H-distribution via Sobolev spaces

Abstract

H-distributions associated to weakly convergent sequences in Sobolev spaces are determined. It is shown that a weakly convergent sequence $(u_n)$ in $W^{-k,p}({\mathbb{R}}^d)$ has the property that $\theta u_n$ converges strongly in $W^{-k,p}({\mathbb{R}}^d)$ for every $\theta \in \mathcal{S}({\mathbb{R}}^d)$ if and only if all H-distributions related to this sequence are equal to zero. Results are applied on a weakly convergent sequence of solutions to a family of linear first order PDEs.