An Infinitesimal Circular Morera Theorem

Abstract

We prove an infinitesimal circular version of Morera's theorem. Let $D\subset \mathbb{C}$ be a domain and let $f\in C(D)$. If, at every $a\in D$, ${\int }_{|\zeta -a|=r}f(\zeta )d\zeta =o(r^2)$ as $r\to 0^{+}$, then $f$ is holomorphic in $D$. In particular, exact vanishing of all sufficiently small centered circular integrals implies holomorphicity. The proof uses a local distributional $\partial$-primitive, a circular identity for weak $\partial$-derivatives, and a pointwise asymptotic mean-value criterion for harmonicity.

6 pages