{"title":"大贝索夫-布尔干姆雷空间及其在算子有界性中的应用","authors":"Yijin Zhang, Dachun Yang, Yirui Zhao","doi":"10.1007/s13324-024-00932-z","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(1<q\\le p \\le r\\le \\infty \\)</span> and <span>\\(\\tau \\in (0,\\infty ]\\)</span>. Besov–Bourgain–Morrey spaces <span>\\({\\mathcal {M}}\\dot{B}^{p,\\tau }_{q,r}({\\mathbb {R}}^n)\\)</span> in the special case where <span>\\(\\tau =r\\)</span>, extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent <span>\\(\\theta \\in [0,\\infty )\\)</span>, the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces <span>\\({\\mathcal {M}}\\dot{B}^{p,\\tau }_{q),r,\\theta }({\\mathbb {R}}^n)\\)</span>. The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of <span>\\({\\mathcal {M}}\\dot{B}^{p,\\tau }_{q),r,\\theta }({\\mathbb {R}}^n)\\)</span> related to Muckenhoupt <span>\\(A_1({\\mathbb {R}}^n)\\)</span>-weights, the authors then obtain an extrapolation theorem of <span>\\({\\mathcal {M}}\\dot{B}^{p,\\tau }_{q),r,\\theta }({\\mathbb {R}}^n)\\)</span>. Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of <span>\\({\\mathbb {R}}^n\\)</span>, the authors establish the sharp boundedness on <span>\\({\\mathcal {M}}\\dot{B}^{p,\\tau }_{q),r,\\theta }({\\mathbb {R}}^n)\\)</span> of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 4","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Grand Besov–Bourgain–Morrey spaces and their applications to boundedness of operators\",\"authors\":\"Yijin Zhang, Dachun Yang, Yirui Zhao\",\"doi\":\"10.1007/s13324-024-00932-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\(1<q\\\\le p \\\\le r\\\\le \\\\infty \\\\)</span> and <span>\\\\(\\\\tau \\\\in (0,\\\\infty ]\\\\)</span>. Besov–Bourgain–Morrey spaces <span>\\\\({\\\\mathcal {M}}\\\\dot{B}^{p,\\\\tau }_{q,r}({\\\\mathbb {R}}^n)\\\\)</span> in the special case where <span>\\\\(\\\\tau =r\\\\)</span>, extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent <span>\\\\(\\\\theta \\\\in [0,\\\\infty )\\\\)</span>, the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces <span>\\\\({\\\\mathcal {M}}\\\\dot{B}^{p,\\\\tau }_{q),r,\\\\theta }({\\\\mathbb {R}}^n)\\\\)</span>. The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of <span>\\\\({\\\\mathcal {M}}\\\\dot{B}^{p,\\\\tau }_{q),r,\\\\theta }({\\\\mathbb {R}}^n)\\\\)</span> related to Muckenhoupt <span>\\\\(A_1({\\\\mathbb {R}}^n)\\\\)</span>-weights, the authors then obtain an extrapolation theorem of <span>\\\\({\\\\mathcal {M}}\\\\dot{B}^{p,\\\\tau }_{q),r,\\\\theta }({\\\\mathbb {R}}^n)\\\\)</span>. Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of <span>\\\\({\\\\mathbb {R}}^n\\\\)</span>, the authors establish the sharp boundedness on <span>\\\\({\\\\mathcal {M}}\\\\dot{B}^{p,\\\\tau }_{q),r,\\\\theta }({\\\\mathbb {R}}^n)\\\\)</span> of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.</p></div>\",\"PeriodicalId\":48860,\"journal\":{\"name\":\"Analysis and Mathematical Physics\",\"volume\":\"14 4\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-06-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Mathematical Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13324-024-00932-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-024-00932-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Grand Besov–Bourgain–Morrey spaces and their applications to boundedness of operators
Let \(1<q\le p \le r\le \infty \) and \(\tau \in (0,\infty ]\). Besov–Bourgain–Morrey spaces \({\mathcal {M}}\dot{B}^{p,\tau }_{q,r}({\mathbb {R}}^n)\) in the special case where \(\tau =r\), extending what was introduced by J. Bourgain, have proved useful in the study related to the Strichartz estimate and the non-linear Schrödinger equation. In this article, by cleverly mixing the norm structures of grand Lebesgue spaces and Besov–Bourgain–Morrey spaces and adding an extra exponent \(\theta \in [0,\infty )\), the authors introduce a new class of function spaces, called generalized grand Besov–Bourgain–Morrey spaces \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\). The authors explore their various real-variable properties including pre-dual spaces and the Gagliardo–Peetre and the ± interpolation theorems. Via establishing some equivalent quasi-norms of \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) related to Muckenhoupt \(A_1({\mathbb {R}}^n)\)-weights, the authors then obtain an extrapolation theorem of \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\). Applying this extrapolation theorem, the Calderón product, and the sparse family of dyadic grids of \({\mathbb {R}}^n\), the authors establish the sharp boundedness on \({\mathcal {M}}\dot{B}^{p,\tau }_{q),r,\theta }({\mathbb {R}}^n)\) of the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator.
期刊介绍:
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.