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On the L1 differentiability of certain Markov semigroups 一类马尔可夫半群的L1可微性
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-01-25 DOI: 10.1016/j.exmath.2025.125655
A. Bendikov , L. Saloff-Coste
{"title":"On the L1 differentiability of certain Markov semigroups","authors":"A. Bendikov ,&nbsp;L. Saloff-Coste","doi":"10.1016/j.exmath.2025.125655","DOIUrl":"10.1016/j.exmath.2025.125655","url":null,"abstract":"<div><div>We discuss the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-analyticity (and lack thereof) for Gaussian symmetric Markov semigroups on compact connected groups in finite and infinite dimension. In the infinite dimensional case, a positive result gives some sufficient conditions for quantitative <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-differentiability.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125655"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Laplacian of a finely superharmonic function 关于精细超调和函数的拉普拉斯算子
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-04-02 DOI: 10.1016/j.exmath.2025.125683
Mohamed El Kadiri
{"title":"On the Laplacian of a finely superharmonic function","authors":"Mohamed El Kadiri","doi":"10.1016/j.exmath.2025.125683","DOIUrl":"10.1016/j.exmath.2025.125683","url":null,"abstract":"<div><div>We define the fine Laplacian <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi></mrow></math></span> of a finely superharmonic function <span><math><mi>u</mi></math></span> on a regular (or more generally a Borel) finely open set <span><math><mrow><mi>U</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, as a Borel measure on <span><math><mi>U</mi></math></span>. If <span><math><mi>U</mi></math></span> is a Euclidean open set, then <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi></mrow></math></span> is the classical Laplacian <span><math><mrow><mi>Δ</mi><mi>u</mi></mrow></math></span> of <span><math><mi>u</mi></math></span> in the distributional sense. Moreover, if <span><math><mi>u</mi></math></span> is finely of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> on <span><math><mi>U</mi></math></span>, then <span><math><mrow><msub><mrow><mi>Δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mi>u</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mfrac><mrow><msup><mrow><mi>∂</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mi>∂</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></mrow></math></span>, where <span><math><mfrac><mrow><msup><mrow><mi>∂</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mi>∂</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfrac></math></span>\u0000 (<span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></math></span>) are the fine partial derivatives of order 2 of <span><math><mi>u</mi></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125683"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On extremal problems of Delsarte type for positive definite functions on LCA groups LCA群上正定函数的Delsarte型极值问题
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-02-18 DOI: 10.1016/j.exmath.2025.125663
E.E. Berdysheva , M.D. Ramabulana , Sz.Gy. Révész
{"title":"On extremal problems of Delsarte type for positive definite functions on LCA groups","authors":"E.E. Berdysheva ,&nbsp;M.D. Ramabulana ,&nbsp;Sz.Gy. Révész","doi":"10.1016/j.exmath.2025.125663","DOIUrl":"10.1016/j.exmath.2025.125663","url":null,"abstract":"<div><div>A unifying framework for some extremal problems on locally compact Abelian groups is considered, special cases of which include the Delsarte and Turán extremal problems. A slight variation of the extremal problem is introduced and the different formulations are studied for equivalence. Extending previous work, a general result on existence of extremal functions for the new variant is proved under a certain general topological condition.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125663"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Orthogonal Fourier Analysis on domains 域的正交傅立叶分析
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2024-11-09 DOI: 10.1016/j.exmath.2024.125629
Mihail N. Kolountzakis
{"title":"Orthogonal Fourier Analysis on domains","authors":"Mihail N. Kolountzakis","doi":"10.1016/j.exmath.2024.125629","DOIUrl":"10.1016/j.exmath.2024.125629","url":null,"abstract":"<div><div>In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125629"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Harmonic extension through cylindrical and conical surfaces 通过圆柱和圆锥表面的谐波扩展
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2024-10-18 DOI: 10.1016/j.exmath.2024.125622
Stephen J. Gardiner, Hermann Render
{"title":"Harmonic extension through cylindrical and conical surfaces","authors":"Stephen J. Gardiner,&nbsp;Hermann Render","doi":"10.1016/j.exmath.2024.125622","DOIUrl":"10.1016/j.exmath.2024.125622","url":null,"abstract":"<div><div>This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125622"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A lonely weak tile 孤独无力的瓷砖
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2024-12-07 DOI: 10.1016/j.exmath.2024.125636
Gergely Kiss , Itay Londner , Máté Matolcsi , Gábor Somlai
{"title":"A lonely weak tile","authors":"Gergely Kiss ,&nbsp;Itay Londner ,&nbsp;Máté Matolcsi ,&nbsp;Gábor Somlai","doi":"10.1016/j.exmath.2024.125636","DOIUrl":"10.1016/j.exmath.2024.125636","url":null,"abstract":"<div><div>The notion of weak tiling was a key ingredient in the proof of Fuglede’s spectral set conjecture for convex bodies, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised by Kolountzakis, Lev and Matolcsi, by giving an example of a set <span><math><mi>T</mi></math></span> which tiles its complement weakly, but <span><math><mi>T</mi></math></span> is neither spectral, nor a proper tile.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125636"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Periodicity and decidability of translational tilings by rational polygonal sets 有理多边形集平动平铺的周期性和可判定性
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2024-10-19 DOI: 10.1016/j.exmath.2024.125620
Jaume de Dios Pont , Jan Grebík , Rachel Greenfeld , José Madrid
{"title":"Periodicity and decidability of translational tilings by rational polygonal sets","authors":"Jaume de Dios Pont ,&nbsp;Jan Grebík ,&nbsp;Rachel Greenfeld ,&nbsp;José Madrid","doi":"10.1016/j.exmath.2024.125620","DOIUrl":"10.1016/j.exmath.2024.125620","url":null,"abstract":"<div><div>The periodic tiling conjecture asserts that if a region <span><math><mrow><mi>Σ</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span> tiles <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> by translations then it admits at least one <em>fully periodic</em> tiling. This conjecture is known to hold in <span><math><mi>R</mi></math></span>, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for <em>polygonal sets</em>: bounded open sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that <em>any</em> translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125620"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Are MSF wavelets minimally supported? 是否最低限度地支持MSF小波？
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-11-29 DOI: 10.1016/j.exmath.2025.125745
Marcin Bownik , Ziemowit Rzeszotnik , Darrin Speegle
{"title":"Are MSF wavelets minimally supported?","authors":"Marcin Bownik ,&nbsp;Ziemowit Rzeszotnik ,&nbsp;Darrin Speegle","doi":"10.1016/j.exmath.2025.125745","DOIUrl":"10.1016/j.exmath.2025.125745","url":null,"abstract":"<div><div>Larson’s problem Larson (2007, Problem 3) asks “Must the support of the Fourier transform of a wavelet contain a wavelet set?”. We give an affirmative answer to a non-measurable variant of this question by proving that the Fourier transform of a wavelet must contain a possibly non-measurable wavelet set. We also provide background results on Larson’s problem and propose two new related problems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125745"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145645748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bounds on energy and potentials of discrete measures on the sphere 球面上离散测度的能量和势的边界
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-07-07 DOI: 10.1016/j.exmath.2025.125712
S.V. Borodachov , P.G. Boyvalenkov , P.D. Dragnev , D.P. Hardin , E.B. Saff , M.M. Stoyanova
{"title":"Bounds on energy and potentials of discrete measures on the sphere","authors":"S.V. Borodachov ,&nbsp;P.G. Boyvalenkov ,&nbsp;P.D. Dragnev ,&nbsp;D.P. Hardin ,&nbsp;E.B. Saff ,&nbsp;M.M. Stoyanova","doi":"10.1016/j.exmath.2025.125712","DOIUrl":"10.1016/j.exmath.2025.125712","url":null,"abstract":"<div><div>We establish upper and lower universal bounds for potentials of weighted designs on the sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to <span><math><mi>p</mi></math></span>-frame energy.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125712"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Maximality and completeness of orthogonal exponentials on the cube 正交指数在立方体上的极大性和完备性
IF 0.9 4区 数学
Expositiones Mathematicae Pub Date : 2026-02-01 Epub Date: 2025-04-02 DOI: 10.1016/j.exmath.2025.125682
Mihail N. Kolountzakis , Nir Lev , Máté Matolcsi
{"title":"Maximality and completeness of orthogonal exponentials on the cube","authors":"Mihail N. Kolountzakis ,&nbsp;Nir Lev ,&nbsp;Máté Matolcsi","doi":"10.1016/j.exmath.2025.125682","DOIUrl":"10.1016/j.exmath.2025.125682","url":null,"abstract":"<div><div>It is possible to have a packing by translates of a cube that is maximal (i.e. no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension 1 there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"44 1","pages":"Article 125682"},"PeriodicalIF":0.9,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147395455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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