Schrödinger equation on the Heisenberg group

Jacek Zienkiewicz¹

¹ Institute of Mathematics University of Wrocław Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Studia Mathematica, Tome 161 (2004) no. 2, pp. 99-111

Résumé

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 $$ \int _{{{\mathbb H}}^n} |\phi (x)| \mathop {\rm sup}_{0 t \leq 1} |e^{\sqrt{-1}\, tL}f(x)|^2\, dx \leq C_{\phi }{\| f\| }^2_{W^s}. $$

Détail

DOI : 10.4064/sm161-2-1

Mots-clés : full laplacian heisenberg group mathbb arbitrary dimension mathbb i l mathbb every phi mathbb have int mathbb phi mathop sup leq sqrt leq phi