对数Bloch空间及其预对偶

摘要：

本文考虑单位圆盘$\D,$上解析函数空间$\bk^1_{\log^α}$，它由要求$\int_\D|f'(z)|ϕ(|z|)\,dA(z)<\infty,$定义，其中$ϕ(r)=\log^α(1/(1-r))$是“$\log^α$ -Bloch”空间的预对偶和相应的小Bloch空间的对偶.本文证明了具有$a_n\downarrow 0$的函数$f(z)=\sum_{n=0}^\infty a_nz^n$在$\bk^1_{\log^α}$中当且仅当$\sum_{n=0}^\infty \log^α(n+2)/(n+1)<\infty$，并应用这一结果得到了判别正系数函数在$\bk^1_{\log^α}.$中的Libera变换的隶属度的一个准则。

英文摘要：

We consider the space $\bk^1_{\log^α}$, of analytic functions on the unit disk $\D,$ defined by the requirement $\int_\D|f'(z)|ϕ(|z|)\,dA(z)<\infty,$ where $ϕ(r)=\log^α(1/(1-r))$ and show that it is a predual of the "$\log^α$-Bloch" space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\sum_{n=0}^\infty a_nz^n$ with $a_n\downarrow 0$ is in $\bk^1_{\log^α}$ iff $\sum_{n=0}^\infty \log^α(n+2)/(n+1)<\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\bk^1_{\log^α}.$ Some properties of the Cesáro and the Libera operator are considered as well.

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