Qiteng Guo | Searcharxiv

arXiv2026

An Infinitesimal Circular Morera Theorem

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 $a\in D$, $\int_{\vert{}\zeta-a\vert{}=r}f(\zeta)\,d\zeta=o(r^2)$ as $r\to0^+$, 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.

math.CV↗

arXiv2026

Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman

For $0<\alpha\leq1$, define $(\mathcal T_\alpha f)(z) :=\int_{B(z,\alpha(1-|z|))}f(\zeta)\,dA(\zeta)$ for $z\in\mathbb D$, where $dA$ is planar Lebesgue measure. We prove that $\mathcal T_\alpha$ is injective on $C(\mathbb D)\cap L^\infty(\mathbb D)$ for $0<\alpha<1$, and that $\mathcal T_1$ is injective on $L^1(\mathbb D)$. In contrast, for each $0<\alpha<1$ there is an injective linear map from $C_c^\infty((0,\alpha))$ into the kernel of $\mathcal T_\alpha$ on $C^\infty(\mathbb D)$; every nonzero function in its image is necessarily unbounded near $\partial\mathbb D$. Under the area-measure interpretation, these results give a complete answer to Hayman--Lingham Problem~7.29, attributed there to L.~Zalcman. The proof combines generalized Abel equations, an Euler--Poisson--Darboux energy argument, and Volterra continuation.

math.FA↗