Multipliers of integrals of Cauchy - Stieltjes type

Abstract

Let ${\rm {\mathbb G}}$ be a domain with closed rectifiable Jordan curve $\ell $ . Let $K({\rm {\mathbb G}})$ be the space of all analytic functions in ${\rm {\mathbb G}} $ representable by a Cauchy - Stieltjes integral. Let ${\rm {\mathfrak M}}(K)$ be the class of all multipliers of the space $K({\rm {\mathbb G}}).$ In this paper we prove that if $f$ is bounded analytic function on ${\rm {\mathbb G}}$ and $${\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop{ess\sup}\limits_{\eta \in \ell } \int_{\ell} \frac{|f(\zeta)-f(\eta)|}{|\zeta -\eta |} |d\zeta |{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} <\infty {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,$$ then $f\in {\rm {\mathfrak M}}(K)$ . If ${\rm {\mathbb G}}={\rm {\mathbb D}}$ is the unit disc, this theorem was proved for the first time by V. P. Havin. In particular for a smooth curve $\ell $ we prove that if $f'\in E^{p} ({\rm {\mathbb G}}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p>1,$ then $f\in {\rm {\mathfrak M}}(K),$ where $E^{p} ({\rm {\mathbb G}})$ are the spaces of Smirnov.