# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

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

### Tom 101 / 2004

Colloquium Mathematicum 101 (2004), 221-236 MSC: 60J35, 60J75, 46E35, 46B70. DOI: 10.4064/cm101-2-6

#### Streszczenie

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.

#### Autorzy

• Victoria KnopovaV. M. Glushkov Institute of Cybernetics
National Academy of Sciences of Ukraine
40, Acad. V. M. Glushkov Ave.
03187 Kiev, Ukraine
e-mail

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