{"title":"Weighted Bourgain–Morrey-Besov–Triebel–Lizorkin spaces associated with operators","authors":"Tengfei Bai, Jingshi Xu","doi":"10.1002/mana.202400223","DOIUrl":"https://doi.org/10.1002/mana.202400223","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> be a space of homogeneous type and <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> be a nonnegative self-adjoint operator on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L^2(X)$</annotation>\u0000 </semantics></math> satisfying a Gaussian upper bound on its heat kernel. First, we obtain the boundedness of the Hardy–Littlewood maximal function and its variant on weighted Bourgain–Morrey spaces. The Hardy-type inequality on sequence Bourgain–Morrey spaces are also given. Then, we introduce the weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces associated with the operator <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math>. We obtain characterizations of these spaces in terms of Peetre maximal functions, noncompactly supported functional calculus, and heat kernel. Atomic decompositions and molecular decompositions of weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces are also proved. Finally, we apply our results to prove the boundedness of the fractional power of <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> and the spectral multiplier of <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> on Bourgain–Morrey Besov and Triebel–Lizorkin spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"886-924"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simultaneous approximation by neural network operators with applications to Voronovskaja formulas","authors":"Marco Cantarini, Danilo Costarelli","doi":"10.1002/mana.202400281","DOIUrl":"https://doi.org/10.1002/mana.202400281","url":null,"abstract":"<p>In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by the sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"871-885"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400281","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}