Holomorphic functions on rotation invariant families of curves passing through the origin (Q1326646)

Let \(\Delta\) denote the unit disc in \(\mathbb{C}\) with boundary \(b\Delta\) and let \(\Gamma\subset \mathbb{C}\) be a smooth Jordan curve which is symmetric with respect to the real axis. Let \(D\) denote the bounded domain bounded by \(\Gamma\) and let \(\Omega= \bigcup_{s\in b\Delta} s\Gamma\) be the union of all sets obtained by rotating \(\Gamma\) about the origin. Let \(f\in C(\Omega)\) and assume that for each \(s\in b\Delta\), (1) \(f_{| s\Gamma}\) has a continuous extension to \(s\overline D\) which is holomorphic on \(sD\). In previous paper the author has proved that if \(0\in \mathbb{C}\backslash \overline D\) then (1) impies that \(f\) is holomorphic on \(\text{Int}(\Omega)\) [Trans. Am. Math. Soc. 280, 247-254 (1983; Zbl 0575.30033)], whereas if \(0\in D\), then in general (1) does not imply that \(f\) is holomorphic on \(\text{Int}(\Omega)\) [Trans. Am. Math. Soc. 306, No. 1, 401-410 (1988; Zbl 0639.30031)]. In the present paper the author considers the critical case when \(0\in b\Delta\). In this case \(\Omega\) is a disc centered at the origin. The main result of the paper is as follows: Theorem 1. Suppose in addition to the above, \(\Gamma\) passes through the origin and is of class \(C^ 2\) in a neighborhood of the origin. Let \(f\in C(\Omega)\) satisfy (1) and assume that \(f\) is infinitely smooth at the origin. Then \(f\) is holomorphic on \(\text{Int}(\Omega)\). In Section 2 of the paper the author considers this set in greater detail when \(\Gamma\) is a circle passing through the origin and shows that it is closely related to the set of continuous functions satisfying the Morera condition along circles passing through the origin, whose characterization is obtained in Section 3.