Communications of the
Korean Mathematical Society CKMS

In this paper, we prove $L^{p}$ estimates of a class of singular integral operators on product domains along surfaces defined by mappings that are more general than polynomials and convex functions. We assume that the kernels are in $L(\log L)^{2}(\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$. Furthermore, we\ prove $L^{p}$ estimates of the related class of Marcinkiewicz integral operators. Our results extend as well as improve previously known results.

Keywords: Singular integral operators on product domains, Marcinkiewicz integral operators on product domains, $L^{p}$ estimates, maximal functions, Hardy Littlewood maximal function, convex

MSC numbers: Primary 42B20; Secondary 42B15, 42B25