Dirichlet and Bergman spaces of holomorphic functions on the unit ball of Cn

Let B denote the unit hall in C-n, n >= 1, and let tau and (del) over tilde denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces D-gamma, gamma>(n-1), and weighted Bergman spaces A(alpha)(p), 0<p<infinity, alpha>n, of holomorphic functions f on B for which D-gamma(f) and parallel to f parallel to(A alpha p) respectively are finite, where D-gamma(f)=integral(B)(1-vertical bar z vertical bar(2))(gamma)vertical bar(del) over tildef(z)vertical bar(2)d tau(z), and parallel to f parallel to(A alpha p)(p) =integral(B)(1-vertical bar z vertical bar(2))(alpha)vertical bar f(z)(p)d tau(z). The main result of the paper is the following theorem.