Pseudodifferential operators with rough negative definite symbols
The author studies pseudodifferential operators with negative definite symbols \(p(x,\xi )\) with minimal assumptions regarding their smoothness in \(\xi\). He proves an asymptotic expansion formula for the composition of such operators and finds new conditions under which a pseudodifferential operator of the above type generates a Feller semigroup. The latter is found possible for a class of symbols which are only continuous in \(\xi\). The smoothness conditions with respect to \(x\) are also improved compared to earlier results in this direction [see \textit{N.\,Jacob}, Forum Math.\ 4, No.\,5, 433--446 (1992; Zbl 0759.60078); \textit{W.\,Hoh}, Stochastics Stochastics Rep.\ 55, No.\, 3--4, 225--252 (1995; Zbl 0880.47029)]. The technique builds on a method by \textit{O.\,A.\thinspace Olejnik} and \textit{E.\,V.\thinspace Radkevich} [Itogi Nauki, Ser.\,Mat., Mat.\ Analiz 1969 (1971; Zbl 0217.41502)].
scientific article; zbMATH DE number 5774256
scientific article