An Update On The $L^p$-$L^q$ Norms of Spectral Multipliers on Unimodular Lie Groups

TL;DR

This paper updates spectral multiplier theorems on unimodular Lie groups by relating spectral projectors to heat kernels, providing sharp $L^p$-$L^q$ estimates that extend previous results and have broad applications.

Contribution

It introduces explicit $L^p$-$L^q$ norm estimates for spectral projectors via heat kernel analysis, broadening the scope of multiplier theorems on unimodular Lie groups.

Findings

Heat kernel estimates are sharp and applicable to a wide class of operators.

Spectral projector bounds lead to automatic Sobolev embeddings.

Time asymptotics for heat kernels follow from the new estimates.

Abstract

This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators.