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 (R m × R n × 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. 1. Introduction. Let R N (N = m or n), N ≥ 2, be the N-dimensional Euclidean space and S N−1 the unit sphere in R N. For nonzero points x ∈ R m and y ∈ R n ,w e denote x � = x/|x| and y � = y/|y| .F orm ≥ 2a ndn ≥ 2, let Ω(x � ,y � ) ∈ L 1 (S m−1 × S n−1 ) be a homogeneous function of degree zero satisfying