Oscillating singular integral operators on graded Lie groups revisited

TL;DR

This paper extends the Euclidean theory of oscillating singular integrals to graded Lie groups, revealing deep connections between geometric measure theory and Fourier analysis via Rockland operators.

Contribution

It introduces criteria based on the oscillating Fefferman condition and uses infinitesimal representations of Rockland operators to analyze decay of Fourier transforms.

Findings

Extended oscillating singular integral theory to graded Lie groups

Established criteria using Fefferman condition and Fourier transform analysis

Linked geometric measure theory with Fourier analysis on Lie groups

Abstract

In this work, we extend the Euclidean theory of oscillating singular integrals due to Fefferman and Stein in \cite{Fefferman1970,FeffermanStein1972} to arbitrary graded Lie groups. Our approach reveals the strong compatibility between the geometric measure theory of a graded Lie group and the Fourier analysis associated with Rockland operators. Our criteria are presented in terms of the oscillating Fefferman condition of the kernel of the operator and its group Fourier transform. One of the novelties of this work is that we use the infinitesimal representation of a Rockland operator to measure the decay of the Fourier transform of the kernel.