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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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  • $\begingroup$ I'd be very surprised, if this is not discussed in your favorite reference for distributions and Sobolev spaces of negative order. $\endgroup$
    – Deane Yang
    Commented Oct 25, 2011 at 19:34
  • $\begingroup$ You might also want to post this question on math.stackexchange.com. It's not really suitable for MO. $\endgroup$
    – Deane Yang
    Commented Oct 25, 2011 at 19:35
  • $\begingroup$ 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? $\endgroup$
    – Spencer
    Commented Oct 25, 2011 at 19:40
  • $\begingroup$ I posted it on math.stackexchange.com, but did not get a useful answer. So I decided to try it here. $\endgroup$
    – Orbicular
    Commented Oct 25, 2011 at 20:28
  • $\begingroup$ @Spencer: S is a smooth function. $\endgroup$
    – Orbicular
    Commented Oct 25, 2011 at 20:33

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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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