Boundedness of fractional integrals on special John–Nirenberg–Campanato and Hardy-type spaces via congruent cubes

Abstract

Let $p\in [1,\infty ]$, $q\in [1,\infty )$, $s\in \mathbb {Z}_+:=\mathbb {N}\cup \{0\}$, $\alpha \in \mathbb {R}$, and $\beta \in (0,1)$. 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, $JN_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$, 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 $JN_{(p,q,s)_\alpha }^{\textrm{con}}(\mathbb {R}^n)$ to $JN_{(p,q,s)_{\alpha +\beta /n}}^{\textrm{con}}(\mathbb {R}^n)$ if and only if $I_{\beta }$ has the vanishing moments up to order s. The obtained result is new even when $p=\infty$ and $s\in \mathbb {N}$, namely, the Campanato space. Moreover, the authors show that $I_{\beta }$ can be extended to a unique continuous linear operator from the Hardy-kind space $HK_{(p,q,s)_{\alpha +\beta /n}}^{\textrm{con}}(\mathbb {R}^n)$, the predual of $JN_{(p',q',s)_{\alpha +\beta /n}}^{\textrm{con}}(\mathbb {R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, to $HK_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$ 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 $(HK_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n))^{*} =JN_{(p',q',s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$, the good properties of molecules of $HK_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$, and the crucial criterion for the boundedness of linear operators on $HK_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)$.