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Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces 与贝索夫空间结构相混合的布尔干涉-莫雷空间
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050152
Yirui Zhao, Yoshihiro Sawano, Jin Tao, Dachun Yang, Wen Yuan
{"title":"Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces","authors":"Yirui Zhao, Yoshihiro Sawano, Jin Tao, Dachun Yang, Wen Yuan","doi":"10.1134/s0081543823050152","DOIUrl":"https://doi.org/10.1134/s0081543823050152","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Bourgain–Morrey spaces <span>(mathcal{M}^p_{q,r}(mathbb R^n))</span>, generalizing what was introduced by J. Bourgain, play an important role in the study related to the Strichartz estimate and the nonlinear Schrödinger equation. In this article, via adding an extra exponent <span>(tau)</span>, the authors creatively introduce a new class of function spaces, called Besov–Bourgain–Morrey spaces <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>, which is a bridge connecting Bourgain–Morrey spaces <span>(mathcal{M}^p_{q,r}(mathbb R^n))</span> with amalgam-type spaces <span>((L^q,ell^r)^p(mathbb R^n))</span>. By making full use of the Fatou property of block spaces in the weak local topology of <span>(L^{q'}(mathbb R^n))</span>, the authors give both predual and dual spaces of <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Applying these properties and the Calderón product, the authors also establish the complex interpolation of <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Via fully using fine geometrical properties of dyadic cubes, the authors then give an equivalent norm of <span>(|kern1pt{cdot}kern1pt|_{mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n)})</span> having an integral expression, which further induces a boundedness criterion of operators on <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Applying this criterion, the authors obtain the boundedness on <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span> of classical operators including the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"2 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054770","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 Embedding of Besov Spaces of Zero Smoothness into Lorentz Spaces 论零光滑度贝索夫空间嵌入洛伦兹空间
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050127
D. M. Stolyarov
{"title":"On Embedding of Besov Spaces of Zero Smoothness into Lorentz Spaces","authors":"D. M. Stolyarov","doi":"10.1134/s0081543823050127","DOIUrl":"https://doi.org/10.1134/s0081543823050127","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We show that the zero smoothness Besov space <span>(B_{p,q}^{0,1})</span> does not embed into the Lorentz space <span>(L_{p,q})</span> unless <span>(p=q)</span>; here <span>(p,qin (1,infty))</span>. This answers in the negative a question posed by O. V. Besov. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889093","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 Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series 论多重三角傅里叶级数部分和的后继普林塞姆收敛性
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050097
S. V. Konyagin
{"title":"On Pringsheim Convergence of a Subsequence of Partial Sums of a Multiple Trigonometric Fourier Series","authors":"S. V. Konyagin","doi":"10.1134/s0081543823050097","DOIUrl":"https://doi.org/10.1134/s0081543823050097","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A. N. Kolmogorov’s famous theorem of 1925 implies that the partial sums of the Fourier series of any integrable function <span>(f)</span> of one variable converge to it in <span>(L^p)</span> for all <span>(pin(0,1))</span>. It is known that this does not hold true for functions of several variables. In this paper we prove that, nevertheless, for any function of several variables there is a subsequence of Pringsheim partial sums that converges to the function in <span>(L^p)</span> for all <span>(pin(0,1))</span>. At the same time, in a fairly general case, when we take the partial sums of the Fourier series of a function of several variables over an expanding system of index sets, there exists a function for which the absolute values of a certain subsequence of these partial sums tend to infinity almost everywhere. This is so, in particular, for a system of dilations of a fixed bounded convex body and for hyperbolic crosses. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"7 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889033","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
Interpolation of Operators in Hardy-Type Spaces 哈代型空间中的算子插值法
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050103
V. G. Krotov
{"title":"Interpolation of Operators in Hardy-Type Spaces","authors":"V. G. Krotov","doi":"10.1134/s0081543823050103","DOIUrl":"https://doi.org/10.1134/s0081543823050103","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A number of statements similar to the Marcinkiewicz interpolation theorem are presented. The difference from the classical forms of this theorem is that the spaces of integrable functions are replaced by certain classes of functions that are extensions of various Hardy spaces. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"12 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054756","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 Regularity of Characteristic Functions of Weakly Exterior Thick Domains 论弱外厚域特征函数的规律性
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050085
Winfried Sickel
{"title":"On the Regularity of Characteristic Functions of Weakly Exterior Thick Domains","authors":"Winfried Sickel","doi":"10.1134/s0081543823050085","DOIUrl":"https://doi.org/10.1134/s0081543823050085","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Let <span>(E)</span> be a domain in <span>(mathbb R^d)</span>. We investigate the regularity of the characteristic function <span>(mathcal X_E)</span> depending on the behavior of the <span>(delta)</span>-neighborhoods of the boundary of <span>(E)</span>. The regularity is measured in terms of the Nikol’skii–Besov and Lizorkin–Triebel spaces. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"72 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054773","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
Truncations and Compositions in Function Spaces 函数空间中的截断和组合
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050140
Hans Triebel
{"title":"Truncations and Compositions in Function Spaces","authors":"Hans Triebel","doi":"10.1134/s0081543823050140","DOIUrl":"https://doi.org/10.1134/s0081543823050140","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The paper deals with some recent assertions about truncations <span>(fmapsto |f|)</span> and compositions <span>(fmapsto gcirc f)</span> in the spaces <span>(A^s_{p,q}(mathbb R^n))</span>, <span>(Ain{B,F})</span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054750","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
Hierarchical Schrödinger Operators with Singular Potentials 具有奇异势的分层薛定谔算子
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050024
Alexander Bendikov, Alexander Grigor’yan, Stanislav Molchanov
{"title":"Hierarchical Schrödinger Operators with Singular Potentials","authors":"Alexander Bendikov, Alexander Grigor’yan, Stanislav Molchanov","doi":"10.1134/s0081543823050024","DOIUrl":"https://doi.org/10.1134/s0081543823050024","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the operator <span>(H=L+V)</span> that is a perturbation of the Taibleson–Vladimirov operator <span>(L=mathfrak{D}^alpha)</span> by a potential <span>(V(x)=b|x|^{-alpha})</span>, where <span>(alpha&gt;0)</span> and <span>(bgeq b_*)</span>. We prove that the operator <span>(H)</span> is closable and its minimal closure is a nonnegative definite self-adjoint operator (where the critical value <span>(b_*)</span> depends on <span>(alpha)</span>). While the operator <span>(H)</span> is nonnegative definite, the potential <span>(V(x))</span> may well take negative values as <span>(b_*&lt;0)</span> for all <span>(0&lt;alpha&lt;1)</span>. The equation <span>(Hu=v)</span> admits a Green function <span>(g_H(x,y))</span>, that is, the integral kernel of the operator <span>(H^{-1})</span>. We obtain sharp lower and upper bounds on the ratio of the Green functions <span>(g_H(x,y))</span> and <span>(g_L(x,y))</span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"118 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889094","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
Trace and Extension Theorems for Homogeneous Sobolev and Besov Spaces for Unbounded Uniform Domains in Metric Measure Spaces 公度量空间中无边界均匀域的同质索波列夫和贝索夫空间的踪迹和扩展定理
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050061
Ryan Gibara, Nageswari Shanmugalingam
{"title":"Trace and Extension Theorems for Homogeneous Sobolev and Besov Spaces for Unbounded Uniform Domains in Metric Measure Spaces","authors":"Ryan Gibara, Nageswari Shanmugalingam","doi":"10.1134/s0081543823050061","DOIUrl":"https://doi.org/10.1134/s0081543823050061","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In this paper we fix <span>(1le p&lt;infty)</span> and consider <span>((Omega,d,mu))</span> to be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure <span>(mu)</span> supporting a <span>(p)</span>-Poincaré inequality such that <span>(Omega)</span> is a uniform domain in its completion <span>(overlineOmega)</span>. We realize the trace of functions in the Dirichlet–Sobolev space <span>(D^{1,p}(Omega))</span> on the boundary <span>(partialOmega)</span> as functions in the homogeneous Besov space <span>(Hkern-1pt B^alpha_{p,p}(partialOmega))</span> for suitable <span>(alpha)</span>; here, <span>(partialOmega)</span> is equipped with a non-atomic Borel regular measure <span>(nu)</span>. We show that if <span>(nu)</span> satisfies a <span>(theta)</span>-codimensional condition with respect to <span>(mu)</span> for some <span>(0&lt;theta&lt;p)</span>, then there is a bounded linear trace operator <span>(T colon, D^{1,p}(Omega)to Hkern-1pt B^{1-theta/p}(partialOmega))</span> and a bounded linear extension operator <span>(E colon, Hkern-1pt B^{1-theta/p}(partialOmega)to D^{1,p}(Omega))</span> that is a right-inverse of <span>(T)</span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140890037","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
Integral Representations and Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain 赫尔德域上正平滑度函数空间的积分表示和嵌入
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050036
O. V. Besov
{"title":"Integral Representations and Embeddings of Spaces of Functions of Positive Smoothness on a Hölder Domain","authors":"O. V. Besov","doi":"10.1134/s0081543823050036","DOIUrl":"https://doi.org/10.1134/s0081543823050036","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We prove embedding theorems for spaces of functions of positive smoothness defined on a Hölder domain of <span>(n)</span>-dimensional Euclidean space. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"102 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054652","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 Best Recovery of a Family of Operators on the Manifold $$mathbb R^ntimesmathbb T^m$$ 论流形 $$mathbb R^ntimesmathbb T^m$$ 上一族算子的最佳恢复力
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050115
G. G. Magaril-Il’yaev, E. O. Sivkova
{"title":"On the Best Recovery of a Family of Operators on the Manifold $$mathbb R^ntimesmathbb T^m$$","authors":"G. G. Magaril-Il’yaev, E. O. Sivkova","doi":"10.1134/s0081543823050115","DOIUrl":"https://doi.org/10.1134/s0081543823050115","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Given a one-parameter family of operators on the manifold <span>(mathbb R^ntimesmathbb T^m)</span>, we solve the problem of the best recovery of an operator for a given value of the parameter from inaccurate data on the operators for other values of the parameter from a certain compact set. We construct a family of best recovery methods. As a consequence, we obtain families of best recovery methods for the solutions of the heat equation and the Dirichlet problem for a half-space. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"281 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054749","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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