Cesàro-type operators on mixed norm spaces

Given a positive Borel measure $\mu$ on $[0,1)$ and a parameter $\beta >0$, we consider the Cesàro-type operator ${\mathcal{C}}_{\mu ,\beta }$ acting on the analytic function $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ on the unit disc of the complex plane $\mathbb{D}$, defined by $${\mathcal{C}}_{\mu ,\beta }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}{\mu }_n\left(\sum _{k=0}^n\frac{\mathrm{\Gamma }(n-k+\beta )}{(n-k)!\mathrm{\Gamma }(\beta )}{a}_k\right)z^n={\int }_0^1\frac{f(tz)}{(1-tz{)}^{\beta }}d\mu (t),$$ where ${\mu }_n={\int }_0^1{t}^{n}d\mu (t)$. This operator generalizes the classical Cesàro 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 $0<p,q\le \mathrm{\infty }$ and $\gamma >0$. Our results extend and unify several known characterizations for the boundedness of Cesàro-type operators acting on spaces of analytic functions.