Rhaly operators acting on Hardy, Bergman, and Dirichlet spaces

In this article we address the question of characterizing the sequences of complex numbers $(\eta )=\{ \eta_n\}_{n=0}^\infty$ whose associated Rhaly operator $\mathcal R_{(\eta )}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty$), on the Bergman spaces $A^p_\alpha$, and on the Dirichlet spaces $\mathcal D^p_\alpha$ ($1\le p<\infty$, $\alpha >-1$). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(\eta )}$ on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $F_{(\eta )}$ defined by $F_{(\eta )}(z)=\sum_{n=0}^\infty \eta _nz^n$ ($z\in \mathbb D$). \par We prove that if $2\le p<\infty$ and $\eta_n=\og \left (\frac{1}{n}\right )$, then $\mathcal R_{(\eta )}$ is bounded on $H^p$. However, there exists a sequence $(\eta )$ with $\eta_n=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R_{(\eta )}$ is not bounded on $H^p$ for $1\le p<2$. \par We deal also with the derivative-Hardy spaces. For $p>0$ the derivative-Hardy space $S^p$ consists of those functions $f$, analytic in the unit disc $\mathbb D$, such that $f^\prime \in H^p$. We prove that if $1\le p<\infty$ and $1<q<\infty$ then $\mathcal R_{(\eta )}$ is a bounded operator from $S^p$ into $S^q$ if and only if it is compact and this happens if and only if $F_{(\eta )}\in S^q$.