The Cauchy–Stieltjes integrals in the theory of analytic functions

Abstract

We study various Stieltjes integrals as Poisson–Stieltjes, conjugate Poisson–Stieltjes, Schwartz–Stieltjes, and Cauchy–Stieltjes ones and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert–Stieltjes integral. These results hold for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands of the class \( \mathcal{C}\mathrm{\mathcal{B}}\mathcal{V} \) (countably bounded variation).