Algebras of singular integral operators with kernels controlled by multiple norms

TL;DR

This paper investigates algebras of singular integral operators on Euclidean spaces and Lie groups, focusing on those with kernels controlled by multiple norms, extending classical Calderón-Zygmund theory to multi-parameter settings.

Contribution

It characterizes new algebras of singular integral operators arising from compositions of Calderón-Zygmund operators with different homogeneities, relevant in sub-elliptic and elliptic problems.

Findings

Operators are pseudo-local and bounded on L^p for 1<p<∞

Algebras include operators related to sub-Laplacians and their inverses

Extend Calderón-Zygmund theory to multi-parameter structures

Abstract

The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calder\'on-Zygmund operators with different homogeneities, such as operators that occur in sub-elliptic problems and those arising in elliptic problems. For example, one would like to describe the algebras containing the operators related to the Kohn-Laplacian for appropriate domains, or those related to inverses of H\"ormander sub-Laplacians, when these are composed with the more standard class of pseudo-differential operators. The algebras we study can be characterized in a number of different but equivalent ways, and consist of operators that are pseudo-local and bounded on $L^{p}$ for $1<p<\infty$. While the usual class of Calder\'on-Zygmund operators is invariant under a one-parameter family of dilations, the operators we study fall outside this class, and reflect a multi-parameter structure.