Title

Multipliers of Cauchy-Stieltjes type transforms

Department

Mathematics

Document Type

Article

Publication Source

Houston Journal of Mathematics

Publication Date

1996-12-01

Volume

22

Issue

2

First Page

357

Last Page

374

Abstract

Let H (Δ) denote the collection of analytic functions on Δ = {z : |z| < 1} and let M denote the collection of finite complex Borel measures on Γ = {cursive Greek chi : |cursive Greek chi| = 1}. For F ∈ H(Δ), let IF represent the collection of functions defined on Δ of the form fμ(z) = ∫Γ F(cursive Greek chiz)dμ(cursive Greek chi), for some μ ∈ M. For α > 0, let Fα(z) = 1/(1-z)α and for α real, let Hα(z) = ∑∞n=0(n + 1)α-1zn. Let ℱα denote IFα and let Hα denote IHα. A function g ∈ H(Δ) is called a multiplier of IF if g · IF ⊂ IF. Let MF denote the collection of multipiers of IF. For a general kernel F, it is proved that, if z ∈ MF and g ∈ H(Δ̄||z||MF) where Δ̄||z||MF = {z : |z| < ||z||MF}, then g ∈ MF. For α real, let Mα denote MHα. For α < 1, the estimation of the L1 [0, 2π] norm of appropriate trigonmetric series is shown to imply that if g ∈ H(Δ) and ∑∞n=0 |ĝ(n)|(n + 1)1-α < +∞, then g ∈ Mα. This result implies that, for -∞ < α < β < 1, there exists a function fβ such that fβ ∈ Mβ \ Mα. Suppose g ∈ Mα for some real a and g(a) = 0 for some a ∈ Δ. It is shown that under these conditions that g(z)/z-a is also an element of Mα.

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