By Rita A. Hibschweiler
Featuring new effects in addition to examine spanning 5 a long time, Fractional Cauchy Transforms offers an entire remedy of the subject, from its roots in classical advanced research to its present country. Self-contained, it contains introductory fabric and classical effects, comparable to these linked to complex-valued measures at the unit circle, that shape the foundation of the advancements that stick with. The authors concentrate on concrete analytic questions, with practical research offering the overall framework.
After interpreting easy houses, the authors research necessary potential and relationships among the fractional Cauchy transforms and the Hardy and Dirichlet areas. They then learn radial and nontangential limits, via chapters dedicated to multipliers, composition operators, and univalent capabilities. the ultimate bankruptcy offers an analytic characterization of the relations of Cauchy transforms whilst regarded as capabilities outlined within the supplement of the unit circle.
About the authors:
Rita A. Hibschweiler is a Professor within the division of arithmetic and facts on the college of latest Hampshire, Durham, USA.
Thomas H. MacGregor is Professor Emeritus, country college of recent York at Albany and a examine affiliate at Bowdoin collage, Brunswick, Maine, united states.
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Additional resources for Fractional Cauchy Transforms
For almost all θ, G 1 (θ) = − i [F1 (θ) + F2 (θ)] and hence Re G1(θ) = Im F1(θ) – Im F2(θ) = Im F(θ), where F is defined in (d). Since F 0 L1 ([–π, π]), it follows that Re G1 0 L1 ([–π, π]). 32) A Characterization of Cauchy Transforms 233 for | z | < 1. 33) for z 0 ⎟∞ \ T. Assume that z 0 ⎟∞ \ D . 33) yields g 1 (z) = − g 1 (1 / z ) = − ∫ T = ∫ T 1 z dυ (ζ ) + i Im g (0) 1 1 1 ζ− z ζ+ ζ+z dυ1 (ζ ) + i Im g 1 (0) . 32) holds for all z 0 ⎟ \ T. The argument will continue with a similar analysis of the function g2 defined by g 2 (z) = f (z) − f (1 / z ) (z 0 ⎟∞ \ T).
1 − ζz Hence there is a measure υ 0 M with || υ || = || µ || and g (z) = z ∫ T 1 dυ(ζ ) . 1 − ζz Thus g is of the form z ⋅ h , where h 0 F1 and it follows that g 0 Hp for 0 < p < 1. This yields f2 0 Hp ( D′) for 0 < p < 1. The argument given previously for f1 applies to g and yields lim || f 2 || p (1 − p) < ∞. 7) p →1− The limit r →1− © 2006 by Taylor & Francis Group, LLC A Characterization of Cauchy Transforms 221 exists for almost all θ 0 [–π,π]. We denote this limit by F(θ). We shall show that F is Lebesgue integrable on [–π,π].
There is no comparable characterization of F1, or more generally, of Fα for α > 0. 21 is implicit. 1) which defines Fα for α > 0 actually defines a function which is analytic in ⎟ \ T when α is a positive integer. 8 for α = 2, 3, … . NOTES The Hardy spaces Hp (Ω) are defined for various domains Ω δ ⎟. A reference for the definitions and results is Duren [1970; see Chapter 10]. I. Smirnov. The result in Lemma 5, namely, that the © 2006 by Taylor & Francis Group, LLC A Characterization of Cauchy Transforms 235 composition of a subharmonic function with an analytic function is subharmonic, is a classical fact.