Complete mapping criteria for generalized Cesàro operators between Hardy spaces

Abstract

Let $μ$ be a finite positive Borel measure on $[0,1)$ and let $γ>0$. 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 $0<p\le1$ with $q=\infty$, the boundedness of $ \mathcal C_{μ,γ}: H^p \to H^q$ is equivalent to $μ$ being a $(γ+1/p-1/q)$-Carleson measure, without further restrictions on $γ$. 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 $F_{μ,γ}\in H^r$ and to $\sum_{n\ge0}(n+1)^{rγ-2}μ_n^r<\infty$, where $F_{μ,α}=\mathcal C_{μ,α}(1)$ and $μ_n=\int_{[0,1)}t^n\,dμ(t)$. For $1<p<\infty$, boundedness and compactness from $H^p$ to $H^\infty$ are characterized by the shifted condition $F_{μ,γ+1}\in H^{p'}$, where $p'=p/(p-1)$. We therefore obtain a complete boundedness classification of the generalized Cesàro operators $\mathcal C_{μ,γ}$ between Hardy spaces $H^p$ and $H^q$ for the full range $0<p,q\le\infty$.