Holomorphic extensions from open families of circles

For a circle $\Gamma =\{ z\in \mathbb{C}\colon \vert z-c\vert=\rho \}$ write $\Lambda (\Gamma )=\{ (z,w)\colon (z-a)(w-\overline{a}) =\rho ^{2}, 0<\vert z-a\vert<\rho \}$. 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.