nexplaining the decisions of the planning system to the\nuser. \\intextcite{DBLP:journals\/aim\/Ortiz18} presented a holistic\napproach to building conversational assistants which includes a\nplanning component to assist the interaction with human\nusers. However, none of these previous approaches employ AI planning\nto generate customized dialog plans.\n\n\n\\citeauthor{garoufi2010automated}~\\shortcite{garoufi2010automated}\nshowed recently how to generate whole sentences word-by-word with AI\nplanning techniques and described an approach to generate multi-turn\nnavigational commands to be followed by a human user in a\nconversational interface to achieve a specified goal.\n\n\\intextcite{black2014automated} developed a formalism that describes\npersuasion dialogues as state transitions in a planning domain.  In\nthis setup, an automated planner was able to find optimal persuasion\nstrategies. However, the execution monitoring is greatly simplified\ndue to the abstract nature of simulated dialogues.\n\n\\intextcite{petrick2013planning} considered execution monitoring\nissues that arise from non-determinism of real-world dialogue systems.\nTheir proposed execution monitor expects deterministic behavior of\nactions in the world and rebuilds the plan if inconsistencies between\nmeasurements and the expectations are observed (via various sensors).\nIn contrast, our execution monitor accounts for non-determinism\nexplicitly within the generated plan. Therefore, our plans can be\nstatic and easier to debug.\n\nAnother direction of research is focused on using (deep) reinforcement\nlearning\nfor end-to-end dialogue\ngeneration \\cite{bhuwan2017,peng2018}. In contrast to our model-based\napproach, training these data centric systems is costly and\nrequires many interactions with human users.\n\nThe work that perhaps is the closest in spirit with ours is that by\n\\cite{Williams2007,Thomson2007,bui2010} who focus on using (factored)\nPOMDPs to manage spoken dialogue systems in various domains. However,\nthese systems, which are quite sensitive to the uncertainty inherent in\nthe spoken utterances, require considerable training to learn\ngood policies and thus drive the dialogue.\n\n\n\n\\hide{\n\n\\jo{\\textbf{Summary of articles 6 - 10} None of them is directly related to planning. I suggest we leave them all out of the related work except number 7 - \\cite{DBLP:journals\/aim\/Ortiz18} which is already cited in the first paragraph of this article. \\\\\n\n\\begin{enumerate}\n\\item[6] Method for assigning a topic label to utterances in a chat log. Starts from noisy hand-crafted labelling and iteratively updates using reinforcement learning. They use rewards designed such that local topic continuity is enforced and global split into topics is such that utterances between topics are different. Meat is in sections 3.1 - 3.4. \n\n\\item[7] From Christian: General argument for using all of the sub-modules possible in order to have an effective dialogue agent (including a need for a general planning component). Also points out that all of the sub-systems need to be designed with the holistic view in mind as it effects how they can inter-operate.\n\nI would add that the author works in Nuance Communications and mentions a few practical examples. \n\n\\item[8] Emphasizes the need for policy driven dialogues as opposed to end-to-end. It uses reinforcement learning for learning the policy which could be posed as an alternative\/complementary approach to our work. Something like action preconditions must also be specified (execution mask in section II D). Based on state of the world (parsed entities ...) and the learned policy the algorithm selects next action and updates the policy according to the reward it receives (small negative for each step and big positive when user's goal is achieved). It employs a user simulator for training. \n\n\\item[9] Historical overview of methods how to do conversational agents. Mostly concerned with end-to-end systems. \n\n\\item[10] From Christian: Proposal for a user simulator. Also useful for the domain they have setup (movie booking with required \/ optional intents, etc). Code is available, so we could potentially use this as an example to model explicitly, or for automated model acquisition. \n\\end{enumerate}\n\n}\n\n\\subsection{Citations from Christian's Mendeley}\n\n\\cm{I haven't yet read through all of these, and there are some repeats from Adi's text, but so far these are the papers that I've identified that may be relevant. Most have links, but I have a PDF for all of them.}\n\n\\begin{enumerate}\n  \\item Srivastava, B. (2018). On Chatbots Exhibiting Goal-Directed Autonomy in Dynamic Environments, 588\u2013590. Retrieved from http:\/\/arxiv.org\/abs\/1803.09789\n  \\item Feng, W., Zhuo, H. H., \\& Kambhampati, S. (2018). Extracting Action Sequences from Texts Based on Deep Reinforcement Learning, 4064\u20134070. https:\/\/doi.org\/arXiv:1803.02632v2\n  \\item Camilleri, G. (n.d.). Dialogue systems and planning, 1\u20138.\n  \\item Steedman, M., \\& Petrick, R. (2007). Planning dialog actions. SIGdia 2007, (September), 265\u2013272. Retrieved from http:\/\/homepages.inf.ed.ac.uk\/steedman\/papers\/dialog\/sigdial07.pdf\n  \\item Foster, M. E., \\& Petrick, R. P. A. (2017). Separating representation, reasoning, and implementation for interaction management: Lessons from automated planning. Lecture Notes in Electrical Engineering, 999 LNEE, 93\u2013107. https:\/\/doi.org\/10.1007\/978-981-10-2585-3$\\_$7\n  \\item Takanobu, R., Huang, M., Zhao, Z., Li, F., Chen, H., Zhu, X., \\& Nie, L. (2017). A Weakly Supervised Method for Topic Segmentation and Labeling in Goal-oriented Dialogues via Reinforcement Learning.\n  \\item Ortiz, C. L. (2018). Holistic Conversational Assistants, 88\u201390.\n  \\item Weisz, G., Budzianowski, P., Su, P.-H., \\& Ga\u0161i\u0107, M. (2018). Sample Efficient Deep Reinforcement Learning for Dialogue Systems with Large Action Spaces, 1\u201314. Retrieved from http:\/\/arxiv.org\/abs\/1802.03753\n  \\item Mathur, V., \\& Singh, A. (2018). The Rapidly Changing Landscape of Conversational Agents, 1\u201314. Retrieved from http:\/\/arxiv.org\/abs\/1803.08419\n  \\item Li, X., Lipton, Z. C., Dhingra, B., Li, L., Gao, J., \\& Chen, Y.-N. (2016). A User Simulator for Task-Completion Dialogues. Retrieved from http:\/\/arxiv.org\/abs\/1612.05688\n  \\item Romero, O. J., Zhao, R., \\& Cassell, J. (2017). Cognitive-inspired conversational-strategy Reasoner for socially-aware agents. IJCAI International Joint Conference on Artificial Intelligence, 3807\u20133813.\n  \\item Petrick, R. P. a, \\& Foster, M. E. (2004). Planning for Social Interaction in a Robot Bartender Domain. Proceedings of the Twenty-Third International Conference on Automated Planning and Scheduling, 389\u2013397. Retrieved from http:\/\/www.aaai.org\/ocs\/index.php\/ICAPS\/ICAPS13\/paper\/viewFile\/6039\/6208\n  \\item Casanueva, I., Budzianowski, P., Su, P.-H., Ultes, S., Rojas-Barahona, L., Tseng, B.-H., \\& Ga\u0161i\u0107, M. (2018). Feudal Reinforcement Learning for Dialogue Management in Large Domains. Retrieved from http:\/\/arxiv.org\/abs\/1803.03232\n  \\item Black, E., Coles, A., \\& Bernardini, S. (2014). Automated planning of simple persuasion dialogues. Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 8624 LNAI, 87\u2013104. https:\/\/doi.org\/10.1007\/978-3-319-09764-0$\\_$6\n  \\item Garoufi, K., \\& Koller, A. (2010). Automated planning for situated natural language generation. Acl, (July), 1573\u20131582.\n  \n  \\ob{I've browsed through the following 6 papers. There's nothing that is really similar to what we do and nothing that uses automated planning in our sense. \n  \n  Further notes under each paper, either very briefly from me, or from Christian.}\n  \\item Feng, W., Zhuo, H. H., \\& Kambhampati, S. (2018). Extracting Action Sequences from Texts Based on Deep Reinforcement Learning. Retrieved from http:\/\/arxiv.org\/abs\/1803.02632\n  \\ob{\nFeng et al (2018) extract sequences of actions from natural text (as in \"The important thing to remember is not to heat up the rice, but keep it cold.\" -> keep(rice, cold)) using Reinforcement Learning. \n\nIn our context, this could be useful for authoring an action-based dialogue system using natural language. }\n  \\item Raiman, J., \\& Raiman, O. (2018). DeepType : Multilingual Entity Linking by Neural Type System Evolution.\n  \\ob{Integrates symbolic information (a type system) into NN reasoning. The system itself extracts the necessary structures. I'd say it's rather off-topic.}\n  \\item Ilievski, V., Musat, C., Hossmann, A., \\& Baeriswyl, M. (2017). Goal-Oriented Chatbot Dialog Management Bootstrapping with Transfer Learning. Retrieved from https:\/\/arxiv.org\/pdf\/1802.00500.pdf\n  \\ob{An approach to help RL learning of dialogue policies by transfer learning from data rich domains.\n  \n Christian's notes:\n  \n General approach is to train a GO chatbot via RL, and then use the overlap with another domain to seed the learning for the new domain. They look at both the partial overlap setting, as well as domain extension (first domain is a subset of the next).\n\nSection 4 seems to be the entire meat of the approach, and there isn't much. I guess it's something that need to be tried, but they don't do anything smart with the weights that carry over, do assume they already have the annotated overlap, don't operate on the NLU \/ NLG level (just the idealized semantic frame level), etc, etc.\n\nThe one thing that may be useful to take away from this paper are the domains that are used. Generally the accuracy is very poor, and a model-based approach should be able to accomplish far greater things.\n}\n  \\item Azaria, A., Krishnamurthy, J., \\& Mitchell, T. M. (2016). Instructable Intelligent Personal Agent. Proceedings of the 30th Conference on Artificial Intelligence (AAAI 2016), 2681\u20132689. Retrieved from http:\/\/ai2-website.s3.amazonaws.com\/publications\/LearnByInst.pdf\n  \n  \\ob{Approach to learn new commands - a mapping NL sentence -> action sequence. No planning though.    \n  \n Christian's notes:\n  \nIntroduces a (traditional NLP) way of interacting with a domain-specific agent so that new commands can be learned. The scope is fairly narrow, but the general idea is a wonderful target -- interacting through natural language to teach an agent new skills.\n\nThe sentences are parsed using a specifically set up and learned grammar, and the key component is to identify the main entities so they can be lifted for generalization (e.g., other email targets).\n\nA significant part of the work focuses on the user study, and all of the qualitative metrics that were lifted from the interaction.\n  }\n \n \\item Peng, B., Li, X., Gao, J., Liu, J., \\& Wong, K.-F. (2018). Integrating planning for task-completion dialogue policy learning. Retrieved from http:\/\/arxiv.org\/abs\/1801.06176\n    \\ob{This certainly falls rather under RL than planning. What they call \"planner\" is a user-simulator that they use on some iterations instead of querying a real user to aid their RL and make it less expensive. No planning in our sense though.\n    \n    \n    Further Christian's notes:\n    \nTheir general approach is to do a bit of look-ahead during online conversations to improve the policy. The use of the word \"planner\" is extremely misleading here.\n\nEssentially, their planner is a policy that is trained to mimic the end-users. This way, an on-the-fly simulation can be interleaved with the real dialogue, and mini segments of offline training can be done online.\n\nNot quite sure what the model is that is being learned, but it appears as though the agent already has most of the components needed for model-based reasoning. The proposed approach also bootstraps with assumed offline generated training.\n\nAnother key point is that they do very little NL work (and provide little details as to how it's done). The NLU and NLG components are entirely missing, and the dialogue agent policy that is being built seems less than compelling -- major reason for this, is that (as far as I can tell), this problem would be trivial for a model-based planner to tackle. No need for RL, no need for deep nets. Etc.\n\nAnyways, there's a good set of references, and simulation is criticized with valid references (which kind of contradicts what it is that they're doing).\n\n  }\n  \\item Shah, P., Hakkani-Ur, D., Ur, G., \\& Rastogi, A. (2017). Building a Conversational Agent Overnight with Dialogue Self-Play, (i). Retrieved from https:\/\/arxiv.org\/pdf\/1801.04871.pdf\n    \\ob{Christian's notes:\n    \n      This work is essentially dedicated to a new way for collecting\n      data on annotated conversations. Instead of having humans\n      converse with one another and then annotate with the slots and\n      frames (or having others annotate), they flip this on its head\n      and generate dialogue at the slot\/frame level (using\n      simulators), then have simple NLG create a crude approximation\n      of a natural utterance, and \/then\/ have a turker paraphrase it\n      (sanity checks done with crowdsourcing as well).\n\n      It's basically a new way to do data collection, and seems to be\n      smart about it. We could possibly use the domains in our own\n      testing, but this falls under the camp of just simple dialogue\n      generation end2end.  }\n\\end{enumerate}\n\n}\n\n\n\n\\section{Summary}\n\\label{sec:summary}\n\nDialogue agents capable of handling multi-turn, goal-oriented conversations\nare becoming increasingly important across a range of domains, \nincluding human resources, career coaching, and personal assistants.\nWe have presented an approach to constructing dialogue plans automatically.\nGiven a library of individual actions available, \nour system produces dialogue plans customized to achieving a given goal.\nDialogue plans are further plugged into a dialogue system that can \norchestrate their execution during a conversation with a user.\nWe have shown that our approach is viable and scalable.\nOur work has been applied in one product, and is in the process of \nbeing integrated into a second one.\n\nFuture work includes building dialogue agents in additional domains.\nDynamically interleaving dialogue agents, to allow the user\nto temporarily change the topic in the middle of a dialogue,\nis another important topic for future work.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \\label{SEC-I}\n\nSoft sintered Mn-Zn ferrites are widely used in high frequency applications based on switching circuits. Their relatively high resistivity and permeability make them optimal for the $30$kHz-$1$MHz bandwidth. \nMiniaturization and the increase of working frequencies make thermal and loss managements the keys for future design of electronic switching devices. It is thus necessary to establish reliable methods to measure, understand and predict the losses in soft ferrites at medium and high frequencies. \n\nLosses are commonly measured using the flux-metric method \\cite{Sato1987-1, Saotome1997-1, Fiorillo2009-1}. However, when excitation frequencies are above $10$ kHz, measurements  can be dramatically affected by spurious phase contributions due to stray capacitances and inductances of the circuits and probes. A possible alternative is the calorimetric method; in adiabatic conditions, the power dissipated in the magnetic material is directly related to the heating rate, reducing the measurement to that of a temperature ramp. We discussed in detail this technique in a recent paper  \\cite{Loyau2009-1} where an experimental set-up  using a platinum probe as temperature sensor was presented. A further improvement of the technique has been proposed \\cite{Loyau2010-1}   by measuring the temperature from the bulk DC resistivity of the ferrite core \\cite{Loyau2010-1}. Whereas the temperature deduced from bulk resistivity represents a measurement averaged over the whole sample volume (as in the case of flux-metric measurements), the platinum probe measurement gives information on dissipation taking place in a thin strip of the material just near its surface. Consequently the comparison between the two measurements can be used  to estimate the loss inhomogeneity, and thus, the relevance of skin effect for a given sample geometry and excitation frequency. It will be shown that, global low level electromagnetic  parameters, obtained by impedance spectroscopy, can be used as local input parameters for the FEM computation of magnetic loss in linear regime. In this way we shall calculate the volume or surface averaged loss values, corresponding to the two types of calorimetric measurements and compare them with experimental results. \n\n\\section{Experimental and computation details} \\label{SEC-II}\n\nPower losses in magnetic materials under AC excitation are associated with all the irreversible phenomena taking place during the core magnetisation (i.e. Barkhausen jumps, eddy currents, domain wall resonance, spin dumping, magnetoacoustic emission, etc.). Under  adiabatic conditions \\cite{Loyau2009-1}, all the dissipated power is converted into heat and induces a temperature increase of the sample. Thus the temperature field through the core $T(x, y, z, t)$ can be described by means of the linearised heat equation giving:\n\n\\begin{equation} \\label{EQ:dT\/dt}\n\\frac{\\partial T(x, y, z, t)}{\\partial t} = \\frac{1}{\\rho_m c_p} p_s(x, y, z)\n\\end{equation}\n\nwhere $p_s(x, y, z)$ is the local heat source density (in Wm$^{-3}$), $c_p$ is the specific heat (in JK$^{-1}$kg$^{-1}$) of the sample and $\\rho_m$ its density (in kgm$^{-3}$). Now, if $V_0$ is the volume of the sample, the magnetic loss per cycle measured using the flux-metric method, at a given excitation frequency $f$, corresponds to the average loss over the sample volume, namely:\n\n\\begin{equation} \\label{EQ:W}\n \\langle W \\rangle = \\frac{c_p}{f V_0} \\int_V \\frac{\\partial T(x, y, z, t)}{\\partial t} dV.\n\\end{equation}\n\nWhereas a measure of the average temperature of the whole sample will result in a correct extrapolation of $\\langle W \\rangle$, any local measurement will be related with dissipations taking place just in a small volume near the position of the temperature probe. In the latter case  $\\langle W \\rangle$ can be deduced properly only when volume losses are uniformly distributed through the sample. \n\nTwo different techniques where used to measure the temperature ramp: the first  using a platinum probe pasted (Pt500) on the sample \\cite{Loyau2009-1} and therefore sensible to the temperature change in a narrow strip just near the surface, and a second technique  based on the measurement of previously calibrated bulk resistivity of the ferrite core \\cite{Loyau2010-1} which returns a global average of the temperature ramp. The latter leads $\\langle W \\rangle$ notwithstanding the inhomogeneities of the loss field due to the geometry of the sample and to the skin effect. In both cases, the sample was a square cross section bar (I rod: $0.64\\times0.64\\times2.54$ cm$^3$) made of 3E27 Ferroxcube Mn-Zn ferrite. Complex impedance spectra of the permeability $\\mu = \\mu' + i \\mu''$ and of the electrical conductivity $\\sigma = \\sigma' + i \\sigma'' = \\sigma' + i \\epsilon \\omega$ (where $\\epsilon$ is the electrical permittivity and $\\omega =  2 \\pi f$ is the angular frequency) where performed with a HP 4195 impedance analyser between 100 kHz and 10 MHz. The losses where measured in a vacuum chamber between 100 kHz and 2 MHz as described in ref. \\cite{Loyau2009-1,Loyau2010-1} from $10$ second long temperature recordings.\n\n\n\n\nIn order to validate the interpretation of the measurements performed with the two techniques, an approach based on the solution of Maxwell equations was used. For this purpose, the code FEMM\\cite{FEMM} was used taking  measurements of complex magnetic permeability and conductivity as intrinsic material parameters. All the following analysis is based on three strong assumptions: (\\textit{i}) the peak induction values are sufficiently low to consider the magnetic behaviour as linear. Thus the concept of real and imaginary magnetic permeability and their relationship with energy losses is valid. This assumption is confirmed by the fact that all the loss measurements presented in this paper scale roughly as the square of the peak induction. This means that the term $\\mu''\/\\mid \\mu \\mid^2$ remains constant in the investigated induction range (from $3$mT to $50$mT);\n(\\textit{ii}) all the measurements of complex magnetic permeability are performed on a toroidal ferrite sample with section small enough to make eddy current loss and skin effect negligible; \n(\\textit{iii}) all the measurements of complex conductivity are performed on bar shaped samples which section is sufficiently reduced to neglect skin effect. \n\nFrom (\\textit{ii}) the measured $\\mu$ is a parameter related to the local phase lag between $\\textbf{H}$ and $\\textbf{B}$ due to all dissipative phenomena but eddy current losses. On the other hand, the complex electrical conductivity $\\sigma$ results from the mixed electrical properties of the grain and of the grain boundaries \\cite{Goodenough2002-1}. Its real part $\\sigma'$, is related to ohmic and dielectric loss effect whereas the imaginary part $\\sigma''$, is due to the effect of mixed permittivity.\n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.5]{fig1a.eps}\n\\label{fig1a}\n}\n\\centering\n\\subfigure[]{\\includegraphics[scale=0.5]{fig1b.eps}\n\\label{fig1b}\n}\n\\caption[Optional caption for list of figures]{Total loss field $w_{tot}(x, y)$ calculated from Eq. (\\ref{EQ:EDDY}) and (\\ref{EQ:magloss}) for $f=100$kHz \\subref{fig1a} and f=$2$MHz \\subref{fig1b}. The dashed rectangle indicates the region used to calculate the surface loss as measured by the platinum probe. The thickness of the rectangle is defined as the product of the heat propagation speed in the material and the measure duration ($10$ seconds). Grey scales indicate the local loss in mJkg$^{-1}$ where the loss varies from $0.04$mJkg$^{-1}$ (dark regions) to  $0.05$mJkg$^{-1}$ (white regions) in \\subref{fig1a} , and  from $0.45$mJkg$^{-1}$  to $0.85$mJkg$^{-1}$  in \\subref{fig1b} . The ratio between the maximum  and the minimum values is $1.25$ at $f=100$kHz \\subref{fig1a} and $2$ at f=$2$MHz \\subref{fig1b}}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.65]{fig2.eps}\n\\caption{Comparison between the bulk measurements of the average total loss and the calculated ones. Peak inductions are: $3$mT, $10$mT, $30$mT, $40$mT and $50$mT  }\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.65]{fig3.eps}\n\\caption{Comparison between bulk and surface loss measurements and the same for computations (over the whole volume and the one delimited by the dashed line in Fig.\\ref{fig1}.  The peak induction is of $50$mT  }\n\\label{fig3}\n\\end{figure}\n\n\n\n\nGiven the geometry of the problem, with the magnetic field always perpendicular (along $z$ axis) to the core section, the calculation of the loss field reduces to a two dimensional problem in the $xy$ plane. Now, under sinusoidal excitation when $H_z(x, y, t) = H_0(x, y) e^{i \\omega t}$, substituting the measured $\\sigma$ and $\\mu$ in the diffusion equation for the magnetic field we have,\n\n\\begin{equation} \\label{EQ:Diffusion2}\n  \\frac{1}{\\mu} \\triangledown^2 H_z(x, y, t) = i \\omega \\sigma H_0(x, y) .\n\\end{equation}\n\nFrom this equation and from the fact that $ \\textbf{j} = \\triangledown \\times \\textbf{H}$ we can calculate the eddy-current loss field across the core section,\n\n\\begin{equation}\\label{EQ:EDDY}\n p_{ec}(x, y) = \\frac{1}{2} \\rho' \\lvert \\textbf{j}(x, y) \\rvert^2\n\\end{equation}\n\nwhere $\\rho' = Re(1\/\\sigma)$. This expression includes all the eddy current related losses. The effect of ohmic and dielectric dissipation in both, the grains and their boundaries, are described by the real part of the resistivity $\\rho'$.\nOn the other hand, the complex part of the permeability $\\mu''$, measured on the very small sample, will account for all the magnetisation process related dissipation phenomena ( excess losses, spin damping, etc.). So we can write the magnetic loss field as:\n\n\\begin{equation}\\label{EQ:magloss}\n p_{mag}(x, y) = \\frac{1}{2} \\omega \\mu'' \\lvert \\textbf{H}_z (x, y) \\rvert^2= \\frac{1}{2} \\omega \\frac{\\mu''}{\\lvert \\mu \\rvert^2 } \\lvert \\textbf{B}_z(x, y) \\rvert^2\n\\end{equation}\n\nThe total power loss will be the sum of the two terms of Eq. (\\ref{EQ:EDDY}) and (\\ref{EQ:magloss}), and the loss per cycle and unit mass will be $ w_{tot}(x, y)= p_{tot}(x, y) \/ (f \\rho_m)$. By averaging over the core section and multiplying by the length, we obtain $\\left\\langle W\\right\\rangle_{v}=\\ell\\int_{-a\/2}^{a\/2}\\int_{-a\/2}^{a\/2}w(x,y)\\mathrm{d}x\\mathrm{d}y$ corresponding to the bulk (resistance) measurement, whereas $\\left\\langle W\\right\\rangle _{s}=\\ell\\int_{0}^{a\/2}\\int_{0}^{d}w(x,y)\\mathrm{d}x\\mathrm{d}y$ yields the value corresponding to surface (PT 500) measurement. The thickness $d$ is defined as the product  of the heat diffusion velocity in the material and the duration of the measurement ($10$ seconds). \n\n\\section{Results and discussion} \\label{SECC-IV}\n\n\nIn order to determine the magnetic field $H_z(x, y)$ across the section area we must specify the induction conditions under which losses have been both calculated and measured. Here we consider the case where a sinusoidal magnetic induction of average value $\\langle B_0 \\rangle$ is forced across the core section area $S$. This implies that the magnetic field verifies:\n\n\\begin{equation}\\label{EQ:Induction}\n \\langle B_0 \\rangle = \\frac{1}{S} \\int_S \\mu H_0(x, y) dS\n\\end{equation}\n\nThe magnetic field $H_z(x, y)$ has been determined solving Eq.(\\ref{EQ:Diffusion2}) coupled with Eq.(\\ref{EQ:Induction}) with a two dimensional finite element code in a frequency range between $100$kHz and $2$MHz on the sample geometry. Calculation results are shown in Fig. \\ref{fig1} where the loss field $w_{tot}(x, y)$ is plotted for $f=100$kHz in Fig.\\ref{fig1a} and $f=2$MHz in Fig.\\ref{fig1b}. \n\n\n\nIn Fig.\\ref{fig2} comparison between the bulk measurements of loss and the calculated average total loss $\\left\\langle W\\right\\rangle_{v}$ is shown for peak inductions ranging from $3$mT to $50$mT. The agreement between measurements and calculations is particularly good for frequencies above $300$kHz. The over estimation of computed losses may be due to the compensation error in the measurement of $\\mu''$ performed with the impedance analyser. Indeed, at low frequency the resistive part of the impedance is nearly zero, so even a small absolute error can yield a strong systematic error. \nFig.\\ref{fig3}  shows the comparison between the surface measured loss and corresponding computation $\\left\\langle W\\right\\rangle_{s}$. Calculations and measurements are in good agreement. With the sample geometry studied here, as the average magnetic path is not much different from the extremal paths in the sample, all the loss inhomogeneities can be ascribed to the skin effect. So, either measurements and finite elements calculations, show that skin effect is negligible in a frequency regime below $600$-$700$kHz. Above these frequencies skin effect becomes relevant, and a large part of the dissipation phenomena is concentrated near the surface of the sample. \nIn conclusion, it has been shown that the calorimetric loss measurement method, exploiting bulk and surface temperature recordings, is able to give indirect information on the eddy current losses in Mn-Zn ferrites. Using FEM computation --for which material parameters input where obtained by impedance spectroscopy-- it is possible to compute the loss fields and to obtain very good agreement with measurements without introducing any fitting parameter. Hence, this model could be integrated in magnetic components design software in order take into account the magnetic losses including eddy currents for the dimensioning. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nLet $1 \\leqslant p<\\infty$ and $\\omega$ be a nonnegative integrable function  on the unit disk $\\mathbb{D}$.  $L^p(\\omega)$ denotes the Banach space with norm\n\\begin{equation*}\n\\|f\\|_{L^p(\\omega)}:=\\bigg(\\int_\\mathbb{D} |f(z)|^p \\omega(z) dA(z)\\bigg)^\\frac{1}{p}.\n\\end{equation*}\nThe weighted harmonic (analytic) Bergman space $L^p_h(\\omega)$ ($L^p_a(\\omega)$)  is the subspace of $L^p(\\omega)$ which is consisting of  harmonic (analytic) functions on $\\mathbb D$. The goal of this paper is to provide a framework to study operator properties (boundedness, compactness, Schatten classes and inveritibility) of Toeplitz operators on $L_h^2(\\omega)$ with nonnegative symbols.\n\nWeighted analytic function spaces and their Toeplitz operators have captured people's attentions for a long time. It is now well known (\\cite{Zhu2007}) that several results on unweighted Bergman space can be extended to the standard weighted Bergman space $L_a^2(\\omega_\\alpha)$,  where $\\omega_\\alpha(z)=(1+\\alpha)(1-|z|^2)^\\alpha$ and $-1<\\alpha<\\infty$.\nIn recent papers \\cite{PR2015} and \\cite{PR2016}, Pel\\'{a}ez and R\\\"{a}tty\\\"{a} characterized the bounded and Schatten class Toeplitz operators (induced by a positive Borel measure) on a weighted Bergman space, here the weight is a radial function satisfying the doubling property $\\int_r^1 \\omega(s)ds \\leqslant C\\int_{\\frac{1+r}{2}}^1 \\omega(s)ds.$  The first results of non-radial weighted Bergman space seems to be due to Luecking (\\cite{Luecking1985}) who investigated  the structure of weighted Bergman space with B\\'{e}koll\\'{e}-Bonami weight. Based on Luecking's representation and duality theorems in \\cite{Luecking1985},  Chac\\'{o}n (\\cite{Cha}) and Constantin  (\\cite{Con2007}, \\cite{Con}) studied the boundedness and compactness of Toeplitz operators on certain weighted Bergman spaces. In \\cite{MW2014}, Mitkovski and Wick established a reproducing kernel thesis for operators on Bergman type space, and their definitions include weighted versions of Bergman spaces on more complicated domains.\n\n\nWe will be primarily interested in weighted harmonic Bergman space $L_h^2(\\omega)$.  Our choice of the weight $\\omega$ is motivated by the  characterization of boundedness of $P_h$ acting on $L^2(\\omega)$, where $P_h$ is the unweghted harmonic Bergman (orthogonal) projection from $L^2(dA)$ to $L_h^2(dA)$. It is well known that  $L_h^2(dA)$ is a reproducing kernel Hilbert space and\n\\begin{equation*}\nP_hf(z)=\\int_\\mathbb{D} f(\\lambda) \\bigg[\\frac{1}{(1-\\overline{\\lambda }z)^2}+\\frac{1}{(1-\\overline{z}\\lambda)^2}-1\\bigg]dA(\\lambda).\n\\end{equation*}\n\nClearly, $P_h$ is a Calder\\'{o}n-Zygmund operator on the homogeneous space $(\\mathbb{D},d,dA)$,  where $d$ is the Euclidean distance and $dA$ is the Lebesgue area measure on  $\\mathbb D$, normalized so that the measure of $\\mathbb D$ is $1$.\n For the  definitions of Calder\\'{o}n-Zygmund operator and homogeneous space, we refer to \\cite{AV2014}.\n\n\n\n\nThe most successful understanding of the (one) weight theory of Calder\\'{o}n-Zgumund operator was spurred by Munckenhoupt's work in 1970s (\\cite{MW1974}), which led to the introduction of the class of $\\mathcal{A}_p$ weight and developments of weighted inequality. We will restrict our attentions to $\\mathcal{A}_2$ weight on $(\\mathbb{D}, d, dA)$.\n\nLet $0<\\omega\\in L^1(\\mathbb{D}, dA)$, it is called a Muckenhoupt $\\mathcal{A}_2$ weight if\n$$[\\omega]_{\\mathcal{A}_2}:=\\sup_{a\\in\\mathbb{D},\\  0<r<1} \\frac{|B(a,r)|_{\\omega}|B(a,r)|_{\\omega^{-1}}}{|B(a, r)|^2} <+\\infty,$$\nwhere $$B(a,r)=\\{z\\in\\mathbb{D}: d(a,z)=|z-a|<r\\},$$\n$$|B(a,r)|_\\omega=\\int_{B(a, r)}\\omega(z) dA(z)$$ and $|\\cdot|$ is the normalized Lebesgue measure on $\\mathbb{D}$.\n\nIt follows form the remarkable $\\mathcal{A}_2$ theorem (\\cite{AV2014},\\cite{Hy2012}) that \\emph{$P_h$ is  bounded  from $L^2(\\omega)$ to $L_h^2(\\omega)$ provided $\\omega$ is a Muckenhoupt $\\mathcal{A}_2$ weight.}  As mentioned above, we will focus on  the weighted harmonic Bergman space $L_h^2(\\omega)$ with  $\\omega\\in \\mathcal{A}_2$.  Little is known about this nature function  space. However,  we will see in Section 2  that $L_h^2(\\omega)$ is  a reproducing kernel Hilbert space with  the reproducing kernel  $K^\\omega_z(\\lambda)$, i.e.,\n$f(z)=\\langle f,K^\\omega_z\\rangle_{L^2(\\omega)}$ for all  $f$  in $L_h^2(\\omega)$.\n\nFor a positive finite Borel measure $\\nu$ on $\\mathbb D$,  we densely define the Toeplitz operator $T_\\nu$ on  $L_h^2(\\omega)$ by\n $$T_\\nu f(z)=\\big\\langle T_\\nu f, K^{\\omega}_z \\big\\rangle_{L^2(\\omega)}=\\int_{\\mathbb D}f(\\lambda)\\overline{K^{\\omega}_z(\\lambda)}d\\nu(\\lambda) \\ \\ \\  \\big(z\\in \\mathbb D\\big).$$\nFor a bounded function $\\varphi$, using the integral representation for the projection operator (from $L^2(\\omega)$ to $ L_h^2(\\omega)$), we can express the Toeplitz operator $T_\\varphi$\n(on $L_h^2(\\omega)$) as the following:\n$$T_\\varphi f (z)=\\int_{\\mathbb D}f(\\lambda) \\overline{K^{\\omega}_z(\\lambda)} \\varphi(\\lambda)  \\omega(\\lambda)dA(\\lambda) \\ \\  \\ \\big(z\\in \\mathbb D\\big).$$\n\nAlthough we follow Luecking's methods in \\cite{Luecking1985} and \\cite{Luecking AJM} for the weighted Bergman spaces, some new difficulties arise in the study of the space $L_h^2(\\omega)$ and the corresponding operators. For instance,  harmonic\nfunctions do not share many powerful tools with analytic functions. One can use the Cauchy formula to estimate the local values of  analytic functions easily. However, because of the tedious remainder, the harmonic version Cauchy formula (known as Cauchy-Pompeiu formula) is not valid now.\nWe instead must rely on some known estimations  on  harmonic functions.  In addition,  just as the weighted Bergman space, one cannot write down an explicit formula for  the reproducing kernel for $L_h^2(\\omega)$. To overcome this obstacle, we will use the  reproducing kernel for the unweighted space $L_h^2$ to help us  study the representation theory of $L_h^2(\\omega)$. However, the properties of reproducing kernel for $L_h^2$ are much more complicated than that of Bergman space $L_a^2$.\n\nUsing some properties of harmonic functions and $\\mathcal {A}_2$ weights, we establish two different atomic decompositions for functions in $L_h^2(\\omega)$ (Theorems \\ref{Atomic Decomposition} and \\ref{Atomic Decomposition cor}), which extend the representation theorems in \\cite{Luecking1985} to the harmonic case.\n\n\nIn Section 3,  we  characterize the boundedness, compactness and  Schatten $p$ class of Toeplitz operators $T_\\nu$ on $L_h^2(\\omega)$  by means of  Berezin transform and Carleson measure. We are pleasure to  mention here that Miao (\\cite{Miao}) have obtained characterizations for the Toeplitz operators with nonnegative symbols to be bounded, compact and in Schatten classes on unweighted harmonic Bergman space $L_h^2$.\n\nSection 4 of this paper is devoted to studying the invertibility of Toeplitz operators on the standard weighted  harmonic Bergman space $L^2_h(\\omega_\\alpha)$.  A little surprising to us, the results illustrate that  the invertibility of Toeplitz operators on Bergman space $L_a^2$ can imply a reverse Carleson  inequality for $L_h^2(\\omega_\\alpha)$, see  Theorems \\ref{invertibility1} and  \\ref{invertibility2}. Based on this inequality, we  generalize  the result on the invertibility of  Bergman Toeplitz operators  with nonnegative symbols (\\cite{Zhao}) to the case of $L_h^2(\\omega_\\alpha)$. As a consequence, we obtain a relationship of the invertibility  between  Toeplitz operators with bounded nonnegative symbols on $L_a^2(\\omega_\\alpha)$ and $L_h^2(\\omega_\\alpha)$, see Corollary \\ref{invertibility3}.\n\nIn the final section, we establish a reverse Carleson type inequality for $L_h^2(\\omega)$ with $\\omega\\in\\mathcal{A}_2$. Indeed, we obtain a sufficient condition for $\\chi_G dA$ to be a reverse Carleson measure for $L_h^2(\\omega)$,  where $G$ is a measurable set in $\\mathbb D$, see Theorem \\ref{the weight w}, which  extends Theorem 3.9 for the weighted  (analytic) Bergman space in \\cite{Luecking1985} to the harmonic setting.\n\n\nThroughout the paper, positive constants will be explicitly denoted by $C, C_0, C_1,\\cdots$, which may depend on some fixed numbers and\nchange at each occurrence.\n\n\\section{The Space $L_h^2(\\omega)$ and its Representation}\n\nIn this section, we present some elementary structures of $L_h^2(\\omega)$ with $\\omega \\in \\mathcal A_2$.  To study the\nharmonic Bergman spaces, we need the following important properties of harmonic functions.\n\\begin{lemma}\\label{subharmonic behavior of harmonic functions} (\\cite{Kuran})\nSuppose that $f$ is a harmonic function on the disk $\\mathbb D$ and $0<p<\\infty$. There exists a positive constant\n$C=C(p)$ such that for every ball  $B(a, r)=\\{z\\in \\mathbb D: |z-a|<r\\}$ in $\\mathbb D$,\n$$|f(a)|^p\\leqslant \\frac{C}{|B(a,r)|} \\int_{B(a,r)} |f(z)|^p dA(z).$$\n In particular, if $p\\geqslant 1$, then the constant $C\\equiv 1$.\nUsing this result one can get the following useful inequalities easily: given $0<p<\\infty$ and $0<r<1$, there exist  positive constants\n$C_1=C_1(p)$ and $C_2=C_2(p)$  such that\n$$|f(a)|^p \\leqslant \\frac{C_1}{(1-r)^{2}}\\cdot \\frac{1}{|D(a,r)|} \\int_{D(a,r)} |f(z)|^p dA(z) \\ \\  \\  \\big(a \\in \\mathbb D\\big)$$\nand\n$$|\\partial f(a)|^p \\leqslant \\frac{C_2}{(1-r)^{2+p}}\\cdot \\frac{1}{|D(a,r)|^{\\frac{p+2}{2}}} \\int_{D(a,r)} |f(z)|^p dA(z) \\ \\  \\  \\big(a \\in \\mathbb D\\big)$$\nfor all $f$ harmonic on $\\mathbb D$, where $\\partial f= \\frac{\\partial f}{\\partial z}$.\n\\end{lemma}\n\n\n\n\n\\emph{Remark 1. }From the above inequalities, it is easy to  show that point evaluations are bounded linear functionals on $L_h^2(\\omega)$ with $\\omega \\in \\mathcal A_2$. As a consequence, $L_h^2(\\omega)$ is a reproducing kernel  Hilbert space.\n\n\n\\emph{Remark 2.} It is not clear whether  $L^2_h(\\omega)$ is complete for a general weight. However, if  $p$ is an analytic polynomial on $\\mathbb D$ and  $\\omega(z)=|p(z)|^2$, Douglas and Wang \\cite{DW2011} showed  that $L_a^2(\\omega)$ is complete, whose proof heavily depends on some  particular properties of polynomials.\n\nIt is clear that $L_h^2(\\omega)$ coincides with its dual space with respect to the $L^2(\\omega)$ inner product.  The next result illustrates that the dual space of $L_h^2(\\omega)$ can be identified with $L^2_h(\\omega^{-1})$ via the unweighted inner product,  which generalizes Luecking's result for $L_a^2(\\omega)$ (see Theorem 2.1 in \\cite{Luecking1985}) to the  setting  of $L_h^2(\\omega)$.\n\\begin{lemma}\\label{duality}\n  Suppose that $\\omega$ is an $\\mathcal{A}_2$ weight. Then the dual space of $L_h^2(\\omega)$ can be identified with $L_h^2(\\omega^{-1})$. The pairing is given by\n$$\\langle f, g\\rangle=\\int_{\\mathbb D} f(z)\\overline{g(z)}dA(z).$$\nConsequently, there exists a bounded,  bijective and linear operator $\\mathscr{F}: L_h^2(\\omega^{-1})\\rightarrow (L_h^2(\\omega))^*$ such that\n$$\\mathscr{F}(f)(g)=\\langle g, \\overline{f}\\rangle_{L^2(dA)}$$\nfor $f\\in L_h^2(\\omega^{-1})$ and $g\\in L_h^2(\\omega)$.\n\\end{lemma}\n\\begin{proof}  Let $\\omega$ be an $\\mathcal {A}_2$ weight. Recall that orthogonal projection $P_h:L^2(dA)\\to L^2_h(dA)$ is a Calder\\'{o}n-Zygmund operator on $(\\mathbb{D},d,dA)$. Then $P_h$ is bounded on both $L^2(\\omega)$ and $L^2(\\omega^{-1})$.\nThus for  each  $f\\in L^2(\\omega)$ and $g\\in L^2(\\omega^{-1})$, we have\n$$\\langle P_h f , g\\rangle=\\langle f, P_h g \\rangle.$$\nNote that each $f\\in L_h^2(\\omega)$  (or $f\\in L_h^2(\\omega^{-1})$) has the following form:\n$$f(z)=\\sum_{n=0}^\\infty a_nz^n +\\sum_{n=1}^\\infty b_n \\overline{z^n},$$\nthus we obtain\n\\begin{align*}\nP_hf(z)&= \\int_{\\mathbb D} f(\\lambda)R_z(\\lambda)dA(\\lambda)\\\\\n&=\\lim_{s\\rightarrow 1^{-}}\\int_{s\\mathbb D} f(\\lambda)\\Big[\\frac{1}{(1-\\overline{z}\\lambda)^2}+\\frac{1}{(1-\\overline{\\lambda}z)^2}-1\\Big]dA(\\lambda)\\\\\n&=f(z)\n\\end{align*}\nfor $z\\in \\mathbb D$.\nThen the rest of this proof are exactly the same as the one of Theorem 2.1 in \\cite{Luecking1985}, we omit the details.\n\\end{proof}\n\n\n\n\n\nLet $a\\in \\mathbb{D}$ and $0<r<1$. A pseudohyperbolic disk $D(a, r)$ is defined by\n\\begin{equation*}D(a, r)=\\bigg\\{z\\in \\mathbb D: \\rho(z, a)=\\Big|\\frac{z-a}{1-\\overline{a}z}\\Big|<r\\bigg\\}.\n\\end{equation*}\n\n\n\nWe will frequently use the following  property of $\\mathcal{A}_2$ weights on pseudohyperbolic disks. For the sake of complete, we include a proof of this fact as follows.\n\n\\begin{lemma}\\label{two disks}\nLet $0<r \\leqslant \\frac{1}{4}$ and $z\\in \\mathbb D$. If $\\xi\\in D(z,r)$, then we have\n\\begin{equation*}\n|D(z,r)|_\\omega < 8 [\\omega]_{\\mathcal{A}_2} |D(\\xi,r)|_\\omega.\n\\end{equation*}\n\\end{lemma}\n\\begin{proof} Observe that $D(z, r) \\subset D(\\xi, 2r)$. Now it suffices to show the following doubling inequality:\n\\begin{equation*}\\label{E:doubling}\n|D(\\xi, 2 r)|_\\omega < 8 [\\omega]_{\\mathcal{A}_2}|D(\\xi,r)|_\\omega \\ \\ \\   \\big(\\xi \\in \\mathbb D \\big).\n\\end{equation*}\n\nSince $\\omega$ is an $\\mathcal {A}_2$weight, we have\n\\begin{equation*}\n\\frac{|D(\\xi, 2r)|_\\omega|D(\\xi,2r)|_{\\omega^{-1}}}{|D(\\xi, 2r)|^2} \\leqslant [\\omega]_{\\mathcal{A}_2}.\n\\end{equation*}\nRecall that a pseudohyperbolic disk $D(z,r)$ is a Euclidean disk with center and radius given by\n\\begin{equation*}\n\\mathfrak{C}=\\frac{1-r^2}{1-r^2|z|^2}z,\\quad \\mathfrak{R}=\\frac{1-|z|^2}{1-r^2|z|^2}r.\n\\end{equation*}\nCombining the above with Cauchy-Schwarz's inequality gives\n\\begin{align*}\n|D(\\xi, 2r)|_\\omega &\\leqslant [\\omega]_{\\mathcal{A}_2} \\frac{|D(\\xi, 2 r)|^2}{|D(\\xi, 2r)|_{\\omega^{-1}}}\\\\\n&\\leqslant [\\omega]_{\\mathcal{A}_2}\\frac{|D(\\xi, r)|^2}{|D(\\xi, r)|_{\\omega^{-1}}} \\cdot \\frac{|D(\\xi, 2 r)|^2}{|D(\\xi, r)|^2}\\\\\n&\\leqslant 4[\\omega]_{\\mathcal{A}_2} |D(\\xi, r)|_\\omega  \\cdot \\bigg(\\frac{1-r^2|\\xi|^2}{1-4r^2|\\xi|^2}\\bigg)^2  \\\\\n&< 8 [\\omega]_{\\mathcal{A}_2} |D(\\xi, r)|_\\omega,\n\\end{align*}\nwhere the last inequality is due to $r \\leqslant\\frac{1}{4}$.\nThis completes the proof of Lemma \\ref{two disks}.\n\\end{proof}\n\nWe now turn to the representation theory of the space $L_h^2(\\omega)$.\nThese results and their proof strategies are motivated by Luecking's works on weighted Bergman space (\\cite{Luecking1985}, \\cite{Luecking AJM}).\n\nBefore studying the representation theory of $L_h^2(\\omega)$, we need to recall the concept of $\\epsilon$-lattice in the unit disk.\nLet $\\epsilon \\in (0, 1)$, a sequence $\\{a_n\\}_{n=1}^{\\infty}$ in the unit disk is  called an $\\epsilon$-lattice in the pseudo-hyperbolic metric if\n\\begin{itemize}\n  \\item $\\mathbb D=\\bigcup_{n=1}^\\infty D(a_n, \\epsilon)$ and\n  \\item $\\inf\\limits_{n \\neq m} \\Big|\\frac{a_n-a_m}{1-\\overline{a_n} a_m}\\Big|\\geqslant \\frac{\\epsilon}{2}.$\n\\end{itemize}\n\n\nNow, we are ready to state the atomic decomposition for $L^2_h(\\omega)$.\n\\begin{thm}\\label{Atomic Decomposition}\nLet $\\omega$ be an $\\mathcal {A}_2$ weight. Then there is an $\\epsilon$-lattice $\\{a_n\\}_{n=1}^{\\infty}$ such that for each $f\\in L_h^2(\\omega)$ we have\n$$f(z)=\\sum_{n=1}^\\infty c_n (1-|a_n|^2)^2 |D(a_n, \\epsilon)|_\\omega^{-\\frac{1}{2}} R_{a_n}(z) $$\nfor some sequence $\\{c_n\\}$ in $\\ell^2(\\mathbb N)$, where\n$$R_\\lambda(z)=\\frac{1}{(1-\\overline{z}\\lambda)^2}+\\frac{1}{(1-\\overline{\\lambda}z)^2}-1$$ is the reproducing kernel for $L_h^2$ at $\\lambda\\in\\mathbb{D}$.\n\\end{thm}\n\\emph{Remarks.}  We have the following estimate of the module of  $R_\\lambda$: there exists an $r_0 \\in (0, \\frac{1}{4}]$  such that if $0<r\\leqslant r_0$, then\n$$\\frac{\\frac{1}{2}}{(1-|\\lambda|)^2} \\leqslant |R_\\lambda(z)| \\leqslant \\frac{3}{(1-|\\lambda|)^2}$$\nfor all $z \\in D(\\lambda, r)$. For the proof of this fact, we refer to Lemma 2.2 in \\cite{Choi}. In what follows,  we will use $r_0$  to denote the constant provided in this remarks.\n\n\n\nTo prove Theorem \\ref{Atomic Decomposition}, we need to establish a harmonic version of Luecking's theorems in \\cite{Luecking1985} and \\cite{Luecking AJM}.\n\\begin{thm}\\label{L:key00}\nLet $\\omega$ be an $\\mathcal {A}_2$ weight. Then there exists an $\\epsilon$-lattice $\\{a_n\\}_{n=1}^\\infty$  for some  $0<\\epsilon<\\frac{1}{16}$\nsuch that\n \\begin{equation*}\n \\sum_{n=1}^\\infty |f(a_n)|^2 |D(a_n, \\epsilon)|_\\omega \\approx \\|f\\|_{L^2(\\omega)}^2\n \\end{equation*}\n for all $f$ in $L_h^2(\\omega)$.  That is, there exist two positive  constant $C_1$ and $C_2$ such that\n \\begin{equation*}\\label{E:doublecarleson}\n C_1\\|f\\|_{L^2(\\omega)}^2 \\leqslant \\sum_{n=1}^\\infty |f(a_n)|^2 |D(a_n, \\epsilon)|_\\omega \\leqslant C_2\\|f\\|_{L^2(\\omega)}^2.\n \\end{equation*}\n for all $f$ in $L_h^2(\\omega)$.\n\\end{thm}\n\nOnce Theorem \\ref{L:key00} is established, we can quickly present a proof of Theorem \\ref{Atomic Decomposition} as follows.\n\n\\begin{proof}[Proof of Theorem \\ref{Atomic Decomposition}] Note that both $\\omega$ and $\\omega^{-1}$ are $\\mathcal{A}_2$ weights, it follows from Theorem \\ref{L:key00} that we can choose $\\epsilon\\in (0, \\frac{1}{16})$ and an $\\epsilon$-lattice $\\{a_n\\}_{n=1}^\\infty$ such that\n$$\\|g\\|_{L^2(\\omega^{-1})}^2\\approx  \\sum_{n=1}^\\infty |g(a_n)|^2 |D(a_n, \\epsilon)|_{\\omega^{-1}}.$$\nBy Cauchy-Schawrz inequality and the definition of $\\mathcal{A}_2$ weight,\n\\begin{equation*}\n|D(a_n, \\epsilon)|^2 \\leqslant |D(a_n, \\epsilon)|_\\omega \\cdot |D(a_n, \\epsilon)|_{\\omega^{-1}}\\leqslant [\\omega]_{\\mathcal{A}_2}|D(a_n, \\epsilon)|^2.\n\\end{equation*}\nTherefore,\n$$\\|g\\|_{L^2(\\omega^{-1})}^2\\approx \\sum_{n=1}^\\infty |g(a_n)|^2 (1-|a_n|^2)^4 |D(a_n, \\epsilon)|_\\omega^{-1} .$$\nThis implies that the linear operator $\\mathscr{L}: L_h^2(\\omega^{-1})\\rightarrow \\ell^2(\\mathbb Z)$ defined  by\n$$\\mathscr L (g):=\\bigg\\{g(a_n) (1-|a_n|^2)^2 |D(a_n, \\epsilon)|_\\omega^{-\\frac{1}{2}}\\bigg\\}_{n=1}^\\infty$$\nis bounded below and so its range is closed. It follows from  the closed range theorem that $\\mathscr L ^*$  is surjective.\n\nFrom the proof of Lemma \\ref{duality},  we have\n$$g(a_n)=\\langle g, R_{a_n}\\rangle_{L^2(dA)}\\ \\ \\ \\  (*)$$\nfor each $g\\in L_h^2(\\omega^{-1})$ and every $n\\geqslant 1$. Let $\\{c_n\\}_{n=1}^\\infty \\in \\ell^2(\\mathbb N)$. Using $(*)$ to  obtain that\n$$\\mathscr L ^*(\\{c_n\\})(z)=\\sum_{n=1}^\\infty c_n R_{a_n}(z) (1-|a_n|^2)^2 |D(a_n, \\epsilon)|_\\omega^{-\\frac{1}{2}},$$\nwhich gives the desired result. This completes the proof of Theorem \\ref{Atomic Decomposition}.\n\\end{proof}\n\nWe  now turn to the proof of Theorem \\ref{L:key00}. Let $0<\\epsilon<\\frac{1}{16}$ and $\\{a_n\\}_{n=1}^\\infty\\subset\\mathbb{D}$ be an $\\epsilon$-lattice. Define a measure $\\mu=\\mu_\\epsilon$ on $\\mathbb{D}$ by\n\\begin{equation*}\\label{E:mu}\n\\mu(z)=\\sum_{n=1}^\\infty\\delta_{a_n}(z)\\big|D(a_n,\\frac{\\epsilon}{4})\\big|_\\omega,\n\\end{equation*}\nwhere $\\delta_{a_n}$ is the Dirac measure concentrated at $a_n$.  Indeed, the conclusion of Theorem \\ref{Atomic Decomposition} tells us that $\\mu$ is a Carleson and reverse  Carleson  measure for $L_h^2(\\omega)$. Firstly, we establish a sufficient condition for a general (positive) measure to be the $L_h^p(\\omega)$ Carleson  measure, where $0<p<\\infty$.\n\\begin{prop}\\label{mu Carleson measure}\nSuppose that $\\nu$ is a positive Borel measure on $\\mathbb D$. If there exist an $0<r\\leqslant r_0$ and a constant $C>0$ independent of $z \\in \\mathbb D$ such that $$\\nu (D(z, r)) \\leqslant C |D(z, r)|_\\omega$$ for all $z\\in \\mathbb D$, then  $\\nu$ is a Carlseon measure for $L_h^p(\\omega)$ ($0<p<\\infty$), i.e.,\nthere is a  positive  constant $C_p$ such that\n$$\\int_{\\mathbb D}|f(z)|^pd\\nu(z)\\leqslant C_p \\|f\\|_{L^p(\\omega)}^p$$\nfor $f\\in L_h^p(\\omega)$. Consequently, $\\mu$ is a  Carleson  measure for $L_h^2(\\omega)$, i.e.,\nthere is an absolute constant $C>0$  such that\n$$\\int_{\\mathbb D}|f(z)|^2d\\mu(z)=\\sum_{n=1}^\\infty |f(a_n)|^2 \\big|D(a_n, \\frac{\\epsilon}{4})\\big|_\\omega \\leqslant C \\|f\\|_{L^2(\\omega)}^2$$\nfor all $f$ in $L_h^2(\\omega)$.\n\\end{prop}\n\\begin{proof}\nFix an $r\\leqslant r_0$. By Lemma \\ref{subharmonic behavior of harmonic functions}, we obtain\n$$|f(z)|^{\\frac{p}{2}}\\leqslant \\frac{C}{|D(z, r)|} \\int_{D(z, r)}|f(\\xi)|^{\\frac{p}{2}} dA(\\xi) \\ \\  \\  \\big(z\\in \\mathbb D\\big),$$\nwhere $0<p<\\infty $ and $C=C(p, r)$. Cauchy-Schwartz inequality and $\\mathcal{A}_2$ condition give us that\n\\begin{align*}\n|f(z)|^p & \\leqslant \\frac{C^2}{|D(z, r)|^2} \\bigg(\\int_{D(z, r)}|f(\\xi)|^{\\frac{p}{2}}dA(\\xi)\\bigg)^2\\\\\n& \\leqslant  \\frac{C^2}{|D(z, r)|^2}  \\bigg(\\int_{D(z, r)}|f(\\xi)|^p\\omega(\\xi)dA(\\xi)\\bigg)\\cdot \\bigg(\\int_{D(z,r)}\\frac{1}{\\omega(\\xi)}dA(\\xi)\\bigg)\\\\\n&\\leqslant C^2 [\\omega]_{\\mathcal{A}_2} \\frac{\\int_{D(z, r)}|f(\\xi)|^p\\omega(\\xi)dA(\\xi)}{|D(z, r)|_\\omega}.\n\\end{align*}\nIntegrating the above over the unit disk to obtain\n\\begin{align*}\n\\int_{\\mathbb D} |f(z)|^p d\\nu(z)&\\leqslant C \\int_{\\mathbb D} |D(z, r)|_\\omega^{-1}\\int_{D(z,r)}|f(\\xi)|^p \\omega(\\xi)dA(\\xi)d\\nu(z)\\\\\n&=C \\int_{\\mathbb D} |D(z, r)|_\\omega^{-1}\\int_{\\mathbb D}|f(\\xi)|^p \\chi_{D(z, r)}(\\xi)\\omega(\\xi)dA(\\xi)d\\nu(z)\\\\\n&=C \\int_{\\mathbb D} \\int_{\\mathbb D} |D(z, r)|_\\omega^{-1}|f(\\xi)|^p \\omega(\\xi) \\chi_{D(\\xi, r)}(z) dA(\\xi)d\\nu(z),\n\\end{align*}\nhere the constant $C$ depends only on $p$ and $r$.  Note that $\\xi \\in D(z, r)$, we have by Lemma \\ref{two disks} that\n$$|D(z, r)|_\\omega \\geqslant C |D(\\xi, r)|_\\omega$$\nfor some absolute  constant $C$.  Therefore,\n\\begin{align*}\n\\int_{\\mathbb D} |f(z)|^2d\\nu(z)&\\leqslant C \\int_{\\mathbb D} \\int_{\\mathbb D}|D(z, r)|_\\omega ^{-1}|f(\\xi)|^p \\omega(\\xi) \\chi_{D(\\xi, r)}(z) dA(\\xi)d\\nu(z)\\\\\n&\\leqslant C \\int_{\\mathbb D} \\int_{\\mathbb D} |D(\\xi, r)|_\\omega^{-1}|f(\\xi)|^p \\omega(\\xi) \\chi_{D(\\xi, r)}(z)d\\nu(z)dA(\\xi).\n\\end{align*}\nNow using our hypothesis on $\\nu$ to get\n\\begin{align*}\n\\int_{\\mathbb D} |f(z)|^p d\\nu(z) &\\leqslant C \\int_{\\mathbb D} |D(\\xi, r)|_\\omega^{-1}\\nu(D(\\xi, r)) |f(\\xi)|^p \\omega(\\xi)   dA(\\xi)\\\\\n& \\leqslant C_1 \\int_{\\mathbb D} |f(\\xi)|^p \\omega(\\xi)dA(\\xi),\n\\end{align*}\nthe constant $C_1 >0$ comes from the assumption  and  $C_1$ is independent of $f\\in L_h^p(\\omega)$.\n\nFor the second conclusion of this proposition,  it sufficient for us to show the following inequality:\n\n$$\\mu(D(a, \\frac{1}{4})) \\leqslant C \\int_{D(a, \\frac{1}{4})} \\omega(z)dA(z)\\ \\ \\ \\ \\  \\big(a\\in \\mathbb D\\big) $$\nfor some  absolute  constant $C>0$.  Indeed, by the definition of $\\mu$, we have\n$$\\mu(D(a,\\frac{1}{4}))=\\sum_{\\rho(a_n, a)<\\frac{1}{4}} \\int_{D(a_n, \\frac{\\epsilon}{4})}\\omega dA=\\sum_{\\rho(a_n, a)<\\frac{1}{4}}\\big|D(a_n, \\frac{\\epsilon}{4})\\big|_\\omega.$$\nIf $\\rho(a, a_n)<\\frac{1}{4}$, then for each $z\\in D(a_n, \\frac{\\epsilon}{4})$ we get\n$$\\rho(z, a)\\leqslant \\rho(z, a_n)+\\rho(a_n, a)<\\frac{\\epsilon}{4}+\\frac{1}{4}<\\frac{1}{2}.$$\nWhich shows that\n$$D(a_n, \\frac{\\epsilon}{4}) \\subset D(a, \\frac{1}{2})$$\nfor every $n \\geqslant 1$ provided $\\rho(a, a_n)<\\frac{1}{4}$.\n\nSince $D(a_n, \\frac{\\epsilon}{4}) \\cap D(a_m, \\frac{\\epsilon}{4})=\\varnothing$ for $n \\neq m$, we obtain\n$$\\bigcup_{\\rho(a_n, a)<\\frac{1}{4}} D(a_n, \\frac{\\epsilon}{4})\\subset D(a, \\frac{1}{2})$$\nand\n$$\\mu(D(a, \\frac{1}{4})) \\leqslant \\big|D(a, \\frac{1}{2})\\big|_\\omega \\leqslant C \\big|D(a, \\frac{1}{4})\\big|_\\omega$$\nfor every $a\\in \\mathbb D$, the constant $C>0$ (independent of $a$) comes from Lemma \\ref{two disks}. This completes the proof of  Proposition \\ref{mu Carleson measure}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn order to finish the proof of Theorem \\ref{L:key00},  we need to show that there is an $\\epsilon \\in (0, \\frac{1}{16})$ such that $\\mu=\\mu_\\epsilon$ is a reverse $L_h^2(\\omega)$ Carleson measure. More precisely, we\nwill prove the following proposition.\n\\begin{prop}\\label{mu reverse Carleson measure}\nThere exists an  $\\epsilon \\in (0, \\frac{1}{16})$ such that\n$$\\int_{\\mathbb D}|f(z)|^2 d\\mu(z)=\\sum_{n=1}^\\infty |f(a_n)|^2 |D(a_n, \\epsilon)|_\\omega \\geqslant  C \\|f\\|_{L^2(\\omega)}^2$$\nfor all $f$ in $L_h^2(\\omega)$, where $C>0$ is an absolute constant.\n\\end{prop}\n\nThe rest of this section is devoted to the proof of the above \\emph{reverse Carleson inequality}.  To do so, we\nneed to prove the following two lemmas related to harmonic functions, which extend\nthe results in \\cite{Luecking1985} and \\cite{Luecking AJM} for weighted Bergman spaces to the present situation.\n\\begin{lemma}\\label{gradient estimation}\nLet $f$ be a harmonic function on $\\mathbb D$  and $\\epsilon \\in (0, \\frac{1}{16})$.   Then there exists a constant $C_1>0$ (independent of $z, \\epsilon$ and $f$) such that\n$$|f(z)-f(0)|\\leqslant C_1 \\epsilon \\int_{D(0, \\frac{1}{4})} |f(\\lambda)| dA(\\lambda)$$\nwhen $|z|<\\epsilon.$ As a consequence, there exists a constant $C_2>0$ (independent of $z, \\epsilon$ and $f$) such that\n$$|f(w)-f(\\xi)|^2 \\leqslant  \\frac{C_2 \\epsilon^2}{|D(\\xi, \\frac{1}{4})|_\\omega}\\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda) dA(\\lambda)$$\nwhen $\\xi\\in D(w, \\epsilon)$.\n\\end{lemma}\n\\begin{proof}\n Observe that\n$$f(z)-f(0)=\\int_{0}^1 \\frac{\\partial}{\\partial t}\\big[f(tz)\\big]dt=\\int_0^1 \\big[\\nabla f(tz) \\cdot z \\big] dt$$\nfor $|z|<\\epsilon<\\frac{1}{16}$.  Thus we have\n$$|f(z)-f(0)|\\leqslant \\sup_{|\\xi|\\leqslant \\epsilon} |\\nabla f(\\xi)|\\cdot |z|.$$\n\nRecall that\n\\begin{align*}\n|\\nabla f|^2&=2\\big(|\\partial f|^2+|\\overline{\\partial} f|^2\\big)=2\\big(|\\partial f|^2+|\\partial \\overline{f}|^2\\big)\\\\\n&\\leqslant \\big[\\sqrt{2} (|\\partial f|+|\\partial \\overline{f}|)\\big]^2,\n\\end{align*}\nwhere $\\overline{\\partial} f= \\frac{\\partial f}{\\partial \\overline{z}}.$\nBy Lemma \\ref{subharmonic behavior of harmonic functions}, there is an absolute constant $C>0$ such that\n\\begin{align*}\n|\\nabla f(\\xi)| & \\leqslant \\frac{2\\sqrt{2}C}{(1-\\frac{1}{16})^3}\\cdot \\frac{1}{|D(\\xi, \\frac{1}{16})|^{\\frac{3}{2}}} \\int_{D(\\xi, \\frac{1}{16})} |f(\\lambda)| dA(\\lambda)\\\\\n&=\\frac{2\\sqrt{2}C}{\\big(1-\\frac{1}{16}\\big)^3}\\cdot \\bigg(\\frac{1-(\\frac{1}{16})^2|\\xi|^2}{\\frac{1}{16}(1-|\\xi|^2)}\\bigg)^3\\int_{D(\\xi, \\frac{1}{16})} |f(\\lambda)| dA(\\lambda).\n\\end{align*}\nNote that $|\\xi| \\leqslant \\epsilon<\\frac{1}{16}$ and if $\\lambda \\in D(\\xi, \\frac{1}{16})$, then\n$$|\\lambda|<|\\xi|+\\frac{1}{16}|1-\\overline{\\lambda}\\xi|<\\frac{1}{4},$$\nso we have $D(\\xi, \\frac{1}{16}) \\subset D(0, \\frac{1}{4})$ and\n$$|\\nabla f(\\xi)| \\leqslant C_1 \\int_{D(0, \\frac{1}{4})} |f(\\lambda)| dA(\\lambda)$$\nfor all $|\\xi|\\leqslant \\epsilon$,\nwhere the constant $C_1$  is independent of $z, \\xi$ and $\\epsilon$.\nTherefore,\n$$|f(z)-f(0)| \\leqslant \\sup_{|\\xi|\\leqslant \\epsilon} |\\nabla f(\\xi)| \\cdot |z|\\leqslant C_1 |z| \\int_{D(0, \\frac{1}{4})} |f(\\lambda)| dA(\\lambda)$$\nfor  $|z|<\\epsilon$ with $\\epsilon \\in (0, \\frac{1}{16})$, this proves the first part of the lemma.\n\nLet $\\varphi_\\xi$ be the M\\\"{o}bius map, then $f\\circ \\varphi_\\xi$ is harmonic on $\\mathbb D$. Changing of variable gives that\n\\begin{align*}\n\\big|f(\\varphi_\\xi(z))-f(\\xi)\\big| \\leqslant \\frac{ C_3 \\epsilon}{|D(\\xi, \\frac{1}{4})|}\\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)| dA(\\lambda)\n\\end{align*}\nfor some absolute constant $C_3>0$.  By Cauchy-Schwartz inequality  we obtain\n\\begin{align*}\n\\big|f(\\varphi_\\xi(z))-f(\\xi)|^2 &\\leqslant \\frac{ C_3^2 \\epsilon^2}{|D(\\xi, \\frac{1}{4})|^2}\n\\bigg(\\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2\\omega(\\lambda) dA(\\lambda)\\bigg)\\cdot \\bigg(\\int_{D(\\xi, \\frac{1}{4})} \\frac{1}{\\omega(\\lambda)}dA(\\lambda)\\bigg)\\\\\n&\\leqslant \\frac{C_3^2 [\\omega]_{\\mathcal{A}_2} \\epsilon^2}{\\big|D(\\xi, \\frac{1}{4})\\big|_\\omega}\\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda) dA(\\lambda).\n\\end{align*}\n Let $w=\\varphi_\\xi(z)$, then $|\\varphi_\\xi(w)|=|z|<\\epsilon$, which gives that\n$$|f(w)-f(\\xi)|^2 \\leqslant \\frac{C_2\\epsilon^2}{\\big|D(\\xi, \\frac{1}{4})\\big|_\\omega}\\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda) dA(\\lambda)$$\nif $\\xi \\in D(w, \\epsilon)$, as desired.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{double intrgral estimation}\nLet $f$ be a harmonic function and $\\epsilon \\in (0, \\frac{1}{16})$.  Let $\\mu$ be the measure defined  in Section 2. Then there exists a constant $C>0$ (independent of $\\epsilon$) such that\n$$\\int_{\\mathbb D} \\int_{\\mathbb D} \\chi_{\\epsilon} (z, \\xi) |D(\\xi, \\epsilon)|_\\omega^{-1} |f(z)-f(\\xi)|^2 \\omega(z) dA(z)d\\mu(\\xi)\n\\leqslant C\\epsilon^2 \\|f\\|_{L^2(\\omega)}^2$$\nfor all $f\\in L_h^2(\\omega)$, where\n$$\\chi_{\\epsilon}(z, \\xi)=\n\\begin{cases}\n1,& \\mathrm{if}\\  z \\in D(\\xi, \\epsilon),\\\\\n0,&  \\mathrm{otherwise}.\n\\end{cases}$$\n\\end{lemma}\n\\begin{proof} By Lemma \\ref{gradient estimation}, we have\n$$\\chi_{\\epsilon}(z, \\xi)|f(z)-f(\\xi)|^2 \\frac{\\omega(z)}{|D(\\xi, \\epsilon)|_\\omega}\n\\leqslant \\bigg(\\frac{C \\epsilon^2} {\\big|{D(\\xi, \\frac{1}{4})}\\big|_\\omega } \\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda)dA(\\lambda) \\bigg)\n\\frac{\\chi_{\\epsilon}(z, \\xi)\\omega(z)}{|D(\\xi, \\epsilon)|_\\omega}.$$\nIntegrating over $z\\in \\mathbb D$ on both sides gives\n$$\\int_{\\mathbb D}\\chi_{\\epsilon}(z, \\xi)|f(z)-f(\\xi)|^2 \\frac{\\omega(z)}{|D(\\xi, \\epsilon)|_\\omega}dA(z)\\leqslant \\frac{C \\epsilon^2} {\\big|D(\\xi, \\frac{1}{4})\\big|_\\omega} \\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda)dA(\\lambda).$$\nNow integrate with respect to $d\\mu(\\xi)$ to obtain\n$$\\int_{\\mathbb D}\\int_{\\mathbb D}|f(z)-f(\\xi)|^2 \\frac{\\chi_{\\epsilon}(z, \\xi) \\omega(z)}{|D(\\xi, \\epsilon)|_\\omega}dA(z)d\\mu(\\xi)\\leqslant \\int_{\\mathbb D}\\frac{C \\epsilon^2} {\\big|D(\\xi, \\frac{1}{4})\\big|_\\omega} \\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda)dA(\\lambda)d\\mu(\\xi).$$\nUsing Fubini's Theorem on the right side and noting $\\chi_{D(\\xi, \\frac{1}{4})}(\\lambda)=\\chi_{D(\\lambda, \\frac{1}{4})}(\\xi)$, we have\n\\begin{align*}\n\\int_{\\mathbb D}\\frac{C \\epsilon^2} {|D(\\xi, \\frac{1}{4})|_\\omega} \\int_{D(\\xi, \\frac{1}{4})} |f(\\lambda)|^2 \\omega(\\lambda)dA(\\lambda)d\\mu(\\xi)\n=C\\epsilon^2 \\int_{\\mathbb D} \\bigg(\\int_{\\mathbb D} \\frac{\\chi_{D(\\lambda, \\frac{1}{4})}(\\xi)}{|D(\\xi, \\frac{1}{4})|_\\omega}d\\mu(\\xi)\\bigg)\n|f(\\lambda)|^2\\omega(\\lambda)dA(\\lambda)\n\\end{align*}\n\nSince $\\lambda \\in D(\\xi, \\frac{1}{4})$, Lemma \\ref{two disks} tells us  that there is an absolute constant $C>0$ such that\n$$\\big|D(\\xi, \\frac{1}{4})\\big|_\\omega \\geqslant C  \\big|D(\\lambda, \\frac{1}{4})\\big|_\\omega.  $$\nThus we obatin\n$$\\int_{\\mathbb D} \\frac{\\chi_{D(\\lambda, \\frac{1}{4})}(\\xi)}{|D(\\xi, \\frac{1}{4})|_\\omega}d\\mu(\\xi) \\leqslant C^{-1} \\frac{\\mu(D(\\lambda, \\frac{1}{4}))}\n{|D(\\lambda, \\frac{1}{4})|_\\omega}.$$\nBy Lemma \\ref{mu Carleson measure}, we have\n$$\\mu(D(\\lambda, \\frac{1}{4})) \\cdot \\big|D(\\lambda, \\frac{1}{4})\\big|_\\omega^{-1}\\leqslant C_1$$\nfor some constant $C_1>0$ (independent of $\\epsilon$), completing the proof.\n\\end{proof}\n\n\nNow we are ready to prove the reverse Carleson inequality in Proposition \\ref{mu reverse Carleson measure}.\n\\begin{proof}[Proof of Proposition \\ref{mu reverse Carleson measure}] Recall that $\\mu$ satisfies the condition: $$\\mu(D(a, \\frac{1}{4})) \\leqslant C \\big|D(a, \\frac{1}{4})\\big|_\\omega $$ for all $a\\in \\mathbb D$. Applying Lemma  \\ref{double intrgral estimation} to  $\\epsilon \\in (0, \\frac{1}{16})$ we have\n$$\\bigg[\\int_{\\mathbb D} \\int_{\\mathbb D}  \\frac{\\chi_{\\epsilon} (z, \\xi)} {|D(\\xi, \\epsilon)|_\\omega} |f(z)-f(\\xi)|^2 \\omega(z) dA(z)d\\mu(\\xi)\n\\bigg]^{\\frac{1}{2}}\\leqslant C \\epsilon \\|f\\|_{L^2(\\omega)}.$$\nThe triangle inequality  gives\n$$I-J \\leqslant C \\epsilon \\|f\\|_{L^2(\\omega)},$$\nwhere\n $$I:=\\bigg[\\int_{\\mathbb D} \\int_{\\mathbb D}  \\frac{\\chi_{\\epsilon} (z, \\xi)} {|D(\\xi, \\epsilon)|_\\omega} |f(z)|^2 \\omega(z) dA(z)d\\mu(\\xi)\n\\bigg]^{\\frac{1}{2}}$$\nand\n$$J:=\\bigg[\\int_{\\mathbb D} \\int_{\\mathbb D}  \\frac{\\chi_{\\epsilon} (z, \\xi)} {|D(\\xi, \\epsilon)|_\\omega} |f(\\xi)|^2 \\omega(z) dA(z)d\\mu(\\xi)\\bigg]^{\\frac{1}{2}}.$$\n\nFor the first integral $I$, we note that for each $z\\in \\mathbb D$,\n\\begin{align*}\n\\int_{\\mathbb D} \\frac{\\chi_{\\epsilon} (z, \\xi)} {|D(\\xi, \\epsilon)|_\\omega } d\\mu(\\xi)&=\\int_{D(z, \\epsilon)} \\frac{d\\mu(\\xi)} {|D(\\xi, \\epsilon)|_\\omega } \\geqslant C_1 \\frac{\\mu(D(z, \\epsilon))}{|D(z, \\epsilon)|_\\omega },\n\\end{align*}\nwhere the $``\\geqslant\"$  follows from Lemma \\ref{two disks} and $C_1$ is an absolute constant.\n\nSince $\\mathbb D=\\bigcup_{n=1}^\\infty D(a_n, \\epsilon)$,  we can select a disk $D(a_j, \\epsilon)$ such that $z\\in D(a_j, \\epsilon)$ for each $z\\in \\mathbb D$. Applying  Lemma \\ref{two disks} to get\n\\begin{align*}\n\\mu(D(z, \\epsilon))&=\\sum_{n=1}^\\infty \\delta_{a_n}(D(z, \\epsilon))\\int_{D(a_n, \\frac{\\epsilon}{4})}\\omega dA\\\\\n&\\geqslant \\big|D(a_j, \\frac{\\epsilon}{4})\\big|_\\omega \\geqslant C_2 |D(a_j, \\epsilon)|_\\omega\n\\geqslant C_3 |D(z, \\epsilon)|_\\omega,\n\\end{align*}\n where $C_2 $  and $C_3$ are  absolute positive constants. Therefore, we have\n$$\\int_{\\mathbb D} \\frac{\\chi_{\\epsilon} (z, \\xi)} {|D(\\xi, \\epsilon)|_\\omega} d\\mu(\\xi)\\geqslant \\widetilde{C}$$\nfor some absolute constant $\\widetilde{C}>0$. These give us  that $$I\\geqslant \\widetilde{C} \\|f\\|_{L^2(\\omega)}.$$\n\nFor the second integral, we observe that\n$$\\int_{\\mathbb D}\\chi_{\\epsilon}(z, \\xi) \\omega(z)dA(z)=|D(\\xi, \\epsilon)|_\\omega,$$\nto get  $$J=\\bigg(\\int_{\\mathbb D} |f(\\xi)|^2 d\\mu(\\xi)\\bigg)^{\\frac{1}{2}}.$$\nThus we have\n$$\\widetilde{C} \\|f\\|_{L^2(\\omega)}-\\bigg(\\int_{\\mathbb D} |f(\\xi)|^2d\\mu(\\xi)\\bigg)^{\\frac{1}{2}} \\leqslant I-J \\leqslant C \\epsilon \\|f\\|_{L^2(\\omega)}.$$\nEquivalently,\n$$(\\widetilde{C}- C\\epsilon)\\|f\\|_{L^2(\\omega)}\\leqslant \\bigg( \\int_{\\mathbb D} |f(\\xi)|^2d\\mu(\\xi)\\bigg)^{\\frac{1}{2}}$$\nfor each $0<\\epsilon<\\frac{1}{16}$.\n\nSince $C, \\widetilde{C}$  are both independent of $\\epsilon$, we can choose $$0<\\epsilon< \\min \\bigg\\{\\frac{1}{16}, \\frac{\\widetilde{C}}{2C}, r_0\\bigg\\}$$ such that\n\\begin{align*}\n\\|f\\|_{L^2(\\omega)}^2 \\leqslant \\frac{1}{(\\widetilde{C}- C\\epsilon)^2} \\int_{\\mathbb D} |f(\\xi)|^2d\\mu(\\xi).\n\\end{align*}\nRecall the definition of $\\mu$, we conclude that\n\\begin{align*}\n\\|f\\|_{L^2(\\omega)}^2 & \\leqslant \\frac{4}{\\widetilde{C}^2} \\sum_{n=1}^\\infty |f(a_n)|^2 |D(a_n, \\frac{\\epsilon}{4})|_\\omega\\\\\n& \\leqslant \\frac{4}{\\widetilde{C}^2} \\sum_{n=1}^\\infty |f(a_n)|^2 |D(a_n, \\epsilon)|_\\omega\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\nThe proof of  Theorem  \\ref{Atomic Decomposition} implies the following result immediately.\n\\begin{thm}\\label{Atomic Decomposition cor}\nSuppose that $\\omega$ satisfies the $\\mathcal{A}_2$ condition. Then there is an $\\epsilon$-lattice $\\{a_n\\}_{n=1}^{\\infty} \\subset \\mathbb D$ such that any $f\\in L_h^2(\\omega)$ has the following form:\n$$f(z)=\\sum_{n=1}^\\infty c_n K^{\\omega}_{a_n}(z) |D(a_n, \\epsilon)|_\\omega^{\\frac{1}{2}}$$\nfor some sequence $\\{c_n\\}$ in $\\ell^2(\\mathbb N)$, where $K^{\\omega}_{a_n}$ is the reproducing kernel for $L_h^2(\\omega)$.\n\\end{thm}\n\\begin{proof}  We consider the linear map $\\mathscr S: L_h^2(\\omega)\\rightarrow \\ell^2(\\mathbb N)$:\n$$\\mathscr S f=\\bigg\\{f(a_n) |D(a_n, \\epsilon)|_\\omega^{\\frac{1}{2}}\\bigg\\}_{n=1}^\\infty.$$\n Since $L_h^2(\\omega)$ is a Hilbert space, we apply  Propositions \\ref{mu Carleson measure} and \\ref{mu reverse Carleson measure}\n to deduce that\n $\\mathscr S^*: \\ell^2(\\mathbb N)\\rightarrow L_h^2(\\omega)$ is surjective and\n $$\\big\\langle \\mathscr S^* (\\{c_n\\}), \\  f \\big\\rangle_{L^2(\\omega)}=\\bigg\\langle \\sum_{n=1}^\\infty c_n K^{\\omega}_{a_n}(z) |D(a_n, \\epsilon)|_\\omega^{\\frac{1}{2}}, \\ f \\bigg\\rangle_{L^2(\\omega)}$$\n for $\\{c_n\\}\\in \\ell^2(\\mathbb N)$ and $f\\in L^2_h(\\omega)$. Therefore,\n $$\\mathscr S^*(\\{c_n\\})(z)=\\sum_{n=1}^\\infty c_n K^{\\omega}_{a_n}(z) |D(a_n, \\epsilon)|_\\omega^{\\frac{1}{2}}.$$\n This completes the proof of this theorem.\n \\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Boundedness and Compactness of $T_\\nu$ on $L^2_h(\\omega)$}\n\n\n Recall Toeplitz operator $T_\\nu$ initially defined on a dense subspace of $L^2_h(\\omega)$ is given by\n $$T_\\nu f(z)=\\int_{\\mathbb D}f(\\lambda)\\overline{K^{\\omega}_z(\\lambda)}d\\nu(\\lambda) \\ \\ \\  \\big(z\\in \\mathbb D\\big).$$\nIn this section, we will characterize the boundedness and compactness of $T_\\nu$ on $L_h^2(\\omega)$ via Berezin transform and Carleson measure for $L_h^2(\\omega)$. Firstly, we define the Berezin transform $\\widetilde{\\nu}$ of $\\nu$ as follows:\n$$\\widetilde{\\nu}(z)=\\frac{1}{\\|R_z\\|_{L^2(\\omega)}^2}\\int_{\\mathbb D}|R_z(\\lambda)|^2d\\nu(\\lambda), $$\nwhere $$R_z(\\lambda)=\\frac{1}{(1-\\overline{\\lambda}z)^2}+\\frac{1}{(1-\\overline{z}\\lambda)^2}-1$$ is the reproducing kernel for  $L_h^2$. The first main result of this section is Theorem \\ref{Boundedness1}.\n\n\\begin{thm}\\label{Boundedness1}\nLet $\\nu$ be a  positive finite Borel measure  on $\\mathbb D$ and $\\omega$  be an  $\\mathcal{A}_2$ weight. The following  conditions are equivalent:\\\\\n(1) $T_\\nu$ extends to a bounded linear operator on $L_h^2(\\omega)$;\\\\\n(2) $\\nu$ is a Carlseon measure for $L_h^2(\\omega)$;\\\\\n(3) There exist an $0<r\\leqslant r_0$ and a  constant $C>0$ independent of $z \\in \\mathbb D$ such that $$\\nu(D(z, r)) \\leqslant C |D(z, r)|_\\omega$$ for all $z\\in \\mathbb D$;\\\\\n(4) Berezin transform $\\widetilde{\\nu}$  is bounded.\n\\end{thm}\n\nTo prove Theorem \\ref{Boundedness1}, we need the following useful lemma.\n\\begin{lemma}\\label{norm of kernel}\nLet $\\omega \\in \\mathcal{A}_2$. If  $0<r\\leqslant r_0$, then there is a constant $C=C(r)$ such that\n$$\\frac{|D(\\lambda, r)|_\\omega}{2(1-|\\lambda|)^4} \\leqslant \\|R_\\lambda\\|_{L^2(\\omega)}^2 \\leqslant C \\frac{|D(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4}$$\nfor all $\\lambda \\in \\mathbb D$.\n\\end{lemma}\n\\begin{proof} Let $\\lambda\\in\\mathbb{D}$. By the remarks below Theorem \\ref{Atomic Decomposition},  there exists an $r_0 \\in (0, \\frac{1}{4}]$  such that if $0<r\\leqslant r_0$, then\n$$\\frac{\\frac{1}{2}}{(1-|\\lambda|)^2} \\leqslant |R_\\lambda(z)| \\leqslant \\frac{3}{(1-|\\lambda|)^2}$$\nfor all $z \\in D(\\lambda, r)$.\nIt follows that for each $r \\in (0, r_0]$ we have\n\\begin{align*}\n\\|R_\\lambda\\|_{L^2(\\omega)}^2 &=\\int_{\\mathbb D} |R_\\lambda(z)|^2 \\omega(z)dA(z)\\\\\n&\\geqslant \\int_{D(\\lambda, r)} |R_\\lambda(z)|^2 \\omega(z)dA(z)\\\\\n&\\geqslant \\frac{1}{4} \\int_{D(\\lambda, r)} \\frac{\\omega(z)}{(1-|\\lambda|)^4} dA(z)\\\\\n&=\\frac{|D(\\lambda, r)|_\\omega}{4(1-|\\lambda|)^4}.\n\\end{align*}\nTo show the other inequality, observe that\n$$|z-\\lambda|<r(1-|\\lambda|)<r|1-\\overline{z}\\lambda| \\  \\   \\ \\  \\big(z, \\lambda \\in \\mathbb D \\big),$$ we have $$S(\\lambda, r):=\\{z\\in \\mathbb D: |z-\\lambda|<r(1-|\\lambda|)\\}\\subset D(\\lambda, r).$$\nThus, we have  by Lemma 2.1 in \\cite{Con} that\n$$ C_1\\frac{|S(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4} \\leqslant \\|K_\\lambda\\|_{L^2(\\omega)}^2 \\leqslant C_2 \\frac{|S(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4}\\leqslant  C_2 \\frac{|D(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4}$$\nfor some positive constants $C_1=C_1(r)$ and $C_2=C_2(r)$, here $K_\\lambda(z)=\\frac{1}{(1-\\overline{\\lambda}z)^2}$ is the reproducing kernel for $L_a^2$ at $\\lambda$.\n Recall that\n$$R_\\lambda(z)=2\\mathrm{Re} (K_\\lambda(z))-1,\n$$\nwe have\n\\begin{align*}\n\\|R_\\lambda\\|_{L^2(\\omega)}^2 \\leqslant \\frac{4C|D(\\lambda, r)|\\omega}{(1-|\\lambda|)^4} + 2\\|\\omega\\|_{L^1}.\n\\end{align*}\nConsequently, to complete the proof we need only to show that there is a constant $C_3$ depending only on  $r$ such that\n$$\\|\\omega\\|_{L^1} \\leqslant C_3 \\frac{ |D(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4} $$\nfor every $\\lambda \\in \\mathbb D.$ Indeed, we may assume that $\\|\\omega\\|_{L^1}=1$. Then it is easy to see that\n$$C (1-|\\lambda|)^2\\leqslant  |D(\\lambda, r)|$$\nfor some constant $C=C(r)$.\nThus we have\n\\begin{align*}\nC (1-|\\lambda|)^2 &\\leqslant |D(\\lambda, r)|=\\int_{D(\\lambda, r)} \\omega(z)^{\\frac{1}{2}} \\omega(z)^{-\\frac{1}{2}}dA(z)\\\\\n&\\leqslant |D(\\lambda, r)|_\\omega^{\\frac{1}{2}}\\cdot |D(\\lambda, r)|_{\\omega^{-1}}^{\\frac{1}{2}}\\\\\n& \\leqslant |D(\\lambda, r)|_\\omega^{\\frac{1}{2}}\\cdot \\|\\omega^{-1}\\|_{L^1}^{\\frac{1}{2}}.\n\\end{align*}\nThis shows that\n$$\\frac{|D(\\lambda, r)|_\\omega}{(1-|\\lambda|)^4}\\geqslant C_3,$$\nto complete the proof of Lemma \\ref{norm of kernel}.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{Boundedness1}.\n\\begin{proof}[Proof of Theorem \\ref{Boundedness1}] Our strategy is $(4)\\Rightarrow(3)\\Rightarrow(2)\\Rightarrow(1)\\Rightarrow(4)$.\n\n$(4)\\Rightarrow (3)$: Note that there exists an $r \\in (0, r_0]$  such that\n$$\\frac{\\frac{1}{2}}{(1-|z|)^2} \\leqslant |R_z(\\lambda)| \\leqslant \\frac{3}{(1-|z|)^2}$$\nfor $\\lambda\\in D(z, r)$.  From the definition of $\\widetilde{\\nu}$, there is a constant $C>0$ such that\n $$\\frac{1}{\\|R_z\\|_{L^2(\\omega)}^2}\\int_{D(z,r)}|R_z(\\lambda)|^2d\\nu(\\lambda)\\leqslant \\widetilde{\\nu}(z)\\leqslant C$$\n for all $z\\in \\mathbb D.$  Combining these with  Lemma \\ref{norm of kernel} gives us that\n $$\\nu(D(z, r))\\leqslant C|D(z,r)|_\\omega\\ \\ \\  \\big(z\\in \\mathbb D \\big)$$\n for some positive constant $C=C(r)$.\n\n$(3)\\Rightarrow (2)$: This was proved in Proposition \\ref{mu Carleson measure}.\n\n\n$(2)\\Rightarrow (1)$: Assume that $\\nu$ is a Carleson measure. By Condition $(2)$ and Lemma \\ref{norm of kernel}, we obtain Condition $(3)$. Now\nProposition \\ref{mu Carleson measure} implies that $\\nu$ is also a Carleson measure for $L_h^1(\\omega)$.  Then for  $f, g$ are harmonic on a neighborhood of $\\overline{\\mathbb D}$ (by Theorem \\ref{Atomic Decomposition}, these functions are dense in $L_h^2(\\omega)$),  we have\n$$\\big|\\langle T_\\nu f, g\\rangle \\big|\\leqslant \\int_{\\mathbb D} |f(z)g(z)|d\\nu(z)\\leqslant C \\|fg\\|_{L^1(\\omega)}\\leqslant C\\|f\\|_{L^2(\\omega)}\\|g\\|_{L^2(\\omega)},$$ which  means that $T_\\nu$ is bounded.\n\n\n$(1)\\Rightarrow (4)$: Suppose that $T_\\nu$ is bounded on $L_h^2(\\omega)$. We consider the partial sum $\\sigma_N=\\sum_{n=1}^N t_n K^\\omega_{a_n}$, where $N\\geqslant 1$, $\\{t_n\\}$ are complex numbers  and $\\{a_n\\}\\subset \\mathbb D$.  Direct calculation shows that\n$$\\|\\sigma_N\\|_{L^2(\\nu)}\\leqslant C \\|\\sigma_N\\|_{L^2(\\omega)}$$\nfor some constant $C>0$.  This implies that if $\\lim\\limits_{N\\rightarrow \\infty}\\|\\sigma_N-g\\|_{L^2(\\omega)}=0$ for some $g\\in L_h^2(\\omega)$, then\n$$\\lim_{N \\rightarrow \\infty} \\langle f, \\sigma_N \\rangle_{L^2(\\nu)}=\\langle f, g\\rangle_{L^2(\\nu)}$$\nfor every $f\\in L^2_h(\\omega)$. Applying Theorem \\ref{Atomic Decomposition cor} to obtain\n$$\\langle T_\\nu f, g\\rangle_{L^2(\\omega)}=\\langle f, g\\rangle_{L^2(\\nu)}$$\nfor every $f, g \\in L_h^2(\\omega)$. In particular, we have\n$$\\langle T_\\nu R_z, R_z \\rangle_{L^2(\\omega)}=\\langle R_z,   R_z\\rangle_{L^2(\\nu)}=\\widetilde{\\nu}(z)\\|R_z\\|_{L^2(\\omega)}^2,$$\nto get $\\widetilde{\\nu}(z)\\leqslant \\|T_\\nu\\|$ for all $z\\in \\mathbb D$. The proof of Theorem \\ref{Boundedness1} is complete now.\n\\end{proof}\n\nFrom the above theorem, it is natural to characterize the compactness of $T_\\nu$ via  vanishing Carleson measure. In fact, we will characterize the compact Toeplitz operators with positive measures as the symbols  via not only  vanishing Carlson measure (for the $\\mathcal{A}_2$ weighted harmonic Bergman space) but also  Berezin transform.\n\\begin{thm}\\label{Comapctness1}\nLet $\\nu$ be a  positive finite Borel measure  on $\\mathbb D$ and $\\omega\\in \\mathcal{A}_2$. The following  conditions are equivalent:\\\\\n(1) $T_\\nu$ is compact on $L_h^2(\\omega)$;\\\\\n(2) $\\nu$ is a vanishing Carlseon measure for $L_h^2(\\omega)$, i.e.,\n$$\\lim_{|z|\\rightarrow 1^-} \\frac{\\nu(D(z, r))}{|D(z, r)|_\\omega}=0$$\nfor some $r\\in (0, 1)$;\\\\\n(3) The Berezin transform $\\lim\\limits_{|z|\\rightarrow 1^-}\\widetilde{\\nu}(z)=0.$\n\\end{thm}\n\\begin{proof}\nWe will show that $(2)\\Rightarrow (1)\\Rightarrow (3)\\Rightarrow (2)$.\n\n$(2)\\Rightarrow (1):$  To prove (1), we need only to show that the inclusion operator $i: L_h^2(\\omega)\\rightarrow L^2(\\nu)$ is compact, i.e.,\n$$\\lim_{n\\rightarrow \\infty}\\int_{\\mathbb D} |f_n(z)|^2 d\\nu(z)=0$$\nwhenever $\\|f_n\\|_{L^2(\\omega)}\\rightarrow 0 \\ (n\\rightarrow \\infty)$, where $\\{f_n\\}_{n=1}^\\infty$ is a bounded sequence in $L_h^2(\\omega)$ which\nconverges to zero uniformly on each compact subset of $\\mathbb D$.\n\n From the proof of Proposition \\ref{mu Carleson measure}, there exists a positive constant $C=C(r)$ such that\n\\begin{align*}\n\\int_{\\mathbb D} |f_n(z)|^2 d\\nu(z) &\\leqslant C \\int_{\\mathbb D}|D(\\xi, r)|_\\omega ^{-1}\\nu(D(\\xi, r)) |f_n(\\xi)|^2 \\omega(\\xi)   dA(\\xi)\\\\\n&=C \\bigg(\\int_{|\\xi|\\leqslant s }+\\int_{|\\xi|>s } \\bigg)\\frac{\\nu(D(\\xi, r))}{ |D(\\xi, r)|_\\omega} |f_n(\\xi)|^2 \\omega(\\xi) dA(\\xi),\n\\end{align*}\nwhere $s\\in (0, 1)$. Under the assumption in (2), we can choose an $s_0\\in (0, 1)$ to make the second  integral as small as we like; Fix $s_0$,\nit is easy to show the first integral converges to zero,  since $f_n\\rightarrow 0 \\ (n\\rightarrow \\infty)$ uniformly on compact subsets.  This proves $(2)\\Rightarrow (1)$.\n\n$(1)\\Rightarrow (3):$  Observe that\n$$\\widetilde{\\nu}(z)=\\bigg\\langle T_\\nu \\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}, \\ \\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}\\bigg\\rangle_{L^2(\\omega)} \\leqslant \\bigg\\|T_\\nu  \\bigg(\\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}\\bigg)\\bigg\\|_{L^2(\\omega)}.$$\nSo, it sufficient for us to show that $\\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}$ converges to zero weakly in $L_h^2(\\omega)$ as $|z|\\rightarrow 1^-.$\nNote  that $\\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}$ is an unit vector in $L_h^2(\\omega)$, we need only to show it converges to zero uniformly on compact subsets of $\\mathbb D$ as $|z|\\rightarrow 1^-$.\nObserve that Lemma \\ref{norm of kernel} implies that there exists a positive  constants $C=C(r_0)$ such that\n$$\\bigg|\\frac{R_z(\\lambda)}{\\|R_z\\|_{L^2(\\omega)}}\\bigg|^2\n\\leqslant   C |R_z(\\lambda)|^2 \\int_{D(z, r_0)}{\\omega^{-1}}dA.$$\nIt is clear that $\\frac{R_z}{\\|R_z\\|_{L^2(\\omega)}}$ converges to zero uniformly (as $|z|\\rightarrow 1^-$) on each disk $|\\lambda|\\leqslant s<1$, since $|D(z, r_0)|\\rightarrow 0$ as $|z|\\rightarrow 1^-$ and $\\omega^{-1}\\in L^1(dA)$.\n\n\n$(3)\\Rightarrow (2):$  By the definition of  $\\widetilde{\\nu}$ and Lemma \\ref{norm of kernel},\nthere exists a constant $C=C(r_0)>0$  such that\n\\begin{align*}\n\\widetilde{\\nu}(z) & =\\frac{1}{\\|R_z\\|_{L^2(\\omega)}^2}\\int_{\\mathbb D}|R_z(\\lambda)|^2d\\nu(\\lambda)\\\\\n& \\geqslant C \\frac{(1-|z|)^4}{|D(z, r_0)|_\\omega} \\int_{D(z, r_0)}|R_z(\\lambda)|^2d\\nu(\\lambda)\\\\\n& \\geqslant  \\frac{C}{4} \\frac{\\nu(D(z, r_0))}{|D(z, r_0)|_\\omega}.\n\\end{align*}\nThen the desired result follows, to complete the proof of Theorem \\ref{Comapctness1}.\n\\end{proof}\n\nIn the rest of this section, we will consider the special class of compact Toeplitz operators to give a characterization of  $\\nu$ for $T_\\nu$ to be in the Schatten class $\\mathcal S^p\\ (p\\geqslant 1)$. The following theorem is the third main result in Section 3.\n\n\\begin{thm}\\label{Schatten p class}\nLet $\\nu$ be a  positive finite Borel measure  on $\\mathbb D$ and $\\omega\\in \\mathcal{A}_2$. Then for $ 1\\leqslant p< \\infty$,\n$T_\\nu \\in \\mathcal S^p$ if and only if\n$$\\sum_{n= 1}^\\infty\\bigg(\\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}\\bigg)^p<+\\infty,$$\nwhere $\\big(\\{a_n\\}_{n=1}^\\infty,  \\ \\epsilon\\big)$ is the $\\epsilon$-lattice obtained by Theorem \\ref{Atomic Decomposition}.\n\\end{thm}\n\nIn order to prove the above result, we need one more lemma, which is a straightforward consequence of Lemmas  \\ref{subharmonic behavior of harmonic functions} and \\ref{norm of kernel}.\n\\begin{lemma}\\label{Schatten class lemma}\nLet $\\omega \\in \\mathcal{A}_2$ and $0<r\\leqslant r_0$. There exists a constant $C=C(r)>0$ such that\n$$C^{-1} \\leqslant  K^\\omega_z(z) \\cdot |D(z, r)|_\\omega \\leqslant C$$\nfor $z\\in \\mathbb D$, where $K^\\omega_z$ is the reproducing kernel of $L_h^2(\\omega)$.\n\\end{lemma}\n\\begin{proof} Note that $K^{\\omega}_{\\lambda}(\\lambda)=\\|K^\\omega_\\lambda\\|_{L^2(\\omega)}^2$\nfor each $\\lambda\\in \\mathbb D$.  Applying  Lemma   \\ref{subharmonic behavior of harmonic functions} to the function $K^\\omega_\\lambda(z)$, we get a constant $C$ depends only on $r$ such that\n$$|K^\\omega_\\lambda(z)|^2\\leqslant \\frac{C \\| K^\\omega_\\lambda\\|_{L^2(\\omega)}^2}{|D(z, r)|_\\omega}=\\frac{C  K^\\omega_\\lambda(\\lambda)}{ |D(z, r)|_\\omega}.$$\nTaking $\\lambda=z$ to get the inequality on the right hand side in Lemma \\ref{Schatten class lemma}.\n\nFor the reverse inequality, note that for each $z\\in \\mathbb D$  we have\n\\begin{align*}\n\\frac{1}{(1-|z|)^2}& \\leqslant \\frac{2}{(1-|z|^2)^2}-1\\\\\n&=R_z (z)=\\big\\langle R_z, K^\\omega_z\\big\\rangle_{L^2(\\omega)}\\\\\n&\\leqslant \\|R_z\\|_{L^2(\\omega)}\\cdot \\|K^\\omega_z\\|_{L^2(\\omega)}\\\\\n& \\leqslant C \\frac{|D(z, r)|_\\omega^{\\frac{1}{2}}}{(1-|z|)^2}\\cdot \\sqrt{K^\\omega_z(z)},\n\\end{align*}\nwhere the constant $C$ comes from Lemma \\ref{norm of kernel}. This finishes the proof.\n\\end{proof}\n\nWe are ready to prove Theorem \\ref{Schatten p class}. The method of its proof  is quite standard.\n\\begin{proof}[Proof of Theorem \\ref{Schatten p class}] Suppose that the  series converges.  We consider the $\\epsilon$-lattice $\\{a_n\\}_{n=1}^\\infty$ given by Theorem \\ref{Atomic Decomposition}, recall that $\\epsilon<r_0$ (see the proof of Proposition \\ref{mu reverse Carleson measure}). For an arbitrary orthonormal basis\n$\\{e_n\\}_{n=1}^\\infty$ of $L_h^2(\\omega)$,\nwe have\n\\begin{align*}\n\\sum_{n=1}^\\infty \\langle T_\\nu e_n, e_n \\rangle &=\\sum_{n=1}^\\infty \\int_{\\mathbb D} |e_n(z)|^2 d\\nu(z)=\\int_{\\mathbb D} K^\\omega_z(z)d\\nu(z)\\\\\n& \\leqslant \\sum_{n=1}^\\infty \\int_{D(a_n, \\epsilon)} K^\\omega_z(z)d\\nu(z)\\\\\n& \\leqslant C \\sum_{n=1}^\\infty \\int_{D(a_n, \\epsilon)} |D(z, \\epsilon)|_\\omega ^{-1}d\\nu(z),\n\\end{align*}\nthe constant $C$ comes from Lemma \\ref{Schatten class lemma}, which depends only on $\\epsilon$. Note that $\\rho(z, a_n)<\\epsilon$ for every $n\\geqslant 1$,  by Lemma \\ref{two disks} and its proof we can choose a constant $C_1=C_1(\\epsilon)$ such that\n$$\\sum_{n=1}^\\infty \\langle T_\\nu e_n, e_n \\rangle \\leqslant\n C_1 \\sum_{n=1}^\\infty \\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}.$$\n This shows that $T_\\nu \\in \\mathcal S^1$ provided $\\sum_{n=1}^\\infty \\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}<+\\infty$.\n\nOn the other hand, if $\\sup\\limits_{n\\geqslant 1}\\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}<+\\infty$,\nthen by the proof Theorem 7.4 in \\cite{Zhu2007} (or the proof of $(3)\\Rightarrow (2)$ in Theorem \\ref{Boundedness1}), we deduce that $T_\\nu$ is bounded on $L_h^2(\\omega)$, i.e., $T_\\nu \\in \\mathcal S^\\infty$. Now applying the interpolation theorem for the Schatten classes (see Theorem 2.6 in \\cite{Zhu2007} if needed), we  obtain that $T_\\nu \\in \\mathcal S^p$ for each $p\\in (1, +\\infty)$ if\n$$\\sum_{n=1}^\\infty \\bigg( \\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}\\bigg)^p<+\\infty.$$\n\n\n\nConversely, we assume that $T_\\nu \\in \\mathcal S^p$ for $1\\leqslant p<\\infty$.  We recall  by Theorem \\ref{Atomic Decomposition} that each $f\\in L_h^2(\\omega)$ has the following form:\n\\begin{align*}\nf(z)&=\\sum_{n=1}^\\infty c_n (1-|a_n|)^2 R_{a_n}(z)|D(a_n, \\epsilon)|_\\omega^{-\\frac{1}{2}}\\\\\n&:=\\sum_{n=1}^\\infty c_n h_n(z),\n\\end{align*}\nwhere $\\{c_n\\}_{n=1}^\\infty \\in \\ell^2(\\mathbb N)$.\n Cauchy-Schwartz inequality shows that\n$$\\|f\\|_{L^2(\\omega)}^2 \\leqslant \\bigg(\\sum_{n=1}^\\infty |c_n|^2 \\bigg)\\cdot \\bigg(\\sum_{n=1}^\\infty \\frac{(1-|a_n|)^4}{|D(a_n, \\epsilon)|_\\omega} \\|R_{a_n}\\|_{L^2(\\omega)}^2\\bigg).$$\n\nSince $\\epsilon<r_0$, we have by Lemma \\ref{norm of kernel} that\n$$\\|f\\|_{L^2(\\omega)}^2 \\leqslant C \\sum_{n=1}^\\infty |c_n|^2$$\nwhere the constant  $C>0$ depends only on $\\epsilon$.\n\nFix an orthonormal basis $\\{e_n\\}_{n=1}^\\infty$ for $L_h^2(\\omega)$ and define a linear operator $\\mathscr A$ on $L_h^2(\\omega)$ by\n$$\\mathscr A \\bigg( \\sum_{n=1}^\\infty c_n e_n \\bigg)=\\sum_{n=1}^\\infty c_n h_n.$$\nThen $\\mathscr A$ is a bounded surjective linear operator on $L_h^2(\\omega)$. Thus the  $\\mathscr A^*$ is well-defined on $L_h^2(\\omega)$ and\n$\\mathscr A^*T_\\nu \\mathscr A \\in \\mathcal S^p$, so that\n$$\\sum_{n=1}^\\infty \\big| \\langle \\mathscr A^*T_\\nu \\mathscr A e_n, e_n \\rangle_{L^2(\\omega)}\\big|^p<+\\infty.$$\nOn the other hand,\n\\begin{align*}\n\\sum_{n=1}^\\infty \\big| \\langle \\mathscr A^*T_\\nu \\mathscr A e_n, e_n \\rangle_{L^2(\\omega)}\\big|^p&=\\sum_{n=1}^\\infty \\big| \\langle T_\\nu \\mathscr A e_n,  \\mathscr A e_n \\rangle_{L^2(\\omega)}\\big|^p\\\\\n&=\\sum_{n=1}^\\infty \\big| \\langle T_\\nu h_n,  h_n\\rangle_{L^2(\\omega)}\\big|^p \\ \\ \\ \\big(\\mathrm{by \\ the \\  definition \\ of} \\ \\mathscr A \\big)\\\\\n&=\\sum_{n=1}^\\infty \\bigg( \\int_{\\mathbb D} |h_n(z)|^2  d\\nu(z)\\bigg)^p\\\\\n&\\geqslant \\sum_{n=1}^\\infty \\bigg( \\int_{D(a_n, \\epsilon)} |h_n(z)|^2  d\\nu(z)\\bigg)^p.\n\\end{align*}\nRecall\n$$|h_n(z)|^2=\\frac{(1-|a_n|)^4 |R_{a_n}(z)|^2}{|D(a_n, \\epsilon)|_\\omega},$$\n we have  by Lemma \\ref{norm of kernel} that\n$$|h_n(z)|^2\\geqslant \\frac{1}{4|D(a_n, \\epsilon)|_\\omega}$$\n if $\\rho (z, a_n)<\\epsilon<r_0$ for each $n\\geqslant 1$. Therefore,\n\\begin{align*}\n\\sum_{n=1}^\\infty \\big| \\langle \\mathscr A^*T_\\nu \\mathscr A e_n, e_n \\rangle_{L^2(\\omega)}\\big|^p \\geqslant 4^{-p} \\sum_{n=1}^\\infty\n\\bigg( \\frac{\\nu(D(a_n, \\epsilon))}{|D(a_n, \\epsilon)|_\\omega}\\bigg)^p.\n\\end{align*}\nThis completes the proof of Theorem \\ref{Schatten p class}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Invertibility of Toeplitz operators on $L_h^2(\\omega_\\alpha)$}\n\n A fundamental and interesting  problem is to determine when a Toeplitz operator is invertible on the Hardy or Bergman\nspace.  In this section, we study the invertibility problem of Toeplitz operators on the standard weighted harmonic Bergman space $L_h^2(\\omega_\\alpha)$ with\n $\\omega_\\alpha=(1+\\alpha)(1-|z|^2)^\\alpha, \\alpha>-1$. Recall that the reproducing kernel for $L_h^2(\\omega_\\alpha)$ is given by\n$$R^\\alpha_z(\\lambda)=K^\\alpha_z(\\lambda)+\\overline{K^\\alpha_z(\\lambda)}-1 \\ \\ \\  \\big(z, \\lambda \\in \\mathbb D\\big),$$\nwhere\n $$K^\\alpha_z(\\lambda)=\\frac{1}{(1-\\overline{z}\\lambda)^{2+\\alpha}}$$\nis the reproducing kernel for the weighted Bergman space $L_a^2(\\omega_\\alpha)$, see \\cite{Shu} if needed.\n\nFor the (unweighted) Bergman space setting, the second author and Zheng provided a necessary and sufficient condition\n for the Toeplitz operators with nonnegative symbols to be invertible  on $L_a^2$ (\\cite{Zhao}). The main tool used in \\cite{Zhao} is Luecking's result  on the reverse Carleson measure for Bergman space (\\cite{Lue}),  which also holds for the harmonic Bergman space. More precisely, Luecking established the following result.\n\n\\begin{lemma}(\\cite{Luecking1983})\\label{reverse Carleson measure1}\nSuppose that $G$ is a measurable subset of $\\mathbb D$. Then the following are equivalent:\\\\\n(i) There exists a  $\\delta\\in (0, 1)$ such that\n$$ |G\\cap K| \\geqslant \\delta |\\mathbb D \\cap K|$$\nfor every ball $K$ whose center lies on $\\partial \\mathbb D$;\\\\\n(ii) $\\chi_{G}dA$ is a reverse Carleson measure for $L_h^2(\\omega_\\alpha)$. That is,  there is a constant $C>0$ such that\n$$\\int_{\\mathbb D}|f(z)|^2 \\omega_\\alpha(z) dA(z)\\leqslant C \\int_{G}|f(z)|^2 \\omega_\\alpha (z) dA(z)$$\nfor all $f\\in L_h^2(\\omega_\\alpha)$.\n\\end{lemma}\n\nMotivated by the ideas  and techniques  used in \\cite{Zhao}, we are able to characterize the invertiblility of Toeplitz operator $T_\\varphi$ ($\\varphi\\geqslant 0$) on $L_h^2(\\omega_\\alpha)$  in terms of the reverse Carleson measure and Berezin transform.\n\n\\begin{thm}\\label{invertibility1} Let $\\varphi\\geqslant 0$ be in $L^\\infty(\\omega_\\alpha)$. The following conditions are equivalent:\\\\\n$(1)$ The Toeplitz operator $T_\\varphi$ is invertible on $L_h^2(\\omega_\\alpha)$;\\\\\n$(2)$ The Berezin transform $\\widetilde{\\varphi}$ is invertible in $L^\\infty(\\omega_\\alpha)$,  where\n$$\\widetilde{\\varphi}(z):=\\frac{1}{\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\int_{\\mathbb D}\\varphi(\\lambda)|R_\\alpha(z, \\lambda)|^2 \\omega_\\alpha(\\lambda) dA(\\lambda)$$\nand\n$$\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}=R^\\alpha_z(z)=\\frac{2}{(1-|z|^2)^{2+\\alpha}}-1;$$\\\\\n$(3)$ There exists $r>0$ such that  $$G:=\\{z \\in \\mathbb D: \\varphi(z)>r \\},$$ $\\chi_{G}dA$ is a reverse Carleson measure for $L_h^2( \\omega_\\alpha)$;\\\\\n$(4)$ There exists a constant $C>0$ such that\n$$\\int_ {\\mathbb D}|f(z)|^2 \\varphi(z)\\omega_\\alpha(z) dA(z)\\geqslant C \\int_{\\mathbb D}|f(z)|^2\\omega_\\alpha(z) dA(z)$$\nfor $f\\in L_h^2(\\omega_\\alpha)$.\n\\end{thm}\nBefore giving the proof the main theorem of this section, we need another lemma, which was proved in \\cite{Luecking1983}, \\cite{Miao1998} and \\cite{Sledd}.\n\\begin{lemma}\\label{D-K}\nSuppose that  the ball $K$ has radius $0<t<1$  and center $u=(1, 0)\\in \\mathbb R^2$.\nLet $f$ be the harmonic  function\n $$f(\\lambda)=f_s(\\lambda):=\\sqrt{1+\\alpha}R^\\alpha_{z_0}(\\lambda)(1-|z_0|^2)^{\\frac{2+\\alpha}{2}},$$\nwhere $z_0=(1-st)u$, $0<s<1$.\nThen for each $\\epsilon>0$, there exist  $s=s(\\epsilon)$  and a positive constant $C=C(\\epsilon)$ (independent of $K$) such that\\\\\n$$\\int_{B\\setminus K} |f(\\lambda)|^2 (1-|\\lambda|)^\\alpha dA(\\lambda)<\\epsilon$$\nand\n$$\\int_{G\\cap K}|f(\\lambda)|^2 (1-|\\lambda|)^\\alpha dA(\\lambda)\\leqslant C \\bigg(\\frac{|G\\cap K|}{|\\mathbb D \\cap K|}\\bigg)^\\beta,$$\nwhere\n$$\\beta=\\begin{cases}\n1,& \\mathrm{if}\\  0\\leqslant \\alpha<\\infty,\\\\\n1-\\frac{1}{\\gamma}, & \\mathrm{if} \\ -1<\\alpha<0,\n\\end{cases}$$\n$\\gamma$ is a number in $(1, -\\frac{1}{\\alpha})$ if $-1<\\alpha<0$.\n\\end{lemma}\n\n\nNow we are ready to prove  Theorem \\ref{invertibility1}.\n\n\\begin{proof}  [Proof of Theorem \\ref{invertibility1}]We will show that $(1)\\Rightarrow (2)\\Rightarrow (3)\\Rightarrow (4)\\Rightarrow (1)$. Without\nloss of generality, we may assume that $0\\leqslant \\varphi \\leqslant 1$.\n\n$(1)\\Rightarrow (2)$: This is trivial.\n\n$(2)\\Rightarrow (3)$: Suppose that $\\widetilde{\\varphi}$ is bounded below  by some positive constant $\\delta$. By Lemma  \\ref{reverse Carleson measure1}, it sufficient to show that\nthere exists a  $\\delta'\\in (0, 1)$ such that\n$$ |G\\cap K| \\geqslant \\delta' |\\mathbb D \\cap K|$$\nfor all  balls $K$ whose centers lie on $\\partial \\mathbb D$.\n\nSince $\\omega_{\\alpha}dA$ is a rotation invariant measure, it is no loss of generality to assume that $K$ has its center at the point $(1, 0)$. It is also clear that we need only to prove the inequality for sufficient small radius $t$, say $t<1$.\n\nNow we consider the subset $G=\\{\\lambda \\in \\mathbb D: \\varphi(\\lambda)>\\frac{\\delta}{4}\\}$.  For each $z \\in \\mathbb D$,\n\\begin{align*}\n\\delta &\\leqslant \\widetilde{\\varphi}(z)=\\frac{1}{\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\int_{\\mathbb D}\\varphi(\\lambda)|R^\\alpha_z (\\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda)\\\\\n&=\\frac{1}{\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\bigg(\\int_{G}+\\int_{\\mathbb D \\setminus G} \\bigg) \\varphi(\\lambda)|R^\\alpha_z( \\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda)\\\\\n&\\leqslant (1-|z|^2)^{2+\\alpha}\\int_{G} \\varphi(\\lambda)|R^\\alpha_z(\\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda) +\\frac{\\delta}{4}.\n\\end{align*}\nLet $L_z$ be the following integral:\n\\begin{align*}\n&L_z=\\frac{1}{\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\int_{G\\cap K} \\varphi(\\lambda)|R^\\alpha_z(\\lambda)|^2\\omega_\\alpha(\\lambda)  dA(\\lambda).\n\\end{align*}\nThen for each $z\\in \\mathbb D$, we have\n\\begin{align*}\nL_z&=\\frac{1}{\\|R^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\bigg(\\int_{G}-\\int_{G \\backslash K}\\bigg)\\varphi(\\lambda)|R^\\alpha_z( \\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda)\\\\\n&\\geqslant \\frac{1}{2}(1-|z|^2)^{2+\\alpha}\\bigg(\\int_{G}-\\int_{G \\backslash K}\\bigg)\\varphi(\\lambda)|R^\\alpha_z(\\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda)\\\\\n& \\geqslant \\frac{3\\delta}{8}-\\frac{1}{2}(1-|z|^2)^{2+\\alpha} \\int_{G \\backslash K}\\varphi(\\lambda)|R^\\alpha_z(\\lambda)|^2\\omega_\\alpha(\\lambda)  dA(\\lambda)\\\\\n&\\geqslant \\frac{3\\delta}{8}-\\frac{1}{2}(1-|z|^2)^{2+\\alpha} \\int_{G \\backslash K}|R^\\alpha_z(\\lambda)|^2\\omega_\\alpha(\\lambda)  dA(\\lambda)\\ \\ \\  \\big(\\mathrm{using}\\ 0\\leqslant \\varphi \\leqslant 1\\big).\n\\end{align*}\n\nFor the $\\delta$ above, Lemma \\ref{D-K} guarantees that we can select  $z_0\\in \\mathbb D$ to define a function $f$ (as the one in Lemma \\ref{D-K}) satisfies the following three inequalities:\n\\begin{align*}\nL_{z_0}& \\geqslant \\frac{3\\delta}{8}-\\frac{1}{2}(1-|z_0|^2)^{2+\\alpha} \\int_{G \\backslash K}|R^\\alpha_{z_0}(\\lambda)|^2\\omega_\\alpha(\\lambda)  dA(\\lambda)\\\\\n&=\\frac{3\\delta}{8}-\\frac{1}{2}\\int_{G \\backslash K} |f(\\lambda)|^2(1-|\\lambda|^2)^\\alpha dA(\\lambda),\n\\end{align*}\n$$\\int_{G \\backslash K} |f(\\lambda)|^2(1-|\\lambda|^2)^\\alpha dA(\\lambda)<\\frac{\\delta}{4}$$\nand\n$$\\int_{G\\cap K}|f(\\lambda)|^2 (1-|\\lambda|)^\\alpha dA(\\lambda)\\leqslant C \\bigg(\\frac{|G\\cap K|}{|\\mathbb D \\cap K|}\\bigg)^\\beta,$$\nwhere the constant $C$ depends only on $\\delta$ and $\\alpha$.  Therefore,\n\\begin{align*}\n\\frac{\\delta}{4} &\\leqslant L_{z_0} \\leqslant (1-|z_0|^2)^{2+\\alpha}\\int_{K\\cap G}|R^\\alpha_{z_0}(\\lambda)|^2\\omega_\\alpha(\\lambda) dA(\\lambda)\\\\\n&=\\int_{G\\cap K}|f(\\lambda)|^2 (1-|\\lambda|^2)^\\alpha dA(\\lambda)\\\\\n&\\leqslant C \\bigg(\\frac{|G\\cap K|}{|\\mathbb D \\cap K|}\\bigg)^\\beta.\n\\end{align*}\nNow we get $(3)$ by Lemma \\ref{reverse Carleson measure1}.\n\n$(3)\\Rightarrow (4)$: Observer that\n\\begin{align*}\n\\int_{\\mathbb D} |f(z)|^2  \\varphi(z)  \\omega_{\\alpha}(z) dA(z)&\\geqslant \\int_{G}\\varphi(z) |f(z)|^2 \\omega_{\\alpha}(z) dA(z)\\\\\n&> r\\int_{G} |f(z)|^2 \\omega_{\\alpha}(z) dA(z)\\\\\n&\\geqslant \\frac{r}{C} \\int_{\\mathbb D} |f(z)|^2 \\omega_{\\alpha} (z) dA(z),\n\\end{align*}\nthe last inequality follows from the definition of the reverse Carleson measure.\n\n$(4)\\Rightarrow (1)$: Use the same arguments as the proof of Corollary 3 in \\cite{Lue}, we obtain that\n$\\|I-T_\\varphi\\|<1$, which implies that $T_\\varphi$ is invertible on $L_h^2(\\omega_{\\alpha})$.  This completes the whole proof of the theorem.\n\\end{proof}\n\nLet $\\mathbb T_\\varphi$  denote the Toeplitz operator with symbol $\\varphi$ on the weighted Bergman space $L_a^2(\\omega_\\alpha)$.\n Combining the main result in \\cite{Lue} and the techniques used in the proof of Theorem \\ref{invertibility1}, we can generalize Theorem 3.2 in \\cite{Zhao} to the case of standard weighted Bergman space.\n\n\n\\begin{thm}\\label{invertibility2}\n Let $\\varphi\\geqslant 0$ be in $L^\\infty(\\omega_\\alpha)$. Then the following  are equivalent:\\\\\n$(1)$ The Toeplitz operator $\\mathbb T_\\varphi$  is invertible on $L_a^2(\\omega_\\alpha)$;\\\\\n$(2)$ The Berezin transform $\\widehat{\\varphi}$ is invertible in $L^\\infty(\\omega_\\alpha)$, where\n$$\\widehat{\\varphi}(z):=\\frac{1}{\\|K^\\alpha_z\\|^2_{L^2(\\omega_\\alpha)}}\\int_{\\mathbb D}\\varphi(\\lambda)|K^\\alpha_z( \\lambda)|^2\\omega_\\alpha(\\lambda)dA(\\lambda);$$\n$(3)$ There exists $r>0$ such that  $$G:=\\{z \\in \\mathbb D: \\varphi(z)>r \\},$$ $\\chi_{G}dA$ is a reverse Carleson measure for $L_a^2(\\omega_\\alpha)$;\\\\\n$(4)$ There exists a  $\\delta\\in (0, 1)$ such that\n$$ |G\\cap K| \\geqslant \\delta |\\mathbb D \\cap K|$$\nfor every ball $K$ whose center lies on $\\partial \\mathbb D$;\\\\\n$(5)$ There exists a constant $C>0$ such that\n$$\\int_ {\\mathbb D}|f(z)|^2 \\varphi(z) \\omega_\\alpha(z) dA(z)\\geqslant C \\int_{\\mathbb D}|f(z)|^2\\omega_\\alpha(z) dA(z)$$\nfor $f\\in L_a^2(\\omega_\\alpha)$.\n\\end{thm}\n\\begin{proof}\nFrom the proof of Theorem \\ref{invertibility1}, it sufficient to show that one can replace the harmonic function $f$ defined in Lemma\n\\ref{D-K} by a suitable analytic function $g$. Indeed, we  construct the desired function $g$ as follows. Suppose that $K$ has radius $0<t<1$  and center $u=(1, 0)\\in \\mathbb R^2$. Define\n $$g(\\lambda)=\\sqrt{\\alpha+1}K^\\alpha_{z_0}(\\lambda)(1-|z_0|^2)^{\\frac{2+\\alpha}{2}},$$\nwhere $z_0=(1-st)u$, $0<s<1$. Then it is not hard to check that the two inequalities in Lemma\n\\ref{D-K} both hold for $g$. Now the rest of the  proof parallels exactly one given in Theorem \\ref{invertibility1}.\n\\end{proof}\n\nBased on the theorem above, we can establish a relationship of the invertibility  between  Toeplitz operators with nonnegative symbols on $L_a^2(\\omega_\\alpha)$ and $L_h^2(\\omega_\\alpha)$.\n\\begin{cor}\\label{invertibility3}\nLet $\\varphi$ be a nonnegative bounded measurable function on $\\mathbb D$. The following four conditions are equivalent:\\\\\n$(1)$  $\\mathbb T_\\varphi$  is invertible on $L_a^2(\\omega_\\alpha)$;\\\\\n$(2)$  $T_\\varphi$  is invertible on $L_h^2(\\omega_\\alpha)$;\\\\\n$(3)$  $\\widehat{\\varphi}$ is invertible in $L^\\infty(\\omega_\\alpha)$;\\\\\n$(4)$  $\\widetilde{\\varphi}$ is invertible in $L^\\infty(\\omega_\\alpha)$.\n\\end{cor}\n\n\n\n\n\n\n\n\n To end this section, we study Toeplitz operators with bounded analytic symbols on $L_h^2(\\omega_\\alpha)$. Unlike the  Bergman space  $L_a^2(\\omega_\\alpha)$ setting, the spectrum of analytic Toeplitz operator $T_\\varphi$ on $L_h^2(\\omega_\\alpha)$ is not\nequal to the closure of $\\varphi(\\mathbb D)$.  Let $H^\\infty$ denote space of  analytic functions in $L^\\infty(\\omega_\\alpha)$, we have the following result.\n\\begin{thm}\\label{Analytic}\nSuppose that $\\varphi \\in H^\\infty$. If $\\varphi$ is invertible, then the Toeplitz operator $T_\\varphi$\nis invertible on $L_h^2(\\omega_\\alpha)$. However, the converse is not true.\n\\end{thm}\n\nLet $\\varphi \\in H^\\infty$  be  invertible. However, $T_\\varphi T_{1\/\\varphi}=T_{1\/\\varphi}T_\\varphi$ does not hold on the harmonic Bergman space in the general case (see Theorem 5 in \\cite{Choe}). For this reason, we need to find some relationships between $T_\\varphi$ and $\\mathbb T_\\varphi$ to study the invertibility problem. To do so, let us introduce some notations first.\n\nFor $\\varphi \\in L^\\infty(\\omega_\\alpha)$, we define $$\\varphi^\\star(z)=\\varphi(\\overline{z}) \\ \\  \\mathrm{and}\\ \\\n\\varphi^*(z)=\\overline{\\varphi(\\overline{z})}.$$\n\nNext, we define the unitary operator $W: L_h^2(\\omega_\\alpha)=zL_a^2(\\omega_\\alpha)\\oplus \\overline{L_a^2(\\omega_\\alpha)}\\rightarrow zL_a^2(\\omega_\\alpha)\\oplus L_a^2(\\omega_\\alpha)$ by\n$$W=\\left( {\\begin{array}{*{20}{c}}\n   {{I}} & {{O}}  \\\\\n   {{O}} & {{U}}  \\\\\n\\end{array}} \\right),$$\nwhere $U$ is a unitary operator on $L^2(\\omega_\\alpha)$, defined by $Uf(z)=f(\\overline{z})$.\n\nIt is clear that $W^*$ maps $zL_a^2(\\omega_\\alpha)\\oplus L_a^2(\\omega_\\alpha)$ to $L_h^2(\\omega_\\alpha)$ and\n$$W^*=\\left( {\\begin{array}{*{20}{c}}\n   {{I}} & {{O}}  \\\\\n   {{O}} & {{U}}  \\\\\n\\end{array}} \\right).$$\n\nFor $f$ and $g$ in $L^2(\\omega_\\alpha)$, let $f\\otimes g$ be the rank one operator defined by\n$$(f\\otimes g)h=\\langle h, g\\rangle_{L^2(\\omega_\\alpha)}f$$\nfor $h\\in L^2(\\omega_\\alpha)$.\n\nLet $P$ denote the orthogonal projection from $L^2(\\omega_\\alpha)$ to $L_a^2(\\omega_\\alpha)$, we define the Hankel operator with symbol $\\varphi$ acting on $L_a^2(\\omega_\\alpha)$ by $\\mathbb H_{\\varphi} f=P(\\varphi U f)$.\nUsing the notations  above,  we obtain the following matrix representation of $WT_\\varphi W^*$, which  will be used to study the invertibility problem.\n\\begin{lemma}\\label{matrix representation}\nLet $\\varphi \\in L^\\infty(\\omega_\\alpha)$. On the space $zL_a^2(\\omega_\\alpha)\\oplus L_a^2(\\omega_\\alpha)$ we have\n$$W T_\\varphi W^*=\\left( {\\begin{array}{*{20}{c}}\n   {{\\mathbb{T}_\\varphi-(1\\otimes \\overline{\\varphi})}} & {{\\mathbb{H}_\\varphi- (1\\otimes \\varphi^*)}}  \\\\\n   {{\\mathbb H_{\\varphi^\\star}}} & {{\\mathbb {T}_{\\varphi^\\star}}}  \\\\\n\\end{array}} \\right).$$\n\\end{lemma}\n  \\begin{proof}The proof is exactly the same as the proof of  Theorem 2.1 in \\cite{Guo}, we omit the details here.\n  \\end{proof}\n\nWe are now ready to prove Theorem \\ref{Analytic}. \\\\\n\\begin{proof} [Proof of Theorem \\ref{Analytic}.] For $\\varphi \\in H^\\infty$ and $f\\in zL_a^2(\\omega_\\alpha)$ we have\n$$\\mathbb H_{\\varphi^\\star} f=P(\\varphi^\\star Uf)=P(\\varphi(\\overline{z})f(\\overline{z}))=0$$\nand $$(1\\otimes \\overline{\\varphi}) f= \\langle f, \\overline{\\varphi}\\rangle_{L^2(\\omega_\\alpha)} 1=0,$$\nsince $f(0)=0$.\n\n Consequently, the matrix representation of $W T_\\varphi W^*$ with $\\varphi \\in H^\\infty$ is given by\n$$W T_\\varphi W^*=\\left( {\\begin{array}{*{20}{c}}\n   {{\\mathbb{T}_\\varphi}} & {{\\mathbb{H}_\\varphi- (1\\otimes \\varphi^*)}}  \\\\\n   {{O}} & {{\\mathbb {T}_{\\varphi^\\star}}}  \\\\\n\\end{array}} \\right).$$\nNote that $\\varphi \\in H^\\infty$ is invertible implies that $\\varphi^\\star$ is also invertible,  so that  $\\mathbb T_{\\varphi^\\star}$ is invertible on $L_a^2(\\omega_\\alpha)$.\n\nOn the other hand, $\\mathbb T_\\varphi$ is invertible on $zL_a^2(\\omega_\\alpha)$ follows from  $\\varphi$ is invertible. The above matrix representation tells  us that $W T_\\varphi W^*$ is invertible on $zL_a^2(\\omega_\\alpha)\\oplus L_a^2(\\omega_\\alpha)$. Thus $T_\\varphi$ is invertible on $L_h^2(\\omega_\\alpha)$.\n\nTo show there exists a function $\\varphi$ in $H^\\infty$ such that $\\varphi$ is not invertible in $L^\\infty(\\omega_\\alpha)$ but $T_\\varphi$\nis invertible on $L_h^2(\\omega_\\alpha)$, we consider the function $\\varphi(z)=z.$ It is easy to show that\n$$\\mathrm{Ker}(T_z)=\\mathrm{Ker}(T^*_z)=\\mathrm{Ker}(T_{\\overline{z}})=\\{0\\}.$$\nObserve that $T_z$ is Fredholm,  we conclude that $T_z$ is invertible on $L_h^2(\\omega_\\alpha)$. This finishes the proof.\n\\end{proof}\n\n\n\\section{A Reverse Carleson type Inequality for $L_h^2(\\omega)$}\nIn the preceding section, we study the invertibility problem  of Toeplitz operators via the reverse Carleson measures for standard weighted harmonic Bergman spaces. In this section, we  establish a sufficient condition for $\\chi_G dA$ to be a reverse Carleson measure for the space $L_h^2(\\omega)$,  where $\\omega\\in \\mathcal{A}_2$ and $G$ is a measurable set in $\\mathbb D$.\n\nFor $a\\in\\mathbb{D}$, $0<r<1$, recall that\n\\begin{equation*}\nS(a,r)=\\{z\\in\\mathbb{D}:|z-a|<r(1-|a|)\\}.\n\\end{equation*}\nThe main result in this section is Theorem \\ref{the weight w}, which is a harmonic version of Theorem 3.9 in \\cite{Luecking1985}.\n\n\\begin{thm}\\label{the weight w}\nSuppose that $G\\subset\\mathbb{D}$  and  $\\omega$ satisfies $\\mathcal{A}_2$ condition. If there exist  $\\delta\\in(0,1)$ and $r\\in(0,1)$ such that for all $a\\in\\mathbb{D}$\n\\begin{equation*}\n|G\\cap S(a, r)|\\geqslant \\delta |S(a, r)|,\n\\end{equation*}\nthen there exists a positive constant $C=C(r, \\delta)$ such that\n$$ \\int_{\\mathbb D}|f(z)|^2\\omega(z)dA(z)\\leqslant C \\int_{G}|f(z)|^2\\omega(z) dA(z)$$\nfor all $f\\in L^2_h(\\omega)$.\n\\end{thm}\n\nTo prove the above theorem, we will adopt some ideas and techniques in \\cite{Luecking1985}. Firstly, we need to introduce a new (weight) function $\\omega^*$ and discuss some properties of $\\omega^*$. In the rest of this section,  we use ``$r$\" and ``$\\delta$\" to denote the numbers provided in Theorem \\ref{the weight w}.\n\nNow we define a positive  function $\\omega^*$ on the open unit disk as follows:\n$$\\omega^*(z)=\\omega_r^*(z):=\\frac{|S(z, r)|_\\omega}{|S(z, r)|}.$$\nIt is clear that $\\omega^*\\in L^1(dA)$, and so $\\omega^*$ is a weight. Moreover, $\\omega^*$ has\nthe following important property.\n\\begin{lemma}\\label{measure estimate0}\nLet $z\\in \\mathbb D$,  there exist constants $C_1$ and $C_2$ depending only on $r$ such that\n$$C_1\\omega^*(a)\\leqslant \\omega^*(z)\\leqslant C_2 \\omega^*(a)$$\nfor all  $a\\in S(z, r)$. Consequently, we have\n$$\\int_{\\mathbb D}\\frac{\\omega^*(a)}{|S(a, r)|} \\chi_{S(a, r)}(z) dA(a)\\leqslant C_3\\omega^*(z)\\ \\ \\ \\big(z \\in \\mathbb D\\big), $$\n where $C_3=C_3(r)$ is a constant.\n\\end{lemma}\n\\begin{proof}\nBy Lemma 2.2  in \\cite{Con}, there exists a positive  constant $C$ depending only on $r$  such that\n$$ C^{-1}|S(a, r)|_\\omega \\leqslant |S(z, r)|_\\omega \\leqslant C|S(a, r)|_\\omega $$\n Moreover, it is well known that  $|S(z, r)|$ is equivalent to $|S(a, r)|$ (with constants independent of $a$ and $z$) if  $a\\in S(z, r)$.\nThis gives the first conclusion of the lemma.  Based on this result,  we have\n\\begin{align*}\n\\int_{\\mathbb D}\\frac{\\omega^*(a)}{|S(a, r)|}\\chi_{S(a, r)}(z)dA(a)\n&\\leqslant C \\int_{\\mathbb D}\\frac{\\omega^*(z)}{|S(a, r)|}\\chi_{S(a, r)}(z)dA(a)\\ \\ \\  \\big(\\mathrm{using}\\ z\\in S(a,r)\\big)\\\\\n &\\leqslant C \\int_{\\mathbb D}\\frac{\\omega^*(z)}{|S(a, r)|}\\chi_{D(a, r)}(z)dA(a) \\ \\ \\  \\big(\\mathrm{since}\\ S(a,r)\\subset D(a,r)\\big) \\\\\n &=C\\int_{\\mathbb D}\\frac{\\omega^*(z)}{|S(a, r)|}\\chi_{D(z, r)}(a)dA(a)\\\\\n &=C\\int_{D(z, r)}\\frac{\\omega^*(z)}{|S(a, r)|}dA(a)\\\\\n& \\leqslant C_3 \\omega^*(z),\n\\end{align*}\nwhere $C_3$ is a positive constant depending only on $r$, as desired.\n\\end{proof}\nAnother property of $\\omega^*$ is given by the following inequality, which will be used to estimate the integral of $|f|^2 \\omega$ over the subset $G$.\n\\begin{lemma}\\label{norm of w less then norm of w*} Let $\\omega$ be an $\\mathcal {A}_2$ weight. Then there exists a constant $C>0$ depending only on  $r$ such that\n$$\\|f\\|_{L^2(\\omega)}^2 \\leqslant C \\|f\\|_{L^2(\\omega^*)}^2$$\nfor all $f\\in L_h^2(\\omega)$.\n\\end{lemma}\n\\begin{proof}\nBy definitions, we have\n\\begin{align*}\n\\|f\\|_{L^2(\\omega^*)}^2&=\\int_{\\mathbb D} |f(z)|^2\\omega^*(z)dA(z)\\\\\n&=\\int_{\\mathbb D} \\omega(\\xi)\\bigg(\\int_{\\mathbb D} |f(z)|^2 \\frac{\\chi_{S(z, r)}(\\xi)}{|S(z, r)|}dA(z)\\bigg)dA(\\xi),\n\\end{align*}\n We next deal with the bracketed expression. Observe\n$$ S\\Big(\\xi, \\frac{r}{2(1+r)}\\Big)\\subset \\big\\{z\\in \\mathbb D: |z-\\xi|<r(1-|z|)\\big\\},$$\nwe obtain\n\\begin{align*}\n\\int_{\\mathbb D} |f(z)|^2 \\frac{\\chi_{S(z, r)}(\\xi)}{|S(z, r)|}dA(z)\n&=\\int_{\\{z\\in \\mathbb D: |z-\\xi|<r(1-|z|\\}} \\frac{|f(z)|^2 }{|S(z, r)|}dA(z)\\\\\n&\\geqslant \\int_{S(\\xi, \\frac{r}{2(1+r)})} \\frac{|f(z)|^2 }{|S(z, r)|}  dA(z)\\\\\n& \\geqslant \\frac{C}{|S(\\xi, r)|} \\int_{S(\\xi, \\frac{r}{2(1+r)})} |f(z)|^2 dA(z) \\ \\ \\  \\ \\bigg(\\mathrm{since}\\ z\\in S\\big(\\xi,\\frac{r}{2(1+r)}\\big)\\bigg)\\\\\n&=C\\bigg[\\frac{1}{|S(\\xi, \\frac{r}{2(1+r)})|} \\int_{S(\\xi, \\frac{r}{2(1+r)})} |f(z)|^2 dA(z)\\bigg] \\cdot \\frac{|S(\\xi, \\frac{r}{2(1+r)})|}{|S(\\xi, r)|}\\\\\n&\\geqslant \\frac{C}{16} |f(\\xi)|^2,\n\\end{align*}\nthe last inequality follows form the subharmonicity  of $|f|^2$ (see Lemma \\ref{subharmonic behavior of harmonic functions}) and\n$C$ depends only on $r$.  Thus we get\n$$\\|f\\|_{L^2(\\omega^*)}^2\\geqslant \\frac{C}{16}  \\int_{\\mathbb D} |f(\\xi)|^2 \\omega(\\xi)dA(\\xi)=\\frac{C}{16} \\|f\\|^2_{L^2(\\omega)},$$\nwhich finishes the proof of Lemma \\ref{norm of w less then norm of w*}.\n\\end{proof}\n\n\nIn order to complete the proof the main theorem in this section, the following two key lemmas are also needed.\n\\begin{prop}\\label{the measure of F}\nLet $G$ be the subset which satisfies the assumption in Theorem \\ref{the weight w}. For $\\eta\\in (0, 1)$, we  define a subset $F$  as the following:\n $$F:=\\{z\\in \\mathbb D: \\omega(z)\\geqslant \\eta \\omega^*(z)\\}.$$\nThen one can choose  $\\eta$ (depending only on $\\delta$ and $r$) sufficiently small such that\n$$|F\\cap S(a, r)|\\geqslant (1-\\frac{\\delta}{2})|S(a, r)|$$\nand\n$$\\Big| G\\cap S(a, r)\\cap F\\Big|\\geqslant \\frac{\\delta}{2}|S(a, r)|$$\nfor all $a\\in \\mathbb D$,\n\n\\end{prop}\n\\begin{proof} We first claim that for any $\\delta'\\in (0, 1)$,\nthere exists  $\\eta'=\\eta'(\\delta')>0$ such that\n$$\\big|\\{z\\in S(a, r): \\omega(z)<\\eta' \\omega^*(a)\\}\\big|<\\delta' |S(a, r)|$$\nfor all $a\\in \\mathbb D$.\n\nIndeed, for each $\\kappa \\in (0, 1)$ and $a\\in \\mathbb D$, we have\n\\begin{align*}\n&\\big|\\{z\\in S(a, r): \\omega(z)<\\kappa \\omega^*(a)\\}\\big|\\cdot \\frac{1}{\\kappa \\omega^*(a)}\\\\\n&< \\int_{\\{z\\in S(a, r): \\omega(z)<\\kappa \\omega^*(a)\\}}\\frac{1}{\\omega(z)} dA(z)\\\\\n&\\leqslant |S(a, r)|_{\\omega^{-1}}\\\\\n&\\leqslant [\\omega]_{\\mathcal{A}_2} |S(a, r)|^2\\cdot |S(a, r)|_\\omega^{-1}.\n\\end{align*}\nSo, we obtain\n$$\\big|\\{z\\in S(a, r): \\omega(z)<\\kappa\\omega^*(a)\\}\\big| <  ([\\omega]_{\\mathcal{A}_2}  \\kappa) |S(a, r)|$$\nfor all $a\\in \\mathbb D$ and $\\kappa\\in (0, 1)$.\nBy this inequality, we can choose any  $0<\\eta' \\leqslant [\\omega]_{\\mathcal{A}_2}^{-1}\\delta'$ to finish the proof of the claim.\n\nLemma \\ref{measure estimate0} guarantees  that there is a constant $C=C(r)$ such that\n$$\\{z\\in S(a, r): \\omega(z)<C \\tau \\omega^*(z)\\}\\subset \\{z\\in S(a, r): \\omega(z)< \\tau  \\omega^*(a)\\}\\ \\ \\ \\big( a\\in \\mathbb D \\big) $$\nfor every $\\tau\\in (0, 1)$. By the claim and its proof, there exists a   $\\tau=\\tau(\\delta)<C^{-1}$ such that\n$$\\big|\\{z\\in S(a, r): \\omega(z)< \\tau \\omega^*(a)\\}\\big|<\\frac{\\delta}{2}|S(a, r)|  \\ \\ \\ \\big( a\\in \\mathbb D \\big) .$$\n\n\nTherefore, we can take  $\\eta=\\eta(\\delta, r)=C\\tau<1$ such that\n$$\\big|\\{z\\in S(a, r): \\omega(z)< \\eta \\omega^*(z)\\}\\big|<\\frac{\\delta}{2}|S(a, r)|  \\ \\ \\ \\big( a\\in \\mathbb D \\big).$$\nUsing this $\\eta$ to define the corresponding  $F$, so that\n$$|F\\cap S(a, r)|=\\big|\\{z\\in S(a, r): \\omega(z)\\geqslant \\eta \\omega^*(z)\\}\\big| \\geqslant (1-\\frac{\\delta}{2})|S(a, r)|$$\nfor all $a\\in \\mathbb D$.\n\n\nBy assumption\n$$|G\\cap S(a, r)|\\geqslant \\delta | S(a, r)|,$$\nwe have\n\\begin{align*}\n\\delta |S(a, r)|&\\leqslant |G\\cap S(a, r)|\\\\\n&=\\Big| \\big[G\\cap S(a, r)\\cap F\\big]\\cup \\big[G\\cap S(a, r)\\cap (\\mathbb D \\setminus F)\\big]\\Big|\\\\\n&\\leqslant \\Big| G\\cap S(a, r)\\cap F\\Big|+|S(a, r)\\cap (\\mathbb D \\setminus F)|\\\\\n&= \\Big| G\\cap S(a, r)\\cap F\\Big|+|S(a, r)|-|S(a, r)\\cap F|\\\\\n&\\leqslant \\Big| G\\cap S(a, r)\\cap F\\Big|+|S(a, r)|-(1-\\frac{\\delta}{2})|S(a, r)|.\n\\end{align*}\nThis gives that\n$$\\Big| G\\cap S(a, r)\\cap F\\Big|\\geqslant \\frac{\\delta}{2}|S(a, r)|$$\nfor all $a\\in \\mathbb D$, as desired.\n\\end{proof}\n\n\\begin{lemma}\\label{the weight function w*} If  $G_0$ is  a measurable  subset of $\\mathbb D$  that satisfies\n $$|G_0\\cap S(a, r)|\\geqslant \\delta_0 |S(a, r)| \\ \\ \\  \\big( a \\in \\mathbb D \\big)$$\nfor some $\\delta_0>0$,\nthen\nthere exists constant $C=C(\\delta_0, r)>0$ such that\n$$ \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)\\leqslant C \\int_{G_0}|f(z)|^2\\omega^*(z) dA(z)$$\nfor all $f\\in L_h^2(\\omega^*)$.\n\\end{lemma}\n\nThe proof of the preceding lemma is some what long and it requires a number of technical lemmas. Let us assume this result  for the moment and we will prove it at the end of this section. Now we give the proof of Theorem \\ref{the weight w}. \\vspace{2mm}\\\\\n\\begin{proof}[Proof of Theorem \\ref{the weight w}]  By  Proposition \\ref{the measure of F} and Lemma \\ref{the weight function w*},\n we have\n$$ \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)\\leqslant C_1 \\int_{G\\cap F}|f(z)|^2\\omega^*(z) dA(z)\\leqslant C_1 \\eta^{-1}\\int_{G}|f|^2\\omega dA$$\nfor all $f \\in L_h^2(\\omega^*)$, where $C_1$ is a constant depending only on $r$ and $\\eta=\\eta(\\delta, r)<1$ is chosen by Proposition \\ref{the measure of F}.\n\nFrom Lemma \\ref{norm of w less then norm of w*}, it is clear that\n$$ \\int_{\\mathbb D}|f(z)|^2\\omega (z)dA(z)\\leqslant C_1 \\eta^{-1} \\int_{G}|f(z)|^2\\omega (z) dA(z)$$\nfor all $f\\in L_h^2(\\omega)$,\nwhich gives the desired inequality in  Theorem \\ref{the weight w}.\n\\end{proof}\n\n\nNow we turn to the proof of Lemma \\ref{the weight function w*}.  Before giving the proof, we need to introduce some notations and prove three technical lemmas.\n\nLet $0<\\theta<\\frac{1}{2}$, we define the subset\n$$E_\\theta(a)=E_\\theta(f, a):=\\Big\\{z\\in S(a, r): |f(z)| >\\theta |f(a)| \\Big\\}$$\nand  the following operator:\n$$B_\\theta f(a):=\\frac{1}{|E_\\theta(a)|}\\int_{E_\\theta(a)}|f(z)|^2 dA(z)\\ \\ \\ \\big(a\\in \\mathbb D \\big).$$\nIt is clear that\n$$B_\\theta f(a)\\geqslant \\frac{1}{|S(a, r)|}\\int_{S(a, r)}|f(z)|^2 dA(z) \\ \\ \\ \\big(a\\in \\mathbb D \\big).$$\n\nFor $\\epsilon \\in (0, 1)$, we consider the  following two subsets, which are very useful to establish our main result.\nDefine $$A=A_\\epsilon:=\\bigg\\{a\\in\\mathbb D: |f(a)|^2 \\leqslant \\frac{\\epsilon}{|S(a, r)|}\\int_{S(a, r)}|f(z)|^2 dA(z)\\bigg\\}$$\nand\n$$B=B_\\epsilon:=\\bigg\\{a\\in\\mathbb D: |f(a)|^2 \\leqslant \\epsilon^2 B_\\theta f(a)\\bigg\\}.$$\n\n\n\n\n\nA useful estimation for the Lebesgue measure of $\\big\\{z\\in S(a, r): |f(z)|>\\theta|f(a)|\\big\\}$ with $f$  harmonic is the following inequality.\n\\begin{lemma}\\label{measure estimate3}\nFix $\\epsilon\\in (0, 1)$. For any $\\delta'\\in (0,1 )$, there exists $ \\theta \\in (0, \\frac{1}{2})$ such that\n$$\\bigg|\\Big\\{z\\in S(a, r): |f(z)|> \\theta |f(a)| \\Big\\}\\bigg|>\\big(1-\\frac{\\delta'}{2}\\big)|S(a, r)|$$\nfor every   $f$  harmonic on $\\mathbb D$ and  satisfies\n$$|f(a)|^2> \\frac{\\epsilon^2}{|S(a, r)|}\\int_{S(a, r)}|f(z)|^2dA(z)\\ \\ \\ (a\\in \\mathbb D).$$\n\\end{lemma}\n\\begin{proof}\nSee the proof of Lemma 2 in  \\cite{Luecking1983}.\n\\end{proof}\n The next lemma provides  an estimation for the integral of $|f|^2\\omega^*$ over the set $A$.\n\\begin{lemma}\\label{the integral over A}\nLet $\\epsilon \\in (0, 1)$, then there exists a constant\n$C$ (independent of $\\epsilon$) such that\n$$\\int_{A}|f(z)|^2\\omega^*(z)dA(z)\\leqslant C\\epsilon \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)$$\nfor all $f\\in L_h^2(\\omega^*)$.\n\\end{lemma}\n\\begin{proof} For $a\\in A$,  we have\n$$|f(a)|^2 \\leqslant \\frac{\\epsilon}{|S(a, r)|}\\int_{\\mathbb D}\\chi_{S(a, r)}(z)|f(z)|^2 dA(z).$$\nMultiplying the above inequality by $\\omega^*(a)$ and integrating over $A$  to obtain\n\\begin{align*}\n\\int_{A}|f(a)|^2 \\omega^*(a) dA(a) & \\leqslant \\int_{A}\\frac{\\epsilon}{|S(a, r)|}\\int_{\\mathbb D}\\chi_{S(a, r)}(z)|f(z)|^2 dA(z)\\omega^*(a)dA(a)\\\\\n&=\\epsilon \\int_{\\mathbb D} |f(z)|^2 \\bigg(\\int_A \\frac{\\chi_{S(a, r)}(z)}{|S(a, r)|}\\omega^*(a) dA(a)\\bigg) dA(z).\n\\end{align*}\nBy the second conclusion of Lemma \\ref{measure estimate0}, we have\n\\begin{align*}\n\\int_A \\frac{\\chi_{S(a, r)}(z)}{|S(a, r)|} \\omega^*(a) dA(a)& \\leqslant C\\omega^*(z),\n\\end{align*}\nwhere $C$  depends only on $r$. This completes the proof.\n\\end{proof}\n\nThe proof of Lemma \\ref{the weight function w*} requires the following inequality, which can be proved by the preceding  lemma.\n\\begin{lemma}\\label{the integral over B}\nLet $\\epsilon \\in (0, 1)$, then there exists a constant\n$C=C(r)$ such that\n$$\\int_{B}|f(z)|^2\\omega^*(z)dA(z)\\leqslant C\\epsilon \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)$$\nfor all $f\\in L_h^2(\\omega^*)$.\n\\end{lemma}\n\\begin{proof} Observe that\n\\begin{align*}\n\\int_{B}|f(z)|^2\\omega^*(z)dA(z)&=\\int_{B\\cap A}|f(z)|^2\\omega^*(z)dA(z)+\\int_{B\\setminus A}|f(z)|^2\\omega^*(z)dA(z)\\\\\n&\\leqslant \\int_{ A}|f(z)|^2\\omega^*(z)dA(z)+\\int_{B\\setminus A}|f(z)|^2\\omega^*(z)dA(z).\n\\end{align*}\nBased on Lemma \\ref{the integral over A}, it sufficient for us to show the following  inequality holds for some constant $C=C(r)$:\n$$J:=\\int_{B\\setminus A}|f(z)|^2\\omega^*(z)dA(z)\\leqslant C\\epsilon \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z).$$\nRecall that for each $a\\in B$ we have\n$$|f(a)|^2 \\leqslant \\frac{\\epsilon^2}{|E_\\theta(a)|}\\int_{E_\\theta (a)}|f(z)|^2dA(z).$$\nFrom the above inequality we have\n\\begin{align*}\nJ&=\\int_{B\\setminus A}|f(a)|^2\\omega^*(a)dA(a)\\\\\n&\\leqslant \\epsilon^2 \\int_{B \\setminus  A} \\bigg( \\frac{1}{|E_\\theta(a)|}\\int_{E_\\theta(a)}|f(z)|^2dA(z)\\bigg) \\omega^*(a)dA(a)\\\\\n&= \\epsilon^2 \\int_{\\mathbb D} \\bigg( \\int_{B\\setminus A} \\frac{\\omega^*(a)}{|E_\\theta(a)|} \\chi_{E_{\\theta}(a)}(z) dA(a)\\bigg)|f(z)|^2dA(z)\\\\\n&\\leqslant  \\epsilon^2 \\int_{\\mathbb D} |f(z)|^2 \\bigg( \\int_{B \\setminus A} \\omega^*(a) \\frac{\\chi_{S(a, r)}(z)}{|E_\\theta(a)|} dA(a)\\bigg) dA(z).\n\\end{align*}\nThe last ``$\\leqslant $\" is due to  $E_\\theta(a) \\subset S(a, r)$.\n\n\nTo finish the proof, we need the following claim.\\\\\n\\textbf{Claim.} There is a positive constant $C=C(r)$ such that\n$$|E_\\theta(a)|\\geqslant C \\epsilon |S(a, r)|\\ \\ \\ \\mathrm{or}\\ \\ \\ |E_\\theta(a)|\\geqslant C |S(a, r)|$$\nfor each $a\\notin A.$\n\nIf the above claim is true, then we get\n\\begin{align*}\n\\int_{B \\setminus A} \\omega^*(a) \\frac{\\chi_{S(a, r)}(z)}{|E_\\theta(a)|} dA(a)\n& \\leqslant C^{-1} \\epsilon \\int_{B \\setminus A}\\omega^*(a) \\frac{\\chi_{S(a, r)}(z)}{|S(a, r)|}dA(a).\n\\end{align*}\nUse Lemma \\ref{measure estimate0} again, we have\n\\begin{align*}\n\\int_{B \\setminus A}\\omega^*(a) \\frac{\\chi_{S(a, r)}(z)}{|S(a, r)|}dA(a) \\leqslant C_1 \\omega^*(z),\n\\end{align*}\nthe constant $C_1$ depends only on $r$. From the definition of $J$, we obtain\n$$J\\leqslant C\\epsilon \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)$$\nfor some positive constant $C=C(r)$.\n\nNow we turn to prove the claim.  For each $a\\notin A$, we have\n\\begin{align*}\n|f(a)|^2 &>\\frac{\\epsilon}{|S(a, r)|}\\int_{S(a, r)}|f(z)|^2 dA(z)\\\\\n&=\\frac{\\epsilon}{r^2(1-|a|)^2}  \\int_{S(a, r)} |f(z)|^2dA(z).\n\\end{align*}\nUsing the change of variables $z=a+r(1-|a|)\\lambda$ to get\n\\begin{align*}\n|f(a)|^2 &> \\epsilon \\int_{\\mathbb D} |f(a+r(1-|a|)\\lambda)|^2dA(\\lambda).\n\\end{align*}\nLet $g(\\lambda):=f(a+r(1-|a|)\\lambda)$, then $g$ is also harmonic on $\\mathbb D$ and\n$$|g(0)|^2> \\epsilon \\int_{\\mathbb D} |g(\\lambda)|^2dA(\\lambda).$$\n\nApplying Lemma \\ref{gradient estimation} to the function $g$ to get a constant $C_0=C_0(r)$ such that\n\\begin{align*}\n|g(z)-g(0)| & \\leqslant C_0 |z| \\int_{D(0, \\frac{r}{4})} |g(\\lambda)| dA(\\lambda)\\leqslant C_0 |z| \\int_{\\mathbb D} |g| dA\n\\end{align*}\nwhenever $|z|\\leqslant \\frac{r}{16}.$  Cauchy-Schwartz inequality gives that\n$$|g(z)-g(0)| \\leqslant C_0 |z| \\bigg(\\int_{\\mathbb D} |g(\\lambda)|^2 dA(\\lambda)\\bigg)^{\\frac{1}{2}}\\leqslant C_0 \\epsilon^{-\\frac{1}{2}}|g(0)|\\cdot |z| .$$\nprovided $|z|\\leqslant \\frac{r}{16}$.\n\nIf\n$$|z| < \\min\\Big\\{\\frac{r}{16}, \\frac{\\epsilon^{\\frac{1}{2}}}{2C_0}\\Big\\},$$\nthen\n$$|g(z)|\\geqslant |g(0)|-|g(z)-g(0)|\\geqslant \\frac{|g(0)|}{2}.$$\nRecall that $0<\\theta< \\frac{1}{2}$, we have\n$|g(z)|>\\theta |g(0)|$ for $|z|<\\min\\big\\{\\frac{r}{16}, \\frac{\\epsilon^{\\frac{1}{2}}}{2C_0}\\big\\}.$ This means\nthat $$B\\big(0, \\frac{r}{16}\\big) \\subset \\Big\\{z\\in \\mathbb D: |g(z)|>\\theta |g(0)| \\Big \\}$$\nor\n$$ B\\big(0, \\frac{\\epsilon^{\\frac{1}{2}}}{2C_0}\\big) \\subset \\Big\\{z\\in \\mathbb D: |g(z)|>\\theta |g(0)| \\Big \\}.$$\n\nOn the other hand, observe that\n\\begin{align*}\n|E_\\theta(a)|&=\\int_{\\{z\\in S(a, r): |f(z)|>\\theta |f(a)|\\}} dA(z)\\\\\n&=\\int_{\\big\\{|\\frac{z-a}{r(1-|a|)}|<1: |f(z)|>\\theta |f(a)|\\big\\}} dA(z)\\\\\n&=|S(a, r)|\\int_{\\big\\{|\\lambda|<1: |f(a+r(1-|a|)\\lambda)|>\\theta |f(a)|\\big\\}} dA(\\lambda)\\\\\n&=|S(a, r)| \\int_{\\{|\\lambda|<1: |g(\\lambda)|>\\theta |g(0)|\\}} dA(\\lambda)\\\\\n&=|S(a, r)|\\cdot \\Big| \\{\\lambda \\in \\mathbb D: |g(\\lambda)|>\\theta |g(0)|\\}\\Big|.\n\\end{align*}\nTherefore, we obtain\n$$|E_\\theta(a)|\\geqslant \\Big(\\frac{r}{16}\\Big)^2 |S(a, r)| $$\nor\n$$|E_\\theta(a)|\\geqslant \\frac{\\epsilon}{4C_0^2} |S(a, r)|.$$\nThis gives the proof of the claim, and so we complete the proof of Lemma \\ref{the integral over B}.\n\\end{proof}\n\n\n\nWe are now  ready to prove Lemma \\ref{the weight function w*}.\n\\begin{proof}[Proof of  Lemma \\ref{the weight function w*}] Suppose that $|G_0\\cap S(a, r)|\\geqslant \\delta_0 |S(a, r)|$.\nFrom Lemmas \\ref{the integral over A} and \\ref{the integral over B}, we can choose\n$\\epsilon$ small enough so that\n$$\\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)<2\\int_{\\mathbb D \\setminus B}|f(z)|^2\\omega^*(z)dA(z).$$\nOn the other hand, if $a\\notin B$, then\n\\begin{align*}\n|f(a)|^2 & >\\epsilon^2 B_\\theta f(a)\\geqslant \\frac{\\epsilon^2}{|S(a, r)|}\\int_{S(a, r)}|f(z)|^2dA(z).\n\\end{align*}\nFor the $\\delta_0$ above, we  apply Lemma \\ref{measure estimate3} to choose a $\\theta\\in (0, \\frac{1}{2})$ such that\n$$\\bigg|\\Big\\{z\\in S(a, r): |f(z)|> \\theta |f(a)| \\Big\\}\\bigg|>(1-\\frac{\\delta_0}{2})|S(a, r)|.$$\nSince  $|G_0\\cap S(a, r)|\\geqslant \\delta_0 |S(a, r)|$, we have\n$$\\bigg|\\Big\\{z\\in S(a, r)\\cap G_0: |f(z)|> \\theta |f(a)| \\Big\\}\\bigg|>\\frac{\\delta_0}{2}|S(a, r)|,$$\nand so\n$$\\frac{1}{|S(a, r)|}\\int_{S(a, r)\\cap G_0}|f(z)|^2dA(z)>\\frac{\\theta^2 \\delta_0}{2} |f(a)|^2\\ \\ \\  \\big(a\\notin B\\big).$$\nMultiplying the above inequality by $\\omega^*(a)$ and integrating over $\\mathbb D \\backslash B$  to get\n\\begin{align*}\n\\frac{\\theta^2 \\delta_0}{2} \\int_{\\mathbb D \\backslash B} \\omega^*(a) |f(a)|^2 dA(a)&<\\int_{\\mathbb D \\backslash B}\\frac{\\omega^*(a)}{|S(a, r)|}\\int_{S(a, r)\\cap G_0}|f(z)|^2dA(z) dA(a)\\\\\n&=\\int_{G_0}|f(z)|^2\\bigg(\\int_{\\mathbb D \\backslash B} \\frac{ \\omega^*(a)}{|S(a, r)|} \\chi_{S(a, r)}(z) dA(a)\\bigg)dA(z)\\\\\n&\\leqslant C \\int_{G_0}|f(z)|^2\\omega^*(z) dA(z),\n\\end{align*}\nthe last ``$\\leqslant$\" follows from Lemma \\ref{measure estimate0} and $C$ depends only on $r$.\n\nTherefore,\n  $$\\int_{G_0}|f(z)|^2\\omega^*(z) dA(z) > \\frac{\\delta_0 \\theta^2}{4C} \\int_{\\mathbb D}|f(z)|^2\\omega^*(z)dA(z)$$\n  for all $f\\in L_h^2(\\omega^*)$. This completes the proof of Lemma \\ref{the weight function w*}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\textbf{Acknowledgments} \\   The authors are grateful to  Professor Kunyu Guo for helpful suggestions. Both authors\n would like to express their  deepest gratitude to Professor Dechao Zheng, X. Zhao's Ph.D thesis adviser, for his valuable guidance, encouragement and  support  in many years. \\vspace{3mm}\\\\\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{The general formalism}\n\\label{sec_general}\nConsider a system in a bosonic SPT phase. By definition, this means it is gapped and\nnon-degenerate in the bulk, and (disregarding symmetry considerations) can be\ncontinuously connected to a product state without a phase transition.\nHowever, in a system with boundary, we can define an\neffective low-energy theory for the boundary, which may be gapless \nnotwithstanding the gap in the bulk. A key property of SPT phases is that\nthe boundary theory of an SPT phase in $d$ spatial dimensions can always be\nrealized at the microscopic level in a\nstrictly $(d-1)$-dimensional system (see Appendix \\ref{sec_edge_construction} for a careful proof of this\nwell-known fact.)  This is in contrast to, for example, integer quantum Hall\nstates in which the boundary is chiral and cannot be realized as a stand-alone\nsystem \\cite{wen_gapless}. For SPT phases, the anomalous nature of the edge arises not from the\nboundary theory itself but from the way it is acted upon by the symmetry.\n\n\nWe assume that the symmetry in the bulk is unitary and on-site, that is, for a\nlattice system with $N$ sites, the symmetry group $G$ is\nis represented as a unitary tensor-product $U(g) = [u(g)]^{\\otimes N}$ of\noperators acting on each site. (We may need to group several sites together into\na single effective site in order to satisfy this condition.)\nWe now consider the low-energy Hilbert space of states with energies below\nsome cutoff that is less than the bulk gap; these states are edge excitations.\nProjecting the unitary representation of the symmetry group onto this low-energy Hilbert space,\nwe obtain a unitary representation, acting only\non the boundary degrees of freedom, that may not be on-site.\nOn the contrary, it appears to be a characteristic of non-trivial SPT phases that\nthe symmetry is realized on the boundary in a\nfundamentally non-on-site way \\cite{spto_higher,*spto_higher_prb,spto_2d,Wen13}. Nevertheless, the key assumption that\nwe make in this paper is that the boundary symmetry, albeit not on-site, is\nnevertheless still \\emph{local} in the sense of Ref.~\\onlinecite{wen_lu} (e.g.\\ it can be\nrepresented as a finite-depth quantum circuit.) This seems a natural assumption,\nbut we expect it to be violated by SPT phases not captured by the cohomological\nclassification (see Section \\ref{sec_beyond} for further discussion).\n\nFor a non-on-site symmetry, there is the possibility that there is an\nobstruction to implementing the symmetry on a manifold with boundary in a\nconsistent way. We intend to show that, by classifying these obstructions, one\nrecovers the cohomological classification of SPT phases.\n A simple example of this idea is the well-known connection\nbetween (1+1)-D SPT's and the projective symmetry transformation of the edge\n\\cite{pollmann-arxiv-2009,chen_gu_wen,schuch},\nwhich we shall now review.\n\n\\begin{figure}\n  \\includegraphics{fig1}\n\\caption{Obtaining a 2-cocycle on the (0+1)-D edge of a (1+1)-D SPT.}\n\\end{figure}\n\\subsection{(1+1)-D SPT's}\nThe boundary of a one-dimensional system simply comprises a pair of points $a$\nand $b$. Let $U(g)$ be the representation of the symmetry group\n$G$ on this boundary.\nAssuming that we chose the system size such that the end-points $a$ and $b$ are\nwell-separated (i.e.\\ by a distance large compared to all intrinsic length\nscales), locality of $U(g)$ simply implies that it\nmust act on $a$ and $b$ separately; that is, it must be a tensor product $U(g) =\nU_a(g) \\otimes U_b(g)$. We can think of $U_a(g)$ as the \\emph{restriction} of\n$U(g)$ to the point $a$. Importantly, however, this restriction is uniquely\ndefined only modulo phase factors. Indeed, $U(g)$ is left invariant under\n$U_a(g) \\to \\beta(g) U_a(g), U_b(g) \\to \\beta(g)^{-1} U_b(g)$ for any\n$\\mathrm{U}(1)$-valued function $\\beta(g)$. Thus, while $U(g)$ is always a representation of the\nsymmetry group $G$, that is $U(g_1) U(g_2) = U(g_1 g_2)$, the non-uniqueness of the restriction procedure implies that\n$U_a(g)$ need only be a \\emph{projective} representation of $G$, which is to say\nthat\n$U_a(g_1) U_a(g_2) = \\omega(g_1, g_2) U_a(g_1 g_2)$ for some\n$\\mathrm{U}(1)$-valued function $\\omega(g_1, g_2)$. The function $\\omega$\ndescribes the obstruction to consistently (i.e.\\ non-projectively) implementing\nthe symmetry on the point $a$. \n\nSince multiplication of the $U_a$'s must be\nassociative, one can derive a consistency condition on $\\omega$ by\nevaluating $U_a(g_1) U_a(g_2) U_a(g_3)$ in two different ways, namely\n\\begin{equation}\n  \\label{omega_associativity_comm}\n  \\omega(g_1, g_2) \\omega(g_1 g_2, g_3) = \\omega(g_2, g_3) \\omega(g_1, g_2 g_3).\n\\end{equation}\nA function $\\omega$ satisfying \\eqnref{omega_associativity_comm} is known as a\n2-cocycle. Furthermore, due to the fact that $U_a(g)$ is only defined up to a\n$g$-dependent phase factor $\\beta(g)$, it follows that we have an equivalence\nrelation on 2-cocycles:\n\\begin{equation}\n  \\omega(g_1, g_2) \\sim \\omega(g_1, g_2) \\beta(g_1) \\beta(g_2)\n  \\beta(g_1 g_2)^{-1}.\n\\end{equation}\nThe group of $2$-cocycles quotiented by the above equivalence relation is, by\ndefinition, the second cohomology group $H^2(G, \\U(1))$. One can then show that\ntwo models are in the same SPT phase if and only if they correspond to the same\nelement of $H^2(G, \\U(1))$. Therefore, SPT phases in (1+1)-D are classified by\n$H^2(G, \\U(1))$.\n\n\n\\begin{figure}\n  \\includegraphics{fig2}\n\\caption{Obtaining a 3-cocycle on the (1+1)-D edge of a (2+1)-D system.}\n\\end{figure}\n\n\\subsection{(2+1)-D SPT's}\n\\label{sec_2p1d}\n\nWhen presented as it was above, the (1+1)-D case suggests an obvious\ngeneralization to higher dimensions: we consider the symmetry $U(g)$ acting on\nthe boundary $C$, then \\emph{restrict} it to a subregion $M$, which in general will\nbe a manifold with boundary ($C$ itself has no boundary as it is the\nboundary of a higher-dimensional manifold), to see if the symmetry is \nimplemented consistently or not.\n\nFirst, we need to give a more general definition of what it\nmeans to restrict a local unitary $U$ acting on a spatial manifold $C$ to a\nsub-manifold $M$, which for the case discussed above was obvious due to the\ntensor-product structure. Specifically, we say that a local unitary $U_M$ acting\non the region $M$ is the restriction of\n$U$ to the region $M$ if it acts the same as $U$ in the \\emph{interior} of $M$, well\naway from the boundary $\\partial M$. We observe two properties about this\nrestriction:\n\\begin{enumerate}[(a)]\n  \\item It always exists for any local unitary. This can easily be seen from,\n    for example, the quantum circuit description.\n  \\item It is defined modulo local unitaries acting in the vicinity of the\n    boundary $\\partial M$.\n\\end{enumerate}\nThe second property is the higher-dimensional generalization of the restriction\nbeing defined only up to phase factors. Thus, in general, if $U(g)$ is a\nrepresentation of the symmetry group $G$, then $U_M(g)$ need only satisfy\n\\begin{equation}\n  \\label{first_coboundary_nonabelian}\n  U_M(g_1) U_M(g_2) = \\Omega(g_1, g_2) U_M(g_1 g_2)\n\\end{equation}\nwhere $\\Omega(g_1, g_2)$ is a local unitary acting in the vicinity of $\\partial\nM$ which represents the obstruction to a consistent representation on $M$ due to\nthe fact that it is a manifold with boundary. Thus, we have reduced the problem\nof classifying local unitary representations $U(g)$ on a $d$-dimensional manifold to that\nof classifying local unitary obstructions $\\Omega(g_1, g_2)$ on a $(d-1)$-dimensional\nmanifold. The idea now is to perform more such reductions, each time reducing\nby 1 the dimensionality of the manifold acted upon, until we get down to the\nsimplest case of 0 dimensions (i.e.\\ points).\n\nFor $(2+1)$-D SPT's, this reduction can be completed as follows. In this case\nthe boundary has only one spatial dimension, and so $\\Omega(g_1, g_2)$ as constructed\nabove already acts on just a pair of points $a$ and $b$. We observe that\n\\eqnref{first_coboundary_nonabelian}, together with the associativity of the\noperators $U_M(g)$, implies that $\\Omega$ must satisfy\n\\begin{equation}\n  \\label{omega_associativity}\n  \\Omega(g_1, g_2) \\Omega(g_1 g_2, g_3) = \\; ^{U_M({g_1})} \\Omega(g_2, g_3)\n\\Omega({g_1},{g_2}{g_3}),\n\\end{equation}\nwhich is a non-abelian analogue of \\eqnref{omega_associativity_comm}, and where we\nhave introduced the conjugation notation $^x y = xyx^{-1}$.  Now we perform a\nsecond restriction, from $\\partial M = \\{ a, b \\}$ to the single point\n$a$. The restriction\n$\\Omega \\to \\Omega_a$ is defined only up to phase factors, and so we conclude\nthat $\\Omega_a$ satisfies \\eqnref{omega_associativity} only up to phase factors:\n\\begin{equation}\n  \\label{hacker}\n  \\Omega_a(g_1, g_2) \\Omega_a (g_1 g_2, g_3) = \\omega(g_1, g_2, g_3) \\;\n  ^{U_M({g_1})} \\Omega_a(g_2, g_3) \\Omega_a(g_1,\n  g_2 g_3),\n\\end{equation}\nwhere $\\omega(g_1, g_2, g_3) \\in \\U(1)$. We show in Appendix\n\\ref{sec_crossed_module} that\n$\\omega$ must satisfy the 3-cocycle condition\n\\begin{equation}\n  \\label{three_cocycle_condition}\n  \\omega(g_1, g_2, g_3) \\omega(g_1 g_2, g_3, g_4)^{-1} \\omega(g_1, g_2 g_3,\n  g_4) \\omega(g_1, g_2, g_3 g_4)^{-1} \\omega(g_2, g_3, g_4) = 1.\n\\end{equation}\nFurthermore, as $\\Omega_a(g, g^{\\prime})$ is only defined up to phase factors\n$\\beta(g, g^{\\prime})$, we\nmust identify\n\\begin{equation}\n  \\label{three_coboundary}\n  \\omega(g_1, g_2, g_3) \\sim \\omega(g_1, g_2, g_3) \\; \\beta(g_1, g_2) \\beta(g_1\n  g_2, g_3) \\beta(g_2, g_3)^{-1} \\beta(g_1, g_2 g_3)^{-1}.\n\\end{equation}\nWe show in Appendix \\ref{sec_crossed_module} that, up to equivalence, the choice of restriction\n$U(g) \\to U_M(g)$ does not affect the 3-cocycle. \nThe group of 3-cocycles quotiented by the equivalence relation\n\\eqnref{three_coboundary} is, by\ndefinition, the third cohomology group $H^3(G, \\U(1))$. Hence, we recover the\ncohomological classification of (2+1)-D SPT's.\n\n\n\n\\subsection{Higher dimensions}\n\\label{sec_higher_dims}\nIn higher dimensions it is not clear whether we can still do the reduction\nprocedure in complete generality as in the (2+1)-D case. Nevertheless, we can still\nperform the reduction if we make some simplifying assumptions about\nthe action of the symmetry on the boundary. (The non-linear sigma models\ndiscussed in Section \\ref{sec_nlsm} are a non-trivial example in which the symmetry on the\nedge takes the required form.) Specifically, we consider a symmetry group $G$ acting on a Hilbert space equipped with a set of\nbasis states labeled by the variables $\\alpha(x)$ associated with each\nspatial location in a closed $(d-1)$-dimensional space $C_1$. We can take the spatial coordinate\n$x$ to be either discrete (i.e.,\na lattice) or continuous. The class of symmetry actions that we\nconsider are those that can be written in the form\n\\begin{equation}\n  \\label{simple_symmetry}\n  U(g) = N(g) S(g),\n\\end{equation}\nsuch that:\n\\begin{enumerate}[(a)]\n\\item $S(g)$ is the on-site part of the symmetry which can be written\n  in the form \n  \\begin{equation}\n    S(g) = \\sum_{ \\alpha } \\ket{ g \\alpha  }\\bra{ \\alpha },\n  \\end{equation}\n  where $\\alpha \\to g \\alpha$ is some on-site action of the symmetry on the classical\n  labels $\\alpha$; and\n\\item in the same basis, the non-on-site part $N(g)$ is diagonal, namely\n  \\begin{equation}\n    N(g) = \\sum_{\\alpha} e^{i \\mathcal{N}^{(1)}(g)[\\alpha]} \\ket{\\alpha}\\bra{\\alpha}\n  \\end{equation}\n  where $\\mathcal{N}(g)$ are functionals of the configuration\n  $\\alpha$. We require these functionals to be sufficiently local that $N(g)$, and hence\n  $U(g)$, are local unitaries.\n\\end{enumerate}\nThe requirement that $U(g)$ be a representation, $U(g_1) U(g_2) = U(g_1 g_2)$,\ncan be written in terms of the functionals $\\mathcal{N}(g)$ as\n\\begin{equation}\n  \\label{first_coboundary}\n  g_1 \\mathcal{N}^{(1)}(g_2) + \\mathcal{N}^{(1)}(g_1) - \\mathcal{N}^{(1)}(g_1 g_2) = 0 \\quad\n  \\mbox{(mod $2\\pi$)},\n\\end{equation}\nwhere we have defined the action of group elements on functionals in the obvious\nway: $(g \\mathcal{F})[\\alpha] = \\mathcal{F}[g^{-1} \\alpha]$. Henceforth we will take\nthe (mod $2\\pi$) to be implied, or in other words we consider the functionals to take values in\n$\\mathbb{R}\/(2\\pi\\mathbb{Z})$.\n\nNow as before, we can restrict $U(g)$ to a subregion $M_1$ with boundary, which\n(since $S(g)$ can be trivially restricted)\namounts to restricting the functionals $\\mathcal{N}^{(1)}(g)$. Then\n\\eqnref{first_coboundary} need be satisfied by the restricted functionals\n$\\widetilde{\\mathcal{N}}^{(1)}(g)$ only up to boundary terms,\n\\begin{equation}\n  \\label{first_coboundary_boundary}\n  g_1 \\widetilde{\\mathcal{N}}^{(1)}(g_2) + \\widetilde{\\mathcal{N}}^{(1)}(g_1) -\n  \\widetilde{\\mathcal{N}}^{(1)}(g_1 g_2) =\n  \\mathcal{N}^{(2)}(g_1, g_2)\n\\end{equation}\nwhere the $\\mathcal{N}^{(2)}(g_1, g_2)$ are functionals which depend only on the\nvalue of $\\alpha$ near the boundary $\\partial M_1$ and describe the obstruction.\nThis corresponds to \\eqnref{first_coboundary_nonabelian}. \n\nIn order to continue the reduction process, we find it useful to define the\ngroup coboundary operators $\\delta_k$ which map functionals depending on $k$\ngroup elements into functionals depending on $k+1$ group elements, as follows:\n\\begin{multline}\n  \\label{coboundary_operator}\n  (\\delta_k \\mathcal{N}^{(k)})(g_1, \\cdots, g_{k+1})\n  = g_1 \\mathcal{N}^{(k)}(g_2, \\cdots g_n) + (-1)^{k+1} \\mathcal{N}^{(k)}(g_1,\n  \\cdots, g_k) \\\\+ \\sum_{i=1}^k (-1)^i \\mathcal{N}^{(k)}(g_1, \\cdots, g_{i-1}, g_i\n  g_{i+1}, g_{i+2}, \\cdots, g_{k+1}).\n\\end{multline}\nIn particular, $(\\delta_1 \\mathcal{N}^{(1)})(g_1, g_2)$ corresponds to the left-hand side of\n\\eqnref{first_coboundary}. The important property which the coboundary operators\nsatisfy is that they form a chain complex, i.e. $\\delta_{k+1} \\circ \\delta_k =\n0$.\n\n\n\\begin{figure}\n \n \n \n  \\includegraphics{fig3}\n  \\caption{The reduction process to obtain a 4-cocycle $\\omega$ on the (2+1)-D\n    edge of a (3+1)-D system, assuming a symmetry representation on the edge of\n  the form \\eqnref{simple_symmetry}.}\n\\end{figure}\n\nWe can now formulate the reduction process for symmetries acting on a manifold\nof arbitrary spatial dimension $d$. At the $k$-th step of the process, we have\na set of functionals $\\mathcal{N}^{(k)}$ acting on a closed $d-k$-dimensional\nmanifold $C_k$ and indexed by $k$ group elements, satisfying $\\delta_k\n\\mathcal{N}^{(k)} = 0$. We then consider restrictions\n$\\widetilde{\\mathcal{N}}^{(k)}$ of these functionals onto the manifold $M_k$,\nwhere $M_k$ is a submanifold of $C_k$ with boundary. As\n$\\widetilde{\\mathcal{N}}^{(k)}$ must act the same as $\\mathcal{N}^{(k)}$ in the\n\\emph{interior} of $M_k$, it follows that $\\mathcal{N}^{(k+1)} \\equiv \\delta_k\n\\widetilde{\\mathcal{N}}^{(k)}$ acts on \nthe boundary $\\partial M_k \\equiv C_{k+1}$.  Furthermore, as $\\delta_{k+1} \\circ\n\\delta_k = 0$, it follows that $\\delta_{k+1} \\mathcal{N}^{(k+1)} = 0$. Thus, we\njust iterate these reduction steps, terminating when we reach $\\omega = \\mathcal{N}^{(d+1)}$, which is simply a\nmapping from $d+1$ group elements to $\\mathrm{U}(1)$ satisfying\n$\\delta_{d+1} \\mathcal{N}^{(d+1)}$; this the definition of a $\\U(1)$\n$(d+1)$-cocycle. Due to the ambiguity in the choice of\nrestrictions, it follows that $\\omega$ is only defined up to\n\\begin{equation}\n  \\label{general_equivalence_relation}\n  \\omega \\sim \\omega + \\delta_{d+1} \\lambda\n\\end{equation}\nwhere $\\lambda$ is some element of $\\U(1)$ depending on $d+1$ group elements. The\ngroup of $(d+1)$-cocycles quotiented by the equivalence relation\n\\eqnref{general_equivalence_relation} is, by definition, the cohomology group $H^{d+1}(G, \\U(1))$. Thus, we recover the cohomological\nclassification of SPT phases in arbitrary dimensions.\n\nFinally, let us discuss the case of symmetry\ngroups that contain anti-unitary operations. It is perhaps unclear in general what is meant\nby restriction of an anti-unitary operation (although see\nRef.~\\onlinecite{gauging_time_reversal}). Nevertheless, if we consider only\nsymmetries that can be represented as a suitable generalization of \\eqnref{simple_symmetry},\nthe same arguments as above can be applied with only minor modifications.\nSpecifically, we consider symmetries of the form\n\\begin{equation}\n  U(g) = N(g) S(g) K^{n(g)},\n\\end{equation}\nwhere $N(g)$ and $S(g)$ are as before, $K$ is complex conjugation in the $\\{\n\\ket{\\alpha} \\}$ basis, and $n(g)$ is $0$ for unitary elements of $G$ and $1$ for\nanti-unitary elements. If we define the action of $G$ on functionals as\n$g \\mathcal{F}[\\alpha] = (-1)^{n(g)} \\mathcal{F}[g^{-1} \\alpha]$, all of the steps in\nthe above derivation can be carried through without change, except that there is a residual non-trivial action of $G$ on $\\U(1)$.\nThus, the classification is\n$H^{d+1}(G, \\U(1))$, but with $\\U(1)$ considered\nas a non-trivial $G$-module, with anti-unitary elements acting by complex\nconjugation.\n\n\n\n\\section{Example: ``Chiral'' symmetry on the edge of a (2+1)-D SPT}\n\\label{sec_chiral_example}\nIt was shown in Ref. \\onlinecite{chen_wen_chiral} that the action of the symmetry on the gapless\nedge of some non-trivial (2+1)-D SPT's is ``chiral'', as expressed (for example)\nin the fact that it acts differently on the left- and right-moving fields. Let\nus show how this corresponds to a local but not on-site symmetry and calculate\nthe corresponding 3-cocycle. We will focus on the simplest case where the\nsymmetry is just $\\mathbb{Z}_2$, but similar arguments can be made for $\\mathbb{Z}_n$\nor $\\U(1)$ symmetries.\n\nWe assume the low-energy theory of the (1+1)-D edge is described by a massless\nboson field $\\varphi$ with compactification radius $2\\pi$, i.e. a bosonic Luttinger liquid,\nwith Lagrangian density\n\\begin{equation}\n  \\mathcal{L} = \\frac{g}{2\\pi}\\left[\\frac{1}{v}(\\partial_t \\varphi)^2 - v (\\partial_x\n  \\varphi)^2\\right].\n\\end{equation}\nWe introduce the dual boson field $\\theta$ according to $\\partial_x \\theta\n= 2\\pi \\Pi$, where $\\Pi$ is the canonical momentum conjugate to $\\varphi$.\nThe commutation relation for $\\theta$ and $\\varphi$ is, therefore,\n\\begin{equation}\n[ \\varphi(x),\\theta(x')] = -2\\pi i \\Theta(x-x')\n\\label{eqn:LL-comm-rel}\n\\end{equation}\nwhere $\\Theta(x)$ is the unit step function.\nNote that this definition, together with the fact that total angular\nmomentum is quantized to integers, implies that $\\theta$ is also an angular\nvariable defined modulo $2\\pi$. \n\nNow, suppose that the fields $\\varphi$ and $\\theta$ transform under $\\mathbb{Z}_2$\naccording to\n\\begin{align}\n  \\varphi &\\to \\varphi + n \\pi, \\nonumber \\\\\n\\theta &\\to \\theta + m \\pi, \\label{various}\n\\end{align}\nHere $(n,m) = (1,0)$ corresponds to a normal on-site\n$\\pi$ rotation of the boson field. On the other hand, as we shall see, $(n,m) = (1,1)$ is the\nnon-on-site symmetry that we would expect at the edge of a non-trivial\n$\\mathbb{Z}_2$ SPT. Also, $m \\neq 0$ corresponds to a\nsuperficially ``chiral'' symmetry in the sense that the left- and right-moving\nfields $\\phi_{L,R} = \\varphi \\pm \\theta$ transform differently under\n$\\mathbb{Z}_2$, but in the $\\mathbb{Z}_2$ case [though not for $\\mathbb{Z}_n$ or\n$\\U(1)$] this chirality is not physically meaningful because $\\theta \\sim\n\\theta + 2\\pi$ so $m$ is actually only defined modulo 2.\n\nFrom the commutation relations (\\ref{eqn:LL-comm-rel}),\none can show that \\eqnref{various} is effected by the unitary\noperator $U = (-1)^{nL + mW} = N^m S^n$, where\nwhere $L$ is the total angular momentum and $W$ is the total winding number, and\nwe define \n\\begin{align}\n  N &= \\exp\\left(-\\frac{i}{2} \\int \\partial_x \\varphi \\, dx\\right) \\\\\n  S &= \\exp\\left(-\\frac{i}{2} \\int \\partial_x \\theta dx \\right). \n\\end{align}\nWe now define the restriction $U_{[a,b]} = N_{[a,b]}^m\nS_{[a,b]}^n$ to a finite interval $[a,b]$, where\n\\begin{align}\n  N_{[a,b]} &= \\exp\\left(-\\frac{i}{2} \\int_a^b \\partial_x \\varphi \\, dx\\right) \\\\\n  S_{[a,b]} &= \\exp\\left(-\\frac{i}{2} \\int_{a-\\epsilon}^{b+\\epsilon} \\partial_x \\theta \\, dx \\right),\n\\end{align}\nwhere we have made use of our freedom to redefine the restriction near the\nboundary of $[a,b]$ to shift the endpoints of the second integral by some\nsmall $\\epsilon > 0$. This ensures that $N_{[a,b]}$ and $S_{[a,b]}$ commute.\nHence, we find that $U_{[a,b]}^2 = N_{[a,b]}^{2m} S_{[a,b]}^{2n}$, where\n\\begin{align}\n  N_{[a,b]}^2 &= \\exp\\left(-i \\int_a^b \\partial_x \\varphi dx\\right) \\\\\n            &= e^{i \\varphi(a)} e^{-i \\varphi(b)} \\\\\nS_{[a,b]}^2 &= \\exp\\left(- i \\int_{a-\\epsilon}^{b+\\epsilon} \\partial_x \\theta dx\\right) \\\\\n            &=  e^{i \\theta(a-\\epsilon)} e^{- i \\theta(b+\\epsilon)}.\n\\end{align}\nThus, as expected, we find that\n$\\Omega \\equiv U_{[a,b]}^2 = [e^{in \\theta(a-\\epsilon)}\ne^{i m\\varphi(a)}] \n[e^{- in \\theta(b+\\epsilon)} e^{-im\\varphi(b)} ] \\equiv \\Omega_a \\Omega_b$ still acts non-trivially at the endpoints $a$\nand $b$ even though $U^2 = 1$.\n\nIn the present example, \\eqnref{omega_associativity} takes the form\n\\begin{equation}\n  U_{[a,b]} \\Omega U_{[a,b]}^{-1} = \\Omega,\n\\end{equation}\nand this equality can readily be verified directly from the forms of $U_{[a,b]}$ and $\\Omega$ given\nabove. On the other hand, the restriction $\\Omega_a$\nsatisfies this equation in general only up to a phase factor. Indeed, we find\n\\begin{equation}\n  U_{[a,b]} \\Omega_a U_{[a,b]}^{-1} = e^{- i n \\theta(a-\\epsilon)} e^{-im \\varphi(a) - imn \\pi} = (-1)^{mn} \\Omega_a.\n\\end{equation}\nHence, we find that the $3$-cocycle associated with the realization of $\\mathbb{Z}_2$\nis given by $\\omega(X,X,X) = (-1)^{mn}$ and $\\omega(g_1, g_2, g_3) = 1$ for $(g_1, g_2, g_3)\n\\neq (X,X,X)$, where $X$ is the generator of $\\mathbb{Z}_2$. For $m = n = 1$\nthis corresponds to a non-trivial 3-cocycle, and the corresponding\nrepresentation of $\\mathbb{Z}_2$ would appear at the boundary of a\nnon-trivial (2+1)-D $\\mathbb{Z}_2$ SPT.\n\n\n\\section{Proof of separation of phases in (2+1)-D.}\n\\label{separation_proof}\nIn this section, we will outline how one can use the ideas given above to prove for\n(2+1)-D systems that systems characterized by different elements of the\ncohomology group $H^3(G, \\U(1))$ must be separated by a bulk phase transition;\nthe details are left to the appendices.\n(Unfortunately, the proof cannot be applied in higher dimensions due to the\nlack of a completely general characterization of anomalous symmetry.)\n\nFirst, as we want to make statements about bulk properties, we need to\nreformulate the ideas of Section \\ref{sec_general} in a slightly different way, in terms of\nproperties of the ground state in the bulk rather than the low-energy physics at\nthe edge. \nWe\nshow in Appendix \\ref{sec_completing} that, given a general ground state $\\ket{\\Psi}$ in some\nSPT phase in $d$ spatial dimensions ($d \\leq 2$), and a region $A$ in the bulk, one can find a\nrepresentation $V_{\\partial A}(g)$ of the symmetry group, which acts \\emph{inside} $A$, but only near the\nboundary $\\partial A$, such that $U_A(g) \\ket{\\Psi} = V_{\\partial A}(g)\n\\ket{\\Psi}$. Here\n$U_{A}(g)$ is the restriction of the symmetry onto the region $A$ (which can\nbe defined consistently since we are assuming the symmetry is represented\non-site in the bulk.) The physical interpretation of this result is simply that,\nas $\\ket{\\Psi}$ is invariant under $U(g)$, therefore $U_A(g) \\ket{\\Psi}$ can\ndiffer from $\\ket{\\Psi}$ only near the boundary $\\partial A$. This representation\n$V_{\\partial A}(g)$ can be anomalous in the same way as the representation of\nthe symmetry on a physical edge, and the anomaly can be classified using the\nmethod of Section \\ref{sec_general}.\n\nThe final result that we need is that the element of $H^{3}(G, \\U(1))$ is\nindependent of the choice of region $A$, \\emph{even} in the presence of\nspatial inhomogeneity; this is also proved in Appendix \\ref{sec_completing}. (Actually, as\ndiscussed in that appendix, we only prove this for certain regions $A$, but that\nis sufficient for the following discussion.) This allows us to prove that two\nsystems $\\mathcal{S}$ and $\\mathcal{S}^{\\prime}$ characterized by different elements of $H^{d+1}(G, \\U(1))$ must be\nseparated by a phase transition \\cite{top_ent_entr}. Indeed, consider two systems connected\nwithout a phase transition. Then, without closing the gap, one can create an\ninterpolated system that looks like $\\mathcal{S}$ on some region $A$ and like\n$\\mathcal{S}^{\\prime}$ on another region $A^{\\prime}$ (see Appendix\n\\ref{sec_completing} for a\ncareful proof of this fact.) It therefore follows that the same element of\n$H^{d+1}(G, \\U(1))$ must be obtained in both cases. By a similar argument, one\nalso finds that a spatial boundary between two different SPT phases must either\nbe gapless or break the symmetry.\n\n\n\\section{Non-linear sigma models}\n\\label{sec_nlsm}\nIt has been found \\cite{boson_iqhe,cenke_neel_order,vishwanath_senthil,adr_classification} that a quite general way to reproduce the essential features of various SPT phases\nis through the field theory of a quantum non-linear sigma model (NL$\\sigma$M), where topological\nproperties of the SPT phase arise out of the bulk theta term included in the\naction. Here, we will show in such models, the presence of the theta term indeed\nleads to an obstruction to on-site representation of the symmetry on a spatial\nedge, in such a way as to\nallow a straightforward calculation of the corresponding element of the\ncohomology group.\n\nFor example, consider in $D$ space-time dimensions (i.e.\\ $D = d+1$) a NL$\\sigma$M for\nthe $(D+1)$-component vector field $\\mathbf{n}$, constrained to have unit norm,\ni.e.\\ $\\mathbf{n}$ lies on a unit $D$-sphere.\nThe (Euclidean) action can be written as the sum of a dynamical\ncontribution $S_{\\mathrm{dyn}}$ and a topological contribution\n$S_{\\mathrm{top}}$:\n\\begin{align}\n  S^{\\mathrm{bulk}} &= S_{\\mathrm{dyn}}^{\\mathrm{bulk}} +\n  S_{\\mathrm{top}}^{\\mathrm{bulk}} \\label{S_bulk}, \\\\\n  S_{\\mathrm{dyn}}^{\\mathrm{bulk}} &=\\frac{1}{g}\\int d^D x \\; \\partial_\\mu \\mathbf{n} \\cdot\n  \\partial_\\mu \\mathbf{n} \\label{S_dyn} \\\\\n  S_{\\mathrm{top}}^{\\mathrm{bulk}} &= i\\Theta \\frac{1}{V_D} \\int\n  \\mathbf{n}^{*}(\\omega_V)\n  \\label{S_top},\n\\end{align}\nwhere $V_{D}$ is the volume of the unit $D$-sphere, and $\\mathbf{n}^{*}(\\omega_V)$ is the\npullback through the map $\\mathbf{n}$ of the volume form on the unit $D$-sphere.\nWritten componentwise, this\namounts to\n\\begin{equation}\n  S_{\\mathrm{top}}^{\\mathrm{bulk}} = i\\Theta \\frac{1}{V_D} \\int d^{D} x \\; \\epsilon^{a_1, \\cdots, a_{D+2}} \n  n^{a_1} \\partial_0 n^{a_2} \\partial_1 n^{a_2} \\cdots \\partial_{D-1} n^{a_{D}},\n\\end{equation}\nwhere $\\epsilon^{a_1, \\cdots, a_{D+1}}$ is the $(D+1)$-dimensional Levi-Civita symbol.\nThe theta term\n$S_{\\mathrm{top}}^{\\mathrm{bulk}}$ measures a topologically invariant ``generalized\nwinding number'' in $\\pi_D(S^D) \\cong \\mathbb{Z}$, and for spacetimes without boundary is quantized to integer\nmultiples of $i\\Theta$. Hence, we implement the requirement that SPT phases be\ntrivial in the bulk by setting $\\Theta$ to be an integer multiple of $2\\pi$,\nthus ensuring that $S_{\\mathrm{top}}^{\\mathrm{bulk}}$ makes no contribution to the partition\nfunction $\\int \\mathcal{D}[\\mathbf{n}] e^{-S}$. In fact, although we have given\na specific form of $S_{\\mathrm{dyn}}^{\\mathrm{bulk}}$ for concreteness, it will not be important\nfor our analysis as the topological features of the system are entirely captured\nby $S_{\\mathrm{top}}^{\\mathrm{bulk}}$.\n\n\nAlthough the inclusion of $S^{\\mathrm{bulk}}_{\\mathrm{top}}$ has no effect on the partition\nfunction in the bulk, it does play a crucial role once we introduce a spatial\nedge. In that case $S_{\\mathrm{top}}^{\\mathrm{bulk}}$ depends (mod $2\\pi i$) only on the values of\n$\\mathbf{n}$ on the boundary (to see this, note that any two extensions into the\nbulk can be connected at the boundary to give a closed surface, on which\n$e^{-S_{\\mathrm{top}}} = 1$); the action on the boundary is referred to as the\n\\emph{Wess-Zumino-Witten} action $S_{WZW}$.\nThus, we can integrate out the gapped bulk to\ngive an effective\naction for the low-energy excitations on the edge of the form\n\\begin{equation}\n  \\exp(-S^{\\mathrm{edge}}) = \\exp(-S^{\\mathrm{edge}}_{\\mathrm{dyn}} -\n  S_{\\mathrm{WZW}}),\n\\end{equation}\nwhere $S^{\\mathrm{edge}}_{\\mathrm{dyn}} = \\int d^d x\n\\mathcal{L}^{\\mathrm{edge}}_{\\mathrm{dyn}}$ is some unimportant dynamical term\nderived from $S^{\\mathrm{bulk}}_{\\mathrm{dyn}}$.\nNote that one can then write \n$S_{\\mathrm{WZW}} = \\int d^d x \\mathcal{L}_{\\mathrm{WZW}}$ for some local\nLagrangian density $\\mathcal{L}_{\\mathrm{WZW}}$ defined on the edge. \nHowever, there is no canonical way to\ndo so.\n\nNow let us consider the symmetry group $G$ in the bulk corresponding to some\ninvertible action $\\mathbf{n} \\to\ng\\mathbf{n}$ for $g \\in G$. We demand that $S_{\\mathrm{dyn}}^{\\mathrm{bulk}}$ and\n$S_{\\mathrm{top}}$ be \\emph{locally}\ninvariant under the symmetry, i.e.\\ that the integrands in Eqs. (\\ref{S_dyn}) and\n(\\ref{S_top}) must be invariant, not just the integral. Then we expect that\n$S^{\\mathrm{edge}}_{\\mathrm{dyn}}$ is also locally invariant under the symmetry.\n  $S_{WZW}$ must also be globally invariant (at least,\n  modulo $2\\pi i$) but in general we do not expect it to be locally\n  invariant. Indeed, because\nthere is no canonical choice for $\\mathcal{L}_{WZW}$, one expects\nthat the symmetry will transform $\\mathcal{L}_{WZW}$ to a different Lagrangian\nthat nevertheless integrates to the same\naction (modulo $2\\pi i$) in a spacetime without boundary.\n\nWe will now show that, after quantization, the lack of local invariance of\n$S_{WZW}$ implies the non-on-site\nnature of the unitary\nrepresentation of the symmetry on the edge.\nWe assume that after quantization the Hilbert space is spanned by a basis of\nstates labeled by spatial configurations of $\\mathbf{n}$ at a fixed time. We\ncan calculate the imaginary-time propagator $e^{-\\beta H}$ (or equivalently, the\nHamiltonian $H$) by a path integral\n\\begin{equation}\n  \\label{propagator}\n  \\bra{ \\mathbf{n}^{\\prime} } e^{-\\beta H} \\ket{ \\mathbf{n} } = \\int\n  \\mathcal{D}[\\mathbf{n}(\\tau)] e^{-S^{\\mathrm{edge}}\\{0,\\beta\\}},\n\\end{equation}\nwhere\n\\begin{equation}\n  S^{\\mathrm{edge}}\\{0,\\beta\\} =\n  \\int d^{D-2} x \\int_0^\\beta d\\tau \\;\n  (\\mathcal{L}^{\\mathrm{edge}}_{\\mathrm{dyn}} + \\mathcal{L}_{WZW})\n\\end{equation}\nis the action evaluated on a spacetime with temporal boundaries at $\\tau=0$ and\n$\\tau=\\beta$. Now, so far we only know that $S_{\\mathrm{WZW}}$ is globally\ninvariant (modulo $2\\pi i$) on a space-time manifold without boundary. Since\n$S_{\\mathrm{WZW}}$ is not \\emph{locally} invariant, in the presence of a\ntemporal boundary we can only conclude \nthat it will transform as\n$S_{\\mathrm{WZW}}\\{0,\\beta\\}\n\\to g S_{\\mathrm{WZW}}\\{0,\\beta\\}$ ($g \\in G$), where the\ndifference can be expressed in terms of the field configurations at the\ntemporal boundaries:\n\\begin{equation}\n  \\label{symdiff}\n  g S_{\\mathrm{WZW}}\\{0,\\beta\\} - S_{\\mathrm{WZW}}\\{0,\\beta\\} =\n  i\\mathcal{N}(g)[\\mathbf{n}(\\tau)] - i\\mathcal{N}(g)[\\mathbf{n}(0)] \\quad (\\mbox{mod $2\\pi i$}),\n\\end{equation}\nwhere $\\mathcal{N}(g)$ is a functional of the field configuration at a fixed time. \n\n\\eqnref{symdiff} implies that the edge Hamiltonian is not invariant under the naive\non-site implementation of the symmetry, $S(g) = \\int \\mathcal{D}[\\mathbf{n}] \\,\n\\ket{g \\mathbf{n}} \\bra{\\mathbf{n}}$. Indeed, combined with \\eqnref{propagator},\nwe find\n\\begin{align}\n  \\bra{\\mathbf{n}^{\\prime}} S(g)^{\\dagger} e^{-\\beta H} S(g) \\ket{\\mathbf{n}} &= e^{i\n\\mathcal{N}(g)[\\mathbf{n}^{\\prime}] - i \\mathcal{N}(g)[\\mathbf{n}]} \\bra{\\mathbf{n}^{\\prime}}\n  e^{-\\beta H} \\ket{\\mathbf{n}} \\\\\n  &= \\bra{\\mathbf{n}^{\\prime}} N(g)^{\\dagger} e^{-\\beta H} N(g) \\ket{\\mathbf{n}}.\n\\end{align}\nwhere\n\\begin{equation}\n  \\label{Ng_nlsm}\n  N(g) = \\int \\mathcal{D}[\\mathbf{n}] e^{i \\mathcal{N}(g)[\\mathbf{n}]}\n  \\ket{\\mathbf{n}} \\bra{\\mathbf{n}}.\n\\end{equation}\nHence, we see that the correct implementation of the symmetry on the edge, which does\ncommute with the Hamiltonian, is $U(g) = N(g) S(g)$. In general,\nthere is no reason to expect $N(g)$ to be on-site, as we shall see.\nHowever, as we show in Section \\ref{nlsm_cochains}, it is necessarily local.\nThus, the symmetry on the edge is a local but non-on-site symmetry precisely of\nthe form considered in Section \\ref{sec_higher_dims}, and we can calculate the appropriate element\nof the cohomology group using the reduction procedure of that section.\n\nWe can also consider anti-unitary symmetries by a straightforward extension of\nthe above considerations. Specifically, an anti-unitary symmetry is \nimplemented in the action by $\\mathbf{n} \\to g \\mathbf{n}$, $i \\to (-1)^{n(g)}i$.\nThen we find that the representation of the symmetry on the edge is $U(g) = N(g)\nS(g) K^{n(g)}$, with $N(g)$ and $S(g)$ as before and $K$ complex conjugation in\nthe $\\mathbf{n}$ basis.\n\n\\subsection{Calculating the cocycle in nonlinear sigma models using $\\U(1)$\ncochains on the target manifold}\n\\label{nlsm_cochains}\nA particularly compact and elegant way of calculating the cocycle for\nNL$\\sigma$Ms\nis by\ninterpreting the theta term in terms of a $\\U(1)$ cochain defined on the target\nmanifold $T = S^{D}$. First we need to state some defintions. \nWe refer to $k$-dimensional oriented integration domains on a\nmanifold $T$ as $k$-chains. Given a $k$-chain $A$, we denote the opposite\norientation by $-A$, and we can also define a sum operation on $k$-chains in the\nnatural way, so that the $k$-chains can be viewed as an additive group. (If one\nwanted to be rigorous, one would define $k$-chains as formal linear combinations\nof oriented $k$-simplices with integer coefficients.) A\n$\\U(1)$ $k$-\\emph{cochain} is a linear mapping from $k$-chains to $\\U(1)$ [which\nwe here write additively as $\\mathbb{R}\/(2\\pi\\mathbb{Z})$]. (Note that we are\nhere referring to \\emph{topological} cochains on a manifold; these should\nbe distinguished from the \\emph{group} cochains that are used to construct the group cohomology of\nsome group $G$.) In particular, each\ndifferential $k$-form $\\omega$ induces a $\\U(1)$ $k$-cochain by integration,\n\\begin{equation}\n  \\omega(A) = \\left( \\int_A \\omega \\right) \\operatorname{mod} 2\\pi.\n\\end{equation}\nwhere in an abuse of notation we will denote the $k$-form and the $\\U(1)$ $k$-cochain\nby the same symbol.\nAny $\\U(1)$ $k$-cochain $\\omega$ on the target manifold $T$ can be used to define a\nlocal  $\\U(1)$-valued functional $F_\\omega$ for a $T$-valued field\n$\\mathbf{n}$ on\na $k$-dimensional space(-time) manifold $M$ via\n\\begin{equation}\n  \\label{F_omega}\n  F_\\omega[\\mathbf{n}] = \\omega(\\mathbf{n}(M)),\n\\end{equation}\nwhere $\\mathbf{n}(M)$ is the image of $M$, viewed as a chain, under the mapping\n$\\mathbf{n}$. If $\\omega$ is derived from a differential $k$-form, this is equivalent to\ndefining $F_\\omega$ as the integral of the pullback, $F_\\omega[\\mathbf{n}] =\n\\left(\\int_M \\mathbf{n}^{*}(\\omega)\\right) \\operatorname{mod} 2\\pi$. In\nparticular, the\ntopological theta term action of \\eqnref{S_top} is a special case of \\eqnref{F_omega}.\n\nWe define the coboundary operator $d$ which maps $k$-cochains to\n$(k+1)$-cochains according to\n\\begin{equation}\n  (d\\omega)(A) = \\omega(\\partial A),\n\\end{equation}\nwhere $\\partial A$ is the boundary of $A$.\nWe call a $k$-cochain $\\omega$ \\emph{exact} if it can be written as $\\omega =\nd\\kappa$ for some $(k-1)$-cochain $\\kappa$. Our central tool is the following\nresult.\n  \\begin{lemma}\n    \\label{lemma:central}\n  A $\\U(1)$ $k$-cochain $\\omega$ on a manifold $T$ is exact if and only if\n  $\\omega(C) = 0$ for all closed (i.e.\\ boundaryless) $k$-chains $C$.\n  \\begin{proof}\n    See Appendix \\ref{appendix_universal}.\n  \\end{proof}\n  \\end{lemma}\nThe property that $\\omega(C) = 0$ for closed $C$ in turn is equivalent to requiring of the induced functional\n$F_{\\omega}$ that it vanish on all closed space-time manifolds. If this is satisfied, then\none expects that for a space-time manifold $M$ with boundary, $F_{\\omega}[\\mathbf{n}]$ should depend\nonly on the values of $\\mathbf{n}$ on the boundary $\\partial M$. Indeed, given\n$\\omega = d\\kappa$, one finds that\n\\begin{align}\n  F_\\omega[\\mathbf{n}] &= (d\\kappa)(\\mathbf{n}(M)) \\\\\n                       &= \\kappa(\\partial \\mathbf{n}(M))  \\\\\n                       &= \\kappa(\\mathbf{n}(\\partial M)) \\\\\n                       &\\equiv F_{\\kappa}[\\mathbf{n}(\\partial M)].\n\\end{align}\nGiven the above considerations, one can show that the procedure for obtaining the edge symmetry from the theta\nterm, and then the cocycle from the edge symmetry, can be reduced to a\nsimple prescription in terms of the $\\U(1)$ cochains defined on the target manifold, with no reference to\nthe space-time manifold at all, which we now describe.\n\n\\begin{table}\n  \\begin{tabular}{c|c|ccccc|}\n    \\multicolumn{6}{r}{$\\leftarrow d$} \\\\\n    \\cline{2-7}\n    & & 3              & 2              &   1            & 0 & \\\\\n    \\cline{2-7}\n  $\\delta$ & 0 & $\\omega^{(0)}$ & $\\kappa^{(0)}$ &              &              & \\\\\n  $\\downarrow$ &  1 & 0          & $\\omega^{(1)}$ & $\\kappa^{(1)}$ &              & \\\\\n               & 2 &            & 0              & $\\omega^{(2)}$ & $\\kappa^{(2)}$ & \\\\\n               & 3 &            &                & 0              & $\\omega^{(3)}$ & \\\\\n               & 4 &            &                &                & 0 & \\\\\n    \\cline{2-7}\n  \\end{tabular}\n  \\caption{A tabular representation of the reduction process to extract a\n    $\\U(1)$ group 3-cocycle $\\nu = \\omega^{(3)}$ starting from a symmetric topological term in (2+1)-D represented by a\n  topological $\\U(1)$ cochain $\\omega^{(0)}$. Each cell in the table is specified by a row\n  label $l$ and a column label $k$, and corresponds to a set of $k$-cochains\n  labeled by $l$ group elements. Going left in the table corresponds to applying\n  the \\emph{topological} coboundary operator $d$, whereas going down\ncorresponds to applying the \\emph{group} coboundary operator $\\delta$ defined by\n\\eqnref{coboundary_operator_cochains}. These two operations commute, so the table can be\ninterpreted as a commutative diagram.}\n\\end{table}\n\nWe start from a topological action $S_{\\mathrm{top}}$ on a spacetime-manifold\n$M$ with $d$-dimensional target manifold $T$, written as $S_{\\mathrm{top}}[\\mathbf{n}] =\nF_{\\omega^{(0)}}[\\mathbf{n}]\n= \\omega^{(0)}(\\mathbf{n}(M))$, where $\\omega^{(0)}$ is an exact $\\U(1)$ $d$-cochain on\n$T$ which is invariant under the action\nof the symmetry, $g \\omega^{(0)} = \\omega^{(0)}$. Here we defined the action of the\nsymmetry on a cochain by $g \\omega(A) = (-1)^{n(g)} \\omega(gA)$, where $n(g)$ is\n1 for anti-unitary elements and 0 for unitary elements, and the action of $g$ on\nchains is derived from its action on $\\mathbf{n}$. Hence, we have $\\delta_0\n\\omega^{(0)}\n= g\\omega^{(0)} - \\omega^{(0)} = 0$, where we have introduced the \\emph{group} coboundary operators\n$\\delta_k$ (\\emph{not} the same as the topological coboundary operator $d$ defined above) in\nthe same way as\n\\eqnref{coboundary_operator} above, namely:\n\\begin{multline}\n  \\label{coboundary_operator_cochains}\n  (\\delta_k \\omega^{(k)})(g_1, \\cdots, g_{k+1})\n  = g_1 \\omega^{(k)}(g_2, \\cdots g_n) + (-1)^{k+1} \\omega^{(k)}(g_1,\n  \\cdots, g_k) \\\\+ \\sum_{i=1}^k (-1)^i \\omega^{(k)}(g_1, \\cdots, g_{i-1}, g_i\n  g_{i+1}, g_{i+2}, \\cdots, g_{k+1}),\n\\end{multline}\n\nGiven a set of exact $(d-k)$-cochains $\\omega^{(k)}$ indexed by $k$ group\n  elements which satisfy $\\delta_k \\omega^{(k)} = 0$, we\n  can write $\\omega^{(k)} = d\\kappa^{(k)}$ for some set of $(d-k-1)$-cochains\n  $\\kappa^{(k)}$. Now, $\\delta_k \\omega^{(k)} = 0$ implies that, for\n  closed chains $C$, $(\\delta_k \\kappa^{(k)})(C) = (\\delta_k\n\\omega^{(k)})(\\partial C) = 0$. Hence, we can define $\\omega^{(k+1)} = \\delta_k\n\\kappa^{(k)}$ which is exact and satisfies $\\delta_{k+1} \\omega^{(k+1)} = 0$.\nThe sequence terminates when we reach $\\omega^{(D)}$, which is a set of\n  $0$-cochains indexed by $k$ group elements. Now a $0$-cochain is essentially\n  just a scalar $\\U(1)$ function defined on the target manifold $T$. But the fact\n  that $\\omega^{(D)}$ evaluates to zero for the closed $0$-chain $a - b$ (where\n  $a$ and $b$ are any two points) implies that the $\\omega^{(D)}$ are\n  \\emph{constant} $U(1)$ functions. Thus, $\\omega^{(D)}$ defines a mapping from\n  $D$ group elements to $\\U(1)$ satisfying $\\delta_D \\omega^{(D)} = 0$, which\n  defines an element of the group cohomology group $H^{D}(G, \\U(1))$.\n\n\n\\subsection{Examples}\nThe possible symmetry transformations that leave the Lagrangian of\n\\eqnref{S_bulk}\ninvariant in space-time dimensions $D = 2,3,4$ were constructed in\nRef.~\\onlinecite{adr_classification} \nfor a variety of different symmetry groups. Our framework allows in\nprinciple for the element of the cohomology group $H^D(G, U(1))$ to be calculated in all of these\ncases. Let us consider a few examples.\n\n\\subsubsection{$Z_2^T$ in (1+1)-D}\n\\label{Z2t_1p1}\nWe write the symmetry group as $Z_2^T = \\{ 1, \\mathbb{T} \\}$.\nThe target manifold is $S^2$ and we work in spherical coordinates $\\mathbf{n} =\n(\\cos \\theta, \\sin \\theta \\cos \\varphi, \\sin \\theta \\sin \\varphi)$. The action\nof $\\mathbb{T}$ on $\\mathbf{n}$ is $\\mathbb{T} \\mathbf{n} = - \\mathbf{n}$, or in terms\nof the spherical coordinates $\\theta \\to \\pi - \\theta, \\varphi \\to \\varphi + \\pi$. The initial\n$\\U(1)$ cochain can be written in terms of a 2-form \n\\begin{equation}\n  \\label{a_omega_0}\n  \\omega^{(0)} = \\Theta \\frac{1}{4\\pi} \\sin \\theta  (d\\theta \\wedge d \\varphi).\n\\end{equation}\nAs $\\omega^{(0)}$ integrates to 0 (mod $2\\pi$) over the whole $2$-sphere, it\nfollows that it can written as $\\omega^{(0)} = d\\kappa^{(0)}$ for some $\\U(1)$\n$1$-cochain $\\kappa^{(0)}$. We can write\n$\\kappa^{(0)}$ explicitly as\n\\begin{equation}\n  \\kappa^{(0)} = \\Theta \\frac{1}{4\\pi} (1-\\cos \\theta) d\\varphi\n\\end{equation}\nTreating $\\kappa^{(0)}$ as a differential 1-form and taking the exterior\nderivative, one recovers \\eqnref{a_omega_0}. When written as a $1$-form,\n$\\kappa^{(0)}$ appears to have a singularity at $\\theta = \\pi$. To show that, as\na $\\U(1)$ $1$-cochain, $\\kappa^{(0)}$ is actually well-defined and satisfies\n$d\\kappa^{(0)} = \\omega^{(0)}$ globally, it is sufficient to check that\n$\\int_C \\kappa^{(0)} = 0$ (mod $2\\pi$) for an loop $C$ of infinitesimal size encircling the\napparent singularity at $\\theta = \\pi$, which is indeed the case.\n\nNow, following the general prescription of Section \\ref{nlsm_cochains}, we define $\\omega^{(1)} =\n\\delta_0 \\kappa^{(0)}$. The only non-trivial component is\n\\begin{align}\n  \\omega^{(1)}(\\mathbb{T}) &= \\mathbb{T} \\kappa^{(0)} - \\kappa^{(0)} \\\\\n                           &= -\\frac{\\Theta}{4\\pi} (1 + \\cos \\theta) - \\frac{\\Theta}{4\\pi}(1-\\cos \\theta) \\\\\n&= -\\frac{\\Theta}{2\\pi} d\\varphi,\n\\end{align}\nfrom which we immediately read off that $\\omega^{(1)} = d\\kappa^{(1)}$, where\n$\\kappa^{(1)} = -\\frac{\\Theta}{2\\pi}\\varphi$ (which is well-defined as a $U(1)$ $0$-cochain\nbecause $\\varphi$ is defined modulo $2\\pi$). Thus, we can define the cocycle\n$\\nu = \\delta_1 \\kappa^{(1)}$, and the only non-zero component is\n\\begin{align}\n  \\nu(\\mathbb{T},\\mathbb{T}) &= \\mathbb{T} \\kappa^{(1)} + \\kappa^{(1)} \\\\\n                                      &= \\frac{\\Theta}{2\\pi} \\{\\varphi + \\pi - \\varphi \\}\\\\\n                                      &= \\frac{\\Theta}{2}.\n\\end{align}\nThus, if $\\Theta$ is an odd multiple of $2\\pi$, this $2$-cocycle corresponds to\na non-trivial SPT phase, with the zero-dimensional boundary transforming\nprojectively under the symmetry, i.e.\\ as a Kramers doublet with $\\mathbb{T}^2 =\n-1$. On the other hand, if $\\Theta$ is an even multiple of $2\\pi$, we have a\ntrivial SPT phase with the edge transforming as $\\mathbb{T}^2 = 1$. Thus, by\ndifferent choices of $\\Theta$ one recovers both elements of the cohomology group\n$H^2(Z_2^T, \\U(1)) \\cong Z_2$.\n\n\\subsubsection{$Z_2$ in (2+1)-D}\nWe write the symmetry group as $Z_2 = \\{ 1, X \\}$. \nThe target manifold is $S^3$ and we work in generalized spherical coordinates\n$\\mathbf{n} = (\\cos \\theta, \\sin \\theta \\mathbf{n}_2)$, where $\\mathbf{n}_2 \\in S^2$. \n The action of $X$ on $\\mathbf{n}$ is $X \\mathbf{n} = - \\mathbf{n}$, or in terms\n of the generalized spherical coordinates $\\theta \\to \\pi - \\theta, \\mathbf{n}_2 \\to -\\mathbf{n}_2$. The initial\n$U(1)$ cochain is \n\\begin{equation}\n  \\omega^{(0)} = \\Theta \\frac{1}{V_3} \\sin^2 \\theta  (d\\theta \\wedge \\omega_{V,2})\n\\end{equation}\nwhere $V_3$ is the volume of the $3$-sphere, and $\\omega_{V,2}$ is the volume\nform for $\\mathbf{n}_2$. We then find that $\\omega^{(0)} = d\\kappa^{(0)}$, where\n\\begin{equation}\n  \\kappa^{(0)} = \\Theta \\frac{1}{V_3} \\left(\\int_0^\\theta \\sin^2 x \\, dx\\right) \\omega_{V,2}.\n\\end{equation}\nWe observe that $V_3$ can be expressed as ($V_2 = 4\\pi$)\n\\begin{equation}\n  \\label{relation}\nV_3 = V_2 \\int_0^\\pi \\sin^2 \\theta \\, d\\theta.\n\\end{equation}\nFrom this one can show that $\\kappa^{(0)}$ is well-defined despite the apparent\nsingularity at $\\theta = \\pi$. Now the only non-trivial element of $\\omega^{(1)}\n= d\\kappa^{(1)}$ is\n\\begin{align}\n  \\omega^{(1)}(X) &= X \\kappa^{(0)} - \\kappa^{(0)} \\\\\n  &= \\Theta \\frac{1}{V_3} \\omega_{V,2} \\int_0^{\\pi} \\sin^2 x dx \\\\\n  &= \\Theta \\frac{1}{4\\pi} \\omega_{V,2}\n\\end{align}\n(here we used \\eqnref{relation} and the fact that $\\omega_{V,2}$ is odd under $\\mathbf{n}_2 \\to\n-\\mathbf{n}_2$.) In fact, this is identical to \\eqnref{a_omega_0}. The reduction process\nthen proceeds nearly identically to that in Section \\ref{Z2t_1p1} above and one finds\nthat the only non-zero component of\nthe $3$-cocycle is\n\\begin{equation}\n  \\nu(X, X, X) = \\frac{\\Theta}{2}.\n\\end{equation}\nThus, one recovers both elements of $H^3(Z_2, U(1)) \\cong Z_2$ for $\\Theta$ an\nodd or even multiple of $2\\pi$ respectively.\n\n\\section{Lattice models of SPT phases}\n\\label{sec_discrete}\nIn Ref.~\\onlinecite{spto_higher,*spto_higher_prb}, the classification of SPT phases in $d$ spatial dimensions was\nbased on an explicit construction of a field theory for a $d+1$-dimensional\ndiscrete spacetime for each element of the cohomology group $H^{d+1}(G, \\U(1))$.\nAlthough a discrete spacetime is perhaps hard to interpret physically, the\nconstruction of Ref.~\\onlinecite{spto_higher,*spto_higher_prb} can also be used to derive a ground-state\nwavefunction on a spatial lattice; a gapped Hamiltonian with this wavefunction\nas its ground state constitutes an (albeit\nunrealistic) lattice Hamiltonian realizing the SPT phase. Hence, it is worthwhile to show\nthat the symmetry on the edge of such of a lattice model is indeed classified\nunder our scheme by the same element of the cohomology group that was\nused to construct the wavefunction. We do this in Appendix \\ref{sec_app_discrete}. In\nparticular, this shows that every element of the cohomology group $H^{d+1}(G,\n\\U(1))$ can be realized in an explicit lattice model.\n\n\\section{Beyond the cohomological classification}\n\\label{sec_beyond}\nIt is now well established\n\\cite{vishwanath_senthil,beyond_cohomology_walker_wang,window,kapustin_cobordism,thorngren_cobordism}\nthat in (3+1)-D there exists an SPT phase with\nrespect to time-reversal symmetry that is beyond the standard\ncohomological classification. The reason why this phase is outside the\ncohomological classification can be readily understood, as follows. Deriving the cohomological classification using\narguments such as those presented in this paper requires at the very least the\nassumption that the symmetry can be implemented locally on a standalone\nrealization of the edge. We will now argue that the beyond-cohomology phase\nviolates this assumption.\n\nIndeed, one possible surface termination for the beyond-cohomology phase is a gapped\n``three-fermion'' topological phase $\\mathcal{F}$ in which all three non-trivial particle\nsectors are fermions. Any purely (2+1)-D realization of this phase is\nnecessarily chiral; that is, its conjugate $\\overline{\\mathcal{F}}$ under time\nreversal cannot be connected to $\\mathcal{F}$ without a phase transition. (One way to\nsee this is to note that $\\mathcal{F}$ and $\\overline{\\mathcal{F}}$ have opposite edge chiral central charges\n$c_{-} = \\pm 4$ and hence a spatial boundary between them must be gapless. If we\nmake the spatial variation from $\\mathcal{F}$ to\n$\\overline{\\mathcal{F}}$ sufficiently slow, this gapless spatial boundary must be interpreted as a\nbulk phase transition \\cite{Chen2013}.) Suppose that a state $\\ket{\\Psi}$ within the\nphase $\\mathcal{F}$ could be invariant under a local anti-unitary operation $T$.\nThen one can always write $T = U \\mathbb{T}$, where $\\mathbb{T}$ is the normal\non-site representation of time-reversal, and $U$ is a local unitary. But then,\nsince $\\mathbb{T} \\ket{\\Psi}$ is in the conjugate phase\n$\\overline{\\mathcal{F}}$, we see that $U$ is a local unitary connecting\n$\\mathcal{F}$ and $\\overline{\\mathcal{F}}$, which is a contradiction.\n\n\\section{Fermionic systems}\n\\label{sec_fermions}\nThe restriction arguments given in Section \\ref{sec_general} are quite general and\ntherefore can be equally well applied to fermionic systems, at least in (2+1)-D.\n(Generalizing to higher dimensions would require one to find an appropriate\nfermionic equivalent of the special form of the symmetry considered in Section\n\\ref{sec_higher_dims}.) Here we will discuss in general terms the issues arising\nwhich result in the fermionic classification differing from the bosonic one,\nwith reference to a particular example of a Fermion SPT protected by a $Z_2$ symmetry.\nAs the general classification is somewhat complicated, we\nwe leave the details to Appendix \\ref{appendix_fermionic}. It would be\ninteresting to see whether it can be related to the ``supercohomology''\nclassification proposed in Ref.~\\onlinecite{supercohomology}. We will consider\nonly cases in which the symmetry is unitary and on-site; Thus, our\nclassification will not include the well-known cases of topological insulators\nand superconductors\\cite{hasan_kane_review,moore_birth_2010,qi_zhang_review}, which are protected by non-unitary symmetries.\n\nThe first issue that needs to be considered is the privileged role of fermion parity. Any local fermionic system must be invariant under the fermion parity $(-1)^F$,\nwhere $F$ is the total fermion number. Therefore, the fermionic symmetry\ngroup $G_f$ characterizing a fermion SPT always contains fermion parity. This\nmust commute with all the other elements of $G_f$ if they describe local\nsymmetries. If we now consider the (1+1)-D edge of a (2+1)-D SPT, by assumption\nit is realizable as a strictly (1+1)-D local fermion system. As this (1+1)-D system must\nalways be invariant under the fermion parity of the edge, we expect that, in the\nrealization of $G_f$ on the (1+1)-D edge, the parity element is represented as\nthe actual fermion parity of the edge. (This can be verified by using the\ntechniques of Appendix \\ref{sec_edge_construction} to construct the edge representation.) That is, by\ncontrast to the bosonic case, the fermionic symmetry group contains an element\nthat is \\emph{always realized on-site on the boundary}. Furthermore, even when we restrict and\nconsider the action of the symmetry on a finite interval, \\emph{the restricted\noperations must be local, and therefore must still commute with the fermion\nparity} (whereas there is no analogous requirement in the bosonic case.)\n\nThe other main difference from the fermionic case occurs when defining the\nrestriction of the obstruction operator $\\Omega(g_1, g_2)$, which acts on a pair\nof points $a$ and $b$, to a single point $a$.\nAt this point, one encounters a subtlety that was glossed over in the bosonic\ntreatment.  $\\Omega(g_1, g_2)$ is clearly local in the sense (``locality\npreserving'') that it\nmaps local operators (including fermion creation and\nannihilation operators) to local operators under Heisenberg evolution. (We can\ndeduce this from the fact that it is true for the $U_M(g)$'s and that the\nlocality preserving property is invariant under multiplication.)\nThis does not necessarily imply \\cite{gross_index,hastings_torus_trick} that it is a local unitary in\nthe sense (``locally generated'') that it can be written as the time evolution of a local fermionic\nHamiltonian in a domain\ncontaining only the two points $a$ and $b$. In other words, we might not be able\nto write $\\Omega(g_1, g_2) = \\Omega_a(g_1, g_2) \\Omega_b(g_1, g_2)$, where\n$\\Omega_a$ and $\\Omega_b$ are fermionic local unitaries acting only near the\npoints $a$ and $b$. For example, the following unitary is locality preserving\nbut not locally generated:\n\\begin{equation}\n  \\label{paradise_lost}\n  \\Omega = (c_a + c_a^{\\dagger}) (c_b + c_b^{\\dagger}),\n\\end{equation}\nwhere $c_{a,b}$ are the annihilation operators for fermions at points $a$ and\n$b$ respectively. If $\\Omega$ is not locally generated, this presents an\nobstacle for defining the restriction $\\Omega \\to \\Omega_a$. This problem was\nnot present in the bosonic case because for bosonic systems locality preserving\nactually does imply locally generated for unitaries acting on a pair of points (for\nunitaries acting on higher-dimensional regions this is no longer the case\n\\cite{gross_index,hastings_torus_trick}.)\nNevertheless, it is clear that, for the $\\Omega$ given by \\eqnref{paradise_lost} there is still a\nnatural definition of ``restriction'' $\\Omega_a = (c_a + c_a^{\\dagger})$, even\nthough $\\Omega_a$ is not really local. More generally, it is always true that\n$\\Omega(g_1, g_2) = \\Omega_a(g_1, g_2) \\Omega_b(g_1, g_2)$, where $\\Omega_a(g_1,\ng_2)$ is\neither a fermionic local unitary acting near the point $a$, or it is such\na local unitary multiplied by $c_a + c_a^{\\dagger}$ (and similarly for\n$\\Omega_b$). In the latter case, however, the \\emph{restricted operations\n$\\Omega_a(g_1, g_2)$ and $\\Omega_b(g_1, g_2)$ will anti-commute rather than commute.} This\nanti-commutation leads to fermionic corrections to the 3-cocycle condition\n[\\eqnref{three_cocycle_condition}], to the the equivalence relation\n[\\eqnref{three_coboundary}], and to the product rule for ``stacking'' SPT phases; see Appendix \\ref{appendix_fermionic} for more details.\n\n\\subsection{Example: Fermionic SPT with $Z_2$ symmetry}\nIn order to illustrate the ideas discussed above, let us consider a (1+1)-D\nfield theory which we expect to describe the edge of a\n(2+1)-D fermionic SPT protected by a $Z_2$ symmetry. (This $Z_2$ is in addition\nto the always-present fermion parity; thus the full fermionic symmetry group is\n$G_f = Z_2 \\times Z_2^f$.) This theory is the fermionic analogue of\nthe bosonic edge we considered in Section \\ref{sec_chiral_example}. The low-energy physics is \ndescribed by a gapless Dirac point (which can emerge, for example, from a microscopic\n lattice model of non-interacting electrons with a Fermi surface.) Thus, we define\nthe fermionic fields $\\Psi_R(x)$ and $\\Psi_L(x)$ corresponding to left- and\nright-moving fermions (in terms of the original lattice operators, these will be\nlocal on a length scale set by the energy cutoff.) We can define the\ncorresponding number operators $N_{L,R} = \\int \\Psi_{L,R}^{\\dagger} \\Psi_{L,R} dx$. The Hamiltonian is\n\\begin{equation}\n  H = J (N_L + N_R).\n\\end{equation}\nThe fermion parity is $(-1)^{N_R + N_L}$ and sends $\\Psi_L \\to -\\Psi_L, \\Psi_R\n\\to -\\Psi_R$. We assume that the additional $Z_2$ symmetry is given by $U = (-1)^{N_R}$;\nthus, it acts only on the right-movers and sends $\\Psi_L \\to \\Psi_L, \\Psi_R \\to\n-\\Psi_R$. This forbids perturbations like $\\Psi_L^{\\dagger} \\Psi_R$ which would open up\na gap, suggesting that the gapless edge is protected by the symmetry. Indeed,\nwe will show that the symmetry corresponds to a non-trivial fermionic cocycle.\n\nWe can define the restriction of the $Z_2$ symmetry to a finite interval $[a,b]$\naccording to\n\\begin{equation}\n  U_{[a,b]} = \\exp\\left(-i \\pi \\int_a ^b \\Psi_R^{\\dagger} \\Psi_R dx\\right).\n  \\label{cslewis}\n\\end{equation}\nIf we invoke the bosonization correspondences $\\Psi_R^{\\dagger} \\Psi_R \\sim\n\\partial_x \\phi_R(x)\/(2\\pi)$, $\\Psi_R(x) \\sim e^{i \\phi_R}$, we see that $\\Omega\n\\equiv U_{ [a,b]\n}^2 \\sim \\Psi_R(a) \\Psi_R^{\\dagger}(b) \\equiv \\Omega_a \\Omega_b$. Thus,\n$U_{ [a,b] }^2$ acts only\non the endpoints as expected. \nHowever, this is an example of the possibility\ndiscussed above, of the operators $\\Omega_a$ carrying non-trivial fermion\nparity.\n\nThe parity of $\\Omega_a$, which we call $\\sigma$ ($\\sigma = -1$ in the current\ncalculation) constitutes one aspect of the non-trivial fermionic\ncocycle. The other aspect comes from the relation $U_{[a,b]} \\Omega\nU_{[a,b]}^{\\dagger} = \\Omega$. The restricted operations $\\Omega_a$ satisfy this\nrelation only up to a phase factor $\\omega$. To calculate this phase factor we need to\nregularize the integral \\eqnref{cslewis} by introducing a soft cutoff; that is,\nwe replace \\eqnref{cslewis} by\n\\begin{equation}\n  U_{[a,b]} = \\exp\\left(-i\\pi \\int_{-\\infty}^{\\infty} f(x) \\Psi_R^{\\dagger}\n  \\Psi_R dx\n  \\right),\n\\end{equation}\nwhere $f$ is a smooth function such that $f(x) = 1$ for $x \\in\n[a+\\epsilon,b-\\epsilon]$, and $f(x) = 0$ for $x < (a - \\epsilon)$ or $x > (b +\n\\epsilon)$. Using the bosonization correspondence to express $U_{[a,b]}^2$ in\nterms of $\\partial_x \\phi_R$ and integrating by parts gives\n\\begin{equation}\n\\Omega_a \\sim \\exp\\left(i \\int_{a-\\epsilon}^{a+\\epsilon} f'(x) \\phi_R dx\\right).\n\\end{equation}\nUsing the fact that $\\phi_R \\to \\phi_R + f(x) \\pi$ under $U_{[a,b]}$ gives\n\\begin{align}\n  \\omega \\equiv U_{[a,b]} \\Omega_a U_{[a,b]}^{\\dagger} \\Omega_a^{\\dagger} \n  &= \\exp\\left(i\\pi \\int_{a-\\epsilon}^{a + \\epsilon}\n  f'(x) f(x) dx\\right)  \\\\\n  &= \\exp\\left(\\frac{i\\pi}{2} \\int_{a-\\epsilon}^{a+\\epsilon} \\frac{d}{dx} [f(x)]^2\n  \\right) \\\\\n  &= i.\n\\end{align}\n\nThe numbers $(\\omega,\\sigma) = (i,-1)$ constitute the fermionic 3-cocycle for the $Z_2$\nsymmetry. We see that taking four copies of the same edge leads to a trivial\nfermionic 3-cocycle (in agreement with the results of\nRef.~\\onlinecite{lu_vishwanath} showing that four\ncopies of the theory under consideration can be gapped out without breaking the\nsymmetry.) Furthermore, if one applies the fermionic 3-cocycle condition\n[see Appendix \\ref{appendix_fermionic}] one sees that the only allowable values of the fermionic\n3-cocycle are the ones obtained by taking copies in this way, namely $(1,1),\n(i,-1), (-1,1)$ and $(-i,-1)$. Thus, we have recovered all the elements of a $\\mathbb{Z}_4$ classification\nfor fermionic SPT's with $G_f = Z_2 \\times Z_2^f$ (which is the same result\nobtained from supercohomology\\cite{supercohomology}). By contrast,\nRefs.~\\onlinecite{ryu_zhang_fermions,qi_fermions,yao_fermions,gu_levin_fermions}\nobtained a $\\mathbb{Z}_8$ classification for the same $G_f$. The odd-numbered\nphases in this classification have an odd number of gapless Majorana modes at\nthe edge, each\nof which is ``half'' of the gapless Dirac mode considered here. The explanation for the\ndiscrepancy in the classification is that the\nsymmetry in these odd-numbered phases does not act locally on the edge, and hence\nthey are not captured by our approach.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nSuppose we have a system whose bulk ground state is invariant\nunder a group $G$ of symmetries that commute with the Hamiltonian.\nLet us further suppose that there is an energy gap to all bulk excitations and a concomitant finite correlation length\nand that we can solve the Hamiltonian (with a sufficiently powerful computer, for instance)\nfor systems much larger than the correlation length. Armed with this information,\nwe wish to determine if the system is in a symmetry-protected topological phase and, if so, which one.\nIn a 1D system on a finite interval, we can identify an SPT phase by the presence\nof gapless excitations at the ends of the system that transform under a\nprojective representation of the symmetry (or, alternatively, the presence of such states in\nthe bipartite entanglement spectrum) \\cite{pollmann-arxiv-2009,chen_gu_wen,schuch,fidkowski_fermions}.\nBut how do we identify an SPT phase in higher dimensions?\nOne approach is to gauge the symmetry $G$.\\cite{Etingof10,levin_gu,Chen14,wang_levin}\nIn 2D, the resulting theory has anyonic excitations in the bulk \\cite{levin_gu}.\nBy determining the statistics of these excitations,\none can deduce the SPT phase of the ungauged system. In 3D, the gauged theory has anyonic excitations\non its surface \\cite{Chen14} and extended excitations (e.g. vortex lines) in its bulk \\cite{wang_levin}.\nThe topological properties of surface and bulk excitations of the gauged model\ncan be used to identify the underlying SPT phase.\nBut this approach involves modifying the system drastically, and it cannot be used if\nall that we are given are the low-energy eigenstates of the original Hamiltonian. Moreover, it may be more difficult,\nas a practical matter, to solve the gauged model and deduce its quasiparticles' topological\nproperties than it is to solve the original Hamiltonian.\n\nHere, we take a different approach, which identifies an SPT directly from the realization\nof the symmetry group $G$ on its boundary states. We consider $d$-dimensional SPTs\nfor which the restriction of $G$ to the low-energy Hilbert space has a local\naction on the $(d-1)$-dimensional boundary of the system.\nIn such a phase, there may be an obstruction to restricting the action of the\nsymmetry to a $(d-1)$-dimensional proper submanifold of the boundary. To analyze such\nan obstruction, we construct a new functional of two group elements by taking\na suitably defined coboundary of the restriction. This localizes the obstruction to the $(d-2)$-dimensional boundary of\nthe $(d-1)$-dimensional proper submanifold of the boundary. We then continue in the same fashion, either\nrestricting a functional of $k$ group elements on a closed $(d-k)$-dimensional manifold\nto a $(d-k)$-dimensional submanifold with boundary or\nconstructing the coboundary of a functional of\n$k$ group elements on a $(d-k)$-dimensional submanifold with boundary, thereby obtaining\na functional of $k+1$ group elements on a $(d-k-1)$-dimensional closed manifold.\nThese functionals are operators that act on the local Hilbert spaces of the corresponding\nsubmanifolds.\nThe resulting sequence of maps between functionals terminates after we reach functionals\nof $d$ group elements acting on a single point; the coboundary of such a functional must be an ordinary phase.\nEquivalence classes of such sequences are classified by the\ncohomology group $H^{d+1}(G,U(1))$ in $d=1,2$ and, with an additional assumption, in $d\\geq 3$.\nConsequently, given the low-energy states of the boundary (or large eigenvalue eigenstates\nof the reduced density matrix for a bipartition of a system without a real boundary),\nwe can, in principle, determine the corresponding element of $H^{d+1}(G,U(1))$.\nThe Hamiltonian need not take any special form -- in fact, it is not even necessary to know\nthe Hamiltonian. As we have shown, this procedure gives the expected results when applied to\ndiscrete \\cite{spto_higher,*spto_higher_prb} and continuous \\cite{boson_iqhe,cenke_neel_order,vishwanath_senthil,adr_classification}\nnon-linear sigma models.\n\nThe obstructions classified by these arguments prevent a model from being continuously deformed into a model\nin which the symmetry is realized on the boundary in an on-site manner. (By assumption, the symmetry\ncan be realized in an on-site manner in the full bulk theory -- by grouping multiple sites into a single site, for instance.)\nAs a result of the incorrigibly non-on-site nature of the symmetry, if we try to gauge it, the resulting gauge\ntheory will be\nanomalous\\cite{Ryu12,Wen13,for_dummies,kapustin_cobordism,kapustin_anomalies,kapustin_anomalies_2}.\nOnly the action of the symmetry on the whole system, bulk and edge together,\ncan be gauged in an anomaly-free fashion. A simple example is a $2+1$-dimensional\n$U(1)$ SPT, which is very similar to the $\\mathbb{Z}_2$ case discussed in Section \\ref{sec_chiral_example}.\nSuch a state is a bosonic integer quantum Hall state \\cite{boson_iqhe}. If the theory is gauged,\nthe edge effective Lagrangian takes the form ${\\cal L}_{\\rm edge} = \\frac{g}{2\\pi}(\\partial_\\mu \\varphi - n A_\\mu)^2\n+ \\frac{m}{2\\pi} A_\\mu \\epsilon_{\\mu\\nu} \\partial_\\nu \\varphi$.\nCharge is no longer conserved at the edge since an electric field along the edge will cause charge to flow\nfrom the bulk to the edge. Following Laughlin \\cite{Laughlin81},\nwe can understand this in an annular geometry. By adiabatically\nincreasing the flux through the center of the annulus by $2\\pi$, the charge at the outer edge is\nincreased by $2nm$, the integer (necessarily even in a bosonic SPT)\nthat characterizes the Hall conductance. The $3$-cocycle obtained by our construction reflects this\ncharge pumped to the edge, as may be seen by noting that\na $U(1)$ transformation applied to a finite interval along the edge is equivalent to applying equal and opposite\ngauge fields at the ends of the interval.\\footnote{By restricting the rotation to an interval, we cannot introduce a net\ntwist, but we can separate equal and opposite twists and focus on the vicinity of just one of them.}\nSince they are equal and opposite, such gauge fields cannot increase\nthe total charge on the edge, but if we focus on the charge to the left of an arbitrary point in the middle of the interval,\nthen this increases by $2nm$ when the gauge field winds by $2\\pi$. The restriction $\\Omega_a$ \ndefined in Eq. \\ref{first_coboundary_nonabelian} measures such a\ncharge\\footnote{\n  The precise relation is slightly subtle, however, because $\\Omega_a$ measures\n  the charge to the left of the endpoint $a$, the very place at which\n  charge is accumulating due to the Hall effect. The precise manner in which $\\Omega_a$ takes into\naccount the charge located at the endpoint itself depends on the restriction $U \\to\nU_M$. However, as long as we use the same restriction both to define $\\Omega_a$\nand to implement the winding, the value of the measured ``charge'' is independent of\nthe choice of restriction. Perhaps the easiest case to interpret is where $U_M$ \nis defined so that the winding of the phase interpolates linearly over a transition region\nbetween no winding outside the interval\nand the desired winding in the interior of the interval. Then $\\Omega_a$ measures the charge to the left\nof a point $x$, averaged over all $x$ in the transition region. This is $nm$, rather than the charge $2nm$\nmeasured to the left of a point in the interior of the interval.\n}.\nMeanwhile, $U_M$ applies gauge fields at the ends of the interval.\nThen, according to the definition (\\ref{hacker}), the cocycle measures accumulated charge\nin response to this change in gauge field.\nWe expect that similar reasoning can relate our constructions to anomalies in higher dimensions and for\ndiscrete symmetries.\n\nIn this paper, we have confined our attention to ``internal'' symmetries. It would be interesting\nto extend them to space group symmetries. States of free fermions protected by\ninversion symmetry \\cite{Turner09,Hughes11};\ntime-reversal symmetry combined with a point group symmetry \\cite{Fu11}; or a rotational symmetry alone\nhave been classified \\cite{Jadaun13}. With the methods described here, it might be possible\nto extend these ideas to interacting fermion systems and to bosonic systems in which\na space group symmetry, projected to the low-energy\nboundary theory, maps sites to sites and then has an additional ``internal'' action that is non-on-site.\nHowever, care must be taken to consider a boundary that respects\nthe space group symmetry and to consider a sequence of submanifolds (which are, presumably, not connected\nmanifolds) that also respect the symmetry.\n\nWe have given a prescription that, in principle, allows one to identify an SPT phase, given\nits ground state wavefunction, and we have shown how to apply it to some long-wavelength effective\nfield theories and exactly soluble lattice models. But how useful can this prescription be in practice,\ngiven an arbitrary -- perhaps experimentally-motivated model? This remains to be\nseen. However, ground state wavefunctions with tensor network descriptions are\nnatural candidates for the reduction procedure\\cite{spto_2d}.\nA numerical implementation would open an important\navenue for future research.\n\nConversely, we have shown in Section \\ref{sec_discrete}\nand Appendix \\ref{sec_app_discrete} that each SPT phase in the cohomological classification has\na representative wavefunction which is the ground state of some lattice Hamiltonian. However, these Hamiltonians are certainly\nnot expected to describe any experimentally realizable systems; finding more realistic\nHamiltonians giving rise to SPT phases is an\nimportant open problem.\n\nAs noted above, our construction leads to $H^{d+1}(G,U(1))$ in $d\\geq 3$ provided we make\nan additional assumption: there exists a local basis for the Hilbert space of the $(d-1)$-dimensional boundary\nin which the symmetry acts on the boundary in an on-site manner, except for a diagonal part\nwhich cannot be made on-site.\nThis assumption holds in a system that is described by a $d$-dimensional non-linear sigma\nmodel with $\\theta$-term at long wavelengths\n\\cite{boson_iqhe,cenke_neel_order,vishwanath_senthil,adr_classification}\nsince the symmetry acts in an on-site manner\non all gradient energy terms in\nthe $(d-1)$-dimensional boundary effective action and non-on-site only on the Wess-Zumino term,\nwhich only enters the phase of the ground state wavefunction.\nHowever, it remains an interesting open question whether there\nare SPT phases in three dimensions that violate this assumption and, consequently,\nrealize the aforementioned non-trivial sequence but in a\nmanner that is not classified by group cohomology. Such an exception to a cohomological classification,\nif it exists, would be distinct from the so-called ``beyond cohomology'' SPT phases\n\\cite{vishwanath_senthil}, which occur due to the violation of\na different assumption -- that the symmetry is realized locally (but not necessarily on-site)\nat the boundary of the system. In ``beyond cohomology'' SPT phases,\nthe symmetry is realized in an inherently non-local manner at the boundary of the system.\nOur methods do not enable us to classify such phases;\nonce the condition of locality is relaxed, a very different approach may be necessary.\n\nThis comment also applies to the most famous SPT phase, the 3D time-reversal-invariant\ntopological insulator \\cite{Moore07,Fu07,Roy09},\nwhere time-reversal acts in an inherently non-local manner at a 2D surface. However, there\nare fermionic SPT phases in which the symmetry is realized locally on the boundary, and these can be classified\nalong the lines discussed in Section \\ref{sec_fermions}. Carrying out this classification to completion and\nrelating it to the notion of ``supercohomology''\\cite{supercohomology} is an\nimportant goal for future work. We remark that dimensional reductions\nhave previously been employed in the classification of fermionic\nSPT phases \\cite{topfieldtheory_insulators,tenfold_way}, and it would be interesting if a connection could be drawn with the\nreduction procedure described here.\n\nFinally, we note that symmmetry-enriched topological (SET) phases \\cite{Etingof10,Mesaros13,Hung13,Lu13,Xu13,Essin13}\ngeneralize SPT phases to systems with topological order.\nIn SET phases, symmetry realization interacts non-trivially with the fusion and braiding properties of anyons,\nas already occurs in topological phases at the (2+1)-D boundary of a (3+1)-D SPT. \nThe possible symmetry fractionalization patterns in (2+1)-D correspond to different\nprojective representations of the anyons and are classified by ${H^2}(G,A)$, where $A$ is\nthe group of Abelian anyons. It is possible that an extension of our methods can\nalso be applied to the analysis of symmetry fractionalization in (3+1)-D SET\nphases which have topological excitations occupying closed loops.\n\n\\begin{acknowledgments}\nWe thank M.~Cheng, C.~Xu, A.D.~Rasmussen, and B.~Ware for helpful discussions.\nD.~Else was supported by a gift from the Microsoft Corporation.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\nIn the series of papers \\cite{CC1,CC2,CC3} by Cheeger and Colding, they investigate much properties of Ricci limit spaces. Especially, the study of the infinitesimal structure on such spaces is pretty important to understand the geometry of that. On a non-collapsing Ricci limit space $(Y,d,\\nu)$, $\\nu$-almost every point has unique tangent cone that is isometric to $N$-dimensional Euclidean space when the sequence of Rimannian manifolds approximating $Y$ are of $N$-dimension \\cite{CC1}. It is also known that the limit measure $\\nu$ is the $N$-dimensional Hausdorff measure multiplied by a constant. For collapsing Ricci limit spaces, the uniqueness of the dimension of tangent cones at almost every point had been an open problem. However Colding and Naber give the affirmative answer to the problem, that is, there exists a unique $k$ less than $N$ so that $\\nu(Y\\setminus \\mathcal{R}_k)=0$, where $\\mathcal{R}_k$ is the set of all points whose tangent cone is unique and isometric to $\\mathbb{R}^k$(see \\cite{CNholder}). Combining the results of Cheeger-Colding and of Colding-Naber leads the mutually absolutely continuity between the limit measure $\\nu$ and the $k$-dimensional Hausdorff measure $\\nu$-almost every point on $\\mathcal{R}_k$. \n\nOn the other hand, $\\mathsf{RCD}$ spaces are one of another generalization of Riemannian manifolds with Ricci curvature bounded from below. $\\mathsf{RCD}$ spaces are defined as a kind of convexity of the functional on the Wasserstein spaces on those(see Section \\ref{sec:pre}). In that sense, $\\mathsf{RCD}$ spaces are defined by the intrinsic way while Ricci limit spaces are defined by the extrinsic way. It is known that all Ricci limit spaces are $\\mathsf{RCD}$ spaces by the stability of $\\mathsf{RCD}$ condition under the Gromov-Hausdorff convergence. Unfortunately, no one knows whether generic $\\mathsf{RCD}$ spaces can be approximated by a family of Riemannian manifolds with uniformly bounded Ricci curvature or not. Hence the study on the geometry and analysis of $\\mathsf{RCD}$ spaces are difficult because we are not able to use mathematical techniques developed on Riemannian manifolds. \n However by using techniques from the study of optimal transportation problem on metric measure spaces instead of differential geometric ones, many geometric and analytic properties on $\\mathsf{RCD}$ spaces are discovered, some of them are new even for Riemannian manifolds. On the other hand, the following fundamental question is still open, though the Ricci limit case is already answered positively; \n\\begin{quest}\\label{quest:main}\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. Is there an integer $l$ with $1\\leq l\\leq[N]$ such that $m$-almost points $x\\in X$ has unique tangent cone isomorphic to $\\mathbb{R}^l$? \n\\end{quest}\nMondino and Naber prove that $m\\left(X\\setminus \\cup_{1\\leq i \\leq[N]}\\mathcal{R}_i\\right)=0$ \\cite{MN}. Question \\ref{quest:main} are able to be reformulated that whether there is an integer $l$ such that $m(X\\setminus \\mathcal{R}_l)=0$ or not. As mentioned above, this problem is completely solved in Ricci limit cases. Adding some conditions, we have partial positive answers for that problems \\cite{KBishop,KL}. One of the main theorem in the present paper holds without any additional assumptions. \n\\begin{thm}[Proposition \\ref{prop:kfunctions} and Proposition \\ref{prop:lk}]\\label{thm:posi}\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $k\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. Assume $\\mathcal{R}_k\\neq\\emptyset$ for $1\\leq k\\leq[N]$. Then $m\\left(\\cup_{k\\leq i\\leq[N]}\\mathcal{R}_i\\right)>0$. In particular, $m(\\mathcal{R}_{[N]})>0$ if $\\mathcal{R}_{[N]}\\neq\\emptyset$. \n\\end{thm} \n\\begin{rem}\n In \\cite{KL}, the author and Lakzian proved that $m(X\\setminus \\mathcal{R}_1)=0$ provided $\\mathcal{R}_1\\neq\\emptyset$. Thus we only need to consider the case when $\\mathcal{R}_k\\neq\\emptyset$ for $k\\geq 2$.  \n\\end{rem}\nLet $k$ be the largest number such that $\\mathcal{R}_k\\neq\\emptyset$. It follows from Theorem \\ref{thm:posi} that $m(\\mathcal{R}_k)>0$. If $m(X\\setminus \\mathcal{R}_k)=0$, then Question \\ref{quest:main} has the affirmative answer. But still, there exists a possibility that $m(\\mathcal{R}_i)>0$ for some $i$ less than $k$. Hence Theorem \\ref{thm:posi} is far from the goal of our Question. However it must be a first step to that. \n\nBy Theorem \\ref{thm:posi}, we are able to define the dimension of $\\mathsf{RCD}$ spaces. \n\\begin{defn}[Definition \\ref{def:dimmm}]\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$, $N\\in(1,\\infty)$. The dimension of $(X,d,m)$ is defined as the largest number $k$ so that $\\mathcal{R}_k\\neq\\emptyset$ and $\\mathcal{R}_l=\\emptyset$ for any $l>k$. $\\mathsf{dim}\\,(X,d,m)$ denotes the dimension of $(X,d,m)$. Equivalently, the dimension of $(X,d,m)$ is also the largest number $k$ such that $m(\\mathcal{R}_k)>0$. \n\\end{defn} \nThe concept of dimension here coincides with that introduced by Colding and Naber for Ricci limit spaces \\cite{CNholder}. See \\cite{KaLi}, \\cite{HRicLp} for the proof. Also the \\emph{analytic dimension} defined by Han \\cite{HanRictensor} coincides(see Remark \\ref{rem:analydim}). \nAnother main theorem is as follows. \n\\begin{thm}[Theorem \\ref{thm:lscdimmm}]\n Let $(X_n,d_n,m_n,x_n)$ be a sequence of pointed $\\mathsf{RCD}^*(K,N)$ spaces for $K\\in\\mathbb{R}$, $N\\in(1,\\infty)$ and converging to $(X_{\\infty},d_{\\infty},m_{\\infty},x_{\\infty})$. Then \n \\begin{align}\n  \\mathsf{dim}\\,(X_{\\infty},d_{\\infty},m_{\\infty})\\leq \\liminf_{n\\rightarrow\\infty}\\mathsf{dim}\\,(X_n,d_n,m_n).\\notag\n \\end{align}\n\\end{thm}\n\\begin{rem}\n The lower semicontinuity of the dimension for tangent cones was already proven for Ricci limit case (\\cite{KaLi, HRicLp}). \n\\end{rem}\n\n\n %\n\n %\n\n\\section{Preliminaries}\\label{sec:pre}\nA triplet $(X,d,m)$ consisting of a complete separable metric space $(X,d)$ and a locally finite positive Borel measure $m$ on $X$ is called a \\emph{metric measure space}. Two metric measure spaces $(X,d,m)$ and $(Y,r,\\nu)$ are \\emph{isomorphic} if there exists an isometry $f:\\mathrm{supp}\\, m\\rightarrow\\mathrm{supp}\\, \\nu$ with $f_*m=\\nu$. \nA continuous curve $\\gamma:[0,1]\\rightarrow X$ is \\emph{absolutely continuous} if there exists an $L^1(0,1)$ function $g$ such that \n\\begin{align}\n  d(\\gamma_s,\\gamma_t)\\leq \\int_s^tg(r)\\,dr,\\qquad\\text{for any }s\\leq t\\in[0,1].\\notag\n\\end{align}\nFor a continuous curve $\\gamma:[0,1]\\rightarrow X$, the metric derivative $\\vert\\dot\\gamma\\vert$ is defined by \n\\begin{align}\n \\vert\\dot\\gamma_t\\vert:=\\lim_{s\\rightarrow t}\\frac{d(\\gamma_t,\\gamma_s)}{\\vert s-t\\vert}\\notag\n\\end{align}\nas long as the right-hand side makes sense.  \nIt is known that every absolutely continuous curve has the metric derivative for almost every point \\cite{AGSbook}. We call an absolutely continuous curve $\\gamma:[0,1]\\rightarrow X$ a \\emph{geodesic} if $\\vert\\dot\\gamma_t\\vert=d(\\gamma_0,\\gamma_1)$ for almost every $t\\in [0,1]$. A metric space $(X,d)$ is called a \\emph{geodesic space} if for any two points, there exists a geodesic connecting them. \n\nWe denote the set of all Lipschitz functions on $X$ by ${\\bf \\mathrm{LIP}}(X)$. For $f\\in{\\bf \\mathrm{LIP}}(X)$, the local Lipschitz constant at $x$, $\\vert \\nabla f\\vert(x)$, is defined as\n\\begin{align} \n \\vert \\nabla f\\vert(x):=\\limsup_{y\\rightarrow x}\\frac{\\vert f(x)-f(y)\\vert}{d(x,y)}\\notag\n\\end{align} \nif $x$ is not isolated, otherwise $\\vert \\nabla f\\vert(x)=\\infty$. For $f\\in L^2(X,m)$, we define the \\emph{Cheeger energy} $\\mathsf{Ch}(f)$ as \n\\begin{align}\n \\mathsf{Ch}(f):=\\frac{1}{2}\\inf\\left\\{\\liminf_{n\\rightarrow \\infty}\\int_X\\vert \\nabla f_n\\vert^2\\,dm\\;;\\;f_n\\in{\\bf \\mathrm{LIP}}(X),\\;f_n\\rightarrow f\\text{ in }L^2(X,m)\\right\\}.\\notag\n\\end{align}\nSet $D(\\mathsf{Ch}):=\\{f\\in L^2(X,m)\\;;\\;\\mathsf{Ch}(f)<\\infty\\}$. It is known that for any $f\\in D(\\mathsf{Ch})$, there exists an $L^2$-function $\\vert \\nabla f\\vert_w$ such that $\\mathsf{Ch}(f)=(1\/2)\\int\\vert \\nabla f\\vert_w^2\\,dm$, which is called a \\emph{minimal weak upper gradient}. For simplicity, we denote the minimal weak upper gradient of $f$ just by $\\vert \\nabla f\\vert$. We define the \\emph{Sobolev space} $W^{1,2}(X,d,m):=L^2(X,m)\\cap D(\\mathsf{Ch})$ equipped with the norm $\\Vert f\\Vert_{1,2}^2:=\\Vert f\\Vert_2^2+2\\mathsf{Ch}(f)$. It is known that $W^{1,2}$ is a Banach space. We say that $(X,d,m)$ is \\emph{infinitesimally Hilbertian} if $W^{1,2}$ is a Hilbert space. \n\nWe denote the set of all Borel probability measures on $X$ by $\\mathcal{P}(X)$. We define $\\mathcal{P}_2(X)$ as the set of all Borel probability measures with finite second moment, that is, $\\mu\\in\\mathcal{P}_2(X)$ if and only if $\\mu\\in\\mathcal{P}(X)$ and there exists a point $o\\in X$ such that $\\int_Xd(x,o)^2\\,d\\mu(x)<\\infty$. We call a measure $q\\in\\mathcal{P}(X\\times X)$ a \\emph{coupling} between $\\mu$ and $\\nu$ if $(p_1)_*q=\\mu$ and $(p_2)_*q=\\nu$, where $p_i:X\\times X\\rightarrow X$ are natural projections for $i=1,2$. For two probability measures $\\mu_0,\\mu_1\\in\\mathcal{P}_2(X)$, we define the \\emph{$L^2$-Wasserstein distance} between $\\mu_0$ and $\\mu_1$ as \n\\begin{align}\n W_2(\\mu_0,\\mu_1):=\\inf\\left\\{\\int_{X\\times X}d(x,y)^2\\,dq(x,y)\\;;\\;q\\in\\mathsf{Cpl}(\\mu_0,\\mu_1)\\right\\}^{1\/2},\\notag\n\\end{align}\nwhere $\\mathsf{Cpl}(\\mu_0,\\mu_1)$ is the set of all couplings of $\\mu_0$ and $\\mu_1$. The pair $(\\mathcal{P}_2(X),W_2)$ is called the \\emph{$L^2$-Wasserstein space}, which is a complete separable geodesic metric space if so is $(X,d)$. We explain how geodesics in $X$ relates to those in $\\mathcal{P}_2(X)$. We denote the space of all geodesics in $X$ by $\\mathsf{Geo}(X)$, equipped with the sup distance. Define the \\emph{evaluation map} $e_t:\\mathsf{Geo}(X)\\rightarrow X$ for $t\\in [0,1]$ by $e_t(\\gamma)=\\gamma_t$. Let $(\\mu_t)_t\\in \\mathsf{Geo}(\\mathcal{P}_2(X))$ be a geodesic connecting $\\mu_0,\\mu_1$ in $\\mathcal{P}_2(X)$. Then there exists a probability measure $\\pi\\in\\mathcal{P}(\\mathsf{Geo}(X))$ such that $(e_t)_*\\pi=\\mu_t$, by which we say that the geodesic $(\\mu_t)_t$ can be lifted to $\\pi$. \n\\subsection{The curvature-dimension condition}\n For given $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$, we define the \\emph{distortion coefficients}, $\\sigma^{(t)}_{K,N}$ for $t\\in[0,1]$, by \n \\begin{align}\n  \\sigma^{(t)}_{K,N}(\\theta):=\\begin{cases}\n                                            \\infty&\\text{if }K\\theta^2\\geq N\\pi^2,\\\\\n                                            \\frac{\\sin(t\\theta\\sqrt{K\/N})}{\\sin(\\theta\\sqrt{K\/N})}&\\text{if }0<K\\theta^2<N\\pi^2,\\\\\n                                            t&\\text{if }K\\theta^2=0,\\\\\n                                            \\frac{\\sinh(t\\theta\\sqrt{-K\/N})}{\\sinh(\\theta\\sqrt{-K\/N})}&\\text{if }K\\theta^2<0.\n                                           \\end{cases}\\notag\n \\end{align}\n Let $(Y,d)$ be a geodesic metric space and $f:Y\\rightarrow\\mathbb{R}\\cup\\{\\pm\\infty\\}$ a function on $Y$. \n\n\n \\begin{defn}[\\cite{EKS}]\n  A function $f:Y\\rightarrow \\mathbb{R}\\cup\\{\\pm\\infty\\}$ is said to be \\emph{$(K,N)$-convex} for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$ if for any two points $y_0,y_1\\in Y$, there exists a geodesic $(y_t)_t$ connecting them such that \n  \\begin{align}\n   &\\exp\\left(-\\frac{1}{N}f(y_t)\\right)\\notag\\\\\n   &\\geq \\sigma^{(1-t)}_{K,N}(d(y_0,y_1))\\exp\\left(-\\frac{1}{N}f(y_0)\\right)+\\sigma^{(t)}_{K,N}(d(y_0,y_1))\\exp\\left(-\\frac{1}{N}f(y_1)\\right)\\notag\n  \\end{align} \n  holds for any $t\\in[0,1]$. \n \\end{defn} \n\n Let $(X,d,m)$ be a geodesic metric measure space. Consider $\\mu=\\rho m\\ll m$ a probability measure that is absolutely continuous with respect to $m$ and its Radon-Nikodym derivative being $\\rho$. We define the relative entropy functional $\\mathrm{Ent}_m$ by \n \\begin{align}\n  \\mathrm{Ent}_m(\\mu):=\\int_{\\{\\rho>0\\}}\\rho\\log\\rho\\,dm,\\notag\n \\end{align}\n whenever $(\\rho\\log\\rho)_+$ is integrable, otherwise we define $\\mathrm{Ent}_m(\\mu)=\\infty$. \n\n \\begin{defn}[\\cite{EKS}, cf. \\cite{AMSnon}]\n  Let $(X,d,m)$ be a geodesic metric measure space. We say that $(X,d,m)$ satisfies the \\emph{entropic curvature-dimension condition} $\\mathsf{CD}^e(K,N)$ for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$ if the relative entropy functional $\\mathrm{Ent}_m$ is $(K,N)$-convex. Moreover if $(X,d,m)$ is infinitesimally Hilbertian, $(X,d,m)$ is called an $\\mathsf{RCD}^*(K,N)$ space. \n \\end{defn}\n\n Under the infinitesimal Hilbertianity condition, $\\mathsf{CD}^e(K,N)$ is equivalent to $\\mathsf{CD}^*(K,N)$ \\cite{EKS}. \n\n \\subsection{Tangent cones and regular sets on $\\mathsf{RCD}$ spaces}\n Let $(X,d,m)$ be a metric measure space. Take a point $x_0\\in \\mathrm{supp}\\, m$ and fix it. We call a quadruple $(X,d,m,x_0)$ a \\emph{pointed metric measure space}. We say that a pointed metric measure space $(X,d,m,x_0)$ is \\emph{normalized} if \n \\begin{align}\n  \\int_{B_1(x_0)}1-d(x_0,\\cdot)\\,dm=1.\\label{eq:normal}\n \\end{align}\n  For $r\\in(0,1)$, define $d_r:=d\/r$ and \n \\begin{align}\n  m^x_r:=\\left(\\int_{B_r(x)}1-d_r(x,\\cdot)\\,dm\\right)^{-1}m.\\notag\n \\end{align}\n Note that the pointed metric measure space $(X,d_r,m^x_r,x)$ is normalized. \n \n Let $C(\\cdot):[0,\\infty)\\rightarrow [1,\\infty)$ be a nondecreasing function. Define $\\mathcal{M}_{C(\\cdot)}$ the family of pointed metric measure spaces $(X,d,m,\\bar x)$ that satisfy \n \\begin{align}\n  m(B_{2r}(x))\\leq C(R)m(B_r(x))\\notag\n \\end{align}\n for any $x\\in \\mathrm{supp}\\,m$, and any $0<r\\leq R<\\infty$. Gigli, Mondino, and Savar\\'e have proven that there exists a distance function $\\mathcal{D}_{C(\\cdot)}:\\mathcal{M}_{C(\\cdot)}\\times \\mathcal{M}_{C(\\cdot)}\\rightarrow [0,\\infty]$, which induces the same topology as the Gromov-Hausdorff one on $\\mathcal{M}_{C(\\cdot)}$ (\\cite{GMS}). It is known that every $\\mathsf{RCD}^*(K,N)$ space for given $K\\in\\mathbb{R}$, $N\\in (1,\\infty)$ belongs to $\\mathcal{M}_{C(\\cdot)}$ for a common function $C(\\cdot):(0,\\infty)\\rightarrow [1,\\infty)$ (see \\cite{Stmms2,FS}), more precisely, they satisfy \n \\begin{align}\n  \\frac{m(B_R(x))}{m(B_r(x))}\\leq \\frac{\\int_0^R\\mathbb{S}_{K,N}^{N-1}(t)\\,dt}{\\int_0^r\\mathbb{S}_{K,N}^{N-1}(t)\\,dt}\\notag\n \\end{align}\n for any $x\\in\\mathrm{supp}\\, m$, $0<r\\leq R$, where \n \\begin{align}\n  \\mathbb{S}_{K,N}(t):=\\begin{cases}\n                           \\sin\\left(t\\sqrt{\\frac{K}{N-1}}\\right)&\\text{if }K>0,\\\\\n                           t&\\text{if }K=0,\\\\\n                           \\sinh\\left(t\\sqrt{\\frac{-K}{N-1}}\\right)&\\text{if }K<0. \n                          \\end{cases}\\notag\n \\end{align}\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. By a simple calculation, we have $(X,d_r,m_r^x)$ for some $x\\in \\mathrm{supp}\\, m$ being an $\\mathsf{RCD}^*(r^2K,N)$ space. Take a point $x\\in\\mathrm{supp}\\, m$ and fix it. Consider the family of normalized metric measure spaces $\\{(X,d_r,m^x_r,x)\\}_{r\\in(0,1)}$. The following is one of a generalization of Gromov's compactness theorem. \n\n \\begin{thm}[\\cite{GMR}]\n  The family of normalized metric measure spaces $\\{(X,d_r,m^x_r)\\}_{r\\in(0,1)}$ is sequentially compact with respect to the pointed measured Gromov-Hausdorff topology. Moreover every limit space $(X,d_{r_n},m^x_{r_n},x)\\rightarrow (Y,d_Y,m_Y,y)$ is a normalized $\\mathsf{RCD}^*(0,N)$ space for a non-increasing sequence $\\{r_n\\}_n$ with $r_n\\rightarrow 0$. \n \\end{thm} \n\n We define the \\emph{tangent cones} at a point $x\\in\\mathrm{supp}\\, m$ by \n \\begin{align}\n  \\mathsf{Tan}(X,d,m,x):=\\left\\{(Y,d_Y,m_Y,y)\\;;\\;(X,d_{r_n},m^x_{r_n},x)\\rightarrow(Y,d_Y,m_Y,y)\\right\\},\\notag\n \\end{align}\n where $\\{r_n\\}_n$ is a non-increasing sequence converging to 0. For simplicity, we just denote by $\\mathsf{Tan}(X,x)$ instead of $\\mathsf{Tan}(X,d,m,x)$ if there is no confusion.  \n\n \\begin{defn}\\label{def:regularsets}\n  Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. We call a point $x\\in\\mathrm{supp}\\, m$ a $k$-\\emph{regular point} if $\\mathsf{Tan}(X,x)=\\{(\\mathbb{R}^k,d_E,\\underline{\\mathcal{L}}^k,0)\\}$, where $\\underline{\\mathcal{L}}^k$ is the normalized Lebesgue measure at $0$. We denote the set of $k$-regular points by $\\mathcal{R}_k$. \n \\end{defn}\n\n Mondino and Naber proved the following \\cite{MN}. \n\n \\begin{thm}[\\cite{MN}*{Corollary 1.2}]\\label{thm:MN}\n  Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. Then \n  \\begin{align}\n   m\\left(X\\setminus \\bigcup_{1\\leq k\\leq N}\\mathcal{R}_k\\right)=0.\\notag\n  \\end{align}\n \\end{thm}\n\n Note that even though by using Theorem \\ref{thm:MN}, we do not know the uniqueness of regular sets in general. For Ricci limit spaces, we know $m(X\\setminus \\mathcal{R}_l)=0$ for some $1\\leq l\\leq N$ (\\cite{CNholder}), and in the following restricted case, the uniqueness of regular sets is known; \n \\begin{enumerate}\n  \\item $\\mathcal{R}_1\\neq\\emptyset$ (\\cite{KL}),\n  \\item A Bishop type inequality holds (\\cite{KBishop}). \n \\end{enumerate}\n\n \\subsection{Smoothing effects of the heat flows and the modified heat flows on $\\mathsf{RCD}$ spaces}\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. By the infinitesimal Hilbertianity, $\\mathsf{Ch}$ is actually a strongly local Dirichlet form (\\cite{AGSRiem}). Let $\\Delta$ be the generator of $\\mathsf{Ch}$, called the \\emph{Laplacian} and $\\{h_t\\}_{t>0}$ the associated semigroup, called \\emph{the heat flow}. Thus for any $L^2$-function $f$, we have \n \\begin{align}\n  \\frac{d}{dt}h_tf=\\Delta h_tf\\notag\n \\end{align}\n in $L^2$-sense. It is known that $\\mathsf{RCD}(K,\\infty)$ condition is equivalent to the \\emph{Bakry-\\'Emery curvature-dimension condition $\\mathsf{BE}(K,\\infty)$}, which requires \n \\begin{align}\n  \\vert\\nabla h_tf\\vert^2\\leq e^{-2Kt}h_t(\\vert \\nabla f\\vert^2)\\quad m\\text{-a.e., }\\notag\n \\end{align}\n for any $t>0$ and any $f\\in W^{1,2}(m)$ (see \\cite{AGSRiem}). \n We define the following set of test functions; \n \\begin{align}\n  \\mathrm{TestF}(X):=\\left\\{f\\in\\mathcal{D}(\\Delta)\\cap L^{\\infty}(m)\\;;\\;\\vert \\nabla f\\vert\\in L^{\\infty}(m)\\text{ and }\\Delta f\\in W^{1,2}(m)\\right\\}.\\notag\n \\end{align}\n By $\\mathsf{BE}(K,\\infty)$, $h_tf\\in \\mathrm{TestF}(X)$ for any $t>0$ and any $f\\in L^2\\cap L^{\\infty}(m)$. In order to obtain more regular functions, we define \\emph{the modified heat flow} $\\hat h_t$ as \n \\begin{align}\n  \\hat h_tf:=\\frac{1}{t}\\int_0^{\\infty} h_sf\\phi(st^{-1})\\,ds,\\notag\n \\end{align}\n where $\\phi\\in C^{\\infty}_c(0,1)$ with $\\int_0^1\\phi(s)\\,ds=1$ is a given nonnegative function. It is known that for a Lipschitz function $f$, $\\hat h_tf$ satisfies $\\hat h_tf\\in \\mathrm{TestF}(X)$, $\\Delta \\hat h_t f\\in L^{\\infty}(m)$, and $\\hat h_t f\\rightarrow f$ in $W^{1,2}$ (cf. \\cite{Gnon}). The latter property plays a crucial role in the proof of the main theorem. \n\n \\subsubsection{Smooth cut-off functions}\\label{subsection:cutoff}\n In \\cite{MN}, Mondino and Naber define a smooth cut-off function on an $\\mathsf{RCD}$ space as follows. Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in \\mathbb{R}$, $N\\in(1,\\infty)$. For every $x\\in X$, $R>0$, and $0<r<R$, there exists a Lipschitz function $\\varphi^x_r:X\\rightarrow \\mathbb{R}$ such that \n \\begin{enumerate}\n  \\item $0\\leq \\varphi^x_r\\leq 1$ on $X$, $\\varphi^x_r\\equiv 1$ on $B_r(x)$ and $\\mathrm{supp}\\, \\varphi^x_r\\subset B_{2r}(x)$,\n  \\item $r^2\\vert \\Delta \\varphi^x_r\\vert+r\\vert \\nabla \\varphi^x_r\\vert\\leq C(K,N,R)$, where the constant $C$ depends only on $K$, $N$, and $R$. \n \\end{enumerate} \n Note that on the rescaled space $(X,d_r,m^x_r)$, the smooth cut-off function $\\varphi^{d_r,x}_1:=\\varphi^x_r$ satisfies $\\vert \\Delta^{d_r}\\varphi^{d_r,x}_1\\vert+\\vert\\nabla^{d_r}\\varphi^{d_r,x}_1\\vert\\leq C$ (see Lemma \\ref{lem:rescaleeffect}).\n\n \\subsection{Convergence results}\n Under the measured Gromov-Hausdorff convergence, the behavior of functions and their differentials are big issues. Gigli-Mondino-Savar\\'e \\cite{GMS}, Ambrosio-Stra-Trevisan \\cite{AST}, and Ambrosio-Honda \\cite{AH} study such behaviors. In this subsection, we just list their theorems we use later. For this purpose, we give some fundamental concepts of convergences of various objects. For a complete separable metric space $(X,d)$, we denote by $\\mathcal{M}_{loc}(X)$ the set of all Borel measures which are finite on every bounded sets. It says that a sequence $\\mu_i\\in\\mathcal{M}_{loc}(X)$ converges to $\\mu\\in\\mathcal{M}_{loc}(X)$ \\emph{weakly} if \n \\begin{align}\n  \\lim_{i\\rightarrow\\infty}\\int_X\\phi\\,d\\mu_i=\\int_X\\phi\\,d\\mu\\notag\n \\end{align}\n holds for any $\\phi\\in C_{bs}(X)$, the set of all continuous functions with bounded supports. \n ~\\\\\n \\par In \\cite{GMS}, the convergence of metric measure spaces are discussed in the fully general setting. In particular one of the main consequences in \\cite{GMS} is the coincidence of concepts of \\emph{the measured Gromov-Hausdorff convergence} and \\emph{the measured Gromov convergence} under the uniform doubling condition. In this paper we also use the so called ``the extrinsic convergence concept\" along \\cite{GMS}, \\cite{AH}, and \\cite{AST}. \n\n \\subsubsection{The pmG convergence}\n Let $(X_i,d_i,m_i,x_i)$ be a sequence of pointed metric measure spaces and $(X_{\\infty},d_{\\infty},m_{\\infty},x_{\\infty})$ a pointed metric measure space. We say that $(X_i,d_i,m_i,x_i)$ converges to $(X_{\\infty},d_{\\infty},m_{\\infty},x_{\\infty})$ in \\emph{the pointed measured Gromov convergence $($pmG$)$ sense} if there exist a complete separable metric space $(X,d)$, isometric embeddings $\\iota_{i}:X_i\\rightarrow X$ for $i\\in \\mathbb{N}\\cup\\{\\infty\\}$ such that $\\iota_i(x_i)\\rightarrow\\iota_{\\infty}(x_{\\infty})$ and $\\mathfrak{n}_i:={(\\iota_i)}_*m_i$ converges to $\\mathfrak{n}_{\\infty}:={(\\iota_{\\infty})}_*m_{\\infty}$ weakly in $\\mathcal{M}_{loc}(X)$. Hereafter we identify $(X_i,d_i,m_i)$ and $(\\iota_i(X_i), d_{\\iota_i(X_i)},\\mathfrak{n}_i)$. Hence $(X_i,d_i,m_i,x_i)$ converges to $(X_{\\infty},d_{\\infty},m_{\\infty},x_{\\infty})$ is equivalent to $m_i$ converges to $m_{\\infty}$ weakly in $\\mathcal{M}_{loc}(X)$ and $x_i\\rightarrow x_{\\infty}$. In this setting, we are able to consider the convergence of a sequence of functions on varying spaces. \n\n \\subsubsection{Convergence of a sequence of functions on varying spaces}\n For $1<p<\\infty$, we say that $f_i\\in L^p(X_i,m_i)$ converges to $f_{\\infty}\\in L^p(X_{\\infty},m_{\\infty})$ weakly if $f_im_i\\rightarrow f_{\\infty}m_{\\infty}$ weakly in $\\mathcal{M}_{loc}(X)$ and $\\limsup_{i\\rightarrow\\infty}\\Vert f_i\\Vert_{L^p(m_i)}<\\infty$. Moreover we say that $f_i\\in L^p(m_i)$ converges to $f_{\\infty}\\in L^p(m_{\\infty})$ \\emph{strongly} if $f_i\\rightarrow f_{\\infty}$ weakly and $\\limsup_{i\\rightarrow\\infty}\\Vert f_i\\Vert_{L^p(m_i)}\\leq \\Vert f_{\\infty}\\Vert_{L^p(m_{\\infty})}$. By a usual argument, we also have \n \\begin{align}\n  \\Vert f_{\\infty}\\Vert_{L^2(m_{\\infty})}\\leq \\liminf_{i\\rightarrow\\infty}\\Vert f_i\\Vert_{L^2(m_i)}\\notag\n \\end{align} \n if $f_i$ $L^2$-weakly converges to $f_{\\infty}$. We are able to define the $L^1$-strongly convergence. $f_i\\in L^1(X,m_i)$ $L^1$-strongly converges to $f\\in L^1(X,m_{\\infty})$ if $\\mathrm{sign}(f_i)\\sqrt{\\vert f_i\\vert}$ $L^2$-strongly converges to $\\mathrm{sign}(f)\\sqrt{\\vert f\\vert}$.  \n\n \\subsubsection{Convergence of a sequence of functions in $W^{1,2}$}\n From now on, we always assume $(X,d,m_i)$, $i\\in\\mathbb{N}\\cup\\{\\infty\\}$ are $\\mathsf{RCD}^*(K,N)$ spaces for $K\\in\\mathbb{R}$, $N\\in(1,\\infty)$. Note that the concepts of pmG convergence coincides with that of pmGH one in this setting. We denote by $\\mathsf{Ch}^i$ the Cheeger energy on each $L^2(m_i)$. We say that $f_i\\in W^{1,2}(m_i)$ converges to $f_{\\infty}\\in W^{1,2}(m_{\\infty})$ \\emph{weakly} if $f_i$ $L^2$-weakly converges to $f_{\\infty}$ and $\\sup_i\\mathsf{Ch}^i(f_i)<\\infty$. Moreover we say $f_i$ are \\emph{$W^{1,2}$-strongly} converges to $f_{\\infty}$ if $f_i$ $L^{2}$-strongly converges to $f_{\\infty}$ and $\\lim_{i\\rightarrow\\infty}\\mathsf{Ch}^i(f_i)=\\mathsf{Ch}^{\\infty}(f_{\\infty})$. We denote by $\\Delta_i$ the Laplacian corresponding to $\\mathsf{Ch}^i$, that is, the generator of that and by $\\mathcal{D}(\\Delta_i)\\subset W^{1,2}(m_i)$ its domain. \n\n \\begin{thm}\\label{thm:convgrad}\n  Assume $(X,d,m_i,x_i)\\xrightarrow{pmG} (X,d,m_{\\infty},x_{\\infty})$ and assume $f_i\\in \\mathcal{D}(\\Delta_i)$ $L^2$-strongly converges to $f$ and $\\sup_i\\Vert \\Delta_if_i\\Vert_{L^2(m_i)}<\\infty$, then $f\\in\\mathcal{D}(\\Delta_{\\infty})$, $\\Delta_if_i$ $L^2$-weakly converges to $\\Delta_{\\infty} f$, and $f_i$ $W^{1,2}$-strongly converges to $f$. \n \\end{thm}\n\n The following theorems play important roles.\n\n \\begin{thm}[\\cite{AH}*{Theorem 7.4}]\\label{thm:precptSobolev}\n  Assume $f_i\\in W^{1,2}(m_i)$ satisfies \n  \\begin{align}\n   \\sup_i\\int_X\\vert f_i\\vert^2\\,dm_i+\\mathsf{Ch}^i(f_i)<\\infty\\notag\n  \\end{align}\n  and for some $($hence all$)$ $\\bar x\\in X$ \n  \\begin{align}\n   \\lim_{R\\rightarrow\\infty}\\limsup_{i\\rightarrow\\infty}\\int_{X\\setminus B_R(\\bar x)}\\vert f_i\\vert^2\\,dm_i=0.\\notag\n  \\end{align}\n  Then $(f_i)$ has a $L^2$-strongly convergent subsequence to a function $f\\in W^{1,2}(m_{\\infty})$. \n \\end{thm}\n\n\n \\begin{thm}[\\cite{AH}*{Theorem 5.4, Theorem 5.6}]\\label{thm:convinnerproduct}\n  Assume that $(X_i,d_i,m_i,p_i)$ pmG converges to $(X_{\\infty},d_{\\infty},m_{\\infty},p_{\\infty})$ and assume $(X_i,d_i,m_i)$ are $\\mathsf{RCD}^*(K,N)$ spaces for $K\\in\\mathbb{R}$, $N\\in(1,\\infty)$. Then \n  \\begin{align}\n   \\lim_{i\\rightarrow\\infty}\\int_{X_i}\\Gamma^i(v_i,w_i)\\,dm_i=\\int_{X_{\\infty}}\\Gamma^{\\infty}(v, w)\\,dm_{\\infty}\n  \\end{align}\n  whenever $v_i$ strongly converges in $W^{1,2}$ to $v$ and $w_i$ weakly converges in $W^{1,2}$ to $w$. Moreover $\\Gamma^i(v_i,w_i)$ $L^1$-strongly converges to $\\Gamma^{\\infty}(v,w)$ whenever $w_i$ also strongly converges in $W^{1,2}$ to $w$.  \n \\end{thm}\n\\section{The existence of a point in a regular set}\\label{sec:main}\nLet $\\Psi(\\epsilon; a)$ denote a function depending on $a\\in\\mathbb{R}$ and tending 0 as $\\epsilon\\rightarrow 0$. The following lemma plays a key role in the proof of main results. \n\\begin{lem}[\\cite{HRicLip}*{Lemma 3.1}]\\label{lem:telescope}\n Let $Z$ be a proper geodesic space, $z\\in Z$, and $\\nu$ a Radon measure on $Z$. \n For $s,\\delta>0$, assume a nonnegative Borel function $F$ defined on $B_s(z)$ satisfies \n \\begin{align}\n  \\frac{1}{\\nu(B_s(z))}\\int_{B_s(z)}F\\,d\\nu\\leq \\delta\\notag\n \\end{align}\n and assume there exists a real number $\\kappa\\geq 1$ such that \n \\begin{align}\n  0<\\nu(B_{2t}(w))\\leq 2^\\kappa\\nu(B_t(w))\\notag\n \\end{align}\n holds for any $w\\in B_s(z)$, any $0<t\\leq s$. \n Then there exists a compact set $K\\subset \\overline{B}_{s\/10^2}(z)$ with $\\nu(K)\/\\nu(B_{s\/10^2}(z))\\geq 1-\\Psi(\\delta;\\kappa)$ such that \n \\begin{align}\n  \\frac{1}{\\nu(B_t(x))}\\int_{B_t(x)}F\\,d\\nu\\leq \\Psi(\\sqrt{\\delta};\\kappa),\\notag\n \\end{align}\n where $\\Psi(x;a_1,\\cdots,a_n)$ is a function tending to 0 as $x\\rightarrow 0$ of order $x$ and depending only on $a_1,\\cdots,a_n$. \n\\end{lem}\nRecall the local structure theorem by Mondino and Naber \\cite{MN}. Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $N\\in(1,\\infty)$. Then there exists a real number $\\beta=\\beta(N)>2$ such that the following theorem holds. \n \\begin{thm}[\\cite{MN}*{Theorem 4.1}]\\label{thm:MNexcess}\n  Take a point $\\bar x\\in \\mathcal{R}_k$ and fix it. For any sufficiently small $\\epsilon>0$, there exists a large number $\\bar R=\\bar R(\\epsilon)\\gg 1$ such that for any $R\\geq \\bar R$, there exists a small number $0<r=r(\\bar x,\\epsilon, R)\\ll 1$ the following holds; there exist pairs of points $\\{p_i,q_i\\}_{i=1,\\ldots,k}\\subset B^{d_r}_{R^{\\beta}}(\\bar x)$ and $\\{p_i+p_j\\}_{1\\leq i<j\\leq k}\\subset B^{d_r}_{2R^{\\beta}}(\\bar x)\\setminus B^{d_r}_{R^{\\beta}}(\\bar x)$ such that \n  \\begin{align}\n   &\\sum_{i=1}^k\\Xint-_{B^{d_r}_R(\\bar x)}\\vert \\nabla e_{p_i,q_i}\\vert^2\\,dm^{\\bar x}_r+\\sup_{y\\in B^{d_r}_R(\\bar x)}e_{p_i,q_i}(y)\\leq \\epsilon\\label{eq:derexcess}\\\\\n   &\\sum_{1\\leq i<j\\leq k}\\Xint-_{B^{d_r}_R(\\bar x)}\\left\\vert \\nabla \\left(\\frac{d_r^{p_i}+d_r^{p_j}}{\\sqrt2}-d_r^{p_i+p_j}\\right)\\right\\vert^2\\,dm^{\\bar x}_r+\\sup_{B^{d_r}_R(\\bar x)}\\left\\vert \\frac{d_r^{p_i}+d_r^{p_j}}{\\sqrt 2}-d_r^{p_i+p_j}\\right\\vert\\leq \\epsilon, \\label{eq:derindep}\n  \\end{align}\n  where $d_r^{p_i}(\\cdot):=d_r(p_i,\\cdot)$. \n \\end{thm}\nFrom now on, without loss of generality, we assume the following. \n\\begin{assump}\n Let $(X,d,m)$ be an $RCD^*(K,N)$ space and $\\bar x\\in \\mathcal{R}_k$. There exist pairs of points $\\{p_i,q_i\\}_{i=1,\\ldots,k}\\subset B_{R^{\\beta}}(\\bar x)$ and $\\{p_i+p_j\\}_{1\\leq i<j\\leq k}\\subset B_{2R^{\\beta}}(\\bar x)\\setminus B_{R^{\\beta}}(\\bar x)$ such that (\\ref{eq:derexcess}) and (\\ref{eq:derindep}) holds for sufficiently small $\\epsilon>0$ and for replacing $d_r$ and $m^{\\bar x}_r$ by just $d$ and $m$ respectively. \n\\end{assump}\nIt is known that $\\vert \\nabla d^{p}\\vert\\equiv 1$ $m$-a.e.(see \\cite{GP}*{Proposition 3.1}). By using the calculation in \\cite{MN,GP}, we have \n\\begin{align}\n \\vert\\Gamma(d^{p_i},d^{p_j})\\vert\\leq 10\\left\\vert \\nabla\\left(\\frac{d^{p_i}+d^{p_j}}{\\sqrt 2}-d^{p_i+p_j}\\right)\\right\\vert^2.\\notag\n\\end{align}\nThus we obtain that \n\\begin{align}\n \\Xint-_{B_R(\\bar x)}\\left\\vert \\Gamma(d^{p_i},d^{p_j})-\\delta_{ij}\\right\\vert\\,dm\\leq 10\\epsilon.\\notag\n\\end{align}\nDefine a function $f_i:=d^{p_i}\\varphi^{\\bar x}_R$ such that $f_i\\in W^{1,2}(X,d,m)$ with bounded support, where $\\varphi^{\\bar x}_R$ is a smooth cut-off function introduced in subsection \\ref{subsection:cutoff}. Then we have \n\\begin{align}\n \\Xint-_{B_R(\\bar x)}\\vert \\Gamma(f_i,f_j)-\\delta_{ij}\\vert\\,dm\\leq 10\\epsilon.\\notag \n\\end{align}\nLet us consider smoothing functions $g^i:=\\hat h_t f_i$, $i=1,\\ldots, k$ such that \n\\begin{align}\n \\Xint-_{B_R(\\bar x)}\\vert \\Gamma(g^i,g^j)-\\delta_{ij}\\vert\\,dm\\leq 20\\epsilon.\\notag\n\\end{align} \nLet $A$ be a subset defined as \n\\begin{align}\n A:=\\left\\{y\\in B_{R\/10^2}(\\bar x)\\;;\\;\\Xint-_{B_s(y)}\\left\\vert \\Gamma(g^i,g^j)-\\delta_{ij}\\right\\vert\\,dm\\leq\\Psi(\\sqrt{\\epsilon};K,N)\\quad\\text{for any $s<R\/10^2$}\\right\\}.\\label{eq:effective}\n\\end{align} \nBy Lemma \\ref{lem:telescope}, $m(A)\/m(B_{R\/10^2}(\\bar x))\\geq 1-\\Psi(\\sqrt{\\epsilon};K,N)$, in particular $m(A)>0$. Since $m(X\\setminus \\mathcal{R})=0$, we have $m(A\\cap \\mathcal{R})>0$. Take a point $y\\in A\\cap \\mathcal{R}$ and assume $y\\in \\mathcal{R}_l$ for some $1\\leq l\\leq N$. Without loss of generality, $B_s(y)\\subset B_R(\\bar x)$. Note that, for such $y$, we have \n\\begin{align}\n \\Xint-_{B^{d_s}_1(y)}\\vert \\Gamma(g^i,g^j)-\\delta_{ij}\\vert\\,dm^y_s\\leq\\Psi(\\sqrt{\\epsilon};K,N).\\notag\n\\end{align} \nSince $f^i$ is a Lipschitz function, $g^i\\in\\mathrm{TestF}(X)$ and $\\Delta g^i\\in L^{\\infty}(m)$.  \nLet us consider the effect of rescaling of metric to the Cheeger energy, Laplacian, and the Gamma operator. \n\\begin{lem}\\label{lem:rescaleeffect}\n Denote the Cheeger energy, Laplacian, heat semigroup, Gamma operator in the rescaled space $(X,d_s,m)$ by $\\mathsf{Ch}^{d_s}$, $\\Delta^{d_s}$, $\\{h_t^{d_s}\\}_{t>0}$, and $\\Gamma^{d_s}$ respectively. Then \n \\begin{align}\n  \\mathsf{Ch}^{d_s}=s^2\\mathsf{Ch},\\quad \\Delta^{d_s}h^{d_s}_t=s^2\\Delta h_{s^2t}, \\quad \\Gamma^{d_s}=s^2\\Gamma.\\notag\n \\end{align}\n\\end{lem}\n\\begin{proof}\n Since \n \\begin{align}\n  \\frac{\\vert f(x)-f(y)\\vert}{d_s(x,y)}=s\\frac{\\vert f(x)-f(y)\\vert}{d(x,y)}\\notag\n \\end{align}\n holds, it is easy to get $\\vert \\nabla^{d_s}f\\vert=s\\vert \\nabla f\\vert$. By the definition of $\\mathsf{Ch}$, we have $\\mathsf{Ch}^{d_s}=s^2\\mathsf{Ch}$. A similar calculation let us obtain $\\Gamma^{d_s}=s^2\\Gamma$. For an appropriate function $f$, consider the heat equation $\\partial_th_t^{d_s}f=\\Delta^{d_s}h^{d_s}_tf$. Since $h_{s^2t}f$ satisfies \n \\begin{align}\n  \\partial_th_{s^2t}f=s^2\\Delta h_{s^2t}f=\\Delta^{d_s}h_{s^2t}f\\notag\n \\end{align}\n and $\\lim_{t\\rightarrow+0}h_{s^2t}f=f$ in $L^2$ sense. By the uniqueness of the heat flow, we obtain $h^{d_s}_tf=h_{s^2t}f$. Hence $\\Delta^{d_s}h^{d_s}_t=s^2\\Delta h_{s^2t}$ holds for any $t>0$. \n\\end{proof}\nHence \n\\begin{align}\n \\Xint-_{B^{d_s}_1(y)}\\vert \\Gamma^{d_s}(s^{-1}g^i,s^{-1}g^j)-\\delta_{ij}\\vert\\,dm^y_s\\leq \\Psi(\\sqrt{\\epsilon};K,N)\\notag\n\\end{align}\nholds. Define new functions $k^i_s:=(g^i_s-g^i_s(y))\\varphi_1^{d_s,y}:=s^{-1}(g^i-g^i(y))\\varphi^{d_s,y}_1$, $i=1,\\ldots,k$. \n\\begin{prop}\\label{prop:kfunctions}\n There exists a limit function $K^i$ on $\\mathbb{R}^l\\in \\mathsf{Tan}(X,y)$ for each $i$ such that $K^i\\in\\mathcal{D}(\\Delta)$ and $k^i_s$ converges to $K^i$ in $W^{1,2}$-strongly and $\\Delta^{d_s}k^i_s$ converges to $\\Delta K^i$ in $L^2$-weakly. \n\n\\end{prop}\n\\begin{proof}\n\\par \\underline{Step 1: The functions $k^i_s$ belong $W^{1,2}(m^y_s)$. } Let us calculate the $L^2(m^y_s)$-norm of $k^i_s$. We denote by $D$ the normalized constant so that $Dm=m^y_s$ for abbreviation. Since the heat flow preserves the $L^2$-norm of a function, note that $\\Vert \\hat h_t f\\Vert_{L^2(m)}\\leq \\Vert f\\Vert_{L^2(m)}$.   \n\\begin{align}\n \\int_X\\vert k^i_s\\vert^2\\,dm^y_s&=D\\int_{B^{d_s}_2(y)}\\vert (g^i_s-g^i_s(y))\\varphi^{d_s,y}_1\\vert^2\\,dm\\label{eq:lem:indepharm}\\\\\n &\\leq D\\int_{B^{d_s}_2(y)}\\vert \\hat{h}_t(s^{-1}f_i-s^{-1}f_i(y))\\vert^2\\,dm\\notag\\\\\n &\\leq D\\int_{B^{d_s}_2(y)}\\vert (d_s^{p_i}-d_s^{p_i}(y))\\vert^2\\,dm\\notag\\\\\n &\\leq 4m^y_s(B^{d_s}_2(y))<\\infty.\\notag\n\\end{align}\nAlso by the locality property of Dirichlet form $\\mathsf{Ch}$, we obtain \n\\begin{align}\n \\int_X\\vert\\nabla^{d_s}k^i_s\\vert^2\\,dm^y_s\n &=\\int_X\\Gamma^{d_s}(k^i_s,k^i_s)\\,dm^y_s\\notag\\\\\n &=\\int_{B^{d_s}_2(y)}\\Gamma^{d_s}(k^i_s,k^i_s)\\,dm^y_s\\notag.\n\\end{align}\nBy the Leibniz rule and the locality property, we have $\\vert\\nabla^{d_s}k^i_s\\vert\\leq \\vert(g^i_s-g^i_s(y))\\vert\\vert\\nabla^{d_s} \\varphi^{d_s,y}_1\\vert+\\varphi^{d_s,y}_1\\vert \\nabla^{d_s} g^i_s\\vert$. Hence \n\\begin{align}\n &\\int_{B^{d_s}_2(y)}\\Gamma^{d_s}(k^i_s,k^i_s)\\,dm^y_s\\leq 2\\int_{B^{d_s}_2(y)}\\vert g^i_s-g^i_s(y)\\vert^2\\vert\\nabla^{d_s}\\varphi^{d_s,y}_1\\vert^2+(\\varphi^{d_s,y}_1)^2\\vert \\nabla^{d_s}g^i_s\\vert^2\\,dm^y_s\\notag\\\\\n &\\leq 8C^2m^y_s(B^{d_s}_2(y))+2\\int_{B^{d_s}_2(y)}\\Gamma^{d_s}(g^i_s,g^i_s)\\,dm^y_s\\notag\\\\\n &=8C^2m^y_s(B^{d_s}_2(y))+2D\\int_{B^{d_s}_2(y)}\\Gamma(g^i,g^i)\\,dm\\notag\\\\\n &\\leq 8C^2m^y_s(B^{d_s}_2(y))+2De^{K_-t}\\int_{B^{d_s}_2(y)}\\Gamma(f^i,f^i)\\,dm\\notag\\\\\n &\\leq 8C^2m^y_s(B^{d_s}_2(y))+2e^{K_-t}m^y_s(B^{d_s}_2(y))<\\infty\\notag\n\\end{align}\nholds. \n\\smallskip\n\\par \\underline{Step 2: The existence of limit functions.} By Step 1, we obtain \n\\begin{align}\n &\\Vert k^i_s\\Vert^2_{L^2(m^y_s)}\\leq 4m^y_s(B^{d_s}_2(y))\\notag\\\\\n &2\\mathsf{Ch}^{d_s}(k^i_s)\\leq (8C^2+2e^{K_-t})m^y_s(B^{d_s}_2(y)).\\notag\n\\end{align}\nSince $y\\in \\mathcal{R}_l$, $\\limsup_{s\\rightarrow 0}m^y_s(B^{d_s}_2(y))<\\infty$. Therefore \n\\begin{align}\n \\sup_{s\\in (0,1)}\\left(\\Vert k^i_s\\Vert^2_{L^2(m^y_s)}+\\mathsf{Ch}^{d_s}(k^i_s)\\right)\\leq (4C^2+e^{K_-t}+4)\\sup_{s\\in(0,1)}m^y_s(B^{d_s}_2(y))<\\infty.\\notag \n\\end{align}\nMoreover by (\\ref{eq:lem:indepharm}), \n\\begin{align}\n \\int_{X\\setminus B^{d_s}_R(y)}\\vert k^i_s\\vert^2\\,dm^y_s=0\\notag\n\\end{align}\nwhenever $R\\geq 3$. Thus applying Theorem \\ref{thm:precptSobolev} leads the existence of a convergent subsequence of $\\{k^i_s\\}_{s\\in (0,1)}$ and the limit function $K^i\\in W^{1,2}(\\mathbb{R}^l)$ for each $i$ such that $k^i_s$ $L^2$-strongly converges to $K^i$. \n\\end{proof}\n\\begin{prop}\\label{prop:lk}\n $l\\geq k$. \n\\end{prop}\n\\begin{proof}\n Recall that we have limit functions $K^i$, $i=1,\\ldots,k$ in $\\mathbb{R}^l$. By Theorem \\ref{thm:convinnerproduct}, we have \n \\begin{align}\n  &\\Xint-_{B_1(0)}\\vert\\Gamma(K^i,K^i)-1\\vert\\,d\\underline{\\mathcal{L}^l}\\leq \\Psi(\\sqrt{\\epsilon};K,N)\\label{eq:almost1}\\\\\n  &\\Xint-_{B_1(0)}\\vert\\Gamma(K^i,K^j)\\vert\\,d\\underline{\\mathcal{L}^l}\\leq \\Psi(\\sqrt{\\epsilon};K,N)\\;\\quad\\text{for }i\\neq j.\\label{eq:almostperpendicular}\n \\end{align}\n (\\ref{eq:almost1}) implies the non-triviality of each $K^i$. Again applying Lemma \\ref{lem:telescope} guarantees the existence a measurable subset $B\\subset B_1(0)$ of positive measure such that for any $p\\in B$, \n \\begin{align}\n  \\Xint-_{B_s(p)}\\vert \\Gamma(K^i,K^j)\\vert\\,d\\underline{\\mathcal{L}^l}\\leq \\Psi(\\epsilon^{1\/4};K,N)\\notag\n \\end{align} \n holds for any $s\\in(0,1)$. Since $\\mathcal{L}^l(B)>0$, at almost every point $p\\in B$, we obtain \n \\begin{align}\n  \\Gamma(K^i,K^j)(p)=\\lim_{s\\rightarrow 0}\\Xint-_{B_s(p)}\\Gamma(K^i,K^j)\\,d\\underline{\\mathcal{L}^l}\\leq \\Psi(\\epsilon^{1\/4};K,N)\\notag\n \\end{align}\n by Lebesgue's differentiation theorem. Hence $\\{\\nabla K^i(p)\\}_{i=1}^k$ is linearly independent in $T_p\\mathbb{R}^l\\cong \\mathbb{R}^l$. Therefore $l$ should be at least $k$. \n\\end{proof}\nThe following corollary is easy to prove. \n\\begin{cor}\n Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$, $N\\in\\mathbb{N}$. Assume $\\mathcal{R}_N\\neq \\emptyset$. Then $\\mathrm{dim}_H (X,d)=N$. \n\\end{cor}\n\\begin{proof}\n By Propositions \\ref{prop:kfunctions}, \\ref{prop:lk}, we have $m(\\mathcal{R}_N)>0$. In \\cite{GP,KeM,DPMR}, they prove that on $\\mathcal{R}_N$, $m\\sim\\mathcal{H}^N$. Thus $\\mathrm{dim}_H(X,d)\\geq N$. On the other hand, Sturm proved $\\mathrm{dim}_H(X,d)\\leq N$ in \\cite{Stmms2}. We have the conclusion.  \n\\end{proof}\n\\section{Dimension on $\\mathsf{RCD}$ spaces}\n We define a version of dimension of an $\\mathsf{RCD}$ space. \n\n \\begin{defn}\\label{def:dimmm}\n  Let $\\mathcal{X}:=(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$. The dimension of $(X,d,m)$ is the largest number $k$ so that $\\mathcal{R}_k\\neq\\emptyset$ and $\\mathcal{R}_l=\\emptyset$ for any $l>k$. For simplicity, $\\mathsf{dim}\\, \\mathcal{X}$ denotes the dimension of $(X,d,m)$. \n \\end{defn}\n\n\n \\begin{rem}\\label{rem:positivity}\n  By Theorem \\ref{thm:posi}, $\\mathsf{dim}\\, \\mathcal{X}=k$ implies $m(\\mathcal{R}_k)>0$. In general, $\\mathsf{dim}\\, \\mathcal{X}\\leq \\mathrm{dim}_H(X,d)$. Even in the case of collapsing Ricci limit spaces, we do not know the equality holds or not. \n \\end{rem}\n\n \\begin{rem}\n  Let $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ be $\\mathsf{RCD}^*(K,N)$ spaces for $K\\in\\mathbb{R}$, $N\\in(1,\\infty)$. Assume that $(X,d_X,m_X)$ and $(Y,d_Y,m_Y)$ are isomorphic to each other. Then by the definition, $\\mathsf{dim}\\, (X,d_X,m_X)=\\mathsf{dim}\\, (Y,d_Y,m_Y)$ holds. Moreover it follows that $\\mathsf{dim}\\, (X,d,m_1)=\\mathsf{dim}\\, (X,d,m_2)$ if $(X,d,m_1)$ and $(X,d,m_2)$ are $\\mathsf{RCD}^*(K,N)$ spaces and $\\mathrm{supp}\\, m_1=\\mathrm{supp}\\, m_2=X$. Indeed, it suffices to prove that $\\mathsf{dim}\\, \\mathcal{X}_1\\leq \\mathsf{dim}\\, \\mathcal{X}_2$ by the symmetry of the condition, where $\\mathcal{X}_i=(X,d,m_i)$. Put $k=\\mathsf{dim}\\, \\mathcal{X}_1$ and denote by $\\mathcal{R}^i_l$ the $l$-dimensional regular set with respect to $\\mathcal{X}_i$. Take a point $x\\in\\mathcal{R}^1_k$ and fix it. Since $(X,d_r,(m_1)^x_r,x)\\rightarrow (\\mathbb{R}^k,d_{\\mathbb{R}^k},\\underline{\\mathcal{L}}^k,0)$ in the measured Gromov-Hausdorff sense implies $(X,d_r,x)\\rightarrow (\\mathcal{R}^k,d_{\\mathbb{R}^k},0)$ in the Gromov-Hausdorff one, we have that the set of metric tangent cones at $x$ is unique and isometric to $(\\mathbb{R}^k,d_{\\mathbb{R}^k},0)$. Hence $\\mathsf{Tan}(X,m_2,x)=\\{(\\mathbb{R}^k,d_{\\mathbb{R}^k},\\nu,0)\\;;\\;\\nu\\text{ is a limit measure on }\\mathbb{R}^k\\}$, where a limit measure means $(m_2)^x_{r_i}\\rightarrow \\nu$, for a non-increasing sequence $\\{r_i\\}$ tending to 0, in the sense of pointed measured Gromov convergence. It remains to prove that $\\nu$ must be $\\underline{\\mathcal{L}}^k$, that is, $x\\in\\mathcal{R}^2_k$. However, $(\\mathbb{R}^k,d_{\\mathbb{R}^k},\\nu)$ is an $\\mathsf{RCD}^*(0,N)$ space with $k$-th straight lines. Thus the splitting theorem \\cite{Gsplit} tells us that $\\nu=\\underline{\\mathcal{L}}^k$. This completes the proof. \n \\end{rem}\n\n Before stating the main result in this section, we give the following definitions and results developed by Ambrosio and Honda \\cite{AHlocal}. For an open set $A\\subset X$, $\\mathrm{LIP}_c(A,d)$ denotes the set of all Lipschitz functions whose supports are compact and contained in $A$. \n\n \\begin{defn}\n  Let $U$ be an open subset in $X$. \n  \\begin{enumerate}\n   \\item Define $W^{1,2}_0(U,d,m):=\\overline{\\mathrm{Lip}_c(U,d)}^{\\Vert\\cdot\\Vert_{W^{1,2}}}$.\n   \\item Define $\\hat{W}_0^{1,2}(U,d,m):=\\{f\\in W^{1,2}\\;;\\;f=0\\;m\\text{-a.e. on }X\\setminus U\\}$.\n  \\end{enumerate}\n \\end{defn}\n\n  It follows that $W^{1,2}_0(B_R(x),d,m)=\\hat W_0^{1,2}(B_R(x),d,m)$ for any $x\\in X$ and for every $R>0$ excepting at most countably many positive numbers (see \\cite{AHlocal}*{Lemma 2.12}). \n\n  Analogously \\emph{the local Cheeger energy} is defined as \n  \\begin{align}\n   \\mathsf{Ch}_{(x,R)}(f):=\\begin{cases}\n                            \\mathsf{Ch}(f)&\\text{if }f\\in W^{1,2}_0(B_R(x),d,m)\\\\\n                            +\\infty&\\text{otherwise}.\n                           \\end{cases}\\notag\n  \\end{align}\n\n Assume $(X,d_n,m_n,x_n)\\rightarrow(X,d_{\\infty},m_{\\infty},x_{\\infty})$ and $\\mathrm{supp}\\, m_n\\ni z_n\\rightarrow z\\in\\mathrm{supp}\\, m_{\\infty}$ in the pmG sense. The \\emph{Mosco convergence} for local Cheeger energies, denoted by \n \\begin{align}\n  \\mathsf{Ch}^n_{\\mathsf{loc}}\\rightarrow \\mathsf{Ch}^{\\infty}_{\\mathsf{loc}}\\quad\\text{at }(z,R)\\in X\\times R,\\notag\n \\end{align}\n is defined as follows; \n \\begin{enumerate}\n  \\item For every $f_n\\in L^2(B_{R_n}(z_n),m_n)$ $L^2$-weakly converging to $f\\in L^2(B_R(z),m_{\\infty})$, \n  \\begin{align}\n   \\mathsf{Ch}^{\\infty}_{(z,R)}(f)\\leq \\liminf_{n\\rightarrow\\infty}\\mathsf{Ch}^n_{(z_n,R_n)}(f_n)\\notag\n  \\end{align}\n  holds, where $z_n\\rightarrow z$ and $R_n\\rightarrow R$. \n  \\item For every $f\\in L^2(B_R(z),m_{\\infty})$ there exist $f_n\\in L^2(B_{R_n}(z_n),m_n)$ $L^2$-strongly converging to $f$ such that \n  \\begin{align}\n   \\mathsf{Ch}^{\\infty}_{(z,R)}(f)=\\lim_{n\\rightarrow\\infty}\\mathsf{Ch}^n_{(z_n,R_n)}(f_n)\\notag\n  \\end{align}\n  holds, where $R_n$ and $z_n$ are the same as above. \n \\end{enumerate}\n\n \\begin{thm}[\\cite{AHlocal}*{Theorem 3.4}]\\label{thm:localMosco}\n  The following are equivalent; \n  \\begin{enumerate}\n   \\item $\\mathsf{Ch}^n_{\\mathsf{loc}}\\rightarrow\\mathsf{Ch}^{\\infty}_{\\mathsf{loc}}$ at $(z,R)$.\n   \\item $W^{1,2}_0(B_R(z),d,m)=\\hat W_0^{1,2}(B_R(z),d_{\\infty},m_{\\infty})$.\n  \\end{enumerate}\n \\end{thm}\n\n We define the new Sobolev space as follows. \n\n \\begin{defn}\n  $W^{1,2}(B_R(z),d,m)$ consists of $f\\in L^2(B_R(z),m)$ those which satisfy the following two conditions; \n  \\begin{enumerate}\n   \\item $\\phi f\\in W^{1,2}(X,d,m)$ for any $\\phi\\in\\mathrm{LIP}_c(B_R(z))$,\n   \\item $\\Gamma(f)\\in L^1(B_R(z))$.\n  \\end{enumerate}\n \\end{defn}\n\n We also define the local $W^{1,2}$-convergence. \n\n \\begin{defn}\n  Let $f_n\\in W^{1,2}(B_R(z_n),d_n,m_n)$ and $f\\in W^{1,2}(B_R(z),d_{\\infty},m_{\\infty})$. We say that $f_n$ \\emph{$W^{1,2}$-weakly converges} to $f$ on $B_R(z)$ if $f_n$ $L^2$-weakly converges to $f$ on $B_R(z)$ with $\\sup_n\\Vert f_n\\Vert_{W^{1,2}}<\\infty$. Furthermore, the \\emph{$W^{1,2}$-strongly convergence} on $B_R(z)$ is defined by requiring the $W^{1,2}$-weakly convergence on $B_R(z)$ and $\\lim_n\\Vert\\Gamma_n(f_n)\\Vert_{L^1(B_R(z_n))}=\\Vert\\Gamma(f)\\Vert_{L^1(B_R(z))}$. \n \\end{defn}\n\n The following Theorem plays a key role in the main result in this section. \n\n  \\begin{thm}[\\cite{AHlocal}*{Corollary 4.3}]\\label{thm:localsum}\n   Let $f_n,g_n\\in W^{1,2}(B_R(z_n),d_n,m_n)$ be $W^{1,2}$-strongly convergent sequences to $f,g\\in W^{1,2}(B_R(z),d_{\\infty},m_{\\infty})$ on $B_R(z)$ respectively. Then \n   \\begin{align}\n    \\lim_{n\\rightarrow\\infty}\\int_{B_R(z_n)}\\Gamma^n(f_n,g_n)\\,dm_n=\\int_{B_R(z)}\\Gamma^{\\infty}(f,g)\\,dm_{\\infty}.\\notag\n   \\end{align} \n  \\end{thm}\n\n \\begin{rem}\\label{rem:sumLp}\n  For two sequences $f_i,g_i$ $L^p$-strongly converging to $f,g$ respectively, the sum of these functions $f_i+g_i$ also $L^p$-strongly converges to $f+g$ for $p\\in[1,\\infty)$. See \\cite{HRicLp} for the proof. In \\cite{HRicLp}, the definition of $L^p$-convergence looks different from that in this paper, but it is actually equivalent. The proof of equivalence is also found in the same article. \n \\end{rem}\n \n \n\n \n \n \n \n \n \n\n\n\n\n\n\n\n\n\n Combining Theorem \\ref{thm:localMosco} and Theorem \\ref{thm:convinnerproduct} leads the following. \n\n  \\begin{thm}\\label{thm:lscdimmm}\n   Let $\\mathcal{X}_n:=(X,d,m_n)$, $x_n\\in\\mathrm{supp}\\, m_n$ $n\\in\\mathbb{N}\\cup\\{\\infty\\}$ be $\\mathsf{RCD}^*(K,N)$ spaces for $K\\in\\mathbb{R}$ and $N\\in(1,\\infty)$ and assume $(\\mathcal{X}_n,x_n)$ converges to $(\\mathcal{X}_{\\infty},x_{\\infty})$ in the pmG sense. Then \n   \\begin{align}\n    \\mathsf{dim}\\, \\mathcal{X}_{\\infty}\\leq\\liminf_{n\\rightarrow\\infty}\\mathsf{dim}\\,\\mathcal{X}_n.\\notag\n   \\end{align}\n  \\end{thm}\n\n \\begin{proof}\n  Let $k$ be the dimension of $\\mathcal{X}_{\\infty}$. By Remark \\ref{rem:positivity}, we have $m_{\\infty}(\\mathcal{R}_k)>0$. Then for any $\\epsilon>0$, there exist a subset $A\\subset\\mathcal{R}_k$ with $m_{\\infty}(A)>0$ and a family of functions $\\{g^i\\}_{i=1}^k$ such that \n  \\begin{align}\n   \\Xint-_{B_s(y)}\\vert\\Gamma^{\\infty}(g^i,g^j)-\\delta_{ij}\\vert\\,d(m_{\\infty})^y_s<\\epsilon\\notag\n  \\end{align}\n  holds for any $s\\in(0,1)$ and $y\\in A$, after rescaling the metric and the measure if needed. By Theorem \\ref{thm:localMosco}, there exist a positive number $s\\in(0,1)$, functions $g^i_n\\in L^2(B_{s_n}(y_n))$ $L^2$-strongly converging to $g^i$ such that \n  \\begin{align}\n   \\mathsf{Ch}_{(y,s)}(g^i)=\\lim_{n\\rightarrow \\infty}\\mathsf{Ch}_{(y_n,s_n)}(g^i_n)\\notag\n  \\end{align} \n  holds for $i=1,\\ldots,k$. Thus we have \n  \\begin{align}\n   \\lim_{n\\rightarrow\\infty}\\Xint-_{B_s(y_n)}\\Gamma^n(g^i_n,g^j_n)\\,dm_n=\\Xint-_{B_s(y)}\\Gamma^{\\infty}(g^i,g^j)\\,dm_{\\infty}<\\epsilon\\label{eq:small1}\n  \\end{align}\n  for $i\\neq j$ by Theorem \\ref{thm:localsum}. Note that by the Bishop-Gromov inequality, $m_n(B)\\rightarrow m_{\\infty}(B)$ for any metric ball $B$. \n  Since it is also clear that $\\chi_{B_1(y)}$ $W^{1,2}$-strongly converges to itself, \n  \\begin{align}\n   \\lim_{n\\rightarrow\\infty}\\Xint-_{B_{s}(y_n)}\\vert \\Gamma^n(g^i_n,g^j_n)-1\\vert\\,dm_n=\\Xint-_{B_s(y)}\\vert\\Gamma^{\\infty}(g^i,g^j)-1\\vert\\,dm_{\\infty}\\label{eq:small2}\n  \\end{align} \n  holds by Remark \\ref{rem:sumLp}. The same argument as in Section \\ref{sec:main} leads $m_n(\\mathcal{R}_k)>0$. Therefore we have $\\mathsf{dim}\\, \\mathcal{X}_n\\geq k$ for sufficiently large $n$. This completes the proof. \n \\end{proof}\n\n  The following is an easy consequence from Theorem \\ref{thm:lscdimmm}\n\n \\begin{cor}\n  Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space of dimension $k$. And define a set $\\underline{\\mathcal{WE}}_i$ as \n  \\begin{align}\n   \\underline{\\mathcal{WE}}_i:=\\left\\{x\\in X\\;;\\;\\mathbb{R}^i\\times Z\\in\\mathsf{Tan}(X,x)\\text{ with }\\mathrm{Diam}\\, Z>0\\right\\}.\\notag\n  \\end{align}\n  Then $\\underline{\\mathcal{WE}}_k=\\emptyset$. \n \\end{cor}\n\n \\begin{rem}\\label{rem:analydim}\n Recently, the study of a version of differential structure on metric measure spaces is developed by Gigli \\cite{Gnon}. Roughly speaking, the $L^2$-sections of cotangent bundles over a metric measure space are considered. Hence the dimension of such spaces makes sense. Indeed, Han define the \\emph{analytic dimension} on a metric measure space with such a differential structure \\cite{HanRictensor}. Combining our result and Theorem 3.3 in \\cite{GPequiv} leads the equivalence of analytic dimension and our dimension defined in this section. \n \\end{rem}\n\n \\begin{rem}\n  After finishing this paper, the author find the paper by De Philippis and Gigli on arXiv \\cite{DPG}. They introduce a notion of \\emph{non-collapsed $\\mathsf{RCD}$ spaces} and study that. Moreover they study much about the Hausdorff dimension not only non-collapsed $\\mathsf{RCD}$ spaces but also the usual ones. They prove that $\\dim_H(X,d)\\leq N-1$ for $\\mathsf{RCD}(K,N)$ spaces with $N\\in\\mathbb{N}$ unless $\\dim_H(X,d)=N$. Combining their result with ours leads the following; Let $(X,d,m)$ be an $\\mathsf{RCD}^*(K,N)$ space for $N\\in\\mathbb{N}$. Then $\\dim (X,d,m)=N-1$ if and only if $\\dim_H(X,d)=N-1$. \n \\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nThe author would like to thank Professor Shouhei Honda for their helpful comments and fruitful discussions. He is supported by Grant-in-Aid for Young Scientists (B) 15K17541. \n\\input{citeforstra}\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}