Zalcman’s problem and related two-radii theorems

Abstract

Let G be the group of conformal automorphisms of the unit disc ${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1\}$. For $r>0$, we put $B_{r}=\{z\in {\mathbb {D}}:|z|<\tanh r\}$. Denote by $\overline{B}_{r}$ the closure of the disc $B_r$, and by $\partial B_{r}$ its boundary. Let $\chi _r$ be the characteristic function (indicator) of $B_r$. Assume that $r_1, r_2\in (0,+\infty )$ are fixed and $R>\max \, \{r_1,r_2\}$. We study the holomorphicity problem for functions $f\in C(B_R)$ satisfying the condition

$$\begin{aligned} \int \limits _{\partial B_{r_j}} f(g(z))\, dz=0 \end{aligned}$$

for all $g\in G$ such that $g \overline{B}_{r_j}\subset B_R$, $j=1,2$. We find the exact conditions for holomorphicity in terms of size $B_R$ and properties of zeros of generalized spherical transforms of functions $\chi _{r_1}$ and $\chi _{r_2}$. In particular, a strengthening of the Berenstein–Pascuas theorem (Israel J Math 86:61–106, 1994) on two radii is obtained.