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 $f(z)={\sum }_{n=0}^{\infty }\hat{f}(n)z^n$ in the unit disc $\mathbb{D}$, we define the fractional derivative $D^{\mu }(f)(z)=\sum _{n=0}^{\infty }\frac{\hat{f}(n)}{{\mu }_{2n+1}}{z}^n$ 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 $\Vert f{\Vert }_{{\mathcal{B}}^{\mu }}={\sup }_{z\in \mathbb{D}}\hat{\mu }(z)|D^{\mu }(f)(z)|<\infty$, where $\hat{\mu }(z)={\int }_{|z|}^1\mu (s)ds$. 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 $\mu \in \mathcal{D}=\hat{\mathcal{D}}\cap \check{\mathcal{D}}$. A radial weight $\mu \in \hat{\mathcal{D}}$ if ${\sup }_{0\le r<1}\frac{\hat{\mu }(r)}{\hat{\mu }\left(\frac{1+r}{2}\right)}<\infty$ and a radial weight $\mu \in \check{\mathcal{D}}$ if there exist $K=K(\mu )>1$ such that ${\inf }_{0\le r<1}\frac{\hat{\mu }(r)}{\hat{\mu }\left(1-\frac{1-r}{K}\right)}>1.$