Fractional Cauchy transform

268 Pages·2005·1.582 MB·English

by  Rita A. Hibschweiler, Thomas H. MacGregor

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 136
FRACTIONAL
CAUCHY
TRANSFORMS
© 2006 by Taylor & Francis Group, LLC
CHAPMAN & HALL/CRC
Monographs and Surveys in Pure and Applied Mathematics
Main Editors
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W.A. Strauss, Brown University
J. van der Hoek, University of Adelaide
© 2006 by Taylor & Francis Group, LLC
CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 136
FRACTIONAL
CAUCHY
TRANSFORMS
Rita A. Hibschweiler
Thomas H. MacGregor
Boca Raton London New York Singapore
© 2006 by Taylor & Francis Group, LLC
C5602_Discl.fm Page 1 Thursday, August 18, 2005 8:51 AM
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Library of Congress Cataloging-in-Publication Data
Hibschweiler, Rita A.
Fractional Cauchy transforms / Rita A. Hibschweiler, Thomas H. MacGregor.
p. cm.
Includes bibliographical references and index.
ISBN 1-58488-560-2 (acid-free paper)
1. Cauchy transform. 2. Functional analysis. I. MacGregor, T. H. (Thomas H.) II. Title.
QA320.H44 2005
515'.723--dc22 2005052867
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© 2006 by Taylor & Francis Group, LLC
CONTENTS
Page

List of Symbols.................................................................................................

Preface..............................................................................................................

Acknowledgements...........................................................................................

Dedication.........................................................................................................

Chapter 1. Introduction

Definition of the families F ..............................................................1
α
Relations between F and H1.............................................................2
1
The Riesz-Herglotz formula.............................................................4
Representations with real measures and h1.......................................5
The F. and M. Riesz theorem............................................................7
The representing measures for functions in F ..................................9
α
The one-to-one correspondence between measures and functions
in the Riesz-Herglotz formula.....................................................11
The Banach space structure of F ....................................................11
α
Norm convergence and convergence uniform on compact sets......13
Notes ...............................................................................................15

Chapter 2. Basic Properties of F
α

Properties of the gamma function and the binomial coefficients....17
A product theorem..........................................................................22
Membership of f and f′in F ..........................................................24
α
The inclusion of F in F when 0 < α < β.........................................26
α β
The union of F for α > 0.................................................................26
α
Mappings from F to F ...................................................................28
α β
An integral condition for membership in F when α > 1.................29
α
Besov spaces and their relationship to F ........................................30
α
Infinite Blaschke products and membership in F ...........................34
α
The inner function S and its membership in F ...............................38
α
Hadamard products and relations with F and H∞...........................41
1
Notes ...............................................................................................46

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Chapter 3. Integral Means and the Hardy and Dirichlet Spaces

Subordination..................................................................................47
Littlewood’s inequality...................................................................48
Estimates on integral means...........................................................49
Relations between F and Hp...........................................................52
α
Growth of the integral means of f when f 0 F , α > 1 and
α
0 < p < 1.....................................................................................55
The Dirichlet spaces and F .............................................................57
α
Inner functions and area..................................................................60
Inner functions and F .....................................................................61
0
Membership of inner functions in Dirichlet spaces, Besov spaces
and F ..........................................................................................64
α
Notes ...............................................................................................65

Chapter 4. Radial Limits

Boundary functions and weak Lp inequalities.................................68
Nontangential limits........................................................................72
Radial limits and nontangential limits of bounded functions..........73
Local integrability conditions and radial limits..............................75
α-capacity.......................................................................................78
Nondecreasing functions and α-capacity........................................81
Nontangential limits and exceptional sets of zero α-capacity.........84
Growth and countable exceptional sets...........................................85
Growth and exceptional sets of measure zero.................................86
Notes ...............................................................................................90

Chapter 5. Zeros

The moduli of the zeros of a function in F , when α > 1.................93
α
Zeros in a Stolz angle.....................................................................95
Functions with large growth when α > 1........................................99
Functions with zeros tending slowly to T.....................................104
Notes .............................................................................................106

Chapter 6. Multipliers: Basic Results

The multiplication operator and the definition of M ....................107
α
The Banach space structure of M .................................................110
α
The inclusion of M in M when 0 < α < β......................................112
α β
Multipliers and H∞........................................................................113
© 2006 by Taylor & Francis Group, LLC
Nontangential limits......................................................................115
Radial variations...........................................................................115
An assumption on integrability sufficient to imply a Lipschitz
condition...................................................................................121
A sufficient condition for a multiplier when 0 < α < 1.................127
Notes .............................................................................................130

Chapter 7. Multipliers: Further Results

A lemma about Toeplitz operators and M ....................................135
1
An integral condition on the second difference............................139
Smoothness conditions.................................................................144
Conditions on the Taylor coefficients
when 0 < α < 1; ..................................................................145
when α = 1; .........................................................................147
when α > 1. .........................................................................160
The condition f′0 H1...................................................................161
Blaschke products.........................................................................162
The singular inner function S........................................................169
Strict monotonicity of multiplier spaces.......................................181
A survey of facts about M ...........................................................181
0
Notes .............................................................................................183

Chapter 8. Composition

Composition with conformal automorphisms...............................186
Factorization in the case of finitely many zeros...........................187
Factorization with infinite Blaschke products when α = 1............190
The extreme points of P................................................................192
A product theorem for subordination families..............................192
The closed convex hull of subordination classes..........................193
Composition operators when α > 1...............................................194
Excluded rays and wedges and membership in F ........................197
α
The hereditary property of composition operators for
increasing α...............................................................................199
A Taylor series condition for a composition operator..................200
Notes .............................................................................................202

Chapter 9. Univalent Functions

The closed convex hull of starlike and of convex mappings........204
Prawitz’s inequality......................................................................207
A sufficient condition for membership in F .................................207
2
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Spirallike and close-to-convex mappings.....................................208
Univalent functions which do not belong to F .............................210
2
The maximum modulus and membership in F .............................212
α
Univalent self-maps of the disk and composition operators.........214
Notes .............................................................................................216

Chapter 10. A Characterization of Cauchy Transforms

The one-to-one mapping between measures and functions..........217
Necessary conditions for representation as a Cauchy
transform...................................................................................222
The composition of a subharmonic function and an
analytic function.......................................................................223
The subharmonicity of the special function u ..............................225
p
A key lemma for representation by a real measure.......................227
Sufficient conditions for representation as a Cauchy transform...231
Notes .............................................................................................234

References .............................................................................................237

© 2006 by Taylor & Francis Group, LLC
List of Symbols

Chapter 1.
D the open unit disk......................................................1
⎟ the set of complex numbers......................................1
T the unit circle............................................................1
M the set of complex-valued Borel measures on T.......1
M* the set of probability measures on T.........................1
F the set of fractional Cauchy transforms of order α....1
α
µ the total variation of the measure µ...........................3
Hp the Hardy space.........................................................3
f the norm of f in Hp....................................................3
Hp
F(θ) the radial limit of f in the direction eiθ......................3
f (eiθ) the radial limit of f in the direction eiθ......................3
P the set of normalized analytic functions with
positive real part....................................................4
F* the set of functions in F represented by a
α α
probability measure...............................................5
hp the set of harmonic functions in D with bounded
integral means.......................................................5
A (α) the binomial coefficients...........................................6
n
f the norm of f in F ...................................................11
F α
α
H the space of functions analytic in D.........................13

Chapter 2.
⎧ 1+z⎫
S the function S(z) = exp ⎨− ⎬ ..........................17
⎩ 1−z⎭
Γ the gamma function.................................................17
C the space of continuous functions on T...................23
f the norm of f in C...................................................23
C
B the Besov space of order α......................................30
α
f the norm of f in B ..................................................30
B α
α
f ∗g the Hadamard product of f and g............................41
A the Banach space of functions analytic in D and
continuous in D .................................................44
© 2006 by Taylor & Francis Group, LLC
Chapter 3.
M (r, f ) the integral mean of | f |p.........................................48
p
1
F the function F (z) = ..................................49
α α (1−z)α
M (r, f ) the integral mean of | f | for p = 0............................56
0
D the Dirichlet space of order α..................................57
α
A(Ω) the Lebesgue measure of Ω ⊂⎟.............................57

Chapter 4.
m(E) the Lebesgue measure of E⊂⎥...............................68
S(θ, γ) the Stolz angle with vertex eiθ and opening γ..........72
P the kernel for defining α-capacity...........................78
α
C (E) the α-capacity of E⊂ [–π, π]..................................78
α

Chapter 5.
N the Nevanlinna class...............................................92

Chapter 6.
M the set of multipliers of F .....................................107
α α
M the multiplication operator induced by f...............108
f
f the norm of f in M ................................................109
M α
α
P (z, α) the weighted partial sums of a power series..........113
n
P(r, θ) the Poisson kernel.................................................118

Chapter 7.
T the Toeplitz operator with symbol φ....................134
φ
D(θ,φ) the second difference of f......................................139

Chapter 8.
C the composition operator induced by φ.................185
φ
E F the set of extreme points of F ...............................192
H F the closed convex hull of F ...................................192

Chapter 9.
U the set of analytic univalent functions in D............203
S the set of normalized members of U......................203
S* the set of starlike mappings in S............................203
K the set of convex mappings in S............................203

© 2006 by Taylor & Francis Group, LLC