Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
This is a special case of 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. – Akhil Mathew May 24 '11 at 22:29

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


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

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.