diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdjal" "b/data_all_eng_slimpj/shuffled/split2/finalzzdjal" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdjal" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\t\\IEEEPARstart{U}{NMANNED} aerial vehicles (UAVs) have recently gained significant interest in a broad range of applications. High mobility and flexible deployment are among the features allowing the UAVs to expand their scope of action \\cite{RZhang_Survey_Accessing, Azari_Survey, Rahmati_TWC, Behzad_PIMRC_RL}. For effective operation, in many of the applications, the UAVs need to maintain a reliable internet connection during their flights. This connection is mainly provided by ground base stations (GBSs). However, due to location-dependant and time-varying characteristics of the communication channels, and the fact that the cellular networks are designed to serve terrestrial users, service is not available in all parts of the sky \\cite{RZhang_cellular_connected_Tcom}. A cost-effective solution to bring this connectivity to the aerial users is to design appropriate trajectories for the UAVs.\n\t\n\t\n\tThe trajectory optimization problem for cellular-connected UAVs has been extensively studied \\cite{RZhang_cellular_connected_Tcom,Guevenc_ICC,Sensing_CommLeter,Behzad_PIMRC_Optimization,Saad_TWC_cellularConnected, RZhang_3Dmap_Globecom}. In \\cite{RZhang_cellular_connected_Tcom}, the authors studied the trajectory design problem for a cellular-connected UAV, and based on graph theory and convex optimization, they proposed an algorithm to minimize the mission time of the UAV. This minimization is subject to a maximum tolerable outage duration. In \\cite{Guevenc_ICC}, by using dynamic programming, the authors proposed a sub-optimal trajectory design algorithm to minimize the flight time of a cellular-connected UAV. In \\cite{Sensing_CommLeter}, the authors proposed an algorithm to optimize the UAV trajectory with the goal of maximizing its energy efficiency. In \\cite{Behzad_PIMRC_Optimization}, the authors minimized the propulsion-power consumption of a fixed-wing UAV while a certain connectivity constraint on the instantaneous signal to interference plus noise ratio (SINR) is satisfied. In \\cite{Saad_TWC_cellularConnected}, the authors studied the interference management problem for the uplink communication of cellular-connected UAVs. The authors in \\cite{Saad_TWC_cellularConnected} obtained the trajectory of the UAVs to achieve a tradeoff between maximizing energy efficiency and minimizing the interference caused on the terrestrial cellular network. \n\t\n\t\n\t\n\t\n\tThe studies in \\cite{RZhang_cellular_connected_Tcom,Guevenc_ICC,Sensing_CommLeter,Behzad_PIMRC_Optimization,Saad_TWC_cellularConnected, RZhang_3Dmap_Globecom} optimize the trajectory of the UAVs subject to a certain connectivity constraint on the received signal. However, due to the channel randomness, there is no guarantee to satisfy these certain constraints in practice. To take the channel randomness into account, we have to consider probabilistic connectivity constraints. These constraints can be defined in terms of the outage probability and give more reliability to the path planning algorithms. In addition to this concern, the proposed algorithms in \\cite{RZhang_cellular_connected_Tcom,Guevenc_ICC,Sensing_CommLeter,Behzad_PIMRC_Optimization,Saad_TWC_cellularConnected, RZhang_3Dmap_Globecom} require instantaneous channel state information. By considering a probabilistic connectivity metric, we can design algorithms that work without having this instantaneous information. To achieve this goal, the UAVs need to have a global model of the outage probability in the environment. Since each UAV has its specific task, and depending on the task, it flies over a particular area, it can not build the global model using its own experience. In contrast, the UAVs need to work together to build this global model.\n\n\n\n\tTo address the issues mentioned above, in this paper, we study the radio mapping and path planning problem for a cellular-connected UAV network. Since the onboard energy of the UAVs is limited, the UAVs can not fly for a long time. To prolong the lifetime of the UAVs, we minimize the flight time of each UAV, ensuring that a probabilistic connectivity constraint is satisfied throughout their flights. This problem is non-convex and challenging to solve. We first reformulate the problem into a mathematically more tractable form. Then, to solve the reformulated problem, we propose a two-step algorithm. In the first step, by using Federated Learning (FL) \\cite{FL2, Federated_Google}, the UAVs collaboratively build a global model of the outage probability in the environment. In this method, each UAV uses its data to update the global model locally. Accordingly, the UAVs do not need to share their collected information with a centralized node to do the training task, and the training process is executed in a distributed manner. As a result, the training process will be faster. \n\tIn the second step, we use the resulting global model to design the trajectory of the UAVs. To achieve this goal, we have to ensure that the connectivity constraint is satisfied. We propose a path planning algorithm based on rapidly-exploring random trees. In our trajectory design algorithm, we do not need the instantaneous channel gains, which was considered in previous works \\cite{RZhang_cellular_connected_Tcom,Guevenc_ICC,Sensing_CommLeter,Behzad_PIMRC_Optimization,Saad_TWC_cellularConnected, RZhang_3Dmap_Globecom}. Moreover, our algorithm allows the UAVs to update the model based on their newly collected data. As a result, the model is trained based on online status of the network, which can remarkably increase the model's accuracy. The simulation results show the effectiveness of this two-step approach for UAV networks. \n\t\n\nThe rest of the paper is organized as follows. In section II, we present the system model and problem formulation. In section III, we propose a two-step algorithm for radio mapping and path planning. Section IV presents the simulation results, and section V concludes the paper. \n\t\n\n\t\n\t\n\n\n\n\n\t\\section{System Model and Problem Formulation}\n\t\n\tWe consider a cellular network, including $J$ GBSs and $U$ cellular-connected UAVs. The UAVs have their missions and fly from their initial locations towards their destinations. \n\tWe use indices $j$ and $u$ to denote the GBSs and UAVs, respectively. The position of the $u$-th UAV at time $t$ is represented by ${\\bf{q}}_u (t) = (x_u(t), y_u (t),h)$, where without loss of generality we assume that the altitude of the UAVs is fixed throughout their flights. The initial and final position of the $u$-th UAV are shown by ${\\bf{q}}_u^I$ and ${\\bf{q}}_u^F$, respectively. The velocity of the $u$-th UAV at time $t$ and its maximum speed are represented by ${\\bf{v}}_u(t) = \\frac{d {\\bf{q}}_u(t)}{dt}$ and $v_{\\text{max}}$, respectively. We denote the position of the $j$-th GBS by ${\\bf{q}}^G_j$. Moreover, the flight time of the $u$-th UAV is represented by $T_u$. \n\t\n\tThe channel gain between the $u$-th UAV and the $j$-th GBS is expressed as \n\t\\begin{equation}\n\t\\label{channel}\n\th_{u,j} (t) = \\frac{\\rho_{u,j} (t)}{\\text{PL}_{u,j} (t)},\n\t\\end{equation}\n\twhere $\\rho_{u,j} (t)$ is the small-scale fading term and ${\\text{PL}}_{u,j} (t)$ is the average path-loss (PL) between the $u$-th UAV and the $j$-th GBS at time $t$. The average PL depends on the probability of having a line of sight (LoS) link between the UAV and the GBS. Let $\\Gamma_{u,j} (t) \\triangleq \\left( \\frac{4\\pi f_c \\rVert {\\bf{q}}_{u} (t) - {\\bf{q}}^G_j \\rVert}{c} \\right )^2$, where $f_c$ is the carrier frequency and $c$ is the light speed. The average PL is given by\n\t\\begin{equation}\n\t\\label{PL}\n\t{\\bf{PL}}_{u,j} (t) = \\Gamma_{u,j} (t) \\left(\\eta_{\\text{LoS}} \\xi_{u,j} + \\eta_{\\text{N-LoS}} (1-\\xi_{u,j}) \\right),\n\t\\end{equation} \n\twhere $\\eta_{\\text{LoS}}$ and $\\eta_{\\text{N-LoS}}$ are additional losses for the LoS and Non-LoS links, respectively, and $\\xi_{u,j}$ is the probability of having a LoS link between the $u$-th UAV and the $j$-th GBS. This probability can be expressed as $\\xi_{u,j} = \\left(1 + a\\text{exp}(-b(\\psi_{u,j}(t)-a))\\right )^{-1}$, where $\\psi_{u,j}(t)$ is the elevation angle between the $u$-th UAV and the $j$-th GBS at time $t$, and $a$ and $b$ are environment-related parameters \\cite{Rahmati_infocom}. According to experimental measurements presented in\\cite{Comm_Mag}, for moderate altitudes (less than 100 meters), $\\xi_{u,j} \\approx 1$ . As a result, \\eqref{PL} is simplified to\n\t\\begin{equation*}\n\t{\\bf{PL}}_{u,j} (t) = \\Gamma_{u,j} (t) \\eta_{\\text{LoS}}.\n\t\\end{equation*}\n\t\n\tThe received signal to interference plus noise ratio (SINR) of the $u$-th UAV from the $j$-th GBS at time $t$ is given by \n\t\\begin{equation}\n\t\\gamma_{u,j} (t) = \\frac{p_j(t) h_{u,j} (t)}{\\sum_{j' \\neq j} p_{j'}(t) h_{u,j'} (t) + \\sigma },\n\t\\end{equation}\n\twhere $\\sigma$ is the noise power and $p_j (t)$ is the transmit power of the $j$-th GBS at time $t$.\n\t\n\t\n\tAs discussed earlier, the UAVs need to maintain reliable communication links to the GBSs. This is essential to support the command and data flows between the UAVs and the cellular network. However, the quality of the link is highly affected by the channel randomness. To take the channel's random characteristic into account, we first define the outage probability as the probability that the received SINR of the UAV from each GBS is less than a certain threshold $\\gamma_{\\text{th}}$, i.e.,\n\t\t \\begin{equation}\n\t\\label{Outage}\n\t\\mathbb{P}_{u}^{\\text{outage}} (t) = \\mathbb{P}\\{ \\gamma_{u,j} (t) \\leq \\gamma_{\\text{th}}, \\forall j \\} = \\mathbb{P}\\{ \\max_{j} \\gamma_{u,j} (t) \\leq \\gamma_{\\text{th}}\\}.\n\t\\end{equation}\n\tUsing this outage probability, we can define the connection metric. we say the $u$-th UAV is connected to the cellular network at time $t$ if the outage probability is less than a given threshold $P_0$, i.e., \n\\begin{equation}\n\\mathbb{P}_{u}^{\\text{outage}} (t) \\leq P_{0}.\n\\end{equation}\n To have a reliable connection to the cellular network, the UAV is not allowed to loose its connection to the cellular network for more than a given time duration $\\delta$. In other words, the maximum continuous time interval that the UAVs can be disconnected from the cellular network is $\\delta$. To formulate this constraint, first we define function $\\tau_u (t)$ as\n\\begin{equation}\n\\label{tau}\n\\tau_u (t) \\triangleq \\max \\big\\{t' \\in [0,t]: \\mathbb{P}_{u}^{\\text{outage}} (t') \\leq P_{0} \\big\\}.\n\\end{equation}\nThis function gives the last time instance before $t$ that the $u$-th UAV is connected to the cellular network. Using \\eqref{tau}, the reliable connection requirement of the $u$-th UAV is given by \\cite{Behzad_ICC}\n\\begin{equation}\n\\label{connectivity_constraint}\n\\max_{t \\in [0,T_u]} \\{t - \\tau_u (t) \\} \\leq \\delta.\n\\end{equation}\nIt is worth mentioning that the value of $\\delta$ is a design parameter and differs for different applications.\n\n\\subsection{Problem Formulation}\nThe goal of each UAV is to minimize its flight time while its constraint for having a reliable connection to the cellular network is satisfied. If ${\\bf{q}}_u= \\{ {\\bf{q}}_u(t), \\forall t \\in [0,T_u]\\}$, the optimization problem of the $u$-th UAV can be expressed as\n\t\\begin{align*}\n\t\\label{Problem_original}\n\t\\min_{{\\bf{q}}_u, T_u} & \\hspace{0.5cm} T_u \\numberthis \\\\ \n\t\\text{s.t. } & \\text{C1: } \\max_{t \\in [0,T_u]} \\{t - \\tau_u (t)\\} \\leq \\delta, \\\\\n\t& \\text{C2: } \\begin{matrix} \\left \\rVert \\frac{d{\\bf{q}}_u (t)}{dt} \\right \\rVert \\leq v_{\\text{max}}, \\hspace{0.5cm} \\forall t \\in [0,T_u], \\end{matrix} \\\\\n\t& \\text{C3: } \\begin{matrix} {\\bf{q}}_u(0)={\\bf{q}}_u^{\\text{I}}, \\text{ and } {\\bf{q}}_u(T_u)={\\bf{q}}_u^{\\text{F}}.\\end{matrix}\n\t\\end{align*}\n In \\eqref{Problem_original}, constraint C1 shows the reliable connection requirement of the UAV. Constraint C2 states that the UAV's velocity is limited to its maximum speed, and constraint C3 represents the initial and final location of the UAV. Problem \\eqref{Problem_original} is non-convex due to C1. Moreover, the outage probability used in $\\tau_u (t)$ depends on the network's topology. Even if we consider a simple topology for the network, the UAVs do not have access to the outage probability in the environment. As a result, traditional optimization techniques can not be used to solve \\eqref{Problem_original}. In addition to these concerns, the UAV's flight time, $T_u$, is among the optimization variables. To solve \\eqref{Problem_original}, we have to find $T_u$ and the value of ${\\bf{q}}_u(t)$ for all $t \\in [0, T_u]$, which is a challenging task. In what follows, we reformulate the problem into a more tractable form. With this reformulation, we do not need to solve the continuous-time problem. Instead, we can find the solution of an equivalent discrete-time problem.\n\n\t\n\t\\subsection{Problem Reformulation}\n\tThe goal of each UAV is to solve its corresponding optimization problem. For the sake of brevity, we omit index $u$ from the problem and continue our discussion for the general case. To reformulate problem \\eqref{Problem_original}, we use the fact that any feasible solution must satisfy constraint C1. Hence, instead of solving the problem for all time instances, it is sufficient to consider the problem for a sequence of discrete time instances $t_1$, $t_2$, $\\ldots$, $t_N$, where $| t_n - t_{n-1}| \\leq \\delta$, $n=1, \\ldots, N$, and make sure that the UAV is connected to the cellular network at these time instances. In other words, if \n\t\t\\begin{equation}\n\t\\mathbb{P}_{\\text{outage}}(t_n) \\leq P_0, \\hspace{0.5cm} n=1, \\ldots, N,\n\t\\end{equation}\n\tthe maximum continuous time that the UAV is in outage will be limited to $\\delta$. Therefore, constraint C1 will be satisfied. To reformulate C2, we have $\\left \\rVert \\frac{d{\\bf{q}} (t)}{dt} \\right \\rVert = \n\t\\frac{\\rVert {\\bf{q}} (t_n) - {\\bf{q}} (t_{n-1}) \\rVert}{|t_n - t_{n-1}|}$, Since $| t_n - t_{n-1}| \\leq \\delta$, the equivalent form of C2 is given by\n\t\\begin{equation}\n\t\\label{C2:equivalent}\n\t \\rVert {\\bf{q}} (t_n) - {\\bf{q}} (t_{n-1}) \\rVert \\leq \\delta v_{\\text{max}}, \\hspace{0.5cm} n=1, \\ldots, N.\n\t\\end{equation}\n\tIt can be shown that to minimize the flight time, the UAVs fly with their maximum speed, $v_{\\text{max}}$. Considering this fact and using a similar approach to what has been shown for C2, we can write the objective of \\eqref{Problem_original} as\n\t\\begin{equation}\n\tT = \\frac{1}{v_{\\text{max}}}\\sum_{n=1}^{N} \\left\\rVert {\\bf{q}} (t_n) - {\\bf{q}} (t_{n-1}) \\right \\rVert.\n\t\\end{equation}\n\tLet ${\\bf{q}}[n] \\triangleq {\\bf{q}}(t_n)$, $n=0, 1, \\ldots, N$. Problem \\eqref{Problem_original} is equivalent to the following discrete-time optimization problem \n\t\\begin{align*}\n\t\\label{Problem_reformulated}\n\t\\min_{\\{{\\bf{q}}[n]\\}_{n=0}^{N}, N} & \\hspace{0.5cm} \\sum_{n=1}^{N} \\rVert {\\bf{q}}[n] - {\\bf{q}}[n-1] \\rVert \\numberthis \\\\ \n\t\\text{s.t. } & \\tilde{\\text{C}}\\text{1: } \\mathbb{P}^{\\text{outage}}({\\bf{q}}[n]) \\leq P_0, \\forall n, \\\\\n\t& \\tilde{\\text{C}}\\text{2: } \\begin{matrix} \\rVert {\\bf{q}}[n] - {\\bf{q}}[n-1] \\rVert \\leq \\delta v_{\\text{max}}, \\end{matrix} \\\\\n\t& \\tilde{\\text{C}}\\text{3: } \\begin{matrix} {\\bf{q}}[0]={\\bf{q}}^{\\text{I}}, \\text{ and } {\\bf{q}}[N]={\\bf{q}}^{\\text{F}}.\\end{matrix}\n\t\\end{align*}\n\t\\section{Two-step Algorithm for Radio Mapping and Path Planning}\n\tTo find the solution of the reformulated problem, we still need to know the outage probability. However, this information is not availbale to the UAVs. Hence, we need to obtain this information first. In what follows we propose a two-step algorithm to solve \\eqref{Problem_reformulated}. In this approach, in the first step, we estimate the outage probability. To achieve this goal, we use Federated Learning (FL) which allows the UAVs to collaborate to build a global model of the outage probability based on their locally collected data. Moreover, the UAVs do not need to share their collected data with a central node in this approach. In the second step, we use the derived model to find the solution of the problem ensuring that $\\tilde{\\text{C}}\\text{1}$ is satisfied.\n\n\n\t\t\\subsection{Radio Mapping based on Federated Learning}\n\t\t\n\t\t\\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[trim={0cm 0cm 0cm 0cm}, width=3.2in,keepaspectratio]{FL.pdf}\n\t\t\\vspace{-0.3cm}\n\t\t\\caption{Federated Learning to estimate the outage probability.}\n\t\t\\label{fig:1}\n\t\\end{figure}\t\n\t\n\t\n\tTo estimate $\\mathbb{P}^{\\text{outage}}$(.), as depicted in Fig. \\eqref{fig:1}, we consider a scenario where all UAVs can collaborate to build a global model of the outage probability in the environment. Let $\\mathcal{D}_u$ denote the data set of the $u$-th UAV and let $\\mathcal{D} = \\cup_{u=1}^{U} \\mathcal{D}_u$. We denote $|\\mathcal{D}_u|=m_u$ and $|\\mathcal{D}|=M=\\sum_{u=1}^{U} m_u$, where $|A|$ is the cardinality of set $A$. The data set of the $u$-th UAV includes all pairs of $({\\bf{q}}_u (t), l_{u}(t))$, where ${\\bf{q}}_u (t)=(x_u(t),y_u(t))$ is the 2D coordinate of a visited location by the UAV and \n\t\\begin{equation}\n\tl_u (t) = \\begin{cases}\n\t1 & \\text{if } \\max_{j} \\gamma_{u,j} (t) \\geq \\gamma_{\\text{th}},\\\\\n\t0 & \\text{otherwise},\n\t\\end{cases}\n\t\\end{equation} \n\tis the label of this data point. To find a global model of the outage probability, we consider a neural network with parameter $\\theta$. The output of this neural network for input $\\bf{q} \\in \\mathbb{R}^2$ is denoted by $P(\\theta,{\\bf{q}})$ which gives the outage probability at location ${\\bf{q}}$. To find an appropriate model for the outage probability, we have to solve the following problem\n\\begin{equation}\n\t\\label{FL:problem:main}\n\t\\min_{\\theta} f(\\theta ) = \\frac{1}{M}\\sum_{m=1}^{M} f_m (\\theta),\n\t\\end{equation}\n\twhere $f_m$ is the loss function corresponding to the $m$-th data point. We consider cross-entropy for the loss function as\n\t\\begin{align}\n\t\\label{CrossEntropy}\n\tf_m (\\theta) \\triangleq & f(\\theta; {\\bf{q}}^m,l^m) \\\\\n\t= & - l^m \\log (P(\\theta,{\\bf{q}}^m)) - (1-l_m)\\log (1-P(\\theta,{\\bf{q}}^m)), \\nonumber\n\t\\end{align}\n\twhere $\\left({\\bf{q}}^m, l^m\\right)$ is the $m$-th data point. We can write the objective of \\eqref{FL:problem:main} as\n\t\\begin{equation}\n\tf(\\theta) = \\frac{1}{M} \\sum_{u=1}^{U} \\sum_{({\\bf{q}}^m,l^m) \\in \\mathcal{D}_u} f(\\theta; {\\bf{q}}^m,l^m).\n\t\\end{equation}\n\tIf we define $F_u (\\theta) = \\frac{1}{m_u} \\displaystyle \\sum_{({\\bf{q}}^m,l^m) \\in \\mathcal{D}_u} f(\\theta; {\\bf{q}}^m,l^m)$, then we have\n\t\\begin{equation}\n\t\\label{FL:reformulated_objective}\n\tf(\\theta) = \\sum_{u=1}^{U} \\frac{m_u}{M} F_u (\\theta).\n\t\\end{equation}\n\t\t\\begin{algorithm}[t]\n\t\t\\footnotesize\n\t\t\\label{FL}\n\t\t\\caption{FL-based radio mapping to estimate outage probability.}\n\t\tThe GBSs initialize the global model parameter, $\\theta$,\n\t\trandomly.\\\\\n\t\tset $t=1$ and $\\theta^{(t)}=\\theta$.\\\\\n\t\t\\For{$t=1$ to $T_{FL}$ (max round)}{ \n\t\t\tThe GBSs send the global model parameter, $\\theta^{(t)}$, to the UAVs\\\\\n\t\t\t\\For {each UAV $u$} { \n\t\t\t\t$\\theta_u = \\theta^{(t)} $\\\\\n\t\t\t\tUpdate training data set $\\mathcal{D}_u$ using newly collected data\\\\\n\t\t\t\t\\For{$epoch = 1$ to $H$}{\n\t\t\t\t\tUpdate its local parameter $\\theta_u$ over its data set $\\mathcal{D}_u$ as\\\\\n\t\t\t\t\t\\[\\theta_u = \\theta_u - \\eta \\nabla F_u (\\theta_u)\\] \n\t\t\t\t}\n\t\t\t\tEach UAV sends its local parameter to the cellular network.\\\\}\n\t\t\tThe GBSs collect all local parameters and updates the global model as\n\t\t\t\\[ \\theta^{(t+1)} = \\displaystyle \\sum_{u=1}^{U} \\frac{m_u}{M}\\theta_u \\]\n\t\t}\n\t\\end{algorithm}\n To minimize \\eqref{FL:reformulated_objective}, instead of using a central approach which requires access to data of all UAVs, we use a distributed approach based on FL \\cite{FL2}. In this algorithm, we assume that the global model parameter, $\\theta$, is available to all GBSs. The GBSs send this parameter to the UAVs and the UAVs update their local model parameter as $\\theta_u = \\theta, \\forall u$. The UAVs fly over the area and based on their received signals from the cellular network, they form their data sets $\\mathcal{D}_u, \\forall u$. Using the collected data, each UAV performs $H$ steps of the stochastic gradient descent (SGD) on its local parameter, $\\theta_u$. In other words, in each step, the $u$-th UAV updates its local parameter as\n\t\\begin{equation}\n\t\\theta_u = \\theta_u - \\eta \\nabla F_u (\\theta_u),\n\t\\end{equation} \n\twhere $\\eta$ is the step size and $\\nabla$ is the gradient operator. The updated parameters, $\\theta_u, \\forall u$, are sent back from the UAVs to the GBSs. The GBSs act as aggregators and share this infromation in their network. By averaging the received local parameters, the GBSs evaluate a new parameter as\n\t\\begin{equation}\n\t\\theta= \\displaystyle \\sum_{u=1}^{U} \\frac{m_u}{M}\\theta_u .\n\t\\end{equation}\n\tThe global model is updated using this new $\\theta$. This updated parameter is again sent to all UAVs to perform their local updates with their new data. This procedure is repeated until the neural network is trained. Algorithm 1 presents the FL-based radio mapping for outage probability estimation. \n\t\\begin{algorithm}[t]\n\t\t\\footnotesize\n\t\t\\label{RRT_star}\n\t\t\\caption{RRT$^*$-based path planning for the UAVs }\n\t\t$V = \\{{\\bf{q}}^I\\}$, $E=\\{\\}$, $\\mathcal{T}= (V,E)$\\\\\n\t\t\\For{$n =1$ \\text{ to } $N$ }{\n\t\t\tRandomly sample a point from the space ${\\bf{q}}_{\\text{rand}}$\\\\\n\t\t\tFind the nearest vertex of the graph to ${\\bf{q}}_{\\text{rand}}$, i.e.,\n\t\t\t\\begin{equation*}\n\t\t\t{\\bf{q}}_{\\text{nearest}} = \\displaystyle \\argmin_{ {\\bf{q}}\\in V} \\rVert {\\bf{q}}_{\\text{rand}} - {\\bf{q}} \\rVert\n\t\t\t\\end{equation*} \n\t\t\tFind ${\\bf{q}}_{\\text{new}} = \\argmin_{\\rVert {\\bf{q}} - {\\bf{q}}_{\\text{nearest}} \\rVert \\leq v_{\\text{max}} \\delta} \\rVert {\\bf{q}}_{\\text{rand}} - {\\bf{q}} \\rVert $\\\\\n\t\t\t\\If{$P(\\theta^*, {\\bf{q}}_{\\text{new}}) \\leq P_0$}{ \n\t\t\t\t$Q_{\\text{near}} = \\{{\\bf{q}} \\in V : \\rVert {\\bf{q}} - {\\bf{q}}_{\\text{new}} \\leq v_{\\text{max}} \\delta\\}$\\\\\n\t\t\t\t$V = V \\cup \\{{\\bf{q}}_{\\text{new}}\\}$\\\\\n\t\t\t\t${\\bf{q}}_{\\text{min}} = {\\bf{q}}_{\\text{nearest}}$\\\\\n\t\t\t\t$c_{\\text{min}} = c({\\bf{q}}_{\\text{nearest}}) + \\rVert {\\bf{q}}_{\\text{nearest}} - {\\bf{q}}_{\\text{new}} \\rVert$\\\\\n\t\t\t\t\\For{all ${\\bf{q}}_{\\text{near}} \\in Q_{\\text{near}}$}{\n\t\t\t\t\t\\If{$P(\\theta^*,{\\bf{q}}_{\\text{near}}) \\leq P_0$ and $c({\\bf{q}}_{\\text{near}}) + \\rVert {\\bf{q}}_{\\text{near}} - {\\bf{q}}_{\\text{new}} \\rVert < c_{\\text{min}}$}{\n\t\t\t\t\t\t${\\bf{q}}_{\\text{min}} = {\\bf{q}}_{\\text{near}}$\\\\\n\t\t\t\t\t\t$c_{\\text{min}} = c({\\bf{q}}_{\\text{near}}) + \\rVert {\\bf{q}}_{\\text{near}}-{\\bf{q}}_{\\text{new}}\\rVert$\\\\\n\t\t\t\t\t}\t\t\n\t\t\t\t}\n\t\t\t\t$E = E \\cup \\{({\\bf{q}}_{\\text{min}},{\\bf{q}}_{\\text{new}} )\\}$\\\\\t\t\n\t\t\t\t\\For{all ${\\bf{q}}_{\\text{near}} \\in Q_{\\text{near}}$}{\n\t\t\t\t\t\\If{$P(\\theta^*,{\\bf{q}}_{\\text{near}}) \\leq P_0$ and $c({\\bf{q}}_{\\text{near}}) + \\rVert {\\bf{q}}_{\\text{near}} - {\\bf{q}}_{\\text{new}} \\rVert < c({\\bf{q}}_{\\text{near}})$}{\n\t\t\t\t\t\tFind the parent of ${\\bf{q}}_{\\text{near}}$, i.e.,\\\\\n\t\t\t\t\t\t${\\bf{q}}_{\\text{p}} = \\{{\\bf{q}} : ({\\bf{q}},{\\bf{q}}_{\\text{near}}) \\in E\\}$\t\\\\\n\t\t\t\t\t\t$E = \\{E \\setminus \\{({\\bf{q}}_{\\text{p}},{\\bf{q}}_{\\text{near}})\\} \\} \\cup \\{({\\bf{q}}_{\\text{new}}, {\\bf{q}}_{\\text{near}})\\}$\n\t\t\t\t\t}\n\t\t\t\t}\t\n\t\t\t}\n\t\t} \n\t\\end{algorithm}\n\tThe advantage of this approach is that the network parameter can be locally updated. This does not require high computation resources. Moreover, the UAVs do not need to share their data sets with the cellular network. In addition to these benefits, the UAVs might fly over differnt areas of the network. Hence, they will not have good knowledge of the other parts of the network. This FL-based algorithm allows the UAVs to collaborate to build a global model based on their limited local experiences. \n\t\n\n\n\t\n \n\t\n\t\n\t\n\t\n\n\t\\subsection{Path Planning based on RRT$^{*}$}\n\tAfter the first step, we have a model of the outage probability in the environment. Let $\\theta^*$ denote the parameter of the trained global model. In this step, we have to solve the following optimization problem\n\t\\begin{align*}\n\t\\label{RRT:Problem}\n\t\\min_{\\{{\\bf{q}}[n]\\}_{n=0}^{N}, N} & \\hspace{0.5cm} \\sum_{n=1}^{N} \\rVert {\\bf{q}}[n] - {\\bf{q}}[n-1] \\rVert \\numberthis \\\\ \n\t\\text{s.t. } & \\tilde{\\text{C}}\\text{1: } P(\\theta^*, {\\bf{q}}[n]) \\leq P_0, \\hspace{0.5cm} n=0,1, \\ldots, N, \\\\\n\t& \\tilde{\\text{C}}\\text{2-}\\tilde{\\text{C}}\\text{3, } \n\t\\end{align*}\n\twhere $P(\\theta^*,{\\bf{q}})$ is the output of the trained neural network for position ${\\bf{q}}$. Problem \\eqref{RRT:Problem} is still challenging to solve. The reason is that the feasible region corresponding to constraint $\\tilde{\\text{C}}$1 is not necessarily convex. Moreover, $N$ which is the number of discrete time steps is a variable of the problem. To overcome these difficulties, we use rapidly-exploring random trees (RRTs) \\cite{Karaman}. RRTs are designed to search non-convex spaces. In what follows, we describe our algorithm which works based on a modified version of RRTs called RRT$^{*}$ \\cite{Karaman}. RRT$^{*}$ is an efficient algorithm which can find the shortest path between a pair of initial and final locations in a continuous space.\n\t \n\t\t\\begin{figure}\n\t\t\\begin{subfigure}{.23\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[trim={2cm 0 2.cm 0.8cm},clip, width=1\\linewidth]{GBS.pdf} \n\t\t\t\\caption{}\n\t\t\t\\label{fig2:1}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}{.23\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[trim={1.8cm 0 0.5cm 0},clip, width=1.1\\linewidth]{outage.pdf} \n\t\t\t\\caption{}\n\t\t\t\\label{fig2:2}\n\t\t\\end{subfigure}\n\t\\vspace{-0.1cm}\n\t\t\\caption{ (a) Location of the GBSs (b) Trained outage probability}\n\t\t\\label{fig2}\n\t\\end{figure} \n\t \n\t \n\tWe use RRT$^*$ to find the path that minimizes the UAV's flight time while ensuring that it satisfies the requirement for having a reliable connection to the cellular network. In our algorithm, we use a tree $\\mathcal{T} = (V,E)$ to represent the path between the initial point to any feasible coordinate ${\\bf{q}}$. The set of vertices, $V$, is the set of all coordinates ${\\bf{q}}$ in the space that has been explored by the algorithm. The root of the tree is the initial point ${\\bf{q}}^{I}$. Coordinate ${\\bf{q}}_p$ is called parent of ${\\bf{q}}_c$ if the UAV arrives in ${\\bf{q}}_c$ from ${\\bf{q}}_p$ and it does not visit any other vertex in between. According to this notation, $E$ is the set of all edges $({\\bf{q}}_p,{\\bf{q}}_c)$ which connects a visited coordinate to its parent. Moreover, in our algorithm, we need to define a cost function $c(.)$ for each visited coordinate. If the UAV passes the sequence of edges $\\{({\\bf{q}}_0, {\\bf{q}}_1), ({\\bf{q}}_1,{\\bf{q}}_2), \\ldots, ({\\bf{q}}_{n-1},{\\bf{q}}_n)\\}$ to reach ${\\bf{q}}_n$, where ${\\bf{q}}_0 = {\\bf{q}}^I$, then the cost of coordinate ${\\bf{q}}_n$ is defined as \n\t\\begin{equation}\n\tc({\\bf{q}}_n) = \\sum_{n'=1}^{n} \\rVert {\\bf{q}}_{n'} - {\\bf{q}}_ {n'-1} \\rVert. \n\t\\end{equation}\n\t\t\nWe start our path planning algorithm from the root, ${\\bf{q}}^I$, and iteratively add new vertices to the tree and update its structure including the edges and costs. In each iteration, to add a new vertex to the tree, we sample a random coordinate from the space. This random coordinate is denoted by ${\\bf{q}}_{\\text{rand}}$. Then, the closest vertex of $\\mathcal{T}$ to ${\\bf{q}}_{\\text{rand}}$ is found and is shown as ${\\bf{q}}_{\\text{nearest}}$. To satisfy constraint C2, we have to ensure that the distance between the new sampled coordinate and ${\\bf{q}}_{\\text{nearest}}$ is less than $\\delta v_{\\text{max}}$, i.e., $\\rVert {\\bf{q}}_{\\text{rand}} - {\\bf{q}}_{\\text{nearest}} \\rVert \\leq \\delta v_{\\text{max}}$. To meet this requirement, in case that this inequality is violated, we replace ${\\bf{q}}_{\\text{rand}}$ with a closer point denoted as ${\\bf{q}}_{\\text{new}}$, where\n\t\\begin{equation*}\n{\\bf{q}}_{\\text{new}} = \\argmin_{\\rVert {\\bf{q}} - {\\bf{q}}_{\\text{nearest}} \\rVert \\leq \\delta v_{\\text{max}}} \\rVert {\\bf{q}}_{\\text{rand}} - {\\bf{q}} \\rVert.\n\\end{equation*}\nAfter satisfying C2, we have to make sure that C1 is also satisfied. If $P(\\theta^*, {\\bf{q}}_{\\text{new}}) \\leq P_0$, then this new coordinate ${\\bf{q}}_{\\text{new}} $ is out of outage and hence, it can be added to the tree. Otherwise, we have to repeat this iteration from the begining to find a coordinate satisfying both C1 and C2. Assuming $P(\\theta^*, {\\bf{q}}_{\\text{new}}) \\leq P_0$, we add this point to the tree as a new vertex. To set the cost of this new vertex, we find all vertices of $\\mathcal{T}$ whose distance to ${\\bf{q}}_{\\text{new}}$ is less than $\\delta v_{\\text{max}}$. From these vertices, we choose the one minimizing \n\\begin{equation}\n\\label{min:cost}\n{\\bf{q}}_{\\text{min}}= \\argmin_{{\\bf{q}} : {\\bf{q}} \\in V {\\text{ and }} \\rVert {\\bf{q}} - {\\bf{q}}_{\\text{new}} \\rVert \\leq \\delta v_{\\text{max}}} c({\\bf{q}}) + \\rVert {\\bf{q}} - {\\bf{q}}_{\\text{new}} \\rVert.\n\\end{equation}\nAfter finding ${\\bf{q}}_{\\text{min}}$, we add edge $({\\bf{q}}_{\\text{min}}, {\\bf{q}}_{\\text{neq}})$ to $E$ and rewire the tree to update the parents of the vertices located in a distance less than $\\delta v_{\\text{max}}$ to ${\\bf{q}}_{\\text{new}}$. Algorithm 2 presents the path planning procedure for the UAVs.\n\n\n\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim={0cm 0cm 0cm 0cm}, width=2.4in,keepaspectratio]{L1.pdf}\n\t\\caption{Loss function for different values of $H$.}\n\t\\vspace{-0.3cm}\n\t\\label{fig:3}\n\\end{figure}\t\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[trim={0cm 0cm 0cm 0cm}, width=2.4in,keepaspectratio]{L2.pdf}\n\t\\caption{Loss function for different number of collaborating UAVs.}\n\t\\label{fig:4}\n\\end{figure}\t\n\n\n\t\\begin{figure*}[t]\n\t\\begin{subfigure}{.31\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={2cm 0 0.5cm 0cm},clip, width=1.05\\linewidth]{T3.pdf} \n\t\t\\caption{}\n\t\t\\label{fig5:1}\n\t\\end{subfigure}\n\\hfill\n\t\\begin{subfigure}{.31\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={2cm 0 0.5cm 0},clip, width=1.05\\linewidth]{T2.pdf} \n\t\t\\caption{}\n\t\t\\label{fig5:2}\n\t\\end{subfigure}\n\\hfill\n\t\\begin{subfigure}{.31\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[trim={2cm 0 0.75cm 0},clip, width=1.02\\linewidth]{T1.pdf} \n\t\t\\caption{}\n\t\t\\label{fig5:3}\n\t\\end{subfigure}\n\t\\caption{ Path of the UAV for different pairs of initial and final locations when (a) $P_0=0.4$ (b) $P_0=0.05$ (c) $P_0=0.30$. }\n\t\\label{fig:5}\n\t\\vspace{-0.5cm}\n\\end{figure*} \n\t\\section{Simulation Results}\n\tIn this section, we present the simulation results to show the performance of the proposed algorithm. We consider a $10$km $\\times$ $10$km area as shown in Fig. \\eqref{fig2:1}. The altitude of the UAVs is $100$m. The transmit power of each GBS is $200$mW. For the fading term, we consider a Rayleigh random variable with parameter $1$. Other simulation parameters are: $v_{\\text{max}} = 5\\frac{\\text{m}}{\\text{s}}$, $\\sigma = -140$dB, $(a,b) = (5,0.5)$, $f_c=2$GHz, $\\eta_{\\text{LoS}}=1$, and $\\eta_{\\text{N-LoS}}=20$. Moreover, we consider $\\gamma_{\\text{th}}=0.65$ and $\\delta = 5$s.\n\t\n\tTo estimate the outage probability, we use a fully-connected neural network. The neural network has three hidden layers. Each hidden layer consists of $256$ neurons. The last layer of the neural network (output layer) has two outputs. We use rectified linear unit (ReLU) activation functions for the hidden layers. For the output layer, we use a softmax activation function. The batch size of each UAV is $500$. \n\t\n\tFig. \\eqref{fig2:2} shows the resulting coverage (1-outage) probability in the environment. The number of collaborating UAVs is $10$, the number of training rounds, $T_{FL}$, is $200$, and the number of SGD updates in each round is $10$. As can be seen, in locations close to the GBSs, we expect to have stronger signals, which lead to a lower outage probability. However, as the distance from the GBSs increases, due to the PL, the outage probability highly increases. It is worth mentioning that to obtain this model, the UAVs do not know the true location of the GBSs. The only information they have is their received signals in different locations. Fig. \\eqref{fig2:2} also shows that connectivity is not available in all parts of the sky. So, it is essential to design proper paths for the UAVs. \n\t\n\t\n\tFig. \\eqref{fig:3} shows the average loss for different numbers of training steps per round ($H$). We observe that as the value of $H$ increases, the UAVs have more time to update their models using the same training data. Therefore, the loss will decrease. \n\t\n\tFig. \\eqref{fig:4} presents the average loss for different numbers of collaborating UAVs. We observe that as the number of collaborating UAVs increases, the loss function decreases. However, this effect is negligible compared to the impact of $H$ on the loss function. In fact, by using more UAVs, we get data from different areas of the environment, depending on the flight paths of the UAVs. To use this data, the UAVs need to have enough updates on their model. If the value of $H$ is small, increasing the number of collaborating UAVs does not significantly increase the global model's accuracy. \n\t\n\t\n Fig. \\eqref{fig:5} presents the trajectory of a UAV for three different pairs of initial and final locations. The values of $P_0$ in (a), (b), and (c) are $0.4$, $0.05$ and $0.30$, respectively. To obtain these paths, Algorithm 2 considers coordinates $\\bf{q}$ with $P(\\theta^*,{\\bf{q}}) \\leq P_0$. We observe that to satisfy the connectivity constraint, the UAV needs to take longer paths which increases its flight time.\n\t\n\t\n\t\\section{conslusion}\n In this paper, we studied the radio mapping and path planning problem for a UAV network. We minimized the UAVs' flight time, ensuring that the UAVs satisfy a probabilistic connectivity constraint during their flights. To solve the problem, we proposed a two-step approach. In the first step, using FL, the UAVs build a global model of the outage probability in the environment based on their collected data. In the next step, using this learned model and rapidly-exploring random trees, we develop a path-planning algorithm that satisfies the cellular-connectivity requirements.\t\n\t\n\t\\bibliographystyle{IEEEtran}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt is well known that \na locally Lipschitz function\ncan be estimated point-wise \nby the Riesz potential of its gradient\nin bounded John domains,\n\\cite[Theorem]{R}, \\cite[Theorem 10]{Haj01},\nand hence, especially, in Lipschitz domains and\nin convex domains,\n\\cite[Lemma 7.16]{Gilbarg-Trudinger}.\nBy modifying the Riesz potential,\npoint-wise\nestimates \ncan be generalized for \nfunctions which are defined in more irregular domains than John domains, \n\\cite[Theorem 3.4]{HH-SK}, \\cite[Theorem 4.4]{HH-S1}.\nMore precisely, for every function $u$ whose weak distributional partial derivatives are in $L^1(G)$,\nthe pointwise estimate\n\\begin{equation}\\label{modified}\n|u(x) - u_D| \\le \\int_G \\frac{|\\nabla u(y)|}{\\psi(|x-y|)^{n-1}} \\, dy\n\\end{equation}\nholds\nfor almost every $x \\in G$. \nHere, $G$ is a domain in the $n$-dimensional Euclidean space and the regularity of the boundary is controlled by the function $\\psi$.\nHedberg's method\n\\cite[Lemma, Theorem 1]{Hed72}\ncan be extended so that this point-wise estimate leads to the Sobolev-type inequality where an Orlicz-space is the target space.\nHedberg's method has been used by A. Cianchi and B. Stroffolini for the classical Riesz potential when\nfunctions are Orlicz functions,\n\\cite[Theorem 1, Corollary 1]{CS}, and \nby the authors for the modified Riesz potential\nwith a special Orlicz function,\n\\cite[Theorem 1.1]{HH-SK} and \\cite[Theorem 1.1]{HH-S},\nand with a general Orlicz function in\n\\cite[Corollary 3.4, Corollary 5.4]{HH-S1}.\nFor other papers on Orlicz embeddings of Cianchi we refer to \\cite{Cianchi1996}, \\cite{Cianchi1999}.\n\n \nIn the present paper we show that the optimal Orlicz function for the modified Riesz potential \nin \\eqref{modified} can be found as a function of $\\psi$\nwhich depends on the geometry of the domain $G$.\n Our main theorem is the following theorem where we give the formula to the Orlicz function.\n\n\n\n\n\n\\begin{theorem}\\label{thm:defn_H_intro}\nLet $1\\le p0.\n\\]\n\\end{theorem}\n\n\n\n\n\nWith this function we obtain the following point-wise estimate.\n\n\n\n\n\\begin{theorem}\\label{thm_main}\nLet $G$ be a domain in ${\\varmathbb{R}^n}$, $n\\geq 2$.\nLet $1\\le p 0$ and if $D$ is a bounded or an unbounded $\\varphi$-cigar John domain with a constant $c_J$ in ${\\varmathbb{R}^n}$, $n \\ge 2$\nand if $1\\le p0$.\n\\end{enumerate}\nWe write\n\\begin{equation}\\label{equ:psi}\n\\psi (t) = \\begin{cases}\n\\varphi(t) & \\quad \\text{if} \\quad 0\\le t \\le 1;\\\\\n\\varphi(1) t & \\quad \\text{if} \\quad t \\ge 1.\\\\\n\\end{cases}\n\\end{equation}\nNow, if $\\varphi$ satisfies the conditions (1)--(5), then $\\psi$ does, too, and the constant in (4) is the same for the functions $\\varphi$ and $\\psi$, that is \n$C_{\\varphi}=C_{\\psi}$.\n\n\n\nThe definition of a bounded John domain goes back to F. John\n\\cite[Definition, p. 402]{J} who defined an inner radius and an outer radius domain, and later this domain was renamed as a John domain\nin \\cite[2.1]{MS79}.\n\nWe extend the definition of John domains following J.~V\\\"ais\\\"al\\\"a\n\\cite[2.1]{Vaisala} in the classical case.\nLet $E$ in ${\\varmathbb{R}^n}$, $n\\geq 2$, be a closed rectifiable curve with endpoints $a$ and $b$. The subcurve between $x\\,,y \\in E$ is denoted by $E[x,y]$. For $x \\in E$ we write\n\\[\nq(x) = \\qopname\\relax o{min}\\bigg\\{\\ell\\Big(E[a,x] \\Big), \\ell\\Big(E[x,b] \\Big) \\bigg\\},\n\\]\nwhere $\\ell\\big(E[a,x]\\big)$ is the length of the subcurve $E[a,x]$.\n\n\\begin{definition}\\label{john}\nA bounded or an unbounded domain $D$ in ${\\varmathbb{R}^n}$ is a $\\varphi$-cigar John domain if there exists a constant $c_J >0$ \nsuch that each pair of points $a, b \\in D$ can be joined by a closed rectifiable curve $E$ in $D$ such that\n\\[\n\\qopname\\relax o{Cig} E(a,b) = \\bigcup\\left\\{B \\left(x, \\frac{\\psi(q(x))}{c_J} \\right): x \\in E\\setminus\\{a,b\\} \\right\\} \\subset D\n\\] \nwhere $B(x,r)$ is an open ball centered at $x$ with a radius $r>0$ and the function \n$\\psi$ is defined as in \\eqref{equ:psi}.\n\\end{definition}\n\nThe set\n$\\qopname\\relax o{Cig} E(a,b)$ is called a cigar with core $E$ joining $a$ and $b$. \nWe point out that if $D$ is a $\\varphi$-cigar John domain with $\\varphi(t) = t^p$, $p \\ge 1$, then it is a $\\varphi$-cigar John domain with $\\varphi(t) = t^q$ for every $q \\ge p$.\nFor the case $\\psi (t)=\\varphi (t)=t$ for all $t\\geq 0$,\nin Definition \\ref{john}, we refer to \\cite[2.1]{Vaisala} and \\cite[2.11 and 2.13]{Nakki_Vaisala}.\n\n\nIf $D$ is a bounded domain then the following definition from \\cite[Definition 4.1]{HH-S1}\nfor a $\\psi$-John domain\ngives an equivalent definition\nto a bounded $\\varphi$-cigar John domain.\n\n\\begin{definition}\\label{bounded-john}\nA bounded domain $D$ in ${\\varmathbb{R}^n}\\,, n\\geq 2\\,,$ is a $\\psi$-John domain if there exist a constants $0< \\alpha \\le \\beta<\\infty$ and a point $x_0 \\in D$ such that each point $x\\in D$ can be joined to $x_0$ by a rectifiable curve $\\gamma:[0,\\ell(\\gamma)] \\to D$, parametrized by its arc length, such that $\\gamma(0) = x$, $\\gamma(\\ell(\\gamma)) = x_0$, $\\ell(\\gamma)\\leq \\beta\\,,$ and\n\\[\n\\psi(t) \\leq \\frac{\\alpha}{\\ell(\\gamma)} \\qopname\\relax o{dist}\\big(\\gamma(t), \\partial D\\big) \\quad \\text{for all} \\quad t\\in[0, \\ell(\\gamma)].\n\\]\nThe point $x_0$ is called a John center of $D$ and \n$\\gamma$ is called a John curve of $x$.\n\\end{definition}\n\nIf the function $\\psi$ is defined as in \\eqref{equ:psi} with the function $\\varphi$, then\na bounded domain is a $\\psi$-John domain if and only if it is a $\\varphi$-John domain. If $\\psi(t) =t$, then our definition for bounded $\\psi$-John domains coincides with the definition of the classical John domains. If $\\psi(t) = t^s$, $s\\ge 1$, then our definition for bounded $\\psi$-John domains coincides with the definition of $s$-John domains.\n\n\n\n\\begin{theorem}\\label{thm:John-John}\nLet $D$ be a bounded domain. \nIf $D$ is a $\\psi$-John domain then $D$ is a $\\varphi$-cigar John domain.\nOn the other hand, if $D$ is a $\\varphi$-cigar John domain with a constant $c_J$, then $D$ is a $\\psi$-John domain with constants\n\\[\n\\alpha= \\frac{c_J\\, \\varphi(1) \\left(\\max\\left\\{2, \\frac{c_J \\qopname\\relax o{diam}(D)}{\\varphi(1)} \\right\\} \\right)^2}{\\psi \\left( \\frac1{2c_J} \\psi \\left(\\frac14 \\qopname\\relax o{diam}(D)\\right)\\right)},\n\\]\n\\[\n\\quad\n\\beta = \\max\\left\\{2, \\frac{c_J \\qopname\\relax o{diam}(D)}{\\varphi(1)} \\right\\}.\n\\]\n\\end{theorem}\n\n\nNote that when $\\qopname\\relax o{diam}(D) \\to \\infty$, then $\\alpha \\to \\infty$ with the same speed as $\\qopname\\relax o{diam}(D)$.\n\n\\begin{proof}\nAssume first that $D$ is a $\\psi$-John domain with a John center $x_0$. Let $a, b \\in D$ and let the John curves $\\gamma_1$ and $\\gamma_2$ connect them to $x_0$,\nrespectively. We may assume that $a,b \\in D \\setminus B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D))$, since inside the ball the points can be connect by two straight lines going via the center of the ball $B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D))$.\nLet $E = \\gamma_1 \\circ \\gamma_2$. Then, \n\\[\n\\begin{split}\n&\\qopname\\relax o{Cig} E(a, b)\\\\ &= \\bigcup_{t \\in (0, \\ell(\\gamma_1)]} B \\left( \\gamma_1(t), \\frac{\\psi(t)}{\\alpha\/\\qopname\\relax o{dist}(x_0, \\partial D)} \\right) \\cup \n\\bigcup_{t \\in (0, \\ell(\\gamma_2)]} B \\left( \\gamma_2(t), \\frac{\\psi(t)}{ \\alpha\/\\qopname\\relax o{dist}(x_0, \\partial D)} \\right)\\\\\n\\end{split}\n\\]\nand thus $D$ is a $\\varphi$-cigar John domain.\n\nAssume then that $D$ is a $\\varphi$-cigar John domain. \nLet us carefully choose a suitable John center so that\nthe center is not too close to the boundary of $D$. Let $x, y \\in D$ such that $|x-y|\\ge \\frac12 \\qopname\\relax o{diam}(D)$. Let $E$ be a core of a John cigar that connects$x$ and $y$.\nThen the length of $E$ is at least $\\frac12 \\qopname\\relax o{diam}(D)$. Let $x_0$ be the center of $E$. Then\n\\[\n\\qopname\\relax o{dist}(x_0, \\partial D) \\ge \\frac{\\psi(\\frac14 \\qopname\\relax o{diam}(D))}{c_J}\n\\] \nso we choose \n$r= \\psi \\Big(\\frac14 \\qopname\\relax o{diam}(D) \\Big)\/c_J$, and hence $B(x_0, r) \\subset D$.\nFrom now on this $r$ and the point $x_0$ are fixed in this proof.\n\nFor every $a \\in D \\setminus\\overline{ B(x_0, r)}$ there exists a curve $E$ such that $\\qopname\\relax o{Cig} E(a, x_0) \\subset D$.\nLet $\\ell (E)$ be the length of $E$, then $\\ell (E) \\le 2$ or by the definition\n\\[\n\\qopname\\relax o{diam}(D) \\ge 2 \\frac{\\psi(\\ell(E)\/2)}{c_J}= 2 \\frac{\\varphi(1) \\ell (E)}{2 c_J}\n\\]\ni.e. $\\ell (E) \\le \\max\\left\\{2, \\frac{c_J \\qopname\\relax o{diam}(D)}{\\varphi(1)} \\right\\}= \\beta$.\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=11 cm]{cigar2.eps}\n\\caption{The cigar from $a$ to $x_0$ (the solid line), the core $E$ (the dotted line) and a new carrot given by the constant $c_J M$ (the dashed line).}\n\\end{figure}\n\n\nNote that the length of $E$ inside the ball $B(x_0, r)$ is at least $r$ and thus for the points in $E \\cap \\partial B(x_0, r)$ the distance to the boundary is at least $\\psi(r\/2)$.\nLet us choose that\n \\[\nM = \\frac{\\psi(\\beta)}{\\psi(r\/2)}= \\frac{\\varphi(1)\\beta}{\\psi(r\/2)}.\n \\]\nSince $r \\le \\ell(E) \\le \\beta$ and $\\psi$ is increasing, we have $M \\ge 1$.\n\nLet $z_0 \\in E$ be the first point from $a$ that satisfies $z_0 \\in \\partial B(x_0, r)$. Let us replace $E[z_0, x_0]$ by the radius of the ball $B(x_0, r)$, if necessary. Let us denote this new arc by $E$.\nLet $\\gamma$ be an arc $E$ parametrized by its curve length, such that $\\gamma(0) = a$, $\\gamma(\\ell(E) ) = x_0$. Since\n\\[\n\\frac{\\psi(\\ell(E))}{M c_J} \\le \\frac{\\psi(r\/2)}{c_J}\n\\]\nwe obtain that\n\\[\n\\bigcup_{t \\in(0, \\ell(E) )}B \\left(\\gamma(t), \\frac{\\psi(t)}{M c_J} \\right) \\setminus B(x_0, r) \\subset \\qopname\\relax o{Cig}[a, x_0].\n\\]\nThis yields that\n\\[\n\\bigcup_{t \\in(0, \\ell(E))}B \\left(\\gamma(t), \\frac{\\psi(t)}{M c_J} \\right) \\subset D\n\\]\nand thus \n\\[\n\\psi(t) \\le M c_J \\qopname\\relax o{dist}(\\gamma(t), \\partial D)\n\\le \\frac{M c_J \\beta}{\\ell(E)} \\qopname\\relax o{dist}(\\gamma(t), \\partial D).\n\\]\nThis yields that we may choose $\\alpha = M c_J \\beta$. Thus, $D$ is a $\\psi$-John domain with these $\\alpha$ and $\\beta$.\n\\end{proof}\n\n\n\n\n\\section{Point-wise estimates}\n\n\n\nWe note that by the condition (4) of $\\varphi$ \n\\begin{equation}\\label{varphi_bdd}\n\\psi (t)\\le C_\\varphi \\varphi(1) t \\quad \\text{for all} \\quad t \\ge 0.\n\\end{equation}\n\n\nWe recall a covering lemma from \\cite[4.3. Lemma]{HH-S1} which is valid \nfor a bounded $\\varphi$-John domain. For the previous versions in classical case we refer to \\cite[Theorem 9.3]{Hajlasz-Koskela} and in a special case to \\cite[Lemma 3.5]{HH-SK}.\n\n\n\n\n\n\\begin{lemma}\\cite[4.3. Lemma]{HH-S1}.\\label{lem:covering} \nLet $\\varphi$ satisfies the conditions (1)--(5).\nLet $\\psi :[0,\\infty )\\to [0,\\infty )$ be defined as in \\eqref{equ:psi}.\n Let $D$ in ${\\varmathbb{R}^n}\\,, n\\geq 2\\,,$ be a bounded $\\psi$-John domain \nwith John constants $\\alpha$ and $\\beta$. Let $x_0 \\in D$ the John center. \nThen for every $x\\in D\\setminus B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D))$ there exists a sequence of balls $\\big(B(x_i, r_i)\\big)$ such that $B(x_i, 2r_i)$ is in $D\\,$ for each\n$i=0,1,\\dots\\,,$ and\nfor some constants $K=K(\\alpha, \\qopname\\relax o{dist}(x_0, \\partial D), \\qopname\\relax o{diam} (D),\\varphi )$, $N=N(n)$, and $M=M(n)$ \n\\begin{itemize}\n\\item\n$B_0 = B\\Big(x_0, \\frac12 \\qopname\\relax o{dist}(x_0, \\partial D)\\Big)$;\n\\item\n$\\psi(\\qopname\\relax o{dist}(x, B_i))\\leq K r_i$, and $r_i \\to 0 $ as $i\\to \\infty$;\n\\item\nno point of the domain $D$ belongs to more than $N$ balls $B(x_i, r_i)$; and\n\\item\n$|B(x_i, r_i) \\cup B(x_{i+1}, r_{i+1})| \\leq M |B(x_i, r_i) \\cap B(x_{i+1}, r_{i+1})|$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{remark} (1)\nThe constant $K$ in the previous lemma can be taken to be\n$K=\\max\\{2\\alpha\/\\qopname\\relax o{dist} (x_0,\\partial D), 2\\varphi(1), \\varphi(\\qopname\\relax o{diam} (D))\/\\qopname\\relax o{diam} (D)\\}.$\\\\\n(2) If $D$ is a $\\varphi$-cigar John domain and the John center has been chosen as in Theorem~\\ref{thm:John-John}, then \n\\[\n\\frac{\\alpha}{\\qopname\\relax o{dist}(x_0, \\partial D)}\\le \\frac{c_J^2\\, \\varphi(1) \\left(\\max\\left\\{2, \\frac{c_J \\qopname\\relax o{diam}(D)}{\\varphi(1)} \\right\\} \\right)^2}{\\psi \\left( \\frac1{2c_J} \\psi \\left(\\frac14 \\qopname\\relax o{diam}(D)\\right)\\right) \\psi\\left(\\frac14 \\qopname\\relax o{diam}(D)\\right)}\n\\to \\frac{32 c_J^5}{\\varphi(1)^4}\n\\]\nas $\\qopname\\relax o{diam}(D) \\to \\infty$.\n\\end{remark}\n\n\nWe recall the following definitions.\nLet $G$ be an open set of ${\\varmathbb{R}^n}$.\nWe denote the Lebegue space by $L^{p}(G)$, $1\\le p < \\infty$. By $L^1_p(G)$, $1\\le p < \\infty$, we denote those locally integrable functions whose first weak distributional derivatives belongs to $L^p(G)$ i.e.\\ $L^1_p(G) =\\left \\{ u \\in L^1_{\\textup{loc}}(G): |\\nabla u| \\in L^p(G) \\right\\}$.\nBy $W^{1,p}(G)$, $1\\le p < \\infty$, we denote those functions from $L^p(G)$ whose first weak distributional derivatives belongs to $L^p(G)$ i.e.\\ $W^{1,p}(G) =\\left \\{ u \\in L^p(G): |\\nabla u| \\in L^p(G) \\right\\}$.\n\nTheorem~\\ref{thm:John-John} and Lemma~\\ref{lem:covering} give the following point-wise estimate which we recall from \n\\cite[4.4. Theorem]{HH-S1}. \n\n\n\\begin{theorem}\\label{thm:Riesz}\nLet $\\varphi$ satisfy the conditions (1)--(5).\nLet $\\psi :[0,\\infty )\\to [0,\\infty )$ be as defined in \\eqref{equ:psi}.\n Let $D$ in ${\\varmathbb{R}^n}\\,, n\\geq 2\\,,$ be a bounded $\\varphi$-cigar John domain \nwith a John constant $c_J$ .\nThen there exists a finite constant $C$ and $x_0 \\in D$ such that for every $u\\in L^1_1(D)$ and for almost every $x\\in D$ \nthe inequality\n\\[\n\\big|u(x) - u_{B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D))}\\big| \\leq C \\int_{D} \\frac{|\\nabla u(y)|}{\\psi \\big( |x-y|\\big)^{n-1}} \\, dy\n\\]\nholds.\nHere\n\\begin{equation*}\nC = c\\left(n, c_J, C_\\varphi, C_\\varphi^{\\Delta_2}, \\varphi(1), \\qopname\\relax o{min}\\bigg\\{\\qopname\\relax o{diam}(D), 1\\bigg\\}\\right).\n\\end{equation*}\n\\end{theorem}\n\nWe recall the definitions of $N$-functions and Orlicz spaces.\n\n\\begin{definition}\nA function $H:[0, \\infty) \\to [0, \\infty)$ is an $N$-function if\n\\begin{enumerate}\n\\item[(N1)] $H$ is continuous,\n\\item[(N2)] $H$ is convex,\n\\item[(N3)] $\\lim_{t \\to 0^+}\\frac{H(t)}{t} =0$ and $\\lim_{t \\to \\infty}\\frac{H(t)}{t} =\\infty$.\n\\end{enumerate}\n\\end{definition}\n\nContinuity and $\\lim_{t \\to 0^+}\\frac{H(t)}{t} =0$ yield that $H(0)=0$. Let $00$ such that $f(x)\\le C g(x)$ for all $x$. The notation $f\\approx g$ means that \n$f\\lesssim g\\lesssim f$.\n\nTwo $N$-functions $H$ and $K$ are equivalent, which is written as\n$H\\simeq K$, if there exists $m\\ge 1$ such that \n$H(t\/m)\\le K(t)\\le H(mt)$ for all $t>0$.\nEquivalent $N$-functions give the same space with \ncomparable norms. \nWe point out that $H\\simeq K$ if and only if for the inverse functions $H^{-1}\\approx K^{-1}$.\n\n\nWe assume that $H$ satisfies the $\\Delta_2$-condition, that is, there exists a constant\n$C^{\\Delta_2}_H$ such that\n\\begin{equation}\\label{H_doubling}\nH (2t)\\le C^{\\Delta_2}_H H(t) \\quad \\text{for all} \\quad t>0.\n\\end{equation}\nIf an $N$-function satisfies the $\\Delta_2$-condition then the relations\n$\\simeq$ and $\\approx$ are equivalent. \nThe constant $C^{\\Delta_2}_H$ is called the \n$\\Delta_2$-constant of $H$.\n\n\nLet $G$ in ${\\varmathbb{R}^n}$ be an open set.\nThe Orlicz class is a set of all measurable functions\n$u$\ndefined on $G$\nsuch that\n\\[\n\\int_G H \\Big(|u(x)| \\Big) \\, dx < \\infty\\,.\n\\]\nWe study the Orlicz space $L^H (G)$ which means \nthe space of all measurable functions\n$u$ defined on $G$ such that \n\\[\n\\int_G H \\Big(\\lambda |u(x)| \\Big) \\, dx < \\infty\n\\]\nfor some $\\lambda >0$.\n\nWhenever the function $H$\nsatisfies the $\\Delta_2$-condition, then the space $L^H (G)$ is a vector space and it is \nequivalent to the corresponding Orlicz class.\nWe study these Orlicz spaces and call their functions Orlicz functions.\nThe Orlicz space\n$L^H (G)$ equipped with the Luxemburg norm\n\\[\n\\|u\\|_{L^\\Phi(G)} = \\inf \\left\\{\\lambda >0: \\int_G \\Phi\\left ( \\frac{|u(x)|}{\\lambda}\\right)\\, dx\\le 1 \\right\\}\n\\]\nis a Banach space.\n\n\n\nWe recall the following theorem from \n\\cite[1.3. Theorem]{HH-S1}.\n\n\\begin{theorem}\\label{thm:pointwise-maximal}\nLet $\\varphi$ satisfy the conditions (1)--(5).\nLet $\\psi :[0,\\infty )\\to [0,\\infty )$ be defined as in \\eqref{equ:psi}.\nLet $1\\le p0\\,.\n\\end{equation}\nLet $\\delta :(0,\\infty )\\to [0,\\infty )$ be a continuous function and\nlet $H: [0,\\infty )\\to [0, \\infty)$ be an N-function satisfying the $\\Delta _2$-condition. Suppose that\nthere exists a finite constant $C_H$ such that the inequality \n\\begin{equation}\\label{sum}\nH\\left(h(\\delta (t)) t + \n\\psi (\\delta (t) )^{1-n}(\\delta(t)) ^{n(1-\\frac{1}{p})} \\right)\\le\nC_H t^p \n\\end{equation}\nholds for all $t>0$.\nLet $G$ in ${\\varmathbb{R}^n}$ be an open set.\nIf $\\| f \\|_{L^p (G)} \\le 1$, then \nthere exists a constant $C$ such that \nthe inequality\n\\begin{equation}\\label{main_riesz_inequality}\nH\\left ( \\int_{G} \\frac{|f(y)|}{\\psi( |x-y|)^{n-1}} \\, dy\\right )\n\\le C (M f (x))^p\n\\end{equation}\nholds for every $x\\in {\\varmathbb{R}^n}$. Here the constant $C$ depends on $n$, $p$, $C_\\varphi$, $C_H$, and the $\\Delta_2$-constants of $\\varphi$ and $H$ only.\n\\end{theorem}\n\nOur goal is to find a formula which\nwould give all suitable functions $H$. Examples of some of these\nfunctions were given in \\cite[Section 6]{HH-S1}.\n\nHere we do the preparations to find $H$.\nAssume that there exists $\\alpha \\in[1, {n}\/{(n-1)})$ such that\n$t^\\alpha\/\\varphi(t)$ is increasing for $t>0$. This yields that $t^\\alpha\/\\psi(t)$ is increasing, too. Under this condition inequality \\eqref{h_sum} holds: Since\n\\[\n\\begin{split}\n\\frac{(2^{-k}t )^n}{\\psi (t 2^{-k})^{n-1}} \n&= \\frac{(2^{-k}t )^n}{(2^{-k}t )^{\\alpha(n-1)}} \\cdot \\frac{(2^{-k}t )^{\\alpha(n-1)}}{\\psi (t 2^{-k})^{n-1}}\\\\ \n&\\le (2^{-k}t )^{n- \\alpha(n-1)} \\frac{t^{\\alpha(n-1)}}{\\psi(t)^{n-1}} = 2^{-k(n-\\alpha(n-1))} \\frac{t^n}{\\psi(t)^{n-1}},\n\\end{split}\n\\]\nwe have\n\\[\n\\sum_{k=1}^{\\infty} \\frac{(2^{-k}t )^n}{\\psi (t 2^{-k})^{n-1}} \\le C(n, \\alpha) \\frac{t^n}{\\psi(t)^{n-1}},\n\\quad \\text{where} \\quad C(n, \\alpha) = \\frac{2^{\\alpha(n-1)}}{2^n - 2^{\\alpha(n-1)}}.\n\\]\n\nLet us define the functions $h$ and $\\delta$ such that\n\\[\nh(t) = C(n, \\alpha) \\frac{ t^n}{\\psi(t)^{n-1}} \\quad \\text{and} \\quad \\delta(t) = t^{- \\frac{p}{n}} \\quad \\text{for all} \\quad t>0. \n\\]\nThen,\n\\[\n\\begin{split}\nh(\\delta (t)) t + \n\\psi (\\delta (t) )^{1-n}(\\delta(t)) ^{n(1-\\frac{1}{p})}\n&= h \\left( t^{- \\frac{p}{n}}\\right) t + \\psi \\left(t^{- \\frac{p}{n}} \\right)^{1-n}\\left(t^{- \\frac{p}{n}} \\right) ^{n(1-\\frac{1}{p})}\\\\\n&= \\frac{C(n,\\alpha) t^{-p}}{\\psi\\left(t^{- \\frac{p}{n}} \\right)^{n-1}} t + \\frac{t^{1-p}}{\\psi \\left(t^{- \\frac{p}{n}} \\right)^{n-1}}\\\\\n&= \\frac{(C(n, \\alpha) +1) t^{1-p}}{\\psi \\left(t^{- \\frac{p}{n}} \\right)^{n-1}}.\n\\end{split}\n\\]\nIf we choose \n\\[\nF^{-1}(t) = \\frac{ (C(n, \\alpha) +1) (t^{1\/p})^{1-p}}{\\psi \\left((t^{1\/p})^{- \\frac{p}{n}} \\right)^{n-1}} = \\frac{(C(n, \\alpha) +1) t^{\\frac1p-1}}{\\psi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\n\\]\nand assume that the inverse function of $F^{-1}$ exists, \nthat is \n$(F^{-1})^{-1}=:F$ exists,\nthen \n\\[\nh(\\delta (t)) t + \n\\psi (\\delta (t) )^{1-n}(\\delta(t)) ^{n(1-\\frac{1}{p})} = F^{-1}(t^p)\n\\]\nand thus \n\\[\nF\\left(h(\\delta (t)) t + \n\\psi (\\delta (t) )^{1-n}(\\delta(t)) ^{n(1-\\frac{1}{p})}\\right) = F \\left(F^{-1}(t^p) \\right) = t^p.\n\\]\nUnfortunately,\nthere is a problem with this function $F$ to be a suitable function $H$; namely, the function $F$ is not necessary convex. For example, if $n=2$, $\\varphi(t) = t^{\\frac32}$, and $p=1.9$, then the function $F$ is not convex, see Figure~\\ref{fig:not_convex}. The angle at the point $(1, F^{-1}(1))$ comes from the angle of $\\psi$ at the point $(1, \\psi(1))$. \nOur main theorem, Theorem~\\ref{thm:defn_H_intro} in Introduction, corrects this point: we show that there exists an $N$-function $H$ that is equivalent with $F$.\n\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=6cm]{not_convex.eps}\n\\caption{The function $F$ is not necessary convex.}\\label{fig:not_convex}\n\\end{figure} \n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:defn_H_intro}]\nLet us write that\n\\[\nF^{-1}(t) = \\frac{t^{\\frac1p-1}}{\\psi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\n\\]\nfor $t>0$ and $F^{-1}(0)=0$.\nLet us first show that $F^{-1}$ is strictly increasing.\nAssume then that $0 < s 0$, since $\\alpha < \\frac{n}{n-1}$. We obtain\n\\[\n\\begin{split}\n\\frac{s}{F^{-1}(s)} &= s^{2-\\frac1p} \\psi\\left(s^{-\\frac1n}\\right)^{n-1}\n= s^{2-\\frac1p-\\frac{\\alpha(n-1)}{n}} \\left( \\frac{\\psi\\left(s^{-\\frac1n}\\right)}{\\left(s^{-\\frac1n}\\right)^\\alpha} \\right)^{n-1}\\\\\n&= s^{1-\\frac1p + 1-\\frac{\\alpha(n-1)}{n}} \\left( \\frac{\\psi\\left(s^{-\\frac1n}\\right)}{\\left(s^{-\\frac1n}\\right)^\\alpha} \\right)^{n-1}\n< t^{1-\\frac1p + 1-\\frac{\\alpha(n-1)}{n}} \\left( \\frac{\\psi\\left(t^{-\\frac1n}\\right)}{\\left(t^{-\\frac1n}\\right)^\\alpha} \\right)^{n-1}\n= \\frac{t}{F^{-1}(t)}\n\\end{split}\n\\]\nand thus inequality \\eqref{equ:N\/t_increasing} holds.\n\nLet us then show that $F^{-1}(cs) \\ge 2 F^{-1} (s)$ for all $s \\ge 0$ with $c= 2^\\frac{np}{n-p}$. \nThe inequality $F^{-1}(cs) \\ge 2 F^{-1} (s)$ is equivalent to \n\\[\n2\\frac{\\psi\\left( \\left(\\frac1{cs} \\right)^{\\frac1n} \\right )^{n-1}}{\\left( \\frac1{cs}\\right)^{1- \\frac1p}} \\le\n\\frac{\\psi\\left( \\left(\\frac1s \\right)^{\\frac1n} \\right )^{n-1}}{\\left( \\frac1s\\right)^{1- \\frac1p}}.\n\\]\nBy the condition $(4)$ of $\\varphi$ and the inequality $p< n$, we obtain\n\\[\n\\begin{split}\n2\\frac{\\psi\\left( \\left(\\frac1{cs} \\right)^{\\frac1n} \\right )^{n-1}}{\\left( \\frac1{cs}\\right)^{1- \\frac1p}} &= 2 \\left(\\frac{\\psi\\left( \\left(\\frac1{cs} \\right)^{\\frac1n} \\right )}{\\left( \\frac1{cs}\\right)^{\\frac1n}} \\right)^{n-1} \\left(\\frac1{cs} \\right)^{\\frac{n-1}n - 1 + \\frac1p}\n= \\left(\\frac{\\psi\\left( \\left(\\frac1{cs} \\right)^{\\frac1n} \\right )}{\\left( \\frac1{cs}\\right)^{\\frac1n}} \\right)^{n-1} \\left(\\frac1{s} \\right)^{\\frac{n-1}n - 1 + \\frac1p}\\\\\n&\\le \\left(\\frac{\\psi\\left( \\left(\\frac1s \\right)^{\\frac1n} \\right )}{\\left( \\frac1s\\right)^{\\frac1n}} \\right)^{n-1} \\left(\\frac1s \\right)^{\\frac{n-1}n - 1 + \\frac1p}\n= \\frac{\\psi\\left( \\left(\\frac1s \\right)^{\\frac1n} \\right )^{n-1}}{\\left( \\frac1s\\right)^{1- \\frac1p}}.\n\\end{split}\n\\]\nThe inequality $F^{-1}(cs) \\ge 2 F^{-1} (s)$ yields that $F$ satisfies the $\\Delta_2$-condition. Let us write $F(t) =s$. Then $F^{-1}(s) =t$. Since $F$ is increasing, we have\n\\[\nF(2t) = F(2F^{-1}(s)) \\le F(F^{-1}(cs)) = cs = c F(t). \n\\]\n\nP.\\ H\\\"ast\\\"o has shown in \\cite[Proposition~5.1]{Hasto} that if $f:[0, \\infty) \\to [0, \\infty)$ satisfies the $\\Delta_2$-condition and $x \\mapsto {f(x)}\/{x}$ is increasing, then $f$ is equivalent to a convex function.\nSince $F$ satisfies inequality \\eqref{equ:N\/t_increasing} and the $\\Delta_2$-condition, we obtain that $F$ is equivalent to a convex function $H$.\n\nUsing \n$\\lim_{t \\to 0^+} F^{-1}(t) =0$ and the bijectivity,\nwe obtain\n\\[\n\\begin{split}\n\\lim_{t \\to 0^+}\\frac{F(t)}{t} &= \\lim_{t \\to 0^+}\\frac{t}{F^{-1}(t)}\n= \\lim_{t \\to 0^+} \\frac{t\\, \\psi\\left( \\left(\\frac1t \\right)^{\\frac1n} \\right )^{n-1}}{\\left( \\frac1t\\right)^{1- \\frac1p}} = \\lim_{t \\to 0^+} \\varphi(1)^{n-1} t^{1-\\frac1p + 1 - \\frac{n-1}{n}} =0 \n\\end{split}\n\\]\nand thus also $\\lim_{t \\to 0^+}\\frac{H(t)}{t} =0$. This gives that $H$ is right continuous at the origin. Thus by convexity the function $H$ is continuous on $[0, \\infty)$.\n\nSince $\\varphi(t)\/t^\\alpha$ is decreasing and $\\alpha < \\frac{n}{n-1}$, we obtain \n\\[\n\\begin{split}\n\\lim_{t \\to \\infty}\\frac{F(t)}{t} &=\n\\lim_{t \\to \\infty}\\frac{t}{F^{-1}(t)}\n= \\lim_{t \\to \\infty} t^{2-\\frac1p} \\varphi\\left(t^{-\\frac1n}\\right)^{n-1}\\\\\n& = \\lim_{t \\to \\infty} t^{2-\\frac1p- \\frac{\\alpha(n-1)}{n}} \\left( \\frac{\\varphi\\left(t^{-\\frac1n}\\right)}{\\left(t^{-\\frac1n}\\right)^\\alpha} \\right)^{n-1} \\ge \\lim_{t \\to \\infty} t^{1-\\frac1p+1- \\frac{\\alpha(n-1)}{n}} \\left( \\frac{\\varphi\\left(1\\right)}{1^\\alpha} \\right)^{n-1}= \\infty.\n\\end{split}\n\\]\nSince the functions $F$ and $H$ are equivalent, this yields that \n\\[\n\\lim_{t \\to \\infty}\\frac{H(t)}{t}= \\infty. \n\\] \nThus we have shown that the function $H$ satisfies the conditions (N1) -- (N3).\n\\end{proof}\n\n\\begin{remark}\\label{rem:H-pienilla-arvoilla}\nLater it is crucial to us that \n\\[\nH^{-1}(t) \\approx \\frac{t^{\\frac1p-1}}{\\psi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\n = \\frac{t^{\\frac1p-1}}{\\varphi(1)^{n-1} \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\n = \\varphi(1)^{1-n} t^{\\frac{n-p}{np}}\n\\]\nfor $01} and Theorem\n\\ref{thm:Sobolev-Poincare-p=1}.\n\n\n\\begin{theorem}[Bounded domain, $11}\nAssume that $\\varphi$ satisfies the conditions $(1)$-- $(5)$, $C_\\varphi=1$ in the condition $(4)$, and there exists $\\alpha \\in[1, {n}\/{(n-1)})$ such that\n$t^\\alpha\/\\varphi(t)$ is increasing for $t>0$. \nLet $\\psi$ be defined as in \\eqref{equ:psi}.\nLet $D \\subset {\\varmathbb{R}^n}$, $n \\ge 2$, be a bounded $\\varphi$-cigar John domain with a constant $c_J$. Let $1< p0\\,,\n\\]\nand there exists a constant $C<\\infty$ such that the inequality\n\\[\n\\| u- u_D \\|_{L^H(D)} \\le C \\|\\nabla u \\|_{L^p (D)},\n\\]\nholds\nfor every $u \\in L^1_p (D)$. Here the constant $C$ depends on $n$, $p$, $C_H^{\\Delta_2}$, $C_\\varphi^{\\Delta_2}$, $c_J$ and $\\qopname\\relax o{min}\\{\\qopname\\relax o{diam}(D), 1\\}$ only.\n\\end{theorem}\n\n\\begin{proof}\nAssume that $\\|\\nabla u\\|_{L^p(D)} \\le 1$.\nCorollary \\ref{cor:funktiosta-gradientiin} yields that\n\\[\nH\\left( \\big|u(x) - u_{B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D))}\\big|\\right) \\le C (M |\\nabla u| (x))^p,\n\\]\nwhere the constant $C$ depends on $n$, $p$, $C_H^{\\Delta_2}$, $C_\\varphi^{\\Delta_2}$, $c_J$, and $\\qopname\\relax o{min}\\{\\qopname\\relax o{diam}(D), 1\\}$ only.\nBy integrating over $D$ and using the fact that the maximal operator is bounded whenever $10$ such that\n\\[\nH\\left(1\/{\\lambda} \\right) |D| = \\int_D H\\left(1\/{\\lambda} \\right) \\, dx =1\n\\]\ni.e. $\\lambda = \\|1 \\|_{L^H(D)}$. By solving $\\lambda$ we obtain\n\\[\n\\|1 \\|_{L^H(D)} = \\frac1{H^{-1}\\left( \\frac1{|D|}\\right)}.\n\\]\nSimilarly, we obtain\n\\[\n\\|1 \\|_{L^{H^*}(D)} = \\frac1{(H^*)^{-1}\\left( \\frac1{|D|}\\right)}.\n\\] \nSince\n\\[\nt \\le H^{-1}(t) (H^{*})^{-1}(t) \\le 2t\n\\]\nfor all $t \\ge 0$, see for example \\cite[Lemma 2.6, p.~56]{DieHHR11}, we obtain that\n\\[\n\\|1 \\|_{L^H(D)}\\|1 \\|_{L^{H^*}(D)} = \\frac1{H^{-1}\\left( \\frac1{|D|}\\right) (H^*)^{-1}\\left( \\frac1{|D|}\\right)} \\le |D|.\n\\]\n\nHence, we have shown that\n\\begin{equation*\n\\| u - u_D\\|_{L^H(D)} \\le C \\|\\nabla u\\|_{L^p(D)}\n\\end{equation*}\nfor every $u \\in L^1_p (D)$.\n\\end{proof}\n\n\\begin{example}\\label{exm:H-funktio}\nLet us choose that $\\varphi(t) = t^s$, $s \\in[1, \\frac{n}{n-1})$. \nWe have calculated in Remark~\\ref{rem:H-pienilla-arvoilla} that \nfor every $\\varphi$ the function $H$ satisfies\n$H(t) \\approx t^{\\frac{np}{n-p}}$ whenever $01$, then\n\\[\nH^{-1}(t) \\approx \\frac{t^{\\frac1p-1}}{\\psi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\n = \\frac{t^{\\frac1p-1}}{\\varphi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}} = \n \\frac{t^{\\frac1p-1}}{ \\left(t^{- \\frac{1}{n}} \\right)^{s(n-1)}}\n = t^{\\frac{n-np+sp(n-1)}{np}}\n\\]\nand thus we have that $H(t) \\approx t^{\\frac{np}{n-np+sp(n-1)}}$ for $t> 1$.\n\\end{example}\n\n\\begin{theorem}[Bounded domain, $p=1$]\\label{thm:Sobolev-Poincare-p=1}\nAssume that the function $\\varphi$ satisfies the conditions $(1)$-- $(5)$, $C_\\varphi=1$ in the condition $(4)$, and there exists $\\alpha \\in[1, {n}\/{(n-1)})$ such that\n$t^\\alpha\/\\varphi(t)$ is increasing for $t>0$. \nLet $\\psi$ be defined as in (\\ref{equ:psi})\nLet $D \\subset {\\varmathbb{R}^n}$, $n \\ge 2$, be a bounded $\\varphi$-cigar John domain with a constant $c_J$. Then there exists an $N$-function $H$, that satisfies $\\Delta_2$-condition and \n\\[\nH^{-1}(t) \\approx \\frac{1}{\\psi \\left(t^{- \\frac{1}{n}} \\right)^{n-1}}\\, \\mbox{ for all } t>0\\,,\n\\]\nsuch that the inequality\n\\[\n\\| u- u_D \\|_{L^H(D)} \\le C \\|\\nabla u \\|_{L^1 (D)},\n\\]\nholds\nfor some constant $C$ and\nfor every $u \\in L^1_p(D)$. Here the constant $C$ depends on $n$, $C_H^{\\Delta_2}$, $C_\\varphi^{\\Delta_2}$ and $c_J$ only.\n\\end{theorem}\n\n\n\\begin{proof\nLet us consider functions $u\\in L^1_1 (D)$ such that\n$\\| \\nabla u \\|_{L^1(D)} \\le 1$. The center ball \n$B(x_0, \\qopname\\relax o{dist}(x_0, \\partial D ))$ is written as $B$. In the proof of Theorem~\\ref{thm:John-John} we had chosen $x_0$ so that $\\qopname\\relax o{dist}(x_0, \\partial D ) \\ge \\psi( \\frac14 \\qopname\\relax o{diam}(D))\/c_J$.\nWe show that there exists a constant $C<\\infty$ such that the inequality\n\\begin{equation}\\label{enough}\n\\int_D H(|u(x)-u_B|) \\, dx \\le C\n\\end{equation}\nholds whenever $\\|\\nabla u\\|_{L^1(D)} \\le 1$. This yields the claim as in the proof of Theorem~\\ref{thm:Sobolev-Poincare-p>1}.\n\nSince $H$ is increasing, we first estimate\n\\[\n\\int_D H(|u(x)-u_B|) \\, dx \\le \\sum_{j \\in \\varmathbb{Z}} \\int_{ \\{x\\in D: 2^j < |u(x) - u_B| \\le 2^{j+1}\\}} H(2^{j+1}) \\, dx.\n\\]\nLet us define \n\\[\nv_j(x) = \\max\\bigg\\{0, \\qopname\\relax o{min}\\Big\\{|u(x) - u_B| - 2^j, 2^j\\Big\\}\\bigg\\}\n\\]\nfor all $x\\in D$. \n\\noindent\nIf $x \\in \\{x\\in D :2^j < |u(x) - u_B|\\le 2^{j+1}\\}$, then $v_{j-1}(x) \\ge 2^{j-1}$. \nWe obtain\n\\begin{equation}\\label{equ:main-2}\n\\int_D H(|u(x)- u_B|) \\, dx \\le \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D :v_j(x) \\ge 2^{j}\\}} H(2^{j+2}) \\, dx.\n\\end{equation}\nBy the triangle inequality\nwe have\n\\[\n\\begin{split}\nv_j(x) &= |v_j(x) - (v_j)_B + (v_j)_B| \\le |v_j(x) - (v_j)_B| + |(v_j)_B|.\n\\end{split}\n\\]\nBy the $(1,1)$-Poincar\\'e inequality in a ball $B$, \\cite[Section 7.8]{Gilbarg-Trudinger}, there exists a constant $C(n)$ such that\n\\begin{equation*\n\\begin{split}\n|(v_j)_B|&= (v_j)_B = \\operatornamewithlimits{\\boldsymbol{--} \\!\\!\\!\\!\\!\\! \\int }_B v_j(x) \\, dx \\le \\operatornamewithlimits{\\boldsymbol{--} \\!\\!\\!\\!\\!\\! \\int }_B |u(x) - u_B| \\, dx\\\\ \n&\\le C(n) |B|^{\\frac1n} \\operatornamewithlimits{\\boldsymbol{--} \\!\\!\\!\\!\\!\\! \\int }_B |\\nabla u(x)| \\, dx \\le C(n) |B|^{\\frac1n-1}.\n\\end{split}\n\\end{equation*}\n\nWe continue to estimate the right hand side of inequality \\eqref{equ:main-2}\n\\begin{equation}\\label{equ:main-3}\n\\begin{split}\n&\\int_D H(|u(x)- u_B|) \\, dx \\le \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: |v_j(x) - (v_j)_B|+ C|B|^{-1} \\ge 2^{j}\\}} H(2^{j+2}) \\, dx\\\\\n&\\quad \\le \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D : |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx +\\sum_{2^{j-1} \\le C(n) |B|^{\\frac1n-1} } \\int_{D} H(2^{j+2}) \\, dx\\\\\n&\\quad \\le \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D : |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx +\\sum_{j=-\\infty}^{j_0} \\int_{D} H(2^{j+2}) \\, dx,\n\\end{split}\n\\end{equation}\nwhere $j_0 = \\lceil\\log(C(n) |B|^{\\frac1n-1})\\rceil$.\n\n\nAssume first that $\\qopname\\relax o{diam} (D)$ is so large that $j_0 \\le -2$.\nWhen $t<1$, then $\\psi(t^{-1\/n}) = \\varphi(1) t^{-1\/n}$ by \\eqref{equ:psi} and thus\n\\[\nH^{-1}(t) = \\frac{1}{\\psi(t^{-1\/n})^{n-1}} = \\varphi(1)^{1-n} t^{\\frac{n-1}{n}}.\n\\] \nThus for $t<1$ we obtain that $H(t) \\approx t^{\\frac{n}{n-1}}$.\n This yields that \n\\begin{equation}\\label{equ:main-4a}\n\\begin{split}\n\\sum_{j=-\\infty}^{j_0} \\int_{D} H(2^{j+2}) \\, dx\n&\\approx |D| \\sum_{j=-\\infty}^{\\lceil\\log(C |B|^{\\frac1n-1})\\rceil} 2^{\\frac{n(j+2)}{n-1}} \\le C|D| 2^{\\frac{n}{n-1} \\cdot \\lceil\\log(C |B|^{\\frac1n-1})\\rceil}\\\\\n& \\le C|D| |B|^{\\frac{n}{n-1}( \\frac1n -1)} = C |D| |B|^{-1}\\\\\n&\\le C \\frac{\\qopname\\relax o{diam}(D)^n}{(\\psi( \\frac14 \\qopname\\relax o{diam}(D))\/c_J)^n}.\n\\end{split}\n\\end{equation}\nThis constant does not blow up when $\\qopname\\relax o{diam}(D) \\to \\infty$:\n\\[\n\\frac{\\qopname\\relax o{diam}(D)^n}{(\\psi( \\frac14 \\qopname\\relax o{diam}(D))\/c_J)^n} \\to \\frac{4^n c_J^n}{\\varphi(1)^n} \\quad \\text{as} \\quad \\qopname\\relax o{diam}(D) \\to \\infty.\n\\]\nAssume then that $\\qopname\\relax o{diam}(D)$ is small. \nThis yields that for every $j_0 \\in \\varmathbb{Z}$ the sum $\\sum_{j=-2}^{j_0}H(2^{j+2})$ is finite and depends on $j_0$.\nWe obtain \n\\begin{equation}\\label{equ:main-4b}\n\\sum_{j=-\\infty}^{j_0} \\int_{D} H(2^{j+2}) \\, dx\n\\le \\sum_{j=-\\infty}^{-2} \\int_{D} H(2^{j+2}) + \\sum_{j=-2}^{j_0}H(2^{j+2})< \\infty.\n\\end{equation}\n\n\n\nThen, we will find an upper bound for the sum\n\\begin{equation*}\n\\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D : |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx\\,. \n\\end{equation*}\nSince $\\|\\nabla v_j\\|_{L^1(D)} \\le \\|\\nabla u\\|_{L^1(D)} \\le 1$, Corollary \\ref{cor:funktiosta-gradientiin} yields that\n\\[\n\\begin{split}\n\\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx \n&= \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: H(|v_j(x) - (v_j)_B|) \\ge H(2^{j-1})\\}} H(2^{j+2}) \\, dx \\\\ \n&\\le \\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: C M|\\nabla v_j|(x) \\ge H(2^{j-1})\\}} H(2^{j+2}) \\, dx.\n\\end{split}\n\\]\nWe choose for every $x \\in \\{x\\in D: C M|\\nabla v_j|(x) \\ge H(2^{j-2})\\}$ a ball $B(x, r_x)$,\ncentered at $x$ and with radius $r_x$ depending on $x$, such that\n\\[\nC \\operatornamewithlimits{\\boldsymbol{--} \\!\\!\\!\\!\\!\\! \\int }_{B(x,r_x)} |\\nabla v_j (y)| \\, dy \\ge \\frac12 H(2^{j-1})\n\\]\nwith the understanding that $|\\nabla v_j|$ is zero outside $D$.\nBy the Besicovitch covering theorem\n(or the $5$-covering theorem) we obtain a subcovering $\\{B_k\\}_{k=1}^{\\infty}$ so that we may estimate by the $\\Delta_2$-condition of $H$\n\\[\n\\begin{split}\n&\\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx\n\\le \\sum_{j \\in \\varmathbb{Z}} \\sum_{k=1}^\\infty \\int_{B_k}H(2^{j+2}) \\, dx\\\\\n & \\quad\\le \\sum_{j \\in \\varmathbb{Z}} \\sum_{k=1}^\\infty |B_k| H(2^{j+2}) \\le \\sum_{j \\in \\varmathbb{Z}} \\sum_{k=1}^\\infty C |B_k| \\frac{H(2^{j+2})}{H(2^{j-1})} \\operatornamewithlimits{\\boldsymbol{--} \\!\\!\\!\\!\\!\\! \\int }_{B_k} |\\nabla v_j (y)| \\, dy\\\\\n&\\quad \\le C \\sum_{j \\in \\varmathbb{Z}} \\int_{D} |\\nabla v_j (y)| \\, dy.\n\\end{split}\n\\]\nLet $E_j= \\{x\\in D :2^j < |u(x) - u_B|\\le 2^{j+1}\\}$.\nSince $|\\nabla v_j|$ is zero almost everywhere in $D\\setminus E_j$ and\n$|\\nabla u (x)| = \\sum_j |\\nabla v_j (x)| \\chi_{E_j}(x)$ for almost every $x \\in D$, we obtain\n\\begin{equation}\\label{equ:main-5}\n\\sum_{j \\in \\varmathbb{Z}} \\int_{\\{x\\in D: |v_j(x) - (v_j)_B| \\ge 2^{j-1}\\}} H(2^{j+2}) \\, dx\n\\le C \\int_{D} |\\nabla u (y)| \\, dy \\le C.\n\\end{equation}\nEstimates \\eqref{equ:main-3}, \\eqref{equ:main-4a}, \\eqref{equ:main-4b}\nand \\eqref{equ:main-5} imply inequality\n\\eqref{enough}.\n\\end{proof}\n\n\\begin{remark}\nCorollary \n\\ref{corollary_bdd}\nin Introduction follows from Theorem \\ref{thm:Sobolev-Poincare-p>1}\nand Theorem \\ref{thm:Sobolev-Poincare-p=1}.\n\\end{remark}\n\n\n\n\n\n\\begin{remark}\nIn Theorem~\\ref{thm:Sobolev-Poincare-p=1} the $N$-function $H$ is the best possible in a sense that it cannot be replaced by any $N$-function $K$ that satisfies the $\\Delta_2$-condition and\n\\[\n\\lim_{t \\to \\infty} \\frac{K(t)}{H(t)} = \\infty.\n\\] \nIn \\cite[Theorem 7.2]{HH-S1} we have shown that the corresponding embedding in Theorem~\\ref{thm:Sobolev-Poincare-p=1}\ndoes not hold if \n\\[\n\\lim_{t \\to 0^+} t^n K \\left( \\frac{1}{\\varphi(t)^{n-1}} \\right)=\\infty.\n\\]\nThis is valid for this function $K$. \nBy the definitions of $H^{-1}$ and $\\psi$ we obtain that\n\\[\n\\lim_{t \\to 0^+} t^n K \\left( \\frac{1}{\\varphi(t)^{n-1}} \\right)\n= \\lim_{s \\to \\infty} \\frac1s K \\left( \\frac{1}{\\varphi\\left(s^{-\\frac1n}\\right)^{n-1}} \\right) \n= \\lim_{s \\to \\infty} \\frac{K \\left( H^{-1}(s) \\right)}{H \\left( H^{-1}(s) \\right)}= \\infty,\n\\]\nand thus there does not exists a constant $c$ such that\n\\[\n\\| u- u_D \\|_{L^K(D)} \\le c \\|\\nabla u \\|_{L^1 (D)},\n\\]\nfor every $u \\in L^1_p(D)$.\n\\end{remark}\n\n\\begin{theorem}[Unbounded domain, $1 \\le p0$. \nLet the function $\\psi$ be defined as in \\eqref{equ:psi}.\nLet $D$ in ${\\varmathbb{R}^n}$, $n \\ge 2$, be an unbounded domain that satisfies the following conditions:\n\\begin{itemize}\n\\item[(a)] $D = \\cup_{i=1}^\\infty D_i$, where $|D_1|>0$;\n\\item[(b)] $\\overline D_i \\subset D_{i+1}$ for each $i$;\n\\item[(c)] each $D_i$ is a bounded $\\varphi$-cigar John domain with a constant $c_J$.\n\\end{itemize}\nLet $1\\le p0\\,,\n\\]\nand there exits a constant $C$ such that the inequality\n\\[\n\\inf_{b \\in \\varmathbb{R}} \\| u- b \\|_{L^H(D)} \\le C \\|\\nabla u \\|_{L^p (D)},\n\\]\nholds\nfor every $u \\in L^1_p(D)$. Here the constant $C$ depends on $n$, $p$, $C_H^{\\Delta_2}$, $C_\\varphi^{\\Delta_2}$ and $c_J$ only.\n\\end{theorem}\n\n\nThe proof follows the proof of \\cite[Theorem~4.1]{H-S92}.\n\n\\begin{proof}\nBy Theorems \\ref{thm:Sobolev-Poincare-p>1} and \\ref{thm:Sobolev-Poincare-p=1}\nthere exits a constant $C$ such that the inequality\n\\begin{equation}\\label{equ:S-P-inequ}\n\\| u- u_{D_i} \\|_{L^H(D_i)} \\le C \\|\\nabla u \\|_{L^p (D_i)}\n\\end{equation}\nholds for each $D_i$ and all $u \\in L^1_p(D)$.\nThe constant $C$ does not blow up when the diameter \nof $D_i$ tends to infinity. In the case $11} and \\ref{thm:Sobolev-Poincare-p=1} \nwith $\\varphi (t)=t^s$ give that a bounded $s$-John domain is a $\\left(\\frac{np}{n-np+sp(n-1)}, p \\right)$-Poincar\\'e domain if $1 \\le p 0\\}$ of $Q$ so that the distance from the mushroom to the origin is at least $1$ and at most $4$, see Figure~\\ref{fig:domain}. \nWe assume that a priori the function $\\varphi$ has the properties (1)--(5), but \nwe have to assume here also that $\\varphi(r_m) \\le r_m$.\nWe need copies of the mushrooms. By an isometric mapping we transform these mushrooms onto the side $\\{(x_1, 0, \\ldots, x_n): x_1, x_3, \\ldots, x_n >0\\}$ of $Q$ and\ndenote them by $Q_m^*$ and $P_m^*$. So again the distance from the mushroom to the origin is at least $1$ and at most $4$.\nWe define\n\\begin{eqnarray}\\label{eq:mushroon-domain}\nG=\\textrm{int}\\left(Q \\cup\\bigcup_{m=1}^{\\infty}\\Big(Q_{m}\\cup P_{m}\\cup Q^{*}_{m}\\cup P^{*}_{m}\\Big)\\right).\n\\end{eqnarray}\nSee Figure~\\ref{fig:domain}. We omit a short calculation which shows that $G$ is a $\\varphi$-cigar John domain. \n\n\\begin{figure}[ht!]\n\\includegraphics[width=11 cm]{unbdd_domain_2.eps}\n\\caption{Unbounded $\\varphi$-cigar John domain.}\\label{fig:domain}\n\\end{figure}\n\n\nLet us define a sequence of piecewise linear continuous functions $(u_k)_{k=1}^{\\infty}$ by setting\n\\begin{equation*}\nu_{k}(x):=\n\\begin{cases}\nF(r_k)& \\textrm{in } Q_{k}, \\\\\n-F(r_k) & \\textrm{in } Q^{*}_{k},\\\\\n0 & \\textrm{in} Q_0,\n\\end{cases}\n\\end{equation*}\nwhere the function $F$ will be given in \\eqref{F}.\nThen the integral average of $u_{k}$ over $G$ is zero\nfor each $k$. \n\nThe gradient of $u_k$ differs from zero in $P_m \\cup P_m^*$ only and \n\\[\n|\\nabla u_k(x)| = \\frac{F(r_m)}{r_m}, \\textrm{ when } x\\in P_m \\cup P_m^* \\,.\n\\] \nNote that\n\\begin{equation*}\n\\int_G |\\nabla u_k(x)|^p \\, dx = 2\\int_{P_m}\n\\biggl(\\frac{F(r_m)}{r_m}\\biggr)^p=\n2r_m\\left(\\varphi(r_m)\\right)^{n-1}\\frac{F(r_m)^p}{r_m^p}\\,.\n\\end{equation*}\nWe require that\n\\begin{equation*}\n\\int_G |\\nabla u_k(x)|^p \\, dx =1\\,.\n\\end{equation*}\nHence, we define\n\\begin{equation}\\label{F}\nF(r_m)=\\biggl(\\frac{r_m^{p-1}}{2\\varphi (r_m)^{n-1}}\\biggr)^{1\/p}\\,.\n\\end{equation}\nLet $H$ be an $N$-function. Then,\n\\begin{equation*}\n\\begin{split}\n\\inf_{b\\in\\varmathbb{R}} \\int_G H(|u_k(x) - b | )\\, dx &\\ge \\inf_{b\\in\\varmathbb{R}} \\biggl( |Q_m| \\cdot |H(F(r_m) -b)| + |Q_m^*|\\cdot |H(-F(r_m) -b)| \\biggr)\\\\\n&\\ge r_m^nH(F(r_m))\\,.\n\\end{split}\n\\end{equation*}\nHence, we have \n\\begin{equation*\n\\begin{split}\nr^n_mH(F(r_m))=\nr_m^nH\\biggl(\\biggl(\\frac{r_m^{p-1}}{ 2 \\varphi (r_m^{n-1})}\\biggr)^{1\/p}\\biggr)\n&\\ge r_m^nH\\biggl(\\frac12 \\biggl(\\frac{r_m^{p-1}}{ \\varphi (r_m^{n-1})}\\biggr)^{1\/p}\\biggr).\n\\end{split}\n\\end{equation*}\nThus, there does not exist a\npositive constant $C$ such that the inequality $\\inf_b \\|u-b \\|_{L^H(G)} \\le C \\| \\nabla u\\|_{L^p(G)}$\ncould hold for all $u$ from the appropriate space if\n\\[\n\\lim_{t \\to 0^+} t^nH\\biggl(\\frac12 \\biggl(\\frac{t^{p-1}}{ \\varphi (t)^{n-1}}\\biggr)^{1\/p}\\biggr) = \\infty.\n\\]\nAssume that $\\lim_{t \\to 0^+} t\/\\varphi(t) = \\infty$. If $H(t)= t^q$, then we obtain that the inequality does not hold if \n\\begin{equation}\\label{equ:ehto1}\nq \\ge \\frac{np}{n-p}.\n\\end{equation}\n\nAssume then that we have a sequence $(s_j)$ of positive numbers going to infinity. For each $s_j$ we may choose points $x(j)$\nand $y(j)$ such that the balls $B (x(j), s_j)$ and $B(y(j), s_j)$ are subsets of the first quadrant and \n$B (x(j), 3s_j) \\cap B(y(j), 3s_j)= \\emptyset$. \nWe define a sequence of piecewise linear continuous functions $(v_j)_{j=1}^{\\infty}$ by setting\n\\begin{equation*}\nv_{j}(x):=\n\\begin{cases}\ns_j^{- \\frac{n-p}{p}}& \\textrm{in } B (x_j^1, s_j), \\\\\n-s_j^{- \\frac{n-p}{p}} & \\textrm{in } B (x_j^2, s_j),\\\\\n0 & \\textrm{in } G\\setminus \\left(B (x_j^1, 2s_j) \\cup B (x_j^2, 2s_j) \\right).\n\\end{cases}\n\\end{equation*}\nNow we have\n\\[\n\\int_G |\\nabla u_j|^p \\, dx \\le C s_j^n \\left | \\frac{s_j^{- \\frac{n-p}{p}}}{s_j}\\right|^p \\le C\n\\]\nfor some constant $C$.\nOn the other hand, for any $b\\in\\varmathbb{R}$\n\\[\n\\begin{split}\n\\int_{G} H(|u_j(x)-b|) \\, dx &\\ge C s_j^n H(|s_j^{- \\frac{n-p}{p}} -b|) + C s_j^n H(|-s_j^{- \\frac{n-p}{p}} -b|)\\\\ \n&\\ge C s_j^n H(|s_j^{- \\frac{n-p}{p}}|).\n\\end{split}\n\\]\nThus, there does not exist a\npositive constant $C_1$ such that the inequality $\\inf_b \\|u-b \\|_{L^H(G)} \\le C_1 \\| \\nabla u\\|_{L^p(G)}$\ncould hold for all $u$ from the appropriate space if\n\\[\n\\lim_{s \\to \\infty} s^n H(s^{- \\frac{n-p}{p}}) = \\lim_{s \\to \\infty} s^{\\frac{pn}{n-p} }H\\left(\\frac1s \\right) = \\infty.\n\\]\n\nBy choosing $H(t)= t^q$, we obtain that the inequality does not hold if \n\\begin{equation}\\label{equ:ehto2}\nq < \\frac{np}{n-p}.\n\\end{equation}\n\nIf $\\lim_{t \\to 0^+} t\/\\varphi(t) = \\infty$ and \nif there were an embedding with the Lebesgue space $L^q$ as a target space, then by \\eqref{equ:ehto1} we \nwould have $q < \\frac{np}{n-p}$ and by \\eqref{equ:ehto2} we would have $q \\ge \\frac{np}{n-p}$. Thus the target space cannot be a Lebesgue space.\nThe target space can be $L^q$ only if $\\lim_{t \\to 0^+} t\/\\varphi(t) < \\infty$. \nAnd in this case $q= \\frac{np}{n-p}$. Note that the limit $\\lim_{t \\to 0^+} t\/\\varphi(t)$ exists since $\\varphi$ is increasing and $\\varphi \\ge 0$. If $\\lim_{t \\to 0^+} t\/\\varphi(t)=m >0$, then there exists $t_0 >0$ such that $ \\frac12 m \\varphi(t) \\le t \\le 2m \\varphi(t)$.\n\n\nThus, we have proved the following theorems.\n\n\\begin{theorem}\nLet $\\varphi$ satisfy (1)--(5), and assume that $\\lim_{t \\to 0^+} t\/\\varphi(t) =\\infty$. Let $G$ be the unbounded $\\varphi$-cigar John domain constructed in \\eqref{eq:mushroon-domain}. Let $1\\le p < n$. Then there do not exist numbers $q\\in \\varmathbb{R}$ and $C\\in\\varmathbb{R}$ such that the inequality\n\\[\n\\inf_{b\\in \\varmathbb{R}} \\| u- b \\|_{L^q (G)} \\le C \\| \\nabla u\\|_{L^p(G)} \n\\]\ncould hold for all $u \\in L^1_p(G)$.\n\\end{theorem}\n\n\n\\begin{theorem}\nLet $\\varphi$ satisfy (1)--(5), and assume that $\\lim_{t \\to 0^+} t\/\\varphi(t) = m < \\infty$. Let $G$ be the unbounded $\\varphi$-cigar John domain constructed in \\eqref{eq:mushroon-domain}. Assume that there exist numbers $q\\in \\varmathbb{R}$ and $C\\in\\varmathbb{R}$ such that the inequality\n\\[\n\\inf_{b\\in \\varmathbb{R}} \\| u- b \\|_{L^q (G)} \\le C \\| \\nabla u\\|_{L^p(G)} \n\\]\n holds for all $u \\in L^1_p(G)$. Then $q= \\frac{np}{n-p}$ and there exists $t_0>0$ such that $\\varphi(t) \\approx t$ for all $t \\in(0, t_0]$.\n\\end{theorem}\n\n \n\n\n\n\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Conclusion and Aknowledgements}\nBasing on the performed numerical analysis of the considered explicit examples for input amplitudes we can claim that\n\\begin{itemize}\n\\item\nQuasieikonal unitarization model under suitable choice of $\\lambda _{E}$ allows to describe scattering amplitude beyond the Black Disk Limit.\n\\item\nBoth the QE and QU unitarization schemes can describe a possible (hypothetic) regime of hadron interaction where $1\/2<{\\rm Im}H(s,b)<1$ at $s\\to \\infty$ and $b$ 0. This fact makes it clear that similar information can, in principle at least, be obtained by either frequency domain or the time domain. At excellent discussion of such correlation functions is provided in books such as Ref.~\\cite{Chaikin1995principles}. It is further known that these correlation encodes the interactions present in the magnetic Hamiltonian \\cite{Squires2012introduction}. In general, fast processes correspond to high frequencies and are often best probed in the frequency domain, whereas the converse is true to slow processes. \n\nConnecting $I({\\bm Q}, \\omega,{\\bm \\epsilon},{\\bm \\epsilon}^\\prime)$ to $S^{\\alpha \\beta}({\\bm Q}, \\omega)$ is a topic of considerable active research \\cite{ament2009theoretical,braicovich2010magnetic,letacon2011intense,dean2013persistence,jia2014persistent}. But the connection between these quantities is particularly well-established for Mott insulators based on $5d^5$ electrons, including iridates such as Sr$_2$IrO$_4$ \\cite{kim2017resonant}. The interpretation of such spectral information will be covered in Sec.~\\ref{sec:results}.\n\n\\section{Instrumentation\\label{sec:instrumentation}}\n\nThe primary consideration of tr-\\gls*{RIXS} instrumentations is to achieve sufficient time and energy resolution while ensuring reasonable photon throughput. \\gls*{RIXS} is one of the most photon-hungry X-ray measurement techniques due to its small inelastic scattering cross-section. Recent advances in energy-analyzing spectrometer designs have made possible much-improved energy resolution and faster data collection at synchrotron light sources \\cite{ghiringhelli2006saxes,dvorak2016towards, shvyd2013merix, kim2018quartz}. In this section, we will focus on aspects specific to the time-resolved experiments.\n\n\\subsubsection{Spatial overlap}\n\nAs is in most other pump-probe experiments, the probe X-ray photon footprint needs to be contained within that of the pump laser. Equally important, the penetration depth of the X-ray normal to the sample surface should not exceed that of the pump laser. At normal incidence, the X-ray penetration depth is usually much larger than that of the optical laser. To ensure a good match between the laser pumping volume and the X-ray probing volume, we could resort to one, or both of the following approaches - (1) using samples thinner than the penetration depth of the pump laser; and (2) aligning the X-ray to come in more grazing relative to the sample surface. Approach (1) could be achieved by thinning down bulk samples, or using films. For thicker samples, approach (2) needs to be used. The X-rays are usually at less than $2^\\circ$ incidence angle relative to the sample surface and the laser is typically at e.g.\\ $10^\\circ$. This non-collinear geometry in approach (2) leads to decreased time resolution. Also, the large X-ray footprint due to glancing incidence gives rise to a large laser spot and hence reduced pump fluence.\n\nThe X-ray penetration depth in the soft X-ray regime ($\\hbar\\omega<2000$~eV) is usually a fraction of a micron on resonance. This makes it easier to implement approach (1) with a collinear pump-probe geometry. For harder X-rays ($\\hbar\\omega>4000$~eV), the penetration depth of a few microns requires approach (2) in many cases. \n\n\n\\subsubsection{Energy resolution}\n\nThe RIXS spectrometer works under the principle that energy dispersive optics distributes X-ray photons with different energies into slightly different directions. After prorating certain distance, X-ray photons with different energies are spatially separated, which can be differentiated by detectors with good spatial resolution. The following are some of the most important factors affecting the spectrometer energy resolution: (1) the length of the spectrometer arm; (2) the photon footprint along the energy dispersing direction; and (3) the size of the X-ray detector pixels. In the soft X-ray regime, ruled gratings are used to disperse X-rays, while high-quality diced single crystals are used for hard X-rays. A detailed description of the principles of spectrometer designs, as well as estimates of the energy resolution, are presented in the supplementary materials. State-of-the-art spectrometers often deliver sub-50~meV energy resolution \\gls*{FWHM} relatively comfortably. Some of the more modern spectrometers and novel designs have an energy resolution approaching 10~meV \\cite{dvorak2016towards,kim2018quartz}, though often only in a particular X-ray energy range.\n\nWith these advances in X-ray spectrometers, the incident X-ray needs to be well monochromated to achieve a high combined total energy resolution. The generic {X}-ray bandwidths from existing \\gls*{XFEL}s are much larger than the resolution of the spectrometers. For example, in the \\gls*{SASE} mode, the typical \\gls*{FWHM} of the incident X-ray at 9~keV is $\\sim$20~eV before entering the monochromator. Ideally the energy resolution of the X-ray shining on the sample should match that of the spectrometer. In this scheme, only a tiny fraction of the SASE beam (within much less than 100 meV) is expected to pass the monochromator for an optimized setup. To achieve a higher X-ray intensity at the sample without compromising the total energy resolution, we argue that \\gls*{tr}-\\gls*{RIXS} will benefit greatly from the seeded operation mode of \\gls*{XFEL}s \\cite{amann2012demonstration, Ratner2015experimental}. While the total X-ray flux exiting the undulator is reduced compared with that in the \\gls*{SASE} mode, the incident photon flux within the desired energy bandpass will actually increase. For the current \\gls*{XFEL}s, self-seeding tends to be more reliable in the hard X-ray regime.\n\nAs mentioned in Sec.~\\ref{sec:rixs}, the different X-ray polarization dependences of different excitations can be used to help isolate the desired signal. Since magnetic X-ray scattering rotates the X-ray polarization, it is often useful to place the spectrometer as close as possible to $90^\\circ$, using a horizontal scattering plane and horizontal incident X-ray polarization. In this way, the undesirable Thompson structural scattering is suppressed, and the visibility of the magnetic signal is enhanced. This works especially well for hard X-ray experiments, in which the whole Brillouin zone can be covered by moving the spectrometer only a few degrees. With a horizontal scattering plane, the X-ray footprint will tend to elongate the beam horizontally, so setting the spectrometer to disperse the X-rays vertically is useful to maintain a constant vertical X-ray source size, independent of the scattering angle. \n\n\\subsubsection{Momentum resolution\\label{sec:momentum_resolution}}\n\nAn important advantage of \\gls*{RIXS}, as compared with Raman spectroscopy, is the much larger momentum transfer between the incident and scattered photons. The main contribution to the momentum resolution is the acceptance angle of the \\gls*{RIXS} spectrometer. Typical momentum resolution is on the order of $\\sim 0.01$~\\AA$^{-1}$ for soft X-rays around 1~keV and $\\sim 0.1$~\\AA$^{-1}$ for hard X-rays around 10~keV. The worse momentum resolution in the hard X-ray regime is simply due to the larger photon momentum. In most cases the momentum resolution can be improved with a smaller acceptance angle provided there are enough scattered X-ray photons to make the experiment feasible. Notably, hard X-rays have significantly larger momentum transfer than soft X-rays, which allows access to multiple Brillouin zones even within the constraint of grazing incidence geometry needed for the penetration depth correction noted above.\n\n\\subsubsection{Time resolution\\label{sec:time_resolution}}\n\nIn \\gls*{tr}-\\gls*{RIXS}, the temporal and energy resolution are fundamentally limited by the time-bandwidth product, $\\Delta E \\Delta \\tau \\sim \\hbar$ setting an ultimate limit on the best resolution. For a phase coherent Gaussian pulse with 100~meV energy resolution, the temporal resolution limit is approximately 40~fs. On top of the energy-time indeterminacy, the total time resolution has contributions from both, the durations of the X-ray pulse and of the pump optical laser, similar to other pump-probe experiments at \\gls*{XFEL}s.\n\nAnother potential contribution to the time resolution can come from what is called the ``wave-front-tilt'' effect \\cite{MOP019}. In this process the incident X-ray pulse entering the monochromator is stretched in time upon exit due to different X-ray photon path lengths through the optics. This effect tends to be more severe for longer-wavelength X-rays and for higher-resolution monochomators, and could reach a few picoseconds in the very soft X-ray regime with a single grating. \n\nThe total time resolution can be further limited by the so-called ``jitter'', which is the uncertainty in the relative arrival time between the pump and probe pulses. The ``jitter'' tends to be between 20-100~fs depending on the \\gls*{XFEL} in question and can be corrected for using a ``timing tool'' \\cite{harmand2013achieving, kang2017hard}. There is also a potential geometrical contribution to the total time resolution arising from the relative optical path length difference between the X-ray and the pump laser over the photon footprint. The collinear geometry where the laser is parallel to the X-ray minimizes the geometrical contribution. This geometry is widely adopted in the soft X-ray elastic scattering experiments, and is expected to be used in \\gls*{tr}-\\gls*{RIXS} in the soft X-ray regime. For hard X-rays and in the presence of bulk samples, matching the pump-probe volume may require a substantial angle between the X-ray and the pump laser at the price of time resolution. One can, in principle, compensate for this effect by imparting a spatially dependent delay in the laser pulse. \n\n\\section{First Results\\label{sec:results}}\nThe first reported magnetic \\gls*{tr}-\\gls*{RIXS} experiments were conducted on Sr$_{2}$IrO$_{4}$ \\cite{Dean2016Ultrafast}, a model quasi-two-dimensional square-lattice quantum antiferromagnet \\cite{jackeli2009mott,Wang_prl_2011,Kim_prl_2012}. Long-range magnetic order occurs below $T_{N}\\approx 240$~K and arises from the interplay of several degrees of freedom \\cite{Gao_prb_1998, Kimprl2008}. Each Ir atom has a $5d^5$ electronic configuration and sits at the center of an oxygen octahedron. All five $d$-electrons reside in the t$_{2g}$ level due to the large crystal field splitting. Strong spin-orbit coupling $\\sim$400~meV further splits the t$_{2g}$ level into the $J_\\text{eff} = 3\/2$ state occupied by 4-$5d$ electrons, and a single-occupied $J_\\text{eff} = 1\/2$ state. The Ir atoms form a square net resembling that of the copper ions in the layered high-T$_C$ cuprates, and a modest Coulomb repulsion induces an insulating band gap of 600~meV (see Fig.~\\ref{fig:CFandMG}a and b). Sr$_{2}$IrO$_{4}$ belongs to the Ruddlesden-Popper series of layered iridates Sr$_{n+1}$Ir$_n$O$_{3n+1}$. The band-gap of the spin-orbit Mott insulating state successively reduces with increasing neighboring Ir-O layers, $n$. In the $n\\to\\infty$ end member SrIrO$_{3}$ the gap is closed making the system metallic (c.f.\\ Fig.~\\ref{fig:CFandMG}c and d) \\cite{Hao2017On}.\n\n\\begin{figure*}[tbh]\n\\includegraphics[width=\\linewidth]{Figs\/CFandMG.pdf}\n\\caption{(a) Crystal-field splitting, spin-orbit coupling, $\\lambda$, and Coulomb induced Mott-gap, $\\Delta$, in Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$. (b)-(d) Schematic electronic structure of Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$.}\n\\label{fig:CFandMG}\n\\end{figure*}\n\nThe charge degree of freedom in these materials was studied by time-resolved optical reflectivity measurements \\cite{Dean2016Ultrafast}. Here electrons are excited from the lower (LHB) into the upper Hubbard band (UHB) by 2~$\\mu$m laser pulses with an energy that matches the electronic band gap. Figure~\\ref{fig:spectra}(a) shows the relevant timescales in Sr$_{2}$IrO$_{4}$ that are needed to recombine the charge carriers under different laser fluences. The plot reveals a decay time that is attributed to the excitation of the charge carriers, and a fast and slow recovery process in the sub-picosecond and few-picosecond regime, respectively.\n\nSince the magnitude of the electron bandwidth, the Coulomb repulsion and the spin-orbit coupling share similar energy scales in the iridates, the same 2~$\\mu$m (620~meV) laser pulse also affects the magnetic properties \\cite{Dean2016Ultrafast}. A combined \\gls*{REXS} and \\gls*{RIXS} study at the hard X-ray Ir $L_{3}$ edge probed the energy, momentum and time-dependent response of the transient magnetic state at $T = 110$~K (c.f.\\ Fig.~\\ref{fig:rixs_process}). Figure~\\ref{fig:spectra}(b) and (c) display the time and fluence dependence of a magnetic Bragg peak in Sr$_{2}$IrO$_{4}$. After the initial suppression of peak intensity following the optical excitation, the three-dimensional long-range antiferromagnetic order fully recovers within several hundred picoseconds. Intriguingly, a partial restoration of magnetism is observed already within a few picoseconds. This is further clarified by tr-\\gls*{RIXS}, as discussed below.\n\n\\begin{figure*}[tbh]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Figs\/spectra.pdf}\n\\caption{Time dependence of (a) the relative optical reflectivity ($\\Delta R \/ R$) and (b) and (c) the magnetic Bragg peak in Sr$_{2}$IrO$_{4}$ after 620~meV photo-excitation at different laser fluences. Excitation spectrum at (d) $\\textbf{Q} = (\\pi, 0)$ and (e) $\\textbf{Q} = (\\pi, \\pi)$ in the non-perturbed state and 2~ps after the laser pulse, under the pump fluence of 6~mJ\/cm$^2$. The difference in the RIXS spectrum before and after the optical excitation at $\\textbf{Q} = (\\pi, \\pi)$ is shown in (f). Taken from Ref.~\\cite{Dean2016Ultrafast}.}\n\\label{fig:spectra}\n\\end{figure*}\n\nThe \\gls*{RIXS} spectra displayed in Fig.~\\ref{fig:spectra}(d-f) show the first ever view of magnetic short-range correlations within a photo-excited ultrafast transient state. Magnetic excitations were measured before, and 2~ps after, the arrival of pump laser pulses, and with a pump fluence large enough to fully suppress the 3D magnetic order \\cite{Dean2016Ultrafast}. The low-energy excitation spectrum of Sr$_{2}$IrO$_{4}$ features a dispersing spin-wave below 200~meV (see Fig.~\\ref{fig:spectra}(d)) and an orbital excitation of the $J_{\\text{eff}} = 1\/2$ state around 600~meV. The main result of the study demonstrates that despite the destruction of the long-range magnetic order, magnons are already observed 2~ps after the impact of the laser pulse. Furthermore, the recovery timescale of these predominantly two-dimensional in-plane magnetic fluctuations matches the partial recovery of long-range magnetic order as found by time-resolved \\gls*{REXS} (shown in Fig.~\\ref{fig:time_evolution} and discussed below).\n\nThe various timescales and their fluence dependences are shown in Fig.~\\ref{fig:time_evolution}. Both charge and magnetic degrees of freedom exhibit fast and slow recovery dynamics \\cite{Dean2016Ultrafast}. Most strikingly, the recovery timescale of the 2D magnetic fluctuations matches that of slower charge recovery, providing direct evidence for a coupling between them. The authors further suggest that the slower restoration of 3D magnetic order may be attributed to incoherently oriented IrO$_{2}$ planes along the tetragonal axis.\n\nThis pioneering experiment paves the way for further time-resolved \\gls*{REXS}\/\\gls*{RIXS} experiments that may lead to a full microscopic understanding of the correlated ground state in the Sr$_{n+1}$Ir$_{n}$O$_{3n+1}$ series. While electrons were pumped across the Mott-gap, the bandwidth, Coulomb repulsion and spin-orbit coupling were not tuned directly in this work. A selective pump of the Ir-O bonds in Sr$_{2}$IrO$_{4}$ with mid-infrared ultrafast laser pulses, for instance, could tune the crystal-field environment and modulate the magnetic interaction strength of the system\\footnote{In general, infrared pulses are not the only ones used in a time-resolved XFEL experiments. A plethora of fundamental collective excitations between 0.3$\\sim$3~THz can be optically targeted using optically-pumped organic crystals \\cite{kubacka2014}.}. Observing the time-dependent evolution of the transient state after shaking the Ir-O bonds will not only disentangle the lattice and electronic degrees of freedom, but may also demonstrate how to drive the material into specific magnetic states by pumping another degree of freedom (in this case, the lattice). The role of Coulomb repulsion may be clarified by investigating other members in the series. For example, Sr$_{3}$Ir$_{2}$O$_{7}$ features two closely-coupled Ir-O layers in the crystal structure and shows major modifications in the materials properties \\cite{Biswas_intech_2016,Kim_prl_2012_2}. This material lies on the verge of a metal-to-insulator transition with a heavily reduced Mott-gap (see Fig. \\ref{fig:CFandMG} c) \\cite{Biswas_intech_2016}. Magnetic long-range order emerges below $T_{N}\\approx 285$~K with a magnetic moment orientation perpendicular to the tetragonal basal plane \\cite{Boseggia_prb_2012}. The excitation spectrum strongly deviates from the isotropic Heisenberg model that describes Sr$_{2}$IrO$_{4}$ with a much-increased magnetic gap. Thus, the restoring forces for the magnetic ground state in Sr$_{3}$Ir$_{2}$O$_{7}$ are expected to be different from those in Sr$_{2}$IrO$_{4}$, and a study of their dynamics following a pump may further clarify the underlying processes. \n\n\\begin{figure*}[tbh]\n\\includegraphics[width=\\linewidth]{Figs\/time_evolution.pdf}\n\\caption{(a) Fluence dependence of the decay time that is required to destroy the ground state. (b) Fast recovery of the charge degree of freedom measured via optical reflectivity. (c) Slow recovery of the charge degree of freedom that matches the recovery timescale of the 2D magnetic correlations. (d) Slow recovery of the 3D magnetic long-range order. Taken from \\cite{Dean2016Ultrafast}.}\n\\label{fig:time_evolution}\n\\end{figure*}\n\n\nIn contrast to other members of the series, SrIrO$_{3}$ is a correlated paramagnetic metal close to the metal-to-insulator transition (see Fig. \\ref{fig:CFandMG} d) \\cite{Biswas_intech_2016}. It can be grown as thin films on substrates where the relative lattice mismatch between the film and the substrate provides an additional tuning parameter. The epitaxial strain induces changes in the Ir-O bond angle and could drive the system over the metal-to-insulator transition. Thus, the variation of film thickness and substrate material enables a selective realization of the material properties. This further adds to the possibilities of altering the magnetic ground state by optical means, and may finally allow one to fully disentangle the coupled degrees of freedom in the series.\n\nThe strong experimental push towards the understanding of electronic short-range correlations in transient states parallels several recent theoretical studies of the Hubbard model that is believed to describe the main properties of many strongly correlated materials \\cite{Eckstein2016ultra,Werner2012,Bittner2018EDMFT,Kogoj2014polaron,Du2017thermalization,Wang2017coherent,Tsuji2013PRL,Mentinknatcommun2015,Mentinkrevew2017,Secchi2013AnnalsPhysics}. Particularly relevant in view of the tr-RIXS study on Sr$_{2}$IrO$_{4}$ are the results in Ref. \\cite{Eckstein2016ultra} that report the interaction of short-range spin fluctuations with the charge degree of freedom in photo-excited Mott insulators. Using non-equilibrium dynamical mean-field theory, the study provides evidence for short-range magnons in a two-dimensional paramagnetic system with relaxation times in the femtosecond time domain. The extension of the theoretical model to an antiferromagnetically ordered state may lead to a microscopic understanding of the time difference between 2D fluctuations and 3D order in Sr$_{2}$IrO$_{4}$ \\cite{Dean2016Ultrafast}.\n\n\n\\section{Vision for the future}\\label{sec:future}\nAs demonstrated in previous sections, \\gls*{tr}-\\gls*{RIXS} has unique advantages for investigating quantum magnetism in transient states. Here we discuss future developments of \\gls*{tr}-\\gls*{RIXS} instrumentations and the scientific problems that will be addressed because of these advances.\n\n\\subsection{New XFELs and RIXS spectrometers}\nHitherto, progress in \\gls*{tr}-\\gls*{RIXS} has been slowed by the limited availability of suitable X-ray sources and spectrometers. In the past 10 or so years since the first \\gls*{XFEL}s came online, their primary focus has been on numerous exciting opportunities in more tractable techniques such as \\gls*{XAS} and X-ray diffraction. Efforts to realize more complex experiments are, however, already well under way. A major increase in the availability and quality of \\gls*{XFEL} sources is likely to drive further progress in realizing highly challenging experiments such as \\gls*{tr}-\\gls*{RIXS}. In particular the more established sources such as the Hamburg \\gls*{FLASH}, the Trieste \\gls*{FERMI}, the Stanford \\gls*{LCLS} and the \\gls*{SACLA} will be joined by the European \\gls*{XFEL}, \\gls*{PAL-XFEL}, the Stanford LCLS-II and \\gls*{SwissFEL} \\cite{flash2006, elettra2012, lcls2010, sacla2012,europeanXFEL2017,pal2017, swissfel2010}. Dramatic improvements in brightness, stability, pulse length and pulse repetition rate are expected. The small \\gls*{RIXS} cross-section may particularly benefit from improved time-average spectral brightness that can potentially exceed that of synchrotrons significantly, making feasible new experiments e.g. probing magnetism in single-layer films. These improvements are particularly beneficial when ultra-high energy resolution is pursued. \n\nThe spectrometer required to analyze the X-ray energy also represents a significant challenge, but several teams have successfully performed experiments at \\gls*{XFEL}s \\cite{Rusydi2014,dell2016extreme,Dean2016Ultrafast,Mitrano2018Ultrafast}. To date, \\gls*{XFEL}-based \\gls*{RIXS} tends to lag behind synchrotron experiments in terms of energy and momentum resolution. Many of the technical developments implemented at synchrotrons \\cite{dvorak2016towards, Brookes2018beamline} can potentially be employed at \\gls*{XFEL}s provided further practical complications, such as limited experimental floor space, compatibility with other experiments, finite beam stability, can be avoided or overcome. In many cases shorter spectrometers ($ \\lessapprox 2$~m compared to $\\sim 15$~m at synchrotrons) are more practical at \\gls*{XFEL}s. A recent compact soft X-ray spectrometer design has delivered down to 300~meV resolution at 1~keV incident energy \\cite{chuang2017modular, Mitrano2018Ultrafast}. Hard X-ray experiments with a 1~m spectrometer achieved $70$~meV at 11.2~keV \\cite{Dean2016Ultrafast}. Looking to the future, multiplexing techniques that can efficiently measure the \\gls*{RIXS} incident energy dependence \\cite{Chuang2016Multiplexed}, streaking approaches to measuring the time-delay or analyzing the scattered x-ray polarization \\cite{braicovich2014simultaneous} will be interesting routes to consider.\n\n\n\\subsection{Evolution of tr-RIXS}\nDriven by these technical developments we anticipate considerable advances in \\gls*{tr}-\\gls*{RIXS} in the coming years. A primary focus will be far more detailed and precise characterizations of spin, charge and orbital behaviors in transient states, from which opens routes disentangling the complex interplay between these different degrees of freedom. For example, time-resolved diffraction measurements have recently quantified how terahertz-frequency optical pulses modify the crystal structure of cuprates \\cite{mankowsky2014nonlinear}. \\gls*{RIXS} could potentially be used to quantify how magnetic exchange is modified in such a transient state. One can also consider using \\gls*{RIXS} to evaluate change in spin stripe-magnon interactions in similar transient states \\cite{Miao2017Charge}. These types of coupling are generic to many different types of correlated system including $3d$, $4d$ and $5d$-electron oxides such as cuprates, nickelates and osmates -- systems in which steady-state \\gls*{RIXS} is becoming more insightful \\cite{dean2013persistence, Fabbris2017Doping, Calder2016spin}. By carefully choosing the optical pump, it would be desirable to selectively drive the different forms of order in the material. With the help of RIXS, it is then possible to address the chicken and egg interdependencies of these order parameters. More ambitiously, the application of higher laser pump fluences, and oscillatory Fluoquet-type pluses can potentially access transient states not adiabatically connected to the states at thermal equilibrium. The extent to which these states have novel properties remains to be seen \\cite{Wang2018theoretical}. It is also notable that \\gls*{FT}-\\gls*{IXS} has recently observed the decay of optical phonons into pairs of acoustic phonons \\cite{Teitelbaum2018Direct}, direct measurement of similar processes between spin waves would be very insightful in conceptualization the evolution of magnetic states out of equilibrium.\n\n\n\n \n\\subsection{New types of RIXS at XFELs}\nThe high brilliance of new sources should allow `diffract-then-destroy' types of \\gls*{RIXS} experiment for detecting the collective excitations in quantum materials under extreme conditions at equilibrium. These involve problems in which the state of interest only exists fleetingly, but the experiment can be performed within the short X-ray pulse duration before the state is destroyed. The most prominent example of this approach is in protein crystallography where diffraction data is collected before the sample is destroyed by radiation damage. With expected higher \\gls*{XFEL} peak brightness, similar ideas will also likely become important for ultrafast quantum materials research. Single-shot \\gls*{XFEL} experiments in pulsed magnetic fields have been demonstrated using X-ray diffraction, paving the way for investigations of magnetic excitation under large magnetic fields in in quantum materials \\cite{Gerber2015}. One can also envisage accessing ultra-high pressures transiently, following themes in \\gls*{XFEL}-based geology research and other areas. An as yet unexploited idea would be to try to measure \\gls*{RIXS} at extremely low temperatures, where the sample temperature would be elevated by beam heating under normal circumstances. \n\nIn standard \\gls*{IXS} experiments, one determines the dynamical properties of the sample in the frequency domain through the energy dependence of $S^{\\alpha \\beta}({\\bm Q}, \\omega)$. \\gls*{XFEL}s open the opportunity to determine $S^{\\alpha \\beta}({\\bm Q}, \\omega)$ in the time domain. This approach has been illustrated in recent experiments by Trigo and collaborators \\cite{Trigo2013}. In this study, the structural X-ray diffuse scattering of crystalline germanium was measured as a function of time delay after photo-excitation. Fourier transforming the oscillatory component of the diffuse scattering yielded the low-energy phonon dispersion. In the limit of negligible pump intensity, the phonon energy dispersion obtained should be the same as that accessible by standard \\gls*{IXS}, leading to this technique being dubbed \\gls*{FT}-\\gls*{IXS}. Notably, the energy resolution of \\gls*{FT}-\\gls*{IXS} is determined not by the spectrometer as in conventional \\gls*{IXS}, but by the longest delay possible, so \\gls*{FT}-\\gls*{IXS} has good sensitivity to slow, low-energy excitations. Clever choices of photo-excitations may also open way to picking out modes of particular interest, which have small structure factors in standard \\gls*{IXS} measurements. Combining \\gls*{FT}-\\gls*{IXS} with atomic core-hole resonance may allow access to electronic excitations from the charge, spin and orbital degrees of freedom (see Sec.~\\ref{sec:rixs}) similar to \\gls*{RIXS}. This would represent what might be thought of as Fourier-transform or time-domain \\gls*{RIXS}, but as far as we are aware such experiments have not yet been successfully performed. Some of these low-energy charge and orbital excitations, e.g.\\ amplitudons, phasons, orbitons, etc.\\ are not accessible by neutrons, and could require energy resolution well below that of any spectrometer-based RIXS at XFELs and synchrotrons either today or anticipated over the next decade. Experimentally identifying these charge\/orbital excitations and understanding their evolution in the temperature-doping phase diagram would be particularly interesting for cuprates and Fe-based superconductors, especially since these excitations are theoretically proposed as critical to the formation of superconductivity \\cite{fradkin2015colloquium, lee2009ferro}.\n\nNot only the time structure, but also the large peak intensity of \\gls*{XFEL} pulses will allow access to novel types of RIXS processes. To date, in the vast majority of \\gls*{RIXS} experiments, the X-ray has been assumed to be a linear perturbation to the material \\cite{ament2011resonant}. Under this assumption, the RIXS spectrum is interpreted as originating from the interaction of a single photon with the sample, where the incident photon density is low enough for there to be a negligible probability of one photon encountering the effects of other photons. \\gls*{XFEL}s now deliver peak photon intensity sufficient to access non-linear processes. In general two cases are distinguished, in which the stimulating X-rays arise either from a separate X-ray beam, or they arrive from scattered photons in the sample. The former case is often referred to as stimulated Raman scattering and has only been realized using photons in the ultra violet regime so far \\cite{Ferrori2016}. The latter case takes advantage of the finite energy distribution in the \\gls*{XFEL} beam, leading to an amplified spontaneous emission that emerges from noise around the center of the X-ray pulse \\cite{Rohringer2012}. The last few years have seen $2p$ core hole stimulated emission in crystalline silicon proving the feasibility of solid-state non-linear experiments \\cite{Beye2013stimulated}. As emphasized by Beye and collaborators \\cite{Beye2013stimulated} stimulated processes can enhance the efficiency of \\gls*{RIXS} by several orders of magnitude and confine emission into a well-defined cone allowing the scattered photons to be more efficiently collected. Non-linear \\gls*{RIXS} therefore has potential to circumvent radiation damage problems. Perhaps even more interestingly, one can use the enhanced specificity of such process to access, for example, hidden order parameters and excitations, such as those only accessible via quadrapole resonances \\cite{Wang2017On}.\n\n\\section{Summary}\nRecent years have seen the advent of \\gls*{tr}-\\gls*{RIXS} as a flexible and rich probe of non-equilibrium phenomena. We argue that this technique has great potential for measuring spin, charge, orbital and lattice excitations after photo-excitation as a means of unpicking how these degrees of freedom intertwine to form emergent states and for characterizing the new laser-driven transient states. The increasing number of operating \\gls*{XFEL} facilities and \\gls*{tr}-\\gls*{RIXS} instruments will not only serve as a basis to solve long-standing scientific questions, but also to investigate physics beyond the limit of linear response theory and to enable the development of novel techniques such as stimulated or Fourier-transform \\gls*{RIXS}. Ultra-bright X-ray pulses may also be used to perform future experiments on samples susceptible to radiation damage or at extremely low temperatures and high magnetic fields, accessing phase-space regions that could not be reached via X-ray scattering before. Thus, the prospected capabilities of tr-RIXS in its various forms are pointing towards a bright future.\n\n\\section{Acknowledgements}\nWe acknowledge D.~Zhu, R.~Mankowsky, V.~Thampy, X.~M.~Chen, J.~G.~Vale, D.~Casa, Jungho Kim, A.~H. Said, P.~Juhas, R.~Alonso-Mori, J.~M.~Glownia, A.~Robert, J.~Robinson, M.~Sikorski, S.~Song, M.~Kozina, H.~Lemke, L.~Patthey, S.~Owada, T.~Katayama, M.~Yabashi, Yoshikazu Tanaka, T.~Togashi, Jian Liu, C.~Rayan Serrao, B.~J.~Kim, L.~Huber, C.-L.~Chang, D.~F.~McMorrow, and M.~F\\\"{o}rst for the considerable joint effort in demonstrating \\gls*{tr}-\\gls*{RIXS}. We thank J.~St\\\"{o}hr and G.~Ingold for valuable discussions. This work is supported by the U.S.\\ Department of Energy, Office of Basic Energy Sciences, Early Career Award Program under Award No.\\ 1047478. Work at Brookhaven National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No.\\ DE-SC0012704. The work at Argonne National Laboratory was supported by the U.S.\\ Department of Energy, Office of Basic Energy Sciences, under Contract No.\\ DE-AC0206CH11357. The work at ShanghaiTech U.\\ was partially supported by MOST of China under the grand No.\\ 2016YFA0401000. The work at ICFO received financial support from Spanish MINECO (Severo Ochoa grant SEV-2015-0522), Ram\\'on y Cajal programme RYC-2013-14838, FIS2015-67898-P (MINECO\/FEDER), Fundaci\\'o Privada Cellex, and CERCA Programme \/ Generalitat de Catalunya. D.G.M.\\ acknowledges funding from the Swiss National Science Foundation, Fellowship No.\\ P2EZP2\\_175092.\n\n\n\\bibliographystyle{rsta}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The Golod-Shafarevitch $p$-group theorem}\n\nSet \\mbox{$\\naturals:= \\{0,1,2,3,\\ldots,\\}$}.\n We begin with a technical result.\n\n\\begin{lemma} [\\normalfont Serre] \\label{lem:serre} For all \\mbox{$(\\mathbf{d}_1,\\mathbf{d}_2) \\in \\reals^2$}\nand \\mbox{$(\\mathbf{a}_n)_{n \\in \\naturals} \\in \\reals^\\naturals$}, if all the constraints\n$$\\mathbf{d}_1 \\ge \\min\\{2, \\mathbf{d}_2+1\\}, \\,\\,\\,\n\\mathbf{d}_1^2 \\ge 4\\mathbf{d}_2,\\,\\,\\,\n\\mathbf{a}_0 = 0,\\,\\,\\,\n\\mathbf{a}_1 \\ge 0,\n\\text{ and, for each }\n n \\in \\naturals,\\,\\,\\,\n \\mathbf{a}_{n+2} \\ge \\mathbf{d}_1{\\cdot} \\mathbf{a}_{n+1} - \\mathbf{d}_2{\\cdot} \\mathbf{a}_{n} + 1 $$\nare satisfied, then \\mbox{$(\\mathbf{a}_n)_{n \\in \\naturals}$} is not bounded.\n\\end{lemma}\n\n\\begin{proof} (Serre) Set \\mbox{$ \\lambda:= \\frac{\\mathbf{d}_1- \\sqrt{\\mathbf{d}_1^2 - 4\\mathbf{d}_2}}{2}$} and\n \\mbox{$ \\mu:= \\frac{\\mathbf{d}_1+ \\sqrt{\\mathbf{d}_1^2 - 4\\mathbf{d}_2}}{2}$}.\\vspace{1mm}\nNotice that \\mbox{$\\sqrt{\\mathbf{d}_1^2 - 4\\mathbf{d}_2} \\ge 2- \\mathbf{d}_1$},\nfor this is clear if \\mbox{$2-\\mathbf{d}_1 \\le 0$}, while\n \\mbox{$(2-\\mathbf{d}_1 > 0) \\Rightarrow (\\mathbf{d}_2 \\le \\mathbf{d}_1 -1) \\Rightarrow\n\\bigl(\\mathbf{d}_1^2 - 4\\mathbf{d}_2 \\ge (2-\\mathbf{d}_1)^2\\bigr)$}.\nIt now follows that \\mbox{$ \\mu \\ge 1$}.\n\nLet $n$ range over~\\mbox{$\\naturals$}. Set \\mbox{$\\mathbf{b}_n:= \\mathbf{a}_{n+1} {-} \\lambda{\\cdot} \\mathbf{a}_n$}.\nThen \\vspace{-.5mm}\n$$ \\mathbf{b}_0 = \\mathbf{a}_1 {-} \\lambda{\\cdot}\\mathbf{a}_0 \\ge 0 \\text{\\,\\,\\,\\,\\,and\\,\\,\\,\\,\\,}\n\\mathbf{b}_{n+1} {-} \\mu{\\cdot} \\mathbf{b}_n = \\mathbf{a}_{n+2}\n - (\\lambda+\\mu){\\cdot}\\mathbf{a}_{n+1} + \\mu {\\cdot} \\lambda{\\cdot} \\mathbf{a}_n\n= \\mathbf{a}_{n+2}\n - \\mathbf{d}_1 {\\cdot} \\mathbf{a}_{n+1} + \\mathbf{d}_2 {\\cdot} \\mathbf{a}_n \\ge 1. $$\nBy induction on $n$, we have \\mbox{$n \\le \\mathbf{b}_n \\le \\mu {\\cdot}\\mathbf{b}_n$}. Thus,\n \\mbox{$n \\le \\mathbf{a}_{n+1} {-} \\lambda{\\cdot} \\mathbf{a}_n$}, and, hence,\n \\mbox{$(\\mathbf{a}_n)_{n \\in \\naturals}$} is not bounded.\n\\end{proof}\n\n\nThe following evolved through work of Golod, Shafarevich, Gasch\\\"utz, Vinberg, and Serre,\nalthough it may not have been expressed in this form before.\n\n\n\\begin{theorem} \\label{thm:exact} Let $K$ be a field, and \\mbox{$B = K \\oplus \\mathfrak{b}$} be an\naugmented $K$-algebra. If \\mbox{$\\mathfrak{b} \\ne \\{0\\}$} and\nthere exists an exact left-$B$-module sequence\\vspace{-2mm}\n\\begin{equation}\\label{eq:needed}\n\\textstyle\\bigoplus\\limits_{D_2}B \\xrightarrow{\\partial}\n\\bigoplus\\limits_{D_1}B \\xrightarrow{\\pi} \\mathfrak{b} \\to 0\n\\end{equation}\n for some sets $D_1$ and $D_2$ such that\n \\mbox{$\\vert D_1 \\vert = \\dim_K( \\mathfrak{b}\/ \\mathfrak{b}^2)$} and\n \\mbox{$\\vert D_2 \\vert \\le \\frac{1}{4} \\vert D_1 \\vert ^2$}, then \\mbox{$\\dim_K(B) \\ge \\aleph_0$}.\n\\end{theorem}\n\n\\begin{proof}(Serre)\nLet $n$ range over \\mbox{$\\naturals$}. Clearly,\n we may assume that \\mbox{$\\dim_K(B\/\\mathfrak{b}^n) < \\aleph_0$}.\n\nSince $K$ is a field and \\mbox{$\\vert D_1 \\vert = \\dim_K( \\mathfrak{b}\/ \\mathfrak{b}^2)< \\aleph_0$},\n the surjective map\\vspace{-1mm}\n$$\\textstyle\\bigoplus\\limits_{D_1} (B\/\\mathfrak{b}) \\xrightarrow{( B \/{\\mathfrak{b}})\\otimes_B \\pi}\n\\mathfrak{b}\/ \\mathfrak{b}^2\\vspace{-1mm}$$ is injective.\nHence,\n \\mbox{$\\operatorname{Ker} \\pi \\subseteq \\bigoplus\\limits_{D_1} \\mathfrak{b}$}. By exactness,\n\\mbox{$ \\partial ( \\bigoplus\\limits_{D_2} B) \\subseteq \\bigoplus\\limits_{D_1} \\mathfrak{b} $}. By left $B$-linearity,\n\\mbox{$ \\partial( \\bigoplus\\limits_{D_2} \\mathfrak{b}^n) \\subseteq\n\\bigoplus\\limits_{D_1} \\mathfrak{b}^{n+1}$}.\nOn applying \\mbox{$(B\/\\mathfrak{b}^{n+1} )\\otimes_B -$} to~\\eqref{eq:needed},\n we obtain an exact left-$B$-module sequence\\vspace{-1mm}\n $$\\textstyle\\bigoplus\\limits_{D_2} (B\/\\mathfrak{b}^{n+1}) \\xrightarrow{\\overline{\\partial}}\n\\bigoplus\\limits_{D_1} (B\/\\mathfrak{b}^{n+1}) \\to \\mathfrak{b}\/\\mathfrak{b}^{n+2} \\vspace{-1mm}$$\nsuch that\n\\mbox{$\\textstyle\\overline{\\partial}\\bigl(\\bigoplus\\limits_{D_2} (\\mathfrak{b}^{n}\/\\mathfrak{b}^{n+1})\\bigr)\n = \\{0\\}.$} There is then induced an exact left-$B$-module sequence \\vspace{-2mm}\n$$\\textstyle\\bigoplus\\limits_{D_2} (B\/\\mathfrak{b}^{n}) \\to\n\\bigoplus\\limits_{D_1} (B\/\\mathfrak{b}^{n+1}) \\to \\mathfrak{b}\/\\mathfrak{b}^{n+2}. \\vspace{-2mm}$$\nSince $K$ is a field,\\vspace{-2mm} $$\\textstyle\\vert D_1 \\vert {\\cdot} \\dim_K (B\/\\mathfrak{b}^{n+1}) \\le\n\\vert D_2 \\vert {\\cdot} \\dim_K (B\/\\mathfrak{b}^{n}) + (\\dim_K (B\/\\mathfrak{b}^{n+2}) -1).$$\nBy hypothesis, \\mbox{$\\vert D_2 \\vert \\le \\frac{1}{4} \\vert D_1 \\vert ^2$}. Also,\n \\mbox{$\\vert D_1 \\vert \\ne 0$}, since $\\pi$ is surjective.\nIf \\mbox{$\\vert D_1 \\vert < 2$}, then \\mbox{$\\vert D_1 \\vert = 1$}, \\mbox{$\\vert D_2 \\vert \\le \\frac{1}{4}$}, and\n \\mbox{$\\vert D_2 \\vert = 0 = \\vert D_1 \\vert - 1$}.\nNow, by Lemma~\\ref{lem:serre}, the sequence \\mbox{$( \\dim_K (B\/\\mathfrak{b}^{n}))_{n \\in \\naturals}$} is not bounded.\nHence, \\mbox{$\\dim_K(B) \\ge \\aleph_0$}.\n\\end{proof}\n\n\n\n\\begin{history} Let $p$ be a prime number, and $G$ be a nontrivial,\nfinite $p$-group. Set \\mbox{$K:= \\integers\/p \\integers$} and \\mbox{$B:=KG$}, the group algebra.\nLet \\mbox{$\\mathfrak{b}$} denote the kernel of the $K$-algebra homomorphism\n\\mbox{$B \\to K$} which carries $G$ to~$\\{1\\}$.\nFor each \\mbox{$n \\in \\naturals$}, set \\mbox{$\\mathbf{d}_n:= \\dim_{K} \\bigl(\\operatorname H_n(G,K)\\bigr)$}.\nRecall that \\mbox{$\\operatorname H_1(G,K) = \\mathfrak{b}\/ \\mathfrak{b}^2$}.\n From the theory of minimal resolutions, it is known that\nthere exist exact left-$B$-module sequences of the form\n\\mbox{$ \\cdots \\to B^{\\mathbf{d}_2} \\to B^{\\mathbf{d}_1} \\to \\mathfrak{b} \\to 0.$}\nBy Theorem~\\ref{thm:exact}, \\mbox{$\\mathbf{d}_2 > \\frac{1}{4}\\mathbf{d}_1^2$}.\n\nIt is known that \\mbox{$\\mathbf{d}_1$} is the minimum\nnumber of elements it takes to generate $G$ as a pro-$p$ group,\nand that for any generating set of \\mbox{$\\mathbf{d}_1$} elements,\n\\mbox{$\\mathbf{d}_2$} is the minimum\nnumber of relations it takes to present $G$ as a pro-$p$ group.\n(By the Burnside basis theorem, \\mbox{$\\mathbf{d}_1$} is the minimum\nnumber of elements it takes to generate $G$ as a group.\nFor any generating set of \\mbox{$\\mathbf{d}_1$} elements,\n it takes at least \\mbox{$\\mathbf{d}_2$} relations\nto present $G$ as a group, but it is not known if \\mbox{$\\mathbf{d}_2$} relations is enough.)\n\nFor details about the foregoing, see~\\cite{GS} or~\\cite{S}.\n\n\n The main objective of Golod and Shafarevich in~\\cite{GS}, and the reason for which~\\eqref{eq:ineq} was first developed,\nwas to prove that \\mbox{$\\mathbf{d}_2 > \\frac{1}{4}(\\mathbf{d}_1{-}1)^2$}.\n It followed from this, together with an earlier result of Shafarevich,\n that the class-field-tower problem had a negative solution,\nthat is, there do exist infinite class-field towers.\nGasch\\\"utz and Vinberg~\\cite{V} independently refined\n the inequality to \\mbox{$\\mathbf{d}_2 > \\frac{1}{4}\\mathbf{d}_1^2$}.\n Serre~\\cite{S} gave the above proof of this refined inequality.\nNevertheless, there still remain many applications of~\\eqref{eq:ineq} which have\nnot been superseded.\n\\end{history}\n\n\n\n\\section{The Koszul resolution for an augmented algebra}\\label{sec:one}\n\n\n\n\\begin{notation}\\label{not:augmented} Let $K$ be a field, $X$ be a set, $F$ be the free associative $K$-algebra on $X$,\nand $\\mathfrak{f}$ be the two-sided\nideal of $F$ generated by $X$.\n We write \\mbox{$F = K \\langle X \\rangle$}.\n\nLet $R$ be a family of elements of $\\mathfrak{f}$, possibly with repetitions,\n and $\\mathfrak{r}$ denote the two-sided ideal of $F$\ngenerated by the elements of $R$. Set \\mbox{$B := F\/\\mathfrak{r}$} and\n\\mbox{$\\mathfrak{b}:= \\mathfrak{f}\/\\mathfrak{r}$}.\nIn summary, \\mbox{$B= K \\oplus \\mathfrak{b}$} is an augmented associative $K$-algebra presented with generating set $X$\nand relating set~$R$. We write \\mbox{$B = K \\langle X \\mid R \\rangle$}.\n\nSet\n\\mbox{$K^{(X)} := \\bigoplus\\limits_{x \\in X}Kx$},\\vspace{1mm}\n\\mbox{$F^{(X)}:= F \\otimes_K K^{(X)}$}, and \\mbox{$B^{(X)}:= B \\otimes_K K^{(X)}$}; these are\nthe free left modules on~$X$ over $K$, $F$, and $B$, respectively.\nSimilar notation will apply\\vspace{1mm} with $R$ in place of $X$. At one stage,\nwe shall use the natural $K$-centralizing $K$-bimodule structure of \\mbox{$K^{(R)}$}.\n\\end{notation}\n\n\n\n\\begin{definitions}\\label{defs:augmented} Each element $f$ of $\\mathfrak{f}$ has a unique\nexpression as a left\n$F$-linear combination of the elements of $X$, and we shall write this as\n\\mbox{$f= \\sum\\limits_{x \\in X} \\frac{\\partial f}{\\partial x} {\\cdot} x$}.\n\nWe have an isomorphism of left $F$-modules\n\\begin{equation*}\n\\textstyle \\mathfrak{f} \\xrightarrow{\\sim} F^{(X)}, \\qquad\n f = \\sum\\limits_{x\\in X} \\frac{\\partial f}{\\partial x}\n{\\cdot} x \\mapsto \\textstyle \\sum\\limits_{x\\in X} \\frac{\\partial f}{\\partial x} \\otimes x;\\vspace{-1mm}\n\\end{equation*}\non applying \\mbox{$(F\/\\mathfrak{r})\\otimes_F -$}, we obtain an isomorphism of left\n \\mbox{$ F\/\\mathfrak{r}$}-modules $$\\textstyle \\mathfrak{f}\/\\mathfrak{rf} \\xrightarrow{\\sim} B^{(X)},\n\\qquad f + \\mathfrak{rf} \\mapsto \\textstyle \\sum\\limits_{x\\in X}\n (\\mkern-2mu\\frac{\\partial f}{\\partial x}+\\mathfrak{r} )\\otimes x.$$\n\\vspace{-4mm}\n\nWe have also a surjection of $F$-bimodules\n\\begin{equation*}\n\\textstyle F \\otimes_K K^{(R)} \\otimes_ K F \\onto \\mathfrak{r}, \\qquad\nf_1 \\otimes r \\otimes f_2 \\mapsto f_1{\\cdot} r{\\cdot} f_2;\n\\end{equation*}\non applying \\mbox{$(F\/\\mathfrak{r}) \\otimes_F - \\otimes_F (F\/\\mathfrak{f})$}, we obtain\na surjection of left \\mbox{$F\/\\mathfrak{r}$}-modules\n $$\\textstyle B^{(R)}\n \\onto \\mathfrak{r}\/\\mathfrak{rf}, \\qquad (f+\\mathfrak{r})\\otimes r \\mapsto f{\\cdot} r + \\mathfrak{rf}.$$\n\n\n\nThe cokernel of the composite \\mbox{$ B^{(R)}\n \\onto \\mathfrak{r}\/\\mathfrak{rf} \\into \\mathfrak{f}\/\\mathfrak{rf} \\xrightarrow{\\sim} B^{(X)}$}\nis isomorphic to \\mbox{$\\mathfrak{f}\/\\mathfrak{r}$}, which is \\mbox{$\\mathfrak{b}$}. We then have\nan exact left-$B$-module sequence\\vspace{-1mm}\n\\begin{equation}\\label{eq:exact}\n\\textstyle B^{(R)} \\xrightarrow{b\\otimes r \\mapsto \\sum\\limits_{x \\in X}\nb{\\cdot} (\\mkern-2mu\\frac{\\partial r}{\\partial x}+ \\mathfrak{r})\\otimes x}\n B^{(X)}\n\\xrightarrow{ b \\otimes x \\mapsto b{\\cdot} (x+\\mathfrak{r})} \\mathfrak{b} \\to 0.\n\\end{equation}\nOn splicing~\\eqref{eq:exact} and\n\\mbox{$0 \\to \\mathfrak{b} \\to B \\to B\/\\mathfrak{b} \\to 0$},\nwe obtain what we call \\textit{the Koszul resolution}\\vspace{-1mm}\n\\begin{equation}\\label{eq:exact2}\n\\textstyle 0 \\to \\operatorname{Ker} \\partial \\to B^{(R)} \\xrightarrow{\\partial:b\\otimes r\n\\mapsto \\sum_{x \\in X}\\mkern-4mu b{\\cdot} (\\mkern-2mu\\frac{\\partial r}{\\partial x}+ \\mathfrak{r})\\otimes x}\n B^{(X)}\n\\xrightarrow{ b \\otimes x \\mapsto b{\\cdot} (x+\\mathfrak{r})} B \\to B\/\\mathfrak{b} \\to 0.\n\\end{equation}\nThe part that interests us is\\vspace{-2mm}\n\\begin{equation}\\label{eq:exact3}\n\\textstyle B^{(R)} \\xrightarrow{b\\otimes r \\mapsto \\sum\\limits_{x \\in X} b{\\cdot} (\\mkern-2mu\\frac{\\partial r}{\\partial x}\n+ \\mathfrak{r})\\otimes x}\n B^{(X)}\n\\xrightarrow{ b \\otimes x \\mapsto b{\\cdot} (x+\\mathfrak{r})} B.\n\\end{equation}\n\\end{definitions}\n\n\n\n\nWe remark that $R$ is a family of elements of \\mbox{$\\mathfrak{f}^2$},\nthen\n\\mbox{$\\dim_K( \\mathfrak{b}\/ \\mathfrak{b}^2) = \\vert X \\vert$}; if,\nmoreover,~\\mbox{$1 < \\mbox{$\\dim_K(B) < \\aleph_0$}$}, then\n applying Theorem~\\ref{thm:exact} to~\\eqref{eq:exact} shows that\n \\mbox{$\\vert R \\vert > \\frac{1}{4} \\vert X \\vert ^2 $}.\n\n\\section{The graded case of the Koszul resolution}\n\nContinuing with the notation developed in Section~\\ref{sec:one}, we now hypothesize\n a $\\integers$-graded $K$-algebra structure for~$B$, as follows.\n\nLet \\mbox{$\\deg:X \\to \\{1,2,3,\\ldots\\}$},\n\\mbox{$x \\mapsto \\deg(x)$}, be any map; there is then an induced\n$\\integers$-graded $K$-algebra structure \\mbox{$F= \\bigoplus\\limits_{n \\in \\integers} F_n$}\nwith $F_n = \\{0\\}$ if $n \\le -1$, $F_0 = K$, $\\bigoplus\\limits_{n \\ge 1} F_n = \\mathfrak{f}$, and\n\\mbox{$x \\in F_{\\deg(x)}$} for each \\mbox{$x \\in X$}.\n\n\nWe henceforth restrict to the case where each element of $R$ lies in \\mbox{$\\bigcup\\limits_{n \\ge 1}\\hskip-2pt F_n$}.\nThere is then an induced $\\integers$-graded $K$-algebra structure\n$B= \\bigoplus\\limits_{n \\in \\integers} B_n$ with $B_n = \\{0\\}$~if $n \\le -1$, $B_0 = K$,\n $\\bigoplus\\limits_{n \\ge 1} B_n = \\mathfrak{b}$, and \\mbox{$x + \\mathfrak{r} \\in B_{\\deg(x)}$}\nfor each \\mbox{$x \\in X$}. We choose a map \\mbox{$\\deg: R\\to \\{1,2,3,\\ldots\\}$},\n\\mbox{$r \\mapsto \\deg(r)$}, such that \\mbox{$r \\in F_{\\deg(r)}$};\nthus, as in~\\cite{GS}, different occurrences of $0$ in $R$ may be assigned different degrees.\n\n\nLet $n$ range over \\mbox{$\\integers$}. Set \\mbox{$\\mathbf{b}_n:= \\dim_K(B_n)$}.\nNow \\eqref{eq:exact3} gives an exact sequence of degree-$n$ $K$-modules\n\\begin{equation*}\n\\textstyle \\bigoplus\\limits_{r \\in R} (B_{n-\\deg(r)} \\otimes_K Kr) \\xrightarrow{b \\otimes r \\mapsto \\sum\\limits_{x \\in X}\nb {\\cdot}(\\mkern-2mu\\frac{\\partial r}{\\partial x} + \\mathfrak{r}) \\otimes x}\n \\bigoplus\\limits_{x \\in X} (B_{n-\\deg(x)} \\otimes_K K x) \\xrightarrow{ b \\otimes x \\mapsto b{\\cdot}(x+\\mathfrak{r})}\nB_n.\n\\end{equation*}\n Since $K$ is a field,\n\\begin{equation}\\label{eq:ineqbis}\n\\textstyle \\sum\\limits_{x \\in X} \\mathbf{b}_{n-\\deg(x)} \\le\n (\\sum\\limits_{r \\in R} \\mathbf{b}_{n-\\deg(r)}) + \\mathbf{b}_n, \\vspace{-1mm}\n\\end{equation}\nwhich is the Golod-Shafarevich inequality.\nSet \\mbox{$X_n:= \\{x \\in X : \\deg(x) = n\\}$} and\n\\mbox{$\\mathbf{x}_n := \\left\\vert X_n \\right\\vert$}. Then\n$$\\textstyle\\sum\\limits_{x \\in X} \\mathbf{b}_{n-\\deg(x)}= \\sum\\limits_{i \\in \\integers}\n \\sum\\limits_{x \\in X_i} \\mathbf{b}_{n-\\deg(x)}\n= \\sum\\limits_{i \\in \\integers} \\mathbf{x}_i{\\cdot} \\mathbf{b}_{n-i}.\\vspace{-1mm}$$\nSimilarly, set \\mbox{$R_n:= \\{r \\in R : \\deg(r) = n\\}$} and\n\\mbox{$\\mathbf{r}_n := \\vert R_n\\vert$}; then\n$\\sum\\limits_{r \\in R} \\mathbf{b}_{n-\\deg(r)} = \\sum\\limits_{i \\in \\integers}\n\\mathbf{r}_i {\\cdot} \\mathbf{b}_{n-i}$.\nNow~\\eqref{eq:ineqbis} becomes\\vspace{-1mm}\n\\begin{equation}\\label{eq:ineq2}\n\\textstyle\n\\mathbf{b}_n + \\sum\\limits_{i \\in \\integers} \\mathbf{r}_i {\\cdot}\\mathbf{b}_{n-i} \\ge\n \\sum\\limits_{i \\in \\integers} \\mathbf{x}_i {\\cdot} \\mathbf{b}_{n-i}.\n\\end{equation}\n\n\\section{Hilbert series}\\label{sec:two}\n\n\nLet $t$ be a new variable. We shall express elements of the power-series ring \\mbox{$\\reals[[t]]$}\nin the form \\mbox{$\\sum\\limits_{n \\in \\integers} a_nt^n $}.~~Set\\vspace{-2mm}\n$$\\textstyle P:= \\{ \\sum\\limits_{n \\in \\integers} a_nt^n \\in \\reals[[t]]\n: a_n \\ge 0 \\text{ for all } n \\in \\integers \\}.\\vspace{-1mm}$$\nThen \\mbox{$P$} is both an additive submonoid and a multiplicative submonoid in \\mbox{$\\reals[[t]]$}.\nLet $\\succeq$ be the relation on \\mbox{$\\reals[[t]]$} such that \\mbox{$\\alpha \\succeq \\beta$} if and only if\n\\mbox{$\\alpha - \\beta \\in P$}.\n\nContinuing with the notation developed in Section~\\ref{sec:two}, we henceforth restrict to the case where\n all the cardinals \\mbox{$\\mathbf{x}_n$} and \\mbox{$\\mathbf{r}_n$} are finite.\n We define the \\textit{Hilbert series} of $B$, $X$, and $R$,\n to be the elements of \\mbox{$\\reals[[t]]$} given by\n \\mbox{$\\operatorname{H}(B):= \\sum\\limits_{n \\in \\integers} \\mathbf{b}_nt^n$},\n \\mbox{$\\operatorname{h}(X):= \\sum\\limits_{n \\in \\integers} \\mathbf{x}_nt^n$},\n and \\mbox{$\\operatorname{h}(R):= \\sum\\limits_{n \\in \\integers} \\mathbf{r}_nt^n$}, respectively.\nNotice that the constant terms are $1$, $0$, and $0$, respectively.\nNow~\\eqref{eq:ineq2} says that \\mbox{$\\operatorname{H}(B) + \\operatorname{h}(R) {\\cdot} \\operatorname{H}(B)\n \\succeq \\operatorname{h}(X) {\\cdot} \\operatorname{H}(B) $}. Hence,\n\\mbox{$\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr){\\cdot}\\operatorname{H}(B) \\succeq 0$}.\nBy considering the constant terms, we see that\n\\begin{equation}\\label{eq:GS}\n\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr){\\cdot}\\operatorname{H}(B)\\succeq 1;\n\\end{equation}\nthis is Lemma~2 of~\\cite{GS}.\nIn fact, one can read directly from~\\eqref{eq:ex} that\n $$\\textstyle\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr){\\cdot}\\operatorname{H}(B)\n - \\operatorname{H}( \\operatorname{Ker} \\partial) \\,\\, =\\,\\, \\operatorname{H}(K)\\,\\,=\\,\\, 1.$$\n\n\\begin{key}\\label{key:key} Consider any \\mbox{$\\gamma \\in t{\\cdot}\\reals[[t]]$}.\n\nIf \\mbox{$\\gamma \\succeq \\operatorname{h}(R)$}, then\n$\n\\bigl(1 - \\operatorname{h}(X)\n+ \\gamma \\bigr){\\cdot}\\operatorname{H}(B)\n \\,\\,\\, \\succeq \\,\\,\\,\n\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr){\\cdot}\\operatorname{H}(B)\n \\,\\,\\, \\succeq \\,\\,\\, 1. $\n\nIf it is also the case that \\mbox{$ (1 - \\operatorname{h}(X) + \\gamma \\bigr)^{-1} \\succeq 0$},\nthen \\mbox{$\\operatorname{H}(B)\n \\,\\,\\, \\succeq \\,\\,\\, \\bigl(1 - \\operatorname{h}(X) + \\gamma \\bigr)^{-1} \\,\\,\\, \\succeq \\,\\,\\, 0. $}\n\nIf it is further the case that $X$ is finite and \\mbox{$\\gamma \\ne \\operatorname{h}(X)$}, or, more generally,\nthat \\mbox{$ (1 - \\operatorname{h}(X) + \\gamma \\bigr)^{-1} \\not\\in \\reals[t]$},\nthen \\mbox{$\n\\operatorname{H}(B)\n$} has infinitely many nonzero coefficients, and, hence,\n \\mbox{$\\dim_K(B)= \\aleph_0$}.\n\\end{key}\n\n\\medskip\n\nFinally, we restrict to the case where $X$ is concentrated in degree 1.\n\n\n\\begin{corollary}[{\\normalfont{Golod}}\\textbf]\\label{cor:gol} Let $K$ be a field, $X$ be a finite, nonempty set,\nand \\mbox{$\\varepsilon$} be an element of\n\\mbox{$\\left[0,\\frac{\\vert X \\vert}{2}\\, \\right]$}. For each integer \\mbox{$n \\ge 2$},\nlet $R_n$ be a family of $X$-homogenous elements in\n\\mbox{$ K\\langle X \\rangle $}\n of $X$-degree~$n$\nsuch that \\mbox{$\\ \\vert R_n \\vert \\le \\varepsilon ^2 (\\vert X \\vert - 2\\varepsilon )^{n-2}$}.\n$\\bigl($When \\mbox{$\\varepsilon = \\frac{\\vert X \\vert -1}{2}$}, this says\n \\mbox{$ \\vert R_n \\vert \\le (\\frac{ \\vert X \\vert -1 }{2})^2$}.$\\bigr)$\nThen \\mbox{$\\dim_K\\bigl(K\\langle X \\mid \\bigcup\\limits_{n \\ge 2} R_n \\rangle\\bigr)= \\aleph_0$}.\n\\end{corollary}\n\n\\begin{proof} Set \\mbox{$\\gamma := \\textstyle\n\\sum\\limits_{n \\ge 2} \\bigl(\\varepsilon ^2 (\\vert X \\vert - 2\\varepsilon )^{n-2} t^n\\bigr) =\n\\sum\\limits_{m \\ge 0} \\bigl(\\varepsilon ^2 (\\vert X \\vert - 2\\varepsilon )^{m} t^mt^2\\bigr) =\n \\frac{\\varepsilon ^2 t^2}{1- (\\vert X \\vert - 2\\varepsilon )t}$}. \\vspace{.5mm}\n\nSet\n\\mbox{$\\alpha:= 1- (\\vert X \\vert - \\varepsilon) t$} \\, and\\, \\mbox{$\\beta:= \\varepsilon t$}.\nThen \\mbox{$\\alpha-\\beta = 1 - \\vert X \\vert t$},\\,\\, \\mbox{$\\alpha +\\beta =\n1- (\\vert X \\vert - 2\\varepsilon )t$}\n,\\, \\mbox{$\\gamma\n= \\frac{\\beta^2}{\\alpha +\\beta}$},\n\\begin{align*}\n& \\textstyle 1 - \\vert X \\vert t + \\gamma =\n (\\alpha-\\beta) + \\frac{\\beta^2}{\\alpha +\\beta}\n= \\frac{\\alpha^2}{ \\alpha+\\beta}\n \\quad\\text{and}\\quad \\bigl(1 - \\vert X \\vert t + \\gamma )^{-1} = \\frac{1}{\\alpha} + \\frac{\\beta}{\\alpha^{2}}.\n\\end{align*}\nThe result now follows by~\\ref{key:key}, since \\mbox{$\\vert X \\vert > \\varepsilon \\ge 0$}.\n\\end{proof}\n\n\n\n\n\n\\begin{history} Golod~\\cite{G} used Corollary~\\ref{cor:gol} to construct a finitely generated, non-nilpotent, nil algebra and\nan infinite, residually finite, finitely generated $p$-group, for each prime $p$. \\end{history}\n\n\n\n\n\\section{The Fox resolution for group algebras}\n\nWe now recall the group-algebra analogue of the Koszul resolution.\n\n \\begin{notation} Let\n$G$ be a group.\nLet \\mbox{$\\operatorname{d}(G)$} denote the smallest of those cardinals $\\kappa$ such that $G$ can be generated by $\\kappa$ elements.\nLet \\mbox{$G'$} denote the derived subgroup of $G$, and set \\mbox{$G^{\\text{ab}}:= G\/G'$}, the\nabelianization of $G$.\n\nLet \\mbox{$\\langle X \\mid R \\rangle$} be a presentation for $G$.\n Clearly, \\mbox{$\\operatorname{d}(G^{\\text{ab}}) \\le \\operatorname{d}(G) \\le \\vert X \\vert$}.\n\n\nLet $K$ be a field and set \\mbox{$B:=KG$}, the group algebra. Let \\mbox{$\\mathfrak{b}$} denote the kernel of the $K$-algebra homomorphism\n\\mbox{$\\epsilon:B \\to K$} which carries $G$ to $\\{1\\}$.\nLet $F$ be the group algebra over $K$\nfor the free group on $X$,\n $\\mathfrak{f}$~be the two-sided\nideal of $F$ generated by \\mbox{$\\{x-1 \\mid x \\in X\\}$}, and\n$\\mathfrak{r}$ be the two-sided ideal of $F$\ngenerated by \\mbox{$\\{r-1 \\mid r \\in R\\}$}. Then \\mbox{$B = F\/\\mathfrak{r}$} and\n\\mbox{$\\mathfrak{b} = \\mathfrak{f}\/\\mathfrak{r}$}.\n\n\nSet \\mbox{$K^{(X)} := \\bigoplus\\limits_{x \\in X}Kx$},\n\\mbox{$F^{(X)}:= F \\otimes_K K^{(X)}$}, and \\mbox{$B^{(X)}:= B \\otimes_K K^{(X)}$}, and similarly\nwith $R$ in place of $X$.\n\\end{notation}\n\n\n\n\\begin{definitions}\\label{defs:groupring} It is not difficult to see that the left ideal of $F$ generated by\n\\mbox{$\\{x-1 \\mid x \\in X\\}$} is closed under right multiplication by the elements of\n\\mbox{$X \\cup X^{-1}$}, and, hence, is the whole of \\mbox{$\\mathfrak{f}$}. We have a left-$F$-module map\n \\mbox{$F^{(X)} \\to \\mathfrak{f}$} which sends each \\mbox{$1 \\otimes x$} to \\mbox{$x-1$};\nto construct an inverse, we shall define a left-$F$-module map\n \\mbox{$ \\mathfrak{f} \\to F^{(X)}$} which sends each \\mbox{$x-1$} to \\mbox{$1 \\otimes x$}.\n\nWe view \\mbox{$F^{(X)}$} as an \\mbox{$(F,K)$}-bimodule, and form the\nbimodule-algebra over $K$ suggestively written in matrix form as\n\\mbox{$\\left(\\begin{smallmatrix}\nF &F^{(X)}\\\\ 0 &K\n\\end{smallmatrix}\\right)$}. There then exists a unique $K$-algebra homomorphism\n\\mbox{$\\left(\\begin{smallmatrix}\n\\phi_{1,1} &\\phi_{1,2}\\\\ 0 &\\phi_{2,2}\n\\end{smallmatrix}\\right): F \\to \\left(\\begin{smallmatrix}\nF &F^{(X)}\\\\ 0 &K\n\\end{smallmatrix}\\right)$} which sends each \\mbox{$x \\in X$} to\nthe invertible element\n\\mbox{$\\left(\\begin{smallmatrix}\nx &1 \\otimes x\\\\ 0 &1\n\\end{smallmatrix}\\right)$}. Thus, the \\mbox{$\\phi_{i,j}$} are $K$-module maps, \\mbox{$\\phi_{1,1}(1) = 1$}, \\mbox{$\\phi_{1,2}(1) = 0$},\n\\mbox{$\\phi_{2,2}(1) = 1$}, and, for all $f$, \\mbox{$g \\in F$},\n$$\\phi_{1,1}(f {\\cdot}g) = \\phi_{1,1}(f) {\\cdot}\\phi_{1,1}(g),\\hskip 5pt\n \\phi_{1,2}(f {\\cdot}g) = \\phi_{1,1}(f) {\\cdot} \\phi_{1,2}(g)+\\phi_{1,2}(f) {\\cdot} \\phi_{2,2}(g),\n \\text{ and }\n \\phi_{2,2}(f {\\cdot}g) = \\phi_{2,2}(f) {\\cdot} \\phi_{2,2}(g).$$ In particular, \\mbox{$\\phi_{1,1}$} and \\mbox{$\\phi_{2,2}$} are\n$K$-algebra homomorphisms. Also, for all \\mbox{$x \\in X$},\n$$\\phi_{1,1}(x) = x, \\hskip5pt\\phi_{1,2}(x) = 1\\otimes x, \\text{ and } \\phi_{2,2}(x) = 1.$$\nIn particular, \\mbox{$\\phi_{1,1}$} is the identity map on $F$, and \\mbox{$\\phi_{2,2} = \\epsilon$}.\nWe now see that \\mbox{$\\phi_{1,2}:F \\to F^{(X)}$} is a $K$-module map such that\nfor all $f$, \\mbox{$g \\in F$},\n\\mbox{$\\phi_{1,2}(f {\\cdot}g) = f {\\cdot}\\phi_{1,2}(g) + \\phi_{1,2}(f) {\\cdot} \\epsilon(g)$}.\nRestricting the domain of \\mbox{$\\phi_{1,2}$} to \\mbox{$\\mathfrak{f}$} gives\na left $F$-module map \\mbox{$\\mathfrak{f} \\to F^{(X)}$} which sends each\n\\mbox{$x-1$} to \\mbox{$ 1\\otimes x$}, as desired. Now\neach element $f$ of $\\mathfrak{f}$ has a unique\nexpression as a left\n$F$-linear combination of the elements of \\mbox{$\\{x-1 \\mid x \\in X\\}$}, which we write as\n\\mbox{$f= \\sum\\limits_{x \\in X} \\frac{\\partial f}{\\partial (x-1)}{\\cdot} (x-1)$}.\\vspace{1mm}\n\nWe have an isomorphism of left $F$-modules\n\\begin{equation*}\n\\textstyle \\mathfrak{f} \\xrightarrow{\\sim} F^{(X)}, \\qquad\n f = \\sum\\limits_{x\\in X} \\frac{\\partial f}{\\partial (x-1)}{\\cdot}\n (x-1) \\mapsto \\textstyle \\sum\\limits_{x\\in X} \\frac{\\partial f}{\\partial (x-1)} \\otimes x;\\vspace{-1mm}\n\\end{equation*}\non applying \\mbox{$(F\/\\mathfrak{r})\\otimes_F -$}, we obtain an isomorphism of left\n \\mbox{$ F\/\\mathfrak{r}$}-modules $$\\textstyle \\mathfrak{f}\/\\mathfrak{rf} \\xrightarrow{\\sim} B^{(X)},\n\\qquad f + \\mathfrak{rf} \\mapsto \\textstyle \\sum\\limits_{x\\in X}\n (\\mkern-2mu\\frac{\\partial f}{\\partial (x-1)}+\\mathfrak{r} )\\otimes x.$$\n\\vspace{-4mm}\n\nWe have also a surjection of $F$-bimodules\n\\begin{equation*}\n\\textstyle F \\otimes_K K^{(R)} \\otimes_ K F \\onto \\mathfrak{r}, \\qquad\nf_1 \\otimes r \\otimes f_2 \\mapsto f_1{\\cdot}(r-1){\\cdot}f_2;\n\\end{equation*}\non applying \\mbox{$(F\/\\mathfrak{r}) \\otimes_F - \\otimes_F (F\/\\mathfrak{f})$}, we obtain\na surjection of left \\mbox{$F\/\\mathfrak{r}$}-modules\n $$\\textstyle B^{(R)}\n \\onto \\mathfrak{r}\/\\mathfrak{rf}, \\qquad (f+\\mathfrak{r})\\otimes r \\mapsto f{\\cdot}(r-1) + \\mathfrak{rf}.$$\n\n The cokernel of the composite \\mbox{$ B^{(R)}\n \\onto \\mathfrak{r}\/\\mathfrak{rf} \\into \\mathfrak{f}\/\\mathfrak{rf} \\xrightarrow{\\sim} B^{(X)}$}\nis isomorphic to \\mbox{$\\mathfrak{f}\/\\mathfrak{r}$}, which is \\mbox{$\\mathfrak{b}$}. We then have\nan exact left-$B$-module sequence\\vspace{-1mm}\n\\begin{equation}\\label{eq:group}\n\\textstyle B^{(R)} \\xrightarrow{ \\partial:b\\otimes r \\mapsto \\sum_{x \\in X}\n b{\\cdot}(\\mkern-2mu\\frac{\\partial (r-1)}{\\partial (x-1)}+\\mathfrak{r})\\otimes x}\n B^{(X)}\n\\xrightarrow{ b \\otimes x \\mapsto b{\\cdot}(x-1+\\mathfrak{r})} \\mathfrak{b} \\to 0.\n\\end{equation}\n\\end{definitions}\n\n\n\n\\begin{theorem}\\label{thm:new} Let \\mbox{$G = \\langle X \\mid R \\rangle$} be\na presentation of a nontrivial, finite group.\nIf \\mbox{$\\vert X \\vert = \\operatorname{d}(G^{\\text{\\normalfont ab}})$},\nthen \\mbox{$\\vert R \\vert > \\frac{1}{4}\\vert X \\vert^2$}; equivalently,\nif \\mbox{$\\operatorname{d}(G) = \\operatorname{d}(G^{\\text{\\normalfont ab}}) =\n\\vert X \\vert$}, then\n \\mbox{$\\vert R \\vert > \\frac{1}{4} (\\operatorname{d}(G))^2$}.\n\\end{theorem}\n\n\\vspace{-5mm}\n\n\\begin{proof} Since \\mbox{$ G^{\\text{ab}}$} is a finite abelian group, \\mbox{$G^{\\text{ab}} \\simeq \\bigoplus\\limits_{i=1}^d (\\integers\/I_i)$}\nfor some finite chain \\mbox{$I_1 \\subseteq I_2 \\subseteq \\cdots \\subseteq I_d$} of proper ideals of $\\integers$.\nLet $p$ be a prime number such that \\mbox{$I_d \\subseteq p\\integers$}, and set \\mbox{$K:= \\integers\/p\\integers$}.\nThen \\mbox{$ K\\otimes_{\\integers} G^{\\text{ab}} \\simeq K^d$}, and\n\\mbox{$d = \\dim_K(K\\otimes_{\\integers} G^{\\text{ab}}) \\le \\operatorname{d}(G^{\\text{ab}}) \\le d$}. Thus,\n\\mbox{$\\dim_K(K\\otimes_{\\integers} G^{\\text{ab}}) = \\operatorname{d}(G^{\\text{ab}}) = \\vert X \\vert$}.\n\n It is well known and straightforward\nto prove that \\mbox{$\\mathfrak{b}\/\\mathfrak{b}^2 \\simeq K\\otimes_{\\integers} G^{\\text{ab}} $} with\n \\mbox{$\\textstyle (g{-}1) + \\mathfrak{b}^2 \\leftrightarrow 1\\otimes gG'$}.\nThen \\mbox{$\\dim_K\\mathfrak{b}\/\\mathfrak{b}^2 = \\dim_K(K\\otimes_{\\integers} G^{\\text{ab}}) = \\vert X \\vert$}.\nThe result now follows from Theorem~\\ref{thm:exact} applied to~\\eqref{eq:group}.\n\\end{proof}\n\n\n\\begin{corollary} [{\\normalfont Golod-Shafarevich}] Let $p$ be a prime number,\nand \\mbox{$G = \\langle X \\mid R \\rangle$} be\na presentation of a nontrivial, finite $p$-group.\nIf\n\\mbox{$\\vert X \\vert=\\operatorname{d}(G)$}, then\n \\mbox{$\\vert R \\vert > \\frac{1}{4} (\\operatorname{d}(G))^2$}.\n\\end{corollary}\n\n \\begin{proof} By the Burnside basis theorem, \\mbox{$\\operatorname{d}(G)\n= \\dim_K(K\\otimes_{\\integers} G^{\\text{ab}})$} for \\mbox{$K = \\integers\/p\\integers$}.\nHence, \\mbox{$\\operatorname{d}(G) = \\operatorname{d}(G^{\\text{ab}})$}, and the\n result follows from the second part of Theorem~\\ref{thm:new}.\n \\end{proof}\n\n\nFor \\mbox{$f \\in F{-}\\{0\\}$}, we set\n\\mbox{$\\deg(f):= \\max \\{ i \\in \\naturals : f \\in \\mathfrak{f}^i \\}$}.\nFor each $r \\in R$, we have then defined \\mbox{$\\deg(r{-}1) \\in \\naturals$}, unless $r = 1$ in~$F$,\nin which case we shall choose some value \\mbox{$\\deg(r{-}1) \\in \\naturals$}.\nLet $n$ range over $\\integers$.\nDefine \\mbox{$\\mathfrak{b}^{n}$} to be $B$ if \\mbox{$n\\le 0$}, and,\nas usual, to be \\mbox{$\\mathfrak{b}^{n-1}{\\cdot}\\mathfrak{b}$} if \\mbox{$n \\ge 1$}.\nIn~\\eqref{eq:group}, we find, for each \\mbox{$r \\in R$}, $$\\textstyle \\partial (\\mathfrak{b}^n \\otimes Kr) \\subseteq\n \\bigoplus\\limits_{x \\in X} (\\mathfrak{b}^{n+\\text{deg}(r-1)-1} \\otimes Kx), \\text{ and, hence, }\n \\partial (\\mathfrak{b}^{n-\\text{deg}(r-1)+1} \\otimes Kr) \\subseteq\n \\bigoplus\\limits_{x \\in X} (\\mathfrak{b}^{n} \\otimes Kx).$$\nOn applying \\mbox{$(B\/\\mathfrak{b}^{n})\\otimes_B-$} to~\\eqref{eq:group}, we get\nan exact left-$B$-module sequence\n\\begin{equation*}\n\\textstyle \\bigoplus\\limits_{r \\in R} (B\/\\mathfrak{b}^{n-\\text{deg}(r-1)+1}) \\otimes Kr\n\\to\n \\bigoplus\\limits_{x \\in X} (B\/\\mathfrak{b}^{n}) \\otimes Kx\n \\to (\\mathfrak{b}+\\mathfrak{b}^{n+1})\/\\mathfrak{b}^{n+1} .\n\\end{equation*}\nSet \\mbox{$\\mathbf{a}_n:= \\dim_K(B\/\\mathfrak{b}^{n+1})$}, and\ndefine \\mbox{$\\delta_n$} to be $0$ if $n < 0$, and to be~$1$ if $n \\ge 0$. Since $K$ is a field, \\mbox{$\n\\textstyle \\vert X \\vert \\mathbf{a}_{n-1} \\le (\\sum\\limits_{r \\in R} \\mathbf{a}_{n-\\text{deg}(r-1)})\n+ (\\mathbf{a}_{n} - \\delta_{n}).$}\n\n\n We set \\mbox{$R_n:= \\{ r \\in R : \\deg(r-1) = n\\}$} and \\mbox{$\\mathbf{r}_n := \\vert R_n\\vert$}.\nWe henceforth assume that \\mbox{$\\vert X \\vert$} and the \\mbox{$\\mathbf{r}_n$} are finite.\nWe define \\mbox{$\\operatorname{h}(R):= \\sum\\limits_{n \\in \\integers} \\mathbf{r}_nt^n\\in \\reals[[t]]$}. We\ndefine \\mbox{$\\operatorname{h}(X)$} similarly, and find \\mbox{$\\operatorname{h}(X) = \\vert X \\vert t$}.\nSet\n\\mbox{$\\mathbf{b}_n:= \\dim_K(\\mathfrak{b}^{n}\/\\mathfrak{b}^{n+1}) = \\mathbf{a}_n - \\mathbf{a}_{n-1}$},\nand \\mbox{$\\operatorname{H}(B):= \\sum\\limits_{n \\in \\integers} \\mathbf{b}_nt^n$}.\nNotice that we have \\mbox{$(1-t){\\cdot}\\sum\\limits_{n \\in \\integers} \\mathbf{a}_nt^n = \\operatorname{H}(B)$}\nand \\mbox{$\\sum\\limits_{n \\in \\integers} \\delta_nt^n = (1-t)^{-1}$}. Now \\vspace{-3mm}\n$$\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr){\\cdot} \\operatorname{H}(B) {\\cdot}\n(1-t)^{-1} \\succeq (1-t)^{-1}.$$\nThis is the form of Vinberg's inequality for filtered algebras~\\cite{V}. The method of proof outlined here\n is reminiscent of the proof of Theorem~\\ref{thm:exact} above. Notice that if there exists some\n \\mbox{$\\varepsilon \\in \\,\\left]0,1\\right[\\,\\,$} such that\n\\mbox{$\\bigl(1 - \\operatorname{h}(X) + \\operatorname{h}(R)\\bigr)\\vert_{t \\mapsto \\varepsilon}$} converges\nto a negative value, then \\mbox{$\\operatorname{H}(B)$} cannot be a polynomial, and, in particular,\n \\mbox{$B$} cannot be finite\\d1dimensional.\n\n\\begin{history} Suppose that \\mbox{$G = \\langle X \\mid R \\rangle$} is a group presentation such\n that \\mbox{$\\operatorname{d}(G^{\\text{\\normalfont ab}}) = \\vert X \\vert < \\aleph_0$}.\n Theorem~\\ref{thm:new} says that if \\mbox{$\\vert R \\vert \\le \\frac{1}{4} \\vert X \\vert^2$},\\vspace{1mm}\nthen $G$ is either trivial or infinite. Wilson~\\cite{Wilson} showed that if \\mbox{$\\vert R \\vert < \\frac{1}{4} \\vert X \\vert^2$},\nthen either $G\\simeq \\integers$ or $G$ maps onto a residually finite, infinite $p$-group,\nfor some prime $p$.\nHis proof is based on Vinberg's inequality and the methods of Golod~\\cite{G}.\n A recent introduction to related results can be found in~\\cite{Ershov}.\n\\end{history}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}