Hadamard products with power functions and multipliers of Hardy spaces

AbstractWe 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=∫Δ̄ζndμ(ζ) where μ is a complex-valued measure on the closed unit disk Δ̄. 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 ∫01tndμ(t), which provides more concrete applications. We show that if the sequences {μn} and {νn} are related by an asymptotic expansion νnμn≈∑k=0∞Aknk(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=∞