Extended Cesàro operators between Hardy and Bergman spaces on the complex ball

We characterize those holomorphic symbols $g$ for which the extended Cesàro operator $V_g$ maps the Hardy space $H^p(B)$ into the weighted Bergman space $A^q_\beta(B)$, $0<p<q<\infty$, $\beta>-1$, on the unit ball $B$ of $\mathbb C^d$.