On the distributional divergence of vector fields vanishing at infinity

Abstract

The equation div υ = F has a solution υ in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BVm/(m−1)(ℝm) (the space of functions in Lm/(m−1)(ℝm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F(uj) converges to zero for each sequence {uj} such that the measure norms of ∇j are uniformly bounded and uj ⇀ 0 weakly in Lm/(m−1)(ℝm).