Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z) = (1 − z)−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0 < Reb <2. Let μn = Δ¯ ζ n dμ(ζ ) where μ is a complex-valued measure on the closed unit disk Δ¯. Such sequences are shown to be multipliers of Hp for 1 p ∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μn} is a multiplier of Hp for every p > 0. When the support of μ is [0, 1] we get the multiplier sequence 1 0 tn dμ(t ), which provides more concrete applications. We show that if the sequences {μn} and {νn} are related by an asymptotic expansion νn μn ≈ ∞ k=0 Ak nk (n→∞) and μn is a multiplier of Hp into Hq , then so is νn. We ask whether {(n + 1)iβ } is a multiplier of Hp when β is a nonzero real number. It is clear that the question has an affirmative answer when p = 2. The answer is shown to be negative when p=∞.  2003 Elsevier Science (USA). All rights reserved.