Tirthankar Bhattacharyya, Mainak Bhowmik, Haripada Sau
Consider a bounded symmetric domain $Omega$ with a finite pseudo-reflection group acting on it as a subgroup of the group of automorphisms. This gives rise to quotient domains by means of basic polynomials $theta$ which by virtue of being proper maps map the v Silov boundary of $Omega$ to the v Silov boundary of $theta(Omega)$. Thus, the natural measure on the v Silov boundary of $Omega$ can be pushed forward. This gives rise to Hardy spaces on the quotient domain. The study of Hankel operators on the Hardy spaces of the quotient domains is introduced. The use of the weak product space shows that an analogue of Hartman's theorem holds for the small Hankel operator. Nehari's theorem fails for the big Hankel operator and this has the consequence that when the domain $Omega$ is the polydisc $mathbb D^d$, the {em Hardy space} is not a projective object in the category of all Hilbert modules over the algebra $mathcal A (theta(mathbb D^d))$ of functions which are holomorphic in the quotient domain and continuous on the closure $overline {theta(mathbb D^d)}$. It is not a projective object in the category of cramped Hilbert modules either. Indeed, no projective object is known in these two categories. On the other hand, every normal Hilbert module over the algebra of continuous functions on the v Silov boundary, treated as a Hilbert module over the algebra $mathcal A (theta(mathbb D^d))$, is projective.
考虑一个有界对称域$Omega$，其上有一个有限伪反射群作为自变群的一个子群。这就通过基本多项式$theta$产生了商域，而基本多项式$theta$由于是适当的映射，可以把$Omega$的v Silov边界映射到$theta(Omega)$的v Silov边界。因此，$Omega$的v Silov边界上的自然度量可以向前推进。这就产生了商域上的哈代空间。介绍了商域哈代空间上汉克尔算子的研究。弱积空间的使用表明，哈曼定理的类似物对小汉克尔算子成立。对于大汉克尔算子，奈哈里定理失效了，其结果是，当域$Omega$是多圆盘$mathbb D^d$、在代数$mathcal A (theta(mathbb D^d))$上的所有希尔伯特模块的范畴中，{em Hardy space}并不是一个投影对象，而这些函数在引域中是全态的，并且在闭合$overline {theta(mathbbD^d)}$ 上是连续的。它也不是 cramped Hilbertmodules 范畴中的投影对象。事实上，在这两个范畴中没有已知的投影对象。另一方面，在 v Silov 边界上连续函数代数上的每一个正常希尔伯特模块，作为代数 $mathcal A (theta(mathbb D^d))$ 上的希尔伯特模块，都是投影的。
{"title":"Hankel operators and Projective Hilbert modules on quotients of bounded symmetric domains","authors":"Tirthankar Bhattacharyya, Mainak Bhowmik, Haripada Sau","doi":"arxiv-2409.04582","DOIUrl":"https://doi.org/arxiv-2409.04582","url":null,"abstract":"Consider a bounded symmetric domain $Omega$ with a finite pseudo-reflection\u0000group acting on it as a subgroup of the group of automorphisms. This gives rise\u0000to quotient domains by means of basic polynomials $theta$ which by virtue of\u0000being proper maps map the v Silov boundary of $Omega$ to the v Silov\u0000boundary of $theta(Omega)$. Thus, the natural measure on the v Silov\u0000boundary of $Omega$ can be pushed forward. This gives rise to Hardy spaces on\u0000the quotient domain. The study of Hankel operators on the Hardy spaces of the quotient domains is\u0000introduced. The use of the weak product space shows that an analogue of\u0000Hartman's theorem holds for the small Hankel operator. Nehari's theorem fails\u0000for the big Hankel operator and this has the consequence that when the domain\u0000$Omega$ is the polydisc $mathbb D^d$, the {em Hardy space} is not a\u0000projective object in the category of all Hilbert modules over the algebra\u0000$mathcal A (theta(mathbb D^d))$ of functions which are holomorphic in the\u0000quotient domain and continuous on the closure $overline {theta(mathbb\u0000D^d)}$. It is not a projective object in the category of cramped Hilbert\u0000modules either. Indeed, no projective object is known in these two categories.\u0000On the other hand, every normal Hilbert module over the algebra of continuous\u0000functions on the v Silov boundary, treated as a Hilbert module over the\u0000algebra $mathcal A (theta(mathbb D^d))$, is projective.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The essential norms of Toeplitz operators with symbols in $C+H^infty$ on weighted Hardy spaces are independent of the weights","authors":"Oleksiy Karlovych, Eugene Shargorodsky","doi":"arxiv-2409.03548","DOIUrl":"https://doi.org/arxiv-2409.03548","url":null,"abstract":"Let $1<p<infty$, let $H^p$ be the Hardy space on the unit circle, and let\u0000$H^p(w)$ be the Hardy space with a Muckenhoupt weight $win A_p$ on the unit\u0000circle. In 1988, B\"ottcher, Krupnik and Silbermann proved that the essential\u0000norm of the Toeplitz operator $T(a)$ with $ain C$ on the weighted Hardy space\u0000$H^2(varrho)$ with a power weight $varrhoin A_2$ is equal to\u0000$|a|_{L^infty}$. This implies that the essential norm of $T(a)$ on\u0000$H^2(varrho)$ does not depend on $varrho$. We extend this result and show\u0000that if $ain C+H^infty$, then, for $1<p<infty$, the essential norms of the\u0000Toeplitz operator $T(a)$ on $H^p$ and on $H^p(w)$ are the same for all $win\u0000A_p$. In particular, if $win A_2$, then the essential norm of the Toeplitz\u0000operator $T(a)$ with $ain C+H^infty$ on the weighted Hardy space $H^2(w)$ is\u0000equal to $|a|_{L^infty}$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the shift operator $M_z$, defined on the Bloch space and the little Bloch space and we study the corresponding lattice of invariant subspaces. The index of a closed invariant subspace $E$ is defined as $text{ind}(E) = dim(E/M_z E)$. We construct closed, shift invariant subspaces in the Bloch space that can have index as large as the cardinality of the unit interval $[0,1]$. Next we focus on the little Bloch space, providing a construction of closed, shift invariant subspaces that have arbitrary large index. Finally we establish several results on the index for the weak-star topology of a Banach space and prove a stability theorem for the index when passing from (norm closed) invariant subspaces of a Banach space to their weak-star closure in its second dual. This is then applied to prove the existence of weak-star closed invariant subspaces of arbitrary index in the Bloch space.
{"title":"Shift invariant subspaces of large index in the Bloch space","authors":"Nikiforos Biehler","doi":"arxiv-2409.03562","DOIUrl":"https://doi.org/arxiv-2409.03562","url":null,"abstract":"We consider the shift operator $M_z$, defined on the Bloch space and the\u0000little Bloch space and we study the corresponding lattice of invariant\u0000subspaces. The index of a closed invariant subspace $E$ is defined as\u0000$text{ind}(E) = dim(E/M_z E)$. We construct closed, shift invariant subspaces\u0000in the Bloch space that can have index as large as the cardinality of the unit\u0000interval $[0,1]$. Next we focus on the little Bloch space, providing a\u0000construction of closed, shift invariant subspaces that have arbitrary large\u0000index. Finally we establish several results on the index for the weak-star\u0000topology of a Banach space and prove a stability theorem for the index when\u0000passing from (norm closed) invariant subspaces of a Banach space to their\u0000weak-star closure in its second dual. This is then applied to prove the\u0000existence of weak-star closed invariant subspaces of arbitrary index in the\u0000Bloch space.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the analysis of parametrized nonautonomous evolutionary equations, bounded entire solutions are natural candidates for bifurcating objects. Appropriate explicit and sufficient conditions for such branchings, however, require to combine contemporary functional analytical methods from the abstract bifurcation theory for Fredholm operators with tools originating in dynamical systems. This paper establishes alternatives classifying the shape of global bifurcating branches of bounded entire solutions to Carath'eodory differential equations. Our approach is based on the parity associated to a path of index 0 Fredholm operators, the global Evans function as a recent tool in nonautonomous bifurcation theory and suitable topologies on spaces of Carath'eodory functions.
{"title":"Global bifurcation of homoclinic solutions","authors":"Iacopo P. Longo, Christian Pötzsche, Robert Skiba","doi":"arxiv-2409.03851","DOIUrl":"https://doi.org/arxiv-2409.03851","url":null,"abstract":"In the analysis of parametrized nonautonomous evolutionary equations, bounded\u0000entire solutions are natural candidates for bifurcating objects. Appropriate\u0000explicit and sufficient conditions for such branchings, however, require to\u0000combine contemporary functional analytical methods from the abstract\u0000bifurcation theory for Fredholm operators with tools originating in dynamical\u0000systems. This paper establishes alternatives classifying the shape of global\u0000bifurcating branches of bounded entire solutions to Carath'eodory differential\u0000equations. Our approach is based on the parity associated to a path of index 0\u0000Fredholm operators, the global Evans function as a recent tool in nonautonomous\u0000bifurcation theory and suitable topologies on spaces of Carath'eodory\u0000functions.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the problem of finding a common fixed point of quasinonexpansive mappings defined on a Hilbert space. To approximate the solution to this problem, we present several iterative processes using the parallel method based on Anh and Chung (2014) and Aoyama (2018).
{"title":"Parallel methods for quasinonexpansive mappings in a Hilbert space","authors":"Koji Aoyama, Shigeru Iemoto","doi":"arxiv-2409.03242","DOIUrl":"https://doi.org/arxiv-2409.03242","url":null,"abstract":"This paper is devoted to the problem of finding a common fixed point of\u0000quasinonexpansive mappings defined on a Hilbert space. To approximate the\u0000solution to this problem, we present several iterative processes using the\u0000parallel method based on Anh and Chung (2014) and Aoyama (2018).","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, $mathcal{G}(g,L,M,N)$ denotes a $L-$window Gabor system on a periodic set $mathbb{S}$, where $L,M,Min mathbb{N}$ and $g={g_l}_{lin mathbb{N}_L}subset ell^2(mathbb{S})$. We characterize which $g$ generates a complete multi-window Gabor system and a multi-window Gabor frame $mathcal{G}(g,L,M,N)$ on $mathbb{S}$ using the Zak transform. Admissibility conditions for a periodic set to admit a complete multi--window Gabor system, multi-window Gabor (Parseval) frame, and multi--window Gabor (orthonormal) basis $mathcal{G}(g,L,M,N)$ are given with respect to the parameters $L$, $M$ and $N$.
{"title":"Admissibility Conditions for Multi-window Gabor Frames on Discrete Periodic Sets","authors":"Najib Khachiaa, Mohamed Rossafi","doi":"arxiv-2409.03423","DOIUrl":"https://doi.org/arxiv-2409.03423","url":null,"abstract":"In this paper, $mathcal{G}(g,L,M,N)$ denotes a $L-$window Gabor system on a\u0000periodic set $mathbb{S}$, where $L,M,Min mathbb{N}$ and $g={g_l}_{lin\u0000mathbb{N}_L}subset ell^2(mathbb{S})$. We characterize which $g$ generates a\u0000complete multi-window Gabor system and a multi-window Gabor frame\u0000$mathcal{G}(g,L,M,N)$ on $mathbb{S}$ using the Zak transform. Admissibility\u0000conditions for a periodic set to admit a complete multi--window Gabor system,\u0000multi-window Gabor (Parseval) frame, and multi--window Gabor (orthonormal)\u0000basis $mathcal{G}(g,L,M,N)$ are given with respect to the parameters $L$, $M$\u0000and $N$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a bifurcation result for a Dirichlet problem driven by the fractional $p$-Laplacian (either degenerate or singular), in which the reaction is the difference between two sublinear powers of the unknown. In our argument, a fundamental role is played by a Sobolev vs. H"older minima principle, already known for the degenerate case, which here we extend to the singular case with a simpler proof.
我们证明了由分数 $p$-Laplacian （退化或奇异）驱动的 Dirichlet 问题的分岔结果，其中反应是未知数的两个次线性幂之间的差。在我们的论证中，Sobolev vs. H"older minima "原理发挥了根本性的作用，该原理在退化情况下已经为人所知，在此我们以更简单的证明将其扩展到奇异情况下。
{"title":"On a doubly sublinear fractional $p$-Laplacian equation","authors":"A. Iannizzotto, S. Mosconi","doi":"arxiv-2409.03616","DOIUrl":"https://doi.org/arxiv-2409.03616","url":null,"abstract":"We prove a bifurcation result for a Dirichlet problem driven by the\u0000fractional $p$-Laplacian (either degenerate or singular), in which the reaction\u0000is the difference between two sublinear powers of the unknown. In our argument,\u0000a fundamental role is played by a Sobolev vs. H\"older minima principle,\u0000already known for the degenerate case, which here we extend to the singular\u0000case with a simpler proof.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers paired operators in the context of the Lebesgue Hilbert space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.
{"title":"Paired kernels and truncated Toeplitz operators","authors":"M. Cristina Câmara, Jonathan R. Partington","doi":"arxiv-2409.02563","DOIUrl":"https://doi.org/arxiv-2409.02563","url":null,"abstract":"This paper considers paired operators in the context of the Lebesgue Hilbert\u0000space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The\u0000kernels of such operators, together with their analytic projections, which are\u0000generalizations of Toeplitz kernels, are studied. Inclusion relations between\u0000such kernels are considered in detail, and the results are applied to\u0000describing the kernels of finite-rank asymmetric truncated Toeplitz operators.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove that every continuous $h$-mid-convex with suitable conditions on $h$ is $h$-convex function. Also, we extend Ostrowski theorem, Blumberg-Sierpinski theorem, Bernstein-Doetsch theorem, Mehdi theorem.
{"title":"Some extensions from famous theorems for $h$-mid-convex function","authors":"Amir Garejelo, Farzollah Mirzapour, Ali Morassaei","doi":"arxiv-2409.02450","DOIUrl":"https://doi.org/arxiv-2409.02450","url":null,"abstract":"In this paper, we prove that every continuous $h$-mid-convex with suitable\u0000conditions on $h$ is $h$-convex function. Also, we extend Ostrowski theorem,\u0000Blumberg-Sierpinski theorem, Bernstein-Doetsch theorem, Mehdi theorem.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A central question which originates in the celebrated work in the 1980's of DiPerna and Majda asks what is the optimal decay $f > 0$ such that uniform rates $|omega|(Q) leq f(|Q|)$ of the vorticity maximal functions guarantee strong convergence without concentrations of approximate solutions to energy-conserving weak solutions of the $2$D Euler equations with vortex sheet initial data. A famous result of Majda (1993) shows $f(r) = [log (1/r)]^{-1/2}$, $r<1/2$, as the optimal decay for emph{distinguished} sign vortex sheets. In the general setting of emph{mixed} sign vortex sheets, DiPerna and Majda (1987) established $f(r) = [log (1/r)]^{-alpha}$ with $alpha > 1$ as a sufficient condition for the lack of concentrations, while the expected gap $alpha in (1/2, 1]$ remains as an open question. In this paper we resolve the DiPerna-Majda $2$D gap problem: In striking contrast to the well-known case of distinguished sign vortex sheets, we identify $f(r) = [log (1/r)]^{-1}$ as the optimal regularity for mixed sign vortex sheets that rules out concentrations. For the proof, we propose a novel method to construct explicitly solutions with mixed sign to the $2$D Euler equations in such a way that wild behaviour creates within the relevant geometry of emph{sparse} cubes (i.e., these cubes are not necessarily pairwise disjoint, but their possible overlappings can be controlled in a sharp fashion). Such a strategy is inspired by the recent work of the first author and Milman cite{DM} where strong connections between energy conservation and sparseness are established.
{"title":"A sparse resolution of the DiPerna-Majda gap problem for $2$D Euler equations","authors":"Oscar Domínguez, Daniel Spector","doi":"arxiv-2409.02344","DOIUrl":"https://doi.org/arxiv-2409.02344","url":null,"abstract":"A central question which originates in the celebrated work in the 1980's of\u0000DiPerna and Majda asks what is the optimal decay $f > 0$ such that uniform\u0000rates $|omega|(Q) leq f(|Q|)$ of the vorticity maximal functions guarantee\u0000strong convergence without concentrations of approximate solutions to\u0000energy-conserving weak solutions of the $2$D Euler equations with vortex sheet\u0000initial data. A famous result of Majda (1993) shows $f(r) = [log\u0000(1/r)]^{-1/2}$, $r<1/2$, as the optimal decay for emph{distinguished} sign\u0000vortex sheets. In the general setting of emph{mixed} sign vortex sheets,\u0000DiPerna and Majda (1987) established $f(r) = [log (1/r)]^{-alpha}$ with\u0000$alpha > 1$ as a sufficient condition for the lack of concentrations, while\u0000the expected gap $alpha in (1/2, 1]$ remains as an open question. In this\u0000paper we resolve the DiPerna-Majda $2$D gap problem: In striking contrast to\u0000the well-known case of distinguished sign vortex sheets, we identify $f(r) =\u0000[log (1/r)]^{-1}$ as the optimal regularity for mixed sign vortex sheets that\u0000rules out concentrations. For the proof, we propose a novel method to construct explicitly solutions\u0000with mixed sign to the $2$D Euler equations in such a way that wild behaviour\u0000creates within the relevant geometry of emph{sparse} cubes (i.e., these cubes\u0000are not necessarily pairwise disjoint, but their possible overlappings can be\u0000controlled in a sharp fashion). Such a strategy is inspired by the recent work\u0000of the first author and Milman cite{DM} where strong connections between\u0000energy conservation and sparseness are established.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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