Derivatives of Blaschke products

Integrability of the Derivative of a Blaschke Product

“…For such products B and 1 < p < 2 we show that B ∈ A p if and only if n (1 − |a n |) 2−p < ∞. This is a converse in this situation of a general result due to Kim [22].…”

“…This theorem is a converse of Theorem 3.1 in Kim's paper [22]. The statement is non-trivial only when 1 < p < 2.…”

“…Of course, it adds nothing new when 0 < p ≤ 1 because we already have the Blaschke condition. On the other hand, we know that if B is an arbitrary infinite Blaschke product then B ∈ A p for any p ≥ 2, see Theorem 1.1 of [22].…”

“…It is, thus, desirable to have criteria for membership of B in, say, Hardy and Bergman spaces based just on some information on the zeros of B. The papers cited mainly considered conditions on the moduli of the zeros, as is the case of [30] and [22]. Many results obtained there appear to be sharp but the sharpness issue was not discussed.…”

On the dual space of a weighted Bergman space on the unit ball of Cn

“…The remaining case is essentially a result of Hardy and Littlewood, but we give a proof using Ahern's technique in [5].…”