Smoothing Properties of Semigroups for Dirichlet Operators of Gibbs Measures

In this paper is investigated the special class of elliptic differential second-order operators with an infinite number of variables with the property of a finite radius of dependence for variables. This class is formed by the Dirichlet operators associated with energy forms of Gibbs measures on compact Riemannian manifolds with a finite radius of interaction. Using this property we represent the Dirichlet operator as a finite sum of self-adjoint operators with independent variables and prove that the Dirichlet operator semigroups preserve the specially constructed scales of continuously differentiable functions. We also obtain that these semigroups raise the smoothness of initial functions.