Semigroups generated by certain pseudo-differential operators on the half-space ${\Bbb R}_{0+}^{n+1}$

Victoria Knopova¹

¹ V. M. Glushkov Institute of Cybernetics National Academy of Sciences of Ukraine 40, Acad. V. M. Glushkov Ave. 03187 Kiev, Ukraine

Colloquium Mathematicum, Tome 101 (2004) no. 2, pp. 221-236

Résumé

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.

Détail

DOI : 10.4064/cm101-2-6

Keywords: the paper two fold first investigate psi bessel potential spaces mathbb study their properties secondly consider fractional powers operator form a psi partial partial hskip mathbb where psi operator real continuous negative definite symbol psi colon kern mathbb mathbb define domain operator a alpha prove domain generates p sub markovian semigroup