Semigroups generated by certain pseudo-differential operators on the half-space R0+n+1

Abstract

The aim of the paper is two-fold. First, we investigate the -Bessel po- tential spaces on R n+1 0+ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form A = (Dx0 ) @ @xn+1 ; (x 0 ;xn+1)2R n+1 0+ ; where (Dx0) is an operator with real continuous negative denite symbol : R n !R. We dene the domain of the operator ( A ) and prove that with this domain it generates an Lp-sub-Markovian semigroup.