The existence and boundedness of multilinear Marcinkiewicz integrals on Companato spaces

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, $$\Vert \mu (\vec{f}){\Vert }_{{\mathcal{E}}^{\alpha ,p}}\le C\prod _{j=1}^m\Vert f_j{\Vert }_{{\mathcal{E}}^{{\alpha }_j,p_j}},$$ where ${\mathcal{E}}^{\alpha ,p}$ is the classical Campanato spaces.

27 pages