Multipliers of integrals of Cauchy - Stieltjes type

Abstract

Let $\mathbb{G}$ be a domain with closed rectifiable Jordan curve $\ell$ . Let $K(\mathbb{G})$ be the space of all analytic functions in $\mathbb{G}$ representable by a Cauchy - Stieltjes integral. Let $\mathfrak{M}(K)$ be the class of all multipliers of the space $K(\mathbb{G}).$ In this paper we prove that if $f$ is bounded analytic function on $\mathbb{G}$ and $$\underset{\eta \in \ell }{\mathrm{esssup}}{\int }_{\ell }\frac{|f(\zeta )-f(\eta )|}{|\zeta -\eta |}|d\zeta |<\infty ,$$ then $f\in \mathfrak{M}(K)$ . 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 $\ell$ we prove that if $f^{\prime }\in E^p(\mathbb{G}),p>1,$ then $f\in \mathfrak{M}(K),$ where $E^p(\mathbb{G})$ are the spaces of Smirnov.

arXiv comment8 pages

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