Ces\`aro-type operators on mixed norm spaces

Abstract

Given a positive Borel measure $\mu$ on $[0,1)$ and a parameter $\beta>0$, we consider the Ces\`aro-type operator $\mathcal C_{\mu,\beta}$ acting on the analytic function $f(z)=\sum_{n=0}^\infty a_n z^n$ on the unit disc of the complex plane $\mathbb D$, defined by \[ \mathcal C_{\mu,\beta}(f)(z)= \sum_{n=0}^\infty \mu_n \left( \sum_{k=0}^n \frac{\Gamma(n-k+\beta)}{(n-k)! \Gamma(\beta)} a_k \right) z^n = \int_0^1 \frac{f(tz)}{(1-tz)^\beta} d\mu(t), \] where $\mu_n=\int_0^1 t^n d\mu(t)$. This operator generalizes the classical Ces\`aro operator (corresponding to the case where $\mu$ is the Lebesgue measure and $\beta=1$) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of $\mathcal C_{\mu,\beta}$ on mixed norm spaces $H(p,q,\gamma)$ for $0<p,q\leq\infty$ and $\gamma>0$. Our results extend and unify several known characterizations for the boundedness of Ces\`aro-type operators acting on spaces of analytic functions.