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

For $0<\alpha \le 1$, define $(\mathcal T_\alpha f)(z) :=\int_{B(z,\alpha(1-|z|))}f(\zeta),dA(\zeta)$for$z\in\mathbb D$,where$dA$isplanarLebesguemeasure\mathrm{.}Weprovethat$\mathcal T_\alpha$isinjectiveon$C(\mathbb D)\cap L^\infty(\mathbb D)$for$0<\alpha<1$,andthat$\mathcal T_1$isinjectiveon$L^1(\mathbb D)$\mathrm{.}Incontrast,foreach$0<\alpha<1$thereisaninjectivelinearmapfrom$C_c^\infty((0,\alpha))$intothekernelof$\mathcal T_\alpha$on$C^\infty(\mathbb D)$;everynonzerofunctioninitsimageisnecessarilyunboundednear$\partial\mathbb D$. 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.