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.
Categories: math.CV math.FA