LOGARITHMIC BLOCH SPACE AND ITS PREDUAL

Miroslav Pavlović

Abstract: We consider the space ${𝔅}_{{log}^{\alpha }}^1$, of analytic functions on the unit disk $𝔻$, defined by the requirement ${\int }_{𝔻}|f^{\text{'}}\left(z\right)\left|\varphi \right(\left|z\right|)dA\left(z\right)<\infty$, where $\varphi \left(r\right)={log}^{\alpha }(1/(1-r))$ and show that it is a predual of the “${log}^{\alpha }$-Bloch” space and the dual of the corresponding little Bloch space. We prove that a function $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ with $a_n\downarrow 0$ is in ${𝔅}_{{log}^{\alpha }}^1$ iff ${\sum }_{n=0}^{\infty }{log}^{\alpha }(n+2)/(n+1)<\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in ${𝔅}_{{log}^{\alpha }}^1$. Some properties of the Cesàro and the Libera operator are considered as well.

Keywords: Libera operator; Cesaro operator; Hardy spaces; logarithmic Bloch type spaces; predual

Classification (MSC2000): 30D55

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