Strongly Singular Integrals on Stratified Groups

Abstract

We consider a class of spectral multipliers on stratified Lie groups which generalise the class of Hormander multipliers and include multipliers with an oscillatory factor. Oscillating multipliers have been examined extensively in the Euclidean setting where sharp, endpoint Lp estimates are well known. In the Lie group setting, corresponding Lp 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 Lp(G) bound when G is a stratified Lie group. More importantly we begin to address whether these estimates are sharp.