Analytic extension of the functions from the part of Shilov boundary of circular domains (Q2742937)

Let \(D\subset\mathbb{C}^n\) be a circular domain, \(M\) be a set with positive measure \(\mu\) in a Shilov boundary \(S=\partial D\) and \(L_\mu^p(M)\) be a space of functions integrable on \(M\) with power \(p\). Existence condition of holomorphic extension of a function \(f\) from \(L_\mu^p(M)\) to the domain \(D\) is presented. This condition is: NEWLINE\[NEWLINE\int_Mf\varphi_j d\mu\to 0 \quad\text{ as }\quad\varphi_j\to 0\text{ on }L_\mu^p (S\backslash M),NEWLINE\]NEWLINE where \(\varphi_j\) are polynomials such that \(\int_S\varphi_jg d\mu=0\) for any holomorphic function \(g\) on \(D\).