Semigroups generated by certain pseudo-differential operators on the half-space ${\Bbb R}_{0+}^{n+1}$
Volume 101 / 2004
Colloquium Mathematicum 101 (2004), 221-236
MSC: 60J35, 60J75, 46E35, 46B70.
DOI: 10.4064/cm101-2-6
Abstract
The aim of the paper is two-fold. First, we investigate the $\psi $-Bessel potential spaces on ${\mathbb R}_{0+}^{n+1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $$ -A_\pm =-\psi (D_{x'})\pm {\partial \over \partial x_{n+1}},\hskip 1em (x',x_{n+1})\in {\mathbb R}^{n+1}_{0+}, $$ where $\psi (D_{x'})$ is an operator with real continuous negative definite symbol $\psi \colon \kern .16667em {\mathbb R}^n\to {\mathbb R}$. We define the domain of the operator $-(-A_\pm )^\alpha $ and prove that with this domain it generates an $L_p$-sub-Markovian semigroup.
