In this paper we study the uniqueness problem for the classical Dirichlet form on a weighted real L 2 -space when the underlying space is finite dimensional. The associated operator H, called the Dirichlet operator, when restricted to the domain of smooth functions, takes the form &2&; } { where ; is the logarithmic derivative of the corresponding weighted measure. By the uniqueness problem we mean the following. Let 2+; } { C 0 (R d ) be considered as an operator in L p (R d , \dx) (we assume throughout the paper that >0 almost everywhere w.r.t. the Lebesgue measure, \ # L 1 loc (R d , dx) and ;={\Â\ # L 2 loc (R d , \dx)). We ask whether the extension of this operator generating a C 0 -semigroup on L p (R d , \dx) is unique. In this paper we concentrate on two cases: p=2 and p=1. If p=2 it is well known that the problem is equivalent to the essential selfadjointness of the operator. This problem has a long history and goes back at least to [1] where it was posed in connection with the definition of the generalized Schro dinger operator. We refer to [2] for some uniqueness results, discussion of applications, and historical remarks.
There are two different types of sufficient conditions for the essential selfadjointness of the Dirichlet operators known so far: global and local. The best global condition obtained in [12]
As examples show (see [8]) the condition | ;| # L 4 loc (R d , \dx) ensuring essential selfadjointness cannot be improved in terms of L p . The best known local condition obtained recently in [3] is the following: \ is locally bounded and locally uniformly positive and | ;| # L # loc (R d , \dx) for some #>d. Note that this is equivalent to | ;| # L # loc (R d , dx) since the measure \dx is locally to equivalent to the Lebesgue measure.
Our aim in this paper is to establish a criterion of strong uniqueness for the Dirichlet operator under local assumptions on the logarithmic derivative ; generalizing the result from [12] (in the finite dimensional case).