An $L^p$-spectral multiplier theorem with sharp $p$-specific regularity bound on Heisenberg type groups

TL;DR

This paper establishes a sharp $L^p$-spectral multiplier theorem for sub-Laplacians on Heisenberg type groups, identifying the minimal regularity needed for boundedness, using restriction estimates with spectral truncation.

Contribution

It provides the first sharp regularity condition for $L^p$-spectral multipliers on Heisenberg type groups, extending previous results with optimal bounds.

Findings

Proved the $L^p$-spectral multiplier theorem with sharp regularity condition.

Utilized restriction estimates with spectral truncation along the Laplacian spectrum.

Established optimal bounds depending on the group's topological dimension.

Abstract

We prove an $L^p$-spectral multiplier theorem for sub-Laplacians on Heisenberg type groups under the sharp regularity condition $s>d\left|1/p-1/2\right|$, where $d$ is the topological dimension of the underlying group. Our approach relies on restriction type estimates where the multiplier is additionally truncated along the spectrum of the Laplacian on the center of the group.