Molecules in Coorbit Spaces and Boundedness of Operators

The Characterizations of Anisotropic Mixed-Norm Hardy Spaces on $\mathbb{R}^n$ by Atoms and Molecules

Let $\vec{p}\in(0,\,\infty)^n$, $A$ be an expansive dilation on $\mathbb{R}^n$,and $H^{\vec{p}}_A({\mathbb {R}}^n)$ be the anisotropic mixed-norm Hardy space defined via the non-tangential grand maximal function studied by \cite{hlyy20}. In this paper, the authors establish new atomic and molecular decompositions of $H^{\vec{p}}_A({\mathbb {R}}^n)$. As an application, the authors obtain a boundedness criterion for a class of linear operators from $H^{\vec{p}}_{A}(\mathbb{R}^n)$ to $H^{\vec{p}}_{A}(\mathbb{R}^n)$. Part of results are still new even in the classical isotropic setting (in the cas…

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces I: Decompositions with Applications to Boundedness of Calderón--Zygmund Operators

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$. In this article, the authors introduce the weak Hardy-type space $WH_X({\mathbb R}^n)$, associated with $X$, via the radial maximal function. Assuming that the powered Hardy--Littlewood maximal operator satisfies some Fefferman--Stein vector-valued maximal inequality on $X$ as well as it is bounded on both the weak ball quasi-Banach function space $WX$ and the associated space, the authors then establish several real-variable characterizations of $WH_X({\mathbb R}^n)$, respectively, in terms of various maximal functions, atoms a…

关于将原子映射为局部 Hardy 空间中分子的算子的核条件

On the kernel conditions of operators mapping atoms to molecules in local Hardy spaces

本文研究了局部 Hardy 空间中原子映射为分子的算子与核大小条件的关系，证明了 Calderón--Zygmund 型算子在特定核条件下可将原子映射为分子，反之亦然，并给出了相关算子的有界性。

Boundedness of Fractional Integrals on Special John--Nirenberg--Campanato and Hardy-Type Spaces via Congruent Cubes

Let $p\in[1,\infty]$, $q\in[1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $α\in\mathbb{R}$. In this article, the authors first find a reasonable version $\widetilde{I}_β$ of the (generalized) fractional integral $I_β$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_β$. Then the authors prove that $\widetilde{I}_β$ is bounded from $JN_{(p,q,s)_α}^…

Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, when $α\in(0,1)$, the authors first find a reasonable version $\widetilde{I}_α$ of the fractional integral $I_α$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{I}_α$ is bounded from $\mathcal{L}_{X^β,q,s,d}(\mathbb{R}^n)$ to $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive c…

Real-Variable Characterizations of Orlicz-Slice Hardy Spaces

In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson in…

可变矩阵加权贝索夫空间

Variable Matrix-Weighted Besov Spaces

本文引入了可变矩阵加权贝索夫空间，并通过一系列算子和分解特征研究了其性质。

带矩阵 $A_\infty$ 权的 Besov–Triebel–Lizorkin 型空间

Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_\infty$ Weights

本文利用此前建立的矩阵 $A_{p,\infty}$ 权刻画，系统研究相应非齐次 Besov 型与 Triebel–Lizorkin 型空间，给出多种刻画并证明多类算子的有界性。