Strichartz estimates and its application to the well-posedness of the nonlinear Schrödinger equations on H-type groups

TL;DR

This work addresses the well-posedness of the nonlinear Schrödinger equation on H-type groups ${\mathbb H}^d_p$ by deriving dispersive and Strichartz estimates for the Schrödinger flow on these groups, with a focus on center dimension $p>1$. The authors provide corrected and complete proofs of dispersive-type decay and non-endpoint Strichartz estimates, expressed with derivative losses, and leverage them to obtain local well-posedness in $H^s({\mathbb H}^d_p)$ for $s_*<s<\tfrac{N}{2}$ and global well-posedness in the critical space $H^{s_c}({\mathbb H}^d_p)$ for small data under precise conditions on $p$ and the nonlinearity power $\alpha$. They establish a detailed functional-analytic framework using Besov and Sobolev spaces on ${\mathbb H}^d_p$ and the spherical Fourier transform, and compare their results with prior work on Heisenberg and stratified groups, highlighting improvements in regularity assumptions. The results significantly advance the understanding of nonlinear dispersive equations on non-Euclidean, two-step nilpotent groups by providing a rigorous Strichartz theory and its application to well-posedness.

Abstract

The aim of this article is to give the well-posedness results for the Cauchy problem of the nonlinear Schrödinger equation with power type nonlinearities on H-type groups. To do this, we prove the dispersive estimate and Strichartz estimate. Although these estimates are given by Hierro (2005), its complete proofs cannot be find. We correct the statement of these estimates, give the proofs, and apply to the nonlinear problem. Our well-posedness results are an improvement of the previous result by Bruno et al.

Theorems & Definitions (50)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Definition 1.8
• Remark 1.9
• Theorem 1.10: Strichartz estimates