Cauchy-Stieltjes型积分的乘子
Multipliers of integrals of Cauchy - Stieltjes type
摘要：
设${\rm {\mathbb G}}$是具有闭可求长Jordan曲线$\ell $的域。设$K({\rm {\mathbb G}})$是${\rm {\mathbb G}} $中所有可由Cauchy - Stieltjes积分表示的解析函数的空间.设${\rm {\mathfrak M}}(K)$是空间$K({\rm {\mathbb G}}).$的所有乘子的类，本文证明了：如果$f$是${\rm {\mathbb G}}$和$${\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop{ess\sup}\limits_{η\in \ell } \int_{\ell} \frac{|f(ζ)-f(η)|}{|ζ-η|} |dζ|{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} <\infty {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,$$上的有界解析函数，则$f\in {\rm {\mathfrak M}}(K)$ .当单位圆盘为${\rm {\mathbb G}}={\rm {\mathbb D}}$时，V. P. Havin首次证明了这一定理。特别是对于光滑曲线$\ell $，我们证明，如果$f'\in E^{p} ({\rm {\mathbb G}}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p>1,$，那么$f\in {\rm {\mathfrak M}}(K),$，其中$E^{p} ({\rm {\mathbb G}})$是Smirnov空间。
英文摘要：
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 $${\kern 1pt} {\kern 1pt} {\kern 1pt} \mathop{ess\sup}\limits_{η\in \ell } \int_{\ell} \frac{|f(ζ)-f(η)|}{|ζ-η|} |dζ|{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} <\infty {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,$$ 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.
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