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 $\widetilde{I}{\beta}$ of the (generalized) fractional integral $I_{\beta}$ on the special John--Nirenberg--Campanato space via congruent cubes, $JN_{(p,q,s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, 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 $\widetilde{I}{\beta}$ is bounded from $JN{(p,q,s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ to $JN{(p,q,s){\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$ 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 $HK_{(p,q,s){\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$, the predual of $JN{(p',q',s){\alpha+\beta/n}}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, to $HK{(p,q,s){\alpha}}^{\mathrm{con}}(\mathbb{R}^n)$ 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 $(HK_{(p,q,s){\alpha}}^{\mathrm{con}}(\mathbb{R}^n))^* =JN{(p',q',s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, the properties of molecules of $HK{(p,q,s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, and a crucial criterion for the boundedness of linear operators on $HK{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$.