Weighted Lp norms of Marcinkiewicz functions on product domains along surfaces

We prove a weighted L^p boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space L^p (Rⁿ × R^m, ω₁ (x)dx, ω₂ (y)dy), provided that the weights ω₁ and ω₂ are certain radial weights and that the kernels are rough in the optimal space L(log L)(Sⁿ⁻¹ × S^m−1). In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted L^p boundedness of the related square and maximal functions. Our weighted L^p inequalities extend as well as generalize previously known L^p boundedness results.