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What is a reference for the (classical and well-known) proof of Weyl's lemma that states:

Let $U$ be an open subset of $R^n$. Then if $f\in L^1_{loc} (U)$ and if $\int_U \hspace{1mm}{f\phi_\bar{z}}=0\;\;\;\forall \phi \in C_c^{\infty}(U) $, then $f$ is a.e. equal to a holomorphic function.

Just any quick and good reference would be appreciated. I know Weyl's lemma has a weaker form involving weak Laplacian. Where can I find a proof of that?

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    $\begingroup$ This is a special case of elliptic regularity (en.wikipedia.org/wiki/Elliptic_regularity), which implies that $f$ is smooth and hence must satisfy the Cauchy-Riemann equations. So a book on PDE (e.g. Evans) should have it. $\endgroup$ May 24, 2011 at 22:29

4 Answers 4

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Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press.

Or

Gilbarg, David; Neil S. Trudinger (1988). Elliptic Partial Differential Equations of Second Order. Springer

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Mats Andersson's Topics in Complex Analysis has a wonderful proof. See page 10, Proposition 1.9.

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  • $\begingroup$ Got this in the review queue: I don't see any problem with this answer. The question explicitely asked for a reference and this is exactly what this answer gave. $\endgroup$
    – leoli1
    Aug 7, 2021 at 10:51
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I think the statement is strange, because clearly a holomorphic function is only defined on even dimensional spaces. I do not think the statement would work for $\mathbb{R}^{3}$, for example. For the reference, the standard one I know is Donaldson's book Riemann Surfaces.

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There's a proof in the book Riemann surfaces by H.M. Farkas and I. Kra, I believe they prove the weak version.

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