Refined Strichartz Estimates for sub-Laplacians in Heisenberg and $H$-type groups

TL;DR

This paper develops refined Strichartz estimates for sub-Laplacians on Heisenberg and H-type groups, extending previous results to non-radial data and applying Fourier restriction techniques in a nilpotent group setting.

Contribution

It introduces new refined Strichartz estimates for sub-Laplacians on H-type groups, including non-radial initial data, using spectral projector estimates and Fourier restriction methods.

Findings

Extended Strichartz estimates to non-radial data on Heisenberg groups.

Established refined estimates for wave equations on H-type groups.

Reinterpreted estimates as Fourier restriction theorems for nilpotent groups.

Abstract

We obtain refined Strichartz estimates for the sub-Riemannian Schr\"{o}dinger equation on $H$-type Carnot groups using Fourier restriction techniques. In particular, we extend the previously known Strichartz estimates previously obtained for the Heisenberg group also to non radial initial data. The same arguments permits to obtain refined Strichartz estimates for the wave equation on $H$-type groups. Our proof is based on estimates for the spectral projectors for sub-Laplacians and reinterprets Strichartz estimates as Fourier restriction theorems for nilpotent groups in the context of trace-class operator valued measures.