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

TL;DR

The paper develops refined Strichartz estimates for the sub-Laplacian on $H$-type groups by recasting dispersive equations as Fourier restriction problems on the Heisenberg fan, using spectral projectors of the twisted Laplacian and the TT$^*$ method. It extends previous radial results on the Heisenberg group to non-radial data and establishes analogous refined estimates for the wave equation, with mixed-norm targets $\|u\|_{L^r_\mathfrak{v} L^q_t L^p_\mathfrak{h}}$ and Sobolev gains $\sigma$ determined by the homogeneous dimension $Q=2d+2m$. The approach hinges on sharp bounds for spectral projectors $\Lambda_k$ of the twisted Laplacian, the operator-valued Fourier-restriction framework on the extended group $\mathbb{G}=\mathbb{R}\times G$, and a detailed analysis of the Heisenberg fan $\Sigma$ and its decomposition $\Sigma_k$. Together, these yield TT$^*$-equivalent restriction/extension estimates that imply Strichartz estimates for both Schrödinger and wave equations on $H$-type groups, with endpoint behavior depending on the codimension $m$ and horizontal dimension $d$. The results advance dispersive analysis on sub-Riemannian manifolds by providing sharp, mixed-norm Strichartz bounds beyond dispersion-dominated regimes, and they connect spectral-projector bounds to global space-time integrability properties via a robust Fourier-restriction framework.

Abstract

We obtain refined Strichartz estimates for the sub-Riemannian Schrödinger 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.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 1.5
• Proposition 1.6
• Definition 2.1
• Remark 2.2
• Lemma 3.1
• Remark 3.2