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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?

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I'd be very surprised, if this is not discussed in your favorite reference for distributions and Sobolev spaces of negative order. – Deane Yang Oct 25 '11 at 19:34
You might also want to post this question on It's not really suitable for MO. – Deane Yang Oct 25 '11 at 19:35
It is true that a distribution T whose distributional laplacian is zero, $\Delta T = 0$, is actually $T = T_f$ for some smooth harmonic function $f$. What is S, sorry? – Spencer Oct 25 '11 at 19:40
I posted it on, but did not get a useful answer. So I decided to try it here. – Orbicular Oct 25 '11 at 20:28
@Spencer: S is a smooth function. – Orbicular Oct 25 '11 at 20:33
up vote 2 down vote accepted

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.

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