SYMMETRIZATION, GREEN’S FUNCTIONS, HARMONIC MEASURES

AND DIFFERENCE EQUATIONS

by

ALEXANDER ROBERT PRUSS

B.Sc. (Honours Scholar’s Electives) University of Western Ontario, 1991

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIES

Department of Mathematics We accept this thesis as conforming

to the required standard

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

THE UNIVERSITY OF BRITISH COLUMBIA

April 1996

\begin{matrix}c \\ ◯ \end{matrix} Alexander Robert Pruss, 1996 ii

Ad maiorem Dei gloriam

iii iv

Abstract

Symmetrization methods transform functions or sets into functions or sets having desirable features such as symmetry or some kind of convexity. We consider whether the values of various functionals do or do not increase under the transformation.

First, we answer in the negative a question of W. K. Hayman (1967) on the precise way that Green’s functions increase under circular rearrangement.

Next we study symmetrization techniques in discrete cases, obtaining convolution-rearrangement inequalities of the form

∑_{x,y∈G} f(x)K(d(x, y))g(y) ≤ ∑_{x,y∈M} f^{#}(x)K(d(x, y))g^{#}(y),

for several graphs G, where f^{#} is a carefully chosen rearrangement of a real function f on G. The graphs we work with are primarily the circular graphs ℤ_{n} and the p-regular trees (p ≥ 3). These inequalities allow us to obtain a full analogue of the classical Faber-Krahn inequality in the case of subsets of a p-regular tree.

We show that the convolution-rearrangement inequalities imply results on the effects of discrete Steiner-type symmetrization on harmonic measures and Green’s functions, and we obtain discrete analogues of some of Albert Baernstein II’s (1994) results for partial differential equations.

Then, we consider the collection 𝔅 of holomorphic functions f on the unit disc 𝔻 with f(0) = 0 and \frac{1}{π} ∫∫_{𝔻} |f′(x + iy)|^{2} dx dy ≤ 1. We study qualitative properties (e.g., symmetry or various kinds of convexity) of the extremals of functionals Λ_{Φ} on 𝔅 defined by Λ_{Φ}(f) = \frac{1}{2π} ∫_{0}^{2π} Φ(f(e^{iθ})) dθ, for f ∈ 𝔅 where Φ is a fixed real function on ℂ satisfying various properties.

S.-Y. A. Chang and D. E. Marshall (1985) proved that Λ_{Φ} is bounded on 𝔅 if Φ(z) = e^{|z|^{2}}. While it is not known whether this Λ_{Φ} attains a maximum over 𝔅, we do show that there is a perturbation of Φ for which the corresponding functional does not attain its supremum.

Finally, a study motivated by several conjectures concerning least harmonic majorant functionals and radial rearrangement leads us to obtain, among other things, an improved version of Arne Beurling’s (1933) shove theorem for harmonic measures on slit discs. vi

Table of Contents

Abstract v

Table of Contents vii

List of Figures xi

Acknowledgements xiii

Foreword xv

Introduction xvii

Chapter I. Definitions, background material and introductory results 1 1. Some conventions and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. General definitions for rearrangement of functions and some basic results . . . . . . . . . . 3 3. Hardy spaces, Poisson integrals and further background material . . . . . . . . . . . . . . . . . 13

3.1. Definitions of Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2. Nontangential limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3. The conjugate function and the M. Riesz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4. Disc algebra and BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5. The Nevanlinna class N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4. Subharmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5. Least harmonic majorants, harmonic measures and uniformizers . . . . . . . . . . . . . . . . . . 20

5.1. Dirichlet problem and harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2. Regularity for the Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.3. Least harmonic majorants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.4. Brownian motion and harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.5. The uniformizer and harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.6. Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.7. Riesz’ theorem and representation of least harmonic majorants . . . . . . . . . . . . . 28

6. Some known results in symmetrization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.1. Circular symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2. Symmetric decreasing rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.3. Steiner symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

7. Counterexamples to a question of Hayman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.1. Hayman’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2. The three counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.3. Proofs that the counterexamples truly contradict Hayman’s conjecture . . . . . . 41

8. Radial monotonicity of Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Table of Contents

Chapter II. Discrete symmetrization 49 1. Definitions and basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2. A general framework for proving discrete master inequalities . . . . . . . . . . . . . . . . . . . . . 52 3. The general case of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4. The octahedron edge graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5. The circle graphs ℤ_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6. Regular trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1. The master inequality on regular trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.2. The Faber-Krahn inequality for subsets of regular trees . . . . . . . . . . . . . . . . . . . . . 68

6.2.1. Statement of the Faber-Krahn inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2.2. Some useful well-known results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.2.3. Proof of not necessarily strict inequality in Theorem 6.2 . . . . . . . . . . . . 74 6.2.4. Proof of condition for strict inequality in Theorem 6.2 . . . . . . . . . . . . . . 75

7. Some open problems and two counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.1. How the computer proved Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8. Discrete Schwarz and Steiner type rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.1. Basic definitions and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2. Rearrangement on a product set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3. Symmetrization and preservation of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9. Haliste’s method for exit times, harmonic measures and Green’s functions . . . . . . . 101

9.1. Definitions and statement of results for generalized harmonic measures and Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.1.1. The kernel and the assumptions on it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.1.2. The kernel in our main examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 9.1.3. The random walk on V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.1.4. Generalized harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 9.1.5. Generalized Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.2. Reducing to the case λ = 0 in Assumption 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 9.3. Exit times and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10. A discrete Beurling shove theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 11. A general rearrangement method for difference equations . . . . . . . . . . . . . . . . . . . . . . . . 121

11.1. Our assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.2. A discrete rearrangement theorem for difference inequalities . . . . . . . . . . . . . . . 124 11.3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

11.3.1. Monotonicity of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 11.3.2. Generalized harmonic measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 11.3.3. Exit times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Chapter III. Chang-Marshall inequality, harmonic majorant functionals, and some

nonlinear functionals on Dirichlet spaces 139

1. The Λ_{Φ} and Γ_{Φ} functionals and Dirichlet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 1.1. The Λ_{Φ} functionals on a finite measure space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 1.2. Dirichlet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 1.3. The Γ_{Φ} functionals acting on domains and the Λ_{Φ} acting on holomorphic and

harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2. The Chang-Marshall, Essén and Moser-Trudinger inequalities . . . . . . . . . . . . . . . . . . . 145 3. General results on the Λ_{Φ} functionals on measure spaces . . . . . . . . . . . . . . . . . . . . . . . . 148

3.1. Existence of extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.2. The Λ_{Φ} functionals on balls of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.3. Critically sharp inequalities and nonexistence of extremals . . . . . . . . . . . . . . . . 157

3.3.1. The general results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3.3.2. Application to the Moser-Trudinger inequality . . . . . . . . . . . . . . . . . . . . . 159 3.3.3. Application to the Chang-Marshall inequality . . . . . . . . . . . . . . . . . . . . . 160 3.3.4. Proofs of the results on critically sharp inequalities . . . . . . . . . . . . . . . . 161

4. Properties of extremals of the Λ_{Φ} on Dirichlet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.1. A variational equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.2. Regularity of extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3. The strict analytic radial increase property (SARIP) . . . . . . . . . . . . . . . . . . . . . . 176 4.4. Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

5. Symmetric decreasing rearrangement and Dirichlet norms . . . . . . . . . . . . . . . . . . . . . . . 179 6. Baernstein’s sub-Steiner rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Chapter IV. Radial rearrangement 191

1. Conjectures and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 1.1. The primary conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 1.2. Consequences of a positive answer to Conjecture 1.2 . . . . . . . . . . . . . . . . . . . . . . 193 1.3. Radial rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

2. Some positive results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 3. Transferring the problems to the cylinder and the question of two-sided lengthwise

Steiner symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4. Formulation in terms of Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5. The case where t ↦ \frac{dΦ(e^{t})}{dt} is concave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6. Haliste’s one-sided lengthwise Steiner rearrangement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7. Brownian motion, simple discrete analogues and exit times . . . . . . . . . . . . . . . . . . . . . . 211

7.1. Uniform motion to the right: a counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 7.2. Exit times of Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8. The Beurling shove theorem and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 9. A discrete one-dimensional analogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

9.1. Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9.2. Various useful identities, formulae and some proofs . . . . . . . . . . . . . . . . . . . . . . . . 230 9.3. Proof of the formula for the probability of safe traversal . . . . . . . . . . . . . . . . . . . 237 9.4. The one-dimensional continuous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

10. Horizontal convexity of extremals for some least harmonic majorant functionals . 241 10.1. Step III of the proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 10.2. Step II of the proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 10.3. Step I and the rest of the proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 247

List of notations and symbols 249 1. Rearrangement-type operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 2. Some other operators and relations for sets and functions . . . . . . . . . . . . . . . . . . . . . . . 249 3. Numerical operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 4. Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5. Greek alphabetical index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6. Latin alphabetical index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Table of Contents

Bibliography 255 Appendix A. Source code for cubetern.c 261 Appendix B. Source code for cm.f 267 Index 273

List of Figures

Introduction 0.1. An example of circular symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 0.2. Boundary values for the Dirichlet problem associated with the w_{r} functional on D . xix

Chapter I. 2.1. Steiner symmetrization about the real axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.1. The circularly symmetric domain U_{a} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.2. The unsymmetrized domain U_{abcd} and the symmetrized domain U_{abcd}^{⊚} . . . . . . . . . . . . . . 40 7.3. The unsymmetrized domain U_{ab} and the symmetrized domain U_{ab}^{⊚}, together with the cone

C_{−b,δ} ⊂ U_{ab}^{⊚} used in the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.1. The construction of the set H_{z} for z in the complement of D . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter II. 4.1. The edge graph H_{8} of the octahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.1. The graph ℤ_{11} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2. Symmetrization of subsets of ℤ_{11} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.1. The ordering on the tree T_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2. The extremal subtrees G^{#} of T_{3} with cardinalities from 1 to 12 . . . . . . . . . . . . . . . . . . . . 70 6.3. An extremal subtree G^{#} ⊆ T_{3} with cardinality 21 and the eigenfunction f corresponding

to the first non-zero eigenvalue of −Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4. An extremal subtree G^{#} ⊆ T_{3} with cardinality 14 and once again with the eigenfunction

f corresponding to the first non-zero eigenvalue of −Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.5. Definition of A, B, A′ and B′ in the case where h(w) = h(v) . . . . . . . . . . . . . . . . . . . . . . . 80 6.6. Definition of A, B, A′ and B′ in the case where h(w) = 1 + h(v) . . . . . . . . . . . . . . . . . . . 81 7.1. The cube ℤ_{2}^{3} and the ternary plane ℤ_{3}^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 9.1. Symmetrization on ℤ × ℤ_{11} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2. Steiner symmetrization on ℤ^{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 10.1. An example of the sets H and H_{1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Chapter III. 5.1. The functions f_{1}, f_{2}, f and f^{⊚} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

Chapter IV. 1.1. A multiply connected domain for Example 1.1, with its radial rearrangement . . . . . . 195 1.2. A simply connected domain for Example 1.1. The radial rearrangement of this domain

List of Figures

will be contained in some disc 𝔻(r′) for 0 < r′ < r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.1. The domain W_{T,ε} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.2. The domain W_{T,ε}^{◀} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.1. The decomposition of I into I_{1} and I_{2} in a case where n = 3 . . . . . . . . . . . . . . . . . . . . . . 223 9.1. An example of the original N-tuple and j = 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.1. The graph of f(e^{iθ}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Acknowledgements

I would like to thank Professor Albert Baernstein II for his patience, encouragement, suggestions, advice and for much lengthy electronic mail correspondence with me. I am grateful to Professors Matts Essén and D. E. Marshall for their warm hospitality in Uppsala and Seattle, respectively, and for various discussions by e-mail and in person. I am thankful to Professor Alec Matheson for a number of interesting discussions by e-mail. I wish to thank Professor Dominik Szynal for his hospitality in Lublin.

I wish to express my sincere gratitude to my supervisor, Professor John J. F. Fournier for patiently bearing with my many questions, for proffering useful advice at many points in my research, and for his general and generous care for my mathematical development.

I would also like to express gratitude to Professors Joel Friedman, Gregory F. Lawler and Philip D. Loewen as well as Mr. Ravi Vakil for various discussions.

The research was partially supported by Professor J. J. F. Fournier’s NSERC Grant #4822. xiv

Foreword

Most of Section I.7 is taken and/or adapted from the author’s paper [89] which will appear in the Proceedings of the American Mathematical Society; text and diagrams taken from this paper are copyright \begin{matrix}c \\ ◯ \end{matrix} 1994 The American Mathematical Society. The American Mathematical Society copyright agreement permits such use of the text by the author.

Most of Section II.3.3 is taken and/or adapted from the author’s paper [88] which will appear in the Canadian Mathematical Bulletin; text taken from this paper is copyright \begin{matrix}c \\ ◯ \end{matrix} 1994 The Canadian Mathematical Society. Permission to use the text in the thesis has been secured.

Scattered major portions of the text in Sections III.1–III.4.4 (with a noteworthy and particular exception of the above-discussed Section III.3.3) are taken and/or adapted from the author’s joint paper with Alec Matheson [75] which will appear in the Transactions of the American Mathematical Society; text taken from this paper is copyright \begin{matrix}c \\ ◯ \end{matrix} 1995 The American Mathematical Society. The author thanks Professor Alec Matheson for granting permission for the author to use his own judgement in describing the author’s contribution to the joint work and for granting permission to use the text in the thesis. The American Mathematical Society copyright agreement permits such use of the text by the authors.

Introduction

Most of Section IV.9 is taken from the author’s paper [87] which will appear in the Annales de l’Institut Henri Poincaré – Probabilités & Statistiques. As of the date of submission of this thesis, the copyright in this paper remains vested in the author. Permission of the journal editor to use the text in this thesis has nonetheless been secured. xvi We study symmetrization and certain nonlinear functionals on collections of sets or functions.

Symmetrization theory strives to replace a given object (function or set) by one which is somehow related to it so that (a) the replacement object has some desirable features, such as symmetry or some kind of convexity, and (b) the replacement “improves” the values of various functionals associated with the object. Note that given a symmetrization method which replaces sets with sets, we can often automatically get a method which replaces functions with functions by applying the method to the level sets of the functions and reassembling the symmetrized level sets to form a new function.

A typical kind of desirable feature that fits under (a) is circular symmetry: a subset of the complex plane is said to be circularly symmetric if the intersection of this subset with every circle centred about the origin is a connected arc centred about the positive real axis. Thus, circular symmetry includes reflection symmetry about the real axis and a kind of angular convexity condition. Given a measurable set S in the complex plane, we can define the circular symmetrization S^{⊚} by requiring that S^{⊚} be circularly symmetric and satisfy the following three conditions for every r ≥ 0:

(i) the circle of radius r about the origin is contained in S^{⊚} if and only if it is contained in S

(ii) the angular measure of the intersection of the circle of radius r about the origin with S^{⊚} equals the angular measure of the intersection of the same circle with S

(iii) the intersection of S^{⊚} with the circle of radius r about the origin is open in that circle.

See Figure 0.1. Condition (ii) implies that the area of S^{⊚} equals that of S.

We now describe the kinds of functionals that interest us in connection with circular symmetrization. Let D be a domain containing the origin (i.e., a connected open subset of ℂ with 0 ∈ D). Let ϕ: [0, ∞) → [−∞, ∞) be such that t ↦ ϕ(e^{t}) is convex and increasing (in particular if ϕ is convex and increasing then this condition will hold). Let h be the infimum of all real functions g which are harmonic on D and satisfy g(z) ≥ ϕ(|z|) for all z in D. Then put Γ_{ϕ}(D) = h(0).

An alternate probabilistic description of Γ_{ϕ}(D) is as follows. Let B_{t} be a Brownian motion in the complex plane starting at the origin. Let τ = inf{t ≥ 0 : B_{t} ∉ D} be the first time that the Brownian motion leaves D. Then Γ_{ϕ}(D) is the expected value of ϕ(|B_{τ} |). It is thus a kind of weighted average of ϕ(|z|) as z ranges over the boundary of D, with the weight at a point being given by the probability density that a Brownian motion first impacts the boundary at that point.

An important result on the Γ_{ϕ} functionals is a consequence of a theorem of Baernstein’s groundbreaking paper [7] and says that Γ_{ϕ}(D) ≤ Γ_{ϕ}(D^{⊚}). Hence, circular symmetrization increases the above-defined Γ_{ϕ} functionals. Circular symmetrization also increases convex circular means of Green’s functions. The Green’s function of a sufficiently regular domain D with pole at w ∈ D is defined to be the (unique) function g(·, w; D) vanishing on the boundary of D, harmonic on D\\{w\} and such that z ↦ g(z, w; D) − log \frac{1}{|z−w|} remains harmonic in a neighbourhood of w.

Figure 0.1: An example of circular symmetrization

Baernstein’s theorem [7] then states that for every r ≥ 0 we have

∫_{0}^{2π} ψ(g(re^{iθ}, w; D)) dθ ≤ ∫_{0}^{2π} ψ(g(re^{iθ}, |w|; D^{⊚})) dθ,

for every convex increasing function ψ. This is of particular interest in the case of w = 0. The case of w = 0 and ψ(t) = t implies the above-mentioned result on the increase of the Γ_{ϕ} under circular symmetrization. An interesting question is whether we may take ψ to be any increasing function, not necessarily convex. This was asked by Hayman [59]. The answer turns out to be negative, even in the case of w = 0, as we shall prove in §I.7.

A second set of interesting functionals are the w_{r} harmonic measure functionals. Fix r > 0. Let D be a domain containing the origin. Let D_{r} be the intersection of D with the disc of radius r centred at 0. Then, let h be the harmonic function on D_{r} which equals 1 on the parts of the boundary of D_{r} which lie on the circle of radius r about the origin (i.e., h equals 1 on (∂D_{r}) ∩ {z ∈ ℂ : |z| = r}) and which equals 0 on all the other parts of the boundary of D_{r}. (See Figure 0.2.) Put w_{r}(D) = h(0). A probabilistic interpretation of this functional is that it is the probability that a Brownian motion starting at the origin reaches the circle of radius r centred at 0 before impacting on any other part of the boundary of D. Hence, intuitively it measures how easy it is for a Brownian particle to reach the circle of radius r while staying in D.

Once again, Baernstein [7] has shown that w_{r}(D) ≤ w_{r}(D^{⊚}) and hence circular symmetrization increases the w_{r} functionals. The intuitive reason for this in terms of the probabilistic interpretation is that the circular symmetrization straightens out and consolidates the roads leading from 0 to the circle of radius r; see Figure 0.1.

Figure 0.2: Boundary values for the Dirichlet problem associated with the w_{r} functional on D

In Chapter I we shall set up definitions and give some theorems on general and specific symmetrization methods, as well as cite and summarize some material on the notions needed to define and study our functionals. We shall also give the most important results of Baernstein’s famous paper [7], as well as our answers to the question of Hayman [59] mentioned above. Finally, we shall conclude Chapter I by obtaining a lower bound on the size of the set on which the function g(·, 0; D) is radially decreasing; this will be useful to us in Chapter IV.

In Chapter II, we shall consider symmetrization theory in discrete settings. For instance, we shall prove a generalization of a full analogue of Baernstein’s above-cited results on the increase in Green’s functions and w_{r}-functionals under circular symmetrization in the setting of subsets of the discrete cylinder ℤ × ℤ_{m}. (Note that special cases of our results can be proved by the methods of Quine [90].) Of course the discrete cylinder is a discrete version of the continuous cylinder ℝ × {z ∈ ℂ : |z| = 1}, which is conformally equivalent to the punctured plane ℂ\{0} under a natural exponential conformal equivalence, and Baernstein’s results can be lifted to the continuous cylinder, which is why we can say that our results on the discrete cylinder are analogous to his theorems.

The method of Chapter II proceeds by proving convolution-symmetrization inequalities of the form

∑_{x,y∈G} f(x)K(d(x, y))g(y) ≤ ∑_{x,y∈G} f^{#}(x)K(d(x, y))g^{#}(y),

where G is a specific connected graph such as ℤ_{m}, d is the shortest-distance metric on G, K is a decreasing function and f^{#} and g^{#} indicate symmetrizations of arbitrary functions f and g on G. Symmetrizations inducing such inequalities do not exist for all graphs; indeed, we shall use a computer-based proof to show that they do not exist for the cube G = ℤ_{2}^{3} or for the ternary plane G = ℤ_{3}^{2}. However, we shall prove the inequalities in the special cases of the circular graph G = ℤ_{m}, the p-regular tree G = T_{p} and the octahedron edge graph G = H_{8}. And, of course, the case of the linear graph G = ℤ goes back to Hardy and Littlewood (see [58, Thm. 371]). Our method for proving these kinds of convolution-symmetrization inequalities will be a discrete version of a method of Beckner [18, 19, 20, 21], and Baernstein and Taylor [15], as presented by Baernstein [11]. This method in the discrete setting is in fact completely elementary.

Given appropriate convolution-symmetrization inequalities, we then get several consequences. First, given the case of G = T_{p} we obtain a full Faber-Krahn inequality for subsets of the p-regular tree. Recall that the classical Faber-Krahn [47, 68] inequality stated that out of all domains D in ℝ^{n} of a fixed area, the first non-zero Dirichlet^{1} eigenvalue of the negated Laplacian −Δ = − ∑_{i=1}^{n} \frac{∂^{2}}{∂x_{i}^{2}} is minimized precisely for a ball. Our Faber-Krahn inequality will provide a characterization of the subsets D of T_{p} of a given size which minimize the first non-zero eigenvalue of the negated combinatorial Laplacian with the Dirichlet boundary condition that our functions vanish outside our subset D. Not surprisingly, these subsets look like a discrete tree version of the balls which were optimal in the classical Faber-Krahn case (see Figure II.6.3 on p. 76). Our proofs, are completely elementary, although the proof of the uniqueness of the optimal subsets is quite involved.

Secondly, given a convolution-symmetrization inequality on a constant degree graph G, we can, as already mentioned, obtain analogues of Baernstein’s results. Indeed, we shall obtain results on symmetrization for domains which are subsets of ℤ × G (where G is ℤ, ℤ_{m}, T_{p} or H_{8}); these will concern the increase in certain generalized harmonic measures and generalized Green’s functions. In fact, we shall proceed by two methods. Firstly, we shall prove these results via a probabilistic approach going back to Haliste [56]. Secondly, we shall show even more general results for difference equations on ℤ×G (even for some non-linear ones) by a modified version of a method of Baernstein [11] and Weitsman [99] who proved such results for partial differential equations.

In Chapter III we shall engage in an analysis of various functionals on some collections of holomorphic functions. For instance, let 𝔅 be the collection of all functions f holomorphic on the unit disc {z ∈ ℂ : |z| < 1} which satisfy f(0) = 0 and have

\frac{1}{π} ∫∫ _{𝔻} |f′(x + iy)|^{2} dx dy ≤ 1.

Note that |f′(x + iy)|^{2} is the Jacobian of the mapping f, and hence the displayed condition can be interpreted as saying that the area of the image of f counting multiplicities does not exceed π. If f is one-to-one then this condition simply says that the area of the image of f does not exceed π. Let Φ be a Borel-measurable function on ℂ. Put

Λ_{Φ}(f) = \frac{1}{2π} ∫_{0}^{2π} Φ(f(e^{iθ})) dθ,

1The Dirichlet eigenvalues are obtained by having −Δ act on those functions on our domain D which vanish on the boundary of D.

for f ∈ 𝔅. (This makes sense since functions in 𝔅 have radial limits almost everywhere on the unit circle.) If ϕ is a Borel-measurable function on [0, ∞), then we abuse notation and write Λ_{ϕ} for Λ_{Φ} where Φ(z) = ϕ(|z|).

The Λ_{ϕ} functionals are closely related to the Γ_{ϕ} functionals. Indeed, we have the inequality

Λ_{ϕ}(f) ≤ Γ_{ϕ}(D)  (0.1)

whenever ϕ: [0, ∞) → [−∞, ∞) is such that t ↦ ϕ(e^{t}) is convex and increasing while D is the image of f (assuming f is non-constant). In the displayed inequality, we will actually have equality if f is one-to-one. Moreover, if f ∈ 𝔅 then Area D ≤ π. Hence, there are interesting connections between the Λ_{ϕ} functionals on 𝔅 and the Γ_{ϕ} functionals on B, where B is the collection of all domains containing the origin and having area at most π.

The result that had sparked our interest in the Λ_{ϕ} functionals was the Chang-Marshall inequality [32] (another proof was later given by Marshall [72]) which says that

sup_{f∈𝔅}Λ_{ϕ}(f) < ∞,  (0.2)

if ϕ(t) = e^{t^{2}} for t ∈ [0, ∞). This inequality is rather difficult to prove and very sharp. Indeed, we shall show that if ϕ is a function on [0, ∞) such that sup_{f∈𝔅} Λ_{ϕ}(f) < ∞ then there exists a C < ∞ such that for all t ∈ [0, ∞) we have ϕ(t) ≤ Ce^{t^{2}}. The Chang-Marshall inequality was improved by Essén [44] to assert that

sup_{D∈B} Γ_{ϕ}(D) < ∞,  (0.3)

where ϕ(t) again equals e^{t^{2}}. (That this is actually an improvement follows from (0.1).)

We spend some time in Chapter III examining the Λ_{Φ} functionals for their own sake. We shall in particular be interested in the extremals of these functionals, namely functions f ∈ 𝔅 such that Λ_{Φ}(f) ≥ sup_{g∈𝔅} Λ_{Φ}(g). For instance, as a generalization of a result of Matheson [73], we have a theorem that if Φ is upper semicontinuous on ℂ and Φ(z) = o(e^{|z|^{2}}) as |z| → ∞ (uniformly in arg z), then Λ_{Φ} has an extremal on 𝔅. We also obtain some results on the Λ_{ϕ} functionals in a few other settings such as analogous functionals acting on balls of Hilbert spaces of measurable functions.

The question of existence of an extremal function is a quite interesting one in the case of the Chang-Marshall inequality. Indeed it is still not known whether Λ_{ϕ} has an extremal if ϕ(t) = e^{t^{2}}. We shall prove, however, that even if it does, still there exists an infinitely differentiable convex increasing function ϕ_{1}(t) satisfying ϕ_{1}(t) ≤ e^{t^{2}} for all t ∈ [0, ∞) such that ϕ_{1}(t)/e^{t^{2}} → 1 as t → ∞ but Λ_{ϕ_{1}} has no extremal. On the other hand, we shall also show that there exists an infinitely differentiable convex increasing function ϕ_{2}(t) satisfying ϕ_{2}(t) ≥ e^{t^{2}} for all t ∈ [0, ∞) and again such that ϕ_{2}(t)/e^{t^{2}} → 1 as t → ∞, but this time with Λ_{ϕ_{2}} achieving a maximum. These results will show that the existence or nonexistence of an extremal function in 𝔅 for Λ_{ϕ} if ϕ(t) = e^{t^{2}} depends on the precise nature of the function e^{t^{2}} and not just on its asymptotics, and that this existence or nonexistence is not stable under perturbations of ϕ.

Given an extremal for Λ_{Φ}, we will under some hypotheses on Φ be able to obtain a variational equation that this extremal will have to satisfy. The variational equation is not a differential equation, but involves a pseudo-differential operator, which complicates the analysis. However, the equation is good enough to yield qualitative results. For example, jointly with Alec Matheson [75] it has been shown that if Φ is infinitely differentiable on ℂ then the extremals, if they exist, must extend to be infinitely differentiable functions on the unit circle (recall that they were assumed to be holomorphic inside the unit disc; this did not by itself say much about their regularity on the boundary of the unit disc). We will see that some of the results mentioned in this paragraph continue to work in more general settings such as α-weighted Dirichlet spaces.

Finally, we wrap up Chapter III by connecting the results with symmetrization theory. We shall prove a result on the relation of α-weighted Dirichlet norms (0 < α < 2) and symmetric decreasing rearrangement, and use this result to prove that if f is extremal for Λ_{Φ} where Φ(z) is of the form ϕ(Re z) for ϕ a convex function on ℝ strictly convex at 0, then f is one-to-one and has a Steiner symmetric image. We also present an alternative rearrangement method due to Baernstein replacing Steiner symmetrization but keeping some of its desirable properties. We shall examine the relation between this alternative method and Steiner symmetrization.

In Chapter IV we come back to symmetrization theory. This time, our interest is in replacing a domain D by a domain D̃ of not larger area which is simply connected and hopefully starshaped^{2} as well, satisfying Γ_{ϕ}(D̃) ≥ Γ_{ϕ}(D) for all ϕ such that t ↦ ϕ(e^{t}) is convex and increasing. We are unable to determine if such a replacement exists. If it does exist, then this shows that Essén’s inequality (0.3) is a consequence of the Chang-Marshall inequality (0.2). This is so because of the relation between the Λ_{ϕ} and Γ_{ϕ} functionals for one-to-one functions and simply connected domains. Moreover, if such a D̃ exists then we can obtain results on the existence of extremal domains in B for the Γ_{ϕ} functionals.

In fact, we make an explicit conjecture on what we think D̃ should be, namely a combination of circular symmetrization (which by itself does not always produce simply connected domains) and Marcus’ radial rearrangement [70] (which may sometimes decrease the Γ_{ϕ} if applied by itself, as we shall show). If our conjecture is correct, then moreover we have w_{r}(D) ≤ w_{r}(D̃). While this weaker inequality is also still open, we shall use another method of Haliste [56] to prove that w_{r}(D) ≤ w_{r}(D̃) for our conjectured choice of D̃ if D is simply connected. Of course, the main point in the construction of D̃ was to obtain a simply connected domain so that if D is a priori simply connected, we do not gain much. Although we do gain something: our D̃ is star-shaped, whereas D need not be.

We shall also discuss some other partial results giving support to our conjectures. For instance we shall note that it essentially follows from known results that Γ_{ϕ}(D) ≤ Γ_{ϕ}(D̃) if t ↦ \frac{dϕ(e^{t})}{dt} is concave (which is of course a major restriction). We also consider some generalizations of Beurling’s shove theorem [23]. These will allow us to, for instance, prove our full conjecture on the increase of the Γ_{ϕ} for domains D of the form 𝔻\I, where 𝔻 is the unit disc and I is a finite collection of closed intervals on the negative real axis.

We shall also describe what our conjectures look like when transferred to the continuous cylinder. In particular, our conjectures can be transformed so as to concern Brownian motion on the cylinder. We shall show that the transformed version of the weaker w_{r}(D) ≤ w_{r}(D̃) conjecture does not hold if the lengthwise component of the Brownian motion on the cylinder is replaced by a uniform motion to the right. This would seem to provide evidence against our conjectures, but the counterexample in fact satisfies the original w_{r}(D) ≤ w_{r}(D̃) conjecture as it comes from a simply connected domain D.

2A set D ⊆ ℂ is star-shaped if for all z ∈ D the line segment joining z with 0 lies in D.

Then, we shall discretize and collapse the cylinder to the set ℤ. We formulate a simple analogue of the w_{r}(D) ≤ w_{r}(D̃) conjecture as a conjecture on a random walk on {1, 2, …, N + 1} with a reflecting boundary condition at \frac{1}{2} and with dangers distributed on the points 1, 2, …, N. The latter conjecture then concerns the optimal distribution of the dangers so that the random walk has greatest chance of surviving to reach N + 1 when started at 1. We shall prove this conjecture (which incidentally is a variation on a problem considered by Essén [43]) via an involved but completely elementary method.

Now, our main conjectures were formulated for functionals Γ_{ϕ} defined as infima of the values at 0 of the harmonic majorants of ϕ(|z|). We can likewise consider similar functionals which are infima of the values at 0 of the harmonic majorants of ϕ(Re z) for convex ϕ. In the final section of our thesis we prove a result in this case, analogous to a weak form of our conjectures. The proof of this result connects in a crucial way with the work of Chapter III by using the variational expression for the extremals of the Λ_{Φ}. Moreover, the proof uses either Steiner symmetrization discussed in Chapter I or Baernstein’s alternative to it discussed in Chapter III.

Note on numbering and organization

Theorems are numbered in the form X.Y, where X is the section number and Y the theorem number. A reference to “Theorem X.Y ” refers to theorem Y of section X of the current chapter. A reference to “Theorem A.X.Y ” refers to theorem Y of section X of chapter A. Note that A ∈ {I, II, III, IV}, while X and Y are Arabic numerals. What has just been said about theorems also applies to lemmas, definitions, remarks, etc. Sections are sometimes subdivided into subsections and even sub-subsections. For instance, a reference to §2.1.4 would refer to sub-subsection 4 of subsection 1 of section 2 of the current chapter. A reference to §III.2.3 would refer to subsection 3 of section 2 of chapter III. Note that theorems, lemmas, etc., are not renumbered by subsections or sub-subsections, but their numbering is reset precisely each time one comes to a new section.

Each chapter begins with an overview.

Some possible reading tracks

A reader whose only interest is symmetrization theory can read Chapters I, II and IV as well as sections I.1.2, III.5 (omitting Corollary III.5.2) and III.6 (omitting Corollary III.6.2). The only adverse effect on symmetrization theory that this will have is that some of the motivations in §IV.1.2 for various conjectures will be obscure and some of the proofs in §IV.10 will refer to unread material.

A reader whose only interest lies in discrete symmetrization theory can read §I.1, §I.2, all of Chapter II as well as §IV.9. In doing so, no necessary background material will be omitted, though it may be desirable to read sections I.5 and I.6 to see continuous versions of some of the discrete theorems; these continuous versions are from time to time parenthetically mentioned in the discussion of the discrete results.

A reader whose only interest is in the Λ_{Φ} and Γ_{Φ} functionals as well as in the Chang-Marshall and Essén inequalities can read §I.3–I.5 together with §III.1–III.4.4. The omission of background material on rearrangements will necessitate the dropping of §III.5, §III.6 and §III.10 which do contain some information on the functionals.

If a reader is not interested in discrete symmetrization theory, Chapter II can be omitted with no loss of continuity. Likewise, so can §IV.9, although if the latter is dropped then the proof of Theorem IV.7.1 will have to remain a mystery.

There are a few results throughout the thesis which are of some interest in and of themselves and which are independent of other work:

(i) A reader interested in an answer to Hayman’s problem on circular symmetrization and Green’s functions and already familiar with the basic notions of symmetrization need only read §I.7.

(ii) A reader already familiar with basic notions about Green’s functions and the Riesz representation theorem for subharmonic functions, and whose interest lies in the radial monotonicity properties of the Green’s function need only read §I.8.

(iii) A reader interested only in the Faber-Krahn inequality on regular trees need only read sections 1–3 and 6 in Chapter II.

(iv) A reader interested in the results on the existence/nonexistence of extremal functions for perturbations of the Chang-Marshall or Moser-Trudinger inequality need only read sections 2, 3.1, 3.2 and 3.3 of Chapter III together with the basic definitions in §III.1. A reader already familiar with the Chang-Marshall or Moser-Trudinger inequality and interested only in the existence/nonexistence result for perturbations will only really need a quick glance at the sections other than §III.3.3.

(v) A reader whose only interest lies in the elementary combinatorial rearrangement result for random walks in a dangerous blind alley need only read §IV.9.

Chapter I

Definitions, background material and introductory results

Overview

After having given some basic conventions (§1), we begin the thesis proper by defining the notion of a rearrangement method and stating a few useful and simple results on rearrangements (§2). Of particular importance will be the fundamental Hardy-Littlewood rearrangement inequalities (Theorems 2.3 and 2.4) which, following Kawohl [65], we shall give for all measure-preserving rearrangement methods. A standard construction central to §2 will be the rearrangement of a function based upon the rearrangement of its level sets (equation (2.1)). Then, we shall review basic results on Hardy spaces on the unit disc and on Poisson integrals there (§3). Then, we discuss basic notions concerning subharmonic functions (§4). Then, in §5 we shall review such notions as harmonic measure, least harmonic majorants, the connection between Brownian motion and harmonic measure, the uniformizer, Green’s functions and finally the Riesz decomposition theorem for subharmonic functions (Theorem 5.9). These notions and standard results will be used in the subsequent parts of Chapter I and in Chapters III and IV. Having reviewed these notions, we then consider in §6 the circular and Steiner rearrangements as well as the symmetric decreasing rearrangement, give a few very basic facts, and then discuss various results of Baernstein [7], [9], [14], Beckner [20] and Haliste [56].

After these basic results and reviews, we proceed in §7 to give a negative answer to a question of Hayman [59] concerning circular rearrangements and Green’s functions. Note that §7 is essentially taken from the author’s paper [89]. Then, we discuss the question of how large we can take the set on which the Green’s function is radially decreasing away from its pole (§8). In particular, we shall show that this happens on the largest disc centred on the pole and contained in the domain in question (Theorem 8.1). In fact this will usually happen on a larger set, and we obtain another lower bound on this set (Theorem 8.2). While our result here may be of some interest in and of itself, the reason we give it is because we will use it in §IV.8 where we shall obtain an improved version of Beurling’s shove theorem [23]. The material of §8 is adapted from the author’s paper [84].

Chapter I. Definitions, background material and introductory results

1. Some conventions and notations

We have ℤ^{+} = {1, 2, … }, ℤ^{−} = {−1, −2, … }, ℤ_{0}^{+} = {0}∪ℤ^{+}, ℤ_{0}^{−} = {0}∪ℤ^{−} and ℤ = ℤ_{0}^{+}∪ℤ^{−}.

The terms “positive” and “negative” shall be taken to mean “non-negative” and “non-positive”, respectively. Likewise, the terms “increasing”, “decreasing”, “decrease”, “increase”, “smaller” and “greater” shall be understood in the weaker sense. When we wish to make a stronger statement, we will explicitly add an auxiliary term such as “strictly”.

The term “countable” means “at most countable”.

We shall use the term domain to mean any non-empty connected open subset of ℂ. We call a set in the plane convex if whenever z and w are two points in it, then the line segment joining z and w lies in this set. We call a set star-shaped if whenever z is a point in the set, then the line segment joining z and 0 lies in this set. We call a set horizontally convex if whenever z and w are two points in this set lying on the same horizontal line (i.e., having Im z = Im w), then the line segment joining z and w lies in the same set.

Measurable functions and semicontinuous functions are assumed to be almost everywhere finite unless otherwise provided for.

We shall use | · | for absolute values of numbers, for cardinalities of discrete sets, and for (sometimes normalized) Lebesgue measures in non-discrete settings. We write

supp f = \overline{\{x : f(x) ≠ 0\}}

for the support of a function f defined on a topological space. If f is defined on a discrete set, then supp f = {x : f(x) ≠ 0}.

Let 𝔻(w; r) = {z ∈ ℂ : |w−z| < r} be the disc of radius r centred on w. Put 𝔻(r) = 𝔻(0; r) and let 𝔻 = 𝔻(1) be the unit disc. Put 𝕋(w; r) = {z ∈ ℂ : |z − w| = r} for r ≥ 0. Set 𝕋(r) = 𝕋(0; r) and let 𝕋 = 𝕋(1) be the unit circle. Note that 𝕋(0) = {0}.

We write A^{c} for the complement of a set A and 2^{A} for its power set, namely 2^{A} = {B : B ⊆ A}. Put A\B = {a ∈ A : a ∉ B}. If A is a collection of sets, write

⋃ A = ⋃_{A∈A} A = {x : ∃A ∈ A. x ∈ A}.

If A is a set then write 1_{A} for the function which is 1 on A and 0 outside A. If P is a proposition, then put 1_{{P }} = 1 if P is true and 1_{{P }} = 0 if P is false.

Write δ_{x,y} = 1_{{x=y}} for the Kronecker δ.

Put t^{+} = max(t, 0) and t^{−} = (−t)^{+} so that t = t^{+} − t^{−} for all t ∈ ℝ. For a real function f write f^{±} for the function x ↦ (f(x))^{±}. Write sgn x = x/|x| if x ≠ 0. The choice of sgn 0 will never be relevant. Write arg z for any choice of the argument of a complex number z so that z = |z|e^{i arg z}.

For a subset S of ℂ and a complex number λ we write

λS = {λz : z ∈ S}.

2. Rearrangements and some basic results

Likewise,

λ + S = {λ + z : z ∈ S}.

The terms “analytic” and “holomorphic” are synonymous for us. The term univalent means “one-to-one and holomorphic”.

A real function f on an open interval (a, b) is said to be convex if whenever a < α ≤ β < b and t ∈ [0, 1] then

f(tα + (1 − t)β) ≤ tf(α) + (1 − t)f(β).

It is said to be strictly convex at x ∈ (a, b) if whenever a < α < β < b and t ∈ (0, 1) are such that x = tα + (1 − t)β then

f(x) < tf(α) + (1 − t)f(β).

Given a function Φ on ℝ^{n}, we write Φ_{,j} = \frac{∂Φ}{∂x_{j}} for the partial derivative of Φ with respect to x_{j} (j = 1, …, n).

Given a measure space (I, μ), we say that the measurable functions f_{n} on I converge in measure to a measurable function f on I providing

lim_{n} μ{x ∈ I : |f(x) − f_{n}(x)| > ε} = 0

for every fixed ε > 0.

2. General definitions for rearrangement of functions and some basic results

We are interested in ways of rearranging a set so as to make it more symmetric, or at least so as to give it some special property such as star-shapedness. We wish to formulate the definitions in the greatest generality we can.

Definition 2.1. Let F be a collection of subsets of a set X. We call F a σ-pseudotopology providing:

(i) ∅ ∈ F,

(ii) X ∈ F, and

(iii) whenever A_{1} ⊆ A_{2} ⊆ A_{3} ⊆ ⋯ are members of F, then ⋃_{n=1}^{∞} A_{n} ∈ F.

A σ-pseudotopology is essentially a one-sided monotone class.

Clearly, any topology is a σ-pseudotopology. Likewise, a σ-algebra is a σ-pseudotopology. Now let F and G be σ-pseudotopologies on X and Y, respectively.

Definition 2.2. A map #: F → G is a rearrangement providing:

(i) we have A^{#} ⊆ B^{#} whenever A and B are two sets in F such that A ⊆ B,

(ii) we have

( ⋃_{n=1}^{∞} A_{n})^{#} = ⋃_{n=1}^{∞} A_{n}^{#},

whenever A_{1} ⊆ A_{2} ⊆ A_{3} ⊆ ⋯ are sets in F, and

(iii) ∅^{#} = ∅ and X^{#} = Y.

Remark 2.1. Note that condition (i) actually follows from condition (ii), but we retain the definition as above for clarity.

Given an extended real-valued function f on X, define the level set at height λ to be

f_{λ} =^{def}{x ∈ X : f(x) > λ}.

Likewise, for an extended real-valued g on Y, put

g_{λ} = {y ∈ Y : g(y) > λ}.

Remark 2.2. Let f be any extended real function. Then,

f^{+}(x) = ∫_{0}^{∞} 1_{f_{λ}}(x) dλ

as can be easily seen. Likewise,

f^{−}(x) = ∫_{−∞}^{0}(1 − 1_{f_{λ}}(x)) dλ.

Thus, if f is any extended real function then

f(x) = ∫_{0}^{∞} 1_{f_{λ}}(x) dλ + ∫_{−∞}^{0}(1_{f_{λ}}(x) − 1) dλ.

Note that this last expression always makes sense, because if one of the two terms on the right side is ±∞ then the other vanishes. Remark 2.3. Let f and g be any extended real functions on a set X. Then f ≡ g if and only if f_{λ} = g_{λ} for every λ ∈ ℝ. This follows immediately from the preceding remark.

Definition 2.3. Say that f is F-lower measurable providing f_{λ} ∈ F for every λ ∈ ℝ.

If F is a topology, then F-lower measurability agrees with F-lower semicontinuity. If F is a σ-algebra, then f is F-lower measurable if and only if it is F-measurable.

Now, for an F-lower measurable extended real-valued function f on X let

f^{#}(y) = sup{λ : y ∈ (f_{λ})^{#}},  (2.1)

where #: F → G is a rearrangement. This is a standard and well-known technique for taking a rearrangement of sets and making it into a rearrangement of functions by rearranging the level sets of the function and then reassembling the rearranged level sets into a new function.

Theorem 2.1. Let f be an extended real F-lower measurable function on X and let #: F → G be a rearrangement. Then for every λ ∈ [−∞, ∞] we have

(f^{#})_{λ} = (f_{λ})^{#}.  (2.2)

In particular, f^{#} is G-lower measurable.

Proof. Fix λ ∈ [−∞, ∞) first. Then from the definition of f^{#}(y) we see that f^{#}(y) > λ if and only if there exists a ρ > λ such that y ∈ (f_{ρ})^{#}. Let ρ_{n} = λ + \frac{1}{n}. If there is a ρ > λ such that y ∈ (f_{ρ})^{#} then choose n ∈ ℤ^{+} so that ρ_{n} ∈ (λ, ρ]. Then we have (f_{ρ})^{#} ⊆ (f_{ρ_{n}})^{#} by property (i) of rearrangements since evidently f_{ρ} ⊆ f_{ρ_{n}} as ρ ≥ ρ_{n}. Hence, y ∈ (f_{ρ_{n}})^{#}. Since all the ρ_{n} are strictly larger than λ, it follows that f^{#}(y) > λ if and only if there exists n ∈ ℤ^{+} such that y ∈ (f_{ρ_{n}})^{#}. But f^{#}(y) > λ if and only if y ∈ (f^{#})_{λ}. Thus,

\begin{aligned}(f^{#})_{λ} & = \{y : ∃n ∈ ℤ^{+}. y ∈ (f_{ρ_{n}})^{#}\} \\ & = ⋃_{n=1}^{∞}(f_{ρ_{n}})^{#} \\ & = ( ⋃_{n=1}^{∞} f_{ρ_{n}})^{#} , \end{aligned}  (2.3)

where we have used property (ii) of rearrangements since f_{ρ_{1}} ⊆ f_{ρ_{2}} ⊆ f_{ρ_{3}} ⊆ ⋯ as ρ_{1} ≥ ρ_{2} ≥ ρ_{3} ≥ · · ·. But, it is easy to see from the definition of f_{λ} that

f_{λ} = ⋃_{n=1}^{∞} f_{ρ_{n}},

so that by (2.3) we see that (f^{#})_{λ} = (f_{λ})^{#} for all λ ∈ [−∞, ∞) as desired.

The remaining case is if λ = ∞. But then