The Stieltjes integrals in the theory of harmonic and analytic functions

TL;DR

This paper investigates various types of Stieltjes integrals related to harmonic and analytic functions, establishing theorems on their boundary behavior and limits for a broad class of integrands.

Contribution

It provides new theorems on the existence of finite angular limits of Stieltjes integrals in harmonic and analytic function theory, including for integrands with countably bounded variation.

Findings

Proved existence of finite angular limits for Poisson-Stieltjes and related integrals.

Established boundary behavior results for integrands of class CBV.

Extended classical results to broader classes of integrands.

Abstract

We study various Stieltjes integrals as Poisson-Stieltjes, conjugate Poisson-Stieltjes, Schwartz-Stieltjes and Cauchy-Stieltjes and prove theorems on the existence of their finite angular limits a.e. in terms of the singular Hilbert-Stieltjes integral in the sense of the principal value. These results hold for arbitrary $2\pi -$periodic bounded integrands that are differentiable a.e. and, in particular, for integrands of the class ${\cal {CBV}}$ (countably bounded variation).