{"title":"On the Sense of Convergence in the Dyadic Representation Theorem","authors":"Tuomas Hytönen","doi":"10.1007/s10114-025-3698-0","DOIUrl":"10.1007/s10114-025-3698-0","url":null,"abstract":"<div><p>The dyadic representation of any singular integral operator, as an average of dyadic model operators, has found many applications. While for many purposes it is enough to have such a representation for a “suitable class” of test functions, we show that, under quite general assumptions (essentially minimal ones to make sense of the formula), the representation is actually valid for all pairs (<i>f,g</i>) ∈ <i>L</i><sup><i>p</i></sup>(ℝ<sup><i>d</i></sup>) × <i>L</i><sup><i>p</i>′</sup>(ℝ<sup><i>d</i></sup>), not just test functions.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"472 - 496"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108855","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}
Dorothee D. Haroske, Leszek Skrzypczak, Hans Triebel
{"title":"Mapping Properties of Fourier Transforms, Revisited","authors":"Dorothee D. Haroske, Leszek Skrzypczak, Hans Triebel","doi":"10.1007/s10114-025-3532-8","DOIUrl":"10.1007/s10114-025-3532-8","url":null,"abstract":"<div><p>The paper deals with continuous and compact mappings generated by the Fourier transform between distinguished Besov spaces <i>B</i><span>\u0000 <sup><i>s</i></sup><sub><i>p</i></sub>\u0000 \u0000 </span>(ℝ<sup><i>n</i></sup>) = <i>B</i><span>\u0000 <sup><i>s</i></sup><sub><i>p,p</i></sub>\u0000 \u0000 </span>(ℝ<sup><i>n</i></sup>), 1 ≤ <i>p</i> ≤ ∞, and between Sobolev spaces <i>H</i><span>\u0000 <sup><i>s</i></sup><sub><i>p</i></sub>\u0000 \u0000 </span>(ℝ<sup><i>n</i></sup>), 1 < <i>p</i> < ∞. In contrast to the paper <i>H. Triebel, Mapping properties of Fourier transforms. Z. Anal. Anwend.</i> 41 (2022), 133–152, based mainly on embeddings between related weighted spaces, we rely on wavelet expansions, duality and interpolation of corresponding (unweighted) spaces, and (appropriately extended) Hausdorff-Young inequalities. The degree of compactness will be measured in terms of entropy numbers and approximation numbers, now using the symbiotic relationship to weighted spaces.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"231 - 254"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108856","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":"Uniform Regularity for Degenerate Elliptic Equations in Perforated Domains","authors":"Zhongwei Shen, Jinping Zhuge","doi":"10.1007/s10114-025-3640-5","DOIUrl":"10.1007/s10114-025-3640-5","url":null,"abstract":"<div><p>This paper is concerned with a class of degenerate elliptic equations with rapidly oscillating coefficients in periodically perforated domains, which arises in the study of spectrum problems for uniformly elliptic equations in perforated domains. We establish a quantitative convergence rate and obtain the uniform weighted Lipschitz and <i>W</i><sup>1,<i>p</i></sup> estimates.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"378 - 412"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108860","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":"A Remark on Stein–Tomas Type Restriction Theorems","authors":"Xiaochun Li","doi":"10.1007/s10114-025-3525-7","DOIUrl":"10.1007/s10114-025-3525-7","url":null,"abstract":"<div><p>A local <i>L</i><sup><i>p</i></sup> estimate is proved by using the <i>σ</i>-uniformity, which is motivated by the study of the Stein–Tomas type restriction theorems and Waring’s problem.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"122 - 130"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108453","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":"New Characterization of Morrey-Herz Spaces and Morrey-Herz-Hardy Spaces with Applications to Various Linear Operators","authors":"Kwok Pun Ho, Yoshihiro Sawano","doi":"10.1007/s10114-025-3570-2","DOIUrl":"10.1007/s10114-025-3570-2","url":null,"abstract":"<div><p>This paper is an offspring of the previous study on Herz spaces. A new characterization of Morrey-Herz spaces is given. As applications, the boundedness of various operators is obtained. For example, higher-order commutators generated by singular integral operators and BMO functions are proved to be bounded on Morrey-Herz spaces. The theory of Morrey-Herz-Hardy spaces is also developed.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"327 - 354"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108466","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}
Dorothee D. Haroske, Zhen Liu, Susana D. Moura, Leszek Skrzypczak
{"title":"Embeddings of Generalised Morrey Smoothness Spaces","authors":"Dorothee D. Haroske, Zhen Liu, Susana D. Moura, Leszek Skrzypczak","doi":"10.1007/s10114-025-3553-3","DOIUrl":"10.1007/s10114-025-3553-3","url":null,"abstract":"<div><p>We study embeddings between generalised Triebel–Lizorkin–Morrey spaces <span>(cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span> and within the scales of further generalised Morrey smoothness spaces like <span>(cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span>, <i>B</i><span>\u0000 <sup><i>s,φ</i></sup><sub><i>p,q</i></sub>\u0000 \u0000 </span>(ℝ<sup><i>d</i></sup>) and <i>F</i><span>\u0000 <sup><i>s,φ</i></sup><sub><i>p,q</i></sub>\u0000 \u0000 </span>(ℝ<sup><i>d</i></sup>). The latter have been investigated in a recent paper by the first two authors (2023), while the embeddings of the scale <span>(cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span> were mainly obtained in a paper of the first and last two authors (2022). Now we concentrate on the characterisation of the spaces <span>(cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span>. Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies’ wavelets. Then we prove necessary and sufficient conditions for the embedding <span>(cal{E}_{varphi_{1},p_{1},q_{1}}^{s_{1}}(mathbb{R}^{d})hookrightarrowcal{E}_{varphi_{2},p_{2},q_{2}}^{s_{2}}(mathbb{R}^{d}))</span>. We can also provide some almost final answer to the question when <span>(cal{E}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span> is embedded into <i>C</i>(ℝ<sup><i>d</i></sup>), complementing our recent findings in case of <span>(cal{N}_{varphi,p,q}^{s}(mathbb{R}^{d}))</span>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"413 - 456"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108859","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":"Weighted Estimates for Generalised Conical Square Functions and Applications","authors":"The Anh Bui, Xuan Thinh Duong, Ji Li","doi":"10.1007/s10114-025-3478-x","DOIUrl":"10.1007/s10114-025-3478-x","url":null,"abstract":"<div><p>Let <span>({cal{A}_{t}}_{t>0})</span> be a family of bounded linear operator on <i>L</i><sup>2</sup>(<i>X</i>) where (<i>X, d, μ</i>) is a metric space with metric <i>d</i> and doubling measure <i>μ</i>. Assume that the family <span>({cal{A}_{t}}_{t>0})</span> satisfies suitable off-diagonal estimates from <span>(L^{p_{0}})</span> to <i>L</i><sup>2</sup> for some <i>p</i><sub>0</sub> < 2. This paper aims to prove weighted bound estimates for conical square functions and g-functions associated to the family <span>({cal{A}_{t}}_{t>0})</span>. Some applications such as weighted bounds for bilinear estimates associated to certain differential operators are also obtained.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"191 - 208"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108862","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}
Nijjwal Karak, Pekka Koskela, Debanjan Nandi, Swadesh Kumar Sahoo
{"title":"Conformal Composition for Borderline Fractional Sobolev Spaces","authors":"Nijjwal Karak, Pekka Koskela, Debanjan Nandi, Swadesh Kumar Sahoo","doi":"10.1007/s10114-025-3649-9","DOIUrl":"10.1007/s10114-025-3649-9","url":null,"abstract":"<div><p>We establish a pointwise property for homogeneous fractional Sobolev spaces in domains with non-empty boundary, extending a similar result of Koskela–Yang–Zhou. We use this to show that a conformal map from the unit disk onto a simply connected planar domain induces a bounded composition operator from the borderline homogeneous fractional Sobolev space of the domain into the corresponding space of the unit disk.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"457 - 471"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108854","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":"Bilinear Exotic Calderón-Zygmund Operators","authors":"Jin Bai, Jinsong Li, Kangwei Li","doi":"10.1007/s10114-025-3589-4","DOIUrl":"10.1007/s10114-025-3589-4","url":null,"abstract":"<div><p>We introduce a bilinear extension of the so-called exotic Calderón-Zygmund operators. These kernels arise naturally from the bilinear singular integrals associated with Zygmund dilations. We show that such a class of operators satisfy a <i>T</i>1 theorem in the same form as the standard Calderón-Zygmund operators. However, one-parameter weighted estimates may fail in general, and unlike the linear case, we are not able to provide the end-point estimates in full generality.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"355 - 377"},"PeriodicalIF":0.8,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108467","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}
Fernando Cobos, Luz M. Fernández-Cabrera, Antón Martínez
{"title":"Interpolation of Closed Ideals of Bilinear Operators","authors":"Fernando Cobos, Luz M. Fernández-Cabrera, Antón Martínez","doi":"10.1007/s10114-025-3506-x","DOIUrl":"10.1007/s10114-025-3506-x","url":null,"abstract":"<div><p>We extend the (outer) measure <span>(gamma_{cal{I}})</span> associated to an operator ideal <span>(cal{I})</span> to a measure <span>(gamma_{frak{J}})</span> for bounded bilinear operators. If <span>(cal{I})</span> is surjective and closed, and <span>(frak{J})</span> is the class of those bilinear operators such that <span>(gamma_{frak{J}}(T)=0)</span>, we prove that <span>(frak{J})</span> coincides with the composition bideal <span>(cal{I}circfrak{B})</span>. If <span>(cal{I})</span> satisfies the Σ<sub><i>r</i></sub>-condition, we establish a simple necessary and sufficient condition for an interpolated operator by the real method to belong to <span>(frak{J})</span>. Furthermore, if in addition <span>(cal{I})</span> is symmetric, we prove a formula for the measure <span>(gamma_{frak{J}})</span> of an operator interpolated by the real method. In particular, results apply to weakly compact operators.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"41 1","pages":"209 - 230"},"PeriodicalIF":0.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108139","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}
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