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p. 27 - 50

Estimates for Feller semigroups generated by pseudodifferential operators

Niels Jacob, René L. Schilling

Mathematisches Institut Universität Erlangen-Nürnberg, Bismarckstrasse 1$\frac12$, 91054 Erlangen, Germany

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Abstract: Let $a:{\Bbb R}^n\times{\Bbb R}^n\to{\Bbb C}$ be a continuous function such that $a(x,.):{\Bbb R}^n\to{\Bbb C}$ is negative definite. Suppose further that the operator $-a(x,D)$ extends to the generator of a Feller semigroup $(T_t)_{t\ge 0}$. Here $a(x,D)$ is defined on $C_0^\infty({\Bbb R}^n)$ by $$ a(x,D)u(x)=(2\pi)^{-n/2}\int\limits_{{\Bbb R}^n}e^{ix\cdot\xi}a(x,\xi)\hat u(\xi)\,d\xi. $$ In this survey we will discuss how we can use the symbol $a(x,\xi)$ to obtain results for $(T_t)_{t\ge 0}$. In particular we are interested to compare $T_t$ with more easy operators like the pseudodifferential operator with the symbol $e^{-ta(x,\xi)}$ or a certain pseudodifferential operator with constant coefficients leading to a convolution semigroup.

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