By Rita A. Hibschweiler

Featuring new effects besides study spanning 5 a long time, Fractional Cauchy Transforms presents an entire remedy of the subject, from its roots in classical advanced research to its present nation. Self-contained, it contains introductory fabric and classical effects, equivalent to these linked to complex-valued measures at the unit circle, that shape the root of the advancements that keep on with. The authors concentrate on concrete analytic questions, with useful research offering the final framework. After interpreting uncomplicated houses, the authors examine critical ability and relationships among the fractional Cauchy transforms and the Hardy and Dirichlet areas. They then examine radial and nontangential limits, by means of chapters dedicated to multipliers, composition operators, and univalent features. the ultimate bankruptcy provides an analytic characterization of the kinfolk of Cauchy transforms while regarded as capabilities outlined within the supplement of the unit circle. concerning the authors: Rita A. Hibschweiler is a Professor within the division of arithmetic and information on the college of latest Hampshire, Durham, united states. Thomas H. MacGregor is Professor Emeritus, country collage of recent York at Albany and a examine affiliate at Bowdoin collage, Brunswick, Maine, USA.\

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For α > 0, let Bα denote the set of functions f that are analytic in D and satisfy 1 π 0 −π ∫ ∫ | f ′(re iθ ) | (1 − r ) α −1 dθ dr < ∞. 21) For f 0 Bα, let || f || Bα ≡ | f (0) | + 1 π 0 −π ∫ ∫ | f ′(re iθ ) | (1 − r ) α −1 dθ dr. 22) defines a norm on Bα and that Bα is a Banach space with respect to this norm. Also if 0 < α < β then Bα δ Bβ. The proof of the next lemma uses the following classical result of E. Cesàro (see Pólya and Szegö, vol. I [1972], p. 16): Suppose that {an} and {bn} ∞ (n = 0, 1, …) are sequences such that bn > 0 for n = 0, 1, …, ∑ bnzn n =0 ∞ converges for | z | < 1 and ∑ n =0 © 2006 by Taylor & Francis Group, LLC b n z n diverges for z = 1.

Since © 2006 by Taylor & Francis Group, LLC 26 Fractional Cauchy Transforms 1 2π ∫ π 1 −π (1 − e −iθ z) γ dθ = 1 for |z| < 1 and since 1/(2π) dθ defines a probability measure, I 0 Fγ* for every γ > 0. Suppose that 0 < α < β and f 0 Fα* . Let γ = β – α. 6 yields f = f ·I 0 Fα*+ γ . Therefore Fα* ⊂ Fβ* . 10 If 0 < α < β then Fα δ Fβ and Fα ≠ Fβ. Proof: Let 0 < α < β and assume that f 0 Fα. 9. Then I 0 Fγ* for every γ > 0 and hence || I || Fβ − α = 1. 7 implies that f = f ·I 0 Fβ and || f || Fβ ≤ || f || Fα .

T This shows that || f ||Fα = || µ || . In particular if µ 0 M* and µ represents f in Fα then || f ||Fα = 1. For example, if | ζ | = 1 and f (z) = 1 /(1 − ζz)α (| z | < 1) then f 0 Fα and f can be represented by unit mass at ζ. Therefore ||1 /(1 − ζz)α ||Fα = 1. 29) defines a norm on Fα for α > 0 and with respect to that norm Fα is a Banach space. 30) is similar. 1) is linear. The kernel K of this mapping consists of the measures µ 0 M such that ∫ζ n dµ(ζ ) = 0 for n = 0, 1, … . 7, the collection of cosets M∋K, with the usual addition of cosets and multiplication of cosets by scalars, also is a Banach space.