Consider 1 < $p<\infty$ and an integer $k$. Does interior elliptic regularity for the Laplacian also hold in the Sobolev space $W^{k,p}$ of negative order? 


More precisely I am interested in the following question: Let $u\in W^{-1,p}(R^n)$ be a distributional solution of $\Delta u=Su,$ where $S$ is smooth. Is it then true that $u$ is smooth?



Elliptic regularity in Sobolev spaces of negative order
The smoothness result holds even for solutions from $\mathcal D'(\mathbb R^n)$. See, for example, Theorem IX.26 in Vol.2 of "Methods of Modern Mathematical Physics" by Reed and Simon.


