Boundedness of Fractional Integrals on Ball Campanato-Type Function Spaces

Abstract

Let $X$ be a ball quasi-Banach function space on ${\mathbb{R}}^n$ satisfying some mild assumptions and let $\alpha \in (0,n)$ and $\beta \in (1,\infty )$. In this article, when $\alpha \in (0,1)$, the authors first find a reasonable version ${\tilde{I}}_{\alpha }$ of the fractional integral $I_{\alpha }$ 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 ${\tilde{I}}_{\alpha }$ is bounded from ${\mathcal{L}}_{X^{\beta },q,s,d}({\mathbb{R}}^n)$ to ${\mathcal{L}}_{X,q,s,d}({\mathbb{R}}^n)$ if and only if there exists a positive constant $C$ such that, for any ball $B\subset {\mathbb{R}}^n$, $|B{|}^{\frac{\alpha }{n}}\le C\Vert 1_B{\Vert }_X^{\frac{\beta -1}{\beta }}$, where $X^{\beta }$ denotes the $\beta$-convexification of $X$. Furthermore, the authors extend the range $\alpha \in (0,1)$ in ${\tilde{I}}_{\alpha }$ to the range $\alpha \in (0,n)$ and also obtain the corresponding boundedness in this case. Moreover, ${\tilde{I}}_{\alpha }$ is proved to be the adjoint operator of $I_{\alpha }$. All these results have a wide range of applications. Particularly, even when they are applied, respectively, to Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the dual theorem on ${\mathcal{L}}_{X,q,s,d}({\mathbb{R}}^n)$ and also on the special atomic decomposition of molecules of $H_X({\mathbb{R}}^n)$ (the Hardy-type space associated with $X$) which proves the predual space of ${\mathcal{L}}_{X,q,s,d}({\mathbb{R}}^n)$.

47 pages, Submitted. arXiv admin note: text overlap with arXiv:2203.15165, arXiv:2206.06080, arXiv:2108.01559