Holomorphic extension of functions from subsets of Šilov boundaries of circular strictly star-shaped domains (Q1337852)

Let \(D\) be a bounded circular strictly star-shaped domain in \(\mathbb{C}^ n\), \(S = S(D)\) be its Šilov boundary. Let \(\mu\) be a positive measure on \(S\) which is invariant with respect to rotations \(z \to e^{i \varphi} z\), \(0 \leq \varphi \leq 2\pi\), such that subsets of \(S\) of zero \(\mu\)-measure have no interior on \(S\). Let \[ H^ 2(D) := \left \{f(z) : \limsup_{r \to 1 - 0} \left(\int_ S | f(rz)|^ 2 d\mu\right)^{1/2} < \infty\right\}, \] where \(f\) is a holomorphic function in \(D\). The author obtains a criterion of the existence of a function \(F \in H^ 2(D)\) which radial limit values are equal a given function \(f_ 0 \to L^ 2(M)\) a.e. on \(M\), where \(M\) is a closed subset of \(S\), \(\mu(M) > 0\). This result is formulated in terms of Fourier coefficients.