Multipliers of integrals of Cauchy - Stieltjes type

Let $\mathbb{G}$ be a domain with closed rectifiable Jordan curve $\mathrm{\ell }$ . Let $K(\mathbb{G})$ be the space of all analytic functions in ${\rm {\mathbb G}} $ representable by a Cauchy - Stieltjes integral. Let $\mathfrak{M}(K)$ be the class of all multipliers of the space $K(\mathbb{G})\mathrm{.}$ 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 $\mathbb{G}=\mathbb{D}$ is the unit disc, this theorem was proved for the first time by V. P. Havin. In particular for a smooth curve $\mathrm{\ell }$ 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.