A note on spectral multipliers on Engel and Cartan groups

TL;DR

This paper provides examples of $L^p$-$L^q$ spectral multipliers for operators on Engel and Cartan groups, demonstrating boundedness without relying on flat co-adjoint orbits or sub-Laplacians.

Contribution

It introduces new spectral multiplier examples on Engel and Cartan groups, expanding the understanding of $L^p$-$L^q$ bounds beyond traditional settings.

Findings

Established $L^p$-$L^q$ boundedness for specific operators on Engel and Cartan groups.

Derived Sobolev-type inequalities from the spectral multiplier estimates.

Obtained heat kernel estimates in these non-flat co-adjoint orbit settings.

Abstract

The aim of this short note is to give examples of $L^p$-$L^q$ bounded spectral multipliers for operators involving left-invariant vector fields and their inverses, in the settings of Engel and Cartan groups. The interest in such examples lies in the fact that a group does not have to have flat co-adjoint orbits, and that the considered operator is not related to the usual sub-Laplacian. The discussed examples show how one can still obtain $L^p$-$L^q$ estimates for similar operators in such settings. As immediate consequences, one gets the corresponding Sobolev-type inequalities and heat kernel estimates.