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

Abstract

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.