Schrödinger equation on the Heisenberg group
Volume 161 / 2004
Studia Mathematica 161 (2004), 99-111
MSC: 35S10, 42B25.
DOI: 10.4064/sm161-2-1
Abstract
Let $L$ be the full laplacian on the Heisenberg group ${{\mathbb H}}^n$ of arbitrary dimension $n$. Then for $f \in L^2({{\mathbb H}}^n)$ such that ${(I-L)}^{s / 2} f \in L^2({{\mathbb H}}^n)$ for some $s>{1 / 2}$ and for every $\phi \in C_{\rm c}({{\mathbb H}}^n)$ we have $$ \int _{{{\mathbb H}}^n} |\phi (x)| \mathop {\rm sup}_{0 < t \leq 1} |e^{\sqrt{-1}\, tL}f(x)|^2\, dx \leq C_{\phi }{\| f\| }^2_{W^s}. $$