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 $H^p$, $1<p<\infty$, that is, we describe all finite positive Borel measures $\mu$ defined on the closed ball $\overline{B}_d$ and such that
$$ \|f \|_{H^p} \le c \|f\|_{L^p(\overline{B}_d,\mu)} $$
for all $f\in H^p(B_d) \cap C(\overline{B}_d)$ and a universal constant $c>0$. Given a noninner holomorphic function $b: B_d \to B_1$, we obtain properties of the reverse Carleson measures for the de Branges–Rovnyak space $\mathcal{H}(b)$.