Pseudo differential operators with negative definite symbols of variable order (Q1840661)

In the remarkable paper [A symbolic calculus for pseudodifferential operators generating Feller semigroups, Osaka J. Math. 35, No.~4, 789-820 (1998; Zbl 0922.47045)] a deep inside of the analysis of pseudodifferential operators generating Markov processes is achieved. The author developed a symbolic calculus for (certain) pseudodifferential operators with negative definite symbols, i.e., pseudodifferential operators having a symbol \(p(x,\xi)\) such that \(\xi\mapsto p(x,\xi)\) admits a Lévy-Khinchin representation. In the paper under review this calculus is extended to operators of variable order, i.e., pseudo-differential operators with symbol \((p(x,\xi))^{\alpha(x)}\) where \(p(x,\xi)\) is as above. Such operators occur naturally in the probabilistic context of stable-like processes and sub-ordination in the sense of Bochner. The principal results of the paper are first of all the completely established symbolic calculus allowing a parametrics construction for ``elliptic'' elements and then its application to the construction of a Feller semigroup generated by \(-(p(x,0))^{\alpha(x)}\) which yields the existence of a corresponding Feller process.