Carleson measures for the Bloch space in the ball

Let ${\mathcal H}ol(B_n)$ denote the space of holomorphic functions in the unit ball $B_n$ of ${\mathbb C}^n$, $n\ge 1$. Given $X\subset {\mathcal H}ol(B_n)$ and $0<q<\infty$, a well-known problem is to characterize the positive measures $\mu$ on $B_n$ such that $X\subset L^q (B_n, \mu)$. We obtain such a characterization when $X$ is the Bloch space ${\mathcal B}(B_n)= \left\{ f\in{\mathcal H}ol(B_n):\ \sup_{z\in B_n} |\nabla f(z)|(1- |z|) < \infty \right\}$ and $\mu$ is a radial measure. For $n=1$, this solution was recently obtained by D.~Girela, J.~{'A}.~Pel{'a}ez, F.~Pérez-Gonz'alez and J.~R"atty"a. Also, we solve the problem when $X$ is the growth space ${\mathcal A}^{-\log}(B_n)$ or $X$ is the growth space ${\mathcal A}^{-\beta}(B_n)= \left\{ f\in{\mathcal H}ol(B_n):\ \sup_{z\in B_n} |f(z)|(1- |z|)^\beta < \infty \right\} ,\quad \beta>0.$