Complete mapping criteria for generalized Cesàro operators between Hardy spaces

Abstract

Let $\mu$ be a finite positive Borel measure on $[0,1)$ and let $\gamma >0$. We establish sharp mapping criteria for the generalized Cesàro operator \begin{equation*} \mathcal C_{μ,γ}f(z) =\sum_{n=0}^\infty μ_n \left(\sum_{k=0}^n \frac{Γ(n-k+γ)}{Γ(γ)(n-k)!}a_k\right)z^n,\qquad z\in \mathbb{D},\end{equation*} between Hardy spaces, including the source and target $H^{\infty }$ endpoints. For $0<p<q<\infty$, for $0<p=q<1$, and for $0<p\le 1$ with $q=\infty$, the boundedness of ${\mathcal{C}}_{\mu ,\gamma }:H^p\to H^q$ is equivalent to $\mu$ being a $(\gamma +1/p-1/q)$-Carleson measure, without further restrictions on $\gamma$. For $1\le q<p\le \infty$, set $1/r=1/q-1/p$. In this range, boundedness and compactness are equivalent to \[ \int_0^1 \left(\frac{μ([t,1))}{(1-t)^{γ-1/r}}\right)^r \frac{dt}{1-t}<\infty. \] This condition is also equivalent to $F_{\mu ,\gamma }\in H^r$ and to ${\sum }_{n\ge 0}(n+1{)}^{r\gamma -2}{\mu }_n^r<\infty$, where $F_{\mu ,\alpha }={\mathcal{C}}_{\mu ,\alpha }(1)$ and ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$. For $1<p<\infty$, boundedness and compactness from $H^p$ to $H^{\infty }$ are characterized by the shifted condition $F_{\mu ,\gamma +1}\in H^{p^{\prime }}$, where $p^{\prime }=p/(p-1)$. We therefore obtain a complete boundedness classification of the generalized Cesàro operators ${\mathcal{C}}_{\mu ,\gamma }$ between Hardy spaces $H^p$ and $H^q$ for the full range $0<p,q\le \infty$.