A symbolic calculus and a parametrix construction for pseudodifferential operators with non-smooth negative definite symbols (Q1009464)

The author introduces negative definite functions and a symbolic calculus for non-smooth negative definite symbols. A subclass of these symbols is introduced with the additional property that they are asymptotically constant in the co-variable. This technique and the part of Fredholm theory used in this work can be found in [\textit{H.\,Kumano-go}, ``Pseudo-differential operators'' (Cambridge/MS--London: MIT Press) (1982; Zbl 0489.35003)] for classes of classical symbols. The minimum differentiability that a symbol needs to have with respect to \(x\) depends on the order of the symbol. The author constructs Feller semigroups with non-smooth symbols by using the Hille--Yosida theorem and the (previously developed) symbolic calculus. Feller semigroups are strongly continuous contraction semigroups on \(C_\infty(\mathbb R^n)\) that are positivity preserving. An easy way to identify compact pseudodifferential operators leads to the construction of a parametrix by means of Fredholm theory, without using the sharp Gårding inequality or Friedrichs symmetrization, which are also available within the non-smooth calculus at hand.