Oscillating spectral multipliers on groups of Heisenberg type

TL;DR

This paper proves endpoint estimates for oscillating spectral multipliers on Heisenberg type groups, linking these bounds to smoothness conditions for Mihlin-H"ormander multipliers, and extends understanding in harmonic analysis on these groups.

Contribution

It establishes endpoint estimates for oscillating spectral multipliers on Heisenberg type groups, connecting these bounds to minimal smoothness requirements for multipliers.

Findings

Established endpoint estimates for spectral multipliers

Connected bounds to smoothness conditions for multipliers

Extended analysis techniques to Heisenberg type groups

Abstract

We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author but requires the detailed analysis of the wave equation on these groups due to M\"uller and Seeger. We highlight and develop the connection between sharp bounds for oscillating multipliers and the problem of determining the minimal amount of smoothness required for Mihlin-H\"ormander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.