Complete mapping criteria for generalized Cesàro operators between Hardy spaces

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 $H^{\mathrm{\infty }}$ endpoints. For $0<p<q<\mathrm{\infty }$, for $0<p=q<1$, and for $0<p\le 1$ with $q=\mathrm{\infty }$, the boundedness of ${\mathcal{C}}_{\mu ,\gamma }:H^p\to H^q$ is equivalent to $\mu$ being a $(\gamma +1\mathrm{/}p-1\mathrm{/}q)$-Carleson measure, without further restrictions on $\gamma$. For $1\le q<p\le \mathrm{\infty }$, set $1\mathrm{/}r=1\mathrm{/}q-1\mathrm{/}p$. In this range, boundedness and compactness are equivalent to $${\int }_0^1{\left(\frac{\mu ([t,1))}{(1-t{)}^{\gamma -1\mathrm{/}r}}\right)}^r\frac{dt}{1-t}<\mathrm{\infty }\mathrm{.}$$ This condition is also equivalent to $F_{\mu ,\gamma }\in H^r$ and to ${\sum }_{n\ge 0}(n+1{)}^{r\gamma -2}{\mu }_n^r<\mathrm{\infty }$, where $F_{\mu ,\alpha }={\mathcal{C}}_{\mu ,\alpha }(1)$ and ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$. For $1<p<\mathrm{\infty }$, boundedness and compactness from $H^p$ to $H^{\mathrm{\infty }}$ are characterized by the shifted condition $F_{\mu ,\gamma +1}\in H^{p^{\mathrm{\prime }}}$, where $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$. 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 $0<p,q\le \mathrm{\infty }$.