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

Qiteng Guo, Yixin He

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 $dA$ is planar Lebesgue measure. We prove that ${\mathcal{T}}_{\alpha }$ is injective…