Singular Integrals with Flag Kernels on Homogeneous Groups: I

TL;DR

This paper proves that convolution operators with flag kernels on homogeneous nilpotent Lie groups form an algebra and are bounded on L^p spaces, extending the understanding of singular integrals in this setting.

Contribution

It establishes the algebraic structure and boundedness of flag kernel operators on homogeneous groups, a significant advancement in harmonic analysis.

Findings

Operators form an algebra under composition.

Boundedness on L^p(G) for 1<p<∞.

Extension of singular integral theory to flag kernels.

Abstract

Let $\mathcal K$ be a flag kernel on a homogeneous nilpotent Lie group $G$. We prove that operators $T$ of the form $T(f)= f*\mathcal K$ form an algebra under composition, and that such operators are bounded on $L^{p}(G)$ for $1<p<\infty$.