Rhaly operators acting on Hardy, Bergman, and Dirichlet spaces

In this article we address the question of characterizing the sequences of complex numbers $(\eta )=\{{\eta }_n{\}}_{n=0}^{\mathrm{\infty }}$ whose associated Rhaly operator ${\mathcal{R}}_{(\eta )}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\mathrm{\infty }$), on the Bergman spaces $A_{\alpha }^p$, and on the Dirichlet spaces ${\mathcal{D}}_{\alpha }^p$ ($1\le p<\mathrm{\infty }$, $\alpha >-1$). 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 $F_{(\eta )}(z)={\sum }_{n=0}^{\mathrm{\infty }}{\eta }_n{z}^n$ ($z\in \mathbb{D}$). \par We prove that if $2\le p<\mathrm{\infty }$ and ${\eta }_n=\text{og}\left(\frac{1}{n}\right)$, then ${\mathcal{R}}_{(\eta )}$ is bounded on $H^p$. However, there exists a sequence $(\eta )$ with ${\eta }_n=\text{og}\left(\frac{1}{n}\right)$ 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 $f^{\mathrm{\prime }}\in H^p$. We prove that if $1\le p<\mathrm{\infty }$ and $1<q<\mathrm{\infty }$ 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 $F_{(\eta )}\in S^q$.