The existence and boundedness of multilinear Marcinkiewicz integrals on Companato spaces

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.