Reverse Carleson measures for Hardy spaces in the unit ball

Let $H^p=H^p(B_d)$ denote the Hardy space in the open unit ball $B_d$ of ${\mathbb{C}}^d$, $d\ge 1$. We characterize the reverse Carleson measures for $H^p$, $1<p<\mathrm{\infty }$, that is, we describe all finite positive Borel measures $\mu$, defined on the closed ball ${\bar{B}}_d$, such that $$\mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}\le c\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\bar{B}}_d,\mu )}$$ for all $f\in H^p(B_d)\cap C({\bar{B}}_d)$ and a universal constant $c>0$. Given a non-inner holomorphic function $b:B_d\to B_1$, we obtain properties of the reverse Carleson measures for the de Branges-Rovnyak space $\mathcal{H}(b)$.