On conditions of analytical continuability of functions defined on a piece of boundary of circular domains (Q2739967)

Let \(D\) be a bounded, circular, strongly star domain in \(\mathbb{C}^{n}\) and let \(S\) be the Shilov boundary of \(D\), \(\mu\) be a positive normed measure on \(S\) which is invariant with respect to rotations. Let us denote by \(L_{\mu}^{p}(M)\) the space of functions integrable with order \(p\) by the measure \(\mu\) on the \(\mu\)-measurable set \(M\subset S\). NEWLINENEWLINENEWLINEThe author proves that if \(1<p<\infty\) and \(1/p+1/q=1\), then \(f\in L_{\mu}^{p}(M)\) is holomorphically continued in \(D\) if and only if \(\int_{M}f\varphi_{j} d\mu\to 0\) for any sequence \(\varphi_{j}\in O_{\mu}(S)\) converging to zero in the norm of the space \(L_{\mu}^{q}(S\setminus M)\). Here \(O_{\mu}(S)\) is the space of functions \(\varphi\in C(S)\) such that \(\int\varphi f d\mu=0\) for all \(f\in A(S)\).