On the Local Property for Positivity Preserving Coercive Forms

Abstract

. We show that, under mild conditions, two well-known definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from di#erential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between two di#erent notions of locality that have appeared in the literature of Dirichlet forms. The first is a slightly modified version of the definition of locality found in the book of Bouleau and Hirsch [BH 91; Chapter I, Corollary 5.1.4], while the second comes from the book of Ma and Rockner [MR 92; Chapter V, Proposition 1.2]. But here we do not assume that the form satisfies any normal contraction property, but only that it is positivity preserving (see Definition 0.1 below). Let (E, F , m) be a measure space, and suppose (E , D(E)) is a densely defined, closed, bilinear form on L 2 (E, F , m). Following [MR 92], we call such a form (E , D(E)) coercive if E(u, u) # 0 for all u ..