Factorization theorems for functions in the Bergman spaces

1. For 0 < p < o and 0 _< a o, we consider the Bergman spaces A of functions analytic in the unit disc U of C and satisfying U where dA denotes the normalized Lebesgue measure (1/) dxdy. The spaces A' are denoted simply A. In the present paper, we discuss factorization properties of functions in these spaces. Our first main result is the following: THEOnEM 1. Let p and a be fixed, and let n 2 be an integer. Then there is a constant C depending only on p, a, and n, such that i/] A'" there exist functions ], 1, A with and i=l Theorem 1 has a somewhat interesting history. The analogous theorem for H spaces over U in place of A is elassicM. Rudin [9] showed that this result does not generalize to H spaces over a polydisc, at least in dimensions greater than 3. Somewhat later, Miles [7] and Rosay [8] extended Rudin's counterexample to dimensions 2 and 3. Interest in the present problem arose partly because of connections between A spaces and H spaces in several variables. Indeed, in view of the theorem in [5], a negative result in place of Theorem 1 would have reproved the result of Rudin, Miles and Ilosay. In the other direction, recent work of Coifman, Rochberg, and Weiss [2] on H spaces in several variables has yielded as a corollary a weaker version of Theorem 1. Our second main theorem deals with division by Blaschke products, which products we define here mainly for purposes of fixing notation. For a U, we define first the M6bius transformations amz (1.1) C(z) 1 az