Hadamard products with power functions and multipliers of Hardy spaces

Abstract

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<Re b<2. Let μ n=∫ 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 H p 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 { μ n } is a multiplier of H p for every p>0. When the support of μ is [0,1] we get the multiplier sequence ∫ 0 1t n dμ(t), which provides more concrete applications. We show that if the sequences { μ n } and { ν n } are related by an asymptotic expansion ν n μ n ≈ ∑ k=0 ∞ A k n k (n→∞) and μ n is a multiplier of H p into H q , then so is ν n . We ask whether {( n+1) iβ } is a multiplier of H p 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=∞.