Compact composition operators on weighted Bergman spaces of the unit ball

For p > 0 and alpha greater than or equal to 0, let A(alpha)(p)(B-n) be the weighted Bergman space of the unit ball B-n in C-n, and denote the Hardy space by H-p(B-n). Suppose that phi : B-n --> B-n is holomorphic. We show that if the composition operator C-phi defined by C-phi(f)= f circle phi is bounded on A(alpha)(p)(B-n) and satisfies lim(/z/-->1-)(1-/z/(2)/1-/phi (z)/(2))(alpha +2)//phi ' (z)//(2) = 0, then C-phi is compact on A(beta)(p)(B-n) for all beta greater than or equal to alpha. Along the way we prove some comparison results on boundedness and compactness of composition operators on H-p(B-n) and A(alpha)(p)(B-n), as well as a Carleson measure-type theorem involving these spaces and more general weighted holomorphic Sobolev spaces.