$L^p$-estimates for the wave equation on the Heisenberg group

Abstract

Let $\mathcal{L}$ denote the sub-Laplacian on the Heisenberg group ${\mathbb{H}}_m$. We prove that $e^{i\sqrt{–\mathcal{L}}}$ $/(1–\mathcal{L}{)}^{\alpha /2}$ extends to a bounded operator on $L^p({\mathbb{H}}_m)$, for $1\le p\le \infty$, when $\alpha >(d–1)|1/p–1/2|$.