The existence and boundedness of multilinear Marcinkiewicz integrals on Companato spaces

TL;DR

This paper proves the boundedness of multilinear Marcinkiewicz integrals on Campanato spaces, establishing conditions under which these integrals are finite and deriving associated norm inequalities.

Contribution

It demonstrates the boundedness of m-linear Marcinkiewicz integrals on Campanato spaces and establishes a key norm inequality, extending previous results to these function spaces.

Findings

Boundedness of m-linear Marcinkiewicz integrals on Campanato spaces.

Almost everywhere finiteness of the integrals under certain conditions.

Norm inequality relating the integrals to Campanato space norms.

Abstract

In this paper, we established the boundedness of m-linear Marcinkiewicz integral on Campanato type spaces. We showed that if the $m$-linear Marcinkiewicz integral is finite for one point, then it is finite almost everywhere. Moreover, the following norm inequality holds, $$\|\mu(\vec{f})\|_{\mathcal{E}^{\alpha,p}} \leq C\prod_{j=1}^m\|f_j\|_{\mathcal{E}^{\alpha_j,p_j}},$$ where $\mathcal{E}^{\alpha,p}$ is the classical Campanato spaces.