Holomorphic extensions from open families of circles

Abstract:

For a circle $\Gamma =\{ z\in \mathbb {C} \colon |z-c|=\rho \}$ write $\Lambda (\Gamma )=\{ (z,w)\colon \ (z-a)(w-\overline {a}) =\rho ^{2},\ 0<|z-a|<\rho \}$. A continuous function $f$ on $\Gamma$ extends holomorphically from $\Gamma$ (into the disc bounded by $\Gamma$) if and only if the function $F(z,\overline {z})=f(z)$ defined on $\{(z,\overline {z})\colon \ z\in \Gamma \}$ has a bounded holomorphic extension into $\Lambda (\Gamma )$. In the paper we consider open connected families of circles $\mathcal {C}$, write $U=\bigcup \{ \Gamma \colon \ \Gamma \in \mathcal {C}\}$, and assume that a continuous function on $U$ extends holomorphically from each $\Gamma \in \mathcal {C}$. We show that this happens if and only if the function $F(z, \overline {z})=f(z)$ defined on $\{ (z,\overline {z})\colon z\in U\}$ has a bounded holomorphic extension into the domain $\bigcup \{ \Lambda (\Gamma )\colon \ \Gamma \in \mathcal {Q}\}$ for each open family $\mathcal {Q}$ compactly contained in $\mathcal {C}$. This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.