[Math] a reference for the ( classical and well-known ) proof of Weyl’s lemma

Question

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?

Answer 1

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