Singular integrals of non-convolution type on product spaces

TL;DR

This paper introduces a new class of pseudo differential operators with symbols satisfying specific inequalities, proves their boundedness on L^p spaces, and shows they form an algebra on product spaces.

Contribution

It defines a novel class of singular integrals of non-convolution type on product spaces and establishes their boundedness and algebraic properties.

Findings

Operators are bounded on L^p for 1<p<infinity

Operators form an algebra

Characterization via kernels with specific properties

Abstract

We study a new class of pseudo differential operators whose symbols satisfy the differential inequality with a mixture of homogeneities. On the other hand, by taking singular integral realization, it can be equivalently defined by kernels carrying certain characteristic properties on product spaces. We prove that these operators are bounded on L^p-spaces for 1<p<infinity. Moreover, they form an algebra.