Strong uniqueness of finite-dimensional Dirichlet operators with singular drifts

Abstract

We show [Formula: see text]-uniqueness for any [Formula: see text] and the essential self-adjointness of a Dirichlet operator [Formula: see text], [Formula: see text] with [Formula: see text] and [Formula: see text]. In particular, [Formula: see text] is allowed to be in [Formula: see text] or in [Formula: see text] for some [Formula: see text], while [Formula: see text] is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.