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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
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