Strongly singular integrals on stratified groups

TL;DR

This paper establishes endpoint $L^p$ bounds for a class of spectral multipliers, including oscillatory ones, on stratified Lie groups, extending known results from Euclidean spaces to a broader non-commutative setting.

Contribution

It proves the endpoint $L^p$ estimates for spectral multipliers on stratified Lie groups, advancing the understanding of oscillatory multipliers in non-commutative harmonic analysis.

Findings

Established endpoint $L^p$ bounds for spectral multipliers on stratified Lie groups.

Extended Euclidean space results to the setting of stratified Lie groups.

Initiated investigation into the sharpness of these bounds.

Abstract

We consider a class of spectral multipliers on stratified Lie groups which generalise the class of H\"ormander multipliers and include multipliers with an oscillatory factor. Oscillating multipliers have been examined extensively in the euclidean setting where sharp, endpoint $L^p$ estimates are well known. In the Lie group setting, corresponding $L^p$ bounds for oscillating spectral multipliers have been established by several authors but only in the open range of exponents. In this paper we establish the endpoint $L^p(G)$ bound when $G$ is a stratified Lie group. More importantly we begin to address whether these estimates are sharp.