## Further pseudodifferential operators generating Feller semigroups and Dirichlet forms.(English)Zbl 0780.31007

The author gives examples of symmetric pseudodifferential operators generating Feller semigroups. These operators have the symbol of the form $$L(x,\xi)=\sum^ n_{j=1}b_ j(x)a^ 2_ j(\xi)$$. Here $$a^ 2_ j:\mathbb{R}\to\mathbb{R}$$, $$j=1,\ldots,n$$, are continuous negative definite functions, and $$b_ j:\mathbb{R}^ n\to\mathbb{R}$$, $$j=1,\ldots,n$$, are nonnegative bounded continuous functions satisfying certain additional conditions. The pseudodifferential operators are given by $L(x,D)u(x)=\sum^ n_{j=1}b_ j(x)(2\pi)^{-n/2}\int_{\mathbb{R}^ n} \exp(ix\xi)\;a^ 2_ j(\xi_ j)\;\hat u(\xi)\;d\xi,$ where $$\hat u$$ denotes the Fourier transform of $$u$$. For $$q\geq 0$$ let $H^{a^ 2,q}=\{u\in L^ 2(\mathbb{R}^ n):\int_{\mathbb{R}^ n}\bigl(1+a^ 2(\xi)\bigr)^{2q} |\hat u(\xi)|^ 2\;d\xi<\infty\}$ be the anisotropic Sobolev space related to the continuous negative definite function $$a^ 2(\xi)=\sum^ n_{j=1}a^ 2_ j(\xi)$$. Main results of the paper state that (under some conditions)
1. for each $$x\in\mathbb{R}^ n$$ the function $$\xi\mapsto\sum^ n_{j=1}b_ j(x)a^ 2_ j(\xi)$$ is negative definite,
2. $$-L(x,D)$$ satisfies the positive maximum principle as an operator defined on $$H^{a^ 2,m_ 0+1}(\mathbb{R}^ n)$$,
3. $$-L(x,D):H^{a^ 2,m_ 0+1}(\mathbb{R}^ n)\to H^{a^ 2,m_ 0}(\mathbb{R}^ n)\subset C_ \infty(\mathbb{R}^ n)$$ has a closed extension which is the generator of a Feller semigroup on $$\mathbb{R}^ n$$.
$$(m_ 0$$ is an even integer coming from conditions imposed on $$a^ 2)$$. The proof of these results uses Courrège’s characterization of the operator satisfying the positive maximum principle as pseudo-differential operator [Ph. Courrège, Sur la forme intégro-différentielle des opérateurs de $$C^ \infty_ K$$ dans $$C$$ satisfaisant du principe du maximum. Théorie Potentiel 10 (1965/66), No. 2 (1967; Zbl 0155.174)], the Hille-Yosida-Ray theorem, and a commutator estimate in anisotropic Sobolev spaces.
The author also discusses some probabilistic consequences and the corresponding Dirichlet form.

### MSC:

 31C25 Dirichlet forms 35S05 Pseudodifferential operators as generalizations of partial differential operators 60J45 Probabilistic potential theory

Zbl 0155.174
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