与球拟Banach函数空间相关的Hardy空间上分数积分的有界性

Pub Date : 2022-06-13 DOI:10.3836/tjm/1502179390
Yiqun Chen, H. Jia, Dachun Yang
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引用次数: 7

摘要

设$X$为与$X$相关的Hardy空间${\mathbb R}^n$和$H_X({\mathbb R}^n)$上的球拟巴拿赫函数空间,设$\alpha\in(0,n)$和$\beta\in(1,\infty)$。在本文中,假设(幂)Hardy—Littlewood极大算子满足$X$上的Fefferman—Stein向量值极大不等式,并且在$X$的关联空间上是有界的,证明分数积分$I_{\alpha}$可以推广为一个从$H_X({\mathbb R}^n)$到$H_{X^{\beta}}({\mathbb R}^n)$的有界线性算子,当且仅当存在一个正常数$C$,使得对于任意球$B\subset \mathbb{R}^n$,$|B|^{\frac{\alpha}{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{\beta-1}{\beta}}$,其中$X^{\beta}$表示$X$的$\beta$ -凸度。此外,在$X$和另一个球拟巴拿赫函数空间$Y$上的一些不同的合理假设下,作者还利用外推定理考虑了$I_{\alpha}$从$H_X({\mathbb R}^n)$到$H_Y({\mathbb R}^n)$的映射性质。这些结果具有广泛的应用前景。特别地,当这些结果分别应用于Morrey空间、混合范数Lebesgue空间、局部广义Herz空间和混合范数Herz空间时,这些结果都是新的。这些定理的证明强烈依赖于$H_X({\mathbb R}^n)$的原子和分子表征。
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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 $H_X({\mathbb R}^n)$.
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