Mean Lipschitz conditions on Bergman space

Abstract

For $f$ analytic on the unit disc let $r_t(f)(z)=f(e^{it}z)$ and $f_r(z)=f(rz)$, rotations and dilations respectively. We show that for $f$ in the Bergman space $A^p$ and $0<\alpha\leq 1$ the following are equivalent. \begin{itemize} \item[(i)] $\n{r_t(f)-f}_{A^p}=\og(|t|^{\alpha}), \quad t\to 0$, \item[(ii)] $\n{(f')_r}_{A^p} =\og\left (1-r)^{\alpha-1}\right ), \quad r\to 1^{-}$, \item[(iii)] $\n{f_r-f}_{A^p}=\og((1-r)^{\alpha}),\quad r\to 1^{-}$. \end{itemize} The Hardy space analogues of these conditions are known to be equivalent by results of Hardy and Littlewood and of E. Storozhenko, and in that setting they describe the mean Lipschitz spaces $\Lambda (p, \alpha)$. On the way, we provide an elementary proof of the equivalence of $(ii)$ and $(iii)$ in Hardy spaces, and show that similar assertions are valid for certain weighted mean Lipschitz spaces.