In this paper, we introduce Orlicz spaces on $ mathbb Z^n times mathbb T^n $ and Orlicz modulation spaces on $mathbb Z^n$, and present some basic properties such as inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{Phi}(mathbb Z^n)$ is close to the modulation space $M^{2}(mathbb Z^n)$ for some particular Young function $Phi$. Then, we study a class of pseudo-differential operators known as time-frequency localization operators on $mathbb Z^n$, which depend on a symbol $varsigma$ and two windows functions $g_1$ and $g_2$. Using appropriate classes for symbols, we study the boundedness of the localization operators on Orlicz modulation spaces on $mathbb Z^n$. Also, we show that these operators are compact and in the Schatten--von Neumann classes.
{"title":"Localization operators on discrete Orlicz modulation spaces","authors":"Aparajita Dasgupta, Anirudha Poria","doi":"arxiv-2409.05373","DOIUrl":"https://doi.org/arxiv-2409.05373","url":null,"abstract":"In this paper, we introduce Orlicz spaces on $ mathbb Z^n times mathbb T^n\u0000$ and Orlicz modulation spaces on $mathbb Z^n$, and present some basic\u0000properties such as inclusion relations, convolution relations, and duality of\u0000these spaces. We show that the Orlicz modulation space $M^{Phi}(mathbb Z^n)$\u0000is close to the modulation space $M^{2}(mathbb Z^n)$ for some particular Young\u0000function $Phi$. Then, we study a class of pseudo-differential operators known\u0000as time-frequency localization operators on $mathbb Z^n$, which depend on a\u0000symbol $varsigma$ and two windows functions $g_1$ and $g_2$. Using appropriate\u0000classes for symbols, we study the boundedness of the localization operators on\u0000Orlicz modulation spaces on $mathbb Z^n$. Also, we show that these operators\u0000are compact and in the Schatten--von Neumann classes.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226616","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 study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem, colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in cite{adiprasito2020theorems}. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.
{"title":"No-dimensional Helly's theorem in uniformly convex Banach spaces","authors":"G. Ivanov","doi":"arxiv-2409.05744","DOIUrl":"https://doi.org/arxiv-2409.05744","url":null,"abstract":"We study the ``no-dimensional'' analogue of Helly's theorem in Banach spaces.\u0000Specifically, we obtain the following no-dimensional Helly-type results for\u0000uniformly convex Banach spaces: Helly's theorem, fractional Helly's theorem,\u0000colorful Helly's theorem, and colorful fractional Helly's theorem. The combinatorial part of the proofs for these Helly-type results is\u0000identical to the Euclidean case as presented in cite{adiprasito2020theorems}.\u0000The primary difference lies in the use of a certain geometric inequality in\u0000place of the Pythagorean theorem. This inequality can be explicitly expressed\u0000in terms of the modulus of convexity of a Banach space.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208147","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 study the heat equation associated with the Jacobi--Cherednik operator on the real line. We establish some basic properties of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a solution to the Cauchy problem for the Jacobi--Cherednik heat operator and prove that the heat kernel is strictly positive. Then, we characterize the image of the space $L^2(mathbb R, A_{alpha, beta})$ under the Jacobi--Cherednik heat semigroup as a reproducing kernel Hilbert space. As an application, we solve the modified Poisson equation and present the Jacobi--Cherednik--Markov processes.
{"title":"The heat semigroup associated with the Jacobi--Cherednik operator and its applications","authors":"Anirudha Poria, Ramakrishnan Radha","doi":"arxiv-2409.05376","DOIUrl":"https://doi.org/arxiv-2409.05376","url":null,"abstract":"In this paper, we study the heat equation associated with the\u0000Jacobi--Cherednik operator on the real line. We establish some basic properties\u0000of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a\u0000solution to the Cauchy problem for the Jacobi--Cherednik heat operator and\u0000prove that the heat kernel is strictly positive. Then, we characterize the\u0000image of the space $L^2(mathbb R, A_{alpha, beta})$ under the\u0000Jacobi--Cherednik heat semigroup as a reproducing kernel Hilbert space. As an\u0000application, we solve the modified Poisson equation and present the\u0000Jacobi--Cherednik--Markov processes.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226615","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}
Fernando Albiac, Jose L. Ansorena, Pablo Berná, Miguel Berasategui
Property~(A) is a week symmetry condition that plays a fundamental role in the characterization of greedy-type bases in the isometric case, i.e., when the constants involved in the study of the efficiency of the thresholding greedy algorithm in Banach spaces are sharp. In this note we build examples of Banach spaces with Schauder bases that have Property~(A) but fail to be unconditional, thus settling a long standing problem in the area. As a by-product of our work we hone our construction to produce counterexamples that solve other open questions in the isometric theory of greedy bases.
{"title":"Conditional bases with Property~(A)","authors":"Fernando Albiac, Jose L. Ansorena, Pablo Berná, Miguel Berasategui","doi":"arxiv-2409.04883","DOIUrl":"https://doi.org/arxiv-2409.04883","url":null,"abstract":"Property~(A) is a week symmetry condition that plays a fundamental role in\u0000the characterization of greedy-type bases in the isometric case, i.e., when the\u0000constants involved in the study of the efficiency of the thresholding greedy\u0000algorithm in Banach spaces are sharp. In this note we build examples of Banach\u0000spaces with Schauder bases that have Property~(A) but fail to be unconditional,\u0000thus settling a long standing problem in the area. As a by-product of our work\u0000we hone our construction to produce counterexamples that solve other open\u0000questions in the isometric theory of greedy bases.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"252 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208149","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}
The aim of this paper is twofold. On the one hand, we manage to identify Banach-valued Hardy spaces of analytic functions over the disc $mathbb{D}$ with other classes of Hardy spaces, thus complementing the existing literature on the subject. On the other hand, we develop new techniques that allow us to prove that certain Hilbert-valued atomic lattices have a unique unconditional basis, up to normalization, equivalence and permutation. Combining both lines of action we show that that $H_p(mathbb{D},ell_2)$ for $0
{"title":"Isomorphisms between vector-valued $H_p$-spaces for $0","authors":"Fernando Albiac, Jose L. Ansorena","doi":"arxiv-2409.04866","DOIUrl":"https://doi.org/arxiv-2409.04866","url":null,"abstract":"The aim of this paper is twofold. On the one hand, we manage to identify\u0000Banach-valued Hardy spaces of analytic functions over the disc $mathbb{D}$\u0000with other classes of Hardy spaces, thus complementing the existing literature\u0000on the subject. On the other hand, we develop new techniques that allow us to\u0000prove that certain Hilbert-valued atomic lattices have a unique unconditional\u0000basis, up to normalization, equivalence and permutation. Combining both lines\u0000of action we show that that $H_p(mathbb{D},ell_2)$ for $0<p<1$ has a unique\u0000atomic lattice structure. The proof of this result relies on the validity of\u0000some new lattice estimates for non-locally convex spaces which hold an\u0000independent interest.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208150","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 Schatten class membership of multi-parameter Hankel operators on the Paley--Wiener space of a bounded convex domain $Omega$. For admissible domains, we develop a framework and theory of Besov spaces of Paley--Wiener type. We prove that a Hankel operator belongs to the Schatten class $S^p$ if and only if its symbol belongs to a corresponding Besov space, for $1 leq p leq 2$. For smooth domains $Omega$ with positive curvature, we extend this result to $1 leq p < 4$, and for simple polytopes to the full range $1 leq p < infty$.
我们考虑了有界凸域$Omega$的Paley--Wiener空间上的多参数汉克尔算子的Schatten类成员资格。对于可接纳域，我们建立了一个 Paley--Wiener 型 Besov 空间的框架和理论。我们证明，当且仅当一个汉克尔算子的符号属于相应的贝索夫空间时，该算子才属于夏顿类（Schatten class）$S^p$，条件是1 leq pleq 2$。对于具有正曲率的光滑域$Omega$，我们将这一结果扩展到$1 leq p < 4$，而对于简单多面体，则扩展到整个范围$1 leq p < infty$。
{"title":"Besov spaces and Schatten class Hankel operators on Paley--Wiener spaces of convex domains","authors":"Konstantinos Bampouras, Karl-Mikael Perfekt","doi":"arxiv-2409.04184","DOIUrl":"https://doi.org/arxiv-2409.04184","url":null,"abstract":"We consider Schatten class membership of multi-parameter Hankel operators on\u0000the Paley--Wiener space of a bounded convex domain $Omega$. For admissible\u0000domains, we develop a framework and theory of Besov spaces of Paley--Wiener\u0000type. We prove that a Hankel operator belongs to the Schatten class $S^p$ if\u0000and only if its symbol belongs to a corresponding Besov space, for $1 leq p\u0000leq 2$. For smooth domains $Omega$ with positive curvature, we extend this\u0000result to $1 leq p < 4$, and for simple polytopes to the full range $1 leq p\u0000< infty$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208179","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 investigate a random geometric graph model introduced by Bonato and Janssen. The vertices are the points of a countable dense set $S$ in a (necessarily separable) normed vector space $X$, and each pair of points are joined independently with some fixed probability $p$ (with $0
{"title":"Random Geometric Graphs in Reflexive Banach Spaces","authors":"József Balogh, Mark Walters, András Zsák","doi":"arxiv-2409.04237","DOIUrl":"https://doi.org/arxiv-2409.04237","url":null,"abstract":"We investigate a random geometric graph model introduced by Bonato and\u0000Janssen. The vertices are the points of a countable dense set $S$ in a\u0000(necessarily separable) normed vector space $X$, and each pair of points are\u0000joined independently with some fixed probability $p$ (with $0<p<1$) if they are\u0000less than distance $1$ apart. A countable dense set $S$ in a normed space is\u0000Rado, if the resulting graph is almost surely unique up to isomorphism: that is\u0000any two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the\u0000geometry of the underlying normed space. It turns out that a key question is in\u0000which spaces must step-isometries (maps that preserve the integer parts of\u0000distances) on dense subsets necessarily be isometries. We answer this question\u0000for a large class of Banach spaces including all strictly convex reflexive\u0000spaces. In the process we prove results on the interplay between the norm\u0000topology and weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all\u0000countable dense sets in strictly convex reflexive spaces are strongly non-Rado\u0000(that is, any two graphs are almost surely non-isomorphic). However, we show\u0000that there do exist Rado sets even in $ell_2$. Finally we construct a Banach\u0000spaces in which all countable dense set are strongly non-Rado.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208172","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 obtain Wiman-Valiron type inequalities for random entire functions and for random analytic functions on the unit disk that improve a classical result of ErdH{o}s and R'enyi and recent results of Kuryliak and Skaskiv. Our results are then applied to linear dynamics: we obtain rates of growth, outside some exceptional set, for analytic functions that are frequently hypercyclic for an arbitrary chaotic weighted backward shift.
{"title":"Rate of growth of random analytic functions, with an application to linear dynamics","authors":"Kevin Agneessens, Karl-G. Grosse-Erdmann","doi":"arxiv-2409.04235","DOIUrl":"https://doi.org/arxiv-2409.04235","url":null,"abstract":"We obtain Wiman-Valiron type inequalities for random entire functions and for\u0000random analytic functions on the unit disk that improve a classical result of\u0000ErdH{o}s and R'enyi and recent results of Kuryliak and Skaskiv. Our results\u0000are then applied to linear dynamics: we obtain rates of growth, outside some\u0000exceptional set, for analytic functions that are frequently hypercyclic for an\u0000arbitrary chaotic weighted backward shift.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"453 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226618","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 study the extremality of nonexpansive mappings on a nonempty bounded closed and convex subset of a normed space (therein specific Banach spaces). We show that surjective isometries are extremal in this sense for many Banach spaces, including Banach spaces with the Radon-Nikodym property and all $C(K)$-spaces for compact Hausdorff $K.$ We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal.
{"title":"On extremal nonexpansive mappings","authors":"Christian Bargetz, Michael Dymond, Katriin Pirk","doi":"arxiv-2409.04292","DOIUrl":"https://doi.org/arxiv-2409.04292","url":null,"abstract":"We study the extremality of nonexpansive mappings on a nonempty bounded\u0000closed and convex subset of a normed space (therein specific Banach spaces). We\u0000show that surjective isometries are extremal in this sense for many Banach\u0000spaces, including Banach spaces with the Radon-Nikodym property and all\u0000$C(K)$-spaces for compact Hausdorff $K.$ We also conclude that the typical, in\u0000the sense of Baire category, nonexpansive mapping is close to being extremal.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"73 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208152","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 use abstract bifurcation theory for Fredholm operators of index zero to deal with periodic even solutions of the one-dimensional equation $mathcal{L}u=lambda u+|u|^{p}$, where $mathcal{L}$ is a nonlocal pseudodifferential operator defined as a Fourier multiplier and $lambda$ is the bifurcation parameter. Our general setting includes the fractional Laplacian $mathcal{L}equiv(-Delta)^{s}$ and sharpens the results obtained for this operator to date. As a direct application, we establish the existence of traveling waves for general nonlocal dispersive equations for some velocity ranges.
{"title":"Periodic solutions to nonlocal pseudo-differential equations. A bifurcation theoretical perspective","authors":"Juan Carlos Sampedro","doi":"arxiv-2409.04253","DOIUrl":"https://doi.org/arxiv-2409.04253","url":null,"abstract":"In this paper we use abstract bifurcation theory for Fredholm operators of\u0000index zero to deal with periodic even solutions of the one-dimensional equation\u0000$mathcal{L}u=lambda u+|u|^{p}$, where $mathcal{L}$ is a nonlocal\u0000pseudodifferential operator defined as a Fourier multiplier and $lambda$ is\u0000the bifurcation parameter. Our general setting includes the fractional\u0000Laplacian $mathcal{L}equiv(-Delta)^{s}$ and sharpens the results obtained\u0000for this operator to date. As a direct application, we establish the existence\u0000of traveling waves for general nonlocal dispersive equations for some velocity\u0000ranges.","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":"142226617","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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