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

TL;DR

This paper investigates the boundedness of fractional integrals on specialized function spaces using congruent cubes, establishing new results on their extension and boundedness criteria linked to vanishing moments.

Contribution

It introduces a new version of fractional integrals on John--Nirenberg--Campanato spaces and characterizes their boundedness via vanishing moments, extending to Hardy-type spaces.

Findings

Boundedness of fractional integrals characterized by vanishing moments.

Extension of fractional integrals to Hardy-type spaces established.

New boundedness criteria for operators on specialized function spaces.

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 $HK_{(p,q,s)_\alpha}^{\mathrm{con}}(\mathbb{R}^n)$.