Cesà ro-type operators on mixed norm spaces

Abstract

Given a positive Borel measure $\hat{I}¼$ on $[0,1)$ and a parameter ${\hat{I}}^2>0$, we consider the Cesà ro-type operator ${\mathcal{C}}_{\hat{I}¼,{\hat{I}}^2}$ 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 \[ \mathcal C_{μ,β}(f)(z)= \sum_{n=0}^\infty μ_n \left( \sum_{k=0}^n \frac{Γ(n-k+β)}{(n-k)! Γ(β)} a_k \right) z^n = \int_0^1 \frac{f(tz)}{(1-tz)^β} dμ(t), \] where $\hat{I}{¼}_n={\int }_0^1{t}^{n}d\hat{I}¼(t)$. This operator generalizes the classical Cesà ro operator (corresponding to the case where $\hat{I}¼$ is the Lebesgue measure and ${\hat{I}}^2=1$) and includes other relevant cases previously studied in the literature. In this paper we study the boundedness of ${\mathcal{C}}_{\hat{I}¼,{\hat{I}}^2}$ on mixed norm spaces $H(p,q,{\hat{I}}^3)$ for $0<p,q\le \infty$ and ${\hat{I}}^3>0$. Our results extend and unify several known characterizations for the boundedness of Cesà ro-type operators acting on spaces of analytic functions.