Let B denote the unit ball in C-n, and dV(z) normalized Lebesgue measure on
B. For alpha > -1, define dV(alpha)(z) = (1- \z\(2))(alpha)dV(z). Let H(B)
denote the space of holomorhic functions on B, and for 0 < p < infinity, l
et A(p)(dV(alpha)) denote L-p(dV(alpha)) boolean AND H(B). In this note we
characterize A(p)(dV(alpha)) as those functions in H(B) whose images under
the action of a certain set of differential operators lie in L-p(dV(alpha))
. This is valid for 1 less than or equal to p < infinity. We also show that
the Cesaro operator is bounded on A(p)(dV(alpha)) for 0 < p < infinity. An
alogous results are given for the polydisc.