Cesàro operator induced by a Bergman kernel

Abstract

Let $\mu$ be a positive Borel measure on $[0,1)$ and $\omega$ a radial weight. In this paper we consider the Cesàro-type operator $C_{\mu ,\omega }$ induced by the reproducing kernel $B^{\omega }$ of the weighted Bergman space $A_{\omega }^2$, given by $$C_{\mu ,\omega }(f)(z)={\int }_0^{1}f(tz)B_t^{\omega }(z)d\mu (t),z\in \mathbb{D},$$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $\omega$ satisfies a natural doubling property, we study the boundedness of $C_{\mu ,\omega }$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A_{\nu }^p$. For $0<p,q<\infty$ and a two-sided doubling weight $\nu$, we completely characterize when $C_{\mu ,\omega }:H^p\to H^q$ and $C_{\mu ,\omega }:A_{\nu }^p\to A_{\nu }^q$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $\mu$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{\mu ,\omega }$ acting on $H^{\infty }$, Korenblum spaces and weighted Hardy spaces.