diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgylh" "b/data_all_eng_slimpj/shuffled/split2/finalzzgylh" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgylh" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn Ref.~\\cite{2N3N} a new formulation for the 2N and 3N bound states in three dimensions has been proposed.\nIn this technique momentum vectors are taken as variables, avoiding a traditional partial-wave decomposition. In addition\nspin operators occurring as scalar products of spin and momentum vectors - shortly called spin-momentum\noperators - are evaluated analytically by means of trace operations. In this approach a NN force is\nemployed using its most general operator structure, i.e. as sum of 6 spin-momentum operators multiplied with\nscalar functions of momenta. A spin-\\linebreak momentum operator representation is used as well for\nthe 2N and 3N bound states, as in Refs.~\\cite{deut} and \\cite{oph3}, respectively.\n\nWe extend the technique developed in Ref.~\\cite{2N3N} to NN scattering. This would be an alternative to other\nthree-\\linebreak dimensional approach formulated in a momentum-helicity basis~\\cite{NN3D}. In addition we\nintroduce a new set of spin-\\linebreak momentum operators different from the one used in \\linebreak\nRef.~\\cite{2N3N}. We find one of the spin-momentum operators in Ref.~\\cite{2N3N} violates time reversal \nand, therefore, has to be multiplied with a time-reversal violating scalar function. Here we prefer to work with \noperators, which are also invariant with respect to time reversal. \nThe idea is to apply the general operator structure not only to the NN force but also to\nthe NN T-matrix. The goal is then to find the scalar functions in the expansion of the T-matrix into the\nspin-momentum operators. First, we insert the spin-momentum operators expansions of the NN interaction and\nT-matrix into the Lippmann-Schwinger equation. Next by analytical evaluation we remove the spin dependence\nyielding finally a set of coupled equations for the scalar functions of the T-matrix. Finally we connect\nthe T-matrix to the anti-symmetrized scattering amplitude parameterized by the \\linebreak Wolfenstein parameters.\n\n\\section{Formulation}\\label{formulation}\n\n\\subsection{The general operator structure of NN potential}\n\nThe general operator structure of NN potential reads\n\\begin{equation}\nV^ { t m_t}({\\bf p'}, {\\bf p}) = \\sum_{j=1}^ 6 v_j^ { t m_t} ({\\bf p'}, {\\bf p}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) , \\label{vinw}\n\\end{equation}\nwith $V^ { t m_t}({\\bf p'}, {\\bf p})$ being the NN potential projected on the NN total isospin states $ \\mid t m_t\\rangle$ as\n\\begin{equation}\nV^ { t m_t}({\\bf p'}, {\\bf p}) = \\langle t m_t \\mid V({\\bf p'}, {\\bf p}) \\mid t m_t\\rangle .\n\\end{equation}\nThe scalar functions $v_j^ { t m_t} ({\\bf p'}, {\\bf p})$ depend only on the vector momenta.\nthe\n$w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})$ are a set of spin-momentum operators,\n\\begin{eqnarray}\nw_1 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})& = & 1\\cr\nw_2 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})& = & {\\mbox{\\boldmath$\\sigma$}}_1 \\cdot {\\mbox{\\boldmath$\\sigma$}}_2\\cr\nw_3 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p)}& = & i \\; ( {\\mbox{\\boldmath$\\sigma$}}_1\n+ {\\mbox{\\boldmath$\\sigma$}}_2 ) \\cdot ( {\\bf p} \\times {\\bf p'})\\cr\nw_4 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})& = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot ( {\\bf p} \\times {\\bf p'}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot ( {\\bf p} \\times {\\bf p'})\\cr\nw_5 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})& = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot ({\\bf p'} + {\\bf p}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot ({\\bf p'} + {\\bf p})\\cr\nw_6 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})& = & {\\mbox{\\boldmath$\\sigma$}}_1\n\\cdot ( {\\bf p'} - {\\bf p}) \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot ( {\\bf p'} - {\\bf p}) ,\n\\end{eqnarray}\nwhich is time-reversal invariant. As an example a leading order (LO) chiral NN potential is given as~\\cite{evgeny.report} \\pagebreak\n\\begin{eqnarray}\nV_{LO}({\\bf p'}, {\\bf p}) & = & -\\frac{1}{(2\\pi)^3} \\frac{g_A^2}{4 F_\\pi^2} \\frac{ w_6 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})}{( {\\bf p'} - {\\bf p})^2 +M_\\pi^2} {\\mbox{\\boldmath$\\tau$}_1} \\cdot {\\mbox{\\boldmath$\\tau$}_2} \\cr\n&& + \\frac{C_S}{(2\\pi)^3}w_1 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& + \\frac{C_T}{(2\\pi)^3}w_2 ( {\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) .\n\\end{eqnarray}\n\n\\subsection{The deuteron}\n\nWe briefly describe the formulation for the deuteron. The deuteron has total spin $1$ and isospin $0$.\nIn spin-momentum operator representation the deuteron state is given as~\\cite{deut}\n\\begin{equation}\n\\Psi_{m_d}({\\bf p}) = \\langle {\\bf p} | \\Psi_{m_d}\\rangle = \\sum_{ k=1}^2 \\phi_k(p) \\; b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) | 1 m_d \\rangle , \\label{deutwf}\n\\end{equation}\nwhere $| 1 m_d \\rangle$ is the total-spin state with magnetic quantum number $m_d$, $\\phi_k(p)$ scalar functions depending on the magnitude of momenta only, and $b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})$ spin-momentum operators given as\n\\begin{eqnarray}\nb_1( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) & = & 1 \\cr\nb_2( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) & = & {\\mbox{\\boldmath$\\sigma$}}_1 \\cdot {\\bf p} \\; {\\mbox{\\boldmath$\\sigma$}}_2 \\cdot {\\bf p} - \\frac{1}{3} p^2 .\n\\end{eqnarray}\nThe scalar functions $\\phi_k(p)$ are connected to the standard partial-wave projected deuteron\n s-wave $\\psi_0(p)$ and d-wave $\\psi_2(p)$ by~\\cite{deut}\n\\begin{eqnarray}\n\\psi_0 (p) & = & \\phi_1 (p) \\cr\n\\psi_2 (p) & = & \\frac{4 p^2} { 3 \\sqrt{2}} \\, \\phi_2 (p) .\n\\end{eqnarray}\nInserting $\\Psi_{m_d}({\\bf p})$ of Eq.~(\\ref{deutwf}) and $V^ { t m_t}({\\bf p'}, {\\bf p})$ of Eq.~(\\ref{vinw}) into the Schr\\\"odinger equation for the deuteron in integral form,\n\\begin{equation}\n\\Psi_{m_d}({\\bf p}) = \\frac{1}{ E_d -\\frac{p^2}{m}} \\int d^3 p' V^ {00}({\\bf p}, {\\bf p'}) \\Psi_{m_d}({\\bf p'}),\n\\end{equation}\nyields\n\\begin{eqnarray}\n\\lefteqn{\\sum_{ k=1}^2 \\phi_k(p) \\; b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) | 1 m_d \\rangle} && \\cr\n&& \\qquad = \\frac{1}{ E_d -\\frac{p^2}{m}} \\int d^3 p' \\sum_{j=1}^ 6 v_j^ {00} ({\\bf p}, {\\bf p'}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}, {\\bf p'}) \\cr\n&& \\qquad \\quad \\sum_{ k'=1}^2 \\phi_{k'}(p) \\; b_{k'}( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}) | 1 m_d \\rangle . \\label{dschr}\n\\end{eqnarray}\n\nTo remove the spin dependence from Eq.~(\\ref{dschr}) we project Eq.~(\\ref{dschr}) on $\\langle 1 m_d | b_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})$ from the left and sum up over $m_d$. We obtain\n\\begin{eqnarray}\n\\sum_{ k=1}^2 A^ d_{i k}(p) \\phi_{k} (p) & = & \\frac{1}{ E_d-\\frac{p^2}{m}}\\int d^3 p' \\sum_{j=1}^6 v_j^ {00}( {\\bf p},{\\bf p'}) \\cr\n&& \\sum_{k'=1}^2 B^ d_{ijk'} ( {\\bf p},{\\bf p'})\\phi_{k'}(p') ,\n\\end{eqnarray}\nwhich is a set of two coupled equations for $\\phi_{k} (p)$, with $A^ d_{i k}(p)$ and $B^ d_{ijk'} ( {\\bf p},{\\bf p'})$ being defined as\n\\begin{equation}\nA^ d_{i k} ( p) \\equiv \\sum_{ m_d = -1}^{1 } \\langle 1 m_d|\nb_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})\n b_k( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p})| 1 m_d \\rangle \n\\end{equation}\n\\begin{eqnarray}\nB^ d_{ijk'} ( {\\bf p},{\\bf p'}) & \\equiv & \\sum_{ m_d = -1}^{1 } \\langle 1 m_d|\nb_i( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}) w_j( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p}, {\\bf p'}) \\cr\n&& \\qquad b_{k'}( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'})| 1 m_d \\rangle .\n\\end{eqnarray}\nThe functions $A^ d_{i k}(p)$ and $B^ d_{ijk'} ({\\bf p},{\\bf p'})$ are scalar functions of the vectors ${\\bf p}$ and ${\\bf p'}$, and need to be calculated only once.\nAs example we have e.g.\n\\begin{eqnarray}\nA^d_{11} (p) & = & 3\\cr\nA^d_{22} (p) & = & \\frac{8}{3} p^4 \\cr\nB^d_{141} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p}\\times {\\bf p'})^2 \\cr\nB^d_{151} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p'}+ {\\bf p})^2 \\cr\nB^d_{161} ({\\bf p},{\\bf p}^{\\prime}) & = & ({\\bf p'}- {\\bf p})^2 .\n\\end{eqnarray}\n\n\\subsection{The NN scattering}\n\nThe operator structure given in Eq.~(\\ref{vinw}) for NN force can also be applied to the NN T-matrix as\n\\begin{equation}\nT^ { t m_t}({\\bf p'}, {\\bf p}) = \\sum_{j=1}^ 6 t_j^ { t m_t} ({\\bf p'}, {\\bf p}) \\;\nw_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) , \\label{tinw}\n\\end{equation}\nwith $t_j^ { t m_t} ({\\bf p'}, {\\bf p})$ being the scalar functions to be found. Inserting both\nthe expansion in Eqs.~(\\ref{vinw}) and (\\ref{tinw}) into the Lippmann-Schwinger equation,\n\\begin{eqnarray}\nT^ { t m_t}({\\bf p'}, {\\bf p}) & = & V^ { t m_t}({\\bf p'}, {\\bf p}) \\cr\n&& + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\int d{\\bf p''} \\frac{V^ { t m_t}({\\bf p'}, {\\bf p''}) T^ { t m_t}({\\bf p''}, {\\bf p})}{p^2 + i\\epsilon -p''^2} , \\qquad\n\\end{eqnarray}\nwhere $\\mu$ is the reduced mass of the NN system, leads to\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k=1}^ 6 t_k^ { t m_t} ({\\bf p'}, {\\bf p}) \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})} && \\cr\n&& \\qquad = \\sum_{k=1}^ 6 v_k^ { t m_t} ({\\bf p'}, {\\bf p}) \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& \\qquad \\quad + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\int d{\\bf p''} \\frac{1}{p^2 + i\\epsilon -p''^2} \\cr\n&& \\qquad \\quad \\sum_{j=1}^ 6 v_j^ { t m_t} ({\\bf p'}, {\\bf p''}) \\; w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p''}) \\cr\n&& \\qquad \\quad \\sum_{k'=1}^ 6 t_{k'}^ { t m_t} ({\\bf p''}, {\\bf p}) \\; w_{k'}({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p''}, {\\bf p}) . \\label{ls}\n\\end{eqnarray}\nWe remove the spin dependence from Eq.~(\\ref{ls}) by multiplying from the left with $w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})$ and perform the trace. \nThis leads to \\pagebreak\n\\begin{eqnarray}\n\\lefteqn{\\sum_{k = 1}^6 A_{ik}({\\bf p'}, {\\bf p}) t_k^{tm_t}({\\bf p'}, {\\bf p}) } && \\cr\n&& \\qquad = \\sum_{k = 1}^6 A_{ik}({\\bf p'}, {\\bf p}) v_k^{tm_t}({\\bf p'}, {\\bf p}) \\cr\n&& \\qquad \\quad + 2 \\mu \\lim_{\\epsilon \\rightarrow 0} \\sum_{j,k' = 1}^6 \\int d{\\bf p''}\n \\frac{1} {p^2 + i\\epsilon - p''^2} \\cr\n&& \\qquad \\quad B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})\n v_j^{tm_t}({\\bf p'}, {\\bf p''}) t_{k'}^{tm_t}({\\bf p''}, {\\bf p}) , \\label{nnls}\n\\end{eqnarray}\nwhere $A_{ik}({\\bf p'}, {\\bf p})$ and $B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})$ are defined as\n\\begin{eqnarray}\nA_{ik}({\\bf p'}, {\\bf p}) & \\equiv & Tr \\{w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) \\cr\n&& \\quad \\; w_k({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})\\} \\\\\nB_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p}) & \\equiv & Tr \\{w_i({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) w_j({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p''}) \\cr\n&& \\quad \\; w_{k'}({\\mbox{\\boldmath$\\sigma$}}_1,{\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p''}, {\\bf p})\\}\n\\end{eqnarray}\nAgain the functions\n $A_{ik}({\\bf p'}, {\\bf p})$ and $B_{ijk'}({\\bf p'}, {\\bf p''}, {\\bf p})$ are scalar functions of the momenta ${\\bf p}$ and ${\\bf p'}$, and need to be evaluated only once.\nAs example we show here,\n\\begin{eqnarray}\nA_{24}({\\bf p}', {\\bf p}) & = & A_{42}({\\bf p}', {\\bf p}) = 4 ({\\bf p} \\times {\\bf p}')^2 \\cr\nA_{56}({\\bf p}', {\\bf p}) & = & A_{65}({\\bf p}', {\\bf p}) = 4 (p'^2 - p^2)^2 \\cr\nB_{122}({\\bf p}', {\\bf p}'', {\\bf p}) & = & B_{212}({\\bf p}', {\\bf p}'', {\\bf p}) = B_{221}({\\bf p}', {\\bf p}'', {\\bf p}) = 12 \\cr\nB_{124}({\\bf p}', {\\bf p}'', {\\bf p}) & = & B_{214}({\\bf p}', {\\bf p}'', {\\bf p}) = 4 ({\\bf p} \\times {\\bf p}'')^2 \\cr\nB_{144}({\\bf p}', {\\bf p}'', {\\bf p}) & = & 4 \\{({\\bf p}'' \\times {\\bf p}') \\cdot ({\\bf p} \\times {\\bf p}'')\\}^2\n\\end{eqnarray}\nEquation (\\ref{nnls}) is a set of six coupled equations for $t_k^{tm_t}({\\bf p'}, {\\bf p})$, which can e.g. be solved\nas a system of linear equations.\n\nThe NN scattering observables can be calculated from the anti-symmetrized scattering amplitude $M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})$, \\linebreak which is defined as\n\\begin{eqnarray}\n\\lefteqn{M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})} && \\cr\n&& \\; \\equiv \\langle t m_t | \\langle m_1' m_2'| \\langle {\\bf p'}| M ( 1 - P_{12}) | {\\bf p}\\rangle | m_1 m_2\\rangle | t m_t\\rangle\n\\end{eqnarray}\nand can be parameterized by the Wolfenstein parameters $a^{t m_t}({\\bf p'},{\\bf p})$, $c^{t m_t}({\\bf p'},{\\bf p})$, $m^{t m_t}({\\bf p'},{\\bf p})$, $g^{t m_t}({\\bf p'},{\\bf p})$, $h^{t m_t}({\\bf p'},{\\bf p})$ \\linebreak as~\\cite{book}\n\\begin{eqnarray}\n\\lefteqn{M^ {t m_t}_{ m_1' m_2', m_1 m_2}({\\bf p'},{\\bf p})} && \\cr\n&& \\quad = a^ {tm_t}({\\bf p'},{\\bf p}) \\; \\langle m_1' m_2'| w_1( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad - i \\frac{ c^ {tm_t}({\\bf p'},{\\bf p})}{ | {\\bf p} \\times {\\bf p'}|} \\;\n \\langle m_1' m_2'|w_3 ({\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p})| m_1 m_2\\rangle\\cr\n&& \\qquad + \\frac{ m^ {tm_t}({\\bf p'},{\\bf p})}{ | {\\bf p} \\times {\\bf p'}|^ 2}\n \\langle m_1' m_2'| w_4( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'}, {\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad + \\frac{ g^ {tm_t}({\\bf p'},{\\bf p})+h^ {tm_t}({\\bf p'},{\\bf p})}{ ( {\\bf p} + {\\bf p'})^ 2} \\cr\n&& \\qquad \\quad \\langle m_1' m_2'| w_5 ( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle \\cr\n&& \\qquad + \\frac{ g^ {tm_t}({\\bf p'},{\\bf p})-h^ {tm_t}({\\bf p'},{\\bf p})}{ ( {\\bf p} - {\\bf p'})^ 2} \\cr\n&& \\qquad \\quad \\langle m_1' m_2'| w_6( {\\mbox{\\boldmath$\\sigma$}}_1, {\\mbox{\\boldmath$\\sigma$}}_2, {\\bf p'},{\\bf p}) | m_1 m_2\\rangle . \\quad \\label{wpar}\n\\end{eqnarray}\nThus, finally we connect the scattering amplitude or similarly the Wolfenstein parameters to the scalar function \\linebreak $t_j^ { t m_t} ({\\bf p'}, {\\bf p})$. This can be accomplished by means of Eq.~(\\ref{wpar}) and the relation\nbetween the M- and T-matrix given as\n\\begin{equation}\nM = -\\mu (2\\pi)^2 T .\n\\end{equation}\nWe obtain\n\\begin{eqnarray}\na^ {tm_t} & = & t_1 + (-)^ t \\; \\Big[ \\frac{1}{2} \\tilde t_1 + \\frac{3}{2} \\tilde t_2 \\cr\n&& + \\frac{1}{2} p^ 4 ( 1-x^ 2) \\tilde t_4 + p^ 2 ( 1-x) \\tilde t_5 + p^ 2 (\n1+x) \\tilde t_6\\Big] \\cr\nc^ {tm_t} & = & i p^ 2 \\sqrt{1 - x^ 2} \\left( t_3 - (-)^ t \\tilde t_3 \\right) \\cr\nm^ {tm_t} & = & t_2 + p^ 4 ( 1 - x^ 2) t_4\n + (-)^ t \\; \\Big[ \\frac{1}{2} \\tilde t_1 - \\frac{1}{2} \\tilde t_2 \\cr\n&& + \\frac{1}{2} p^ 4 ( 1- x^ 2) \\tilde t_4 - p^ 2 (1-x)\\tilde t_5 - p^ 2 ( 1+x) \\tilde t_6 \\Big]\\cr\ng^ {tm_t} & = & t_2 + p^ 2 ( 1+x) t_5 + p^ 2( 1-x) t_6 \\cr\n&& + (-)^ t \\; \\Big[ \\frac{1}{2} \\tilde t_1 - \\frac{1}{2} \\tilde t_2\n- \\frac{1}{2} p^ 4 ( 1-x^ 2) \\tilde t_4 \\Big]\\cr\nh^ {tm_t} & = & p^ 2(1+x) t_5 - p^ 2(1-x) t_6 \\cr\n&& + (-)^ t \\; \\Big[- p^ 2(1-x) \\tilde t_5 + p^ 2(1+x) \\tilde t_6 \\Big] . \\label{wpart},\n\\end{eqnarray}\nwhere $x = {\\bf \\hat p'} \\cdot {\\bf \\hat p}$\nNote that in Eq.~(\\ref{wpart}) we drop ${\\bf p}$ and ${\\bf p'}$ for simplicity and apply the following notation,\n\\begin{eqnarray}\nt_j & \\equiv & t_j^ {t m_t} ( {\\bf p'}, {\\bf p}) \\cr\n\\tilde t_j &\\equiv & t_j^ {t m_t} ( {\\bf p'}, -{\\bf p}) .\n\\end{eqnarray}\n\n\\section{Summary}\n\nWe propose a new technique to calculate the 2N system as function of momentum vectors, i.e.\n without employing a partial wave decomposition. The technique\nis useful especially in energy regions of hundreds of MeV or when considering the NN t-matrix as input to a three-body calculation. Based on scalar interactions, the scattering of three-bosons has been successfully carried out up to the GeV regime, formulating the Faddeev equations as functions of vector momenta~\\cite{Liu:2004tv}. The formulation of NN scattering presented here is an important\nstep on the way of performing realistic three-body scattering calculations at higher energies.\n\n Based on the general operator\nstructure of the NN interaction we derive the formulation in a spin-momentum operator representation. Here\nthe NN potential, the T-matrix, and the deuteron state are expanded in a set of scalar products\nof spin operators and momentum vectors. We derive a set of two coupled equations for the deuteron wave function\ncomponents, which are connected to the standard partial wave projected wave function s- and d-wave in a simple\nmanner. In case of the NN scattering we obtain a set of six coupled equations for the scalar functions defining\nthe NN T-matrix, and therefore, the scattering amplitude in the Wolfenstein representation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec1intro}}\n\nMultiple testing (MT) and control of the false discovery rate (FDR, \\cit\n{Benjamini:1995}) has been conducted in a variety of scientific endeavours\nincluding the microarray study in \\cite{Efron:2001} and the analysis of\nspatial data in \\cite{Benjamini:2007}. It is very challenging to develop MT\nprocedures with a desired FDR when the test statistics are strongly dependent.\nRecently, \\cite{Fan:2012} considered MT in the following setting. Let \n\\mathbf{Z}\\sim \\mathsf{N}_{m}\\left( \\boldsymbol{\\mu },\\boldsymbol{\\Sigma \n\\right) $ be an $m$-dimensional Normal random vector (rv) with mean $\\boldsymbol\n\\mu }=\\left( \\mu _{1},...,\\mu _{m}\\right) ^{T}\\in \\mathbb{R}^{m}$ and a\nknown correlation matrix $\\boldsymbol{\\Sigma }$ $\\geq 0$. Given an\nobservation $\\mathbf{z}=\\left( z_{1},...,z_{m}\\right) ^{T}$ of $\\mathbf{Z}$,\nthe $i$th null $H_{i}:\\mu _{i}=0$ versus its alternative $H_{i}^{\\ast }:\\mu\n_{i}\\neq 0$ is tested using $p_{i}=2\\Phi \\left( -\\left\\vert z_{i}\\right\\vert\n\\right) $ and some $t\\in \\left[ 0,1\\right] $ such that $H_{i}$ is rejected\nif and only if $p_{i}\\leq t$, where $\\Phi \\left( \\cdot \\right) $ is the\ncumulative distribution function (cdf) of the standard Normal rv. To tackle this MT\nproblem, \\cite{Fan:2012} proposed the technique of principal factor approximation (PFA) to decompose the dependence embedded in $\\boldsymbol\n\\Sigma }$ and used Theorem 6 in \\cite{Lyons:1988} together with the PFA to\nobtain the strong law of large numbers (SLLN) for the normalized number of conditional rejections\n\\begin{equation}\n\\tilde{R}\\left( t\\right) =m^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{k}\\right)\n=m^{-1}\\sum_{i=1}^{m}1_{\\left\\{ p_{i}\\leq t|\\mathbf{\\tilde{w}}_{k}\\right\\}\n}\\label{eqb1a\n\\end{equation}\nfor arbitrary $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ and $\\boldsymbol{\\Sigma\n\\geq0$, where $1_{A}$ is the indicator of a set $A$ and $\\mathbf{\\tilde{w\n}_{k}\\sim\\mathsf{N}_{k}\\left( 0,\\mathbf{I}\\right) $ are called by\n\\cite{Fan:2012} the ``principal factors'\n(see \\autoref{SecRevisitPFA} for more details). The SLLN for $\\tilde\n{R}\\left( \\cdot\\right) $ (and that for another related quantity) enables\n\\cite{Fan:2012} to derive an almost sure (a.s.) approximation to the false\ndiscovery proportion (FDP, \\cite{Genovese:2002}) and it is considered a\ntheoretical breakthrough in FDR control of MT under dependence.\n\n\nUnfortunately, we found that the arguments given by the authors of \\cite{Fan:2012} do not seem to\nbe fully sufficient to yield the SLLN for $\\tilde{R}\\left( \\cdot\\right) $\nfor arbitrary $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ and $\\boldsymbol{\\Sigma\n\\geq0$, mainly due to their delicate handle of the involved functional\nremainders in the Taylor expansions and their use of some implicit assumptions\nto derive certain auxiliary asymptotic assertions. In this article, we will\nprovide a complete justification of this law with clarifications on all needed\nassumptions. In \\autoref{SecRevisitPFA} we will revisit the technique of PFA\nand provide examples on different speeds of the PFA to components of the original rv. In\n\\autoref{Sec:ReviewFan}, we will briefly review the strategy on the proof used\nin \\cite{Fan:2012} and point out the extra arguments needed to justify this law.\nIn \\autoref{SecLLNDisRejProc}, we present our detailed proof. Our\nreformulation provides an integrated view on how the speed of the PFA to\n$\\mathbf{Z}$, the dependence among $z_{i}$'s, and the magnitudes of $\\mu_{i\n$'s should interact with each other in order to validate this law. We end the\narticle with a short discussion in \\autoref{SecDiscussion}.\n\n\\section{PFA with different component speeds\\label{SecRevisitPFA}}\n\nLet the triple $\\left( \\Omega,\\mathcal{F},\\mathbb{P}\\right) $ be the probability\nspace on which all involved rv's are defined. We restate the technique of PFA\ndeveloped in \\cite{Fan:2012} as follows. Let $\\mathbf{w}=\\left(\nw_{1},...,w_{m}\\right) ^{T}\\sim\\mathsf{N}_{m}\\left( \\mathbf{0\n,\\mathbf{I}\\right) $. The spectral decomposition of $\\boldsymbol{\\Sigma}$\nwith eigenvalues (counting multiplicity) $\\lambda_{1,m}\\geq\\lambda_{2,m\n\\geq\\cdots\\geq\\lambda_{m,m}\\geq0$ and corresponding eigenvectors\n$\\boldsymbol{\\gamma}_{i}=\\left( \\gamma_{i1},...,\\gamma_{im}\\right) ^{T}$ for\n$i=1,...,m$ implie\n\\begin{equation}\n\\mathbf{Z}=\\boldsymbol{\\mu}+\\boldsymbol{\\eta}+\\mathbf{v}\\text{,}\n\\label{eqb103\n\\end{equation}\nwhere $\\boldsymbol{\\eta}=\\left( \\eta_{1},...,\\eta_{m}\\right) ^{T}$ with\n$\\eta_{i}=\\sum\\nolimits_{j=1}^{k}\\sqrt{\\lambda_{j,m}}\\gamma_{ij}w_{j}$ and\n$\\mathbf{v}=\\left( v_{1},...,v_{m}\\right) ^{T}$ with\n$v_{i}=\\sum\\nolimits_{j=k+1}^{m}\\sqrt{\\lambda_{j,m}}\\gamma_{ij}w_{j}$ for any\n$0\\leq k\\leq m$. In (\\ref{eqb103}), $\\boldsymbol{\\eta}=\\mathbf{0}$ is set when\n$k=0$. From (\\ref{eqb103}), we see that $\\omega_{i,m}=\\mathbb{V}\\left[\n\\eta_{i}\\right] =\\sum\\nolimits_{j=1}^{k}\\lambda_{j,m}\\gamma_{ij}^{2}$ and\n$\\sigma_{i,m}=\\mathbb{V}\\left[ v_{i}\\right] =1-\\omega_{i,m}$, where\n$\\mathbb{V}\\left[ \\cdot\\right] $ denotes the variance operator. Note that\n$\\left( w_{1},...,w_{k}\\right) ^{T}$ is exactly $\\mathbf{\\tilde{w}}_{k}$.\n\nFor (\\ref{eqb103}) with $k\\geq1$, \\cite{Fan:2012} pointed out that for a fixed\n$\\delta>0$ there always exists some $1\\leq k\\leq m$ ($k$ can depend on $m$)\nsuch tha\n\\begin{equation}\n\\vartheta_{m}=m^{-1}\\sqrt{\\lambda_{k+1,m}^{2}+...+\\lambda_{m,m}^{2}}\\leq\nCm^{-\\delta}\\text{ for sufficiently large }m\\label{eqb23\n\\end{equation}\nfor some finite constant $C>0$. It should be noted\nthat (\\ref{eqb23}) never holds for any $\\delta>0$ when $k=0$ and\n$\\boldsymbol{\\Sigma}>0$. Further, \\cite{Fan:2012} defined\n\\[\na_{i,m}=\\left( 1-\\omega_{i,m}\\right) ^{-1\/2\n\\]\nwith the implicit assumption that $\\omega_{i,m}<1$ for $1\\leq i\\leq m$.\nNote that $a_{i,m}\\geq1$ for any $1\\leq i\\leq m$ and finite $m$.\n\nLet $cov\\left( \\cdot,\\cdot\\right) $ be the covariance operator and\n$\\mathbf{A}=cov\\left( \\mathbf{v},\\mathbf{v}\\right) =\\left( q_{ij}\\right)\n_{m\\times m}$. The magnitudes of $\\left\\{ a_{i,m}\\right\\} _{i=1}^{m}$ play a\ncrucial role in the asymptotic analysis on $\\tilde{R}\\left( \\cdot\\right) $\nsince they control the speed of PFA via $\\boldsymbol{\\eta}$ to $\\mathbf{Z}$\nand affect the dependence structure, i.e., entries of $\\mathbf{A}$, among\ncomponents of $\\mathbf{v}$. Most importantly, $\\left\\{ a_{i,m}\\right\\}\n_{i=1}^{m}$ affect the remainders of the Taylor expansion used in\n\\cite{Fan:2012}. Let\n$\\mathbf{T}_{m}=\\left( \\gamma_{ij}\\right) _{m\\times m}$, $\\mathbf{D\n_{m}=diag\\left\\{ \\lambda_{1,m},...,\\lambda_{m,m}\\right\\} $ and\n$\\boldsymbol{\\Sigma}=\\boldsymbol{\\Sigma}_{m}$,\nwhere the subscript $m$ denotes the dimension of matrices.\nWe give an example where $\\omega_{i,m}=1$, i.e.,\n$a_{i,m}=\\infty$ for $1\\leq i\\leq k$ for some $k$.\n\n\\begin{lemma}\n\\label{Lm:BlockOrtho}For all even $m\\geq4$ and any $\\boldsymbol{\\mu\n\\in\\mathbb{R}^{m}$, there exists $\\boldsymbol{\\Sigma}_{m}>0$, a block diagonal\n(not diagonal) matrix, such that $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left(\n\\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right) $ obeys (\\ref{eqb103}) and\nthat (\\ref{eqb23}) holds with $k=2^{-1}m$, $C=1$ and $\\delta\\in\\left(\n0,2^{-1}\\right) $ but $a_{i,m}=\\infty$ for $1\\leq i\\leq k$ and $a_{i,m}=1$\nfor $k+1\\leq i\\leq m$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, we construct the needed positive eigenvalues $\\left\\{ \\lambda\n_{i,m}\\right\\} _{i=1}^{m}$ (with $\\lambda_{i,m}\\geq\\lambda_{i+1,m}$). Pick\n$\\delta\\in\\left( 0,2^{-1}\\right) $, $k=2^{-1}m$ and $\\left\\{ \\varepsilon\n_{j}\\right\\} _{j=1}^{k}$ such that $0<\\varepsilon_{j}<\\varepsilon_{j+1}<1$\nfor all $1\\leq j\\leq k-1$. Let $\\lambda_{k+j,m}=1-\\varepsilon_{j}$ and\n$\\lambda_{j,m}=1+\\varepsilon_{j}$ for $1\\leq j\\leq k$. Then $\\sum_{i=1\n^{m}\\lambda_{i,m}=m$. Since $\\min\\left\\{ \\sqrt{2}m^{1\/2-\\delta},1\\right\\}\n=1$ whenever $m\\geq1$ and\n\\[\n\\lambda_{k+1,m}\\leq\\sqrt{2}m^{1\/2-\\delta\n\\]\nimplies\n\\[\nm^{-1}\\sqrt{\\sum\\nolimits_{j=k+1}^{m}\\lambda_{j,m}^{2}}\\leq2^{-1\/2\nm^{-1\/2}\\lambda_{k+1,m}\\leq m^{-\\delta}\\text{,\n\\]\nit follows that the choice of $\\left\\{ \\lambda_{i,m}\\right\\} _{i=1}^{m}$,\n$C$, $\\delta$ and $k$ validates (\\ref{eqb23}).\n\nNext, we construct the desired $\\boldsymbol{\\Sigma}_{m}$ and $\\mathbf{Z}$.\nKeep $k=2^{-1}m$. Let $\\mathbf{Q}_{1}\\in\\mathcal{O}_{k}$ and $\\mathbf{Q\n_{2}\\in\\mathcal{O}_{m-k}$, where $\\mathcal{O}_{n}$ denotes the set of $n\\times\nn$ orthogonal matrices. Define $\\mathbf{T}_{m}=diag\\left\\{ \\mathbf{Q\n_{1},\\mathbf{Q}_{2}\\right\\} $. Then, $\\mathbf{T}_{m}=\\left( \\gamma\n_{ij}\\right) _{m\\times m}$ is orthogonal such that $\\max_{1\\leq i\\leq k\n\\max_{k+1\\leq j\\leq m}\\gamma_{ij}=0$ but $\\sum_{j=k+1}^{m}\\gamma_{ij}^{2}=1$\nfor all $k+1\\leq i\\leq m$. Now setting $\\mathbf{Z}=\\boldsymbol{\\mu\n+\\mathbf{T}_{m}\\sqrt{\\mathbf{D}_{m}}\\mathbf{w}$ for any $\\boldsymbol{\\mu\n\\in\\mathbb{R}^{m}$ gives $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu\n},\\boldsymbol{\\Sigma}_{m}\\right) $ with $\\boldsymbol{\\Sigma}_{m\n=\\mathbf{T}_{m}\\mathbf{D}_{m}\\mathbf{T}_{m}^{T}$. Obviously, $\\mathbf{Z}$\nadmits decomposition (\\ref{eqb103}) with this $k=2^{-1}m$ and\n$\\boldsymbol{\\Sigma}_{m}>0$ is a block diagonal matrix. However, $\\omega\n_{i,m}=1$ for $1\\leq i\\leq k$ and $\\omega_{i,m}=0$ for $k+1\\leq i\\leq m$,\nmeaning that $a_{1}=...=a_{k}=\\infty$ and $a_{k+1}=...=a_{m}=1$. This\ncompletes the proof.\n\\end{proof}\n\nIt can be easily seen that $\\boldsymbol{\\Sigma}_{m}>0$ is a block diagonal\nmatrix if and only $\\mathbf{T}_{m}$ is so. The Normal rv $\\mathbf{Z\n\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right) $\nprovided in \\autoref{Lm:BlockOrtho} has a block diagonal correlation matrix\n$\\boldsymbol{\\Sigma}_{m}$ and has a decomposition $\\mathbf{Z}=\\boldsymbol{\\mu\n}+\\boldsymbol{\\eta}+\\mathbf{v}$ where $\\eta_{i}=0$ a.s. for $1\\leq i\\leq k$\nand $v_{i}=0$ a.s. for $k+1\\leq i\\leq m$. Such Normal rv's $\\mathbf{Z}$\npresent a simpler case for multiple testing which $\\mu_{i}$'s are zero since\n$\\left( z_{1},...,z_{k}\\right) ^{T}$ are independent of $\\left(\nz_{k+1},...,z_{m}\\right) ^{T}$, the latter of which has weakly dependent\ncomponents as defined by \\cite{Fan:2012}.\n\nWe have the following example for which each $a_{i,m}$ for $1\\leq i\\leq m$ is\nfinite for any finite $m$:\n\n\\begin{lemma}\n\\label{Lm:DensestOrtho}For any $m\\geq2$, there exists an orthogonal matrix\n$\\mathbf{T}_{m}=\\left( \\gamma_{ij}\\right) _{m\\times m}$ such that\n$\\gamma_{ij}\\neq0$ for all $1\\leq i\\leq j\\leq m$. Thus, for any\n$\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ there exits $\\boldsymbol{\\Sigma}_{m}>0$\nfor which $\\mathbf{Z}\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu\n,\\boldsymbol{\\Sigma}_{m}\\right) $ admits (\\ref{eqb103}) for any $1\\leq k\\leq\nm$ such that $10$ can be\nchosen to induce (\\ref{eqb23}) with $k= 2^{-1}m $, $C=1$ and\n$\\delta\\in\\left( 0,2^{-1}\\right) $.\n\\end{lemma}\n\n\\begin{proof}\nFirst, we show the existence of such a $\\mathbf{T}_{m}$. Let $$S^{m-1}=\\left\\{\n\\mathbf{x}\\in\\mathbb{R}^{m}:\\left\\Vert \\mathbf{x}\\right\\Vert =1\\right\\} .$$\nDenote by $\\left\\langle \\cdot,\\cdot\\right\\rangle $ the inner product in\nEuclidean space and by $^{\\perp}$ the orthogonal complement with respect to\n$\\left\\langle \\cdot,\\cdot\\right\\rangle $. Pick $\\mathbf{u}=\\left(\nu_{1},...,u_{m}\\right)^{T} \\in S^{m-1}$ such that $0<\\min_{1\\leq i\\leq m}$\n$\\left\\vert u_{i}\\right\\vert <\\max_{1\\leq i\\leq m}$ $\\left\\vert u_{i\n\\right\\vert <1$ and $2u_{i}^{2}\\neq1$ for all $1\\leq i\\leq m$. Define\n$\\Pi=\\left\\{ \\mathbf{x}\\in\\mathbb{R}^{m}:\\left\\langle \\mathbf{x\n,\\mathbf{u}\\right\\rangle =0\\right\\} $. Then $\\Pi$ is a hyperplane in\n$\\mathbb{R}^{m}$ with normal $\\mathbf{u}$. Let $L=\\left\\{ x\\mathbf{u\n:x\\in\\mathbb{R}\\right\\} $. Then $\\Pi=L^{\\perp}$. Let $\\tilde{T}_{m}$ be the\nreflection with respect to $\\Pi$ (that keeps $\\Pi$ invariant but flips\n$\\mathbf{u}$). Then $\\tilde{T}_{m}\\mathbf{x}=\\mathbf{x}-2\\left\\langle\n\\mathbf{x},\\mathbf{u}\\right\\rangle \\mathbf{u}$ for all $\\mathbf{x\n\\in\\mathbb{R}^{m}$. In particular, $\\tilde{T}_{m}\\mathbf{e}_{i}=\\mathbf{e\n_{i}-2\\left\\langle \\mathbf{e}_{i},\\mathbf{u}\\right\\rangle \\mathbf{u\n=\\mathbf{e}_{i}-2u_{i}\\mathbf{u}$, where $\\mathbf{e}_{i}\\in\\mathbb{R}^{m}$ has\nthe only non-zero entry, $1$, at its $i$th entry. By the construction of\n$\\mathbf{u}$, for each $1\\leq i\\leq m$ each entry of $\\tilde{T}_{m\n\\mathbf{e}_{i}$ is non-zero. Consequently, the matrix $\\mathbf{T}_{m}$ with\nthe $m$ columns $\\boldsymbol{\\gamma}_{i}=\\tilde{T}_{m}\\mathbf{e}_{i}=\\left(\n\\gamma_{i1},...,\\gamma_{im}\\right)^{T} $ is orthogonal and none of the\n$\\gamma_{ij}$'s is zero.\n\nNow we construct the needed $\\boldsymbol{\\Sigma}_{m}>0$. Take any $m$ positive\nnumbers $\\left\\{ \\lambda_{i,m}\\right\\} _{i=1}^{m}$ and set $\\mathbf{D\n_{m}=diag\\left\\{ \\lambda_{1,m},...,\\lambda_{m,m}\\right\\} $. Then\n$\\mathbf{Z}=\\boldsymbol{\\mu}+\\mathbf{T}_{m}\\sqrt{\\mathbf{D}_{m}}\\mathbf{w}$\nfor any $\\boldsymbol{\\mu}\\in\\mathbb{R}^{m}$ satisfies $\\mathbf{Z\n\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu},\\boldsymbol{\\Sigma}_{m}\\right) $\nwith $\\boldsymbol{\\Sigma}_{m}=\\mathbf{T}_{m}\\mathbf{D}_{m}\\mathbf{T}_{m}^{T}$.\nFurther, (\\ref{eqb103}) holds and $10$.\n\n\\item For each $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu\n_{m},\\boldsymbol{\\Sigma}_{m}\\right) $, (\\ref{eqb103}) holds for each $1\\leq\nk\\leq m$ and (\\ref{eqb23}) holds with $k= 2^{-1}m $, $C=1$ and\n$\\delta\\in\\left( 0,2^{-1}\\right) $.\n\n\\item $10$. Take the $\\mathbf{u}$ and $\\tilde{T}_{m}$\nconstructed in the proof of \\autoref{Lm:DensestOrtho} but let $u_{m}=u_{0}$\nfor a fixed, small positive constant $u_{0}$ (e.g., $u_{0}=10^{-5}$ can be\nused); take the $\\mathbf{T}_{m}=\\left( \\gamma_{ij}\\right) _{m\\times m}$\ninduced by $\\tilde{T}_{m}$ under the canonical orthonormal basis $\\left\\{\n\\mathbf{e}_{i}\\right\\} _{i=1}^{m}$ such that the $i$th column of\n$\\mathbf{T}_{m}$ is $\\tilde{T}_{m}\\mathbf{e}_{i}$. Then none of the entries $\\gamma_{ij}$ of\n$\\mathbf{T}_{m}$ is zero, $\\gamma_{im}=-2u_{i}u_{0}$ for $1\\leq i\\leq m-1$ but\n$\\gamma_{mm}=1-2u_{0}^{2}$. So,\n\\[\n\\sigma_{m,m}=\\sum\\nolimits_{j=k+1}^{m}\\lambda_{j,m}\\gamma_{mj}^{2}\\geq\n\\lambda_{m,m}\\gamma_{mm}^{2}=\\lambda_{m,m}\\left( 1-2u_{0}^{2}\\right) ^{2\n\\]\nand\n\\[\n\\liminf_{m\\rightarrow\\infty\n}\\sigma_{m,m}\\geq\\left( 1-\\varepsilon_{0}\\right) \\left( 1-2u_{0\n^{2}\\right) ^{2}\\text{.\n\\]\nTherefore,\n\\[\n\\limsup_{m\\rightarrow\\infty}a_{m,m}=\\left( \\liminf_{m\\rightarrow\\infty\n\\sigma_{m,m}^{1\/2}\\right) ^{-1}\\leq\\dfrac{1}{\\left( 1-2u_{0}^{2}\\right)\n\\sqrt{1-\\varepsilon_{0}}}<\\infty\\text{.\n\\]\nThe proof is completed.\n\\end{proof}\n\nIn fact, we can further construct more elaborate sequence of $\\left\\{\n\\mathbf{Z}_{m}\\right\\} _{m}$ with $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left(\n\\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right) $ such that\namong $\\left\\{ a_{i,m}\\right\\} _{i=1}^{m}$ all the following three types of\nbehaviour occur for some $1\\leq i,i^{\\prime},i^{\\prime\\prime}\\leq m$:\n\n\\begin{enumerate}\n\\item $10$.\n\n\\item For each $\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu\n_{m},\\boldsymbol{\\Sigma}_{m}\\right) $, (\\ref{eqb103}) holds for each $1\\leq\nk\\leq m$ and (\\ref{eqb23}) holds with $k= 2^{-1}m $, $C=1$ and\n$\\delta\\in\\left( 0,2^{-1}\\right) $.\n\n\\item $10$ for $1\\leq i\\leq k$ but\n$\\lim_{m\\rightarrow\\infty}u_{1}=0$. Define $\\Pi$ and $L$ as in\n\\autoref{Lm:DensestOrtho} and let $\\tilde{T}_{m}$ be the reflection with\nrespect to $\\Pi$. Then $\\tilde{T}_{m}\\mathbf{e}_{i}=\\mathbf{e}_{i\n-2u_{i}\\mathbf{u}$. Let the matrix $\\mathbf{T}_{m}$ have its $i$th column\n$\\boldsymbol{\\gamma}_{i}=\\tilde{T}_{m}\\mathbf{e}_{i}$. We see that\n\\begin{align*}\n\\sigma_{k+1,m} & =\\lambda_{k+1,m}\\left( 1-2u_{k+1}^{2}\\right) ^{2\n+\\lambda_{m,m}\\left( 2u_{k+1}u_{m}\\right) ^{2}\\\\\n& \\geq\\left( 1-\\varepsilon_{0}\\right) \\left[ \\left( 1-2\\tilde{u}_{0\n^{2}\\right) ^{2}+2\\tilde{u}_{0}^{2}\\right]\n\\end{align*}\nand $\\sigma_{k+1,m}<1$ for any such $m$. However, $\\sigma_{1,m}=4\\lambda\n_{k+1,m}u_{1}^{2}\\tilde{u}_{0}^{2}+2^{-1}\\lambda_{m,m}u_{1}^{2}$, $\\sigma_{m,m}=0$ and $0<\\sigma_{i,m}<\\infty$ for\n$i\\notin\\left\\{ k+1,m\\right\\} $ for any such $m$. Therefore,\n\\[\n\\limsup_{m\\rightarrow\\infty}a_{k+1,m}=\\left( \\liminf_{m\\rightarrow\\infty\n}\\sigma_{k+1,m}^{1\/2}\\right) ^{-1}<\\infty\n\\]\nand $\\lim_{m\\rightarrow\\infty}a_{1,m}=\\left( \\liminf_{m\\rightarrow\\infty\n}\\sigma_{1,m}^{1\/2}\\right) ^{-1}=\\infty$. The proof is completed.\n\\end{proof}\n\nThe examples we constructed demonstrate the different\nbehaviour of $\\left\\{ a_{i,m}\\right\\} _{i=1}^{m}$ in the PFA to\n$\\mathbf{Z}_{m}\\sim\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu}_{m\n,\\boldsymbol{\\Sigma}_{m}\\right) $ as $m$ changes and hence the need for a\ncareful analysis of the terms involving $a_{i,m}$'s and ratios between the\n$a_{i,m}$'s. However, such an analysis has not been explicitly and carefully conducted in\n\\cite{Fan:2012}.\n\n\\section{Review of strategy in \\cite{Fan:2012}\\label{Sec:ReviewFan}}\n\nLet $\\mathbb{E}$ be the expectation operator. Before we start reviewing the\nstrategy in \\cite{Fan:2012} to show the SLLN for $\\tilde{R}\\left(\n\\cdot\\right) $, we quote as \\autoref{LemmaLyons} Theorem 6 of\n\\cite{Lyons:1988} as follows:\n\n\\begin{lemma}\n[\\cite{Lyons:1988}]\\label{LemmaLyons}Let $\\left\\{ X_{n}\\right\\}\n_{n=1}^{\\infty}$ be a sequence of real-valued random variables such that\n$\\mathbb{E}\\left[ \\left\\vert X_{n}\\right\\vert ^{2}\\right] \\leq1$. If\n\\ $\\left\\vert X_{n}\\right\\vert \\leq1$ a.s. and\n\\[\n\\sum\\nolimits_{N=1}^{\\infty}N^{-1}\\mathbb{E}\\left[ \\left\\vert N^{-1\n\\sum\\nolimits_{n=1}^{N}X_{n}\\right\\vert ^{2}\\right] <\\infty\\text{,\n\\]\nthen $\\lim_{m\\rightarrow\\infty}N^{-1}\\sum_{n=1}^{N}X_{n}=0$ a.s..\n\\end{lemma}\n\nBy (\\ref{eqb23}) and the Cauchy-Schwarz inequality\n\\begin{equation}\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert\n\\leq\\vartheta_{m}\\leq Cm^{-\\delta}\\text{,}\\label{eqb15\n\\end{equation}\nwhere $q_{ij}=cov\\left( v_{i},v_{j}\\right) $ and $\\rho_{ij}= q_{ij\n\\sigma_{i,m}^{-1\/2}\\sigma_{j,m}^{-1\/2}$ if $\\sigma_{i,m}\\sigma_{j,m}\\neq0$ and $\\rho\n_{ij}=0$ otherwise. Namely, $\\left\\{ v_{i}\\right\\} _{i=1}^{m}$ are\n\\textquotedblleft weakly dependent\\textquotedblright\\ as termed in\n\\cite{Fan:2012}. This implies that $\\mathbf{Z}$ conditional on\n$\\boldsymbol{\\eta}$ (or equivalently $\\mathbf{\\tilde{w}}_{k}$) has weakly\ndependent components. Hence it may be possible to apply \\autoref{LemmaLyons}\nto $\\left\\{ X_{i}\\right\\} _{i=1}^{m}$ with $X_{i}=1_{\\left\\{ p_{i}\\leq\nt|\\mathbf{\\tilde{w}}_{k}\\right\\} }-\\mathbb{E}\\left[ 1_{\\left\\{ p_{i}\\leq\nt|\\mathbf{\\tilde{w}}_{k}\\right\\} }\\right] $ as $m \\rightarrow \\infty$ to yield the SLLN for $\\tilde{R}\\left(\nt\\right) =m^{-1}\\sum_{i=1}^{m}X_{i}$. Sinc\n\\begin{align}\n\\mathbb{V}\\left[ m^{-1}\\sum\\nolimits_{i=1}^{m}X_{i}\\right] & =m^{-2\n\\sum\\nolimits_{i=1}^{m}\\mathbb{V}\\left( X_{i}\\right) +2m^{-2}\\sum_{1\\leq\ni0$, then\n\\begin{equation}\n\\sum_{m=1}^{\\infty}m^{-1}\\mathbb{V}\\left[ m^{-1}\\sum\\nolimits_{i=1}^{m\nX_{i}\\right] \\leq4\\sum_{m=1}^{\\infty}m^{-2}+2\\sum_{m=1}^{\\infty\nm^{-1-\\delta_{1}}<\\infty\\text{,}\\label{eqb4\n\\end{equation}\nwhere $M>0$ is a generic constant that can assume different (and appropriate)\nvalues at different occurrences. Once (\\ref{eqb4}) is established,\n\\autoref{LemmaLyons} immediately implie\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}\\left\\vert \\tilde{R}\\left( t\\right) -\\mathbb{E\n\\left[ \\tilde{R}\\left( t\\right) \\right] \\right\\vert =0\\text{\na.s..}\\label{eqb3\n\\end{equation}\n\nTo establish (\\ref{eqb2}), \\cite{Fan:2012} applied Taylor expansion with\nrespect to $q_{ij}^{1\/2}$ (when $q_{ij}>0$) to each ter\n\\[\n\\zeta_{ij}=\\mathbb{P}\\left( p_{i}\\leq t,p_{j}\\leq t|\\mathbf{\\tilde{w}\n_{k}\\right), i\\neq j\n\\]\nand this produces a total of $2^{-1}m\\left( m-1\\right) $ functional\nremainders whose various orders are mainly affected by the magnitudes of\n$\\left\\{ a_{i,m}\\right\\} _{i=1}^{m}$, the ratios between certain pairs among $\\left\\{\na_{i,m}\\right\\} _{i=1}^{m}$, and the corresponding functional mean values.\nHowever, it seems that \\cite{Fan:2012} failed to fully (and correctly)\nhandle all these remainders for which $\\left\\vert \\rho_{ij}\\right\\vert $ are\nclose $1$ and $a_{i,m}a_{j,m}^{-1}$ (or $a_{i,m}^{-1}a_{j,m}$) grows\nexponentially fast as $m$ increases.\n\n\\section{SLLN for normalized number of conditional rejections\\labe\n{SecLLNDisRejProc}}\n\nWe are ready to provide the extra arguments for the complete proof of the SLLN for $\\tilde{R}\\left(\n\\cdot\\right) $. We state the key result of \\cite{Fan:2012} in our notations as follows:\n\n\\begin{proposition}\n[\\cite{Fan:2012}]\\label{Prop2Fan2012}Suppose $\\mathbf{Z}_{m}\\sim\\mathsf{N\n_{m}\\left( \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right) $ with\n$\\boldsymbol{\\Sigma}_{m}\\geq0$. Choose an appropriate $k$ such that\n(\\ref{eqb23}) holds for some $\\delta>0$. The\n\\begin{equation}\n\\mathbb{V}\\left[ m^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{^{k}}\\right) \\right]\n=O_{\\mathbb{P}}\\left( m^{-\\delta}\\right) \\label{9\n\\end{equation}\nan\n\\begin{equation}\n\\mathbb{P}\\left\\{ \\lim_{m\\rightarrow\\infty}\\left\\vert m^{-1}R\\left(\nt|\\mathbf{\\tilde{w}}_{^{k}}\\right) -\\mathbb{E}\\left[ m^{-1}R\\left( t|\\mathbf{\\tilde\n{w}}_{^{k}}\\right) \\right] \\right\\vert =0\\right\\} =1\\text{.}\\label{eqb5\n\\end{equation}\n\n\\end{proposition}\n\nTo present our proof, we introduce some notations and sets. For any two positive sequences\n$\\left\\{ x_{m},y_{m}:m\\geq1\\right\\} $, we write $x_{m}\\asymp y_{m}$ if and only if\n$0<\\liminf_{m\\rightarrow\\infty}x_{m}y_{m}^{-1}\\leq\\limsup_{m\\rightarrow\\infty\n}x_{m}y_{m}^{-1}<\\infty$. Define three sets for the magnitudes of $\\left\\{\na_{i,m}\\right\\} _{i=1}^{m}$ a\n\\[\n\\left\\{\n\\begin{tabular}\n[c]{l\n$E_{1}=\\left\\{ i\\in\\mathbb{N}:a_{i,m}<\\infty\\text{ for any } m\\text{ and\n}\\lim_{m\\rightarrow\\infty}a_{i,m}=\\infty\\right\\} $,\\\\\n$E_{2}=\\left\\{ i\\in\\mathbb{N}:a_{i,m}=\\infty\\text{ for some } m\\right\\}\n$,\\\\\n$E_{3}=\\left\\{ i\\in\\mathbb{N}:\\limsup_{m\\rightarrow\\infty}a_{i,m\n< \\infty\\right\\} $,\n\\end{tabular}\n\\ \\ \\ \\right.\n\\]\nwhere it is understood that $a_{i,m}$ is undefined when $i>m$ for each finite\n$m$. Additionally, define the set counting how many pairs of $v_{i}$ and\n$v_{j}$ can be highly dependent but not necessarily linearly dependent a.s. a\n\\[\nS_{\\varepsilon,m}=\\left\\{ \\left( i,j\\right) :1\\leq i1-\\varepsilon\\right\\}\n\\]\nfor $\\varepsilon\\in\\left( 0,1\\right) $ since $m^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert\n=O\\left( m^{-\\delta}\\right) $ does not necessarily impl\n\\[\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert \\rho_{ij}\\right\\vert\n=O\\left( m^{-\\delta}\\right) \\text{.\n\\]\n\n\n\\autoref{Prop2Fan2012} is reformulated as follows:\n\n\\begin{theorem}\n\\label{MainTheoremNormalOneThresholdWeak}Let $\\mathbf{Z}_{m}\\sim\n\\mathsf{N}_{m}\\left( \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right) $\nwith $\\boldsymbol{\\Sigma}_{m}\\geq0$ and choose $k=k_{m}$ such that (\\ref{eqb23})\nholds for some $\\delta>0$. Suppose the following hold:\n\n\\begin{enumerate}\n\\item When $E_{1}\\neq\\varnothing$, there exists some $q\\in\\mathbb{N}$\nindependent of $m$ such tha\n\\begin{equation}\na_{\\left( m\\right) }\\asymp a_{\\left( 1\\right) }^{q}\\text{,} \\label{eqb22b\n\\end{equation}\nwhere $a_{\\left( 1\\right) }=\\min_{i\\in E_{1},1\\leq i\\leq m}a_{i,m}$ and\n$a_{\\left( m\\right) }=\\max_{i\\in E_{1},1\\leq i\\leq m}a_{i,m}$.\n\n\\item For some $\\tilde{\\varepsilon}\\in\\left( 0,1\\right) $,\n\\begin{equation}\n\\limsup_{m\\rightarrow\\infty}m^{-2+\\delta}\\left\\vert S_{\\tilde{\\varepsilon\n}\\right\\vert <\\infty\\text{.}\\label{eqb24b\n\\end{equation}\n\n\\end{enumerate}\n\nThen\n\\begin{equation}\n\\mathbb{V}\\left[ \\tilde{R}\\left( t\\right) \\right] =O\\left( m^{-\\delta\n}\\right) \\label{eqb20\n\\end{equation}\nholds except on the event\n\\begin{equation}\nG_{t,\\varepsilon}=\\cup_{m\\geq1}\\cup_{i\\in E_{1},1\\leq i\\leq m}\\left\\{\n\\omega\\in\\Omega:\\left\\vert \\pm z_{t\/2}-\\eta_{i}-\\mu_{i}\\right\\vert\n<\\varepsilon\\right\\} \\label{eqb19\n\\end{equation}\nfor any $\\varepsilon >0$, where $z_{t\/2}=\\Phi^{-1}\\left( t\/2\\right) $ and $\\Phi^{-1}$ is the inverse\nof $\\Phi$.\n\\end{theorem}\n\n\\begin{proof}\nLet $Y_{i}=1-X_{i}$. Then $cov\\left( Y_{i},Y_{j}\\right) =cov\\left(\nX_{i},X_{j}\\right) $ and (\\ref{eqb38}) is equivalent to\n\\begin{align*}\n\\mathbb{V}\\left[ m^{-1}\\sum\\nolimits_{i=1}^{m}X_{i}\\right] & \\leq\n4m^{-1}+2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in S_{\\tilde{\\varepsilon},m\n}cov\\left( Y_{i},Y_{j}\\right) \\\\\n& +2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in S_{\\tilde{\\varepsilon},m}^{C\n}cov\\left( Y_{i},Y_{j}\\right) \\\\\n& \\leq4m^{-1}+Mm^{-\\delta}+2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in\nS_{\\tilde{\\varepsilon},m}^{C}}cov\\left( Y_{i},Y_{j}\\right) \\text{,\n\\end{align*}\nsince (\\ref{eqb24b}) implies $m^{-2+\\delta}\\left\\vert S_{\\tilde{\\varepsilon\n,m}\\right\\vert 0$, Since $cov\\left( X_{i},X_{j}\\right) =0$ for all $\\,1\\leq j\\leq m$ when $i\\in\nE_{2}$, the inequality in (\\ref{eqb6}) is not affected by such summands\n(asymptotically). Therefore, it suffices to sho\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m\n}cov\\left( Y_{i},Y_{j}\\right) =O\\left( m^{-\\delta_{1}}\\right\n\\label{eqb7}\n\\end{equation}\nwhere\n\\[\nI_{\\tilde{\\varepsilon},m}=\\left\\{ \\left( i,j\\right): \\left( i,j\\right) \\in S_{\\tilde\n{\\varepsilon},m}^{C}\\cap\\left( \\left( E_{1}\\otimes E_{1}\\right) \\cup\\left(\nE_{3}\\otimes E_{3}\\right) \\right) \\right\\}\n\\]\nand $\\otimes$ denotes the Cartesian product. We can break $I_{\\tilde\n{\\varepsilon},m}=I_{\\tilde{\\varepsilon},m}^{+}\\cup$ $I_{\\tilde{\\varepsilon\n,m}^{-}$ and sho\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+\n}cov\\left( Y_{i},Y_{j}\\right) =O\\left( m^{-\\delta_{1}}\\right) \\label{eqb8\n\\end{equation}\nan\n\\begin{equation}\n2m^{-2}\\sum\\nolimits_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{-\n}cov\\left( Y_{i},Y_{j}\\right) =O\\left( m^{-\\delta_{1}}\\right) \\text{,\n\\label{eqb9\n\\end{equation}\nwhere $I_{\\tilde{\\varepsilon},m}^{+}=\\left\\{ \\left( i,j\\right) \\in\nI_{\\tilde{\\varepsilon},m}:\\rho_{ij}\\geq0\\right\\} $ and $I_{\\tilde\n{\\varepsilon},m}^{-}=\\left\\{ \\left( i,j\\right) \\in I_{\\tilde{\\varepsilon\n},m}:\\rho_{ij}<0\\right\\} $. Since the techniques of proving (\\ref{eqb8}) and\n(\\ref{eqb9}) are the same, in the sequel we will just show (\\ref{eqb8}).\n\nLet $h_{1}\\left( x\\right) =\\left( b^{2}-x^{2}\\right) ^{-1\/2}$\nfor $\\vert b \\vert \\neq \\vert x \\vert$. Then\n\\begin{equation}\nh_{1}\\left( x\\right) =h\\left( 0\\right) +h_{1}^{\\prime}\\left(\n\\theta\\right) \\left( x-0\\right) =b^{-1}+\\dfrac{\\theta}{\\left( b^{2\n-\\theta^{2}\\right) ^{3\/2}}x\\label{eqb75\n\\end{equation}\nfor some $\\theta$ strictly between $0$ and $x$, where $h_{1}^{\\prime}\\left(\n\\theta\\right) =\\theta\\left( b^{2}-\\theta^{2}\\right) ^{-3\/2}$. Set\n$o\\left( x\\right) =h_{1}^{\\prime}\\left( \\theta\\right) x$, which does not\nnecessarily satisfy $\\lim_{x\\rightarrow0}\\left\\vert x^{-1}o\\left( x\\right)\n\\right\\vert =0$ if $\\lim_{x\\rightarrow0}b=0$. By Taylor\nexpansion, we have, for $\\left\\vert b\\right\\vert \\neq\\left\\vert x\\right\\vert\n$\n\\begin{align*}\n& \\Phi\\left( \\dfrac{xz+bc}{\\left( b^{2}-x^{2}\\right) ^{1\/2}}\\right) \\\\\n& =\\Phi\\left( c+r_{1}\\left( c,x\\right) \\right) \\\\\n& =\\Phi\\left( c\\right) +\\phi\\left( c\\right) xzb^{-1}+\\phi\\left(\nc\\right) \\left( xz+bc\\right) o\\left( x\\right) +\\dfrac{1}{2}\\left[\n-\\theta_{\\ast}\\phi\\left( \\theta_{\\ast}\\right) \\right] r_{1}^{2}\\left(\nc,x\\right) \\\\\n& =\\Phi\\left( c\\right) +\\phi\\left( c\\right) xzb^{-1}+\\tilde{o}\\left(\nc,x\\right) \\text{,\n\\end{align*}\nfor some $\\theta_{\\ast}$ strictly between $c$ and $c+r_{1}\\left( c,x\\right)\n$, where $\\phi\\left( t\\right) =\\dfrac{d}{dt}\\Phi\\left( t\\right) $,\n$\\tilde{o}\\left( c,x\\right) =r_{2}\\left( c,x\\right) +r_{3}\\left(\nc,x\\right) $ and\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l\nr_{1}\\left( c,x\\right) =xzb^{-1}+\\left( xz+bc\\right) o\\left( x\\right)\n\\text{,}\\\\\nr_{2}\\left( c,x\\right) =-\\dfrac{1}{2 \\sqrt{2 \\pi}}\\theta_{\\ast}e^{-\\theta_{\\ast}^{2\n\/2}r_{1}^{2}\\left( c,x\\right) \\text{,}\\\\\nr_{3}\\left( c,x\\right) =\\phi\\left( c\\right) \\left( xz+bc\\right) o\\left(\nx\\right) \\text{.\n\\end{array}\n\\right. \\label{eqb11\n\\end{equation}\n\n\nWe will now omit the subscript $m$ in $a_{i,m}$ and write $a_{i,m}$ as $a_{i\n$. Set $b_{ij}=a_{i}^{-1}a_{j}^{-1}$, $r_{1,i}=-z_{t\/2}-\\eta_{i}-\\mu_{i}$,\n$r_{2,i}=z_{t\/2}-\\eta_{i}-\\mu_{i}$ and $c_{l,i}=a_{i}r_{l,i}$ with $l=1,2$.\nSince $z_{t\/2}\\leq0$ for $t\\in\\left[ 0,1\\right] $, then $p_{i}\\geq t$ if and\nonly if $c_{2,i}\/a_{i}\\leq v_{i}\\leq c_{1,i}\/a_{i}$. Using\nthe formula from \\cite{Plackett:1954} or page 192 of \\cite{Tong:1990} for\n$0\\leq\\rho_{ij}<1$, we hav\n\\begin{align}\n\\mathbb{E}\\left[ Y_{i}Y_{j}\\right] & =\\mathbb{P}\\left( \\left\\{\np_{i}>t,p_{j}>t|\\mathbf{\\tilde{w}}_{k}\\right\\} \\right) \\label{eqb27}\\\\\n& =\\mathbb{P}\\left( \\left\\{ c_{2,i}\/a_{i}0\\text{.}\\label{eqb81\n\\end{equation}\nIn what follows, we will just work on the event $\\tilde{D}$,\nwhere (\\ref{eqb81}) will help us obtain uniform boundedness of the\nfunctional coefficients in (\\ref{eqb82}).\n\nFrom (\\ref{eqb81}), we se\n\\[\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\n\\sup_{\\left( r_{1,i},r_{2,i},a_{i}\\right)\n}\\left\\{ \\left\\vert \\phi\\left( c_{1,i}\\right) -\\phi\\left( c_{2,i}\\right)\n\\right\\vert a_{i}\\right\\} \\leq M .\n\\]\nThu\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left\\vert \\Delta\\phi\\left( c_{i}\\right)\n\\Delta\\phi\\left( c_{j}\\right) \\right\\vert b^{-2}x^{2}\\leq Mm^{-2\\delta\n}\\text{.}\\label{eqb56\n\\end{equation}\nClearly\n\\begin{align*}\n\\left\\vert g_{ij}\\left( \\cdot\\right) \\right\\vert & =\\left\\vert \\Delta\n\\Phi\\left( c_{i}\\right) \\Delta\\tilde{o}_{j}\\left( x\\right) +\\Delta\n\\tilde{o}_{i}\\left( x\\right) \\Delta\\Phi\\left( c_{j}\\right) +\\Delta\n\\tilde{o}_{i}\\left( x\\right) \\Delta\\tilde{o}_{j}\\left( x\\right)\n\\right\\vert \\\\\n& \\leq2\\left\\vert \\Delta\\tilde{o}_{j}\\left( x\\right) \\right\\vert\n+2\\left\\vert \\Delta\\tilde{o}_{i}\\left( x\\right) \\right\\vert +\\left\\vert\n\\Delta\\tilde{o}_{i}\\left( x\\right) \\Delta\\tilde{o}_{j}\\left( x\\right)\n\\right\\vert \\text{.\n\\end{align*}\nIf $\\Delta\\tilde{o}_{j}\\left( x\\right) $ for $j=1,...,m$ are dominated by a polynomial\n$g^{\\ast}\\left( \\left\\vert x\\right\\vert ,\\left\\vert z\\right\\vert \\right) $\nwith uniformly bounded functional coefficients without the constant term, the\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right) \\right\\vert \\phi\\left( z\\right) dz\\leq m^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}\n}\\tilde{g}\\left( \\left\\vert x\\right\\vert \\right) \\leq Mm^{-\\beta_{0}\\delta\n}\\text{,}\\label{eqb49\n\\end{equation}\nfor $0<\\beta_{0}\\leq2$, where $\\tilde{g}\\left( \\cdot\\right) $ is a\npolynomial in $\\left\\vert x\\right\\vert $ and $\\beta_{0}>0$ is the smallest\ndegree of $\\left\\vert x\\right\\vert $ in $g^{\\ast}\\left( \\left\\vert\nx\\right\\vert ,\\left\\vert z\\right\\vert \\right) $ and we have used\n(\\ref{eqb15}) and\n\\begin{equation}\nm^{-2}\\sum\\nolimits_{1\\leq i\\leq j\\leq m}\\left\\vert q_{ij}\\right\\vert ^{\\beta\n}=O\\left( m^{-\\delta\\min\\left\\{ \\beta,2\\right\\} }\\right) \\label{eqb17\n\\end{equation}\nfor all $\\beta\\in(0,2]$ derived from (\\ref{eqb23}) by H\\\"{o}lder's inequality.\nObviously, (\\ref{eqb49}) implie\n\\begin{align}\n& \\text{ \\ }m^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert \\Delta\\tilde{o\n_{i}\\left( x\\right) \\right\\vert \\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right) \\right\\vert \\phi\\left( z\\right) dz\\label{eqb46}\\\\\n& \\leq m^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left( \\int\\left\\vert \\Delta\\tilde{o\n_{i}\\left( x\\right) \\right\\vert ^{2}\\phi\\left( z\\right) dz\\right)\n^{1\/2}\\nonumber\\\\\n& \\text{ \\ \\ \\ }\\times\\left( \\int\\left\\vert \\Delta\\tilde{o}_{j}\\left(\nx\\right) \\right\\vert ^{2}\\phi\\left( z\\right) dz\\right) ^{1\/2}\\nonumber\\\\\n& \\leq Mm^{-2\\beta_{0}\\delta}\\text{,}\\nonumber\n\\end{align}\nfrom which (\\ref{eqb8}) holds with $\\delta_{1}=2\\beta_{0}\\delta$.\nTherefore, it suffices to justify (\\ref{eqb49}).\n\nBy their definitions\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\n\\tilde{o}\\left( c_{l,i},x\\right) =\\phi\\left( c_{l,i}\\right) x^{2\nzh_{1}^{\\prime}\\left( \\theta_{i}\\right) +\\phi\\left( c_{l,i}\\right)\nbc_{l,i}h_{1}^{\\prime}\\left( \\theta_{i}\\right) x-\\dfrac{1}{2}\\theta\n_{l,i} \\phi \\left(\\theta_{l,i}\\right) r_{1}^{2}\\left( c_{l,i},x\\right) \\text{,}\\\\\n\\Delta\\tilde{o}_{i}\\left( x\\right) =r_{3,1}\\left( c_{i},x\\right)\nx^{2}z+r_{3,2}\\left( c_{i},x\\right) x+r_{4}\\left( c_{i},x\\right) \\text{,\n\\end{array}\n\\right.\n\\]\nwher\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\nr_{3,1}\\left( c_{i},x\\right) =\\phi\\left( c_{1,i}\\right) h_{1}^{\\prime\n}\\left( \\theta_{i}\\right) -\\phi\\left( c_{2,i}\\right) h_{1}^{\\prime}\\left(\n\\theta_{i}\\right) \\text{,}\\\\\nr_{3,2}\\left( c_{i},x\\right) =\\left[ c_{1,i}\\phi\\left( c_{1,i}\\right)\nh_{1}^{\\prime}\\left( \\theta_{i}\\right) -c_{2,i}\\phi\\left( c_{2,i}\\right)\nh_{1}^{\\prime}\\left( \\theta_{i}\\right) \\right] b\\text{,}\\\\\nr_{4}\\left( c_{i},x\\right) =-\\dfrac{1}{2}\\theta_{1,i}\\phi\\left(\n\\theta_{1,i}\\right) r_{1}^{2}\\left( c_{1,i},x\\right) +\\dfrac{1}{2\n\\theta_{2,i}\\phi\\left( \\theta_{2,i}\\right) r_{1}^{2}\\left( c_{2,i\n,x\\right).\n\\end{array}\n\\right.\n\\]\nFrom\n\\[\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in S_{\\tilde{\\varepsilon},m}^{C\n}\\left\\vert \\rho_{ij}\\right\\vert \\leq1-\\tilde{\\varepsilon}\\text{,\n\\]\nit follows that\n\\begin{equation}\nb^{-2}\\tilde{\\rho}\\geq b^{-2}\\left\\vert \\theta\/b\\right\\vert \\left[ 1-\\left(\nx\/b\\right) ^{2}\\right] ^{-3\/2}\\geq\\left\\vert h_{1}^{\\prime}\\left(\n\\theta\\right) \\right\\vert \\geq\\dfrac{\\left\\vert \\theta\\right\\vert }{b^{3\n}\\text{,}\\label{eqb37\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{\\rho}=\\left( 1-\\tilde{\\varepsilon}\\right) \\left[ 1-\\left(\n1-\\tilde{\\varepsilon}\\right) ^{2}\\right] ^{-3\/2}\\text{.}\\label{eqb57\n\\end{equation}\nUsing (\\ref{eqb81}), (\\ref{eqb22b}) and the property of $\\phi\\left(\n\\cdot\\right) $, we hav\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left( r_{l,i},a_{i}\\right)\n}\\left\\vert \\phi\\left( c_{l,i}\\right) h_{1}^{\\prime}\\left( \\theta\n_{i}\\right) \\right\\vert \\leq M\\label{eqb51\n\\end{equation}\nan\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left( r_{l,i},a_{i},a_{j}\\right)\n}\\left\\vert c_{l,i}\\phi\\left( c_{l,i}\\right) h_{1}^{\\prime}\\left(\n\\theta_{i}\\right) a_{i}^{-1\/2}a_{j}^{-1\/2}\\right\\vert \\leq M\\text{.\n\\label{eqb52\n\\end{equation}\nSo\n\\[\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{c_{1,i},c_{2,i}}\\left\\vert\n\\phi\\left( c_{1,i}\\right) h_{1}^{\\prime}\\left( \\theta_{i}\\right)\n-\\phi\\left( c_{2,i}\\right) h_{1}^{\\prime}\\left( \\theta_{i}\\right)\n\\right\\vert \\leq M\n\\]\nan\n\\begin{equation}\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\sup_{\\left( r_{1,i},r_{2,i},a_{i\n,a_{j}\\right) }\\left\\vert \\tau_{ij}\\left( c_{i},\\theta_{i}\\right)\n\\right\\vert \\leq M\\text{.}\\label{eqb18\n\\end{equation}\nwhere\n\\[\n\\tau_{ij}\\left( c_{i},\\theta_{i}\\right) =\\left[ c_{1,i}\\phi\\left(\nc_{1,i}\\right) h_{1}^{\\prime}\\left( \\theta_{i}\\right) -c_{2,i}\\phi\\left(\nc_{2,i}\\right) h_{1}^{\\prime}\\left( \\theta_{i}\\right) \\right] a_{i\n^{-1\/2}a_{j}^{-1\/2}\\text{.\n\\]\nTherefore\n\\begin{align}\n& m^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert r_{3,1}\\left( c_{i},x\\right)\n\\right\\vert x^{2}\\left\\vert z\\right\\vert \\phi\\left( z\\right) dz\\label{eqb53\n\\\\\n& \\leq Mm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}x^{2}=Mm^{-2\\delta}\\nonumber\n\\end{align}\nan\n\\begin{align}\n& m^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\int\\left\\vert r_{3,2}\\left( c_{i},x\\right)\n\\right\\vert \\left\\vert x\\right\\vert \\phi\\left( z\\right) dz\\label{eqb54}\\\\\n& \\leq Mm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\tilde{\\varepsilon},m}^{+}}\\left\\vert x\\right\\vert \\leq Mm^{-\\delta\n}\\text{,}\\nonumber\n\\end{align}\nwhere we have used the inequality (\\ref{eqb15}) and (\\ref{eqb17}).\n\nThe last term $\\varkappa_{m}=m^{-2}\\sum_{1\\leq i\\max\\left\\{\n\\left\\vert r_{1}\\left( c,x\\right) \\right\\vert ,\\left\\vert c\\right\\vert\n\\right\\} \\text{.\n\\]\nNoticing furthe\n\\begin{align*}\n\\limsup_{m\\rightarrow\\infty}\\dfrac{r_{1}\\left( c,x\\right) }{c} &\n\\leq\\limsup_{m\\rightarrow\\infty}\\left( 1+\\left\\vert \\dfrac{z}{c}\\right\\vert\n\\right) \\left\\vert b^{2}h_{1}^{\\prime}\\left( \\theta\\right) \\right\\vert \\\\\n& \\leq\\limsup_{m\\rightarrow\\infty}\\left( 1+\\left\\vert \\dfrac{z\n{c}\\right\\vert \\right) \\left\\vert \\left[ 1-\\left( 1-\\tilde{\\varepsilon\n\\right)^{2} \\right] ^{-3\/2}\\right\\vert \\\\\n& \\leq\\left( 1-\\tilde{\\varepsilon}\\right) ^{-1}\\tilde{\\rho}<\\infty\\text{,\n\\end{align*}\nwe se\n\\[\n\\left\\{\n\\begin{array}\n[c]{c\n\\left\\vert r_{1}\\left( c,x\\right) \\right\\vert \\leq M\\left( 1-\\tilde{\\varepsilon}\n\\right) ^{-1}\\tilde{\\rho}\\left\\vert c\\right\\vert \\text{,}\\\\\nM\\left( 1-\\tilde{\\varepsilon}\\right) ^{-1}\\tilde{\\rho}\\left\\vert c\\right\\vert\n\\geq\\left\\vert \\theta_{\\ast}\\right\\vert >\\left\\vert c\\right\\vert \\text{,\n\\end{array}\n\\right.\n\\]\nan\n\\[\n\\left\\vert r_{2}\\left( c,x\\right) \\right\\vert \\leq\\left\\vert \\theta_{\\ast\n}e^{-\\theta_{\\ast}^{2}\/2}r_{1}^{2}\\left( c,x\\right) \\right\\vert \\leq\nM\\left\\vert c\\right\\vert ^{3}\\exp\\left( -\\left\\vert c\\right\\vert\n^{2}\/2\\right) \\text{,\n\\]\nwhere we used the fact that $t\\phi\\left( t\\right) $ is decreasing in $t>0$.\nFro\n\\begin{align*}\n& r_{1}^{2}\\left( c,x\\right) \\\\\n& =x^{2}b^{-2}z^{2}+2x^{3}b^{-1}h_{1}^{\\prime}\\left( \\theta\\right)\nz^{2}+2x^{2}ch_{1}^{\\prime}\\left( \\theta\\right) z+x^{4}h_{1}^{\\prime\n2}\\left( \\theta\\right) z^{2}\\\\\n& +2x^{3}bch_{1}^{\\prime2}\\left( \\theta\\right) z+b^{2}c^{2}h_{1}^{\\prime\n2}\\left( \\theta\\right) x^{2}\\\\\n& =r_{1,1}\\left( c,z\\right) x^{2}+2r_{1,2}\\left( c,z\\right) x^{3\n+h_{1}^{\\prime2}\\left( \\theta\\right) x^{4}z^{2}\\\\\n& \\leq r_{1,1}^{\\ast}\\left( c,h_{1}^{\\prime}\\left( \\theta\\right) \\right)\nx^{2}+r_{1,2}^{\\ast}\\left( c,h_{1}^{\\prime}\\left( \\theta\\right) \\right)\n\\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left( \\theta\\right) \\left\\vert\nc\\right\\vert x^{4}\\text{,\n\\end{align*}\nwe see\n\\[\n\\int_{B\\left( \\sqrt{c}\\right) }\\left\\vert r_{2}\\left( c,x\\right)\n\\right\\vert \\phi\\left( z\\right) dz\\leq M\\left\\vert c\\right\\vert \\exp\\left(\n-\\left\\vert c\\right\\vert ^{2}\/2\\right) g^{\\ast\\ast}\\left( \\left\\vert\nx\\right\\vert \\right) \\text{,\n\\]\nwher\n\\[\ng^{\\ast\\ast}\\left( \\left\\vert x\\right\\vert \\right) =r_{1,1}^{\\ast}\\left(\nc,h_{1}^{\\prime}\\left( \\theta\\right) \\right) x^{2}+r_{1,2}^{\\ast}\\left(\nc,h_{1}^{\\prime}\\left( \\theta\\right) \\right) \\left\\vert x\\right\\vert\n^{3}+h_{1}^{\\prime2}\\left( \\theta\\right) \\left\\vert c\\right\\vert x^{4\n\\]\nan\n\\[\n\\left\\{\n\\begin{array}\n[c]{l\nr_{1,1}\\left( c,z\\right) =b^{-2}z^{2}+2ch_{1}^{\\prime}\\left( \\theta\\right)\nz+b^{2}c^{2}h_{1}^{\\prime2}\\left( \\theta\\right) \\text{,}\\\\\nr_{1,2}\\left( c,z\\right) =b^{-1}h_{1}^{\\prime}\\left( \\theta\\right)\nz^{2}+bch_{1}^{\\prime2}\\left( \\theta\\right) z\\text{,}\\\\\nr_{1,1}^{\\ast}\\left( c,h_{1}^{\\prime}\\left( \\theta\\right) \\right)\n=b^{-2}\\left\\vert c\\right\\vert +2\\left\\vert c\\right\\vert ^{3\/2}\\left\\vert\nh_{1}^{\\prime}\\left( \\theta\\right) \\right\\vert +b^{2}c^{2}h_{1}^{\\prime\n2}\\left( \\theta\\right) \\text{,}\\\\\nr_{1,2}^{\\ast}\\left( c,h_{1}^{\\prime}\\left( \\theta\\right) \\right)\n=b^{-1}h_{1}^{\\prime}\\left( \\theta\\right) \\left\\vert c\\right\\vert\n+\\left\\vert bh_{1}^{\\prime2}\\left( \\theta\\right) \\right\\vert \\left\\vert\nc\\right\\vert ^{3\/2}\\text{.\n\\end{array}\n\\right.\n\\]\nSince $\\varepsilon a_{i}\\leq\\left\\vert c_{l,i}\\right\\vert $ for $l=1,2$ and $1\\leq\ni\\leq m$ by (\\ref{eqb81}) and $\\lim_{m\\rightarrow\\infty}a_{i}=\\infty$, we ge\n\\[\n\\sup_{m\\geq1}\\max_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\sup_{c,b}\\left\\{ \\left\\vert c\\right\\vert\n^{3}\\exp\\left( -\\left\\vert c\\right\\vert ^{2}\/2\\right) b^{-2}h_{1}^{\\prime\n2}\\left( \\theta\\right) \\right\\} \\leq M\n\\]\nand $M\\left\\vert c\\right\\vert \\exp\\left( -\\left\\vert c\\right\\vert\n^{2}\/2\\right) g^{\\ast\\ast}\\left( \\left\\vert x\\right\\vert \\right) $ is thus\ndominated by a polynomial with uniformly bounded coefficients whose lowest\ndegree in $\\left\\vert x\\right\\vert $ is $2$. Hence\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\int_{B\\left( \\sqrt{c}\\right) }\\left\\vert\nr_{2}\\left( c,x\\right) \\right\\vert \\phi\\left( z\\right) dz\\leq\nMm^{-2\\delta}\\text{.}\\label{eqb55\n\\end{equation}\n\n\nOn the other hand, on $\\mathbb{R}\\backslash B\\left( \\sqrt{c}\\right) $ we\nhav\n\\begin{align*}\n& \\left\\vert r_{1}^{2}\\left( c,x\\right) \\right\\vert \\\\\n& \\leq\\left\\vert r_{1,1}\\left( c,z\\right) \\right\\vert x^{2}+2\\left\\vert\nr_{1,2}\\left( c,z\\right) \\right\\vert x^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right) x^{4}z^{2}\\\\\n& \\leq\\left[ b^{-2}z^{2}+2\\left\\vert ch_{1}^{\\prime}\\left( \\theta\\right)\nz\\right\\vert +b^{2}c^{2}h_{1}^{\\prime2}\\left( \\theta\\right) \\right] x^{2}\\\\\n& +\\left[ \\left\\vert 2b^{-1}h_{1}^{\\prime}\\left( \\theta\\right)\nz^{2}+2bch_{1}^{\\prime2}\\left( \\theta\\right) z\\right\\vert \\right]\n\\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left( \\theta\\right) x^{4}\\\\\n& \\leq\\left[ b^{-2}\\left\\vert c\\right\\vert +2\\left\\vert ch_{1}^{\\prime\n}\\left( \\theta\\right) \\right\\vert \\left\\vert c\\right\\vert ^{1\/2}+b^{2\nc^{2}h_{1}^{\\prime2}\\left( \\theta\\right) \\right] x^{2}\\\\\n& +2\\left( b^{-1}h_{1}^{\\prime}\\left( \\theta\\right) \\left\\vert\nc\\right\\vert +2bch_{1}^{\\prime2}\\left( \\theta\\right) \\left\\vert c\\right\\vert\n^{1\/2}\\right) \\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right) x^{4}\\\\\n& \\leq M\\left( \\left\\vert c\\right\\vert ^{3}+\\left\\vert h_{1}^{\\prime}\\left(\n\\theta\\right) \\right\\vert \\left\\vert c\\right\\vert ^{3\/2}+c^{2}h_{1}^{\\prime\n2}\\left( \\theta\\right) \\right) x^{2}\\\\\n& +M\\left( h_{1}^{\\prime}\\left( \\theta\\right) \\left\\vert c\\right\\vert\n^{2}+h_{1}^{\\prime2}\\left( \\theta\\right) \\left\\vert c\\right\\vert\n^{3\/2}\\right) \\left\\vert x\\right\\vert ^{3}+h_{1}^{\\prime2}\\left(\n\\theta\\right) x^{4}\\\\\n& \\leq M\\left\\vert c\\right\\vert ^{3}\\left( x^{2}+\\left\\vert x\\right\\vert\n^{3}+x^{4}\\right) \\leq M\\left\\vert c\\right\\vert ^{3}x^{2\n\\end{align*}\nan\n\\begin{align*}\n\\left( \\int_{\\mathbb{R}\\backslash B\\left( \\sqrt{c}\\right) }\\left\\vert\nr_{1}\\left( c,x\\right) \\right\\vert \\phi\\left( z\\right) dz\\right) ^{2} &\n\\leq\\int_{\\mathbb{R}\\backslash B\\left( \\sqrt{c}\\right) }\\left\\vert r_{1\n^{2}\\left( c,x\\right) \\right\\vert \\phi\\left( z\\right) dz\\\\\n& \\leq x^{2}\\int_{\\mathbb{R}\\backslash B\\left( \\sqrt{c}\\right) }M\\left\\vert\nc\\right\\vert ^{3}\\phi\\left( z\\right) dz\\leq Mx^{2\n\\end{align*}\nsince $\\lim_{\\left\\vert c\\right\\vert \\rightarrow\\infty}\\left\\vert c\\right\\vert\n^{s^{\\prime}}\\phi\\left( \\left\\vert c\\right\\vert \\right) =0$ for any\n$s^{\\prime}>0$. This mean\n\\[\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\int_{\\mathbb{R}\\backslash B\\left( \\sqrt\n{c}\\right) }\\left\\vert r_{2}\\left( c,x\\right) \\right\\vert \\phi\\left(\nz\\right) dz\\leq Mm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\left\\vert x\\right\\vert\n=Mm^{-\\delta}\\text{,\n\\]\nwhich, with (\\ref{eqb55}), implie\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\int\\left\\vert r_{2}\\left( c,x\\right)\n\\right\\vert \\phi\\left( z\\right) dz\\leq Mm^{-\\delta}\\text{.}\\label{eqb66\n\\end{equation}\nTherefor\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{\\ast}}\\int\\left\\vert r_{4}\\left( c_{i},x\\right)\n\\right\\vert \\phi\\left( z\\right) dz\\leq Mm^{-\\delta}\\text{.}\\label{eqb67\n\\end{equation}\nCombining (\\ref{eqb53}), (\\ref{eqb54}) and (\\ref{eqb67}) give\n\\begin{equation}\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{+}}\\int\\left\\vert \\Delta\\tilde{o}_{i}\\left(\nx\\right) \\right\\vert \\phi\\left( z\\right) dz\\leq Mm^{-\\delta\n\\text{.}\\label{eqb76\n\\end{equation}\nThus (\\ref{eqb49}) and (\\ref{eqb46}) hold with $\\delta_{1}=2\\delta$. Consequently\n\\[\nm^{-2}\\sum_{\\left( i,j\\right) \\in I_{\\varepsilon,m}^{+}}\\int\\left\\vert\ng_{ij}\\left( \\cdot\\right) \\right\\vert \\phi\\left( z\\right) dz=\n\\left( m^{-2\\delta}\\right) \\text{,\n\\]\nwhich, with (\\ref{eqb56}), implies $\\mathbb{V}\\left[ m^{-1}\\sum\n\\nolimits_{i=1}^{m}X_{i}\\right] =O\\left( m^{-2\\delta}\\right) $\nexcept on the event $G_{t,\\varepsilon}$. This completes the proof.\n\\end{proof}\n\nUsing \\autoref{LemmaLyons}, we have the following corollary from\n\\autoref{MainTheoremNormalOneThresholdWeak}.\n\n\\begin{corollary}\n\\label{ASconvNormal}The same conditions of\n\\autoref{MainTheoremNormalOneThresholdWeak} impl\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}\\left\\vert m^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{^{k\n}\\right) -\\mathbb{E}\\left[ m^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{^{k\n}\\right) \\right] \\right\\vert =0\\label{10\n\\end{equation}\noutside the event $G_{t,\\varepsilon}$.\n\\end{corollary}\n\nFrom the proof of \\autoref{MainTheoremNormalOneThresholdWeak}, we see that\namong the functional remainders there are mainly two types that can diverge\nto infinity as $m\\rightarrow\\infty$:\n\n\\begin{enumerate}\n\\item $\\phi\\left( a_{i,m}r_{i}\\right) h^{\\prime}\\left( \\theta_{i}\\right)\na_{i,m}$ for which $\\left\\vert a_{i,m}r_{i}\\right\\vert =O\\left( 1\\right) $,\n$a_{i,m}\\rightarrow\\infty$ and $\\left\\vert h^{\\prime}\\left( \\theta\n_{i}\\right) \\right\\vert \\rightarrow\\infty$, where $r_{i}=\\pm z_{t\/2}-\\eta\n_{i}-u_{i}$.\n\n\\item $\\phi\\left( a_{i,m}r_{i}\\right) h^{\\prime}\\left( \\theta_{i}\\right)\na_{i,m}^{-1}a_{j,m}$ for which $\\left\\vert a_{i,m}r_{i}\\right\\vert\n\\rightarrow\\infty$, $r_{i}=O\\left( 1\\right) $, $a_{i,m},a_{j,m\n\\rightarrow\\infty$, $h^{\\prime}\\left( \\theta_{i}\\right) \\rightarrow\\infty$\nbut $a_{j,m}\\geq e^{Ma_{i,m}^{2}}$ for some $M>0$ large when $m$ is large enough.\n\\end{enumerate}\nSuch terms will very likely inflate $\\mathbb{V}\\left[ \\tilde{R}\\left( t\\right) \\right]\n$ out of the desired order $m^{-\\delta}$ for the SLLN for $\\tilde{R}\\left(\nt\\right) $ to hold. In fact, the event $G_{t,\\varepsilon}$ contains all\n$\\omega\\in\\Omega$ for which the above two cases can happen but it does not\nnecessarily have diminishing probability.\n\nWe briefly describe the roles the two additional conditions (\\ref{eqb22b}) and\n(\\ref{eqb24b}). When $a_{i,m}\\rightarrow\\infty$ but $a_{i.m}<\\infty$ for\nfinite $m$, $a_{i,m}$ represents the rate\nat which $\\tilde{\\eta}_{i}=\\eta_{i}+\\mu_{i}$ stochastically approximates\n$Z_{i}$, and condition (\\ref{eqb22b}) requires that the relative rate at which\n$a_{i,m}$ $\\rightarrow\\infty$ can not be exponential. Condition (\\ref{eqb24b})\ncontrols how many pairs of $\\left( v_{i},v_{j}\\right) $, $i\\neq j$ can be\nhighly correlated and restricts the contribution of such pairs in the variance\nof $m^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{k}\\right) $. It also controls the\n\\textquotedblleft speed\\textquotedblright\\ at which $\\mathbf{\\Sigma}_{m}>0$\ncan approach to singularity. In addition, it ensures the validity and accuracy of the expansion\n(\\ref{eqb27}) (since (\\ref{eqb27}) is undefined when $\\left\\vert \\rho\n_{ij}\\right\\vert =1$), and prevents $h_{1}^{\\prime}\\left( \\theta\\right) $ in\n(\\ref{eqb37}) from diverging to infinity (unexpectedly fast). Further, it induces (\\ref{eqb57}),\nand together with condition (\\ref{eqb22b}) it validates (\\ref{eqb53}),\n(\\ref{eqb54}) and (\\ref{eqb67}).\nThese extra sufficient conditions ensure the uniform boundedness of the\ninvolved functional remainders in such Taylor expansions and induce (\\ref{eqb20})\noutside the event $G_{t,\\varepsilon}$.\n\n\\section{Discussion\\label{SecDiscussion}}\n\nUnder two more conditions rather than none in \\cite{Fan:2012}, we have\nreformulated the SLLN for the normalized number of conditional rejections when\ntesting which components of a Normal rv $\\mathbf{Z}_{m}\\sim\\mathsf{N\n_{m}\\left( \\boldsymbol{\\mu}_{m},\\boldsymbol{\\Sigma}_{m}\\right) $ have zero means when its\ncorrelation matrix $\\mathbf{\\Sigma}_{m}$ is known. Our proof shows that the speed of the\nPFA to the original Normal rv, the degree of dependency between components of\nthe Normal rv, and the magnitudes of the conditional component means should be compatible\nwith each other in order to yield such a law. Relaxation of (\\ref{eqb22b}) and\n(\\ref{eqb24b}) is possible. But such a\nlaw may not hold for arbitrary $\\boldsymbol{\\mu}_{m}$ and arbitrary\n$\\mathbf{\\Sigma}_{m}\\geq0$, since, on the even\n\\[\nH_{t,\\varepsilon,\\tilde{\\varepsilon}}=\\left\\{ \\omega\\in\\Omega:1-\\tilde\n{\\varepsilon}\\leq\\rho_{ij}<1,a_{i,m}^{-2}\\log a_{j,m}\\geq1,i,j\\in E_{1},i\\neq\nj\\right\\} \\cap G_{t,\\varepsilon\n\\]\nwhen $0<\\max\\left\\{ \\varepsilon,\\tilde{\\varepsilon}\\right\\} \\ll1$\n\\[\n\\mathbb{P}\\left\\{ \\lim_{m\\rightarrow\\infty}m^{-2+\\delta}\\sum\\nolimits_{1\\leq\ni0\n\\]\n(see (\\ref{eqb121}) for the definition of $\\Delta\\tilde{o}_{j}\\left(\nx\\right) $) and\n\\begin{equation}\n\\mathbb{P}\\left\\{ \\limsup_{m\\rightarrow\\infty}m^{\\delta}\\mathbb{V}\\left[\nm^{-1}R\\left( t|\\mathbf{\\tilde{w}}_{k}\\right) \\right] =\\infty\\right\\}\n>0\\label{eqb104\n\\end{equation}\ncan happen for any $\\delta >0$, which then invalidates (\\ref{9}) and (\\ref{eqb5}).\nSince estimating the FDP and control of the FDR under other strong but\nunknown types of dependency among the test statistics still remains a highly\nchallenging problem, such an area of research requires\ndevelopment of new methods and clarification of the boundaries beyond which\ncertain methods become fragile or invalid.\n\n\\bibliographystyle{imsart-nameyear}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOne important parameter for the determination of the unitarity triangle\nof the CKM matrix is $V_{ub}$. While its phase can only be determined \nfrom non--leptonic decays, its magnitude will be determined from \nsemileptonic processes. Various methods have been proposed to extract \n$V_{ub}$ from both inclusive as well as exclusive semileptonic decays, \nand the final numbers after the era of the $B$ factories will probably \nbe obtained from a mixture of both exclusive and inclusive methods. \n\nThe disadvantage of exclusive decays is their dependence on form factors\nwhich are nonperturbative quantities. Modelling these form factors \nnecessarily introduces some uncontrollable systematic uncertainty into \nthe determination of $V_{ub}$ and hence a model independent method is\ndesirable. At present the values of $V_{ub}$ extracted from exclusive \nsemileptonic $B\\to\\pi$ decays using lattice QCD, QCD sum rules and \nquark models range from $|V_{ub}| = (2.5 - 4.5)\\times10^{-3}$. Clearly\nthis is not a satisfactory situation, in particular in view of the \ndata to be expected from the $B$ factories. Although there have been\nrecent efforts to update existing models (see\ne.g. \\cite{hphx9711268}--\\nocite{hphx9712399,hphx9801421}\\cite{hphx9801443}), \nthere is still room for some improvement.\n\nThere are constraints on form factors originating from their analyticity \nproperties and the unitarity of the underlying theory. The ideas \nhow to implement these constraints are in fact quite old \n\\cite{prvxd4x3519}--\\nocite{prvxd3x2807,prvxd4x725,prvxd4x2020}\\cite{npxb189x157}\nand have recently attracted renewed attention. In particular, combining these\nunitarity bounds with lattice data \\cite{hphx9509358} has lead to relatively\ntight and model independent bounds on the form factors of $B \\to \\pi$\ntransitions.\n\nIn the present paper we improve the application of unitarity bounds \nin such a way that not only points at which the form factor is known\n(e.g. from lattice data or heavy quark relations) can be included, \nbut also the slope and higher derivatives at some point, \nwhich could be known for instance from \nsum rule considerations. At the point of maximal \nmomentum transfer the derivatives are correlated with the value of the \nform factor due to analyticity and we shall discuss the restrictions\nobtained from this in some detail. \n\nA similar improvement of the unitarity bounds has been proposed by\nBoyd et al.~\\cite{hphx9702300,hphx9705252}, where the form factors for \n$B \\to \\pi \\ell \\nu$ are expanded in a cleverly chosen conformal \nvariable $z$ and some of the relations we use in the present paper appear \nin a similar form in these references. \n\nIn the next section we shall summarize the formalism as described in \n\\cite{npxb189x157}.\nThe improved bounds are considered in section \\ref{secimprov} where we\ngeneralize the method to include known slopes and higher \nderivatives. We make use of the correlation between the derivatives \nand the value of the form factor taken at $q^2_{max}$ to tighten \nthe bounds even without using any additional physics input. In order \nto get useful bounds some physics input is necessary and we chose to \nuse the chiral limit which is valid close to the end point. This has \na strong effect since any knowledge of the form factors in the \nend point region tightens the bounds significantly. \nWith this input we study numerical examples. \n\\section{Bounds on Form Factors}\n\\label{secbounds}\nFor later use we summarize in this section the formalism how to derive \nbounds on the form factors using only perturbative QCD, unitarity and \nthe ana\\-ly\\-ti\\-ci\\-ty of the form factors in the complex plane. \n\nWe choose to describe the hadronic matrix element of the semileptonic \n$\\bar B^0 \\to \\pi^+ \\ell^- \\bar\\nu_\\ell$ decays with the following two \nform factors:\n\\begin{eqnarray}\n\\label{anfang}\n\\langle \\pi^+(p')|V^\\mu|\\bar B^0(p)\\rangle\n&=&\\left(p^\\mu+{p'}^\\mu-\\frac{M^2-m^2}{q^2}q^\\mu\\right)f^+(q^2)\\\\\n&&+\\frac{M^2-m^2}{q^2}q^\\mu f^0(q^2)\\nonumber\n\\end{eqnarray}\nwhere $M^2=m_B^2$, $m^2=m_\\pi^2$, $V^\\mu=\\bar u\\gamma^\\mu b$ and $q=p-p'$.\nIn this notation $q^2$ runs from $q^2_{min}=m_\\ell^2$ to\n$q^2_{max}=(M-m)^2$. Throughout this paper we will neglect the lepton masses\nand therefore set $q^2_{min}=0$. Furthermore, this special choice of\ndecomposition into Lorentz vectors leads to form factors which have to satisfy\nthe kinematical constraint\n\\begin{equation}\n\\label{kinconst}\nf^+(0) = f^0(0).\n\\end{equation}\nThe form factors are real functions of a real variable but it is\nconvenient to think of them as analytic functions in the complex\n$q^2$--plane. \n\nTo derive bounds on $f^+(q^2)$ and $f^0(q^2)$ we consider the\ntwo--point function \n\\begin{eqnarray}\n\\label{2pfunction}\n\\Pi^{\\mu\\nu} &\\equiv& i\\int\\intf{x}\\ex{iq\\cdot\nx}\\langle0|T\\{V^\\mu(x)V^{\\nu\\dagger}(0)\\}|0\\rangle \\nonumber\\\\\n&\\equiv& -(g^{\\mu\\nu}q^2 - q^\\mu q^\\nu)\\Pi_T(q^2) + q^\\mu q^\\nu \\Pi_L(q^2),\n\\end{eqnarray}\nwhere $V^\\mu=\\bar u\\gamma^\\mu b$ as above and with\n$\\Pi_{T\/L}(q^2)$ corresponding to the propagation of a $J^P=1^-\/0^+$\nparticle. This two--point function can be (and has been) reliable evaluated\nin the deep euclidean region where $-q^2 \\equiv Q^2 \\gg \\Lambda^2\\unter{QCD}$,\ni.e. at an energy scale where perturbative QCD is appliccable.\n\nIncluding a sum over all possible intermediate states\n$\\Gamma$ we get the following result for the imaginary part of\n (\\ref{2pfunction}): \n\\begin{eqnarray}\n\\label{states}\n-(g^{\\mu\\nu} q^2 - q^\\mu q^\\nu)\\im{\\Pi_T(q^2)}+q^\\mu q^\\nu\\im{\\Pi_L(q^2)}\n\\nonumber \\\\\n= \\frac 12\\sum_\\Gamma(2\\pi)^4 \\delta^{(4)}(q-p_\\gamma)\\langle0|V^\\mu(0)|\\Gamma\n\\rangle\\langle\\Gamma|V^{\\nu\\dagger}(0)|0\\rangle.\n\\end{eqnarray}\nWe therefore get an equation for the spectral functions $\\im{\\Pi_{T\/L}(q^2)}$\n(i.e. for the absorptive parts of $\\Pi_{T\/L}(q^2)$) which we can relate to the\nreal parts using the substracted dispersion relations\n\\begin{eqnarray}\n\\label{displ}\n\\chi_L(Q^2)\\equiv\\left(-\\pabl{}{Q^2}\\right)(-Q^2 \\Pi_L(Q^2)) =\\frac\n1\\pi\\int_0^\\infty\\into{t} \\frac{t\\im{\\Pi_L(t)}}{(t+Q^2)^2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{dispt}\n\\chi_T(Q^2)\\equiv\\frac12\\left(-\\pabl{}{Q^2}\\right)^2(-Q^2 \\Pi_T(Q^2)) =\\frac\n1\\pi\\int_0^\\infty\\into{t} \\frac{t\\im{\\Pi_T(t)}}{(t+Q^2)^3}.\n\\end{eqnarray}\n\nWe now restrict ourselves to include only contributions of the $B^*$ and\nthe $B\\pi$ states in the sum over all intermediate states in\n (\\ref{states}).\nIt is possible to discard all other intermediate states because \n (\\ref{states}) is a sum of positive terms if the indices are treated\nsymmetrically. Using isospin and crossing symmetry we can use\n(\\ref{anfang}) to express (\\ref{states}) by the two form factors.\nProjecting out now the transversal\/longitudinal parts, one gets the\ninequalities\n\\begin{eqnarray}\n\\label{ineql}\n\\im{\\Pi_L(t)}\\ge\\frac 32\\frac{t_+ t_-}{16\\pi}\\sqrt{(t-t_+)(t-t_-)}\\frac\n{|f^0(t)|^2}{t^3}\\Theta(t-t_+)\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\label{ineqt}\n\\im{\\Pi_T(t)} &\\ge&\\pi\\left(\\frac{m_{B^*}}{f_{B^*}}\\right)^2\n\\delta(t-m_{B^*}^2) \\nonumber\\\\\n&+& \\frac32 \\frac\n1{48\\pi}\\frac{[(t-t_+)(t-t_-)]^{3\/2}}{t^3}|f^+(t)|^2\\Theta(t-t_+),\n\\end{eqnarray}\nwhere $t=q^2$ and $t_\\pm = (M\\pm m)^2$.\n \nIt is now possible to get bounds on the form factors if one inserts\n(\\ref{ineql},\\ref{ineqt}) in (\\ref{displ},\\ref{dispt}). Since the l.h.s. of \n(\\ref{displ},\\ref{dispt}) can be calculated for $Q^2 \\gg \\Lambda^2\\unter{QCD}$\nusing perturbative QCD one gets inequalities which restrict the form factors. \nThese inequalities take the form (in shorthand notation) \n\\begin{eqnarray}\n\\label{shorthand}\nJ(Q^2) \\ge \\frac 1\\pi \\int_{t_+}^\\infty\\into{t} k(t,Q^2)|f(t)|^2\n\\end{eqnarray}\nwhere $J(Q^2)$ denotes the QCD input (i.e. the perturbative calculation of\n$\\chi_{T\/L}$) including the $B^*$--resonance in case of $f^+$\nand $k(t,Q^2)$ is a known kinematical function. The exact value\/structure of \n$J(Q^2)$ and $k(t,Q^2)$ does of course depend on the form factor under\nexamination.\n\nTo translate the inequality (\\ref{shorthand}) in constraints on the form factor\nfor values of $t$ in the range $[0,t_-]$ (which is the kinematical region of \nphysical interest), we map the complex $t$--plane into the unit disc with \nthe conformal transformation\n\\begin{eqnarray}\n\\frac{1+z}{1-z} = \\sqrt{\\frac{t_+-t}{t_+-t_-}}\n\\end{eqnarray}\nso that (\\ref{shorthand}) becomes\n\\begin{eqnarray}\n\\label{main}\nJ(Q^2)\\ge\\int_{|z|=1} \\frac{\\mbox{d}z}{2\\pi i z}|\\phi(z,Q^2) f(z)|^2\n\\end{eqnarray}\nwith $f(z)\\hat = f(t(z))$.\nHere we have used the fact, that $k(t,Q^2)$ is a positive definite\nquantity so that $\\phi(z,Q^2)\\equiv\\sqrt{k(t(z),Q^2)}$ times the squareroot\nof the Jacobian of the transformation. \n\nThe value of the form factor $f(z)$ for any point $z(t)$ is accessible \nby defining an inner product\n\\begin{eqnarray}\n\\langle g|h \\rangle = \\int_{|z|=1} \\frac{\\mbox{d}z}{2\\pi i z} \\bar g(z)h(z)\n\\end{eqnarray}\nand by considering the product $\\langle g_t|\\phi f\\rangle$ where\n\\begin{eqnarray}\ng_t = \\frac 1{1-\\bar z(t)z},\n\\end{eqnarray}\nso that $f(z(t))$ has no poles in the range $[0,t_-]$. Cauchy's \ntheorem now yields\n\\begin{eqnarray}\n\\label{einf}\n\\langle g_t|\\phi f\\rangle = \\phi(z(t),Q^2) f(z(t)).\n\\end{eqnarray}\nOn the other hand, if there is a pole at $t=t_p$ away from the cut in the\ncomplex $t$--plane (i.e. for $t>t_-$), one would obtain\n\\begin{eqnarray}\n\\langle g_t|\\phi f\\rangle = \\phi(z(t),Q^2) f(z(t)) + \\frac{\\mbox{Res}\n\\{\\phi f,z(t_p)\\}}{z(t_p)-z(t)}.\n\\end{eqnarray}\nThe residue can either be approximated or eliminated completely by the \ntransformation \\cite{plxb301x257}\n\\begin{eqnarray}\n\\label{phitrafo}\n\\phi(z,Q^2)\\quad \\to \\quad \\phi_p(z,Q^2)\\equiv \\phi(z,Q^2)\\frac {z-z(t_p)}\n{1-\\bar z(t_p)z}\n\\end{eqnarray}\nwhere $t_p$ is assumed to lie in the range $[t_-,t_+]$ so that $\\phi_p$ is \npositive for $z=z(t)$ with $t$ in $[0,t_-]$. This replacement cancels the\npole of $f(t)$ so that (\\ref{einf}) holds. The crucial property of this\ntransformation is the fact that, since $|(z-z(t_p))\/(1-\\bar z(t_p)z)| =1$\nfor $z$ on the unit circle, $\\langle \\phi f|\\phi f\\rangle=\\langle \\phi_p\nf|\\phi_p f\\rangle$ and the QCD constraints are left unchanged.\n\n\\begin{figure}[ht]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{without.ps} \\\\\n\\caption{Bounds on the form factors $f^0(t)$ (on the left side, $t$ increasing \nto the left) and $f^+(t)$ (on the right side), derived\nwithout any additional constraints and plotted over the whole kinematical\nrange of $[0,t_-]$ ($t$ in $\\mbox{GeV}^2$).\n\\label{figwithout}} \n\\end{figure}\n\nBecause of the positivity of the inner product we have\n\\begin{eqnarray}\n\\Det{\n\\begin{array}{cc}\n\\langle \\phi f |\\phi f \\rangle & \\langle \\phi f |g_t \\rangle \\\\\n\\langle g_t |\\phi f \\rangle & \\langle g_t |g_t \\rangle\n\\end{array}\n} \\ge 0,\n\\end{eqnarray}\nwhich, by eliminating $\\langle\\phi f|\\phi f\\rangle$ with (\\ref{main}), \ngives us the following bounds on the form factors\n\\begin{eqnarray}\n\\label{withoutpoints}\n|f(t)|^2 \\le J(Q^2)\\frac1{1-z^2(t)}\\frac1{|\\phi(z(t),Q^2)|^2}\n\\end{eqnarray}\nwhere we have made use of the fact that $z(t)$ is real for $t$ in $[0,t_-]$.\n\nThese Bounds, which are derived using only perturbative QCD and which do not\ndepend on any model assumptions or input parameters (they are therefore quite\nloose), are plotted in figure \\ref{figwithout}. They were calculated using\n$Q^2=0$ in $J(Q^2)$ at order ${\\cal O}(\\alpha_s)$ \n\nIf we know the value of the form factor $f_{i}\\equiv f(t_{i})$\nat $n$ points $i=1,\\dots, n$, we can define a matrix $\\cal{M}$\n\\begin{eqnarray}\n\\label{mat}\n\\cal{M} = \\left(\n\\begin{array}{ccccc}\n\\langle \\phi f |\\phi f \\rangle & \n\\langle \\phi f |g_t \\rangle &\n\\langle \\phi f |g_{t_1} \\rangle &\n\\cdots &\n\\langle \\phi f |g_{t_n} \\rangle \\\\\n\n\\langle g_t |\\phi f \\rangle & \n\\langle g_t |g_t \\rangle &\n\\langle g_t |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_t |g_{t_n} \\rangle \\\\\n\n\\langle g_{t_1} |\\phi f \\rangle & \n\\langle g_{t_1} |g_t \\rangle &\n\\langle g_{t_1} |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_{t_1} |g_{t_n} \\rangle \\\\\n\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\n\\langle g_{t_n} |\\phi f \\rangle & \n\\langle g_{t_n} |g_t \\rangle &\n\\langle g_{t_n} |g_{t_1} \\rangle &\n\\cdots &\n\\langle g_{t_n} |g_{t_n} \\rangle\n\\end{array}\n\\right)\n\\end{eqnarray}\nand obtain, using again the positivity of the inner product,\n\\begin{eqnarray}\n\\label{ineq}\n\\det{\\cal M} \\ge 0.\n\\end{eqnarray}\nSince $\\langle \\phi f |\\phi f \\rangle$ is eliminated using (\\ref{main})\nand since all components except $f(t)$ \nof this matrix are known --- they are either constants or functions of $t$ ---\nthe inequality (\\ref{ineq}) leads to bounds on the\nform factors $f(t)$. The inclusion of some known values (which i.e. are \npredicted by a model or which one can get by lattice calculations, see\n\\cite{hphx9509358}) therefore will give us more stringent bounds \n\\begin{eqnarray}\nF_{lo}(t,t_i, f_i) \\le f(t) \\le F_{up}(t,t_i, f_i)\n\\end{eqnarray}\nwith $F_{lo}$ and $F_{up}$ calculable functions of $t$ with the parameters \n$t_i,f_i$.\n\nThe upper and lower bounds can be written as\n\\begin{eqnarray}\n\\label{bounds}\nF_{up,lo}(t,Q^2) =\n\\frac{-\\beta(t)\\pm\\sqrt{c(Q^2)\\cdot\\Delta(t)}}{\\alpha\\cdot\\phi(t,Q^2)}.\n\\end{eqnarray} \nThe shape of $F_{up,lo}(t,Q^2)$ as functions of $t$ depend on the values \nfor $t_{i}$ and $f_{i}$. In (\\ref{bounds}), $\\alpha$ is a scaling constant,\n$\\beta(t)$ gives roughly the shape of the form factor while the squareroot in\n(\\ref{bounds}) distinguishes the upper from the lower bound. In the squareroot,\n$c(Q^2)$ is a constant in $t$ which depends only on the energy scale $Q^2$,\nand $\\Delta(t)$\nis a function with zeros in every $t_{i}$. The reason why the function\n$\\Delta(t)$ should behave like this is quite clear: \nsince we fixed the value of the form factor at certain points $t_i$, \nthe upper and lower bound should coincide in these points\n(for an exact definition of the functions appearing in (\\ref{bounds}) \nsee \\cite{hphx9509358}). \n\n\\section{Improving the Bounds}\n\\label{secimprov}\nThe method of including `fixed points' (i.e. to fix the value of the\nform factor at certain kinematical points) which we summarized in \n\\ref{secbounds} is well known and used throughout\nthe literature. However, this formalism can be applied in a slightly different\nway: to include the slope or even higher derivatives of the form factor. \nThis is desirable because it is possible to obtain the slope \ne.g.\\ for low $t$ from QCD sum rules or for $t$ near the kinematical end point\nfrom the chiral limit. \n\nConsider two fixed points $t_1$ and $t_2=t_1 + \\epsilon$ with $\\epsilon$\narbitrary but small. We get the coresponding values of $f(t_i)$ via a\nTaylor expansion: \n\\begin{eqnarray}\nf(t_1) &\\equiv& f_1 \\\\\nf(t_2) &=& f(t_1) + \n \\epsilon \\left. \\frac{d}{dt}f(t) \\right|_{t=t_1} \\, . \n\\end{eqnarray}\nWe can now calculate \nthe bounds depending on these values for $t_1,t_2,f_1$ and $f_2$ \nand take the limit $\\epsilon \\to 0$. It is useful to define a function \n$\\psi (z,Q^2) \\equiv f(z)\\phi(z,Q^2)$, in terms of which we obtain \n\\begin{eqnarray}\n\\label{alptraum}\n\\psi_{up,lo} &=&\n \t\\frac{(1-z_1^2)(\\psi_1(1-2zz_1+z_1^2)+\\psi_1'(z-z_1)(1-z_1^2))}\n\t{(1 - z z_1)^2} \\nonumber\\\\\n&\\pm& \\frac{(1-z_1^2)(z-z_1)^2}{(1-zz_1)^2\\sqrt{1-z^2}} \\times \\\\\n&&\n\t\\sqrt{J-\\psi_1^2(1+z_1^2)+2\\psi_1\\psi_1'z_1(1-z_1^2)\n\t-[\\psi_1']^2(1-z_1^2)^2} \\nonumber\n\\end{eqnarray}\nwhere $z_1\\equiv z(t_1)$, $\\psi_1 \\equiv \\psi (z_1,Q^2)$ \nand \n$$ \n\\psi_1' = \\left. \\frac{\\partial}{\\partial z} \\psi (z,Q^2) \\right|_{z=z_1} \\, .\n$$\n\nIt is obvious how to extend this to higher derivatives (including e.g.\\ \nthe curvature of the form factor), but the corresponding equations become\ntedious. In parts of the numerical analysis presented below the curvature\nhas been included. \n\nIt has been observed (see \\cite{hphx9702300,hphx9705252} and\n\\cite{hphx9603414,hphx9712417}) that the inequality \n(\\ref{main}) also yields restrictions for the derivatives of the form \nfactor in the end point $t_-$. We shall use this input to restrict the \npossible values of the form factor at $t_-$ in terms of the slope and \nthe curvature. In order to do this we rewrite (\\ref{main}) into the \nform\n\\begin{eqnarray} \\label{main1}\n\\frac 1{2\\pi}\\int_0^{2\\pi}\\into{\\Theta} |\\psi(\\ex{i\\Theta})|^2 \\le J(Q^2).\n\\end{eqnarray}\nIn the case of $f^0$ it is now possible to\nperform a Taylor expansion around $z_0 = 0$ ($t=t_-$) which is convergent \nin the full unit disc, since $f^0$ does not have poles or cuts in this \nregion. This is due to the fact that no physical intermediate states can \ncontribute. We therefore get the inequality\n\\begin{eqnarray}\n\\label{ineq1}\nJ(Q^2) \\ge\\psi^2(0)+\\sum_{n=1}^\\infty \\left(\\frac1{n!}\\right)^2\\psi^{(n)^2}(0).\n\\end{eqnarray}\nwhere $\\psi^{(n)}(0)$ denotes the $n$th derivative of $\\psi$ at the point\n$z=0$. Note that the form factor, $z(t)$ and thus also $\\psi$ are real\nin the region $0 \\le t \\le t_-$.\n\nThe form factor $f^+$ is not analytic inside the unit disc, since there \nis a contribution of the $B^*$ in this channel. Hence one expects that the \nTaylor expansion for this form factor converges only within the circle\n$0 \\le |z| \\le |z(m_{B^*}^2)|$. The integral in (\\ref{main1}) runs over \nthe unit circle and thus the integration contour lies outside the radius \nof convergence of the Taylor series. In order to take into account the \n$B^*$--pole one has to expand in a Laurent series or to subtract the pole. \nHowever, closer inspection reveal that these two methods are equivalent. \nWe choose to subtract the pole and thus define a function\n\\begin{eqnarray}\n\\tilde f(z) = f(z) - \\frac{\\mbox{Res}\\{f,z_p\\}}{z-z_p}\\frac{\\phi(z_p)}{\\phi(z)}\n\\end{eqnarray}\nand obtain\n\\begin{eqnarray}\n\\label{ftilde1}\n\\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\psi(\\ex{i\\Theta})|^2 \n&=& \\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\tilde \\psi(\\ex{i\\Theta})\n+\\frac{\\mbox{Res}\\{\\phi f,z_p\\}}{\\ex{i\\Theta} - \\ex{i\\Theta_p}}|^2 \\\\\n\\label{ftilde2}\n&=& \\frac{\\mbox{Res}^2\\{\\phi f,z_p\\}}{1-z_p^2} +\n\\int_0^{2\\pi}\\frac{\\into{\\Theta}}{2\\pi} |\\tilde \\psi(\\ex{i\\Theta})|^2 \n\\end{eqnarray}\nwhere we define $\\tilde \\psi = \\tilde f \\phi$. Note that the terms \nlinear in $\\tilde \\psi$ and $1\/({\\ex{i\\Theta} - \\ex{i\\Theta_p}})$ vanish, \nsince only terms without $\\Theta$ dependence contribute to the integral. \n\nThus we obtain instead of (\\ref{ineq1}) \n\\begin{eqnarray} \n\\label{ineq2}\nJ(Q^2)-\\frac{\\mbox{Res}^2\\{\\phi f,z_p\\}}{1-z_p^2} \\ge\\tilde\\psi^2(0)+\n\\sum_{n=1}^\\infty \\left(\\frac1{n!}\\right)^2\\tilde\\psi^{(n)^2}(0).\n\\end{eqnarray}\n\n\\begin{figure}[htp]\n\\epsfxsize=9cm\n\\leavevmode\n\\centering\n\\epsffile[70 130 540 650]{ellipse.ps} \\\\\n\\caption{Possible combinations of point, slope and curvature at the \nend point $f^+(t_-)$, allowed parameter triplets lie inside the ellipsoid. \nTo clarify the po\\-si\\-tion of the ellipsoid, it's contours were drawn on \nthree faces of the cube. \n\\label{figellipse}}\n\\end{figure}\n\nIf one redefines $J(Q^2)$ to include the residue and if one \nuses the analytic funtion $\\tilde f$, (\\ref{ineq2}) is the same \nas (\\ref{ineq1}).\n\nSince (\\ref{ineq1}) is a sum of positive terms, we can \ncut off the sum at some value of $n$ and get (e.g. for $n\\le2$)\n\\begin{eqnarray}\n\\label{kurz}\n[(\\phi f)(0)]^2 + [(\\phi f)'(0)]^2 + \\frac14 [(\\phi f)''(0)]^2 < J(Q^2)\n\\end{eqnarray}\nwhere we re--substituted $\\psi$.\n\nSuch equations give us ellipsoids in the parameter space of value, slope\nand higher derivatives of the form factor at the kinematical end point, where\nall allowed parameter combinations lie inside the ellipsoid. If we take, for\nexample, $n\\le2$ as in (\\ref{kurz}) we would get a three--dimensional\nellipsoid in point--slope--curvature--space (see figure \\ref{figellipse}).\n\nThe resulting constraints are not as good as the ones that can be obtained \nfor $B \\to D$ transitions \\cite{hphx9603414,hphx9712417}; this is due to \nheavy quark symmetries, which are not as useful in $B \\to \\pi$ decays. \n\nHowever, one may exploit the chiral symmetry for the light degrees of \nfreedom as an additional physics input. These symmetries hold at \nsmall momenta of the pions and are therefore applicable in the end \npoint region where the momentum transfer to the leptons becomes maximal. \nOne may combine heavy quark and chiral symmetry as in \n\\cite{prvxd45x2188}--\\nocite{hphx9602353}\\cite{hphx9707410} and compute \nthe form factors in this limit. One finds \\cite{prvxd45x2188} \n\\begin{eqnarray}\n\\label{chiralform}\nf^+(t) &=& \\frac{f_B}{2f_\\pi}\\left[1+\\frac{g(M^2-m^2+t)}{M^2+m^2+2\\Delta M\n-t}\\right] \\nonumber\\\\\nf^0(t) &=& \\frac{f_B}{2f_\\pi}\\left[1+\\frac{g(M^2-m^2+t)}{M^2+m^2+2\\Delta M-t}\n\\right. \\\\\n&&\\hspace{1.8cm}\n+\\left.\\frac{t}{M^2-m^2}\\left\\{1-\\frac{g(3M^2-m^2+t)}{M^2+m^2+2\\Delta M-t}\n\\right\\}\\right] \\nonumber\n\\end{eqnarray}\nwhich results in the residue\n\\begin{eqnarray}\n\\label{chiralres}\n\\mbox{Res}\\{f,t_p\\} = -\\frac {f_B}{f_\\pi}gM(M+\\Delta)\n\\end{eqnarray}\nat\n\\begin{eqnarray}\nt_p = M^2+m^2+2\\Delta M\n\\end{eqnarray}\nwhere $\\Delta = m_{B^*} - m_B$. \nThe additional parameters which appear in these relations are\nthe chiral coupling constant $g$ which \ndescribes the $BB^*\\pi$--coupling and the ratio of the decay constants \n$f_B \/ f_\\pi$. \n\nThe form factors (\\ref{chiralform}) are valid in the chiral limit, \nwhich holds only in a small piece of the kinmatically allowed region. \nIn order to extend this to the full range, a variety of \nans\\\"atze have been invented to extend this pole--behaviour to small\n$t$ (see e.g. \\cite{zpxc29x637}--\\nocite{hphx9401303} \\nocite{zpxc48x663}\n\\cite{prvlx68x2887}); however, this makes the extraction ov $V_{ub}$ \nmodel dependent. \n\nWe will now use the improved unitarity bounds and \nthe chiral limit to derive bounds on the form factors. These bounds \nhave the advantage to be model independent, however, as we shall see, \nthey are not very tight over the whole range of $t$. This is mainly\ndue to the fact that the parameters $g$ and $f_B\/f_\\pi$ suffer from large \ntheoretical uncertainties. \n\n\\begin{figure}[htp]\n\\epsfxsize=9cm\n\\leavevmode\n\\centering\n\\epsffile[70 130 540 650]{chiralcone.ps} \\\\\n\\caption{Possible combinations of point, slope and curvature at the end point\n$f^+(t_-)$ using the chiral limit.\n\\label{figchiralcone}}\n\\end{figure}\n\nIn the following we shall consider $f^+$ only; here we can avoid \nthe problem of the large uncertainties in $g$ and $f_B\/f_\\pi$\nby considering fractions like\n$f^{+\\prime}(t_-)\/f^{+} (t_-)$ and $f^{+\\prime\\prime}(t_-)\/f^{+} (t_-)$ \nin which the $f_B\/f_\\pi$--dependence drops out and \nthe remaining $g$--dependence is small, since terms involving $g$ \nare suppressed:\n\\begin{eqnarray}\n\\frac{{f^+}'(t_-)}{f^+(t_-)} &=& \\frac {M+\\Delta}{2 M (M-m) (\\Delta+m)}\n\\frac{1}{1+\\frac{\\Delta+m}{g(M-m)}} \\, , \\\\\n\\frac{{f^+}''(t_-)}{f^+(t_-)} &=& \\frac {1}{2 M^2 (M-m) (\\Delta+m)}\n\\frac{1}{1+\\frac{\\Delta+m}{g(M-m)}} \\, .\n\\end{eqnarray}\n\nThus in the chiral limit the ratios \n$f^{+\\prime}(t_-)\/f^{+} (t_-)$ and $f^{+\\prime\\prime}(t_-)\/f^{+} (t_-)$ \nare practically fixed and we can \nexpress the slope $s$ and the curvature $c$ in terms of the value $p$ \nof $f^+$ at $t_-$. This would yield a straight line in a $p,s,c$ plot; \nhowever, varying the coupling $g$ in a generous range between $0.25$ and \n$0.5$ \\cite{hphx9605342} and allowing for derivations of the chiral limit\nof about 15\\%, we obtain the rectangular cone shown in \nfigure~\\ref{figchiralcone}. \n\nIn case of the residue it is not possible to get rid of the $g$-- and\n$f_B\/f_\\pi$--parameter, so one has to take the full \nuncertainties into account\nwhen calculating the bounds. One could also choose to drop the residue \ncompletely and work with the redefined $\\phi_p(z,Q^2)$ of (\\ref{phitrafo}) \nbut we decided to use (\\ref{chiralres}) because \nthe resulting bounds are more stringent.\n\nUsing (\\ref{chiralform}) and their first derivatives at the end point as\ninput parameters in (\\ref{alptraum}) (the extension to higher derivatives\nis trivial) we get upper and lower bounds for $0\\le t\\le t_-$ which depend\nstrongly on the chiral parameters $g$ and $f_B\/f_\\pi$. In addition to\nthat, a combination of figure \\ref{figellipse} and \\ref{figchiralcone} can be\nused to constrain the allowed range for the end point value of $f(t_-)$ and\nits derivatives.\n\nThe combination of unitarity bounds, extended to include derivatives, and \nthe chiral limit now allows us to calculate bounds for the form factors \n$f^+(t)$ and $f^0(t)$.\n\n\\section{Numerical Results and Discussion}\n\\label{seccalc}\nInserting $t_1=t_-$ ($z_1=0$) in (\\ref{alptraum}) one gets\n\\begin{eqnarray}\n\\psi_{up,lo} &=&\n \t(\\psi_1+\\psi_1'z) \\pm \\frac{z^2}{\\sqrt{1-z^2}}\n\t\\sqrt{J-\\psi_1^2-[\\psi_1']^2}\n\\end{eqnarray}\nwhich leads to the bounds\n\\begin{eqnarray}\n\\label{endbounds}\nF_{up,lo} = \\frac{f_1 \\phi_1 + (f_1 \\phi_1)'z}{\\phi(z,Q^2)} \\pm\n\t\t\\frac{z^2}{\\phi(z,Q^2)}\\sqrt{\\frac{J-(f_1 \\phi_1)^2 - \n\t\t[(f_1 \\phi_1)']^2}{1-z^2}}.\n\\end{eqnarray}\nWe used $Q^2=0$ for the QCD calculations and\ninserted all possible combinations of end point value and slope into\n(\\ref{endbounds}). We also allowed \nvariation of $g$ and $f_B\/f_\\pi$ from $0.25\\le g\\le 0.5$ and \n$1\\le f_B\/f_\\pi\\le 1.7$ (\\cite{hphx9605342, hlax9710057}). \nWithin this variation we determined the bounds \nfor fixed parameter values and finally took the loosest ones as our result.\n\nWe scanned the parameter space of possible $ps$--pairs by dividing\nthe allowed $p$--range into small, equally spaced intervals.\nFor each of these possible $p$--values, there exists an allowed $s$--range\nwhich is also divided into small intervalls. A similar procedure was applied \nfor the chiral input parameters.\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile{pieces.ps} \\\\\n\\caption{Bounds on the form factor $f^+(t)$, derived by using a fixed\nend point and varying the slope according to the chiral limit\n\\label{figpieces}} \n\\end{figure}\n\nFigure~\\ref{figpieces} shows the resulting bounds for the form factor $f^+$. \nThe values of the form factor at $t_-$ vary between 3.2 and 13.5 due to the \nuncertainties in $f_B\/f_\\pi$ and $g$. The two solid lines represent the \nresult for the maximal value of the form factor at $t_-$, where the upper \n(lower) line corresponds to the minimal (maximal) possible value for the \nslope at $t_-$. Similarly, we have plotted the corresponding lines for \nintermediate values of the form factor at $t_-$ down to the minimal value, \nrepresented by the long--dashed line. \n\nThe upper and lower bounds coincide at $t=0$ due to the kinematical\nconstraint (\\ref{kinconst}). We have computed $f^0$ in the same way\nas we did for $f^+$, scanning the parameter space for $p$ and $s$ at $t=t_-$. \nThe resulting bounds for \n$f^0$ at $t=0$ are much tighter that those for $f^+ (0)$ and hence the \nkinematical constraint has a significant effect on $f^+$. \nIn order to be conservative we have varied the parameters for both form\nfactors independently; that is \nwe did not use the correlation between the form factors implied by the \nchiral limit, namely that at $t=t_-$ they are both given by the same two \nparameters $f_B\/f_\\pi$ and $g$. \n\nIn figure \\ref{figwith} we have combined the results for $f^+$ and $f^0$, \nplotting the upper bound for the maximal values of $f^+ (t_-)$ and \n$f^0 (t_-)$ with the minmal slopes and the lower bounds for the minmal \nvalues of $f^+ (t_-)$ and $f^0 (t_-)$ with the maximal slopes. \nCompared to the standard method (see figure \\ref{figwithout}) \nthe inclusion of the slope and the chiral limit has significantly \nimproved the bounds. We have to point out, however, that not any \ncurve within these two bounds is allowed as a form factor, since QCD \nimplies relations between slopes and the form factor values. This can \nbe seen in figure~\\ref{figpieces}.\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{wholeslope.ps} \\\\\n\\caption{Bounds on the form factors $f^0(t)$ (on the left side, $t$ increasing \nleftward) and $f^+(t)$ (on the right side, $t$ increasing rightward) derived \nwith restriction on possible $ps$--pairs at $f(t_-)$ and including the\nkinematical constraint, plotted over the whole\nkinematical range of $[0,t_-]$ ($t$ in $\\mbox{GeV}^2$). \n\\label{figwith}} \n\\end{figure}\n\nFinally one can also use model input into the machinery of unitarity \nbounds. In particular, a model can be used to obtain points and curvatures\naway from $t_-$. \n\nIn figure \\ref{figmodel2p} and \\ref{figmodel2a} we have used the \nISGW II model \\cite{hphx9503486} as an input. Fig.\\ref{figmodel2p}\nshows the bounds obtained by the standard formalism using two \npoints ($t=10$ GeV${}^2$ and $t_-$) from this model, while in \nfig.\\ref{figmodel2a} we use these two points but also the derivative\nat $t_-$, obtained again from ISGW II, as input parameters. The dashed line \nis the model prediction itself. \n \nAnother commonly used model is the BSW model \\cite{zpxc29x637}. \nIn fig.\\ref{figpoint} we use this model to determine the value of \nthe form factor $f^+$ at $t_-$, while in fig.\\ref{figslope} also \nthe slope at $t_-$ was taken from the model. It is interesting to \nnote that if one also includes the curvature of $f^+$ at $t_-$ \nin the BSW model --- as in fig.\\ref{figcurve} --- \nthe lower bound shifts significantly upwards, such \nthat the model becomes inconsistent with the bounds. This indicates that \nthe curvature of the BSW model is too large in the end point, since \nthe lower bound comes out to be quite high; its value at $t=0$\nis about one, which is in contradiction not only with the BSW model, \nbut also with sum rule calculations \\cite{hphx9305348,hphx9802394}. \n \n\\begin{figure}[hbp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{model2pkc.ps} \\\\\n\\caption{Bounds on $f^0$ (left) and $f^+$ (right) derived by using the ISGW II\nmodel and using the values at $t=10\\mbox{ GeV}^2$ and $t=t_-$ as input\nparameters. The bounds are drawn solid, the dashed line corresponds to the\nmodel. \n\\label{figmodel2p}} \n\\end{figure}\n\n\\begin{figure}\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{model2akc.ps} \\\\\n\\caption{Same as in figure \\ref{figmodel2p} but including the slope at $t=t_-$\nas well.\n\\label{figmodel2a}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modpoint.ps} \\\\\n\\caption{Bounds on $f^+$ derived by using the BSW model and using the value \nat $t=t_-$ as input parameter. The bounds are drawn solid, the dashed line \ncorresponds to the model. \n\\label{figpoint}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modslope.ps} \\\\\n\\caption{Same as in figure \\ref{figpoint} but including the slope at $t=t_-$\nas well.\n\\label{figslope}} \n\\end{figure}\n\n\\begin{figure}[htp]\n\\epsfxsize=12cm\n\\leavevmode\n\\centering\n\\epsffile[70 250 540 540]{modcurve.ps} \\\\\n\\caption{Same as in figure \\ref{figslope} but including the curvature at \n$t=t_-$ as well.\n\\label{figcurve}} \n\\end{figure}\n\nThe pole form of the BSW model is motivated by the form factor in the \nchiral limit and thus it is no surprise that they have similar curvatures \nof $f^+$ at $t_-$. We take this as an indication \nthat one should not exploit the chiral limit to obtain derivatives\nof the form factors beyond the slope; going beyond the slope requires\nto take into account higher orders in chiral perturabtion theory. \n\n\\section{Conclusions} \n\\label{secconc}\nUsing unitarity and analyticity to derive bounds on form factors \nusing input from perturbative QCD has become a widely used tool, \nsince this method allows to constrain form factors over the whole\nrange of $q^2$ in a model independent way. \n\nThis method has attracted some attention in $B$ decays and has been \ncombined with heavy quark symmetry; for the $b \\to c$ transitions \nthis has lead to stringent constraints on the form factors, i.e.\nthe Isgur--Wise function. \n\nIn $B \\to \\pi$ transitions heavy quark symmetries are not as useful \nand it is necessary to obtain model independent information from \nother sources. The general bounds in $B \\to \\pi$ decays \nare not very tight and hence also not very useful. \nIncluding points where the form factor is known from other sources\nimproves the bounds; this method has been combined with lattice data\nto obtain model independent bounds for the form factors for \n$B \\to \\pi$ transitions. \n\nIn the present paper we have focussed also at the $B \\to \\pi$ decays\nand improved the bounds on the relevant form factors one more step by \nincluding slopes and higher derivatives of form factors, which in \nsome cases may be known from other sources. Again from unitarity \nthe value, the slope, and even higher derivatives of the form factor at \n$t_-$ are constrained to lie inside an ellipsoid which can be used \nas an input into the machinery we have proposed here. \n\nStill this does not tighten the bounds very much and some more physics\ninput beyond perturbative QCD is needed. Close to the kinematical end point \n$t_-$ chiral \nperturbation theory is valid and we used the chiral limit of the \nform factors as an input. With this we arrived at bounds which are\nmuch tighter than the general bounds obtained without any knowledge \non the form factors. The model independent results of our paper are \nshown in fig.\\ref{figpieces} and fig.\\ref{figwith}. \n \nFinally one may also combine models with the unitarity bounds from QCD. \nTaking points, slopes and curvatures from these models one may test \nthe consistency of a given model with QCD. An extensive analysis of \nthe various models is beyond the scope of the present paper; we \nonly considered the ISGW II and the BSW model as examples. Although \nboth models lie within the bounds given in fig.\\ref{figwith} (the \nISGW II touching the lower bound at $t_-$), this still does not \nmean that they are consisten with QCD; the curvature of the BSW model \nat $t_-$ turns out to be incompatibel with the QCD constraints. \n\n\\section*{Acknowledgements}\nWe thank Changhao Jin, who visited Karlsruhe in the early stage of this\nwork, for useful discussions on this subject. This work was supported by \nthe ``Graduiertenkolleg: Elementarteilchenphysik an Be\\-schleu\\-ni\\-gern''\nand the ``Forschergruppe: Quantenfeldtheorie, Computeralgebra und \nMonte Carlo Simulationen'' of the Deut\\-sche For\\-schungs\\-ge\\-mein\\-schaft.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\\section{Introduction}\n\nThe task of program synthesis is to automatically generate a program that is\nconsistent with a specification such as a set of input-output examples, and has\nbeen studied since the early days of Artificial\nIntelligence~\\citep{waldinger1969prow}. There has been a lot of recent progress\nmade on \\emph{neural program induction}, where novel neural architectures\ninspired from computation modules such as RAM, stack, CPU, turing machines, and\nGPU~\\citep{graves2014neural,joulin2015inferring,nram,graves2016hybrid,npi,kaiser2015neural}\nhave been proposed to train these architectures in an end-to-end fashion to\nmimic the behavior of the desired program. While these approaches have achieved\nimpressive results, they do not return explicit interpretable programs, tend not\nto generalize well on inputs of arbitrary length, and require a lot of examples\nand computation for learning each program. To mitigate some of these\nlimitations, \\emph{neural program synthesis}\napproaches~\\citep{JohnsonHMHLZG17,NeurosymbProgsynth,RobustFill} have been\nrecently proposed that learn explicit programs in a Domain-specific language\n(DSL) from as few as five input-output examples. These approaches, instead of\nusing a large number of input-output examples to learn a single program, learn a\nlarge number of different programs, each from just a few input-output examples.\nDuring training, the correct program is provided as reference, but at test time,\nthe learnt model generates the program from only the input-output examples.\n\n\n\n\n\\input{motiv_figure.tex}\n\n\nWhile neural program synthesis techniques improve over program induction\ntechniques in certain domains, they suffer from two key limitations. First,\nthese approaches use supervised learning with reference programs and suffer from\nthe problem of \\textit{Program Aliasing}: For a small number of input-output\nexamples, there can be many programs that correctly transform inputs to outputs.\nThe problem is the discrepancy between the single supervised reference program\nand the multitude of correct programs. Figure \\ref{fig:progsynth} shows an\nexample of this: if maximizing the probability of ground truth program,\npredicting Program B would be assigned a high loss even though the two programs\nare semantically equivalent for the input-output example.\nMaximum likelihood training forces the model to learn to predict ground truth\nprograms, which is different from the true objective of program synthesis:\npredicting \\emph{any} consistent program. To address this problem, we alter the\noptimization objective: instead of maximum likelihood, we use policy gradient\nreinforcement learning to directly encourage generation of \\emph{any} program\nthat is consistent with the given examples.\n\n\nThe second limitation of neural program synthesis techniques based on sequence\ngeneration paradigm~\\citep{RobustFill} is that they often overlook the fact that\nprograms have a strict syntax, which can be checked efficiently. Similarly to\nthe work of~\\citet{NeurosymbProgsynth}, we explore a method for leveraging the\nsyntax of the programming language in order to aggressively prune the\nexponentially large search space of possible programs. In particular, not all\nsequences of tokens are valid programs and syntactically incorrect programs can\nbe efficiently ignored both during training and at test time. A syntax checker\nis an additional form of supervision that may not always be present. To address\nthis limitation, we introduce a neural architecture that retains the benefits of\naggressive syntax pruning, even without assuming access to the definition of the\ngrammar made in previous work~\\citep{NeurosymbProgsynth}. This model\nis jointly conditioned on syntactic and program correctness, and can implicitly\nlearn the syntax of the language while training.\n\n\n\nWe demonstrate the efficacy of our approach by developing a neural program\nsynthesis system for the Karel programming language~\\citep{pattis1981karel}, an\neducational programming language, consiting of control flow constructs such as\nloops and conditionals, making it more complex than the domains tackled by\nprevious neural program synthesis works.\n\nThis paper makes the following key contributions:\n\\begin{itemize}\n\\item We show that Reinforcement Learning can directly optimize for generating\n any consistent program and improves performance compared to pure supervised\n learning.\n\\item We introduce a method for pruning the space of possible programs using a\n syntax checker and show that explicit syntax checking helps generate better\n programs.\n\\item In the absence of a syntax checker, we introduce a model that jointly\n learns syntax and the production of correct programs. We demonstrate this\n model improves performance in instances with limited training data.\n\\end{itemize}\n\n\\section{Related work}\n\nProgram synthesis is one of the fundamental problems in Artificial Intelligence.\nTo the best of our knowledge, it can be traced back to the work\nof~\\citet{waldinger1969prow} where a theorem prover was used to construct LISP\nprograms based on a formal specification of the input-output relation. As formal\nspecification is often as complex as writing the original program, many\ntechniques were developed to achieve the same goal with simpler partial specifications in\nthe form of input-output (IO)\nexamples~\\citep{amarel1970representations,summers1977methodology}. Rule-based\nsynthesis approaches have recently been successful in delivering on the promise\nof Programming By Example~\\citep{lieberman2001your}, the most widely known\nexample being the FlashFill system~\\citep{FlashFill} in Excel. However, such\nsystems are extremely complicated to extend and need significant development\ntime from domain experts to provide the pruning rules for efficient search.\n\nAs a result, the use of Machine Learning methods have been proposed, based on\nBayesian probabilistic models~\\citep{liang2010learning} or Inductive Logic\nprogramming~\\citep{muggleton1991inductive,muggleton2014meta} to automatically\ngenerate programs based on examples. Recently, inspired by the success of Neural\nNetworks in other applications such as vision~\\citep{krizhevsky2012nips} or\nspeech recognition~\\citep{graves2013speech} differentiable controllers were made\nto learn the behaviour of programs by using gradient descent over differentiable\nversion of traditional programming concepts such as memory addressing\n\\citep{graves2014neural}, manipulating stacks\n\\citep{joulin2015inferring,grefenstette2015learning}, register machines\n\\citep{nram}, and data manipulation~\\citep{neelakantan2015neural}. These\napproaches to program induction however tend to struggle with generalization,\nespecially when presented with inputs of a different dimension than the one they\nwere trained with and require a very large amount of training data. Some\nexceptions to this include Neural Programmer Interpreters~\\citep{npi} and its\nextensions~\\citep{nplattice,cai2017making} that learn from program traces rather\nthan only examples. However, they still learn a different model for each program\nand are computationally expensive, unlike our system that uses a single model\nfor learning a large number of programs.\n\nA series of recent works aim to infer explicit program source code with the\nassumption that code structure provides an inductive bias for better\ngeneralization. In particular, explicitly modeling control flow statements such\nas conditionals and loops can often lead to programs capable of generalizing,\nregardless of input size~\\citep{terpret,anc,diffForth}. One remaining\ndrawback of these approaches is the need to restart learning from scratch for\neach new program. Thus they are largely unsuited for the situation of\nsynthesizing a new program on-the-fly from very few examples.\nThe latest developments use large datasets of artificially generated programs\nand learn to map embeddings of IO examples to information about the programs to\ngenerate. \\citet{balog2016deepcoder} produce scores over attributes, to be used\nas heuristics to speed up search-based techniques. \\citet{NeurosymbProgsynth}\nuse their dataset to learn probability over the expansion rules of a predefined\ngrammar, while \\citep{RobustFill} directly predict the source code of the\nprograms. These last two methods use supervised training to maximize the\nlikelihood of a single reference program, while we directly optimize for the\ngeneration of any consistent program.\n\nOur approach to optimize program\ncorrectness is similar in spirit to advances in Neural Machine\nTranslation~\\citep{googleNMT,ranzato2015sequence} that leverage\nreinforcement learning to optimize directly for evaluation\nmetrics. Taking advantage of the fact that programs can be\nsyntactically checked and unit tested against the specification\nexamples, we show how to improve on those\nREINFORCE~\\citep{williams1992simple} based methods. Recently,\n\\citet{guu2017language} proposed a method similar to ours based on\nMaximum Marginal Likelihood to generate programs based on a\ndescription in natural language. From an application point of view,\nour target domain is more complex as our DSL includes control flow\noperations such as conditionals and loops. Moreover, natural language\nutterances fully describe the steps that the program needs to take,\nwhile learning from IO examples requires planning over potentially\nlong executions. Their approach is more akin to inferring a formal\nspecification based on a natural language description, as opposed to\nour generation of imperative programs.\n\nIncorporating knowledge of the grammar of the target domain to enforce\nsyntactical correctness has already proven useful to model arithmetic\nexpressions, molecules~\\citep{kusner2017grammar}, and\nprograms~\\citep{NeurosymbProgsynth, yin17acl}. These approaches define the model over the\nproduction rules of the grammar; we instead operate directly over the terminals\nof the grammar. This allows us to learn the grammar jointly with the model in\nthe case where no formal grammar specification is available. Our approach is\nextremely general and can be applied to the very recently proposed methods for\ninferring and executing programs for visual reasoning~\\citep{JohnsonHMHLZG17} that\nto the best of our knowledge does not directly explictly encourage grammar\nconsistency of the resulting program.\n\n\n\n\n\n\\section{Problem overview}\nBefore describing our proposed methods, we establish the necessary notation,\npresent the problem setting and describe the general paradigm of our approach.\n\n\\subsection{Program Synthesis formulation}\nTo avoid confusion with probabilities, we will use the letter $\\lambda$ to\ndenote programs. $I$ and $O$ will be used to denote respectively input states\nand output states and we will use the shortcut $IO$ to denote a pair of\ncorresponding input\/output examples. A state constitutes what the programs are\ngoing to be operating on, depending on the application domain. In FlashFill-type\napplications~\\citep{NeurosymbProgsynth,RobustFill}, Input and Output states would\nbe strings of characters, while in our Karel environment, states are grids\ndescribing the presence of objects. If we were to apply our method to actual\nprogramming languages, states would represent the content of the machine's\nregisters and the memory.\n\nAt training time, we assume to have access to $N$ training samples, each\ntraining sample consisting of a set of $K$ Input\/Output states and a program\nimplementing the mapping correctly:\n\\begin{equation}\n \\mathcal{D} = \\Bigg\\{\\ \\Big( \\ioset{k}{i}{1..K}, \\ \\lambda_i \\Big)\\ \\Bigg\\}_{i=1..N} \\text{such that: } \\quad \\lambda_i(I^k_i) = O^k_i \\quad \\forall i \\in 1 .. N, \\quad \\forall k \\in 1 .. K\n\\end{equation}\nwhere $\\lambda_i(I^k_i)$ denotes the resulting state of applying the program\n$\\lambda_i$ to the input state $I^k_i$. Our goal is to learn a synthesizer $\\sigma$\nthat, given a set of input\/output examples produces a program:\n\\begin{equation}\n \\sigma : \\quad \\left\\{ IO^k \\right\\}_{k=1..K} \\quad \\longrightarrow \\quad \\hat{\\lambda}\n\\end{equation}\nWe evaluate the programs on a set of test cases for which we have both\nspecification examples and held-out examples:\n\\begin{equation}\n \\mathcal{D}_{\\text{test}}= \\Bigg\\{\\ \\Big( \\ioset{k_\\text{spec}}{j}{1..K} , \\quad \\ioset{k_\\text{test}}{j}{K+1..K^\\prime} \\Big) \\ \\Bigg\\}_{j=1..N_\\text{test}}\n\\end{equation}\nAt test time, we evaluate the performance of our learned synthesizer by\ngenerating, for each sample in the test set, a program $\\hat{\\lambda_j}$. The\nmetric we care about is \\emph{Generalization}:\n{\\small\n\\begin{equation}\n \\label{eq:generalization}\n \\Big\\{\\ j \\in \\{1..N_\\text{test}\\}\\ \\text{such that } \\ \\hat{\\lambda_j}(I^k_j) == O^k_j \\quad \\forall k \\in \\left\\{1..K^\\prime\\right\\} \\Big\\} \\text{ where } \\hat{\\lambda_j} = \\sigma\\left( \\ioset{k_\\text{spec}}{j}{1..K} \\right).\n\\end{equation}\n}\n\n\\subsection{Neural Program Synthesis Architecture}\nSimilar to \\citet{RobustFill} we use a sequential LSTM-based~\\citep{lstm} language model, conditioned on an\nembedding of the input-output pairs. Each pair is encoded independently by a convolutional neural network (CNN)\nto generate a joint embedding. A succinct description of the architecture can be\nfound in section \\ref{subsec:exp_domain} and the exact dimensions are available in\nthe supplementary materials.\n\n\\begin{figure}[t]\n { \\noindent\\resizebox{\\textwidth}{!}{\n \\input{figures\/model\/model.tex}\n }}\n \\caption{Architecture of our model. Each pair of Input-Output is embedded\n jointly by a CNN. One decoder LSTM is run for each example, getting fed in a\n concatenation of the previous token and the IO pair embedding (constant\n across timestep). Results of all the decoders are maxpooled and the\n prediction is modulated by the mask generated by the syntax model. The\n probability over the next token is then obtained by a Softmax transformation.}\n \\label{fig:model_arch}\n\\end{figure}\n\n\nEach program is represented by a sequence of tokens $\\lambda = [s_1, s_2, ...,\ns_L]$ where each token comes from an alphabet $\\Sigma$. We model the program one\ntoken at a time using an LSTM. At each timestep, the input consists of the\nconcatenation of the embedding of the IO pair and of the last predicted token.\nOne such decoder LSTM is run for each of the IO pairs, all using the same\nweights. The probability of the next token is defined as the Softmax of a linear\nlayer over the max-pooled hidden state of all the decoder LSTMs. A schema\nrepresenting this architecture can be seen in Figure~\\ref{fig:model_arch}.\n\nThe form of the model that we are learning is:\n{\\small\n\\begin{equation}\n p_{\\theta}(\\lambda_i \\ | \\ \\ioset{k}{i}{1..K}) = \\prod_{t=1}^{L_i}\\ p_\\theta(s_t \\ | \\ s_1, ...,s_{t-1}, \\ioset{k}{i}{1..K})\n\\end{equation}\n}\nAt test time, the most likely programs are obtained by running a beam search. One\nof the advantages of program synthesis is the ability to execute hypothesized programs. Through execution, we remove syntactically incorrect programs\nand programs that are not consistent with the observed examples. Among the remaining programs, we\nreturn the most likely according to the model.\n\n\\section{Objective Functions}\n\n\\subsection{Maximum Likelihood optimization}\nTo estimate the parameters $\\theta$ of our model, the default solution is to\nperform supervised training, framing the problem as Maximum Likelihood\nestimation. \\citet{RobustFill} follow this approach and use stochastic gradient\ndescent to solve:\n\\begin{equation}\n \\label{eq:mle-obj}\n \\theta^{\\star} = \\argmax_{\\theta} \\prod_{i=1..N} p_{\\theta}(\\lambda_i\\ | \\ \\ioset{i}{i}{1..K}) = \\argmax_{\\theta} \\sum_{i=1..N} \\log \\left( p_{\\theta}( \\lambda_i \\ | \\ \\ioset{k}{i}{1..K}) \\right)\n\\end{equation}\n\nHowever, this training objective exhibits several drawbacks. First, at\ntraining time, the model is only exposed to the training data distribution,\nwhile at test time, it is fed back the token from its own previous predictions.\nThis discrepancy in distribution of the inputs is well known in Natural Language\nProcessing under the name of \\textit{exposure bias}~\\citep{ranzato2015sequence}.\n\nMoreover, this loss does not represent the true objective of program synthesis.\nIn practice, any equivalent program should be as valid a prediction as the\nreference one. This property, that we call \\textit{program aliasing}, is not\ntaken into account by the MLE training.\nIdeally, we would like the model to learn to reason about the necessary steps\nneeded to map the input to the output. As a result, the loss shouldn't penalize\ncorrect programs, even if they do not correspond to the ground truth.\n\n\n\\subsection{Optimizing Expected Correctness}\n\\label{subsec:expected-correctness}\nThe first modification that we propose is to change the target objective to\nbring it more in line with the goal of program synthesis. We replace the\noptimization problem of (\\ref{eq:mle-obj}) by\n\\begin{equation}\n \\label{eq:RL}\n \\theta^{\\star} = \\argmax_{\\theta} \\mathcal{L}_{R}(\\theta), \\qquad \\text{ where } \\mathcal{L}_{R}(\\theta) = \\sum_{i..N} \\left( \\sum_{\\lambda} p_{\\theta}(\\lambda \\ | \\ioset{k}{i}{1..K})\\ R_i(\\lambda)\\right),\n\\end{equation}\nwhere $R_i(\\lambda)$ is a reward function designed to encode the quality of\nthe sampled programs. Note that this formulation is extremely generic and would\nallow to represent a wide range of different objective functions. If we assume\nthat we have access to a simulator to run our programs, we can design $R_i$ so\nas to optimize for generalization on held-out examples, preventing the model to\noverfit on its inputs. Additional property such as program conciseness, or\nruntime efficiency could also be encoded into the reward.\n\nHowever, this expressiveness comes at a cost: the inner sum in (\\ref{eq:RL}) is\nover all possible programs and therefore is not tractable to compute. The\nstandard method consists of approximating the objective by defining a Monte\nCarlo estimate of the expected reward, using $S$ samples from the model. To\nperform optimization, an estimator of the gradient of the expected reward is\nbuilt based on the REINFORCE trick~\\citep{williams1992simple}.\n\\begin{equation}\n \\begin{split}\n \\label{eq:Rl-sample}\n \\mathcal{L}_{R}(\\theta) &\\approx \\sum_{i=1..N}\\sum_{r=1}^{S} \\frac{1}{S}\\ R_i(\\lambda_r), \\qquad \\text{ where } \\lambda_r \\sim p_\\theta(\\ \\cdot \\ | \\ioset{k}{i}{1..K})\\\\\n \\nabla_{\\theta} \\mathcal{L}_R(\\theta) &\\approx \\sum_{i=1..N} \\sum_{r=1}^{S} \\frac{1}{S} \\ R_i(\\lambda_r) \\nabla_{\\theta}\\log\\left( p_\\theta(\\ \\lambda_r \\ | \\ioset{k}{i}{1..K}) \\right)\n \\end{split}\n\\end{equation}\n\nHowever, given that we sample from a unique model, there is a high chance that\nwe will sample the same programs repeatedly when estimating the gradient. This\nis especially true when the model has been pre-trained in a supervised manner. A\ndifferent approach is to approximate the distribution of the learned\ndistribution by another one with a smaller support.\n\nTo obtain this smaller distribution, one possible solution is to employ the $S$\nmost likely samples as returned by a Beam Search. We generate the embedding of\nthe IO grids and perform decoding, keeping at each step the $S$ most likely\ncandidates prefixes based on the probability $p_{\\theta}$ given by the model. At\nstep $t$, we evaluate $p_{\\theta}(s_1 \\dots s_t, \\ioset{k}{i}{1..K})$ for all\nthe possible next token $s_t$ and all the candidates $(s_1 \\dots s_{t-1})$\npreviously obtained. The $S$ most likely sequences will be the candidates at the\n$(t+1)$ step. Figure \\ref{fig:beamsearch} represents this process.\n\n\\begin{figure}[t]\n\t\\noindent\\resizebox{\\textwidth}{!}{\n \\input{figures\/beamsearch\/bs.tex}\n\t\t}\n \\caption{\\label{fig:beamsearch}Approximation using a beamsearch. All\n possibles next tokens are tried for each candidates, the $S$ (here 3) most\n likely according to $p_\\theta$ are kept. When an End-Of-Sequence token\n (green) is reached, the candidate is held out. At the end, the most likely\n complete sequences are used to construct an approximate distribution,\n through rescaling.}\n\\end{figure}\n\n\nBased on the final samples obtained, we define a probability distribution to use\nas an approximation of $p_{\\theta}$ in \\eqref{eq:RL}. As opposed to\n\\eqref{eq:Rl-sample}, this approximation introduces a bias. It however has the\nadvantage of aligning the training procedure more closely with the testing\nprocedure where only likely samples are going to be decoded. Formally, this\ncorresponds to performing the following approximation of the objective function,\n($\\text{BS}(p_\\theta, S)$ being the $S$ samples returned by a beam search with\nbeam size $S$):\n\\begin{equation}\n \\label{eq:rl-bs}\n \\begin{split}\n & \\mathcal{L}_R(\\theta) \\approx \\sum_{i..N} \\left( \\sum_{\\lambda} q_{\\theta}(\\lambda \\ | \\ioset{k}{i}{1..K})\\ R_i(\\lambda)\\right)\\\\\n & \\text{where } \\qquad q_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K}) = \\begin{cases}\n \\frac{p_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K})}{\\sum_{\\lambda_r \\in \\text{BS}(p_{\\theta}, S)} p_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K})} &\\qquad \\text{ if } \\lambda_r \\in \\text{BS}(p_{\\theta}, S)\\\\\n 0 &\\qquad \\text{otherwise}.\n \\end{cases}\n \\end{split}\n\\end{equation}\nWith this approximation, the support of the distribution $q_{\\theta}$ is much\nsmaller as it contains only the $S$ elements returned by the beam search. As a\nresult, the sum become tractable and no estimator are needed. We can simply\ndifferentiate this new objective function to obtain the gradients of the loss\nwith regards to $p_\\theta$ and use the chain-rule to subsequently obtain\ngradient with regards to $\\theta$ necessary for the optimization.\nNote that if $S$ is large enough to cover all possible programs, we recover the\nobjective of \\eqref{eq:RL}.\n\n\nBased on this more tractable distribution, we can define more complex objective\nfunctions. In program synthesis, we have the possibility to prune out several\npredictions by using the specification. Therefore, we can choose to go beyond\noptimizing the expected reward when sampling a single program and optimize the\nexpected reward when sampling a bag of $C$ programs and keeping the best one.\nThis results in a new objective function: {\\small\n \\begin{equation}\n \\label{eq:rl-div-gen}\n \\theta^{\\star} = \\argmax_{\\theta} \\sum_{i=1..N} \\left( \\sum_{ \\{\\lambda_{1},..,\\lambda_{C}\\} \\in {\\text{BS}(p_{\\theta, s})^C}} \\left[ \\max_{j \\in {1..C}} R_i(\\lambda_j)\\right] \\left( \\prod_{r \\in {1..C}} q_{\\theta}\\left(\\lambda_r \\ | \\ioset{k}{i}{1..K}\\right) \\right) \\right),\n \\end{equation}}\nwhere $q_{\\theta}$ is defined as previously. We argue that by optimizing this\nobjective function, the model gets the capability of ``hedging its bets'' and\nassigning probability mass to several candidates programs, resulting in a higher\ndiversity of outputs.\n\nIn the special case where the reward function only takes values in $\\{0, 1\\}$,\nas it is when we are using correctness as a reward, this can be more easily computed as:\n{\\small\\begin{equation}\n \\label{eq:rl-div-zeroone}\n \\theta^{\\star} = \\argmax_{\\theta} \\sum_{i=1..N} \\left(1 - \\left(\\sum_{\\lambda_r \\in \\text{BS}(p_{\\theta}, S)} [R_i(\\lambda_r) == 0]\\ q_{\\theta}(\\lambda_r \\ | \\ioset{k}{i}{1..K}) \\right)^C \\right)\n \\end{equation}}\nThe derivation leading to this formulation as well as a description on how to\nefficiently compute the more general loss\\eqref{eq:rl-div-gen} can be found in\nappendix \\ref{app:div_details}.\nNote that although this formulation brings the training objective function\ncloser to the testing procedure, it is still not ideal. It indeed makes the\nassumption that if we have a correct program in our bag of samples, we can\nidentify it, ignoring the fact that it is possible to have some incorrect program\nconsistent with the IO pairs partial specification (and therefore not prunable).\nIn addition, this represents a probability where the $C$ programs are sampled\nindependently, which in practice we wouldn't do.\n\n\\section{Model}\n\\subsection{Conditioning on the syntax}\n\\label{subsec:handwritten}\nOne aspect of the program synthesis task is that syntactically incorrect\nprograms can be trivially identified and pruned before making a prediction.\nAs a result, if we use \\texttt{stx} to denote the event that the sampled program\nis syntactically correct, what we care about modeling correctly is\n$p( \\lambda \\ | \\ioset{k}{i}{1..K}, \\mathtt{stx} )$.\nUsing Bayes rule, we can rewrite this:\n\\begin{equation}\n \\label{eq:bayes}\n \\begin{split}\n p\\left( \\ \\lambda \\ |\\ \\ioset{k}{i}{1..K}, \\mathtt{stx}\\right) &\\propto p\\left(\\ \\mathtt{stx}\\ |\\ \\ioset{k}{i}{1..K}, \\lambda\\ \\right) \\times p\\left(\\ \\lambda \\ |\\ \\ioset{k}{i}{1..K}\\ \\right)\\\\\n &\\propto p\\left(\\ \\mathtt{stx}\\ |\\ \\lambda\\ \\right) \\times p\\left(\\ \\lambda \\ |\\ \\ioset{k}{i}{1..K}\\right)\n \\end{split}\n\\end{equation}\nWe drop the conditional dependency on the IO pairs in the second line\nof~(\\ref{eq:bayes}) as the syntactical correctness of the program is independent\nfrom the specification when conditioned on the program. We can do the same operation at the token level, denoting by\n$\\mathtt{stx}_{1 \\dotsc t}$ the event that the sequence of the first $t$ tokens $s_1 \\dotsb s_t$ doesn't\ncontain any syntax error and may therefore be a prefix to a valid program.\n\\begin{equation}\n \\label{eq:tokenbayes}\n \\begin{split}\n p\\left( \\ s_t \\ |\\ s_1 \\dotsb s_{t-1}, \\ioset{k}{i}{1..K}, \\mathtt{stx}_{1 \\dotsc t}\\right) \\propto \\quad& p\\left( \\ \\mathtt{stx}_{1 \\dotsc t}\\ | s_1\\dotsb s_t\\right) \\times \\\\\n & \\quad \\ p\\left(\\ s_t\\ |\\ s_1 \\dotsb s_{t-1}, \\ioset{k}{i}{1..K}\\right)\n \\end{split}\n\\end{equation}\nGiven a grammar, it is possible to construct a checker to determine\nvalid prefixes of programs. Example applications include\ncompilers to signal syntax errors to the user and autocomplete features of\nIntegrated Development Environments (IDEs) to restrict the list of suggested\ncompletions.\n\nThe quantity $ p\\left( \\ \\mathtt{stx}_{1 \\dotsc t}\\ |\\ s_1\\dotsb s_t\\right)$ is\ntherefore not a probability but can be implemented as a deterministic process\nfor a given target programming language. In practice, this is implemented by\ngetting at each timestep a mask \\mbox{$M = \\{-\\inf, 0\\}^{ | \\ \\Sigma \\ |}$}\nwhere $M_j = -\\inf$ if the $j$-th token in the alphabet is not a valid token in\nthe current context, and 0 otherwise. This mask is added to the output of the\nnetwork, just before the Softmax operation that normalizes the output to a\nprobability over the tokens.\n\nConditioning on the syntactical correctness of the programs provides several\nadvantages: First, sampled programs become syntactically correct by\nconstruction. At test time, it allows the beam search to only explore useful\ncandidates. It also ensures that when we are optimizing for correctness, the\nsamples used to approximate the distribution are all going to be valid\ncandidates. Restricting the dimension of the space on which our model is defined\nalso makes the problem simpler to learn.\n\n\n\n\n\\subsection{Jointly learned syntax}\n\\label{subsec:jointGrammar}\nIt may not always be feasible to assume access to a syntax checker. In general,\nwe wish to retain the syntax checker's ability to aggressively prune the search\nin program space without requiring access to the syntax itself. To this end, we\npropose to represent the syntax checker as a neural network module and learn it\njointly. Similar to the base model, we implement learned syntax checking using\nan LSTM $g_\\phi$. Comparing to the decoder LSTM, there are two major\ndifferences:\n\\begin{itemize}\n\\item The syntaxLSTM is conditioned only on the program tokens, not on the IO pairs. This ensures that the learned checker models only the syntax of the language.\n\\item The output of the syntaxLSTM is passed through an elementwise $x \\mapsto -\n \\exp(x)$ activation function and added to the decoder LSTM's output. Similar\n to the mask in Section~\\ref{subsec:handwritten}, the exponential activation\n function allows the syntaxLSTM to output high penalties to any tokens deemed\n syntactically incorrect.\n %\n\\end{itemize}\n\nThe addition of the syntaxLSTM doesn't necessitate any change to the training\nprocedure as it is simply equivalent to a change of architecture. However, in\nthe supervised setting, when we have access to syntactically correct programs,\nwe have the possibility of adding an additional term to the loss\n~(\\ref{eq:mle-obj}) to prevent the model from masking valid programs:\n\\begin{equation}\n \\label{eq:stx_loss}\n \\mathcal{L}_{\\mathtt{syntax}} = - \\sum_{i=1 \\dotsb N} \\sum_{t= 1 \\dotsb L} g_{\\phi}\\left(\\ s^i_t\\ | s^i_1 \\dotsb s^i_{t-1}\\right), \\qquad \\text{ where } \\lambda_i = [s^i_1 , s^i_2 \\dotsb s^i_L]\n\\end{equation}\nThis loss penalizes the syntaxLSTM for giving negative scores to each token\nbelonging to a known valid program. We use the reference programs as example of\nvalid programs when we perform supervised training.\n\n\\section{Experiments}\n\\subsection{The domain: Karel}\n\\label{subsec:exp_domain}\nThe Karel programming language is an educational programming\nlanguage~\\citep{pattis1981karel}, used for example in Stanford CS introductory\nclasses~\\citep{cs106} or in the Hour of Code initiative~\\citep{hoc}. It features\nan agent inside a gridworld (See Figure \\ref{fig:progsynth}), capable of moving\n(\\texttt{move}, \\texttt{turn\\{Left,Right\\}}), modifying world state\n(\\texttt{\\{pick,put\\}Marker}), and querying the state of the nearby environment\nfor its own markers (\\texttt{markerPresent}, \\texttt{noMarkerPresent}) or for\nnatural obstacles (\\texttt{frontIsClear}, \\texttt{leftIsClear},\n\\texttt{rightIsClear}).\n Our goal is to learn to generate a program in the Karel\nDSL given a small set of input and output grids. The language supports for\nloops, while loops, and conditionals, but no variable assignment. Compared to\nthe original Karel language, we only removed the possibility of defining\nsubroutines. The specification for the DSL can be found in\nappendix~\\ref{sec:karel-spec}.\n\nTo evaluate our method, we follow the standard practice~\\citep{RobustFill,\n NeurosymbProgsynth,neelakantan2015neural, balog2016deepcoder} and use a\nsynthetic dataset generated by randomly sampling programs from the DSL. We\nperform a few simple heuristic checks to ensure generated programs have\nobservable effect on the world and prune out programs performing spurious\nactions (e.g. executing a \\texttt{turnLeft} just after a \\texttt{turnRight} for\nexample). For each program, we a set of IO pairs are generated by sampling\nrandom input grids and executing the program on them to obtain the corresponding\noutput grids. A large number of them are sampled and 6 are kept for each\nprogram, ensuring that all conditionals in a program are hit by at least one of\nthe examples. The first 5 samples serve as the specification, and the sixth one\nis kept as held-out test pair. 5000 programs are not used for training, and get\nsplit out between a validation set and a test set.\n\n\nWe represent the input and output elements as grids where each cell in the grid\nis a vector with 16 channels, indicating the presence of elements\n(\\texttt{AgentFacingNorth}, \\texttt{AgentFacingSouth}, $\\cdots$,\n\\texttt{Obstacle}, \\texttt{OneMarkerPresent}, \\texttt{TwoMarkersPresent},\n$\\cdots$). The input and output grids are initially passed through independent\nconvolution layers, before being concatenated and passed through two\nconvolutional residual blocks and a fully connected layer, mapping them to a\nfinal 512-dimensional representation. We run one decoder per IO pair and perform\na maxpooling operation over the output of all the decoders, out of which we\nperform the prediction of the next token. Our models are implemented using the\nPytorch framework~\\citep{pytorch}. Code and data will be made available.\n\n\n\\subsection{Results}\nWe trained a variety of models on the full Karel dataset containing 1-million\nexamples as well as a reduced dataset containing only $10,000$ examples. In\ngeneral, the small dataset serves to help understand the data efficiency of the\nprogram synthesis methods and is motivated by the expected difficulty of\nobtaining many labeled examples in real-world program synthesis domains.\n\nThe Karel DSL was previously used by ~\\citet{metainduction} to study the\nrelative perfomances of a range of methods depending on the available amount of\ndata. The task considered was however different as they attempted to perform\nprogram induction as opposed to program synthesis. Rather than predicting a\nprogram implementing the desired transformation, they simply output a\nspecification of the changes that would result from applying the program so a\ndirect number comparison wouldn't be meaningful.\n\nModels are grouped according to training objectives. As a baseline, we use\n\\textbf{MLE}, which corresponds to the maximum likelihood objective\n(Eq.\\ref{eq:mle-obj}), similar to the method proposed by ~\\citet{RobustFill}.\nUnless otherwise specified, the reward considered for our other methods is\ngeneralization: +1 if the program matches all samples, including the held out one\nand 0 otherwise. \\textbf{RL} uses the expected reward objective\n(Eq.\\ref{eq:RL}), using REINFORCE to obtain a gradient\nestimate (Eq.\\ref{eq:Rl-sample}). \\textbf{RL\\_beam} attempts to solve the proxy\nproblem described by Equation~\\eqref{eq:rl-bs} and \\textbf{RL\\_beam\\_div} the\nricher loss function of Equation~\\eqref{eq:rl-div-gen}.\n\\textbf{RL\\_beam\\_div\\_opt} also optimizes the loss of\nequation~\\eqref{eq:rl-div-gen} but the reward additionally includes a term\ninversly proportional to the number of timesteps it takes for the program to\nfinish executing. All RL models are initialized from pretrained supervised\nmodels.\n\n\\setlength{\\floatsep}{5pt}\n\\begin{table}\n \\small\n \\renewcommand{\\arraystretch}{1}\n \\setlength{\\tabcolsep}{0.7ex}\n \\begin{tabular}{ l@{\\hspace*{3ex}}cc@{\\hspace*{3ex}}cc}\n \\toprule\n & \\multicolumn{2}{c}{\\textbf{Full Dataset}} & \\multicolumn{2}{c}{\\textbf{Small Dataset}} \\\\\n \\cmidrule(lr{3ex}){2-3}\\cmidrule(lr{3ex}){4-5}\n \\textbf{Top-1} & \\multicolumn{1}{l}{\\textbf{Generalization}} & \\multicolumn{1}{l}{\\textbf{Exact Match}} & \\multicolumn{1}{l}{\\textbf{Generalization}} & \\multicolumn{1}{l}{\\textbf{Exact Match}} \\\\\n \\midrule\n \\textbf{MLE} & 71.91 & \\textbf{39.94} & 12.58 & 8.93 \\\\\n \\textbf{RL} & 68.39 & 34.74 & 0 & 0 \\\\\n \\textbf{RL\\_beam} & 75.72 & 8.21 & \\textbf{25.28} & \\textbf{17.63} \\\\\n \\textbf{RL\\_beam\\_div} & 76.20 & 31.25 & 23.72 & 16.31 \\\\\n \\textbf{RL\\_beam\\_div\\_opt} & \\textbf{77.12} & 32.17 & 24.24 & 16.63 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{\\textbf{RL\\_beam} optimization of program correctness results in\n consistent improvements in top-1 generalization accuracy over supervised\n learning \\textbf{MLE}, even though the exact match of recovering the\n reference program drops. The improved objective function results in further\n improvements.}\n \\label{tab:rl_results}\n\\end{table}\n\n\\paragraph{Optimizing for correctness (RL):\\ }\\mbox{}\nResults in Table \\ref{tab:rl_results} show that optimizing for the expected\nprogram correctness consistently provides improvements in top-1 generalization\naccuracy. \\textbf{Top-1 Generalization Accuracy} (Eq. \\ref{eq:generalization})\ndenotes the accuracy of the most likely program synthesized by beam search\ndecoding having the correct behaviour across all input-output examples. We\ndidn't perform any pruning of programs that were incorrect on the 5\nspecification examples. The improved performance of RL methods confirms our\nhypothesis that better loss functions can effectively combat the program aliasing problem.\n\nOn the full dataset, when optimizing for correctness, \\textbf{Exact Match\n Accuracy} decreases, indicating that the RL models no longer prioritize\ngenerating programs that exactly match the references. On the small dataset,\n\\textbf{RL\\_beam} methods improves both exact match accuracy and generalization.\n\nComparing RL\\_beam to standard RL, we note improvements across all levels of\ngeneralization. By better aligning the RL objective with the sampling that\nhappens during beam search decoding, consistent improvements can be made in\naccuracy. Further improvements are made by encouraging diversity in the beam of\nsolutions (RL\\_beam\\_div) and penalizing long running programs\n(RL\\_beam\\_div\\_opt).\n\nIn the settings where little training data is available, RL methods\nshow more dramatic improvements over MLE, indicating that data\nefficiency of program synthesis methods can be greatly improved by\nusing a small number of samples first for supervised training and\nagain for Reinforcement Learning.\n\nAs a side note, we were unable to achieve high performance when\ntraining the RL methods from scratch. The necessity of extensive\nsupervised pretraining to get benefits from Reinforcement Learning\nfine-tuning is well-known in the Neural Machine Translation\nliterature~\\citep{ranzato2015sequence,googleNMT,\n wiseman2016sequence,bahdanau2016actor}.\n\nTable \\ref{tab:beam_results} examines the top-1, top-5, and top-50\ngeneralization accuracy of supervised and RL models. RL\\_beam methods performs\nbest for top-1 but their advantage drops for higher-rank accuracy. Inspection of\ngenerated programs shows that the top predictions all become diverse\nvariations of the same program, up to addition\/removal of no-operations\n(\\texttt{turnLeft} followed by \\texttt{turnRight}, full circle obtained by a\nseries of four \\texttt{turnLeft}). The RL\\_beam\\_div objective helps alleviate\nthis effect as does RL\\_beam\\_div\\_opt, which penalizes redundant programs.\nThis is important as in the task of Program Synthesis, we may not necessarily\nneed to return the most likely output if it can be pruned by our specification.\n\n\\begin{table}\n \\begin{floatrow}\n \\capbtabbox[.4\\textwidth]{%\n \\small\n \\renewcommand{\\arraystretch}{1}\n \\setlength{\\tabcolsep}{0.7ex}\n \\begin{tabular}{ l@{\\hspace*{1ex}}c@{\\hspace*{1ex}}c@{\\hspace*{1ex}}c}\n \\toprule\n \\textbf{Generalization} & Top-1 & Top-5 & Top-50 \\\\\n \\midrule\n \\textbf{MLE} & 71.91 & 79.56 & \\textbf{86.37} \\\\\n \\textbf{RL\\_beam} & 75.72 & 79.29 & 83.49 \\\\\n \\textbf{RL\\_beam\\_div} & 76.20 & 82.09 & 85.86 \\\\\n \\textbf{RL\\_beam\\_div\\_opt} & \\textbf{77.12} & \\textbf{82.17} & 85.38 \\\\\n \\bottomrule\n \\end{tabular}\n }{%\n \\caption{\\label{tab:beam_results}\\textbf{Top-k accuracies: } MLE shows\n greater relative accuracy increases as k increases than RL. Methods\n employing beam search and diversity objectives reduce this accuracy\n gap by encouraging diversity in the beam of partial programs.}\n }\n \\hfill\n \\capbtabbox[.55\\textwidth]{%\n \\small\n \\renewcommand{\\arraystretch}{1}\n \\setlength{\\tabcolsep}{0.7ex}\n \\begin{tabular}{ l@{\\hspace*{3ex}}ccc@{\\hspace*{3ex}}ccc @{\\hspace*{6ex}}ccc @{\\hspace*{3ex}}ccc}\n \\toprule\n \\textbf{Top-1 Generalization} & \\textbf{Full Dataset} & \\textbf{Small Dataset} \\\\\n \\midrule\n \\textbf{MLE} & 71.91 & 12.58 \\\\\n \\textbf{MLE\\_learned} & 69.37 & \\textbf{17.02} \\\\\n \\textbf{MLE\\_handwritten} & 72.07 & 9.81 \\\\\n \\textbf{MLE\\_large} & \\textbf{73.67} & 13.14 \\\\\n \\bottomrule\n \\end{tabular}\n }{%\n \\caption{\\label{tab:syntax_results}\\textbf{Grammar prunes the space of\n possible programs}: On the full dataset, handwritten syntax checking\n MLE\\_handwritten improves accuracy over no grammar MLE, although\n MLE\\_large shows that simply adding more parameters results in even\n greater gains. On the small dataset, learning the syntax MLE\\_learned\n outperforms handwritten grammar and larger models.}\n }\n \\end{floatrow}\n\\end{table}\n\n\\paragraph{Impact of syntax:}\\mbox{}\nWe also compare models according to the use of syntax:\n\\textbf{MLE\\_handwritten} denotes the use of a handwritten syntax\nchecker (Sec \\ref{subsec:handwritten}), \\textbf{MLE\\_learned} denotes\na learned syntax (Sec \\ref{subsec:jointGrammar}), while no suffix\ndenotes no syntax usage. Table \\ref{tab:syntax_results} compares\nsyntax models.\n\nOn the full dataset, leveraging the handwritten syntax leads to\nmarginally better accuracies than learning the syntax or using no\nsyntax. Given access to enough data, the network seems to be capable\nof learning to model the syntax using the sheer volume of training\nexamples.\n\nOn the other hand, when the amount of training data is limited,\nlearning the syntax produces significantly better performance. By\nincorporating syntactic structure in the model architecture and\nobjective, more leverage is gained from small training\ndata. Interestingly, the learned syntax model even outperforms the\nhandwritten syntax model. We posit the syntaxLSTM is free to learn a\nricher syntax. For example, the syntaxLSTM could learn to model the\ndistribution of programs and discourage the prediction of not only\nsyntactically incorrect programs, but also the unlikely ones.\n\nTo control for the extra parameters introduced by the syntaxLSTM, we\ncompare against MLE\\_large, which uses no syntax but features\na larger decoder LSTM, resulting in the same number of parameters as\nMLE\\_learned. Results show that the larger number of parameters is not\nenough to explain the difference in performance, which again indicates\nthe utility of jointly learning syntax.\n\n\\paragraph{Analysis of learned syntax:}\\mbox{}\nSection~\\ref{subsec:jointGrammar} claimed that by decomposing our\nmodels into two separate decoders, we could decompose the learning so\nthat one decoder would specialize in picking the likely tokens given\nthe IO pairs, while the other would enforce the grammar of the\nlanguage. We now provide experimental evidence that this decomposition\nhappens in practice.\n\\setcounter{figure}{1}\n\\begin{wrapfigure}[25]{R}{.4\\textwidth}\n \\floatbox[\\nocapbeside]{table}[\\FBwidth]\n {\\caption{Importance of Syntax}\\label{tab:syntaxDecomp}}\n {\n \\noindent\\resizebox{\\linewidth}{!}{\n \\begin{tabular}{@{}c@{\\hspace*{5ex}}cc@{}}\n \\toprule\n \\twoLines{\\% Synctactically}{Correct}& \\twoLines{Joint}{Model} & \\twoLines{Without}{Learned Syntax}\\\\\n \\midrule\n Amongst Top1 & 100 \\% & 0 \\% \\\\\n Amongst Top5 & 100 \\% & 0 \\% \\\\\n Amongst Top50 & 100 \\% & 0 \\% \\\\\n Amongst Top100 & 99.79 \\% & .04 \\% \\\\\n \\bottomrule\n \\end{tabular}}\n }\n \\vspace{5pt}\n \\ffigbox{\\caption{Syntax Comparison}\\label{fig:grammarComp}}\n {\\begin{subfloatrow}\n \\ffigbox[\\FBwidth][][]{\\caption{Manual}\\label{subfig:manGrammar}}\n {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/manual_syntax.png}}\n \\ffigbox[\\FBwidth][][]{\\caption{Learned}\\label{subfig:learnedGrammar}}\n {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/learned_syntax.png}}\n \\ffigbox[\\FBwidth][][]{\\caption{Diff}\\label{subfig:diffGrammar}}\n {\\includegraphics[width=.3\\textwidth]{figures\/grammar\/diff.png}}\n \\end{subfloatrow}\n \\caption{\\label{fig:grammarComp}Syntax Comparison}\n }\n \\end{wrapfigure}\n\nTable~\\ref{tab:syntaxDecomp} shows the percentage of syntactically\ncorrect programs among the most likely predictions of the \\textbf{MLE\n + learned} model trained on the full dataset. Both columns\ncorrespond to the same set of parameters but the second column doesn't\napply the syntaxLSTM's mask to constrain the decoding process. The\nprecipitous drop in syntax accuracy indicates the extent to which the\nprogram decoder has learned to rely on the syntaxLSTM to produce\nsyntactically correct programs.\n\nFigure~\\ref{fig:grammarComp} compares the syntax masks generated by\nthe learned and handwritten syntax models while decoding Program A in\nFigure~\\ref{fig:progsynth}. (\\ref{subfig:manGrammar}) shows output of\nthe handwritten syntax checker; (\\ref{subfig:learnedGrammar}) shows\nsyntaxLSTM output. White cells indicates tokens that are labeled\nsyntactically correct at each decoding step.\n\nFigure \\ref{subfig:diffGrammar} analyzes the difference between the\nhandwritten and learned syntax masks. White indicates similar output,\nwhich occupies the majority of the visualization. Blue cells\ncorrespond to instances where the syntaxLSTM labeled a token correct\nwhen it actually was syntactically incorrect. This type of error can\nbe recovered if the program decoder predicts those tokens as unlikely.\n\nOn the other hand, red cells indicate the syntaxLSTM predicted a valid\ntoken is syntactically incorrect. This type of error is more dangerous\nbecause the program decoder cannot recover the valid token once it is\ndeclared incorrect. The majority of red errors correspond to tokens\nwhich are rarely observed in the training dataset, indicating that the\nsyntaxLSTM learns to model more than just syntax - it also captures\nthe distribution over programs. Given a large enough dataset of real\nprograms, the syntaxLSTM learns to consider non-sensical and unlikely\nprograms as syntactically incorrect, ensuring that generated programs\nare both syntactically correct and likely.\n\n\n\n\\section{Conclusion}\nWe presented two novel contributions to improve state-of-the-art neural program\nsynthesis techniques. Our first contribution uses Reinforcement Learning to\noptimize for generating any consistent program, which we show helps in improving\ngeneralization accuracy of the learned programs for large training datasets. Our\nsecond contribution incorporates syntax checking as an additional conditioning\nmechanism for pruning the space of programs during decoding. We show that\nincorporating syntax leads to significant improvements with limited training\ndatasets.\n\n\\clearpage\n{\\small\n\\bibliographystyle{plainnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nBesov spaces $B^s_{p,q}(\\mathbb{R}^n)$ were introduced by Besov \\cite{besov}. This scale of spaces has been a favorite through the years, with thousands of references available. Perhaps two of its most interesting features is that many earlier classes of function spaces appears in this scale, as Sobolev spaces, and also that there are many equivalent ways to define $B^s_{p,q}(\\mathbb{R}^n)$, in such way you can pick the more suitable one for your purpose. The reader may see Stein \\cite{stein}, Peetre \\cite{peetre}, Triebel \\cite{tbook} for an introduction of Besov spaces on $\\mathbb{R}^n$. For a historical account on Besov spaces and related topics, see Triebel \\cite{tbook} and the more short but useful Jaffard \\cite{jaffard}, Yuan, Sickel and Yang \\cite{ysy} and Besov and Kalyabin \\cite{besov2}.\\\\ \n\nIn the last decades there were a huge amount of activity on the generalisation of harmonic analysis (see Deng and Han \\cite{deng}), and Besov spaces in particular, to less regular {\\it phase spaces}, replacing $\\mathbb{R}^n$ by something with a poorer structure. It turns out that for {\\it small $s > 0$} and $p,q\\geq 1$, a proper definition demands strikingly weak assumptions. There is a large body of literature that provides a definition and properties of Besov spaces on {\\it homogeneous spaces}, as defined by Coifman and Weiss \\cite{cw}. Those are quasi-metric spaces with a doubling measure, which includes in particular Ahlfors regular metric spaces. We refer to the pionner work of Han and Sawyer \\cite{hs} and Han, Lu and Yang \\cite{han2} for Besov spaces on homogeneous spaces and more recently Alvarado and Mitrea \\cite{sharp} and Koskela, Yang, and Zhou \\cite{koskela2} (in this case for metric measure spaces) and Triebel \\cite{fractal}\\cite{fractal2} for Besov spaces on fractals. There is a long list of examples of homogeneous spaces in Coifman and Weiss \\cite{examples}. The d-sets as defined in Triebel \\cite{fractal} are also examples of homogeneous spaces. \n\\vspace{5mm}\n\n\\noindent {\\it We give a very elementary (and yet pratical) construction of Besov spaces $\\mathcal{B}^s_{p,q}$, with $p,q\\geq 1$ and $0 < s < 1\/p$, for measure spaces endowed with a grid, that is, a sequence of finite partitions of measurable sets satisfying certain mild properties. }\n\\vspace{5mm}\n\nThis construction is close to the martingale Besov spaces as defined by Gu and Taibleson \\cite{martingale}, however we deal with \"nonisotropic splitting\" in our grid without applying the Gu-Taibleson recalibration procedure to our grid, which simplifies the definition and it allows a wider class of examples. \\\\\n\nThe main tool in this work is the concept of {\\it atomic decomposition}. An atomic decomposition is the representation of each function in a space of functions as a (infinite) linear combination of fractions of the so-called {\\bf atoms}. The advantage of atomic decompositions in that the atoms are functions that are often far more regular than a typical element of the space. But a distinctive feature (compared with Fourier series with either Hilbert basis or unconditional basis) is that such atomic decomposition is in general {\\it not} unique. However, in a successful atomic decomposition of a normed space of functions we can attribute a \"cost\" to each possible representation, and the norm of the space is equivalent to the minimal cost (or infimum) among all representations. A function represented by a single atom has norm at most one, so the term \"atom\" seems to be appropriated. \\\\\n\n Coifman \\cite{coifman} introduced the atomic decomposition of the real Hardy space $H^p(\\mathbb{R})$ and Latter \\cite{latter} found an atomic decomposition for $H^p(\\mathbb{R}^n).$ The influential work of Frazier and Jawerth \\cite{fj} gave us an atomic decomposition for the Besov spaces $B^s_{p,q}(\\mathbb{R}^n)$. In the context of homogeneous spaces, we have results by Han, Lu and Yang \\cite{han2} on atomic decomposition of Besov spaces by H\\\"older atoms. \\\\\n \n Closer to the spirit of this work we have the atomic decomposition of Besov space $B^s_{1,1}([0,1])$, with $s\\in (0,1)$, by de Souza \\cite{souzao1} using special atoms, that we call {\\bf Souza's atoms} (see also De Souza \\cite{souzao2} and de Souza, O'Neil and Sampson \\cite{sn}). A Souza's atom $a_J$ on an interval $J$ is quite simple, consisting of a function whose support is $J$ and $a_J$ is {\\it constant} on $J$. \\\\\n \n We also refer to the results on the B-spline atomic decomposition of the Besov space of the unit cube of $\\mathbb{R}^n$ in DeVore and Popov \\cite{spline2} (with Ciesielski \\cite{spline1} as a precursor), that in the case $0< s < 1\/p$ reduces to an atomic decomposition by Souza's atoms, and the work of Oswald \\cite{oswald0} \\cite{oswald} on finite element approximation in bounded polyhedral domains on $\\mathbb{R}^n$.\\\\\n \nOn the study of Besov spaces on $\\mathbb{R}^n$ and smooth manifolds, Souza's atoms may seem to have setbacks that restrict its usefulness. They are not smooth, so it is fair to doubt the effectiveness of atomic decomposition by Souza's atoms to obtain a better understanding of a partial differential equation or to represent data faithfully\/without artifacts, a constant concern in applications of smooth wavelets (see Jaffard, Meyer and Ryan \\cite{jaffard2}). \\\\\n\nOn the other hand, in the study of ergodic properties of piecewise smooth dynamical systems the {\\it transfer operators} plays a huge role. Those operators acts on Lebesgue spaces $L^1(m)$, but they are more useful if one can show it has a (good) action on more regular function spaces. Unfortunately in many cases the transfer operator does {\\it not} preserve neither smoothness nor even continuity of a function. Discontinuities are a fact of life in this setting, and they do not go away as in certain dissipative PDEs, since the positive $1$-eigenvectors of this operator, of utmost importance importance in its study, often have discontinuities themselves. The works of Lasota and Yorke \\cite{ly} and Hofbauer and Keller \\cite{hk3} are landmark results in this direction. See also Baladi \\cite{bb} and Broise \\cite{broise} for more details. Atomic decomposition with Souza's atoms, being the simplest possible kind of atom with discontinuities, might come in handy in these cases. That was a major motivation for this work. \\\\\n\nBesides this fact, in an {\\it abstract} homogeneous space higher order smoothness does not seem to be a very useful concept, since we can define $B^s_{p,q}$ just for small values of $s$, so atomic decompositions by Souza's atoms sounds far more attractive. \\\\\n\n\nIndeed, the simplicity of Souza's atoms allow us to throw away the necessity of a metric\/topological structure on the phase space. We define {\\bf Besov spaces} on every measure space with a non atomic finite measure, provided we endowed it with a {\\it good grid}. A {\\bf good grid} is just a sequence of finite partitions of measurable sets satisfying certain mild properties. We give the definition of Besov space on measure space with a (good) grid in Part II. \\\\\n\n\n \nOn the literature we usually see a known space of functions be described using atomic decomposition. This typically comes after a careful study of such space, and it is often a challenging task. More rare is the path we follow here. We {\\it start} by {\\it defining } the Besov spaces {\\it through} atomic decomposition by Souza's atoms. This construction of the spaces and the study of its basic properties, as its completeness and its compact inclusion in Lebesgue spaces, uses fairly general and simple arguments, and it does not depend on the particular nature of the atoms used, except for very mild conditions on its regularity. In Part I we describe this construction in full generality. \n\\vspace{5mm}\n\n\\noindent {\\it We construct Besov-ish spaces. There are far more general than Besov spaces. In particular the nature of the atoms may change with its location and scale in the phase space and the grid itself can be very irregular.} \n\\vspace{5mm}\n\n\nThe main result in Part I is Proposition \\ref{trans}. Due the generality of the setting, its statement is hopeless technical in nature, however this very powerful result has a simple meaning. Suppose we have an atomic decomposition of a function. If we replace each of those atoms by a combination of atoms (possibly of a different class) in such way we do not change the location of the support and also the \"mass\" of the representation is concentrate pretty much on the same original scale, then we obtain a new atomic decomposition, and the cost of this atomic decomposition can be estimate by the cost of the original atomic decomposition. We will use this result many times along this work. This is obviously connected with {\\it almost diagonal operators} as in Frazier and Jawerth \\cite{fj}\\cite{fj2}.\\\\\n\nIn Part II we also offer a detailed analysis of the Besov spaces defined there. Since we defined it using combinations of Souza's atoms, it is not clear {\\it a priori} how rich are those spaces. So \n\\vspace{5mm}\n\n\\noindent {\\it We give a bunch of alternative characterisations of these Besov spaces. We show that using more flexible classes of atoms (piecewise H\\\"older atoms, $p$-bounded variation atoms and even Besov atoms with higher regularity), we obtain the same Besov space. This often allows us to easily verify if a given function belongs to $B^s_{p,q}$. We also have a mean oscillation characterisation in the spirit of Dorronsoro \\cite{mo} and Gu and Taibleson \\cite{martingale}, and we also obtained another one using Haar wavelets.}\n\\vspace{5mm} \n\n\n Haar wavelets were introduced by Haar \\cite{haar} in the real line. The elegant construction of unbalanced Haar wavelets in general measure spaces with a grid by Girardi and Sweldens \\cite{gw} will play an essential role here. If in general homogeneous spaces the Calder\\'on reproducing formula appears to be the bit of harmonic analysis that survives in it and it allows to carry out the theory, in finite measure spaces with a good grid (and in particular {\\it compact } homogenous spaces) full-blown Haar systems are alive and well. Recently a Haar system was used by Kairema, Li, Pereyra and Ward \\cite{ka} to study dyadic versions of the Hardy and BMO spaces in homogeneous spaces. \\\\\n \n We also provided a few applications in part III. In particular we study the boundedness of pointwise multipliers acting in the Besov space. Since it is very easy to multiply Souza's atoms, the proofs of these results are very short and easy to understand, generalising some of the results for Besov spaces in $\\mathbb{R}^n$ by Triebel\\cite{multi} and Sickel \\cite{sickel}. We also study the boundedness of left composition in Besov spaces of measure spaces with a grid, similar to some results for $B^s_{p,q}(\\mathbb{R}^n)$ in Bourdaud and Kateb \\cite{k1} (see also Bourdaud and Kateb \\cite{k0}\\cite{kateb1}).\\\\\n\n\nIt may come as a surprise to the reader that Besov spaces on compact homogeneous spaces as defined by Han, Lu and Yang \\cite{han2} (and in particular Gu-Taibleson recalibrated martingale Besov spaces \\cite{martingale}) are indeed {\\it particular cases} of Besov spaces defined here, provided $0< s< 1\/p$ and $s$ is small. We show this in a forthcoming work \\cite{smania-homo}.\n\\vspace{5mm} \n\n\n\n\n\n\n\\section{Notation} We will use $C_1, C_2, \\dots...$ for positive constants and $\\lambda_1, \\lambda_2, \\dots$ for positive constants smaller than one. Whenever a (non-generic) constant is defined, it appears in blue. Every other appearance of the same constant appears in red, and if you click on it the link will send you to its definition. \n\n\n\\begin{table}[h]\n \\centering\n \\caption{Most important notation\/symbols\/constants}\n \\label{tab:table1}\n \\begin{tabular}{cc}\n \\toprule\n Symbol & Description\\\\\n \\midrule\n $I$ & phase space\\\\\n $m$ & finite measure in the phase space $I$\\\\\n $a_P, b_P$ & an atom supported on P\\\\\n $\\mathcal{A}$ & a class of atoms \\\\\n $\\mathcal{A}(Q)$ & set of atoms of class $\\mathcal{A}$ supported on $Q$ \\\\\n $\\mathcal{A}^{sz}_{s,p}$& class of $(s,p)$-Souza's atoms \\\\\n $\\mathcal{A}^{h}_{s,\\beta,p}$& class of $(s,\\beta,p)$-H\\\"older atoms \\\\\n $\\mathcal{A}^{bv}_{s, \\beta,p}$& class of $(s,\\beta,p)$-bounded variation atoms \\\\\n $\\mathcal{A}^{bs}_{s,\\beta,p,q} $ & class of $(s,\\beta,p,q)$-Besov's atoms \\\\\n $\\mathcal{P}$ & grid of $I$\\\\\n $\\mathcal{P}^k$ & family subsets of $I$ at the $k$-th level of $\\mathcal{P}$\\\\\n $\\Crr{menor} \\leq \\Crr{maior}$ & describes geometry of the grid $\\mathcal{P}$\\\\\n $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ & (s,p,q)-Banach space defined by the class of atoms $\\mathcal{A}$\\\\\n $\\mathcal{B}^s_{p,q}$ or $\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})$ & (s,p,q)-Banach space defined by Souza's atoms\\\\\n $P, Q, W$ & elements of the grid $\\mathcal{P}$\\\\\n $L^p$ or $L^p(m)$ & Lesbesgue spaces of $(I,\\mathbb{A}, m)$ \\\\\n $|\\cdot|_p$ & norm in $L^p$, $p \\in (0,\\infty].$ \\\\\n $p'$ & $1\/p+ 1\/p'=1$, with $p\\in [1,\\infty]$. \\\\\n $\\rho$ & $\\min\\{1,p,q\\}$.\\\\\n $\\hat{t}$ & $\\max\\{t,1\\}.$\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\vspace{1cm}\n\n\\newpage \n\n\\centerline{ \\bf I. DIVIDE AND RULE.}\n\\addcontentsline{toc}{chapter}{\\bf I. DIVIDE AND RULE.}\n\\vspace{1cm}\n\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent In Part I. we are going to assume $s > 0$, $p \\in (0,\\infty)$ and $q\\in (0,\\infty].$ \\end{minipage} \n}\n\\ \\\\ \\\\\n\n\\section{Measure spaces and grids}\\label{partition} Let $I$ be a measure space with a $\\sigma$-algebra $\\mathbb{A}$ and $m$ be a measure on $(I,\\mathbb{A})$, $m(I)< \\infty$. Given a measurable set $J\\subset I$ denote $|J|=m(J)$. We denote the Lebesgue spaes of $(I,\\mathbb{A},m)$ by $L^p$. A {\\bf grid} is a sequence of finite families of measurable sets with positive measure $\\mathcal{P}= (\\mathcal{P}^k)_{k\\in \\mathbb{N}}$, so that at least one of these families is non empty and\n\\begin{itemize}\n\\item[${\\Cll[G]{fin}}$.] Given $Q\\in \\mathcal{P}^k$, let\n$$\\Omega_Q^k=\\{ P \\in \\mathcal{P}^k\\colon \\ P\\cap Q\\neq \\emptyset \\}.$$\nThen\n$$\\Cll{mult1}=\\sup_k \\sup_{Q \\in \\mathcal{P}^k} \\# \\Omega_Q^k< \\infty.$$\n\\end{itemize}\n\n Define $|\\mathcal{P}^k|=\\sup \\{|Q|\\colon Q \\in \\mathcal{P}^k\\}$. To simplify the notation we also assume that $P\\neq Q$ for every $P\\in \\mathcal{P}^i$ and $Q\\in \\mathcal{P}^j$ satisfying $i\\neq j$. We often abuse notation using $\\mathcal{P}$ for both $(\\mathcal{P}^k)_{k\\in \\mathbb{N}}$ and $\\cup_k \\mathcal{P}^k$. \n \\begin{remark} \n There are plenty of examples of spaces with grids. Perhaps the simplest one is obtained considering $[0,1)$ with the Lebesgue measure and the {\\bf dyadic grid} $\\mathcal{D}=(\\mathcal{D}^k)_k$ given by\n $$\\mathcal{D}^k=\\{[i\/2^k,(i+1)\/2^k), \\ 0\\leq i < 2^k \\}.$$\n We can also consider the dyadic grid $\\mathcal{D}_n=(\\mathcal{D}^k_n)_k$ of $[0,1)^n$, endowed with the Lebesgue measure, given by\n $$\\mathcal{D}^k_n=\\{J_1\\times\\cdots \\times J_n, \\ with \\ J_i\\in \\mathcal{D}^k \\}.$$\n and also the corresponding $d$-adic grids replacing $2^k$ by $d^k$ in the above definitions. The above grids are somehow special since they are nested sequence of partitions of the phase space $I$ and all elements on the same level has exactly the same measure. \n \\end{remark}\n \n \\begin{remark} Indeed, any measure space with a finite non-atomic measure can be endowed with a grid made of a nested sequence of partitions and such that all elements on the same level has exactly the same measure, since any such measure space is measure-theoretically the same that a finite interval with the Lebesgue measure.\n \\end{remark}\n \n \\begin{remark} If we consider a (quasi)-metric space $I$ with a finite measure $m$, we would like to construct \"nice\" grids on $(I,m)$. It turns out that if $(I,m)$ is a homogeneous space one can construct a nested sequence of partitions of the phase space $I$ and all elements on the same level are open subsets and have \"essentially\" the same measure. This is an easy consequence of a famous result by Christ \\cite{christ}. See \\cite{smania-homo}.\n \\end{remark}\n \n \\begin{remark} One can constructs grids for smooth compact manifolds and bounded polyhedral domains in $\\mathbb{R}^n$ using successive subdivisions of an initial triangulation of the domain (see for instance Oswald \\cite{oswald0}\\cite{oswald} ).\n \\end{remark}\n \n\n \n \n\n\n\\section{A bag of tricks.}\n\n Following closely the notation of Triebel \\cite{fractal}, consider the set $\\ell_q(\\ell_p^{\\mathcal{P}})$ of all indexed sequences $$x=(x_{P})_{P\\in \\mathcal{P}},$$ with $x_{P}\\in \\mathbb{C}$, satisfying \n$$|x|_{\\ell_q(\\ell_p^{\\mathcal{P}})}= \\Big( \\sum_k \\big(\\sum_{P\\in \\mathcal{P}^k} |x_{P}|^p \\big)^{q\/p} \\Big)^{1\/q} < \\infty,$$\nwith the usual modification when $q=\\infty$. Then $(\\ell_q(\\ell_p^{\\mathcal{P}}),|\\cdot|_{\\ell_q(\\ell_p^{\\mathcal{P}})})$ is a complex $\\rho$-Banach space with $\\rho= \\min \\{1,p,q\\}$, that is, $d(x,y)=|x-y|_{\\ell_q(\\ell_p^{\\mathcal{P}})}^\\rho$ is a complete metric in $\\ell_q(\\ell_p^{\\mathcal{P}}).$\n\n\nThe following is a pair of arguments we will use along this paper to estimate norms in $\\ell_p$ and $\\ell_q(\\ell_p^{\\mathcal{P}})$. Those are very elementary, and we do not claim any originality. We collect them here to simplify the exposition. The reader can skip this for the cases $p, q \\geq 1$, when the results bellow reduce to the familiar H\\\"older's and Young's inequalities. Their proofs were mostly based on \\cite[Proof of Theorem 3.1]{fj}. Recall that for $t \\in (0,\\infty]$ we defined $\\hat{t}=\\max \\{ 1, t\\}$.\n\n\\begin{proposition}[H\\\"older-like trick] \\label{holder}Let $t\\in (0, \\infty)$ and $q\\in (0,\\infty]$. Let $a=(a_k)_k, b=(b_k)_k,c=(c_k)_k$ nonnegative sequences such that for every $k$\n$$a_k^{1\/\\hat{t}} \\leq C^{1\/\\hat{t}} b_k^{1\/\\hat{t}}c_k^{1\/\\hat{t}}.$$\nThen if $q< \\infty$ we have\n$$(\\sum_k a_k^{1\/\\hat{t}})^{\\hat{t}\/t} \\leq C^{1\/t} \\Cll{co}(t,q,b)\\Big( \\sum_k c_k^{q\/t}\\Big)^{1\/q},$$\nand if $q=\\infty$ \n$$(\\sum_k a_k^{1\/\\hat{t}})^{\\hat{t}\/t} \\leq C^{1\/t} \\Crr{co}(t,q,b)\\sup_k c_k^{1\/t}.$$\nwhere \n\\begin{itemize} \n\\item[A.] If $t\\geq 1$ and $q\\geq 1$ then $\\Crr{co}(t,q,b)=(\\sum_k b_k^{q'\/t})^{1\/q'} $ if $q > 1$, \\\\ and $\\Crr{co}(t,1,b)=\\sup_k b_k^{1\/t}$ if $q=1$.\n\\item[B.] If $t\\geq 1$ and $q\\leq 1$ then $\\Crr{co}(t,q,b)=\\sup_k b_k^{1\/t}$.\n\\item[C.] If $t< 1$ and $q\/t\\geq 1$ then $\\Crr{co}(t,q,b)=(\\sum_k b_k^{(q\/t)'})^{1\/(t(q\/t)')} $ if $q< t$, \\\\ and \n$\\Crr{co}(t,q,b)=\\sup_k b_k^{1\/t}$ if $q=t$. \n\\item[D.] If $t< 1$ and $q\/t< 1$ then $\\Crr{co}(t,q,b)= \\sup_k b_k^{1\/t} $.\n\\end{itemize}\n\n\\end{proposition} \n\\begin{proof} We have \n\n\\noindent {\\it Case A.} If $t\\geq 1$ and $q\\geq 1$, by the H\\\"older inequality for the pair $(q,q')$\n$$\\sum_{k} a_k^{1\/\\hat{t}}=\\sum_k a_k^{1\/t} \\leq C^{1\/t} (\\sum_k b_k^{q'\/t})^{1\/q'} \\big( \\sum_k c_k^{q\/t}\\big)^{1\/q}.$$\n\n\n\\noindent {\\it Case B.} If $t\\geq 1$ and $q\\leq 1$ then the triangular inequality for $|\\cdot|^q$ implies\n\\begin{eqnarray*}\n(\\sum_{k} a_k^{1\/\\hat{t}})^q&=&(\\sum_k a_k^{1\/t})^q \\leq C^{q\/t} \\big( \\sum_k b_k^{1\/t}c_k^{1\/t}\\big)^q \\\\\n&\\leq& C^{q\/t} \\sum_k \\big( b_k^{1\/t}c_k^{1\/t}\\big)^q \\\\\n&\\leq& C^{q\/t} (\\sup_k b_k^{q\/t})\\Big( \\sum_kc_k^{q\/t}\\Big)\n\\end{eqnarray*}\n\n\\noindent {\\it Case C.} If $t< 1$ and $q\/t \\geq 1$ then $\\hat{t}\/t=1\/t$ and by the H\\\"older inequality for the pair $(q\/t,(q\/t)')$\n$$\\sum_{k} a_k^{1\/\\hat{t}}\\leq C (\\sum_k b_k^{(q\/t)'})^{1\/(q\/t)'} \\big( \\sum_k c_k^{q\/t}\\big)^{t\/q},$$\n\n\\noindent {\\it Case D.} if $t< 1$ and $q\/t< 1$ then using the triangular inequality for $|\\cdot|^{q\/t}$\n\\begin{align*} (\\sum_k a_k^{1\/\\hat{t}})^{q\/t} &\\leq C^{q\/t} \\big(\\sum_k b_k c_k\\big)^{q\/t}\\\\\n&\\leq C^{q\/t} \\sum_k \\big(b_k c_k\\big)^{q\/t}\\\\\n&\\leq C^{q\/t} (\\sup_k b_k^{q\/t}) \\Big( \\sum_k c_k^{q\/t}\\Big). \\end{align*} \n\\end{proof}\n\n\\begin{proposition}[Convolution trick]\\label{young} Let $p,q\\in (0, \\infty)$. Let $a=(a_k)_{k\\in \\mathbb{Z}}, b=(b_k)_{k\\in \\mathbb{Z}},c=(c_k)_{k\\in \\mathbb{Z}}\\geq 0$ be such that for every $k$\n$$a_k^{1\/\\hat{p}} \\leq C^{1\/\\hat{p}} \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{1\/\\hat{p}}c_i^{1\/\\hat{p}}.$$\nThen\n$$(\\sum_k a_k^{q\/p})^{1\/q} \\leq C^{1\/p} \\Crr{co2}(p,q,b) \\Big( \\sum_k c_k^{q\/p}\\Big)^{1\/q},$$\nwhere $\\Cll{co2}\\geq 1$ satisfies \n\\begin{itemize} \n\\item[A.] If $p\\geq 1$ and $q\\geq 1$ then $\\Crr{co2}(p,q,b)= \\sum_{k\\in \\mathbb{Z}} b_k^{1\/p}$. \n\\item[B.] If $p\\geq 1$ and $q\\leq 1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k^{q\/p})^{1\/q}$.\n\\item[C.] If $p< 1$ and $q\/p\\geq 1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k)^{1\/p}$.\n\\item[D.] If $p< 1$ and $q\/p< 1$ then $\\Crr{co2}(p,q,b)=(\\sum_{k\\in \\mathbb{Z}} b_k^{q\/p})^{1\/q}$\n\\end{itemize}\n\n\\end{proposition} \n\\begin{proof} We have\n\n\n\\noindent {\\it Case A.} If $p\\geq 1$ and $q\\geq 1$, by the Young's inequality for the pair $(1,q)$ \n$$(\\sum_k a_k^{q\/p})^{1\/q} \\leq C^{1\/p} \\big( \\sum_k b_k^{1\/p}\\big) \\Big( \\sum_k c_k^{q\/p}\\Big)^{1\/q}.$$\n\n\\noindent {\\it Case B.} If $p\\geq 1$ and $q\\leq 1$ then the triangular inequality for $|\\cdot|^q$ and the Young's inequality for the pair $(1,1)$ imply \n\\begin{eqnarray*}\n\\sum_{k} a_k^{q\/p}&=& C^{q\/p} \\sum_k \\big( \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{1\/p}c_i^{1\/p} \\big)^q \\\\\n&\\leq&C^{q\/p} \\sum_k \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{q\/p}c_i^{q\/\\hat{p}} \\\\\n&\\leq& C^{q\/p} (\\sum_k b_k^{q\/p})\\Big( \\sum_k c_k^{q\/p}\\Big)\n\\end{eqnarray*}\n\n\\noindent {\\it Case C.} If $p< 1$ and $q\/p \\geq 1$ then by the Young's inequality for the pair $(1, q\/p)$\n$$(\\sum_{k} a_k^{q\/p})^{p\/q} \\leq C (\\sum_k b_k) \\big( \\sum_k c_k^{q\/p}\\big)^{p\/q}.$$\n\n\\noindent {\\it Case D.} If $p< 1$ and $q\/p< 1$ then using the triangular inequality for $|\\cdot|^{q\/p}$ and the Young's inequality for the pair $(1,1)$ \n\\begin{eqnarray*} \\sum_k a_k^{q\/p}&\\leq& C^{q\/p} \\sum_k \\big(\\sum_{i\\in \\mathbb{Z}} b_{k-i} c_i\\big)^{q\/p}\\\\\n&\\leq& C^{q\/p} \\sum_k \\sum_{i\\in \\mathbb{Z}} b_{k-i}^{q\/p} c_i^{q\/p}\\\\\n&\\leq& C^{q\/p} (\\sum_k b_k^{q\/p}) \\Big( \\sum_k c_k^{q\/p}\\Big).\n\\end{eqnarray*} \n\\end{proof}\n\n\\section{Atoms}\n\n Let $\\mathcal{P}$ be a grid. Let $p \\in [1,\\infty), u \\in [1,\\infty]$, and $s> 0$. A family of {\\bf atoms } associated with $\\mathcal{P}$ of type $(s,p,u)$ is an indexed family $\\mathcal{A}$ of pairs $(\\mathcal{B}(Q),\\mathcal{A}(Q))_{Q\\in \\cup_k \\mathcal{P}^k}$ where \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{banach}}$.] $\\mathcal{B}(Q)$ is a complex Banach space contained in $L^{pu}$.\n\\item[${\\Cll[A]{suporte}}$.] If $\\phi \\in \\mathcal{B}(Q)$ then $\\phi(x)=0$ for every $x \\not\\in Q$.\n\\item[${\\Cll[A]{convex}}$.] $\\mathcal{A}(Q)$ is a convex subset of $\\mathcal{B}(Q)$ such that $\\phi \\in \\mathcal{A}(Q)$ if and only if $\\sigma~\\phi~\\in~\\mathcal{A}(Q)$ for every $\\sigma \\in \\mathbb{C}$ satisfying $|\\sigma|=1$.\n\\item[${\\Cll[A]{atom}}$.] We have $$|\\phi|_{pu} \\leq |Q|^{s-\\frac{1}{u'p}}$$ for every $\\phi \\in \\mathcal{A}(Q)$. \n\\end{itemize}\n\n We will say that $\\phi \\in \\mathcal{A}(Q)$ is an $\\mathcal{A}$-atom of type $(s,p,u)$ supported on $Q$ and that $\\mathcal{B}(Q)$ is the local Banach space on $Q$. Sometimes we also assume \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{compact}}$.] For every $Q\\in \\mathcal{P}$ we have that $\\mathcal{A}(Q)$ is a compact subset in the strong topology of $L^p$.\\\\\n\\end{itemize}\nor\n\\begin{itemize}\n\\item[${\\Cll[A]{compactw}}$.] We have $p\\in [1,\\infty)$ and every $Q\\in \\mathcal{P}$ the set $\\mathcal{A}(Q)$ is a sequentially compact subset in the {\\it weak } topology of $L^p$.\\\\\n\\end{itemize}\nor even \\\\\n\\begin{itemize}\n\\item[${\\Cll[A]{finite}}$.] For every $Q\\in \\mathcal{P}$ we have that $\\mathcal{B}(Q)$ is finite dimensional and $\\mathcal{A}(Q)$ contains a neighborhood of $0$ in $\\mathcal{B}(Q)$.\\\\\n\\end{itemize}\n \n \n We provide examples of classes of atoms in Section \\ref{secatom}.\n\n\\section{Besov-ish spaces} \n\n Let $p \\in (0,\\infty)$, $u \\in [1,\\infty]$, $q \\in (0,\\infty]$, $s> 0$, $\\mathcal{P}=(\\mathcal{P}^k)_{k\\geq 0}$ be a grid and let $\\mathcal{A}$ be a family of atoms of type $(s,p,u)$. We will also assume that \n\\begin{itemize}\n\\item[${\\Cll[G]{sum}}.$] We have\n$$\\Cll{decaypart} = \\Crr{co}(p,q, (|\\mathcal{P}^k|^s)_k) < \\infty.$$\nand\n$$\\lim_k |\\mathcal{P}^k|=0.$$\n\\end{itemize}\nThe {\\bf Besov-ish space} $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$ is the set of all complex valued functions $g \\in L^p$ that can be represented by an absolutely convergent series on $L^p$\n\\begin{equation} \\label{rep} g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q\\end{equation}\nwhere $a_Q$ is in $\\mathcal{A}(Q)$, $s_Q \\in \\mathbb{C}$ and with finite {\\bf cost }\n\\begin{equation} \\label{rep2} \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |s_Q|^p)^{q\/p} \\big)^{1\/q} < \\infty.\\end{equation}\nNote that the inner sum in (\\ref{rep}) is finite. By absolutely convergence in $L^p$ we mean that \n$$\\sum_{k=0}^{\\infty} \\big| \\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q \\big|_p^{p\/\\hat{p}} < \\infty.$$ \nThe series in (\\ref{rep}) is called a {\\bf $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$-representation} of the function $g$. \nDefine\n$$|g|_{\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})} = \\inf \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |s_Q|^p )^{q\/p} \\big)^{1\/q} ,$$\nwhere the infimum runs over all possible representations of $g$ as in (\\ref{rep}). \n\nQuite often along this work, when it is obvious the choice of the measure space $I$ and\/or the grid $\\mathcal{P}$ we will write either $\\mathcal{B}^s_{p,q}(\\mathcal{P}, \\mathcal{A})$ or even $\\mathcal{B}^s_{p,q}( \\mathcal{A})$ instead of $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$. Whenever we write just $\\mathcal{B}^s_{p,q}$ it means that we choose the Souza's atoms $\\mathcal{A}^{sz}_{s,p}$, with parameters $s$ and $p$, a class of atoms we properly define in Section \\ref{souzaa}. \n\n\n\n\n\n\n\\begin{proposition}\\label{lp} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Let $t \\in (0,\\infty)$ be such that \n\\begin{equation}\\label{c11} s-\\frac{1}{p}+\\frac{1}{t} \\geq 0, \\ p \\leq t \\leq pu\\end{equation} \nand suppose\n\\begin{equation}\\label{decay} \\Cll{kt}=\\Crr{mult1}^{1+1\/t} \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) < \\infty,\\end{equation} \nThen for every coefficients $s_Q$ satisfying (\\ref{rep2}) and every $\\mathcal{A}$-atoms $a_Q$ on $Q$ the series (\\ref{rep}) converges absolutely on $L^t$. Indeed \n\\begin{align}\\label{flp}\n|\\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q|_t&\\leq \\Crr{mult1}^{1+1\/t} |\\mathcal{P}^k|^{s-\\frac{1}{p} +\\frac{1}{t}} \\Big( \\sum_{Q \\in \\mathcal{P}^k} |s_Q|^p \\Big)^{1\/p}\n\\end{align} \nand\n\\begin{equation}\\label{inclulp} |g|_{t} \\leq \\Crr{kt} | \\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}.\\end{equation}\n\\end{proposition}\n\\begin{proof} Firstly note that if $p\\leq t \\leq pu$\n\\begin{align*} |a_P|^t_t &= \\int |a_P(x)|^t 1_P\\ dm(x)\\\\\n&\\leq ||a_P|^t|_{\\frac{pu}{t}} |1_P|_{\\frac{pu}{pu-t}} \\\\\n&\\leq |a_P|_{pu}^t |P|^{\\frac{pu-t}{pu}}\\\\\n&\\leq |P|^{ts-\\frac{t}{u'p}} |P|^{\\frac{pu-t}{pu}} =|P|^{t(s-\\frac{1}{p} +\\frac{1}{t})}.\\end{align*} \nConsequently\n\\begin{align}\n|\\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q|_t^{t}&\\leq \\int |\\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q|^t \\ dm\\nonumber \\\\\n&\\leq \\sum_{Q \\in \\mathcal{P}^k} \\int_Q |\\sum_{P \\in \\mathcal{P}^k} s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq \\sum_{Q \\in \\mathcal{P}^k} \\int_Q |\\sum_{P \\in \\Omega_Q^k} s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t} \\sum_{Q \\in \\mathcal{P}^k} \\int_Q \\sum_{P \\in \\Omega_Q^k} |s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t} \\sum_{Q \\in \\mathcal{P}^k} \\sum_{P \\in \\Omega_Q^k} \\int |s_P a_P|^t \\ dm \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t}\\sum_{Q \\in \\mathcal{P}^k} \\sum_{P \\in \\Omega_Q^k} \\int |s_P|^t |a_P|^t_t \\ dm \\ \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t} \\sum_{Q \\in \\mathcal{P}^k} \\sum_{P \\in \\Omega_Q^k} |P|^{t(s-\\frac{1}{p} +\\frac{1}{t})} |s_P|^t \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t+1} \\sum_{P \\in \\mathcal{P}^k} |P|^{t(s-\\frac{1}{p} +\\frac{1}{t})} |s_P|^t \\ \\nonumber \\\\\n&\\leq \\Crr{mult1}^{t+1} |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} \\sum_{P \\in \\mathcal{P}^k} |s_P|^t \\ \\nonumber \n\\end{align} \nBy Proposition \\ref{holder} (H\\\"older-like trick) and $p\\leq t$ we have \n\\begin{align}\n|\\sum_k \\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q|_t &\\leq \\Big( \\sum_k\\big( |\\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q|_t^{t}\\big)^{1\/\\hat{t}} \\Big)^{\\hat{t}\/t} \\nonumber \\\\ \n&\\leq \\Crr{mult1}^{1+\\frac{1}{t}} \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) \\Big( \\sum_k \\big( \\sum_{P \\in \\mathcal{P}^k} |s_P|^t \\big)^{q\/t}\\Big)^{1\/q}. \\nonumber \\\\\n&\\leq \\Crr{mult1}^{1+\\frac{1}{t}} \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) \\Big( \\sum_k \\big( \\sum_{P \\in \\mathcal{P}^k} |s_P|^p \\big)^{q\/p}\\Big)^{1\/q}. \\nonumber \n\\end{align} \n\\end{proof}\n\\begin{remark}\\label{sharp} Note that due ${\\Crr{sum}}$ if $t=p$ then $\\Crr{kt} <\\infty$. Sometimes it is convenient to use sharper estimates than (\\ref{flp}) and (\\ref{inclulp}) replacing $|\\mathcal{P}^k|$ by the sequence \n$$C^k =\\max \\{ |Q|\\colon \\ Q\\in \\mathcal{P}^k, \\ s_Q\\neq 0\\}.$$\nFor instance, if $s_Q=0$ for every $Q\\in \\mathcal{P}^k$ with $k\\leq N$, then we can replace $ \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})} )_k ) $ by $\\Crr{co}(t,q, ( |\\mathcal{P}^k|^{t(s-\\frac{1}{p} +\\frac{1}{t})}1_{(N,\\infty)}(k) )_k ) $ in (\\ref{decay}). Here $1_{(N,\\infty)}$ denotes the indicator function of the set $(N,\\infty)$.\n\\end{remark}\n\n\n\n\n\\begin{proposition} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Then $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ is a complex linear space and $|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is a $\\rho$-norm, with $\\rho=\\min\\{1,p,q\\}$. Moreover the linear inclusion\n$$\\imath\\colon (\\mathcal{B}^s_{p,q}(\\mathcal{A}),|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})\\rightarrow (L^p,|\\cdot|_p)$$\nis continuous. \\end{proposition} \n\\begin{proof} Le $f , g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$. Then there are $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations \n$$ f = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k} s_Q' a_Q' \\ and \\ g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q.$$\nLet $sgn \\ 0 =0$ and $sgn \\ z =z\/|z|$ otherwise. Of course\n\\begin{equation} \\label{quo} \\sum_{Q \\in \\mathcal{P}^k} c_Q b_Q = \\sum_{Q \\in \\mathcal{P}^k} s_Q' a_Q' + \\sum_{Q \\in \\mathcal{P}^k} s_Q a_Q,\\end{equation} \nwhere\\footnote{We don't need to worry so much if $|s_Q| + |s_Q'|=0$, since in this case $c_Q=0$ and we can choose $b_Q$ to be an arbitrary atom (for instance $a_Q$) in such way that (\\ref{quo}) holds. For this reason we are not going to explicitly deal with similar situations ( that is going to appear quite often) along the paper.} \n$$b_Q = \\frac{|s_Q'|}{|s_Q| + |s_Q'|} sgn(s_Q') a_Q' + \\frac{|s_Q|}{|s_Q| + |s_Q'|} sgn(s_Q') a_Q,$$\nand\n$$c_Q = |s_Q'| + |s_Q|.$$\nNote that $sgn(s_Q') a_Q', sgn(s_Q) a_Q$ are atoms due ${\\Crr{convex}}$. So by ${\\Crr{convex}}$ we have that $b_Q$ is also an atom, since it is a convex combination of atoms. In particular \n$$\\sum_k \\sum_{Q \\in \\mathcal{P}^k} c_Q b_Q $$\nconverges absolutely in $L^p$ to $f+g$. It remains to prove that this is a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation of $f+g$. Indeed\n$$\\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p)^{q\/p} \\big)^{\\rho\/q} \\leq \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |s_Q'|^p)^{q\/p} \\big)^{\\rho\/q} + \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |s_Q|^p)^{q\/p} \\big)^{\\rho\/q}.$$\nTaking the infimum over all possible $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations of $f$ and $g$ we obtain\n$$|f+g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho \\leq |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho + |g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}^\\rho.$$\nThe identity $ |\\gamma f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}= |\\gamma| |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is obvious. By Proposition \\ref{lp} we have that if $|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}=0$ then $|f|_p=0$, so $f=0$, so $|\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}$ is a $\\rho$-norm, moreover (\\ref{inclulp}) tell us that $\\imath$ is continuous. \n\\end{proof}\n\\begin{proposition}\\label{compa2} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. Suppose that $g_n$ are functions in $\\mathcal{B}^s_{p,q}(\\mathcal{A})$ with $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations \n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a $\\mathcal{A}$-atom supported on $Q$, satisfying \n\\begin{itemize}\n\\item[i.] There is $C$ such that for every $n$ \n\\begin{equation}\\label{estt} (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C.\\end{equation} \n\\item[ii.] For every $Q\\in \\mathcal{P}$ we have that $s_Q=\\lim_n s_Q^n$ exists. \n\\item[iii.] For every $Q\\in \\mathcal{P}$ there is $a_Q\\in \\mathcal{A}(Q)$ such that \n\\begin{enumerate}\n\\item either the sequence $a_Q^n$ converges to $a_Q$ in the strong topology of $L^p$, or\n\\item we have $p\\in [1,\\infty)$ and $a_Q^n$ weakly converges to $a_Q$.\n\\end{enumerate}\n\\end{itemize}\nthen $g_n$ either strongly or weakly converges in $L^p$, respectively, to $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$, where $g$ has the $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation\n\\begin{equation}\\label{repp} g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q a_Q\\end{equation} \nthat satisfies \n\\begin{equation}\\label{estt2} (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C\\end{equation} \n\\end{proposition}\n\\begin{proof}By (\\ref{estt}) it follows that (\\ref{estt2}) holds and that (\\ref{repp}) is indeed a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation of a function $g$. It remains to prove that $g_n$ converges to $g$ in $L^p$ in the topology under consideration. Given $\\epsilon > 0$, fix $N$ large enough such that \n$$\\Crr{mult1}^{1+1\/p} \\Crr{co}(p,q, ( |\\mathcal{P}^k|^{ps}1_{[N,\\infty)}(k) )_k ) (2C^\\rho+1)^{1\/\\rho}< (\\epsilon\/2)^{\\hat{p}\/p}.$$\nWe can write\n$$g_n-g= \\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q) + \\sum_{k> N} \\sum_{Q\\in {\\mathcal P}^k} c_Q^n b_Q^n,$$\nwhere\n$$b_Q^n = \\frac{|s_Q^n|}{|s_Q^n| + |s_Q|} sgn(s_Q^n) a_Q^n + \\frac{|s_Q|}{|s_Q^n| + |s_Q|} sgn(-s_Q) a_Q,$$\nis an atom in $\\mathcal{A}(Q)$, and \n$$c_Q^n = |s_Q^n| + |s_Q|.$$\nNote that the series in the r.h.s. converges absolutely in $L^p$. Of course\n$$ (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|c_Q^n|^p)^{q\/p})^{1\/q}\\leq (2C^\\rho+1)^{1\/\\rho},$$\nSo by (\\ref{inclulp}) in Proposition \\ref{lp} (see also Remark \\ref{sharp}) we have \n$$\n|\\sum_{k> N} \\sum_{Q \\in \\mathcal{P}^k} c_Q^n b_Q^n|_p\\leq \\Crr{mult1}^{1+1\/p} \\Crr{co}(p,q, ( |\\mathcal{P}^k|^{ps}1_{[N,\\infty)}(k) )_k ) (2C^\\rho+1)^{1\/\\rho} < (\\epsilon\/2)^{\\hat{p}\/p}. \n$$\nIn the case $ii.1$, note that if $n$ is large enough then \n$$| \\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q)|_p^{p\/\\hat{p}}< \\epsilon\/2,$$\nand consequently $|g-g_n|_p^{p\/\\hat{p}} < \\epsilon$. So $g_n$ strongly converges to $g$. \n\n\\noindent In the case $ii.2$, given $\\phi\\in (L^p)^\\star$, with $p\\geq 1$ we have that for $n$ large enough\n$$|\\phi(\\sum_{k\\leq N} \\sum_{Q\\in {\\mathcal P}^k} (s_Q^na_Q^n - s_Qa_Q))|\\leq |\\phi|_{(L^p)^\\star} \\epsilon\/2,$$\nand of course\n$$\n|\\phi(\\sum_{k> N} \\sum_{Q \\in \\mathcal{P}^k} c_Q^n b_Q^n|_p)| \\leq |\\phi|_{(L^p)^\\star} \\epsilon\/2,\n$$\nso $g_n$ weakly converges to $g$ in $L^p$. \n\\end{proof}\n\n\\begin{corollary}\\label{compa1} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$, and \n\\begin{enumerate}\n\\item either ${\\Crr{compact}}$ or\n\\item we have $p\\geq 1$ and ${\\Crr{compactw}}$.\n\\end{enumerate}\nThen\n\\begin{itemize}\n\\item[i.] Let $g_n \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ be such that $|g_n|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq C$ for every $n$. Then there is a subsequence that converges either strongly or weakly in $L^p$, respectively, to some $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ with $|g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq~C$.\n\\item[ii.] In both cases $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a complex $\\rho$-Banach space, with $\\rho=\\min\\{1,p,q\\}$, \n\\item[iii] If ${\\Crr{compact}}$ holds then the inclusion \n$$\\imath \\colon (\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}) \\rightarrow (L^p, |\\cdot|_p)$$\nis a compact linear inclusion. \n\\end{itemize} \n\\end{corollary} \n\\begin{proof}[ Proof of i.] There are $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representations\n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a $\\mathcal{A}$-atom supported on $Q$ and\n\\begin{equation} \\label{sed0} (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{q\/p})^{1\/q}\\leq C+\\varepsilon_n,\\end{equation}\nand $1\\geq \\varepsilon_n\\to 0$.\nIn particular, $|s_Q^n|\\leq C+1$. Since the set $\\cup_k \\mathcal{P}^k$ is countable, by the Cantor diagonal argument, taking a subsequence we can assume that $s_Q^n\\to_n s_Q$ for some $s_Q \\in \\mathbb{C}$. Due ${\\Crr{compact}}$ (${\\Crr{compactw}}$) and the Cantor diagonal argument, we can suppose that $a_Q^n$ strongly (weakly) converges in $L^p$ to some $a_Q \\in \\mathcal{A}(Q)$. We set \n$$g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Qa_Q.$$\nBy Proposition \\ref{compa2} we conclude that $g\\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ with $|g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq C$, and that $g_n$ converges to $g$ in $L^p$.\n\\end{proof}\n\\begin{proof}[Proof of ii.] Let $g_n$ be a Cauchy sequence on $\\mathcal{B}^s_{p,q}(\\mathcal{A})$. By Proposition \\ref{lp} we have that $g_n$ is also a Cauchy sequence in $L^p$. Let $g$ be its limit in $L^p$. By Corollary \\ref{compa1}.i have that $g \\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$. Note that for large $m$ and $n$ \n$$|g_n-g_m|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq \\epsilon,$$\nand $g_n - g_m$ converges to $g_n-g$ in $L^p$, so again by Corollary \\ref{compa1}.i we have that \n$$|g_n-g|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}\\leq \\epsilon,$$\nso $g_n$ converges to $g$ in $\\mathcal{B}^s_{p,q}(\\mathcal{A})$.\n\\end{proof}\n\n\\begin{proof}[Proof of iii.] It follows from i. \\end{proof}\n\nThe proof of the following resukt is quite similar. \n\\begin{corollary}\\label{compa12} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$, and \n\\begin{enumerate}\n\\item either ${\\Crr{compact}}$ or\n\\item we have $p\\geq 1$ and ${\\Crr{compactw}}$.\n\\end{enumerate}\nThen for every $f\\in \\mathcal{B}^s_{p,q}(\\mathcal{A})$ there is a $\\mathcal{B}^s_{p,q}(\\mathcal{A})$-representation\n$$f=\\sum_k \\sum_{P\\in \\mathcal{P}^k} c_P a_P$$\nsuch that \n$$|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}=\\Big( \\sum_k \\big(\\sum_{P\\in \\mathcal{P}^k} |c_P|^p\\big)^{q\/p} \\Big)^{1\/q}.$$\n\\end{corollary}\n\n\nWe refer to Edmunds and Triebel \\cite{et} for more information on compact linear transformations between quasi-Banach spaces.\n\n\\begin{corollary} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$ and ${\\Crr{banach}}$-${\\Crr{atom}}$. If for every $Q\\in \\mathcal{P}$ we have that $\\mathcal{B}(Q)$ is finite-dimensional and $\\mathcal{A}(Q)$ is a closed subset of $\\mathcal{B}(Q)$ then \n $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a $\\rho$-Banach space, with $\\rho=\\min\\{1,p,q\\}$.\n\\end{corollary}\n\\begin{proof} Since all norms are equivalent in $\\mathcal{B}(Q)$ we have that ${\\Crr{atom}}$ implies that $\\mathcal{A}(Q)$ is a closed and bounded subset of $\\mathcal{B}(Q)$, so it is compact. By Corollary \\ref{compa1}.ii it follows that $(\\mathcal{B}^s_{p,q}(\\mathcal{A}), |\\cdot|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})})$ is a $\\rho$-Banach space.\n\\end{proof}\n\n\\section{Scales of spaces} \n\n\\label{escala} Note that a family of atoms $\\mathcal{A}$ of type $(s,p,u)$ induces an one-parameter scale \n$$\\tilde{s}\\rightarrow \\mathcal{A}_{\\tilde{s},p},$$\nwhere $\\mathcal{A}_{\\tilde{s},p}$ is the family of atoms of type $(\\tilde{s},p,u)$ defined by \n$$\\mathcal{A}_{\\tilde{s},p}(Q)= \\{ |Q|^{\\tilde{s}-s}a_Q\\colon \\ a_Q\\in \\mathcal{A}\\}.$$\nMoreover a family of atoms $\\mathcal{A}$ of type $(s,p,\\infty)$ induces a two-parameter scale\n$$(\\tilde{s},\\tilde{p})\\rightarrow \\mathcal{A}_{\\tilde{s},\\tilde{p}},$$\nwhere $\\mathcal{A}_{\\tilde{s},\\tilde{p}}$ is the family of atoms of type $(\\tilde{s},\\tilde{p},\\infty)$ defined by \n$$\\mathcal{A}_{\\tilde{p},\\tilde{s}}(Q)= \\{ |Q|^{\\tilde{s}-s+1\/p-1\/\\tilde{p} }a_Q\\colon \\ a_Q\\in \\mathcal{A}\\}.$$\n\n\n\n\\begin{proposition} \\label{compa} Assume ${\\Crr{fin}}$-${\\Crr{sum}}$. Suppose that the $(s,p,\\infty)$-atoms $\\mathcal{A}$ satisfy ${\\Crr{banach}}$-${\\Crr{atom}}$. Let $0\\leq s < \\tilde{s} $ and $q,\\tilde{q} \\in [1,\\infty]$. Suppose\n$$\\Big( \\sum_{k} |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q} < \\infty.$$\nThen \n\\begin{itemize}\n\\item[A.] We have $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\subset {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ and the inclusion is a continuous linear map.\n\\item[B.] Suppose that also satisfies ${\\Crr{compact}}$. Let $g_n \\in \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})$ be such that $|g_n|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$ for every $n$. Then there is a subsequence that converges in ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ to some $g \\in \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{s,p})$ with $|g|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$. \n\\item[C.] Suppose that also satisfies ${\\Crr{finite}}$. The inclusion $\\imath\\colon \\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\mapsto {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ is a compact linear map. \n\\end{itemize}\n\\end{proposition}\n\\begin{proof} Consider a $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})$-representation \n$$f=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Qa_Q,$$\nSince $a_Q$ is an $\\mathcal{A}_{\\tilde{s},p}$-atom, we have that $b_Q=a_Q|Q|^{s-\\tilde{s}}$ is an $\\mathcal{A}_{s,p}$-atom. In particular, we can write\n$$f=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q|Q|^{\\tilde{s}-s}b_Q.$$\nIf $k\\geq k_0$ then \n\\begin{eqnarray} \n(\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{1\/p} &\\leq& |\\mathcal{P}^k|^{\\tilde{s}-s} (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{1\/p} \\nonumber \\\\ \n&\\leq& |\\mathcal{P}^k|^{\\tilde{s}-s} \\Big(\\sum_{k\\geq k_0 } (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{\\tilde{q}\/p}\\Big)^{1\/\\tilde{q}},\\nonumber\n\\end{eqnarray}\nso \n\\begin{eqnarray} \n&& \\Big( \\sum_{k\\geq k_0} (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\ &\\leq& \\Big( \\sum_{k\\geq k_0} |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q} \\Big(\\sum_{k\\geq k_0 } (\\sum_{Q\\in {\\mathcal P}^k}|s_Q|^p )^{\\tilde{q}\/p}\\Big)^{1\/\\tilde{q}}.\\label{uyuy}\n\\end{eqnarray}\n\n\\noindent {\\it Proof of A.} In particular taking $k_0=0$ we conclude that $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})\\subset {\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$ and\n$$|f|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{p,s})}\\leq \\Big( \\sum_{k} |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q} | f|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{p,\\tilde{s}})}.$$\n\n\\noindent {\\it Proof of B.} By definition, there exist $s^n_Q\\in \\mathbb{C}$, such that\n$$g_n=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q^na_Q^n,$$\nwhere $a_Q^n$ is a $\\mathcal{A}_{\\tilde{s},p}$-atom supported on $Q$ and\n\\begin{equation} \\label{sed} (\\sum_{k=0}^{\\infty}(\\sum_{Q\\in {\\mathcal P}^k}|s_Q^n|^p)^{\\tilde{q}\/p})^{1\/\\tilde{q}}\\leq C+\\varepsilon_n,\\end{equation}\nwhere $\\varepsilon_n\\to 0$.\nIn particular, $|s_Q^n|\\leq C+\\varepsilon_n$. Since the set $\\cup_k \\mathcal{P}^k$ is countable,\n by the Cantor diagonal argument, taking a subsequence we can assume that \n $s_Q^n\\to s_Q$ and (due ${\\Crr{compact}}$) that $a_Q^n$ \nconverges in $\\mathcal{B}(Q)$ and $L^p$ to some $a_Q\\in \\mathcal{A}_{\\tilde{s},p}$. By Lemma \\ref{compa2} the sequence $g_n$ converge in $L^p$ to a function $g$ such that $|g|_{\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{\\tilde{s},p})}\\leq C$ and with $\\mathcal{B}^{\\tilde{s}}_{p,\\tilde{q}}(\\mathcal{A}_{s,p})$-representation\n$$g=\\sum_{k=0}^{\\infty}\\sum_{Q\\in {\\mathcal P}^k}s_Q a_Q.$$\nIt remains to show that the convergence indeed occurs in the topology of ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$. For every $k_0\\geq 0$ and $\\delta > 0$ we can write \n$$g_n-g= \\sum_{k< k_0} \\sum_{Q\\in {\\mathcal P}^k} \\delta d_Q^{n} + \\sum_{k\\geq k_0} \\sum_{Q\\in {\\mathcal P}^k} |Q|^{\\tilde{s}-s} c_Q^{n} b_Q^{n},$$\nwhere \n$$d_Q^{n}= \\frac{1}{\\delta}(s_Q^na_Q^n - s_Q a_Q),$$\nand with $b_Q^{n}\\in \\mathcal{A}_{s,p}$ given by \n$$b_Q^{n} = \\frac{ |Q|^{s-\\tilde{s}} |s_Q^n|}{|s_Q^n| + |s_Q|} sgn(s_Q^{n}) a_Q^{n} + \\frac{ |Q|^{s-\\tilde{s}} |s_Q|}{|s_Q^n| + |s_Q|} sgn(-s_Q) a_Q,$$\nand\n$$c_Q^{n} = |s_Q^n| + |s_Q|.$$\nNote that $b_Q^{n} \\in \\mathcal{A}_{s,p}$. Given $\\epsilon > 0$, choose $k_0$ such that \n $b_Q^{n} \\in \\mathcal{A}_{s,p}$. Given $\\epsilon > 0$, choose $k_0$ such that \n$$ \\Big( \\sum_{k\\geq k_0} |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q} (2C+1) \\leq (\\epsilon\/2)^{1\/\\rho}.$$\nBy (\\ref{uyuy}) and (\\ref{sed}) for each $n$ large enough we have\n$$\n \\Big( \\sum_{k\\geq k_0} (\\sum_{Q\\in {\\mathcal P}^k}|c_Q^{n}|^p |Q|^{p(\\tilde{s}-s)}\\Big)^{q\/p} \\Big)^{1\/q} \\leq \\Big( \\sum_{k\\geq k_0} |\\mathcal{P}^k|^{q(\\tilde{s}-s)} \\Big)^{1\/q} (2C+1) < (\\epsilon\/2)^{1\/\\rho}.\n$$\nIn particular\n$$\\big| \\sum_{k\\geq k_0} \\sum_{Q\\in {\\mathcal P}^k} |Q|^{\\tilde{s}-s} c_Q^{n} b_Q^{n}\\big|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho < \\epsilon\/2.$$\nChoose $\\delta > $ such that \n$$\n \\Big( \\sum_{k< k_0} (\\sum_{Q\\in {\\mathcal P}^k}\\delta^p\\Big)^{q\/p} \\Big)^{\\rho\/q}< \\epsilon\/2.\n$$\n\nDue ${\\Crr{finite}}$ there is $\\eta > 0$ such that for every $Q\\in \\mathcal{P}^k$, with $k< k_0$, if $h \\in \\mathcal{B}(Q)$ satisfies $|h|_{\\mathcal{B}(Q)}\\leq \\eta$ then $h \\in \\mathcal{A}_{p,s}(Q)$. Since $\\lim_n s_Q^na_Q^n=s_Q a_Q$ in $\\mathcal{B}(Q)$ we conclude that for $n$ large enough we have \n$$ d_Q^{n}=\\frac{1}{\\delta}(s_Q^na_Q^n - s_Q a_Q) \\in \\mathcal{A}_{s,p}(Q)$$\nfor every $Q\\in \\mathcal{P}^k$, with $k< k_0$. In particular\n$$\\big|\\sum_{k< k_0} \\sum_{Q\\in {\\mathcal P}^k} \\delta d_Q^{n}\\big|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho< \\epsilon\/2.$$\nWe conclude that\n$$| g_n-g|_{{\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})}^\\rho< \\epsilon,$$\nfor $n$ large enough, so the sequence $g_n$ converges (due Corollary \\ref{compa1}) to $g$ in the topology of ${\\mathcal B}^s_{p,q}(\\mathcal{A}_{s,p})$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Transmutation of atoms} \nIt turns out that sometimes a Besov-ish space can be obtained using different classes of atoms. The key result in Part I is the following \n\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{transm.pdf}\n\\caption{ In the transmutation of atoms we replace atoms whose supports are represented in orange (supports with distinct scales are drawn in distinct lines for a better understanding) by linear combination of atoms (whose supports are represented in green), in such way we do not change too much the location of the support and also most of the \"mass\" of the representation is concentrated on the same original scale. }\n\\end{figure}\n\n\n\n\\begin{proposition}[Transmutation of atoms] \\label{trans} Assume \\\\ \n\n\\begin{itemize}\n\\item[{\\bf I.}] Let $\\mathcal{A}_2$ be a class of $(s,p,u_2)$-atoms for a grid $\\mathcal{W}$, satisfying ${\\Crr{banach}}$-${\\Crr{compact}}$ and ${\\Crr{fin}}$-${\\Crr{sum}}$. Let $\\mathcal{G}$ be also a grid satisfying ${\\Crr{fin}}$-${\\Crr{sum}}$.\n\\item[{\\bf II.}] Let $k_i\\in \\mathbb{N}$ for $i\\geq 0$ be a sequence such that there is $\\alpha > 0 $ and $A, B\\in \\mathbb{R}$ satisfying\n$$ \\alpha i +A \\leq k_i\\leq \\alpha i +B $$\nfor every $i$.\n\\item[{\\bf III.}] There is $\\lambda \\in(0,1)$ such that the following holds. For every $Q\\in \\mathcal{G}$ and $P\\in \\mathcal{W}$ satisfying $P\\subset Q$ there are atoms $b_{P,Q} \\in \\mathcal{A}_2(P)$ and corresponding $s_{P,Q}\\in \\mathbb{C}$ such that \n$$h_Q=\\sum_k \\sum_{P\\in \\mathcal{W}^k, P\\subset Q} s_{P,Q} b_{P,Q}.$$\nis a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation of a function $h_Q$, with $s_{P,Q}=0$ for every $Q\\in \\mathcal{G}^i$, $P \\in \\mathcal{W}^k$ with $k < k_i$ and moreover\n\\begin{equation}\\label{ad} \\sum_{P\\in \\mathcal{W}^k, P \\subset Q} |s_{P,Q}|^p \\leq \\Cll{rf} \\lambda^{k-k_i}.\\end{equation}\nfor every $k\\geq k_i$. \\\\ \\\\\n\\end{itemize}\nLet\n$$\\mathcal{H}^k= \\bigcup_{Q\\in \\mathcal{G}} \\{ P \\subset Q\\colon \\ P\\in \\mathcal{W}^k \\ and \\ s_{P,Q}\\neq 0 \\}. $$\nThen \\\\ \\\\\n\\begin{itemize}\n\\item[{\\bf A.}] For every coefficients $(c_Q)_{Q\\in \\mathcal{G}}$ such that \n$$\\big( \\sum_i \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{q\/p}\\big)^{1\/q}<\\infty$$\nwe have that the sequence\n\\begin{equation}\\label{seq33} N\\mapsto \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i }c_Q h_Q\\end{equation}\nconverges in $L^p$ to a function in $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$ that has a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation \n\\begin{equation}\\label{cs} \\sum_k \\sum_{P\\in \\mathcal{H}^k} m_P d_P\\end{equation}\nwhere $m_P \\geq 0$ for every $P$ and \n\\begin{eqnarray} \\label{igual1} &&\\big(\\sum_k \\big( \\sum_{\\substack{ P \\in \\mathcal{W}^k\\\\ P\\in \\mathcal{H}}} |m_{P}|^p \\big)^{q\/p}\\big)^{1\/q} \\nonumber \\\\ \n&\\leq& \\Crr{mult1} \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q} \\big( \\sum_i \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{q\/p}\\big)^{1\/q}\n\\end{eqnarray}\nHere $\\Cll{m1}=\\max \\{\\ell \\in \\mathbb{N}, \\ell < \\alpha\\}+1$ and $b=(b_n)_{n\\in \\mathbb{Z}}$ is defined by \n$$b_n = \\begin{cases} \\lambda^{ \\alpha n} & \\text{ if } n> \\frac{A}{\\alpha} - 1,\\\\\n0 & \\text{ if } n\\leq \\frac{A}{\\alpha} - 1,\n \\end{cases}\n $$\n\n\\item[{\\bf B.}] Suppose that the assumptions of A. hold and that $s_{P,Q}$ are non negative real numbers and $b_{P,Q} > 0$ on $P$ for every $P, Q$. Then $m_P\\neq 0$ and $d_P \\neq 0$ on $P$ imply that $P\\subset supp \\ h_Q$ for some $Q\\in \\mathcal{W}^k$ satisfying $c_Q \\neq 0$ and $s_{P,Q} >0$. If we additionally assume that $c_Q \\geq 0$ for every $Q$ then $m_P\\neq 0$ also implies $d_P > 0$ on $P$.\n\\item[{\\bf C.}] Let $\\mathcal{A}_1$ be a class of $(s,p,u_1)$-atoms for the grid $\\mathcal{G}$ satisfying ${\\Crr{banach}}$-${\\Crr{atom}}$. Suppose that there is $\\lambda < 1$ such that for every atom $a_Q\\in \\mathcal{A}_1(Q)$ we can find $s_{P,Q}$ and $b_{P,Q}$ in III. such that $h_Q=a_Q$. Then $$\\mathcal{B}^s_{p,q}(\\mathcal{A}_1)\\subset \\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$$ and this inclusion is continuous. Indeed\n$$|\\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)} \\leq \\Crr{mult1} \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q} |\\phi|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}_1)}$$\nfor every $\\phi \\in \\mathcal{B}^s_{p,q}(\\mathcal{A}_1).$ \n\\end{itemize}\n\\end{proposition}\n\\begin{proof} \nFor every $P\\in \\mathcal{H}^k$, with $k\\in \\mathbb{N}$, and $N\\in \\mathbb{N}\\cup \\{\\infty\\}$ define\n$$m_{P,N}=\\sum_{i\\leq N} \\sum_{\\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|=\\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|$$\nDue $\\Crr{sum}$ this sum has a finite number of terms. If this sum has zero terms define $m_{P,N}=0$ and let $d_{P,N}$ be the zero function. Otherwise define \n\\begin{equation}\\label{from} d_{P,N}= \\frac{1}{m_{P,N}}\\sum_{i\\leq N} \\sum_{\\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} c_Q s_{P,Q} b_{P,Q}.\\end{equation} \nWe have that $d_{P,N}$ is an $\\mathcal{A}_2(P)$-atom. \n\\vspace{5mm} \n\n\\noindent {\\bf Claim I.} {\\it We claim that for $N\\in \\mathbb{N}$}\n $$\\sum_{k} \\sum_{P\\in \\mathcal{H}^k} m_{P,N} d_{P,N} = \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i}c_Q h_Q. $$\nNote that if $Q\\in \\mathcal{G}^i$ then due (\\ref{flp}), with $t=p$ and (\\ref{ad}) we have\n$$\\sum_k \\big| \\sum_{\\substack{P\\in \\mathcal{H}^k\\\\ P\\subset Q}} s_{P,Q} b_{P,Q}\\big|_p^{p\/\\hat{p}} < \\infty.$$\nConsequently we can do the following manipulation in $L^p$\n\\begin{eqnarray*}\n \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i} c_Q h_Q&=& \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i} \\sum_k \\sum_{\\substack{ P\\in \\mathcal{H}^k\\\\ P\\subset Q}} s_{P,Q} b_{P,Q}\\\\\n &=& \\sum_k \\sum_{ P\\in \\mathcal{H}^k} \\sum_{i\\leq N} \\sum_{Q\\in \\mathcal{G}^i} \\sum_{P\\subset Q} s_{P,Q} b_{P,Q}\\\\\n &=& \\sum_k \\sum_{ P\\in \\mathcal{H}^k} m_{P,N} d_{P,N}.\n \\end{eqnarray*} \nThis concludes the proof of Claim I. \n\\vspace{5mm}\n\n\\noindent {\\bf Claim II.} {\\it For every $N\\in \\mathbb{N}\\cup \\{\\infty\\}$ we claim that \n\\begin{equation}\\label{a222} \\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,N} d_{P,N}\\end{equation}\nis a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation and }\n\\begin{align}& \\big(\\sum_k \\big( \\sum_{ P\\in \\mathcal{H}^k} |m_{P,N}|^p \\big)^{q\/p}\\big)^{1\/q} \\nonumber \\\\\n\\label{ess} &\\leq \\Crr{mult1} \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q}\\big( \\sum_i \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{q\/p}\\big)^{1\/q}.\n\\end{align}\nIndeed \n\\begin{align*} \n\\Big( \\sum_{P \\in \\mathcal{H}^k} |m_{P,N}|^p \\Big)^{1\/\\hat{p}}&= \\Big( \\sum_{P \\in \\mathcal{W}^k} \\big( \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|\\big)^p \\Big)^{1\/\\hat{p}} \\\\\n&\\leq \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\big( \\sum_{P \\in \\mathcal{W}^k} \\big( \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q s_{P,Q}|\\big)^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\big( \\sum_{P \\in \\mathcal{W}^k} \\sum_{ \\substack{Q\\in \\mathcal{G}^i\\\\ P \\subset Q}} |c_Q|^p |s_{P,Q}|^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\sum_{\\substack{ k_i\\leq k \\\\ i\\leq N}} \\big(\\sum_{Q\\in \\mathcal{G}^i} |c_Q|^p \\sum_{\\substack{P \\in \\mathcal{W}^k\\\\ P \\subset Q}} |s_{P,Q}|^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\sum_{\\substack{k_i\\leq k \\\\ i\\leq N}} \\lambda^{(k-k_i)\/\\hat{p}} \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\sum_{\\substack{\\alpha i + A\\leq k}} \\lambda^{(k-\\alpha i -B)\/\\hat{p}} \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{1\/\\hat{p}} . \\numberthis \\label{for}\n\\end{align*}\nIf $\\alpha=1$ and $A=B=0$ then this is a convolution, so we can use Proposition \\ref{young} (the convolution trick) and it easily follows (\\ref{igual1}). In the general case, consider\n$$u_k=\\sum_{P \\in \\mathcal{H}^k} |m_{P,N}|^p \\text{ and } c_i=\\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p$$\nEvery $k\\in \\mathbb{N}$ can be written in an {\\it unique way} as $k=\\alpha j_k+ \\ell_k+r_k$, with $j_k \\in \\mathbb{N}$, $\\ell_k \\in \\mathbb{N}$, $\\ell_k+r_k < \\alpha$ and $r_k \\in [0,1)$. Fix $\\ell \\in [0,\\alpha)\\cap\\mathbb{N}$ and $j\\in \\mathbb{N}$. Then there is at most one $k'\\in \\mathbb{N}$ such that $\\ell_k=\\ell$ and $j_k=j$. Indeed, if $k'= \\alpha j + \\ell+r'$ and $k''= \\alpha j + \\ell+r''$, with $r',r''\\in [0,1)$ and $\\ell+r'$ and $\\ell+r''$ smaller than $\\alpha$, then $k'-k''=r'-r''\\in (-1,1)$, so $k'=k''$ and $r'=r''$. If such $k'$ exists, denote $k(\\ell,j)=k'$ and $r(\\ell,j)= k(\\ell,j)-\\alpha j -\\ell$ and $a_{\\ell,j}=u_{k(\\ell,j)}$. Otherwise let $a_{\\ell,j}=0$. Then (\\ref{for}) implies \n\\begin{align*}\na_{\\ell,j}^{1\/\\hat{p}}&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\sum_{\\substack{\\alpha i + A\\leq \\alpha j+ \\ell+r(\\ell,j)}} \\lambda^{( \\alpha j+ \\ell+r(\\ell,j)-\\alpha i -B)\/\\hat{p}} c_i^{1\/\\hat{p}} \\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\lambda^{-B\/\\hat{p}} \\sum_{\\substack{ i \\leq j+ \\frac{-A+\\ell+r(\\ell,j)}{\\alpha} } } \\lambda^{ \\alpha (j- i)\/\\hat{p}} c_i^{1\/\\hat{p}}\\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\lambda^{-B\/\\hat{p}} \\sum_{\\substack{ i < j+ \\frac{-A}{\\alpha} +1}} \\lambda^{ \\alpha (j- i)\/\\hat{p}} c_i^{1\/\\hat{p}}\\\\\n&\\leq \\Crr{mult1}^{p\/\\hat{p}} \\Crr{rf}^{1\/\\hat{p}} \\lambda^{-B\/\\hat{p}} \\sum_{i \\in \\mathbb{Z} } b_{j-i}^{1\/\\hat{p}} c_i^{1\/\\hat{p}}\n\\end{align*}\nHere $b_n= \\lambda^{ \\alpha n} $ if $n> A\/\\alpha - 1$, and $b_n=0$ otherwise. Fixing $\\ell \\in \\mathbb{N}, \\ \\ell< \\alpha$, Proposition \\ref{young} (the convolution trick) gives us \n\\begin{eqnarray*} K_\\ell= \\big(\\sum_{\\substack{k\\in \\mathbb{N} \\\\ \\ell_k=\\ell}} u_{k}^{q\/p}\\big)^{1\/q} &=&\\big(\\sum_{j} a_{\\ell,j}^{q\/p}\\big)^{1\/q}\\\\\n&\\leq& \\Crr{mult1} \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\big( \\sum_i \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{q\/p}\\big)^{1\/q}.\n\\end{eqnarray*} \nand\n\\begin{align} \\label{bbbb} \\big(\\sum_k \\big( \\sum_{ P\\in \\mathcal{H}^k} |m_{P,N}|^p \\big)^{q\/p}\\big)^{1\/q}&= \\big(\\sum_k u_k^{q\/p}\\big)^{1\/q} \\nonumber \\\\\n&= \\big(\\sum_{0\\leq \\ell < \\alpha} \\sum_{\\substack{k\\in \\mathbb{N} \\\\ \\ell_k=\\ell}} u_{k}^{q\/p}\\big)^{1\/q}= \\big(\\sum_{0\\leq \\ell < \\alpha}K_\\ell^q \\big)^{1\/q}\\nonumber \\\\\n &\\leq \\Crr{mult1} \\Crr{rf}^{1\/p}\\lambda^{-\\frac{B}{p}} \\Crr{co2}(p,q,b) \\Crr{m1}^{1\/q} \\big( \\sum_i \\big( \\sum_{Q\\in \\mathcal{G}^i} |c_{Q}|^p \\big)^{q\/p}\\big)^{1\/q}.\n\\end{align}\nThis implies in particular that the sum in (\\ref{a222}) is a $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation. This proves Claim II. \n \\vspace{5mm}\n \n \\noindent {\\bf Claim III.} {\\it We have that in the strong topology of $L^p$\n$$\\lim_{N\\rightarrow \\infty} \\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,N} d_{P,N} =\\sum_k \\sum_{P\\in \\mathcal{H}^k} m_{P,\\infty} d_{P,\\infty}. $$}\nFor each $P\\in \\mathcal{H}$ the sequence\n\\begin{equation}\\label{seqdf} N\\mapsto m_{P,N}\\end{equation} \nis eventually constant, therefore convergent. The same happens with \n\\begin{equation}\\label{seqdf2} N\\mapsto d_{P,N}.\\end{equation}\n\n\n\\noindent Estimate (\\ref{ess}) and Proposition \\ref{compa2} imply that that (\\ref{seq33}) converges in $L^p$ to a function with $\\mathcal{B}^s_{p,q}(\\mathcal{A}_2)$-representation (\\ref{a222}) with $N=\\infty$. This concludes the proof of Claim III. \n\nThen Claim I, II, and III imply A. taking $m_p=m_{P,\\infty}$ and $d_P=d_{P,\\infty}.$ We have that C. is an immediate consequence of A. Note that (\\ref{from}) and A. give B.\n\\end{proof}\n\\vspace{1cm}\n\n\\section{Good grids} \\label{goodgrids} A $(\\Cll[c]{menor},\\Cll[c]{maior})$-{\\bf good grid} , with $0< \\Crr{menor}< \\Crr{maior} < 1$, is a grid $\\mathcal{P}= (\\mathcal{P}^k)_{k\\in \\mathbb{N}}$ with the following properties:\n\\begin{itemize}\n\\item[${\\Cll[G]{g2}}.$] We have $\\mathcal{P}^0=\\{I\\}$.\n\\item[${\\Cll[G]{g3}}.$] We have $I=\\cup_{Q \\in \\mathcal{P}^k} Q$ (up to a set of zero $m$-measure).\n\\item[${\\Cll[G]{g4}}.$] The elements of the family $\\{ Q \\}_{Q\\in \\mathcal{P}^k} $ are pairwise disjoint. \n\\item[${\\Cll[G]{g5}}.$] For every $Q\\in \\mathcal{P}^k$ and $k > 0$ there exists $P \\in \\mathcal{P}^{k-1}$ such that $Q\\subset P$. \n\\item[${\\Cll[G]{g6}}.$] We have\n$$\\Crr{menor} \\leq \\frac{|Q|}{|P|} \\leq \\Crr{maior}$$\nfor every $Q\\subset P$ satisfying $Q\\in \\mathcal{P}^{k+1}$ and $P\\in \\mathcal{P}^{k}$ for some $k\\geq 0$. \n\\item[${\\Cll[G]{g7}}.$] The family $\\cup_k \\mathcal{P}^k$ generate the $\\sigma$-algebra $\\mathbb{A}$. \n\\end{itemize}\n\n\\section{Induced spaces} \nConsider a Besov-ish space $\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$, where $\\mathcal{P}$ is a good grid. Given $Q\\in \\mathcal{P}^{k_0}$, we can consider the sequence of finite families of subsets $\\mathcal{P}_Q=(\\mathcal{P}_Q^i)_{i\\geq 0}$ of $Q$ given by \n$$\\mathcal{P}^i_Q=\\{ P\\in \\mathcal{P}^{k_0+i}, P\\subset Q\\}.$$\nLet $\\mathcal{A}_Q$ be the restriction of the indexed family $\\mathcal{A}$ of pairs $(\\mathcal{B}(P),\\mathcal{A}(P))_{P\\in \\mathcal{P}}$ to indices belonging to $\\mathcal{P}_Q$. Then we can consider the {\\bf induced} Besov-ish space $\\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q)$. Of course the inclusion\n$$i\\colon \\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q) \\rightarrow \\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})$$\nis well defined and it is a weak contraction, that is\n$$|f|_{\\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A})}\\leq |f|_{\\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q)}.$$\nUnder the degree of generality we are considering here, the {\\bf restriction } transformation\n$$r\\colon \\mathcal{B}^s_{p,q}(I,\\mathcal{P}, \\mathcal{A}) \\rightarrow L^p$$\ngiven by $r(f)= f\\cdot 1_Q$, is a bounded linear transformation, however it is easy to find examples of Besov-ish spaces where $f 1_Q \\not\\in \\mathcal{B}^s_{p,q}(Q,\\mathcal{P}_Q, \\mathcal{A}_Q).$\n\n\n\n\n\n\n\n\n\n\\section{Examples of classes of atoms.}\\label{secatom}\n\nThere are many classes of atoms one may consider. We list here just a few of them. \n\n\n\\subsection{Souza's atoms}\\label{souzaa}\n\nLet $Q\\in \\mathcal{P}$. A {\\bf $(s,p)$-Souza's atom} supported on $Q$ is a function $a\\colon I \\rightarrow \\mathbb{C}$ such that $a(x)=0$ for every $x \\not\\in Q$ and $a$ is constant on $Q$, with \n$$|a|_\\infty\\leq |Q|^{s-1\/p}.$$\n\nThe set of Souza's atoms supported on $Q$ will be denoted by $\\mathcal{A}^{sz}_{s,p}(Q).$ A {\\bf canonical Souza's atom} on $Q$ is the Souza's atom such that $a(x)= |Q|^{s-1\/p}$ for every $x \\in Q$. Souza's atoms are $(s,p,\\infty)$-type atoms.\n\n\n\n\n\n\n\n\n\n\n\n \\subsection{H\\\"older atoms} \\label{hatoms} Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\\cdot,\\cdot)$, such that every $Q\\in \\mathcal{P}$ is a bounded set and there is $\\Cll[c]{hh},\\Cll[c]{hhh} \\in (0,1)$ such that\n$$\\Crr{hh} \\leq \\frac{diam \\ P}{diam \\ Q} \\leq \\Crr{hhh}$$\nfor every $P\\subset Q$ with $P \\in \\mathcal{P}^{k+1}$ and $Q \\in \\mathcal{P}^{k}$. Additionally assume that there is $\\Cll{kki}\\geq 0$ and $D\\geq 0$ such that \n$$\\frac{1}{\\Crr{kki}^{D}}|Q| \\leq (diam \\ Q)^D \\leq \\Crr{kki}^{D} |Q|.$$\n\nLet \n$$0< s < \\frac{1}{p}, \\ s < \\beta.$$ For every $Q\\in \\mathcal{P}$, Let $C^\\alpha(Q)$ be the Banach space of all functions $\\phi$ such that $\\phi(x)=0$ for $x \\not\\in Q$, and \n$$|\\phi|_{C^\\alpha(Q)}= |\\phi|_\\infty + \\sup_{\\substack{x,y \\in Q\\\\ x\\neq y}} \\frac{|\\phi(x)-\\phi(y)|}{d(x,y)^\\alpha} < \\infty.$$\nLet $\\mathcal{A}^{h}_{s,\\beta,p}(Q) \\subset C^{\\beta D}(Q)$ be the convex subset of all functions $\\phi$ satisfying \n$$\\sup_{\\substack{x,y \\in Q\\\\ x\\neq y}} \\frac{|\\phi(x)-\\phi(y)|}{d(x,y)^{D \\beta}} \\leq |Q|^{s-1\/p-\\beta} \\ and \\ |\\phi|_{\\infty} \\leq |Q|^{s-1\/p}.$$\nWe say that $\\mathcal{A}^{h}_{s,\\beta,p}(Q)$ is the set of {\\bf $(s,\\beta,p)$-H\\\"older atoms } supported on $Q$. Of course $\\mathcal{A}^{h}_{s,\\beta,p}$-atoms are $(s,p,\\infty)$-type atoms and $\\mathcal{A}^{sz}_{s,p}(Q)\\subset \\mathcal{A}^{h}_{s,\\beta,p}(Q)$.\n\n\n\\subsection{Bounded variation atoms} Now suppose that $I$ is an interval of $\\mathbb{R}$ with length $1$, $m$ is the Lebesgue measure on it and the partitions in the grid $\\mathcal{P}$ are partitions by intervals. \nLet $Q$ be an interval and $s \\leq \\beta$, $p\\in [1,\\infty)$. A $(s,\\beta,p)$-bounded variation atom on $Q$ is a function $a\\colon \\mathbb{R} \\rightarrow \\mathbb{C}$ such that $a(x)=0$ for every $x \\not\\in Q$,\n$$|a|_\\infty\\leq |Q|^{s-1\/p}.$$ \nand\n$$var_{1\/\\beta}(a,Q) \\leq |Q|^{s-1\/p}.$$ \nHere $var_{1\/\\beta}(\\cdot,Q)$ is the pseudo-norm \n$$var_{1\/\\beta}(a,Q) = \\sup (\\sum_i |a(x_{i+1})-a(x_i)|^{1\/\\beta})^{\\beta},$$\nwhere the sup runs over all possible sequences $x_1 < x_1 < \\dots < x_n$, with $x_i$ in the interior of $Q$.\nWe will denote the set of bounded variation atoms on $Q$ as $\\mathcal{A}^{bv}_{s, p,\\beta}(Q)$. Bounded variation atoms are also $(s,p,\\infty)$-type atoms.\n\\vspace{1cm}\n\n\n\n\\newpage\n\n\\centerline{ \\bf II. SPACES DEFINED BY SOUZA'S ATOMS.}\n\\addcontentsline{toc}{chapter}{\\bf II. SPACES DEFINED BY SOUZA'S ATOMS.}\n\\vspace{1cm}\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent In Part II. we suppose that $s> 0$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. \\end{minipage} \n}\n\\ \\\\ \\\\\n\n\n\\section{Besov spaces in a measure space with a good grid}\n\n\n We will study the Besov-ish spaces $\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$ associated with the measure space with a good grid $(I, \\mathcal{P},m)$. We denote $\\mathcal{B}^s_{p,q}=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$. Note that $\\mathcal{A}^{sz}_{s,p}$ satisfies ${\\Crr{banach}}$-${\\Crr{finite}}$. Note that by Proposition \\ref{lp} there is $\\beta > 1$ such that $\\mathcal{B}^s_{p,q} \\subset L^\\beta$.\n \nIf $p\\in [1,\\infty)$, $q\\in [1,\\infty]$ and $0< s< \\frac{1}{p}$ we wil say that $\\mathcal{B}^s_{p,q}$ is a {\\bf Besov space}. \n\n\n\\section{Positive cone}\n\nWe say that $f$ is $\\mathcal{B}^s_{p,q}$-positive if there is a $\\mathcal{B}^s_{p,q}$-representation\n$$f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} c_Pa_P$$\nwhere $c_P\\geq 0$ and $a_P$ is the standard $(s,p)$-Souza's atom supported on $P$. The set of all $\\mathcal{B}^s_{p,q}$-positive functions is a convex cone in $\\mathcal{B}^s_{p,q}$, denoted $\\mathcal{B}^{s+}_{p,q}$. We can define a ``norm\" on $\\mathcal{B}^{s+}_{p,q}$ as\n$$|f|_{\\mathcal{B}^{s+}_{p,q}}=\\inf \\Big( \\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} c_P^p \\big)^{q\/p} \\Big)^{1\/q},$$\nwhere the infimum runs over all possible $\\mathcal{B}^s_{p,q}$-positive representations of $f$. Of course for every $f,g \\in \\mathcal{B}^{s+}_{p,q}$ and $\\alpha \\geq 0$ we have \n$$|\\alpha f|_{\\mathcal{B}^{s+}_{p,q}}=\\alpha | f|_{\\mathcal{B}^{s+}_{p,q}}, \\ |f+g|_{\\mathcal{B}^{s+}_{p,q}}\\leq |f|_{\\mathcal{B}^{s+}_{p,q}}+ |g|_{\\mathcal{B}^{s+}_{p,q}}, \\ | f|_{\\mathcal{B}^{s}_{p,q}} \\leq | f|_{\\mathcal{B}^{s+}_{p,q}}.$$\nMoreover if $f\\in \\mathcal{B}^s_{p,q}$ is a real-valued function then one can find $f_+,f_-\\in \\mathcal{B}^{s+}_{p,q}$ such that $f=f_+-f_-$ and $$|f_+|_{\\mathcal{B}^{s+}_{p,q}} \\leq |f|_{\\mathcal{B}^{s}_{p,q}} \\ and \\ |f_-|_{\\mathcal{B}^{s+}_{p,q}} \\leq |f|_{\\mathcal{B}^{s}_{p,q}}.$$\n\nAn obvious but important observation is\n\n\\begin{proposition} if $f \\in \\mathcal{B}^{s+}_{p,q}$ then its support \n$$supp \\ f = \\{ x\\in I\\colon \\ f(x)\\neq 0\\}.$$\nis (up to a set of zero measure) is a countable union of elements of $\\mathcal{P}$. \n\\end{proposition} \n\n\n\n\\section{Unbalanced Haar wavelets} \\label{haar} Let $\\mathcal{P}=(\\mathcal{P}^k)_k$ be a good grid. For every $Q \\in \\mathcal{P}^k$ let $\\Omega_Q=\\{ P_1^Q,\\dots,P_{n_Q}^Q\\}$, $n_Q\\geq 2$, be the family of elements $\\mathcal{P}^{k+1}$ such that $P_i^Q \\subset Q$ for every $i$, and ordered in some arbitrary way. The elements of $\\Omega_Q$ will be called {\\bf children} of $Q$. Note that every $Q\\in \\mathcal{P}$ has at least two children. We will use one of the method described (Type I tree with the logarithmic subtrees construction) in Girardi and Sweldens \\cite{gw} to construct an unconditional basis of $L^{\\beta}$, for every $1< \\beta <\\infty$. \n\nLet $\\mathcal{H}_Q$ be the family of pairs $(S_1,S_2)$, with $S_i \\subset \\Omega_Q$ and $S_1\\cap S_2=\\emptyset$, defined as\n$$\\mathcal{H}_Q=\\cup_{j\\in \\mathbb{N}} \\mathcal{H}_{Q,j},$$\nwhere $\\mathcal{H}_{Q,j}$ are constructed recursively in the following way. Let $\\mathcal{H}_{Q,0}=\\{ (A,B)\\}$, where $A=\\{ P_1^Q,\\dots,P_{[n_Q\/2]}^Q \\}$ and $B=\\{ P_{[n_Q\/2]+1}^Q,\\dots,P_{n_Q}^Q \\}$. Here $[x]$ denotes the integer part of $x\\geq 0$. Suppose that we have defined $\\mathcal{H}_{Q,j}$. For each element $(S_1,S_2)\\in \\mathcal{H}_{Q,j}$, {\\it fix} an ordering $S_1=\\{ R_1^1, \\dots, R_{n_1}^1 \\}$ and $S_2=\\{ R_1^2, \\dots, R_{n_2}^2 \\}$. For each $i=1,2$ such that $n_i\\geq 2$, define $T_i^1=\\{ R_1^i,\\dots,R_{[n_i\/2]}^i \\}$ and $T_i^2=\\{ R_{[n_i\/2]+1}^i,\\dots,R_{n_i}^i \\}$ and add $(T_i^1,T_i^2)$ to $\\mathcal{H}_{Q,j+1}$. This defines $\\mathcal{H}_{Q,j+1}.$\n\nNote that since $\\mathcal{P}$ is a good grid we have $\\mathcal{H}_{Q,j}=\\emptyset$ for large $j$ and indeed \n$$\\sup_{Q\\in \\mathcal{P}} \\#\\mathcal{H}_Q < \\infty.$$\n\nDefine $\\mathcal{H}=\\cup_{Q\\in \\mathcal{P}} \\mathcal{H}_Q$. For every $S=(S_1,S_2) \\in\\mathcal{H}_Q$ define\n\n$$\\phi_{S} = \\frac{1}{m_{(S_1,S_2)}}\\Big( \\frac{\\sum_{P \\in S_1} 1_{P}}{\\sum_{P \\in S_1} |P| } - \\frac{\\sum_{R \\in S_2} 1_{R}}{\\sum_{R \\in S_2} |R|} \\Big)$$\nwhere \n$$m_{(S_1,S_2)}= \\big( \\frac{1}{\\sum_{P \\in S_1} |P|} + \\frac{1}{\\sum_{R \\in S_2} |R|} \\big)^{1\/2} .$$\nNote that \n$$\\int_Q \\phi_S \\ dm=0.$$\nSince $1\\leq \\#S_i\\leq 1\/\\Crr{menor}$ we have\n$$\\Crr{menor}|Q|\\leq \\sum_{P \\in S_i} |P|\\leq \\frac{\\Crr{maior}}{\\Crr{menor}} |Q|,$$\nso\n$$\\Big(\\frac{2\\Crr{menor}}{\\Crr{maior}}\\Big)^{1\/2}\\frac{1}{|Q|^{1\/2}} \\leq m_{(S_1,S_2)}\\leq \\Big(\\frac{2}{\\Crr{menor}}\\Big)^{1\/2}\\frac{1}{|Q|^{1\/2}}$$\nConsequently \n\\begin{align}\\label{estphi} \\frac{\\Crr{c1}}{|Q|^{1\/2}} &\\leq \\frac{1}{m_{(S_1,S_2)}} \\min \\{ \\frac{1}{\\sum_{P \\in S_1} |P| }, \\frac{1}{\\sum_{R \\in S_2} |R| } \\} \\nonumber \\\\\n&\\leq |\\phi_{S}(x)|\\leq \\frac{1}{m_{(S_1,S_2)}} \\max \\{ \\frac{1}{\\sum_{P \\in S_1} |P| }, \\frac{1}{\\sum_{R \\in S_2} |R| } \\}\\nonumber \\\\\n&\\leq \\frac{\\Crr{c2}}{|Q|^{1\/2}} .\\end{align}\nfor every $x \\in \\cup_{P\\in S_1\\cup S_2} P.$ Here \n$$\\Cll{c1}= \\frac{\\Crr{menor}^{3\/2}}{\\sqrt{2}\\Crr{maior}} \\ and \\ \\Cll{c2}= \\frac{\\Crr{maior}^{1\/2}}{\\sqrt{2}\\Crr{menor}^{3\/2}}+1. $$\n\n\\begin{figure}\n\\includegraphics[scale=0.6]{haartudo.pdf}\n\\end{figure}\nLet \n$$\\hat{\\mathcal{H}}= \\{ I\\}\\cup \\mathcal{H}$$ and define\n$$\\phi_I= \\frac{1_I}{|I|^{1\/2}}.$$\nThen by Girardi and Sweldens \\cite{gw} we have that \n$$\\{ \\phi_S\\}_{S\\in \\hat{\\mathcal{H}}}$$\nis an unconditional basis of $L^\\beta$ for every $\\beta > 1$. \n \n \\section{Alternative characterizations I: Messing with norms.}\\label{alt1}\n We are going to describe three norms that are equivalent to $|\\cdot|_{\\mathcal{B}^s_{p,q}}$. Their advantage is that they are far more concrete, in the sense that we do not need to consider arbitrary atomic decompositions to define them. \n\n \\subsection{Haar representation} \nFor every $f\\in L^\\beta$, $\\beta > 1$, the series \n\\begin{equation}\\label{fs22} f = \\sum_{S\\in \\hat{\\mathcal{H}}} d_S^f \\phi_S \\end{equation} \nis converges unconditionally in $L^\\beta$, where\n $d_S^f = \\int f \\phi_S \\ dm.$ \n We will call the r.h.s. of (\\ref{fs22}) the {\\bf Haar representation} of $f$. \n Define\n $$ N_{haar}(f)=|I|^{1\/p-s-1\/2} |d_I^f|+ \\Big( \\sum_{k} \\big( \\sum_{\\substack{ _{Q\\in \\mathcal{P}^{k}}}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{_{S\\in \\mathcal{H}_Q}} |d_S^f|^p\\big)^{q\/p} \\Big)^{1\/q},$$\n\n\n\n\\subsection{Standard atomic representation} Note that \n$$k_I^fa_I= d_I^f \\phi_I,$$\nwhere $k_I= |I|^{1\/p-s-1\/2}d_I^f$ and $a_I$ is the canonical Souza's atom on $I$. Let $S\\in \\mathcal{H}$. Then $S\\in \\mathcal{H}_Q$, with $S=(S_1,S_2)$ and some $Q\\in \\mathcal{P}^k$, with $k\\geq 0$. It is easy to see that for every $P\\in S_1\\cup S_2$ the function \n $$a_{S,P}= \\frac{ |Q|^{1\/2}}{\\Crr{c2}} |P|^{s-1\/p} \\phi_S 1_{P}$$\nis a Souza's atom on $P$. Choose \n $$c_{S,P}^f= \\Crr{c2} |Q|^{-1\/2} |P|^{1\/p-s} d_S^f. $$\n Note that\n\\begin{equation} \\label{estcsp} |c_{S,P}^f|\\leq \\Crr{c2} \\max\\{ \\Crr{maior}^{1\/p-s}, \\Crr{menor}^{1\/p-s}\\} |Q|^{1\/p-s-1\/2} |d_S^f|.\\end{equation} \nFor every child $P$ of $Q \\in \\mathcal{P}^k$, $k\\geq 0$, define \n $$\\tilde{a}_P^f= \\frac{1}{\\tilde{k}_P^f} \\sum_{S=(S_1,S_2)\\in \\mathcal{H}_Q} \\sum_{P\\in S_1\\cup S_2} c_{S,P}^f a_{S,P},$$\n where \n\\begin{equation}\\label{hh} \\tilde{k}_P^f = \\sum_{S=(S_1,S_2)\\in \\mathcal{H}_Q} \\sum_{P\\in S_1\\cup S_2} |c_{S,P}^f|.\\end{equation}\nThe (finite) number of terms on this sum depends only on the geometry of $\\mathcal{P}$. \nThen $\\tilde{a}_P^f$ is a Souza's atom on $P$ and\n$$ \\sum_{S\\in \\mathcal{H}_Q} d_S^f \\phi_S = \\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\_{P\\subset Q}}} \\tilde{k}_P^f \\tilde{a}_P^f.$$\n Let $a_P$ be the canonical $(s,p)$-Souza's atom on $P$ and choose $x_P\\in P$. Denote\n$$k_P^f= \\frac{ \\tilde{a}_P^f(x_P)}{|P|^{s-1\/p}} \\tilde{k}_P^f= \\frac{1}{|P|^{s-1\/p}}\\sum_{S\\in \\mathcal{H}_Q} d_S^f \\phi_S(x_P)=\\frac{1}{|P|^{s-1\/p}} \\int f \\sum_{S\\in \\mathcal{H}_Q} \\phi_S(x_P)\\phi_S \\ dm.$$\nIn particular for every $P$ \n $$f\\mapsto k^f_P$$\n extends to a bounded linear functional in $L^1$. We have $|k_P^f|\\leq \\tilde{k}_P^f$ and \n\\begin{align} f &= k_I^f a_I + \\sum_k \\sum_{Q\\in \\mathcal{P}^{k}}\\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\_{P\\subset Q}}} k_P^f a_P \\nonumber \\\\\n \\label{sumf} &= \\sum_i \\sum_{Q\\in \\mathcal{P}^{i}} k_Q^f a_Q.\n\\end{align}\n where this series converges unconditionally in $L^\\beta$. Here $a_P$ is the canonical Souza's atom. We will call the r.h.s. of (\\ref{sumf}) the {\\bf standard atomic representation} of $f$. Let\n $$ N_{st}(f)=|k_I^f|+ \\Big( \\sum_{k\\geq 1} \\big( \\sum_{Q\\in \\mathcal{P}^{k}} |k_Q^f|^p \\big)^{q\/p}\\Big)^{1\/q}.$$\n \n \\subsection{Mean oscillation}\n Define for $p\\in [1,\\infty)$\n$$osc_p(f,Q)= \\inf_{c \\in \\mathbb{C}} (\\int_Q |f(x)-c|^p \\ dm(x) )^{1\/p},$$\nand \n$$osc_\\infty(f,Q)=\\inf_{c \\in \\mathbb{C}} |f-c|_{L^\\infty(Q)}.$$\nDenote for every $p\\in [1,\\infty)$ and $q \\in [1,\\infty]$\n\\begin{equation} \\label{oc} osc^s_{p,q}(f)= \\Big( \\sum_k \\big( \\sum_{Q\\in \\mathcal{P}^k} |Q|^{-sp} osc_p(f,Q)^p \\big)^{q\/p} \\Big)^{1\/q},\\end{equation} \nwith the obvious adaptation for $q=\\infty$. Let $$ N_{osc}(f)=|I|^{-s}|f|_p+osc^s_{p,q}(f).$$\n \n \n\n \\subsection{These norms are equivalent} We have \n \n \\begin{theorem} \\label{alte} Suppose $s> 0$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. Each one of the norms $|f|_{\\mathcal{B}^s_{p,q}}$, $N_{st}(f)$, $N_{haar}(f)$, $N_{osc}(f)$ is finite if and only if $f\\in \\mathcal{B}^s_{p,q}$. Furthermore these norms are equivalent on $\\mathcal{B}^s_{p,q}$. Indeed\n\\begin{equation}\\label{in1} |f|_{\\mathcal{B}^s_{p,q}}\\leq N_{st}(f),\\end{equation}\n\\begin{equation}\\label{in2}N_{st}(f)\\leq \\Cll{e} N_{haar}(f),\\end{equation} \n\\begin{equation}\\label{in3}N_{haar}(f)\\leq \\Crr{c2} N_{osc}(f),\\end{equation}\n\\begin{equation}\\label{in4}N_{osc}(f)\\leq \\Cll{no} |f|_{\\mathcal{B}^s_{p,q}}.\\end{equation}\nwhere\n$$\\Crr{e}=1+\\Crr{c2} \\max\\{ \\Crr{maior}^{1\/p-s}, \\Crr{menor}^{1\/p-s}\\} \\Crr{menor}^{-2-1\/p},$$\n $$\\Crr{no}=\\Crr{mult1}^{1+1\/p} \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{sp} )_k )|I|^{-s}+\\frac{1}{1-\\Crr{maior}^s}.$$\n \\end{theorem}\n \\begin{proof} The inequality (\\ref{in1}) is obvious. To simplify the notation we write $d_S, k_P$ instead of $d_S^f, k_P^f$.\\ \\\\ \\\\\n \\noindent {\\bf Proof of (\\ref{in2}).} The number of terms in the r.h.s. of (\\ref{hh}) depends only on the geometry of $\\mathcal{P}$. Indeed \n \\begin{equation}\\label{numero} \\sup_{Q \\in \\mathcal{P}} \\sum_{S=(S_1,S_2) \\in \\mathcal{H}_Q} \\#(S_1 \\cup S_2) \\leq \\frac{1}{\\Crr{menor}^2}.\\end{equation} \n Consider the standard atomic representation of $f$ given by (\\ref{sumf}). Note that by (\\ref{estcsp})\n\\begin{align*} \n\\sum_{Q\\in \\mathcal{P}^{k}} \\sum_{\\substack{ _{P \\in \\mathcal{P}^{k+1}}\\\\ _{P\\subset Q}}} |k_P|^p &\\leq \\sum_{Q\\in \\mathcal{P}^{k}} \\Big(\\sum_{_{S\\in \\mathcal{H}_Q}} \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}} |c_{S,P}| \\Big)^p \\\\\n&\\leq \\Crr{c2}^p \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\} \\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\Big( \\sum_{_{S\\in \\mathcal{H}_Q}} \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}} |d_S|\\Big)^p \\\\\n&\\leq \\Crr{c2}^p \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\}\\Crr{menor}^{-2p} \\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{_{S\\in \\mathcal{H}_Q}} \\sum_{\\substack{_{P\\in S_1\\cup S_2}\\\\_{S=(S_1,S_2)}}} |d_S|^p \\\\\n&\\leq \\Crr{c2}^p \\max\\{ \\Crr{maior}^{1-sp}, \\Crr{menor}^{1-sp}\\} \\Crr{menor}^{-2p-1} \\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{_{S\\in \\mathcal{H}_Q}} |d_S|^p.\n\\end{align*}\nfor every $k$. Consequently\n\\begin{align*} \n& |k_I|+ \\Big( \\sum_{k\\geq 1} \\big( \\sum_{P \\in \\mathcal{P}^{k}} |k_P|^p\\big)^{q\/p} \\Big)^{1\/q} \\leq \\\\\n&\\leq |I|^{_{\\frac{1}{p}-s-1\/2}} |d_I| + \\Crr{c2} \\max\\{ \\Crr{maior}^{_{\\frac{1}{p}-s}}, \\Crr{menor}^{_{\\frac{1}{p}-s}}\\}\\Crr{menor}^{_{-2-\\frac{1}{p}}} \\Big( \\sum_k \\big( \\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{_{S\\in \\mathcal{H}_Q}} |d_S|^p\\big)^{q\/p} \\Big)^{1\/q}.\n\\end{align*}\nThis completes the proof of (\\ref{in2}). \\\\ \\\\\n\\noindent {\\bf Proof of (\\ref{in3}).} Note that\n$$|d_I|\\leq \\int_I |f| |\\phi_I|\\ dm\\leq |f|_p |I|^{1\/2-1\/p}.$$Given $\\epsilon > 0$ and $Q \\in \\mathcal{P}$, choose $c_Q\\in Q$ such that \n$$\\Big( \\int_Q |f - c_Q|^p \\ dm \\Big)^{1\/p} \\leq (1+\\epsilon) osc_p(f,Q).$$\n Since $\\phi_S$ has zero mean on $Q$ for every $S\\in \\mathcal{H}_Q$ we have \n\\begin{align*} \n&\\big(\\sum_{Q\\in \\mathcal{P}^{k}}|Q|^{1-sp -\\frac{p}{2}} \\sum_{S\\in \\mathcal{H}_Q} |d_S|^p\\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{S\\in \\mathcal{H}_Q} |\\int f \\phi_S \\ dm |^p \\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{S\\in \\mathcal{H}_Q} |\\int_Q f \\phi_S -c_Q \\phi_S\\ dm |^p \\big)^{1\/p} \\\\\n&\\leq\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{S\\in \\mathcal{H}_Q} (\\int_Q |f - c_Q| |\\phi_S| \\ dm )^p \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2}\\big(\\sum_{\\substack{_{Q\\in \\mathcal{P}^{k}}\\\\ _{Q\\subset J}}} |Q|^{_{1-sp -\\frac{p}{2}}} \\sum_{S\\in \\mathcal{H}_Q} ((1+\\epsilon) osc_p(f,Q) |Q|^{1\/p'-1\/2} )^p \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2}(1+\\epsilon) \\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{1-sp +p\/p'-p}} \\sum_{S\\in \\mathcal{H}_Q} osc_p(f,Q)^p \\big)^{1\/p} \\\\\n&\\leq \\Crr{c2} (1+\\epsilon)\\big(\\sum_{Q\\in \\mathcal{P}^{k}} |Q|^{_{-sp}} \\sum_{S\\in \\mathcal{H}_Q} osc_p(f,Q)^p \\big)^{1\/p}.\n\\end{align*}\nSince $\\epsilon$ is arbitrary, this concludes the proof of (\\ref{in3}). \\\\ \\\\\n\\noindent {\\bf Proof of (\\ref{in4}).} Finally note that if $f \\in \\mathcal{B}^s_{p,q}$ and $\\epsilon > 0$ then there is a $\\mathcal{B}^s_{p,q}$-representation of $f$ \n$$f= \\sum_{P \\in \\mathcal{P}} k_P a_P.$$\n such that \n $$ \\big( \\sum_{k=0}^{\\infty} (\\sum_{Q \\in \\mathcal{P}^k} |k_Q|^p)^{q\/p} \\big)^{1\/q} \\leq (1+\\epsilon) |f|_{\\mathcal{B}^s_{p,q}}.$$\nFor each $J \\in \\mathcal{P}^{k_0}$, choose $x_J \\in J$. Then \n\\begin{align} & \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp } osc_p(f,J)^p \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp} \\int_J |f(x) - \\sum_{\\substack{_{Q \\in \\mathcal{P}}\\\\ _{J \\subset Q}}} k_Q a_Q(x_J)|^p \\ dm \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp} \\int |\\sum_{k> k_0} \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} k_R a_R |^p \\ dm \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\Big( \\int | \\sum_{k> k_0}\\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-s} \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} k_R a_R |^p \\ dm \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k> k_0} \\Big( \\int | \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-s} \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} k_R a_R |^p \\ dm \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k> k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp}\\int | \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} k_R a_R |^p \\ dm \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k> k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } \\Big( \\frac{\\sup \\{ |R|\\colon R \\in \\mathcal{P}^k, R \\subset J \\}}{ |J|}\\Big)^{sp} \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} |k_R|^p \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k> k_0} \\Big( \\Crr{maior}^{sp(k-k_0)} \\sum_{J\\in \\mathcal{P}^{k_0} } \\sum_{\\substack{_{R \\in \\mathcal{P}^k}\\\\_{R \\subset J}}} |k_R|^p \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k> k_0} \\Crr{maior}^{s(k-k_0)} \\big( \\sum_{R \\in \\mathcal{P}^k} |k_R|^p \\big)^{1\/p}. \\nonumber \n\\end{align}\nThis is a convolution, so \n\\begin{align*} \\Big( \\sum_{k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp } osc_p(f,J)^p \\Big)^{q\/p} \\Big)^{1\/q} &\\leq \\frac{1+\\epsilon}{1-\\Crr{maior}^s} |f|_{\\mathcal{B}^s_{p,q}}.\\end{align*}\nand since $\\epsilon > 0$ is arbitrary, by Proposition \\ref{lp} we obtain \n\\begin{align*} &|I|^{-s}|f|_p + \\Big( \\sum_{k_0} \\Big( \\sum_{J\\in \\mathcal{P}^{k_0} } |J|^{-sp } osc_p(f,J)^p \\Big)^{q\/p} \\Big)^{1\/q} \\\\\n&\\leq \\Big(\\Crr{mult1}^{1+1\/p} \\Crr{co}(t,q, ( |\\mathcal{P}^k|^{sp} )_k )|I|^{-s}+\\frac{1}{1-\\Crr{maior}^s}\\Big) |f|_{\\mathcal{B}^s_{p,q}}.\\end{align*}\nThis proves (\\ref{in4}).\n \\end{proof}\n \n The following is an important consequence of this section. \n \\begin{corollary} \\label{fou} For each $P\\in \\mathcal{P}$ there exists a linear functional in $L^1$ \n $$f \\mapsto k_P^f$$\n with the following property. The so-called standard $\\mathcal{B}^s_{p,q}$-representation of $f\\in \\mathcal{B}^s_{p,q}$ given by \n$$ f = \\sum_k \\sum_{P \\in \\mathcal{P}^{k}} k_P^f a_P,$$\nsatisfies \n$$\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |k_P^f|^p\\big)^{q\/p}\\Big)^{1\/p}\\leq \\Crr{c2} \\Crr{e}\\Crr{no} |f|_{\\mathcal{B}^s_{p,q}}.$$\n\n\n \n\n \\end{corollary}\n \n \n \\section{Alternative characterizations II: Messing with atoms.}\n Here we move to alternative descriptions of $\\mathcal{B}^s_{p,q}$ which are quite different of those in Section \\ref{alt1}. Instead of choosing a definitive representation of elements of $\\mathcal{B}^s_{p,q}$ we indeed give atomic decompositions of $\\mathcal{B}^s_{p,q}$ using far more general classes of atoms.\n \n \\subsection{Using Besov's atoms}\n\nThe advantage of Besov's atoms is that it is a wide and general class of atoms, that includes even unbounded functions. They can be considered in {\\it every} measure space endowed with a good grid as in Section \\ref{partition}. Moreover in appropriate settings it contains H\\\"older and bounded variations atoms, which will be quite useful in the get other characterizations of $\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})$. The atomic decompositions of $B^s_{p,q}(\\mathbb{R}^n)$ by Besov's atoms were considered by Triebel \\cite{ns} in the case $s>0$, $p=q\\in [1,\\infty]$ and \nby Schneider and Vyb\\'\\i ral \\cite{corjan} in the case $s>0$, $p,q\\in [1,\\infty]$. They are refered there as \"non-smooth atomic decompositions\". \n\n\nLet $s < \\beta$ and $\\tilde{q}\\in [1,\\infty]$. A {\\bf $(s,\\beta,p,\\tilde{q})$-Besov atom} on the interval $Q$ is a function $a \\in \\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})$ such that $a(x)=0$ for $x \\not\\in Q$,\nand\n$$|a|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})}\\leq \\frac{1}{\\Crr{ba}} |Q|^{s-\\beta}.$$\nwhere \n$$\\Cll{ba}=\\Crr{mult1}^{1+1\/p} \\Big( \\sum_k \\Crr{maior}^{k\\beta\\tilde{q}} \\Big)^{1\/\\tilde{q}}\\geq 1.$$\nThe family of $(s,\\beta,p,\\tilde{q})$-Besov atoms supported on $Q$ will be denoted by $\\mathcal{A}^{bv}_{s,\\beta,p,\\tilde{q}}(Q)$. Naturally $\\mathcal{A}^{sz}_{s,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q)$. By Proposition \\ref{lp} we have\n\\begin{eqnarray}\n|a|_{p}&\\leq& \\frac{\\Crr{mult1}^{1+1\/p}}{\\Crr{ba}} \\Big( \\sum_k |\\mathcal{P}^k_Q|^{\\beta\\tilde{q}} \\Big)^{1\/\\tilde{q}} |a|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{\\beta,p})}\\\\\n&\\leq& |Q|^{s-\\beta} |Q|^{\\beta} = |Q|^s= |Q|^{s-\\frac{1}{\\infty}}. \n\\end{eqnarray} \nso a $(s,\\beta,p,\\tilde{q})$-Besov atom is an atom of type $(s,p,1)$. \\\\ \\\\\n\nThe following result says there are many ways to define $\\mathcal{B}^s_{p,q}$ using various classes of atoms. \n\n\\begin{proposition}[Souza's atoms and Besov's atoms]\\label{besova}Let $\\mathcal{P}$ be a good grid. Let $\\mathcal{A}$ be a class of $(s,p,u)$-atoms , with $u\\geq 1$, such that for some $s< \\beta$, $\\tilde{q}\\in [1,\\infty]$, and $\\Crr{56}$, $\\Crr{566} \\geq 0$ we have that for every $Q\\in \\mathcal{P}$ \n$$\\frac{1}{\\Cll{56}} \\mathcal{A}^{sz}_{s,p}(Q) \\subset \\mathcal{A}(Q)\\subset \\Cll{566} \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q).$$\nThen \n $$\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{sz}_{s,p})=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A})=\\mathcal{B}^s_{p,q}(\\mathcal{P},\\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}).$$ \n Moreover\n$$|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})} \\leq \\Crr{56} |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})} \\ and \\ |f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A}^{sz}_{s,p})} \\leq \\frac{\\Crr{566}}{1-\\Crr{maior}^{\\beta-s}}|f|_{\\mathcal{B}^s_{p,q}(\\mathcal{A})}.$$\n\\end{proposition} \n\\begin{proof} The first inequality is obvious. To prove the second inequality, recall that due Proposition \\ref{trans} it is enough to show the following claim\n\\vspace{5mm}\n\n\\noindent {\\it Claim. Let $b_J$ be a $(s,\\beta,p,\\tilde{q})$-Besov atom on $J\\in \\mathcal{P}^j$. Then for every $P\\subset J$ with $P\\in \\mathcal{P}$ there is $m_P \\in \\mathbb{C}$ such that \n$$b_J= \\sum_{P\\subset J} m_P a_P,$$\nwhere $a_P$ is the canonical $(s,p)$-Souza's atom on $P$ and \n$$ \\Big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |m_P|^p\\Big)^{1\/p} \\leq \\frac{2}{\\Crr{ba}} \\Crr{maior}^{(k-j)(\\beta-s)}.$$}\n\\vspace{5mm} \n\n\n\\noindent Indeed, since\n$$|b_J|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(J,\\mathcal{P},\\mathcal{A}^s_{\\beta,p})}\\leq \\frac{1}{\\Crr{ba}} |J|^{s-\\beta},$$\nthere exists a $\\mathcal{B}^\\beta_{p,\\tilde{q}}(J,\\mathcal{P},\\mathcal{A}^s_{\\beta,p})$-representation\n$$b_J=\\sum_{P\\in \\mathcal{P}, P\\subset J} c_P d_P,$$\nwhere $d_P$ is the canonical $(\\beta,p)$-Souza's atom on $P$ and \n$$\\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |c_P|^p\\big)^{1\/p} \\leq \\Big(\\sum_i \\big( \\sum_{P\\in \\mathcal{P}^i, P\\subset J} |c_P|^p \\big)^{\\tilde{q}\/p} \\Big)^{1\/\\tilde{q}}\\leq \\frac{2}{\\Crr{ba}} |J|^{s-\\beta}.$$\nThen \n$$a_P = |P|^{s-\\beta}d_P$$\nis a $(s,p)$-Souza's atom and \n$$b_J=\\sum_{P\\in \\mathcal{P}, P\\subset J} m_P a_P,$$\nwith $m_P = c_P |P|^{\\beta-s}$ and \n\\begin{align} \\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |m_P|^p\\big)^{1\/p} &= \\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} \\Big(\\frac{|P|}{|J|}\\Big)^{p(\\beta-s)} |J|^{p(\\beta-s)}|c_P|^p\\big)^{1\/p} \\nonumber \\\\\n&\\leq \\Crr{maior}^{(k-j)(\\beta-s)} |J|^{\\beta-s} \\big( \\sum_{P\\in \\mathcal{P}^k, P \\subset J} |c_P|^p\\big)^{1\/p} \\nonumber \\\\\n&\\leq \\frac{2}{\\Crr{ba}} \\Crr{maior}^{(k-j)(\\beta-s)}. \\nonumber\n\\end{align}\n\\end{proof} \n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \\subsection{Using H\\\"older atoms} \\label{hatoms2} Suppose that $I$ is a quasi-metric space with a quasi-distance $d(\\cdot,\\cdot)$ and a good grid satisfying the assumptions in Section \\ref{hatoms}. \n\n\\begin{proposition} \\label{hold} Suppose\n$$0< s < \\beta < \\tilde{\\beta},$$\n$p\\in [1,\\infty)$ and $\\tilde{q}\\in [1,\\infty].$ For every $Q\\in \\mathcal{P}$ we have \n\\begin{equation}\\label{final1} \\Cll{at3} \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\tilde{q}}(Q),\\end{equation} for some $\\Crr{at3} > 0$. Moreover\n\\begin{equation}\\label{final2} \\Cll{at4} \\mathcal{A}^{h}_{s,\\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q),\\end{equation} for some $\\Crr{at4} > 0.$ In particular\n\\begin{equation}\\label{inclusion1}\\mathcal{B}^s_{p,q}=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{h}_{s,\\tilde{\\beta},p})=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{h}_{s,\\beta,p}(Q))\\end{equation}\nand the corresponding norms are equivalent.\n\\end{proposition} \n\\begin{proof} Let $\\phi \\in \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)$. Then $\\phi$ has a continuous extension to $\\overline{Q}$. So firstly we assume that $\\phi\\geq 0$ has a continuous extension to $\\overline{Q}$. Define \n$$c_Q= \\min \\phi(Q) |Q|^{1\/p-\\beta}.$$\nand for every $P\\subset W\\subset Q \\in \\mathcal{P}^j$ with $P \\in \\mathcal{P}^{k+1}$ and $W \\in \\mathcal{P}^{k}$ define\n\\begin{equation} \\label{cv} c_P= (\\inf \\phi (P) -\\inf \\phi (W))|P|^{1\/p-\\beta}\\end{equation}\nOf course in this case $c_P\\geq 0$ and\n\\begin{align*} |c_P| &\\leq (\\inf \\phi (P) -\\inf \\phi (W))|P|^{1\/p-\\beta} \\\\\n&\\leq |Q|^{s-1\/p-\\tilde{\\beta}} (diam \\ W )^{\\tilde{\\beta}D} |P|^{1\/p-\\beta}\\\\\n& \\leq \\frac{\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^{\\tilde{\\beta}}}\\Big( \\frac{|P|}{|Q|}\\Big)^{1\/p} |Q|^{s-\\tilde{\\beta}} |P|^{\\tilde{\\beta}-\\beta}\\\\\n& \\leq \\frac{\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^{\\tilde{\\beta}}} \\Cll{super} \\Big( \\frac{|P|}{|Q|}\\Big)^{1\/p} |Q|^{s-\\beta} \\Big(\\frac{|P|}{|Q|}\\Big)^{\\tilde{\\beta}-\\beta}.\n\\end{align*}\nHere $\\Crr{super}= \\sup_{Q\\in \\mathcal{P}}|Q|^{\\tilde{\\beta}-\\beta}.$\nConsequently \n\\begin{align}\\label{adapt} \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} |c_P|^p & \\leq |Q|^{p(s-\\beta)} \\Crr{maior}^{(k+1-j)p(\\tilde{\\beta}-\\beta)} \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} \\frac{ \\Crr{super}^p\\Crr{kki}^{D\\tilde{\\beta}p}}{\\Crr{hh}^{p \\beta}} \\frac{|P|}{|Q|} \\nonumber \\\\\n&\\leq |Q|^{p(s-\\beta)} \\frac{ \\Crr{super}^p \\Crr{kki}^{D\\tilde{\\beta}p}}{\\Crr{hh}^{\\beta p}} \\Crr{maior}^{(k+1-j)p(\\tilde{\\beta}-\\beta)}.\\end{align}\nso\n\\begin{align*}\\Big( \\sum_k \\big( \\sum_{\\substack{P\\in \\mathcal{P}^{k+1}\\\\ P\\subset Q}} |c_P|^p\\big)^{\\tilde{q}\/p} \\Big)^{1\/{\\tilde{q}}} &\\leq \\frac{ \\Crr{super}\\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^\\beta} \\Big(\\frac{1}{1-\\Crr{maior}^{\\tilde{q} (\\tilde{\\beta}-\\beta) }}\\Big)^{1\/\\tilde{q}} |Q|^{s-\\beta}.\\end{align*}\nThis implies that \n$$\\tilde{\\phi}=\\sum_{\\substack{ P\\in \\mathcal{P}\\\\ P\\subset Q}} c_P a_P, $$\nwhere $a_P$ is the canonical $(\\beta,p)$-atom on $P$, is a $\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q,\\mathcal{P}_Q\\mathcal{A}^{sz}_{\\beta,p})$-representation of a function $\\tilde{\\phi}$. From (\\ref{cv}) it follows that\n$$\\sum_{k\\leq N} \\sum_{P\\in \\mathcal{P}^k} c_P a_P(x)= \\min \\phi(W)$$\nfor every $x \\in W\\in \\mathcal{P}^{N}.$ In particular\n$$\\lim_{N\\rightarrow \\infty} \\sum_{k\\leq N} \\sum_{P\\in \\mathcal{P}^k} c_P a_P(x) =\\phi(x).$$\nfor almost every $x$, so $\\tilde{\\phi}=\\phi$. So\n\\begin{equation}\\label{inbv} |\\Crr{at3}\\phi|_{\\mathcal{B}^\\beta_{p,\\tilde{q}}(Q,\\mathcal{P}_Q,\\mathcal{A}^{sz}_{\\beta,p})}\\leq \\frac{1}{2} |Q|^{s-\\beta}\\end{equation} \nwhere\n$$\\Crr{at3}= \\Big( 2\\frac{\\Crr{super} \\Crr{kki}^{D\\tilde{\\beta}}}{\\Crr{hh}^\\beta} \\Big(\\frac{1}{1-\\Crr{maior}^{\\tilde{q}(\\tilde{\\beta}-\\beta)}}\\Big)^{1\/\\tilde{q}}\\Big)^{-1}.$$\n\nIn the general case, note that $\\phi_+(x)=\\max\\{ \\phi,0\\}, \\phi_-(x)=\\min\\{-\\phi,0\\} \\in \\mathcal{A}^{h}_{s,\\tilde{\\beta},p}(Q)$ and $\\phi=\\phi_+-\\phi_-$. Applying (\\ref{inbv}) to $\\phi_+$ and $\\phi_-$ we obtain (\\ref{final1}).\n\nThe second inclusion (\\ref{final2}) in the proposition can be obtained taking $\\tilde{\\beta}=\\beta$ in (\\ref{adapt}) and a few modifications in the above argument. By Proposition \\ref{besova} we have (\\ref{inclusion1}).\n\\end{proof}\n\n\\subsection{Using bounded variation atoms} Now suppose that $I$ is an interval of $\\mathbb{R}$ of length $1$, $m$ is the Lebesgue measure on it and the partitions in $\\mathcal{P}$ are partitions by intervals. \n\n\\begin{proposition} If \n$$0< s\\leq \\beta \\leq \\frac{1}{p}$$\nthen $$\\Crr{at1} \\mathcal{A}^{bv}_{s, \\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q).$$\nfor every $Q\\in \\mathcal{P}.$\nIf $$0< s\\leq \\beta < \\tilde{\\beta} \\leq \\frac{1}{p}$$ \nthen $$\\Crr{at2} \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,q}(Q)$$\nfor every $q \\in [1,\\infty]$. In particular\n$$\\mathcal{B}^s_{p,q}= \\mathcal{B}^s_{p,q}(\\mathcal{A}^{bv}_{s, \\beta,p})=\\mathcal{B}^s_{p,q}(\\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}).$$\nand the corresponding norms are equivalents.\n\\end{proposition} \n\\begin{proof} Suppose\n$$0< s \\leq \\beta \\leq \\tilde{\\beta}\\leq 1$$\nLet $Q\\in \\mathcal{P}^j$ and $a_Q\\in \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)$. We have\n$$|a_Q|_p\\leq ( |Q||Q|^{sp-1} )^{1\/p}=|Q|^s\\leq \\Cll{supe} |Q|^{s-\\beta},$$\nwhere $\\Crr{supe}= \\sup_{Q\\in \\mathcal{P}} |Q|^\\beta.$\nNote that for every $k\\geq j$\n\\begin{align*}\n&\\sum_{W\\in \\mathcal{P}^k} |W|^{-\\beta p} osc_p(a_Q,W)^p \\\\\n &\\leq \n\\sum_{W \\subset Q, W\\in \\mathcal{P}^k} |W|^{1-\\beta p} osc_\\infty(a_Q,W)^p \\\\\n&\\leq \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k} |W|^{\\frac{1-\\beta p}{1-\\tilde{\\beta} p}} \\big)^{1-\\tilde{\\beta} p} \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k} osc_\\infty(a_Q,W)^{1\/\\tilde{\\beta}} \\big)^{\\tilde{\\beta} p} \\\\ \n&\\leq \\Big( \\max_{W \\subset Q, W\\in \\mathcal{P}^k} |W|^{\\frac{(\\tilde{\\beta}-\\beta) p}{1-\\tilde{\\beta} p} } \\Big)^{1-\\tilde{\\beta} p} \\big( \\sum_{W \\subset Q, W\\in \\mathcal{P}^k} |W| \\big)^{1-\\tilde{\\beta} p} (var_{1\/\\tilde{\\beta}}(a_Q,Q))^{p} \\\\\n&\\leq \\Crr{maior}^{(k-j)(\\tilde{\\beta}-\\beta)p}|Q|^{(\\tilde{\\beta}-\\beta)p} |Q|^{1-\\tilde{\\beta} p} |Q|^{sp-1}\\\\\n&\\leq \\Crr{maior}^{(k-j)(\\tilde{\\beta}-\\beta)p} |Q|^{(s-\\beta) p}.\n\\end{align*}\nNote that in the case $\\tilde{\\beta}p=1$ the argument above needs a simple modification. For $k < j$ let $W_Q\\in \\mathcal{P}^k$ be such that $Q\\subset W_Q$. Then \n\\begin{align*}\n&\\sum_{W\\in \\mathcal{P}^k} |W|^{-\\beta p} osc_p(a_Q,W)^p \\leq |W_Q|^{-\\beta p} |Q| |Q|^{sp -1} \\\\\n&\\leq |W_Q|^{-\\beta p} |Q|^{sp} = \\Big( \\frac{|Q|}{|W_Q|}\\Big)^{\\beta p} |Q|^{(s-\\beta)p}\\leq \\Crr{maior}^{(j-k)\\beta p} |Q|^{(s-\\beta)p}.\n\\end{align*}\nBy Theorem \\ref{alte} we have that if $\\beta=\\tilde{\\beta}$ then $\\Cll{at1} \\mathcal{A}^{bv}_{s, \\beta,p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,\\infty}(Q)$ for some $\\Crr{at1} > 0$, and if $\\beta < \\tilde{\\beta}$ we have that for every $q \\in [1,\\infty)$ \n\\begin{align*}\n&|I|^{-s}|a_Q|_p+\\Big( \\sum_{k} \\big( \\sum_{W\\in \\mathcal{P}^k} |W|^{-sp} osc_p(a_Q,W)^p \\big)^{q\/p}\\Big)^{1\/q} \\\\\n&\\leq \\Big( \\Crr{supe} |I|^{-s}+\\big( \\frac{1}{1-\\lambda^{q(\\tilde{\\beta}-\\beta)}} + \\frac{1}{1-\\Crr{maior}^{q\\beta}} \\big)^{1\/q} \\Big) |Q|^{s-\\beta} .\\\\\n\\end{align*}\nso $\\Cll{at2} \\mathcal{A}^{bv}_{s, \\tilde{\\beta},p}(Q)\\subset \\mathcal{A}^{bs}_{s,\\beta,p,q}(Q)$ for some $\\Crr{at2} > 0$.\n\\end{proof}\n\n \n \n\n\n\\section{Dirac's approximations} \nWe will use the Haar basis and notation defined by Section \\ref{haar}. For every $x_0 \\in I$ and $k_0 \\in \\mathbb{N}$ define the finite family \n$$\\mathcal{S}_{x_0}^{k_0} = \\{ S \\colon \\ S=(S_1,S_2) \\in \\mathcal{H}(Q),\\ with \\ Q \\in \\mathcal{P}^k, \\ k < k_0, \\ x_0 \\in \\cup_{a=1,2} \\cup_{P \\in S_{a}} P \\}$$\nLet $N(x_0,k_0)=\\# \\mathcal{S}_{x_0}^{k_0}$. Then we can enumerate the elements $$S^1, S^2, \\dots, S^{N(x_0,k_0)}$$ of $\\mathcal{S}_{x_0}^{k_0}$ \nsuch that $S^i=(S^i_1,S^i_2)$ satisfies\n$$x_0 \\in \\cup_{P \\in S^i_{a_i}} P $$\nfor some $a_i \\in \\{1,2\\}$ and\n$$\\cup_{P \\in S^{i+1}_1\\cup S^{i+1}_2} P \\subset \\cup_{Q \\in S^{i}_1\\cup S^{i}_2} Q$$ \nfor every $i$. \nLet \n$$\\psi_0 = \\frac{1_{I}}{|I|}$$\nand define for $i >0$ \n\\begin{eqnarray*}\n\\psi_{i}&=& (-1)^{a_{i}+1}\\frac{\\sum_{R \\in S^{i}_{3-a_{i}}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} \\big( \\frac{ \\sum_{P \\in S^{i}_1}1_{P}}{\\sum_{P \\in S^{i}_1}|P|} - \\frac{ \\sum_{R \\in S^{i}_2}1_{R}}{\\sum_{R \\in S^{i}_2}|R|} \\big) \\\\\n&=& (-1)^{a_{i}+1} \\frac{\\sum_{R \\in S^{i}_{3-a_{i}}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} m_{S^i} \\phi_i.\n\\end{eqnarray*}\nOne can prove by induction on $j$ that for $j > 0$\n$$\\sum_{i=0}^j \\psi_i = \\frac{\\sum_{P \\in S^{j}_{a_j}}1_{P} }{\\sum_{P \\in S^{j}_{a_j}}|P|}.$$\nand in particular \n$$\\sum_{i=0}^{N(x_0,k_0)} \\psi_i (x_0)= \\frac{1_{Q_{k_0}} }{|Q_{k_0}|},$$\nwhere $x_0 \\in Q_{k_0} \\in \\mathcal{P}^{k_0}.$ In other words \n\\begin{equation}\\label{part} \\frac{1}{ |I|^{1\/2} }\\phi_{I} + \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} \\Big) m_{S^i} \\phi_{S^i}=\\frac{1_{Q_{k_0}} }{|Q_{k_0}|},\\end{equation} \nNote that \n\\begin{align}\n\\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} \\Big) m_{S^i}&= \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|} \\Big) \\Big( \\frac{\\sum_{R \\in S_2^i} |R| + \\sum_{P \\in S_1^i} |P| }{(\\sum_{R \\in S_2^i} |R| ) (\\sum_{P \\in S_1^i} |P|)} \\Big)^{1\/2} \\nonumber \\\\\n\\label{igual} &= \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|} \\Big)^{1\/2} \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}} \n\\end{align}\nMultiplying (\\ref{part}) by $f$ and integrating it term by term, and using (\\ref{igual}) we obtain \n$$\\frac{d_{I}}{|I|^{1\/2}} + \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|} \\Big)^{1\/2} \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}} d _{S^i} =\\int f \\cdot \\frac{1_{Q_{k_0}}}{|Q_{k_0}|} \\ dm.$$\n\n\\noindent If $f \\in L^\\beta$, with $\\beta > 1$, it can be written as \n$$f = \\sum_{S\\in \\hat{\\mathcal{H}}(\\mathcal{P})} d_S\\phi_S$$\nwith $d_S= \\int f \\phi_S \\ dm$, where this series converges unconditionally on $L^\\beta$. Let\n\\begin{equation}\\label{k0} f_{k_0} = d_{I} \\phi_{I}+ \\sum_{k < k_0} \\sum_{Q\\in \\mathcal{P}^k}\\sum_{S\\in \\mathcal{H}(Q)} d_S\\phi_S.\\end{equation}\nThen\n\\begin{align} f_{k_0}(x_0) &= d_{I} \\phi_{I} +\\sum_{i=1}^{N(x_0,k_0)} d_{S^i} \\phi_{S^i}(x_0).\\nonumber \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}}+ \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1}d_{S^i} \\frac{1}{m_{S^i}} \\frac{1}{\\sum_{P \\in S_{a_i}^i} |P| } \\nonumber \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}}+\\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} d_{S^i} \\Big( \\frac{(\\sum_{R \\in S_2^i} |R| ) (\\sum_{P \\in S_1^i} |P|)}{\\sum_{R \\in S_2^i} |R| + \\sum_{P \\in S_1^i} |P| } \\Big)^{1\/2} \\frac{1}{\\sum_{P \\in S_{a_i}^i} |P| } \\nonumber \\\\\n&=\\frac{d_{I}}{|I|^{1\/2}} + \\sum_{i=1}^{N(x_0,k_0)} (-1)^{a_i+1} d_{S^i} \\Big( \\frac{\\sum_{R \\in S^{i}_{3-a_i}}|R|}{\\sum_{P \\in S^{i}_{a_i}} |P|} \\Big)^{1\/2} \\frac{1}{(\\sum_{Q \\in S^{i-1}_{a_{i-1}}} |Q|)^{1\/2}} \\nonumber \\\\\n&= \\int f \\cdot \\frac{1_{Q_{k_0}}}{|Q_{k_0}|} \\ dm. \\nonumber \n\\end{align}\nLet \n\\begin{equation}\\label{reep} f = \\sum_k \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P,\\end{equation}\nbe the series given by (\\ref{sumf}). Note that\n\\begin{equation} f_{k_0} = \\sum_{k\\leq k_0} \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P,\\end{equation}\n\nConsequently \n\n\\begin{proposition}[Dirac's Approximations]\\label{boup} Let $f \\in L^\\beta$, with $\\beta > 1$. Let\n$$ f = \\sum_k \\sum_{P\\in \\mathcal{P}^{k}} k_P a_P.$$\n\\begin{itemize}\n\\item[A.] If this representation is either as in (\\ref{sumf}) or $k_P\\geq 0$ for every $P$ then we have For every $Q \\in \\mathcal{P}$ \n$$\\Big| \\sum_{J\\in \\mathcal{P}, Q \\subset J} k_J |J|^{s-1\/p} \\Big| \\leq |f1_Q|_\\infty.$$\n\\item[B.] In the case of the representation (\\ref{sumf}) we also have \n$$\\sum_{J\\in \\mathcal{P}, Q \\subset J} k_J |J|^{s-1\/p} =\\int f \\cdot \\frac{1_{Q}}{|Q|} \\ dm.$$\n\\end{itemize}\n\\end{proposition} \n\\begin{proof} We have that $A.$ is obvious if $k_P\\geq 0$ for every $P$. In the other case note that \nfor every $x_0 \\in Q\\in \\mathcal{P}^{k_0}$ we have\n $$f_{k_0}(x_0)=\\sum_{J\\in \\mathcal{P}, Q\\subset J} k_J |J|^{s-1\/p}= \\int f \\cdot \\frac{1_{Q}}{|Q|} \\ dm.$$\nso A. and B. follows.\n\\end{proof}\n\n\n\\newpage\n\n\\centerline{ \\bf III. APPLICATIONS.}\n\\addcontentsline{toc}{chapter}{\\bf III. APPLICATIONS.}\n\\vspace{1cm}\n\n\\fcolorbox{black}{white}{\n\\begin{minipage}{\\textwidth}\n\\noindent In Part III. we suppose that $0< s<1\/p$, $p\\in [1,\\infty)$ and $q\\in [1,\\infty]$. \\end{minipage} \n}\n\\ \\\\ \\\\\n\n \n\\section{Pointwise multipliers acting on $\\mathcal{B}^s_{p,q}$}\n\nHere we will apply the previous sections to study pointwise multipliers of $\\mathcal{B}^s_{p,q}$. To be more precise, Let $g\\colon I \\rightarrow \\mathbb{C}$ be a measurable function. We say that $g$ is a pointwise multiplier acting on $\\mathcal{B}^s_{p,q}$ if the transformation\n$$G(f)=gf$$\ndefines a bounded operator in $\\mathcal{B}^s_{p,q}$. We denote the set of pointwise multipliers by $M(\\mathcal{B}^s_{p,q})$. We can consider the norm on $M(\\mathcal{B}^s_{p,q})$ given by\n$$|g|_{M(\\mathcal{B}^s_{p,q})}=\\sup\\{|gf|_{\\mathcal{B}^s_{p,q}}\\ s.t. \\ |f|_{\\mathcal{B}^s_{p,q}}\\leq 1 \\}.$$\nOf course a necessary condition for a function to be a multiplier is that \n\n$$\\mathcal{B}^s_{p,q,selfs}=\\{ g\\in \\mathcal{B}^s_{p,q}\\colon \\sup_{a_Q \\in \\mathcal{A}^{sz}_{s,p}} |ga_Q|_{\\mathcal{B}^s_{p,q}} < \\infty\\}$$ \nDenote\n$$|g|_{\\mathcal{B}^s_{p,q,selfs}}=\\sup_{a_Q \\in \\mathcal{A}^{sz}_{s,p}} |ga_Q|_{\\mathcal{B}^s_{p,q}}.$$ \nThe linear space $\\mathcal{B}^s_{p,q,selfs}$ endowed with $|\\cdot|_{\\mathcal{B}^s_{p,q,selfs}}$is a normed space introduced by Triebel \\cite{ns}. We have \n$$|g|_{\\mathcal{B}^s_{p,q,selfs}}\\leq |g|_{M(\\mathcal{B}^s_{p,q})}$$ \nIn the following three propositions we see that many results of Triebel \\cite{ns} and Schneider and Vyb\\'\\i{ral} \\cite{corjan} for Besov spaces in $\\mathbb{R}^n$ can be easily moved to our setting. The simplest case occurs when $p=q=1$.\n\n\n\n\\begin{proposition} We have that $M(\\mathcal{B}^s_{1,1})=\\mathcal{B}^s_{1,1,selfs}$.\n\\end{proposition} \n\\begin{proof} Let $g \\in \\mathcal{B}^s_{1,1,selfs}.$ Given $f\\in \\mathcal{B}^s_{1,1}$ and $\\epsilon > 0$ one can find a $\\mathcal{B}^s_{1,1}$-representation \n$$f=\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qa_Q,$$\nwhere $a_Q$ is a $(s,1)$-Souza's atom and \n$$\\sum_k \\sum_{Q\\in \\mathcal{P}^k} |c_Q| < (1+\\epsilon) |f|_{\\mathcal{B}^s_{1,1}}$$\nso \n$$\\sum_k \\sum_{Q\\in \\mathcal{P}^k} |c_Q ga_Q|_{\\mathcal{B}^s_{1,1}} < (1+ \\epsilon) |g|_{\\mathcal{B}^s_{1,1,selfs}}|f|_{\\mathcal{B}^s_{1,1}}.$$ \nand consequently \n$$|gf|_{\\mathcal{B}^s_{1,1}}=|\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qga_Q|_{\\mathcal{B}^s_{1,1}}\\leq (1+ \\epsilon) |g|_{\\mathcal{B}^s_{1,1,selfs}}|f|_{\\mathcal{B}^s_{1,1}}.$$\nSince $\\epsilon$ is arbitrary we get\n$$|gf|_{\\mathcal{B}^s_{1,1}}\\leq |g|_{\\mathcal{B}^s_{1,1,selfs}} |f|_{\\mathcal{B}^s_{1,1}}.$$\n\\end{proof} \n\n\n\\begin{lemma}\\label{incint3} Let $W\\in \\mathcal{P}$. The restriction application\n$$r\\colon \\mathcal{B}^s_{p,q}(I, \\mathcal{P}, \\mathcal{A}^{sz}_{s,p})\\rightarrow \\mathcal{B}^s_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})$$\ngiven by $r(f)=1_Wf$ is continuous. Indeed there is $\\Cll{incint} \\geq 1$, that does not depend on $W$, such that \n\\begin{enumerate}\n\\item[A.] For every $f \\in \\mathcal{B}^s_{p,q}$ we have \n\\begin{equation}\\label{incint2} |1_Wf|_{\\mathcal{B}^s_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})}\\leq \\Crr{incint} |f|_{\\mathcal{B}^s_{p,q}}. \\end{equation}\nIn particular $1_W\\in M(\\mathcal{B}^s_{p,q})$.\n\\item[B.] For every $\\mathcal{B}^{s+}_{p,q}$-representation \n$$f=\\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} c_Qa_Q$$\n one can find a $\\mathcal{B}^{s+}_{p,q}(W, \\mathcal{P}_W, \\mathcal{A}^{sz}_{s,p})$-representation \n$$1_Wf= \\sum_{k}\\sum_{Q\\in \\mathcal{P}^k} d_Qa_Q$$\nsuch that \n\\begin{equation}\\label{incint2b}\n\\Big( \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k} (d_Q)^p\\big)^{q\/p} \\Big)^{1\/q} \\leq \\Crr{incint} \\Big( \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k} (c_Q)^p\\big)^{q\/p} \\Big)^{1\/q} . \n\\end{equation} \nMoreover $d_Q\\neq 0$ implies $Q\\subset \\ supp \\ f$.\n\\end{enumerate}\n\\end{lemma} \n\\begin{proof} Let $Q\\in \\mathcal{P}.$ Denote by $a_Q$ the canonical $(s,p)$-Souza's atom supported on $Q$. If $W\\cap Q=\\emptyset$ we can write $1_Wa_Q= 0a_Q.$ If $Q\\subset W$ then $1_Wa_Q= 1a_Q.$ If $W\\subset Q$ then\n$$1_Wa_Q= \\Big(\\frac{|W|}{|Q|}\\Big)^{1\/p-s}a_W,$$\nwhere\n$$ \\Big(\\frac{|W|}{|Q|}\\Big)^{1\/p-s}\\leq \\Crr{maior}^{(k_0(W)-k_0(Q))(1\/p-s)}$$\nIn every case we can write\n$$h_Q=1_Wa_Q = \\sum_k \\sum_{\\substack{P\\in \\mathcal{P}^k\\\\ P\\subset Q}} s_{P,Q} a_P,$$\nwith \n$$ \\sum_{\\substack{P\\in \\mathcal{P}^k\\\\ P\\subset Q}} |s_{P,Q}| ^p\\leq \\Crr{maior}^{(k-k_0(Q))(1-sp)}$$\nBy Proposition \\ref{trans}.A and \\ref{trans}.B there is $\\Crr{incint}$ such that A. and B. hold. \n\\end{proof}\n\n\\begin{proposition} We have that $ \\mathcal{B}^s_{p,q,selfs} \\subset L^\\infty$ and this inclusion is continuous.\n\\end{proposition} \n\\begin{proof} Let $g\\in \\mathcal{B}^s_{p,q,selfs}$. Then $ga_Q\\in \\mathcal{B}^s_{p,q}$ and by Lemma \\ref{incint3} we have\n$$|g1_Q|_{ \\mathcal{B}^s_{p,q}(Q, \\mathcal{P}_Q, \\mathcal{A}^{sz}_{s,p})}\\leq \\Crr{incint}|g1_Q|_{\\mathcal{B}^s_{p,q}} \\leq \\Crr{incint} |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\n\nBy Proposition \\ref{lp} (taking t = $p$) we have $(s,p)$-Souza's atom $a_Q$ we have \n$$|ga_Q|_p\\leq \\Crr{mult1}^{1+1\/p} \\Crr{co}(p,q, ( |\\mathcal{P}^k_Q|^{sp})_k ) \\Crr{incint} |g|_{\\mathcal{B}^s_{p,q,selfs}} < \\Cll{nova} |Q|^s |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\nfor some constant $\\Crr{nova}$. In other words\n$$\\Big( |Q|^{sp-1} \\int_Q |g|^p \\ dm \\Big)^{1\/p} \\leq \\Crr{nova} |Q|^s |g|_{\\mathcal{B}^s_{p,q,selfs}},$$\nso\n$$\\frac{1}{|Q|} \\int_Q |g|^p \\ dm \\leq \\Crr{nova}^p |g|_{\\mathcal{B}^s_{p,q,selfs}}^p, $$\nfor every $Q\\in \\mathcal{P}$. Due to the fact that $\\cup_k\\mathcal{P}^k$ generates the $\\sigma$-algebra $\\mathbb{A}$, by L\\'evy's Upward Theorem (see Williams \\cite{martin}) for almost every $x\\in I$ the following holds. If $x\\in Q_k\\in \\mathcal{P}^k$ then \n$$\\lim_k \\frac{1}{|Q_k|} \\int_{Q_k} |g|^p \\ dm = |g(x)|^p.$$\nSo \n$$|g|_\\infty\\leq \\Crr{nova} |g|_{\\mathcal{B}^s_{p,q,selfs}}.$$\n\\end{proof} \n\n\\begin{figure}\n\\includegraphics[scale=0.5]{narch.pdf}\n\\caption{ Non-Archimedean behaviour. Illustration for Proposition \\ref{sepa}. The filled regions are the supports of the functions $g_i$. the squares are the elements of the grid on which there are atoms contributing for the representation of $f$. Every square intercepts at most one support, so each atom ``sees'' only the function whose support is nearby.}\n\\end{figure}\n\n\\subsection{Non-Archimedean behaviour in $\\mathcal{B}^\\beta_{p,\\tilde{q},selfs}$} If we have a sequence $g_i \\in M(\\mathcal{B}^s_{p,q})$ we can get the naive estimate \n$$|(\\sum_i g_i)f|_{\\mathcal{B}^s_{p,q}}\\leq \\big( \\sum_i |g_i|_{M(\\mathcal{B}^s_{p,q})}\\big)|f|_{\\mathcal{B}^s_{p,q}}.$$\n\nBut it is remarkable that sometimes one can get a far better estimate. To state the result we need to define\n\n$$|g|_{\\mathcal{B}^{s,t}_{p,q,selfs}}=\\sup_{\\substack{ a_Q \\in \\mathcal{A}^{sz}_{s,p}\\\\ Q\\in \\mathcal{P}^j \\\\ j\\geq t}} |ga_Q|_{\\mathcal{B}^s_{p,q}}.$$\nIt is easy to see that this norm is equivalent to $|\\cdot|_{\\mathcal{B}^{s}_{p,q,selfs}}$.\n\n\\begin{proposition} \\label{sepa} Let $\\beta > s$. There is $\\Cll{gen}$ with the following property. Let $g_i \\in \\mathcal{B}^{\\beta,t_i}_{p,\\tilde{q},selfs}$, with $i\\in \\Lambda \\subset \\mathbb{N}$, and $t_i \\in \\mathbb{N}$.\n\nConsider a function $f$ with a $\\mathcal{B}^s_{p,q}$-representation \n$$ f=\\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nsatisfying \n\\begin{itemize}\n\\item[A.] We have \n$$\\sup_{\\substack{ Q\\in \\mathcal{P} \\\\ c_Q\\neq 0}} \\ \\sum_{_{Q\\cap supp \\ g_i \\neq \\emptyset}} |g_i|_{ \\mathcal{B}^{\\beta,t_i}_{p,\\tilde{q},selfs}} \\leq N.$$\n\\item[B.] If $Q\\in \\mathcal{P}^k$ satisfies $c_Q\\neq 0$ and $Q\\cap supp \\ g_i \\neq \\emptyset$ then $k\\geq t_i$. \n\\end{itemize}\nThen we can find a $\\mathcal{B}^s_{p,q}$-representation\n\\begin{equation} \\label{rre} (\\sum_i g_i)f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} d_Q a_Q\\end{equation} \nsuch that \n\\begin{align}\\label{reep} &\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |d_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\\n&\\leq \\Crr{gen} N \\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |c_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \n\\end{align} \n\\end{proposition} \n\\begin{proof} It is enough to prove the result for the case when $\\Lambda$ is finite. Let $Q\\in \\mathcal{P}^{k_0}$ with $c_Q\\neq 0$. There is $\\{i_1,\n\\dots, i_j\\}\\subset \\Lambda$, such that \n\\begin{equation}\\label{aa1} (\\sum_i g_i ) a_Q= \\sum_{\\ell\\leq j} g_{i_\\ell} a_Q.\\end{equation} \nand $Q\\cap supp \\ g_{i_\\ell} \\neq \\emptyset$ for every $\\ell$. In particular $k_0\\geq \\max_\\ell t_{i_\\ell}.$\nBy Lemma \\ref{incint3} for each $\\ell\\leq j$ we can find a $\\mathcal{B}^\\beta_{p,q} $-representation \n\\begin{equation}\\label{hjk} g_{i_\\ell}a_Q= \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} \\tilde{s}_{P,Q}^\\ell b_{P}\\end{equation}\nsuch that $b_P$ is the canonical $(\\beta,p)$-Souza's atom supported on $P$ and \n$$\\Big( \\sum_k \\big( \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} |\\tilde{s}_{P,Q}^\\ell|^p \\big)^{\\tilde{q}\/p} \\Big)^{1\/\\tilde{q}}\\leq 2\\Crr{incint} |g_{i_\\ell}|_{\\mathcal{B}^{\\beta,t_{i_\\ell}}_{p,\\tilde{q},selfs}} .$$\nSince $b_P= |P|^{\\beta -s} a_P$, where $a_P$ is the canonical $(s,p)$-Souza's atoms supported on $P$, we can write\n$$g_{i_\\ell}a_Q= \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} s_{P,Q}^\\ell a_{P},$$\nwith $s_{P,Q}^\\ell=\\tilde{s}_{P,Q}^\\ell|P|^{\\beta -s}$ satisfying \n\\begin{align*}\n &\\big( \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} |s_{P,Q}^\\ell|^p \\big)^{1\/p} \\\\\n&\\leq 2\\Crr{incint} |g_{i_\\ell}|_{\\mathcal{B}^{\\beta,t_{i_\\ell}}_{p,\\tilde{q},selfs}} \\Crr{maior}^{(\\beta-s)(k-k_0)} \\sup_{Q\\in \\mathcal{P}}|Q|^{\\beta-s},\n\\end{align*} \nso we can write\n$$(\\sum_i g_i ) a_Q= \\sum_k \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} s_{P,Q} a_{P},$$\nwith \n$$s_{P,Q}=\\sum_\\ell s_{P,Q}^\\ell$$\n satisfying \n$$\\big( \\sum_{\\substack{ P\\in \\mathcal{P}^k \\\\ P\\subset Q}} |s_{P,Q}|^p \\big)^{1\/p} \\leq 2 N \\Crr{incint} \\Crr{maior}^{(\\beta-s)(k-k_0)} \\sup_{Q\\in \\mathcal{P}}|Q|^{\\beta-s}.$$\nBy Proposition \\ref{trans}.A we can find a $\\mathcal{B}^s_{p,q}$-representation (\\ref{rre}) satisfying (\\ref{reep}). \n\\end{proof}\n\n\\begin{remark}\\label{posrem} If $g$ is $\\mathcal{B}^\\beta_{p,q}$-positive we can define\n$$|g|_{\\mathcal{B}^{s+,t}_{p,q,selfs}}=\\sup_{\\substack{ a_Q \\in \\mathcal{A}^{sz}_{s,p}\\\\ Q\\in \\mathcal{P}^j \\\\ j\\geq t}} |ga_Q|_{\\mathcal{B}^{s+}_{p,q}}.$$\n If we assume additionally that $g_i$ are $\\mathcal{B}^{\\beta,t_i}_{p,q}$-positive, Proposition \\ref{sepa} remains true if we replace all the instances of $|\\cdot|_{\\mathcal{B}^{s,t}_{p,q,selfs}}$ by $|\\cdot|_{\\mathcal{B}^{s+,t}_{p,q,selfs}}$ in its statement. Moreover by Proposition \\ref{trans}.B and Lemma \\ref{incint3}. B we can conclude that\n \\begin{itemize}\n \\item[i.] if $c_Q\\geq 0$ for every $Q$ then $d_Q\\geq 0$ for every $Q$,\n \\item[ii.] If $Q$ is such that $d_Q\\neq 0$ then $Q\\subset supp \\ g_i$, for some $i\\in \\Lambda$. \n\\end{itemize}\n\\end{remark} \n\n\\begin{corollary}\\label{23er} For every $\\beta > s$ and $\\tilde{q}\\in [1,\\infty]$ we have $\\mathcal{B}^\\beta_{p,\\tilde{q},selfs} \\subset M(\\mathcal{B}^s_{p,q})$. Moreover this inclusion is continuous. \n\\end{corollary} \n\n\n\n\n\\subsection{Strongly regular domains }\\label{srd} We may wonder on which conditions the characteristic function of a set $\\Omega$ is a pointwise multiplier in $\\mathcal{B}^{s}_{p,q}$. \n\\begin{definition} A measurable set $\\Omega\\subset I$ is a {\\bf $(\\alpha,\\Cll{rp2},\\Cll{k1} )$-strongly regular domain} if for every $Q\\in \\mathcal{P}^j$, with $j\\geq \\Crr{k1}$, there is family $\\mathcal{F}^k(Q\\cap \\Omega) \\subset \\mathcal{P}^k$ such that \n\\begin{itemize}\n\\item[i.] We have $Q\\cap \\Omega = \\cup_{k} \\cup_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} P$.\n\\item[ii.] If $P,W \\in \\cup_{k} \\mathcal{F}^k(Q\\cap \\Omega)$ and $P\\neq W$ then $P\\cap W=\\emptyset$. \n\\item[iii.] We have\n\\begin{equation} \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} |P|^{\\alpha}\\leq \\Crr{rp2} |Q|^{\\alpha}.\\end{equation} \n\\end{itemize}\n\\end{definition} \n\n\nThe following result can be associated with results in Triebel \\cite{ns} for $B^s_{p,p}(\\mathbb{R}^n)$, specially when we consider the setting of Besov spaces in compact homogenous spaces. See Section \\ref{srd} for details.See also Schneider and Vyb\\'\\i ral \\cite{corjan}.\n\n\\begin{proposition} \\label{pos2} If $\\Omega$ is a $(1-\\beta p,\\Crr{rp2},\\Crr{k1} )$-strongly regular domain then \n\\begin{equation}\\label{fimm} |1_\\Omega|_{\\mathcal{B}^{\\beta+,{\\Crr{k1}}}_{p,\\infty,selfs}}\\leq \\Crr{rp2}^{1\/p}.\\end{equation}\n\\end{proposition} \n\\begin{proof} Given $Q\\in \\mathcal{P}^j$, with $j\\geq \\Crr{k1}$ we can write\n\\begin{equation} \\label{poi} 1_\\Omega a_Q = \\sum_{k} \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} \\Big(\\frac{|P|}{|Q|} \\Big)^{1\/p-\\beta} a_P.\\end{equation} \nwhere $a_P$ is a $(\\beta,p)$-atom. Note that \n$$ \\big( \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} \\Big(\\frac{|P|}{|Q|} \\Big)^{1-\\beta p} \\big)^{1\/p} \\leq \\Crr{rp2}^{1\/p},$$\nso (\\ref{fimm}) holds.\n\\end{proof}\n\n\\begin{proposition}[Pointwise Multipliers I] \\label{pm1} There is $\\Cll{gen2}$ with the following property. Suppose that $\\Omega_i$ are $(1-\\beta p,K_i,t_i)$-strongly regular domains, $i\\in \\Lambda \\subset \\mathbb{N}$, and $\\Theta_i> 0$ for every $i\\in \\Lambda_i$.\nConsider a function $f$ with a $\\mathcal{B}^s_{p,q}$-representation \n$$ f=\\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nsatisfying \n\\begin{itemize}\n\\item[A.] We have \n$$\\sup_{\\substack{ Q\\in \\mathcal{P} \\\\ c_Q\\neq 0}} \\ \\sum_{_{Q\\cap \\Omega_i \\neq \\emptyset}} \\Theta_i K_i^{1\/p} \\leq N.$$\n\\item[B.] If $Q\\in \\mathcal{P}^k$ satisfies $c_Q\\neq 0$ and $Q\\cap \\Omega_i \\neq \\emptyset$ then $k\\geq t_i$. \n\\end{itemize}\n Then we can find a $\\mathcal{B}^s_{p,q}$-representation\n\\begin{equation} \\label{rre2} (\\sum_i \\Theta_i 1_{\\Omega_i})f= \\sum_k \\sum_{P\\in \\mathcal{P}^k} d_Q a_Q\\end{equation} \nsuch that \n\\begin{align}\\label{reep2} &\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |d_Q|^p \\big)^{q\/p} \\Big)^{1\/q} \\nonumber \\\\\n&\\leq \\Crr{gen2} N\\Big(\\sum_k \\big( \\sum_{P\\in \\mathcal{P}^k} |c_Q|^p \\big)^{q\/p} \\Big)^{1\/q}.\n\\end{align} \nMoreover \n\\begin{itemize}\n\\item[i.] If $Q$ satisfies $d_Q\\neq 0$ then $Q\\subset \\Omega_i$ for some $i \\in \\Lambda$. \n\\item[ii.] If $c_Q\\geq 0$ for every $Q$ then $d_Q\\geq 0$ for every $Q$. \n\\end{itemize}\n\\end{proposition} \n\\begin{proof} It follows from Proposition \\ref{sepa}, Proposition \\ref{pos2} and Remark \\ref{posrem}. \n\\end{proof} \n\n\n\n\n\\subsection{Functions on $\\mathcal{B}^{1\/p}_{p,\\infty}\\cap L^\\infty$ } \n\nWe would like to give explicit examples of multipliers in $\\mathcal{B}^s_{p,q}$. One should compare the following result with the study by Triebel\\cite{multi} of the regularity of the multiplication on Besov spaces. See also Maz'ya and Shaposhnikova \\cite{sob} for more information on multipliers in classical Besov spaces.\n\n\n\\begin{proposition}[Pointwise multipliers II] \\label{mult} Let $g\\in \\mathcal{B}^{1\/p}_{p,\\infty}\\cap L^\\infty$. Then the multiplier operator\n$$G\\colon \\mathcal{B}^s_{p,q} \\rightarrow \\mathcal{B}^s_{p,q}$$\ndefined by $G(f)=gf$ is a well defined and bounded operator acting on $(\\mathcal{B}^s_{p,q},|~\\cdot~|_{\\mathcal{B}^s_{p,q}})$.\nIndeed\n$$|G|_{\\mathcal{B}^s_{p,q}}\\leq \\Crr{e}\\Crr{no} \\frac{|g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} }{1-\\Crr{maior}^{1\/p-s}} +|g|_\\infty,$$\nwhere $\\Crr{e}=\\Crr{e}(1\/p,p,\\infty)$ and $\\Crr{no}=\\Crr{no}(1\/p,p,\\infty)$ are as in Corollary \\ref{fou}.\n\\end{proposition} \n\\begin{remark} \\label{pos3} We can get a similar result replacing $\\mathcal{B}^{a}_{p, b}$ by $\\mathcal{B}^{a+}_{p, b}$ everywhere. \n\\end{remark} \n\\begin{proof} Let $a_Q=|Q|^{s-1\/p}1_Q$ be the canonical $(s,p)$-Souza's atom on $Q$ and $b_J=1_J$ be the canonical $(1\/p,p)$-Souza's atom on $J$. Given $\\epsilon > 0$, let \n$$f = \\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q$$\nbe a $\\mathcal{B}^s_{p,q}$-representation of $f$ such that \n$$ \\Big( \\sum_k \\big(\\sum_{Q\\in \\mathcal{P}^k} |c_Q|^p\\big)^{q\/p} \\Big)^{1\/q} \\leq (1+\\epsilon)|f|_{\\mathcal{B}^s_{p,q}}$$\nand\n$$g =\\sum_k \\sum_{J\\in \\mathcal{P}^k} e_J b_J$$\nbe a $\\mathcal{B}^{1\/p}_{p,\\infty}$-representation of $g$ given by Corollary \\ref{fou} (in the case of Remark \\ref{pos3} we can consider an optimal $\\mathcal{B}^{1\/p}_{p,\\infty}$-positive representation of $g$). We claim that \n$$u_1 =\\sum_j \\sum_{J \\in \\mathcal{P}^j} \\big(\\sum_{J\\subset Q, Q\\neq J,Q\\in \\mathcal{P}} \\big( \\frac{|J|}{|Q|}\\big)^{1\/p-s} c_Q e_J \\big) a_J, \\ and $$\n$$u_2 = \\sum_k \\sum_{Q \\in \\mathcal{P}^k} \\big( \\sum_{Q\\subset J, J\\in \\mathcal{P} } c_Q e_J \\big) a_Q,$$\nare $\\mathcal{B}^s_{p,q}$-representations of functions $u_i \\in \\mathcal{B}^s_{p,q}$. Firstly note that the inner sums are finite. Moreover if $J\\in \\mathcal{P}^j$ we denote by \n$Q_k(J)$ the unique element of $\\mathcal{P}^k$, with $k\\leq j$ that satisfies $J \\subset Q_k(J)$ then \n\\begin{align}& \\ \\big( \\sum_{J \\in \\mathcal{P}^j } \\big|\\sum_{k \\leq j} \\Big( \\frac{|J|}{|Q_k(J)|}\\Big)^{1\/p-s} c_{Q_k(J)} e_J\\big|^p \\big)^{1\/p} \\nonumber \\\\\n&\\leq \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)} \\big( \\sum_{J \\in \\mathcal{P}^j }| c_{Q_k(J)} e_J|^p \\big)^{1\/p} \\nonumber \\\\\n&\\leq \\big( \\sum_{J \\in \\mathcal{P}^j }|e_J|^p \\big)^{1\/p} \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)} \\max_{Q\\in \\mathcal{P}^k} |c_Q|\\nonumber \\\\\n&\\leq \\Big( \\max_j \\big( \\sum_{J \\in \\mathcal{P}^j }|e_J|^p \\big)^{1\/p} \\Big) \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)} (\\sum_{Q\\in \\mathcal{P}^k} |c_Q|^p)^{1\/p} \\nonumber \\\\\n&\\leq \\Crr{e}\\Crr{no} |g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} \\sum_{k \\leq j} \\Crr{maior}^{(j-k)(1\/p-s)} (\\sum_{Q\\in \\mathcal{P}^k} |c_Q|^p)^{1\/p} \\nonumber \n\\end{align} \nThe right hand side is a convolution, so we can easily get \n$$ |u_1|_{\\mathcal{B}^s_{p,q}} \\leq (1+\\epsilon) \\Crr{e}\\Crr{no} \\frac{|g|_{\\mathcal{B}^{1\/p}_{p,\\infty}} }{1-\\Crr{maior}^{1\/p-s}} |f|_{\\mathcal{B}^s_{p,q}}.$$\nMoreover by Proposition \\ref{boup}.B, with $s=1\/p$, we obtain\n\\begin{align}& \\Big( \\sum_{Q \\in \\mathcal{P}^k} \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J} c_Q e_J \\big|^p \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\Big( \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\big| \\sum_{J\\in \\mathcal{P}, Q\\subset J} e_J \\big|^p \\Big)^{1\/p} \\nonumber \\\\\n&\\leq \\Big( \\sum_{Q \\in \\mathcal{P}^k} |c_Q|^p \\Big)^{1\/p} |g|_{\\infty}. \\nonumber \n\\end{align} \nSo\n$$ |u_1|_{\\mathcal{B}^s_{p,q}} \\leq (1+\\epsilon) |f|_{\\mathcal{B}^s_{p,q}} |g|_\\infty .$$\nWe claim that $gf= u_1+u_2$. Indeed let\n$$f_{k_0} = \\sum_{k s$. Every $(1-\\beta p, C,0)$-strongly regular domain is a $(1-sp,C',\\Crr{maior}^{(\\beta-s)p})$-regular domain,\nfor some $C'$. \n\\end{proposition}\n\n\\begin{proof} Consider a $(1-\\beta p,\\Crr{rp2},0)$-strongly regular domain $\\Omega$. There are at most $\\Crr{menor}^{-k_0(\\Omega)}$ elements in $\\mathcal{P}^{k_0(\\Omega)}$ and\n$$\\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{-k_0(\\Omega)} \\leq \\frac{|Q|}{|W|}\\leq \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)}$$\nfor every $Q,W\\in \\mathcal{P}^{k_0(\\Omega)}$. Consequently\n$$\\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{-k_0(\\Omega)} \\leq \\frac{|\\Omega|}{|Q|}\\leq \\Crr{menor}^{-k_0(\\Omega)} \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)}$$\nfor each $Q \\in \\mathcal{P}^{k_0(\\Omega)}$.\nFor every $Q\\in \\mathcal{P}^{k_0(\\Omega)}$ there is a family $\\mathcal{F}^k(Q\\cap \\Omega)$ such that \n$$\\sum_k \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)} P= Q\\cap \\Omega$$\nand\n$$ \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\leq C |Q|^{1-\\beta p}.$$\nLet \n$$\\mathcal{F}^k(\\Omega)=\\cup_{Q\\in \\mathcal{P}^{k_0(\\Omega)} } \\mathcal{F}^k(Q\\cap \\Omega).$$\nWe have\n\\begin{align*} \\sum_{Q\\in \\mathcal{P}^{k_0(\\Omega)}}\\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-s p}&=\\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p} |P|^{(\\beta-s)p}\\\\\n&\\leq \\sum_{Q\\in \\mathcal{P}^{k_0(\\Omega)}} \\big(\\max_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{(\\beta-s)p} \\big) \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\\\\n&\\leq \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} \\sum_{Q\\in \\mathcal{P}^{k_0(\\Omega)}} |Q|^{(\\beta-s)p} \\sum_{P\\in \\mathcal{F}^k(Q\\cap \\Omega)}|P|^{1-\\beta p}\\\\\n&\\leq C \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} \\sum_{Q\\in \\mathcal{P}^{k_0(\\Omega)}} |Q|^{(\\beta-s)p} |Q|^{1-\\beta p} \\\\\n& \\leq C \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} \\sum_{Q\\in \\mathcal{P}^{k_0(\\Omega)}} |Q|^{1-s p}\\\\\n&\\leq C \\Crr{menor}^{-k_0(\\Omega)} \\Big(\\frac{\\Crr{maior}}{\\Crr{menor}}\\Big)^{k_0(\\Omega)(1-s p)} \\Crr{maior}^{(k-k_0(\\Omega))(\\beta-s)p} |\\Omega|^{1-s p} \n\\end{align*} \nThis concludes the proof. \\end{proof}\n\\begin{remark} Suppose that there is $\\Cll{compara}$ such that for every $k$ and every $Q, W\\in \\mathcal{P}^k$ we have\n$$\\frac{1}{\\Crr{compara}}\\leq \\frac{|Q|}{|W|}\\leq \\Crr{compara},$$\nand \n$$\\#\\{ P\\in \\mathcal{P}^{k_0(\\Omega)}\\colon \\ P\\cap \\Omega\\neq \\emptyset \\}\\leq \\Cll{numero},$$\nThen it is easy to see that one can choose $C'= \\Crr{numero}\\Crr{compara}C$.\n\n\n\\end{remark} \n\n\nThe following result is similar to results for Sobolev spaces by Faraco and Rogers \\cite{faraco}. See also Sickel \\cite{sickel}.\n\n\\begin{corollary} If $\\{ \\Omega_r\\}_{r\\in \\Lambda}$ is a $(1-ps, \\Crr{domain},\\Crr{domainc})$-regular family then there is $\\Cll{rel}$ such that for every $g\\in \\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ and $r\\in\\Lambda$ we can find a $\\mathcal{B}^{s}_{p,q}$-representation\n\\begin{equation}\\label{pdd} g\\cdot 1_{\\Omega_r}= \\sum_k \\sum_{Q\\in \\mathcal{P}^k, Q \\subset \\Omega_r } d_Q^r a_Q,\\end{equation}\nsuch that \n\\begin{equation}\\label{hiip1} \\Big( \\sum_j \\big( \\sum_r \\sum_{\\substack{ Q\\in \\mathcal{P}^j \\\\ Q \\subset \\Omega_r }} |d_Q^r|^p )^{q\/p} \\Big)^{1\/q} \\leq \\Crr{rel} |g|_{\\mathcal{B}^{s}_{p,q}}.\\end{equation}\nNote that \n$$\\Omega=\\cup_r\\Omega_r$$\nis a $(1-ps, \\Crr{domain},\\Crr{domainc})$-regular domain and \n $F(g)=g1_{\\Omega}$ is a bounded operator in $\\mathcal{B}^{s}_{p,q}\\cap L^\\infty$ satisfying \n\\begin{equation}\\label{G} |F|_{\\mathcal{B}^{s}_{p,q}\\cap L^\\infty}\\leq \\Crr{e}\\Crr{no} \\Big( 1+ \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}\\Big).\\end{equation}\nMoreover\n\\begin{equation}\\label{estG} |1_\\Omega|_{\\mathcal{B}^s_{p,q}}\\leq \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}.\\end{equation}\n\\end{corollary} \n\\begin{proof} Notice that\n$$f=1_{\\cup_r \\Omega_r}= \\sum_k \\sum_{Q\\in \\mathcal{P}^k} c_Q a_Q,$$\nwhere $c_Q=|Q|^{1\/p-s}$ for every $Q\\in \\cup_k \\cup_r \\mathcal{F}^k(\\Omega_r)$ and $c_Q=0$ otherwise.\nLet \n$$g =\\sum_k \\sum_{J\\in \\mathcal{P}^k} e_J a_J$$\nbe $\\mathcal{B}^s_{p,q}$-representations $g$ given by Corollary \\ref{fou}. Consider $u_1, u_2$ as in the proof of Proposition \\ref{mult33}. By Proposition \\ref{boup} we can get exactly the same estimate as in the proof of Proposition \\ref{mult33}.\n\nNote that those $Q\\in \\mathcal{P}^k$ for which the corresponding atom $a_Q$ has a no vanishing coefficient in the definition of $u_1$ belongs to $\\cup_r \\cup_j \\mathcal{F}^j(\\Omega_r)$, and moreover every $J\\in \\mathcal{P}^k$ for which the corresponding atoms $a_J$ has no vanishing coefficients in the definition of $u_2$ is contained in some $Q\\in \\mathcal{F}^j(\\Omega_r)$, for some $j$ and $r$. In particular $J\\subset \\Omega_r$. So (\\ref{pdd}) holds, with\n$$d_Q^r= \\big( \\sum_{Q\\subset J, J\\in \\mathcal{P} } |J|^{s-1\/p} c_Q e_J \\big) + \\big(\\sum_{Q\\subset J, J\\neq Q,J\\in \\mathcal{P}} |J|^{s-1\/p} c_J e_Q \\big) $$\nfor every $Q\\subset \\Omega_r$. \n\nNote also that\n\\begin{eqnarray*}\n&&\\Big( \\sum_k \\big(\\sum_r \\sum_{Q\\in \\mathcal{F}^k(\\Omega_r)} |Q|^{1-sp}\\big)^{q\/p} \\Big)^{1\/q}\\\\\n&\\leq& \\Crr{domain}^{1\/p} \\Big( \\sum_{k\\geq k_0(\\cup_r \\Omega_r)} \\Crr{domainc}^{(k-k_0(\\Omega))q\/p} \\Big)^{1\/q} |\\Omega|^{1\/p-s}\\\\\n&\\leq& \\frac{\\Crr{domain}^{1\/p}}{(1-\\Crr{domainc}^{q\/p})^{1\/q}} |\\Omega|^{1\/p-s}.\n\\end{eqnarray*}\nso (\\ref{estG})\nand consequently (\\ref{G}) hold. \n\\end{proof}\n\n\\begin{remark} Using the methods in Faraco and Rogers \\cite{faraco} one can show that quasiballs in $[0,1]^n$ (and in particular quasidisks in $[0,1]^2$, that is, domains delimited by quasicircles) give examples of regular domains in $[0,1]^n$ endowed with the good grid of dyadic $n$-cubes and the Lebesgue measure $m$.\n\\end{remark}\n\n\n\\section{A remarkable description of $\\mathcal{B}^s_{1,1}$.}\n\nWhen $p=q=1$ (and $s >0$ small), something curious happens. We can skip the good grid and characterise the Besov space $\\mathcal{B}^s_{1,1}$ of a homogeneous space using regular domains. Fix $\\Crr{domain}\\geq 1$ and $\\Crr{domainc} \\in (0,1)$. Let $\\mathcal{W}$ be the family of all $(1-s, \\Crr{domain},\\Crr{domainc})$-regular domains. Of course $\\mathcal{P}\\subset \\mathcal{W}$. Let $\\hat{\\mathcal{W}}$ be a family of sets satisfying \n$$\\mathcal{P}\\subset \\hat{\\mathcal{W}} \\subset \\mathcal{W}$$\nDefine $B^{1-s}$ as the set of all functions $f \\in L^{1\/(1-s)}$ that can be written as\n\\begin{equation}\\label{ree} f=\\sum_{i=0}^{\\infty} c_i \\frac{1_{A_i}}{|A_i|^{1-s}},\\end{equation}\nwhere $A_i\\in \\hat{\\mathcal{W}}$ for every $i\\in \\mathbb{N}$ and \n$$\\sum_i |c_i| < \\infty.$$\nIt is easy to see that \n$$|f|_{1\/(1-s)}\\leq \\sum_i |c_i|.$$\nDefine\n$$|f|_{B^{1-s}}=\\inf \\sum_i |c_i|,$$\nwhere the infimum runs over all possible representations (\\ref{ree}). One can see that $(B^{1-s},|\\cdot|_{B^{1-s}})$ is a normed vector space.\n\n\\begin{proposition} \\label{rema}We have that $B^{1-s}=\\mathcal{B}^{s}_{1,1}(\\mathcal{P})$ and the corresponding norms are equivalent.\\end{proposition}\n\\begin{proof} Note that (\\ref{estG}) says that there is $C$ such that if $A\\in \\mathcal{W}$ then $1_{A} \\in \\mathcal{B}^s_{1,1}(\\mathcal{P})$ and $$|1_A|_{\\mathcal{B}^s_{1,1}(\\mathcal{P})}\\leq C |A|^{1-s}.$$\nIn particular if $f$ has a representation (\\ref{ree}) we conclude that\n$$|f|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}\\leq C |f|_{B^{1-s}}.$$\n In particular $B^{1-s}\\subset \\mathcal{B}^{s}_{1,1}(\\mathcal{P}).$ On the other hand if $g\\in \\mathcal{B}^{s}_{1,1}(\\mathcal{P}).$ then we can write\n$$g = \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k} s_Q \\frac{1_Q}{|Q|^{1-s}}$$\n$$\\sum_{P\\in \\mathcal{P}} |s_Q|= \\sum_{k=0}^{\\infty} \\sum_{Q \\in \\mathcal{P}^k} |s_Q| < \\infty.$$\nand $|g|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}$ is the infimum of $\\sum_{P\\in \\mathcal{P}} |s_Q|$ over all possible representations. In particular $g\\in B^{1-s}$\nand\n$$|g|_{B^{1-s}}\\leq |g|_{\\mathcal{B}^{s}_{1,1}(\\mathcal{P})}.$$\n\\end{proof}\n\n\n\\begin{remark} Let $I=[0,1]$ with the dyadic grid $\\mathcal{D}$ and the Lebesgue measure $m$. We prove in Part IV that $\\mathcal{B}^s_{1,1}(\\mathcal{D})$, with $0< s<1$, is the Besov space $B^s_{1,1}([0,1])$, and its norms are equivalent. Note that every interval $[a,b]\\subset [0,1]$ is a $(1-s, 2,2^{s-1})$-regular domain. So we can apply Proposition \\ref{rema} with $\\hat{\\mathcal{W}}=\\{[a,b], \\ 0\\leq a < b\\leq 1\\}.$ That is, $f$ belongs to $B^s_{1,1}([0,1])$ if and only if it can be written as in (\\ref{ree}), where every $A_i$ is an interval and $\\sum_i |c_i| < \\infty$, and the norm in $B^s_{1,1}([0,1])$ is equivalent to the infimum of $\\sum_i |c_i|$ over all possible such representations. This characterisation of the Besov space $B^s_{1,1}([0,1])$ was first obtained by Souza \\cite{souzao1}.\n\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Left compositions.} The folllowing result generalizes a well-known result on left composition operators acting on Besov spaces of $\\mathbb{R}^n$. See Bourdaud and Kateb \\cite{k1} \\cite{k0}\\cite{kateb1} for recents developments on the study of left compositions on Besov spaces of $\\mathbb{R}^n$. \n\n\\begin{proposition} \\label{expo} Let \n$$g\\colon I \\rightarrow \\mathbb{C}$$ be a Lipchitz function such that $g(0)=0$. Then the left composition\n$$L_g\\colon \\mathcal{B}^s_{p,q} \\rightarrow \\mathcal{B}^s_{p,q}$$\ndefined by $L_g(f)=g\\circ f$ is well defined and\n$$|g\\circ f|_p+ osc^s_{p,q}(g\\circ f)\\leq K (|f|_p +osc^s_{p,q}(f)),$$\nwhere $K$ is the Lipchitz constant of $g$. Consequently there exists $C$ such that \n$$|L_g(f)|_{\\mathcal{B}^s_{p,q}}\\leq C|f|_{\\mathcal{B}^s_{p,q}}$$\nfor every $f\\in \\mathcal{B}^s_{p,q}$.\n\\end{proposition}\n\\begin{proof} Note that \n\\begin{align}\nosc_p(g\\circ f, Q)&=\\inf_{a\\in \\mathbb{C}}\\Big( \\int _Q |g(f(x))-a|^p \\ dm(x) \\Big)^{1\/p} \\nonumber \\\\\n& \\leq \\inf_{a\\in \\mathbb{C}}\\Big( \\int _Q |g(f(x))-g(a)|^p \\ dm(x) \\Big)^{1\/p} \\nonumber \\\\ \n& \\leq K \\inf_{a\\in \\mathbb{C}}\\Big( \\int _Q |f(x)-a|^p \\ dm(x) \\Big)^{1\/p} = K osc_p(f, Q).\n\\end{align}\nSo it easily follows that $osc^s_{p,q}(g\\circ f)\\leq K osc^s_{p,q}(f)$. Of course $|g\\circ f|_p\\leq K|f|_p$. In particular $g\\circ f \\in \\mathcal{B}^s_{p,q}.$\n\\end{proof}\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}