论文标题
分数积分在与球准 - 浴场函数空间相关的耐力空间上的界限
Boundedness of Fractional Integrals on Hardy Spaces Associated with Ball Quasi-Banach Function Spaces
论文作者
论文摘要
令$ x $为$ {\ mathbb r}^n $和$ h_x（{\ mathbb r}^n）$的Ball Quasi-Banach功能空间，与$ x $相关的hardy空间，然后让$α\ in（0，n）$和$β\ in（1，\ infty）$。在本文中，假设（有能力的） - 极限的最大操作员满足了$ x $上的fefferman-stein vector-valued最大不平等，并且在$ x $的关联空间上具有限制，则作者证明，$i_α$可以将$i_α$扩展到$ h_x（$ h_x）的边界linear operator， $ h_ {x^β}（{\ m athbb r}^n）$，并且仅当存在正常常数$ c $，以便对任何球$ b \ subset \ subset \ mathbb {r}^n $，$ | b |^{ \ | \ mathbf {1} _b \ | _x^{\ frac {β-1}β} $，其中$ x^β$ x^β$表示$β$ - 征为$ x $。此外，在$ x $和另一个Ball Quasi-Banach功能空间$ y $上的一些不同的合理假设下，作者还考虑$i_α$的映射属性来自$ h_x（{\ Mathbb r}^n）$ to $ h_y（{\ Mathbb r}^n）$通过使用ExtrapoLation theorem theorem。所有这些结果都有广泛的应用。特别是，当分别将其应用于莫里空间，混合 - 勒布斯格空间，当地的普遍性的HERZ空间和混合 - 纳尔兹的HERZ空间时，所有这些结果都是新的。这些定理的证明很大程度上取决于$ h_x（{\ mathbb r}^n）$的原子和分子特征。
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 $α\in(0,n)$ and $β\in(1,\infty)$. 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 $I_α$ can be extended to a bounded linear operator from $H_X({\mathbb R}^n)$ to $H_{X^β}({\mathbb R}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset \mathbb{R}^n$, $|B|^{\fracα{n}}\leq C \|\mathbf{1}_B\|_X^{\frac{β-1}β}$, where $X^β$ denotes the $β$-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 $I_α$ 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 $H_X({\mathbb R}^n)$.
