We study the Marcinkiewicz integral operator M𝒫f(x)=(∫−∞∞|∫|y|≤2tf(x−𝒫(y))(Ω(y)/|y|n−1)dy|2dt/22t)1/2, where 𝒫 is a polynomial mapping from ℝn into ℝd and Ω is a homogeneous function of degree zero on ℝn with mean value zero over the unit sphere Sn−1. We prove an Lp boundedness result of M𝒫 for rough Ω

Abstract.We prove the L p boundedness of the Marcinkiewicz integral operators 1. Introduction. Marcinkiewicz integrals have been studied by many authors, dating back to the investigations of such operators by Zygmund on the circle and by Stein on R n .We shall be primarily concerned with Marcinkiewicz integrals on the product space R n × R m , since the more general setting of R n 1 × · · · × R n k can be handled similarly (see Section 4).For n, m ≥ 2, x ∈ R n \{0}, y ∈ R m \{0}, we let x = x/|x| and y = y/|y|. Let Ω ∈ L 1 (S n−1 × S m−1 ) be a function satisfying the following cancellation conditions:

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We establish the L p boundedness of singular integrals on product domains with rough kernels in L(log L) 2 and are supported by subvarieties.

In this paper, we establish an L p boundedness result of a class of Marcinkiewicz integral operators on product domains with rough kernels.