Fractional Derivative Description of the Bloch Space

Abstract

We establish new characterizations of the Bloch space \(\mathcal {B}\) which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function \(f(z)=\sum _{n=0}^\infty \widehat{f}(n) z^n\) in the unit disc \(\mathbb {D}\) , we define the fractional derivative \( D^{\mu }(f)(z)=\sum \limits _{n=0}^{\infty } \frac{\widehat{f}(n)}{\mu _{2n+1}} z^n \) induced by a radial weight \(\mu \) , where \(\mu _{2n+1}=\int _0^1 r^{2n+1}\mu (r)\,dr\) are the odd moments of \(\mu \) . Then, we consider the space \( \mathcal {B}^\mu \) of analytic functions f in \(\mathbb {D}\) such that \(\Vert f\Vert _{\mathcal {B}^\mu }=\sup _{z\in \mathbb {D}} \widehat{\mu }(z)|D^\mu (f)(z)|<\infty \) , where \(\widehat{\mu }(z)=\int _{|z|}^1 \mu (s)\,ds\) . We prove that \(\mathcal {B}^\mu \) is continously embedded in \(\mathcal {B}\) for any radial weight \(\mu \) , and \(\mathcal {B}=\mathcal {B}^\mu \) if and only if \(\mu \in \mathcal {D}=\widehat{\mathcal {D}}\cap \check{\mathcal {D}}\) . A radial weight \(\mu \in \widehat{\mathcal {D}}\) if \(\sup _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( \frac{1+r}{2}\right) }<\infty \) and a radial weight \(\mu \in \check{\mathcal {D}}\) if there exist \(K=K(\mu )>1\) such that \(\inf _{0\le r<1}\frac{\widehat{\mu }(r)}{\widehat{\mu }\left( 1-\frac{1-r}{K}\right) }>1.\)