{"title":"与球拟Banach函数空间相关的Hardy空间上分数积分的有界性","authors":"Yiqun Chen, H. Jia, Dachun Yang","doi":"10.3836/tjm/1502179390","DOIUrl":null,"url":null,"abstract":"Let $X$ be a ball quasi-Banach function space on ${\\mathbb R}^n$ and $H_X({\\mathbb R}^n)$ the Hardy space associated with $X$, and let $\\alpha\\in(0,n)$ and $\\beta\\in(1,\\infty)$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_{\\alpha}$ can be extended to a bounded linear operator from $H_X({\\mathbb R}^n)$ to $H_{X^{\\beta}}({\\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\\subset \\mathbb{R}^n$, $|B|^{\\frac{\\alpha}{n}}\\leq C \\|\\mathbf{1}_B\\|_X^{\\frac{\\beta-1}{\\beta}}$, where $X^{\\beta}$ denotes the $\\beta$-convexification of $X$. Moreover, under some different reasonable assumptions on both $X$ and another ball quasi-Banach function space $Y$, the authors also consider the mapping property of $I_{\\alpha}$ from $H_X({\\mathbb R}^n)$ to $H_Y({\\mathbb R}^n)$ via using the extrapolation theorem. All these results have a wide range of applications. Particularly, when these are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all these results are new. The proofs of these theorems strongly depend on atomic and molecular characterizations of $H_X({\\mathbb R}^n)$.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces\",\"authors\":\"Yiqun Chen, H. Jia, Dachun Yang\",\"doi\":\"10.3836/tjm/1502179390\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $X$ be a ball quasi-Banach function space on ${\\\\mathbb R}^n$ and $H_X({\\\\mathbb R}^n)$ the Hardy space associated with $X$, and let $\\\\alpha\\\\in(0,n)$ and $\\\\beta\\\\in(1,\\\\infty)$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_{\\\\alpha}$ can be extended to a bounded linear operator from $H_X({\\\\mathbb R}^n)$ to $H_{X^{\\\\beta}}({\\\\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\\\\subset \\\\mathbb{R}^n$, $|B|^{\\\\frac{\\\\alpha}{n}}\\\\leq C \\\\|\\\\mathbf{1}_B\\\\|_X^{\\\\frac{\\\\beta-1}{\\\\beta}}$, where $X^{\\\\beta}$ denotes the $\\\\beta$-convexification of $X$. Moreover, under some different reasonable assumptions on both $X$ and another ball quasi-Banach function space $Y$, the authors also consider the mapping property of $I_{\\\\alpha}$ from $H_X({\\\\mathbb R}^n)$ to $H_Y({\\\\mathbb R}^n)$ via using the extrapolation theorem. All these results have a wide range of applications. Particularly, when these are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all these results are new. The proofs of these theorems strongly depend on atomic and molecular characterizations of $H_X({\\\\mathbb R}^n)$.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3836/tjm/1502179390\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3836/tjm/1502179390","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ and $H_X({\mathbb R}^n)$ the Hardy space associated with $X$, and let $\alpha\in(0,n)$ and $\beta\in(1,\infty)$. In this article, assuming that the (powered) Hardy--Littlewood maximal operator satisfies the Fefferman--Stein vector-valued maximal inequality on $X$ and is bounded on the associate space of $X$, the authors prove that the fractional integral $I_{\alpha}$ can be extended to a bounded linear operator from $H_X({\mathbb R}^n)$ to $H_{X^{\beta}}({\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset \mathbb{R}^n$, $|B|^{\frac{\alpha}{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{\beta-1}{\beta}}$, where $X^{\beta}$ denotes the $\beta$-convexification of $X$. Moreover, under some different reasonable assumptions on both $X$ and another ball quasi-Banach function space $Y$, the authors also consider the mapping property of $I_{\alpha}$ from $H_X({\mathbb R}^n)$ to $H_Y({\mathbb R}^n)$ via using the extrapolation theorem. All these results have a wide range of applications. Particularly, when these are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all these results are new. The proofs of these theorems strongly depend on atomic and molecular characterizations of $H_X({\mathbb R}^n)$.