On a class of holomorphic functions representable by Carleman formulas in some class of bounded, simply connected domains from their values on an analytic arc
Let \({\mathcal U}\) be a bounded simply connected domain with Jordan rectifiable boundary containing an open analytic arc \(M\) (\(0 < m(M) < m({\mathcal U})\)). The quenching function is defined as an exponent of the solution to the Schwarz boundary value problem with piece-wise constant data (equal to 1 on \(M\) and to 0 on \(\partial {\mathcal U}\setminus M\)). It is proved that if \(f\) is a holomorphic function in \({\mathcal U}\) having angular boundary values almost everywhere on \(M\), Lebesgue-summable on \(M\), and \(\partial {\mathcal U}\) is regular with respect to \(M\), then 1) if \(f\) belongs to Hardy class \({\mathcal H}^1\) near the base \(M\) then it can be represented via Carleman formula \[ f(z) = \lim_{n\rightarrow \infty} \frac{1}{2\pi i} \int_{M} \frac{\omega^n(\zeta)}{\omega^n(z)} \frac{f(\zeta) \,d \zeta}{\zeta - z}, \] where convergence is uniform with respect to any compact subset of \({\mathcal U}\); 2) if the Carleman formula holds for \(f\) point-wise and in addition \(f\) has boundary values at the end-points of \(M\), then \(f\) belongs to Smirnov class \(E^1({\mathcal U}_{\tau})\) for certain Ahlfors-regular exhaustion of \({\mathcal U}\).
scientific article; zbMATH DE number 5118930