Boundedness of oscillating singular integrals on Lie groups of polynomial growth

TL;DR

This paper studies the boundedness of oscillating singular integrals on Lie groups with polynomial growth, extending classical conditions using sub-Riemannian structures and Fourier analysis for graded groups.

Contribution

It introduces kernel criteria for boundedness of oscillating integrals on Lie groups, generalizing Fefferman and Stein's conditions with sub-Laplacian and Rockland operator frameworks.

Findings

Kernel criteria are established for Lie groups of polynomial growth.

Extension of classical oscillating conditions to non-commutative Lie groups.

Application of sub-Riemannian and Fourier analysis techniques.

Abstract

We investigate the boundedness of oscillating singular integrals on Lie groups of polynomial growth in order to extend the classical oscillating conditions due to Fefferman and Stein for the boundedness of oscillating convolution operators. Kernel criteria are presented in terms of a fixed sub-Riemannian structure on the group induced by a sub-Laplacian associated to a H\"ormander system of vector fields. In the case where the group is graded, kernel criteria are presented in terms of the Fourier analysis associated to an arbitrary Rockland operator.