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Homogeneous Besov Space in Dunkl setting 邓克尔环境中的同质贝索夫空间
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-08-01 DOI: arxiv-2408.00340
Mengmeng Dou, Jiashu Zhang
{"title":"Homogeneous Besov Space in Dunkl setting","authors":"Mengmeng Dou, Jiashu Zhang","doi":"arxiv-2408.00340","DOIUrl":"https://doi.org/arxiv-2408.00340","url":null,"abstract":"The purpose of this paper is to characterize the homogeneous Besov space in\u0000the Dunkl setting. We utilize a new discrete reproducing formula, that is, the\u0000building blocks are differences of the Dunkl-Poisson kernel which involves both\u0000the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the\u0000Dunkl setting, new test functions and distributions are introduced, and a new\u0000decomposition is established.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882716","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}
引用次数: 0
A note on ubiquity of geometric Brascamp-Lieb data 关于布拉斯坎普-李布几何数据普遍性的说明
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-31 DOI: arxiv-2407.21440
Neal Bez, Anthony Gauvan, Hiroshi Tsuji
{"title":"A note on ubiquity of geometric Brascamp-Lieb data","authors":"Neal Bez, Anthony Gauvan, Hiroshi Tsuji","doi":"arxiv-2407.21440","DOIUrl":"https://doi.org/arxiv-2407.21440","url":null,"abstract":"Relying substantially on work of Garg, Gurvits, Oliveira and Wigderson, it is\u0000shown that geometric Brascamp--Lieb data are, in a certain sense, ubiquitous.\u0000This addresses a question raised by Bennett and Tao in their recent work on the\u0000adjoint Brascamp--Lieb inequality.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872793","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}
引用次数: 0
Darboux equivalence for matrix-valued orthogonal polynomials 矩阵值正交多项式的达尔布等价性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-30 DOI: arxiv-2407.20994
Ignacio Bono Parisi, Inés Pacharoni, Ignacio Zurrián
{"title":"Darboux equivalence for matrix-valued orthogonal polynomials","authors":"Ignacio Bono Parisi, Inés Pacharoni, Ignacio Zurrián","doi":"arxiv-2407.20994","DOIUrl":"https://doi.org/arxiv-2407.20994","url":null,"abstract":"In this work, we give some criteria that allow us to decide when two\u0000sequences of matrix-valued orthogonal polynomials are related via a Darboux\u0000transformation and to build explicitly such transformation. In particular, they\u0000allow us to see when and how any given sequence of polynomials is Darboux\u0000related to a diagonal matrix of classic orthogonal polynomials. We also explore\u0000the notion of Darboux-irreducibility and study some sequences that are not a\u0000Darboux transformation of classical orthogonal polynomials.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872794","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}
引用次数: 0
On almost everywhere convergence of planar Bochner-Riesz mean 论平面 Bochner-Riesz 平均值的几乎无处收敛性
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-30 DOI: arxiv-2407.20887
Xiaochun Li, Shukun Wu
{"title":"On almost everywhere convergence of planar Bochner-Riesz mean","authors":"Xiaochun Li, Shukun Wu","doi":"arxiv-2407.20887","DOIUrl":"https://doi.org/arxiv-2407.20887","url":null,"abstract":"We demonstrate that the almost everywhere convergence of the planar\u0000Bochner-Riesz means for $L^p$ functions in the optimal range when $5/3leq\u0000pleq 2$. This is achieved by establishing a sharp $L^{5/3}$ estimate for a\u0000maximal operator closely associated with the Bochner-Riesz multiplier operator.\u0000The estimate depends on a novel refined $L^2$ estimate, which may be of\u0000independent interest.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872796","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}
引用次数: 0
Triebel-Lizorkin spaces in Dunkl setting 邓克尔背景下的特里贝尔-利佐尔金空间
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-30 DOI: arxiv-2408.05227
Chuhan Sun, Zhiming Wang
{"title":"Triebel-Lizorkin spaces in Dunkl setting","authors":"Chuhan Sun, Zhiming Wang","doi":"arxiv-2408.05227","DOIUrl":"https://doi.org/arxiv-2408.05227","url":null,"abstract":"We establish Triebel-Lizorkin spaces in the Dunkl setting which are\u0000associated with finite reflection groups on the Euclidean space. The group\u0000structures induce two nonequivalent metrics: the Euclidean metric and the Dunkl\u0000metric. In this paper, the L^2 space and the Dunkl-Calderon-Zygmund singular\u0000integral operator in the Dunkl setting play a fundamental role. The main tools\u0000used in this paper are as follows: (i) the Dunkl-Calder'on-Zygmund singular\u0000integral operator and a new Calderon reproducing formula in L^2 with the\u0000Triebel-Lizorkin space norms; (ii) new test functions in terms of the L^2\u0000functions and distributions; (iii) the Triebel-Lizorkin spaces in the Dunkl\u0000setting which are defined by the wavelet-type decomposition with norms and the\u0000analogous atomic decomposition of the Hardy spaces.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142209772","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}
引用次数: 0
The Hausdorff dimension of planar elliptic measures via quasiconformal mappings 通过准共形映射的平面椭圆度量的豪斯多夫维度
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-30 DOI: arxiv-2407.21145
Ignasi Guillén-Mola
{"title":"The Hausdorff dimension of planar elliptic measures via quasiconformal mappings","authors":"Ignasi Guillén-Mola","doi":"arxiv-2407.21145","DOIUrl":"https://doi.org/arxiv-2407.21145","url":null,"abstract":"In this paper we study the dimension of planar elliptic measures via the\u0000application of quasiconformal mappings. In fact, in our case studies, we find a\u0000quasiconformal mapping that relates the elliptic measure in a domain to the\u0000harmonic measure in its image domain, and we deduce bounds for the Hausdorff\u0000dimension of the elliptic measure by the known results on the harmonic side.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862684","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}
引用次数: 0
Parabolic cylinder functions revisited using the Laplace transform 利用拉普拉斯变换重访抛物柱面函数
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-29 DOI: arxiv-2407.20403
Rodica D. Costin, Georgios Mavrogiannis
{"title":"Parabolic cylinder functions revisited using the Laplace transform","authors":"Rodica D. Costin, Georgios Mavrogiannis","doi":"arxiv-2407.20403","DOIUrl":"https://doi.org/arxiv-2407.20403","url":null,"abstract":"In this paper we gather and extend classical results for parabolic cylinder\u0000functions, namely solutions of the Weber differential equations, using a\u0000systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of\u0000the Weber differential equation begin{equation*}\u0000w''(z)-left(frac{z^2}{4}+aright)w(z)=0 end{equation*} and provide\u0000representations by Laplace integrals extended to include all values of the\u0000complex parameter $a$; we find an integral integral representation for the\u0000function $V$; none was previously available. For the Weber equation in the form begin{equation*} u''(x)+left(frac{x^2}{4}-aright)u(x)=0, end{equation*} we define a new\u0000fundamental system $E_pm$ which is analytic in $ainmathbb{C}$, based on\u0000asymptotic behavior; they appropriately extend and modify the classical\u0000solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for\u0000the approach.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862685","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}
引用次数: 0
Generalized Sublevel Estimates for Form-Valued Functions and Related Results for Radon-like Transforms 形式值函数的广义子级估计和类似拉顿变换的相关结果
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-26 DOI: arxiv-2407.18860
Philip T. Gressman
{"title":"Generalized Sublevel Estimates for Form-Valued Functions and Related Results for Radon-like Transforms","authors":"Philip T. Gressman","doi":"arxiv-2407.18860","DOIUrl":"https://doi.org/arxiv-2407.18860","url":null,"abstract":"Motivated by the testing condition for Radon-Brascamp-Lieb multilinear\u0000functionals established in arXiv:2201.12201, this paper is concerned with\u0000identifying local conditions on smooth maps $u(t)$ with values in the space of\u0000decomposable p-forms on some real vector space V which guarantee uniform\u0000integrability of $||u(t)||^{-tau}$ over a certain natural, noncompact family\u0000of norms. One can loosely regard this problem as a higher-dimensional analogue\u0000of establishing uniform bounds for the size of a sublevel set of a function in\u0000terms of the size of its derivatives. The resulting theorem relies extensively\u0000on ideas from Geometric Invariant Theory to understand what appropriate\u0000derivative bounds look like in this context. Several examples and applications\u0000are presented, including a new local characterization of so-called \"model\"\u0000Radon-like transforms in terms of the semistability of a natural curvature\u0000functional (giving an equivalent but rather different criterion than the one\u0000first established in arXiv:2303.03325).","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872806","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}
引用次数: 0
Wasserstein-infinity stability and mean field limit of discrete interaction energy minimizers 离散相互作用能量最小化的瓦瑟斯坦无穷稳定性和平均场极限
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-25 DOI: arxiv-2407.18395
Ruiwen Shu
{"title":"Wasserstein-infinity stability and mean field limit of discrete interaction energy minimizers","authors":"Ruiwen Shu","doi":"arxiv-2407.18395","DOIUrl":"https://doi.org/arxiv-2407.18395","url":null,"abstract":"In this paper we give a quantitative stability result for the discrete\u0000interaction energy on the multi-dimensional torus, for the periodic Riesz\u0000potential. It states that if the number of particles $N$ is large and the\u0000discrete interaction energy is low, then the particle distribution is\u0000necessarily close to the uniform distribution (i.e., the continuous energy\u0000minimizer) in the Wasserstein-infinity distance. As a consequence, we obtain a\u0000quantitative mean field limit of interaction energy minimizers in the\u0000Wasserstein-infinity distance. The proof is based on the application of the\u0000author's previous joint work with J. Wang on the stability of continuous energy\u0000minimizer, together with a new mollification trick for the empirical measure in\u0000the case of singular interaction potentials.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862774","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}
引用次数: 0
Fractional medians and their maximal functions 分数中值及其最大函数
arXiv - MATH - Classical Analysis and ODEs Pub Date : 2024-07-25 DOI: arxiv-2407.17700
Yohei Tsutsui
{"title":"Fractional medians and their maximal functions","authors":"Yohei Tsutsui","doi":"arxiv-2407.17700","DOIUrl":"https://doi.org/arxiv-2407.17700","url":null,"abstract":"In this article, we introduce the fractional medians, give an expression of\u0000the set of all fractional medians in terms of non-increasing rearrangements and\u0000then investigate mapping properties of the fractional maximal operators defined\u0000by such medians. The maximal operator is a generalization of that in Stromberg.\u0000It turns out that our maximal operator is a more smooth operator than the usual\u0000fractional maximal operator. Further, we give another proof of the embedding\u0000from $BV$ to $L^{n/(n-1),1}$ due to Alvino by using the usual medians.","PeriodicalId":501145,"journal":{"name":"arXiv - MATH - Classical Analysis and ODEs","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141781071","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}
引用次数: 0
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