Bloch space on the unit ball of Cn

In this paper we prove that for 0 < p < infinity, the norm of a function f in the Bergman space L-a(p) on the unit ball B of C-n, n greater than or equal to 1,is equivalent to the quantity integral(B)\<(del)over tilde> f(z)\(2)\f(z)\(p-2)h(\z\)d tau(z) where <(del)over tilde> and tau denote the invariant gradient and invariant measure on B, respectively, and h(\z\) = integral(\z\)(1) (1 - t(2))(n-1)(1 - t(2n))/t(2n-1) dt. If n > 1, this result allows us to extend the characterization J(2) of the Bloch space obtained in [OYZ, Theorem 2] to the range 0 < p < infinity. We also get this kind of description of Bloch functions for n=1. Moreover, we generalize the result obtained in [CKP] and show that f is an element of H(B) is a Bloch function if and only if for some p, 0 < p < infinity, \<(del)over tilde>f(z)\(p)dv(z) is a Bergman-Carleson measure. Finally, we get some results for spaces H-p and BMOA, e.g. an extension of the classical Littlewood-Paley inequality to the case of the unit ball.