Complete mapping criteria for generalized Cesàro operators between Hardy spaces

RESUMO

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_{\mu,\gamma}f(z) =\sum_{n=0}^\infty \mu_n \left(\sum_{k=0}^n \frac{\Gamma(n-k+\gamma)}{\Gamma(\gamma)(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_{\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 \[ \int_0^1 \left(\frac{\mu([t,1))}{(1-t)^{\gamma-1/r}}\right)^r \frac{dt}{1-t}<\infty. \] This condition is also equivalent to $F_{\mu,\gamma}\in H^r$ and to $\sum_{n\ge0}(n+1)^{r\gamma-2}\mu_n^r<\infty$, 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 $F_{\mu,\gamma+1}\in H^{p'}$, where $p'=p/(p-1)$. 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 $0<p,q\le\infty$.