Strichartz inequalities for the wave equation with the full Laplacian on the Heisenberg group

TL;DR

The paper studies dispersive and Strichartz inequalities for the wave equation associated with the full Laplacian $\mathcal L$ on the Heisenberg group $\mathbb H_n$, addressing the non-homogeneous nature of $\mathcal L$ and comparing to the Kohn-Laplacian case. The authors develop a harmonic-analysis framework on $\mathbb H_n$, including the spherical Fourier transform and Laguerre-based kernels, and build a Littlewood-Paley decomposition and Besov spaces $\dot B^{\rho,{q}}_{r}(L)$ tied to $\Delta$ and $\mathcal L$. They prove dispersive bounds for $e^{-it\sqrt{\mathcal L}}$ via stationary phase and spectral localization, yielding $\|e^{-it\sqrt{\mathcal L}} f\|_{L^{\infty}} \le C|t|^{-1/2} \|f\|_{\dot B^{\rho,1}_1({\mathcal L})}$, and use these to derive Strichartz estimates for the linear evolution. They also analyze sharpness, showing the time decay is optimal and the Besov-index range is sharp, and discuss limitations relative to the sublaplacian case, situating the results within the Bahouri–Gérard–Xu / Kohn-Laplacian context.

Abstract

We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood--Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn-Laplacian.

Theorems & Definitions (15)

• Proposition 1
• Theorem 2
• Corollary 3
• Lemma 4
• Proposition 5
• Proposition 6
• Proposition 7
• Lemma 8
• Lemma 9
• Lemma 10