Jacob, Niels Further pseudodifferential operators generating Feller semigroups and Dirichlet forms. (English) Zbl 0780.31007 Rev. Mat. Iberoam. 9, No. 2, 373-407 (1993). 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. Reviewer: Z.Vondraček (Zagreb) Cited in 15 Documents MSC: 31C25 Dirichlet forms 35S05 Pseudodifferential operators as generalizations of partial differential operators 60J45 Probabilistic potential theory Keywords:negative definite functions; Dirichlet forms; symmetric pseudodifferential operators; Feller semigroups; anisotropic Sobolev space; positive maximum principle; probabilistic consequences Citations:Zbl 0155.174 PDFBibTeX XMLCite \textit{N. Jacob}, Rev. Mat. Iberoam. 9, No. 2, 373--407 (1993; Zbl 0780.31007) Full Text: DOI EuDML