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

Abstract

Let $p\in [1,\infty ]$, $q\in [1,\infty )$, $s\in {\mathbb{Z}}_{+}:=\mathbb{N}\cup \{0\}$, and $\alpha \in \mathbb{R}$. In this article, the authors first find a reasonable version ${\tilde{I}}_{\beta }$ of the (generalized) fractional integral $I_{\beta }$ on the special John--Nirenberg--Campanato space via congruent cubes, $J{N}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$, which coincides with the Campanato space ${\mathcal{C}}_{\alpha ,q,s}({\mathbb{R}}^n)$ when $p=\infty$. To this end, the authors introduce the vanishing moments up to order $s$ of $I_{\beta }$. Then the authors prove that ${\tilde{I}}_{\beta }$ is bounded from $J{N}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$ to $J{N}_{(p,q,s{)}_{\alpha +\beta /n}}^{\mathrm{con}}({\mathbb{R}}^n)$ if and only if $I_{\beta }$ has the vanishing moments up to order $s$. The obtained result is new even when $p=\infty$ and $s\in \mathbb{N}$. Moreover, the authors show that $I_{\beta }$ can be extended to a unique continuous linear operator from the Hardy-kind space $H{K}_{(p,q,s{)}_{\alpha +\beta /n}}^{\mathrm{con}}({\mathbb{R}}^n)$, the predual of $J{N}_{(p^{\prime },q^{\prime },s{)}_{\alpha +\beta /n}}^{\mathrm{con}}({\mathbb{R}}^n)$ with $\frac{1}{p}+\frac{1}{p^{\prime }}=1=\frac{1}{q}+\frac{1}{q^{\prime }}$, to $H{K}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$ if and only if $I_{\beta }$ has the vanishing moments up to order $s$. The proof of the latter boundedness strongly depends on the dual relation $(H{K}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n){)}^{\ast }=J{N}_{(p^{\prime },q^{\prime },s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$, the properties of molecules of $H{K}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$, and a crucial criterion for the boundedness of linear operators on $H{K}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$.

38 pages; Front. Math. China. (to appear). arXiv admin note: text overlap with arXiv:2108.01517