A rough multiple Marcinkiewicz integral along continuous surfaces

Abstract

By means of the method of block decompositions for kernel functions and some delicate estimates on Fourier transforms, the $L^p(\boldsymbol{R}^m\times\boldsymbol{R}^n\times\boldsymbol{R})$-boundedness of the multiple Marcinkiewicz integral is established along a continuous surface with rough kernel for some $p>1$. The condition on the integral kernel is the best possible for the $L^2$-boundedness of the multiple Marcinkiewicz integral operator.