A property of Sobolev spaces on complete Riemannian manifolds

Abstract

Let (M, g) be a complete Riemannian manifold with metric g and the Riemannian volume form dv. We consider the ℝk-valued functions T ∈ [W-1,2(M) ∩ L1loc (M)]k and u ∈ [W1,2(M)]k on M, where [W1,2(M)]k is a Sobolev space on M and [W-1,2(M)]k is its dual. We give a sufficient condition for the equality of ⟨T, u⟩ and the integral of (T ∙ u) over M, where ⟨∙, ∙⟩ is the duality between [W-1,2(M)]k. This is an extension to complete Riemannian manifolds of a result of H. Brézis and F. E. Browder.