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Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the following statement:

Suppose $M$ is non-compact (but assume it to be complete if you like), does elliptic regularity hold in the following sense: $\Delta_{g}u=f$ for $f\in C^{\infty}(M)$ implies $u\in C^{\infty}(M)$.

I am explicitly interested in the non-compact sense. My question can maybe be reformulated: Does (smooth) elliptic regularty hold on non-compact manifolds? I would guess that it is still true that $\Delta_{g}u=f$ for $f\in H^{m}_{loc}(M)$ implies $u\in H^{m+2}_{loc}(M)$, since this is a local statement, but then I cannot argue via the Sobolev embedding theorem to get the "smooth" claim, since this theorem is in general not ture for non-compact manifolds. (There are not even canonical global Sobolev spaces on non-compact manifolds in general)

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    $\begingroup$ Can't you invoke the Sobolev embedding at the local level and get your smoothness locally ? $\endgroup$
    – Ayman Moussa
    Commented Feb 5 at 11:47
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    $\begingroup$ I cannot access the book right now, but this must be done in quite some level of generality in Hörmander's The Analysis of Linear Partial Differential Operators III. It's not the best place to learn about those things but there will be a quotable theorem there. $\endgroup$
    – Pierre PC
    Commented Feb 5 at 12:46
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    $\begingroup$ Use a partition of unity to reduce the statement to a local one for a function compactly supported on a coordinate chart. At that point, any elliptic regularity theorem on an open domain in $\mathbb{R}^n$ can be applied, inclusing the Sobolev space one. Such theorems are proved in many expositions of elliptic PDEs, including Gilbarg-Trudinger, the books of Michael Taylor. Expositions of pseudodifferential operators always contain a proof of $C^\infty$ regularity. $\endgroup$
    – Deane Yang
    Commented Feb 5 at 16:14
  • $\begingroup$ Thank you very much, for all your comments. That clears it up. @PierrePC In the first book of Hörmander he has the general statement $\mathrm{WF}(Pu)=\mathrm{WF}(u)$ for arbitrary domains and elliptic $P$, so in particular also for non-compact manifolds, which seems to imply my claim. $\endgroup$
    – B.Hueber
    Commented Feb 5 at 16:53
  • $\begingroup$ @DeaneYang Thanks! If you transform your comment to an answer, I will accept it. $\endgroup$
    – B.Hueber
    Commented Feb 5 at 16:54

2 Answers 2

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Use a partition of unity to reduce the statement to a local one for a function compactly supported on a coordinate chart. At that point, any elliptic regularity theorem on an open domain in Rn can be applied, inclusing the Sobolev space one. Such theorems are proved in many expositions of elliptic PDEs, including Gilbarg-Trudinger, the books of Michael Taylor. Expositions of pseudodifferential operators always contain a proof of $C^\infty$ regularity

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Using a partition of unity as pointed out by Deane Yang you can reduce the problem to the local coordiante system. Then you can reduce it to an elliptic equation on a torus and you can prove regularity using Sobolev spaces which on torus can be expressed in terms of generalized Fourier series. That makes the problem particularly easy. You can find all details in Chapter 6 of:

Warner, Frank W., Foundations of differentiable manifolds and Lie groups. Reprint, Graduate Texts in Mathematics, 94. New York etc.: Springer-Verlag. (1983). ZBL0516.58001.

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    $\begingroup$ I second the recommendation of Warner's book, if you want a proof without having to learn too much PDE theory. Warner does everything from scratch, if I recall correctly, by doing it on the torus and using Fourier series. In general, $\endgroup$
    – Deane Yang
    Commented Mar 3 at 17:13
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    $\begingroup$ To finish my sentence, in general Warner is an underrated book on differential geometry and topology that has nice clean proofs of major theorems, which are hard to find elsewhere. $\endgroup$
    – Deane Yang
    Commented Mar 3 at 19:38

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