Faculté des Sciences, 4 Avenue Ibn Battouta B.P. 1014 RP, Rabat – Maroc
N° d’ordre 2892
THÈSE DE DOCTORAT
Présentée par
Nom et Prénom : Hatim NAQOS
Discipline: Mathématiques
Spécialité: Analyse
Titre:
Asymptotic Behavior of Toeplitz and composition operators
Soutenue le 19 Juillet 2016
Devant le jury
Président :
Ali ALAMI IDRISSI PES, Faculté des sciences de Rabat.
Examinateurs :
Omar EL-FALLAH PES, Faculté des sciences de Rabat.
M'hammed BENLARBI DELAI PES, Faculté des sciences de Rabat.
EL Hassan ZEROUALI PES, Faculté des sciences de Rabat.
Abdelhamid BOUSSEJRA PES, Faculté des sciences de Kenitra.
Acknowledgements
The work of this doctoral thesis was performed in the laboratory ”Analyse et Applications” in the Faculty of Science of Rabat.
My first thanks go to my supervisor, Professor Omar El-FALLAH, Professor at the Faculty of Sciences of Rabat, who governed me during years of thesis and during my Masters project, guiding my first steps in the world of research and making me discover the great wealth of functional analysis and operator theory. Through-out these years. He has always proposed very interesting research topics while giving me great freedom in my choices and my actions, supporting each of my projects. His patience, his pedagogy, his availability, his encour-agement and the confidence granted to me, have contributed to a very large extent, has the complete fulfillment of my work and the culmination of this memory thesis. It is for all these beautiful things I thank him very much.
My sincere thanks to Professor Ali ALAMI IDRISSI, Professor at the Faculty of Sciences of Rabat. It is an honor for me that he chairs the examining board.
I sincerely thank Professor Mhammed BENLARBI DELAI, Professor at the Faculty of Sciences Rabat, for agreeing to perform an report on this thesis and participate on my examining board.
I extend sincere thanks to Professor EL Hassan ZEROUALI, Professor at the Faculty of Sciences of Ra-bat, for agreeing to participate in my thesis committee. For his moral and practical support, his human quality, his availability and his scientific participation.
I wish to thank Professor Abdelhamid BOUSSEJRA, Professor at the University Ibn-Tofail Faculty of Science Kenitra, who agreed without hesitation to be part of the examining board. I thank him very much for evaluating and giving their precious remarks in order to ameliorate my modest work.
I warmly thank Professor Karim KELLAY, Professor at the University of Bordeaux, for agreeing to report on this thesis. for his availability, his patience, his advice, and his scientific rigor. I also want to thank him for his kindness and encouragements.
I am very grateful to Professor Houssam MAHZOULI, Professor at the Faculty of Sciences of Rabat, for his help, his encouragement, and of course for all the information that he shared with me. It is a great pleasure to work with him.
I would like to express here my acknowledgments to all the professors of the laboratory ” Analyse et Ap-plications ”. I also want to thank all members of the laboratory for the benefice discussions we have had on our different areas of mathematical researches. Also, I thank every person who contributes to the elaboration of this work, they welcomed me and encouraged me and made me feel as a part of their big family. I will forever remember in my heart the moments shared with me during these years.
A special thanks goes to the partners that I work with in the laboratory especially Ibrahim MARRHICH and Mohammed EL IBBAOUI.
I want to express my thanks to all my friends. In particular: Driss AADI, Abdelmalek ARABAT, As-maa BANANI, Mohammed ElKASSIMI, Youssef ELMADANI, Tarik ELMESSAOUDI, Ahmed JA, Mouad MAHRI, Ibrahim MARRHICH, Zakariae MOUHCINE, Hajar NEAAIMA, Mohammed RISSOULI, Abdel-ghani SALHI, Nousaiba TANANI and Mohammed ZIYAT . . .
Last but not the least, I thank my parents Elmustapha NAQOS and Najat SABIKI for supporting me throughout all my studie, my grand Mother and my sisters Fatima-ezzahra, Ilham, Oumaima, Rim and Marwa for supporting me spiritually throughout writing this thesis and my life in general.
Beauty is the first test: there is no permanent place in this world for ugly mathematics
Abstract
In this thesis, we study composition operators, induced by a sub-domain of the unit disc whose boundary inter-sects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation 1 − r = γ(|θ|). We obtain an explicit characterization for the membership in Schatten p- classes, in terms of γ. Next, we consider Toeplitz operators on a class of analytic function spaces, including standard Bergman spaces on the unit disc and Fock spaces. We give a complete characterization of those compact operators whose eigenvalues have a prescribed slow decay to zero.
R ´esum ´e
Nous ´etudions les op´erateurs de composition, induites par un sous-domaine du disque unit´e dont la fronti`ere intersecte le cercle unit´e en +1 et qui comporte au voisinage de 1 une ´equation polaire 1 − r = γ(|θ|). Nous obtenons une caract´erisation explicite pour lappartenance au Schatten p-classes, en termes de l’´equation po-laire. Ensuite, nous consid´erons les op´erateurs de Toeplitz sur une classe d’espaces fonctionnels analytiques, y compris les espaces de Bergman standards sur le disque unit´e et les espaces de Fock. Nous donnons une caract´erisation compl`ete de ces op´erateurs compacts dont les valeurs propres ont une d´ecroissance lente vers z´ero.
Mots-clefs — Op´erateur de Composition, Op´erateur de Toeplitz, Espace de Hardy, Espace de Bergman, Espace de Dirichlet
Contents
Acknowledgements I
Abstract IV
R´esum´e V
R´esum´e de La th`ese VIII
Introduction XI
1 Bounded Operators 1
1.1 Definitions . . . 1
1.2 Compact Operators on Hilbert Spaces . . . 2
1.3 Sh(H) Class Operators . . . 7
2 Reproducing Kernel Hilbert Spaces 13 2.1 Definitions and Properties . . . 13
2.2 Berezin Transform . . . 16
3 Ahlfors Distortion Theorem 18 3.1 Harmonic Measure . . . 18
3.2 Extremal Distance . . . 21
3.3 Ahlfors Distortion Theorem . . . 25
4 Spaces Of Holomorphic Functions 30 4.1 Harmonic and Subharmonic Functions . . . 32
4.1.1 Harmonic Functions . . . 32
4.1.2 Sub-Harmonic Functions . . . 33
4.2 Hardy Spaces . . . 35
5 Composition Operators 40
5.1 Boundedness of Composition Operators . . . 40
5.2 Compactness of Composition Operators . . . 43
6 Composition Operators with Univalent Symbol in Schatten Classes 47 6.1 Introduction . . . 47
6.2 Backgroud and Notations . . . 50
6.2.1 Nevanlinna counting functions. . . 50
6.2.2 Pull-back measures . . . 51
6.2.3 Composition operators with univalent symbol . . . 52
6.3 Estimate of harmonic measure. . . 53
6.4 Proof of the main theorem . . . 57
6.5 Final remarks . . . 60
7 Asymptotic Behavior of Eigenvalues of Toeplitz Operators 61 7.1 Introduction . . . 61
7.2 Weights and examples . . . 62
7.2.1 The class W . . . 62
7.2.2 Examples . . . 63
7.3 Lattices . . . 64
7.4 Berezin transform . . . 66
7.5 Boundedness and compactness. . . 69
7.6 Decay of eigenvalues . . . 70
R ´esum ´e de La th `ese
L’´etude des op´erateurs de composition a fait l’objet de beaucoup de math´ematiciens voir par exemple les livres de C. Cowen et B. MacCluer [8] et J. H. Shapiro [39], il se trouve `a l’interface de la th´eorie des op´erateurs et la th´eorie des fonctions holomorphe. E. Nordgren a commenc´e l’´etude de l’op´erateur de composition avec une fonction fixe ϕ agissant sur un espace de fonctions holomorphes au milieu des ann´ees 60 et a donn´e leur nom Composition operators. Au cours des ann´ees qui ont suivi, la bibliographie a ´et´e prolong´ee d’une mani`ere o`u il serait difficile de lire tous les travaux sur le sujet.
Pour une fonction holomorphe ϕ qui prend D (Le disque unit´e du plan complexe) dans lui-mˆeme, on associe l’op´erateur de composition Cϕd´efini par
Cϕf = f ◦ ϕ , f holomorphe sur D.
Nous ´etudions la relation entre les propri´et´es de Cϕ et les propri´et´es du symbole ϕ: le but est de voir la
bornitude, la compacit´e, l’appartenance au classes de Schatten..., de Cϕ comme cons´equences des propri´et´es
analytiques et g´eom´etriques du symbole ϕ.
Le bornitude de l’op´erateur de composition est une question fondamentale, il arrive que certains op´erateurs de composition ne sont pas born´ees parce que l’espace est trop petit ou trop grand. Nous verrons dans les espaces classiques les plus importants, que l’op´erateur de composition induit par un symbole holomorphe du disque unit´e dans lui-mˆeme est un op´erateur born´e. Si tous les op´erateurs de compositions dont le symbole est un automorphisme du disque est born´e dans un espace des fonctions holomorphes, alors la bornitude de nombreux autres op´erateurs de composition suit imm´ediatement. Le principe de la subordination Littlewood (Th´eor`eme 4.6), donne la bornitude des op´erateurs de composition dans nombreux espaces sous la condition ϕ(0) = 0.
Soit H(D) l’espace des fonctions holomorphe sur D. Pour α ≥ 0, l’espace des fonctions holomorphe `a poids Hαest d´efini par
Hα:= f ∈ H(D) : Z D |f0(z)|2dAα(z) < ∞ , avec dAα(z) = (1 + α)(1 − |z|2)αdA(z).
Pour α ∈ [0, 1), Hα := Dαsont les espaces de Dirichlet (l’espace de Dirichlet classique correspond `a α = 0).
On notera ´egalement que, pour α > 1, Hα sont les espaces de Bergman `a poids. Plus pr´ecis´ement, si A2β
d´esigne l’espace de Bergman `a poids d´efini par
A2β := f ∈ H(D) : Z D |f (z)|2dA β(z) < ∞ , (β > −1),
alors Hα = A2α−2. Par le principe de subordination de Littlewood, tout op´erateur de composition est born´e
sur Hα. puisque Cϕ fixe les constants, kCϕkHα ≥ 1. Il en d´ecoule que kCϕkHα = 1 pour les fonctions
holomorphe v´erifiant ϕ(0) = 0.
La compacit´e des op´erateurs de composition signifie que pour un symbole ϕ l’op´erateur Cϕtransforme les
parties born´ees dans des parties relativement compactes. L’´etude de la compacit´e a ´et´e initi´ee par H. J. Schwarts dans [36], qui a obtenu que si kϕk∞ < 1 alors Cϕ est un op´erateur compact sur Hα. D’un autre cˆot´e, si le
symbole ϕ touche le cercle unit´e T dans une partie de la mesure de Lebesgue non nulle, c’est-`a-dire m({eit, |ϕ(eit)| = 1}) > 0,
alors Cϕn’est pas compact sur Hα. La question qui se pose, qu’est-ce qui se passe quand le symbole ϕ touche
le cercle unit´e en un seul point. Ce sont les cas interm´ediaires qui rendent le probl`eme int´eressant. Par exemple, supposons que ϕ prend D dans un polygone inclus dans le cercle unit´e. Donc l’op´erateur Cϕsera compact sur
Hα, mˆeme si kϕk∞= 1. H. J. Schwartz dans [36] a observ´e que l’application ϕ(z) = (1 + z)/2, qui prend le
disque unit´e de mani`ere conforme dans un disque qui toucher le cercle unit´e seulement au point +1, induit sur l’espace de Hardy un op´erateur de composition noncompact.
Notez que si ϕ, ψ sont deux fonctions holomorphes univalent telles que ϕ(D) = ψ(D), alors Cϕ = ChCψ
avec h est un automorphismes du disque, et par cons´equent Cϕ et Cψ sont ´egaux `a un multiplicatif op´erateur
inversible. Ainsi, Cϕest compact si et seulement si Cψ v´erifie la mˆeme propri´et´e. Ceci est appel´e le Principe
Comparaison. Alors, il est naturel de se demander comment caract´eriser ces propri´et´es en fonction de la g´eom´etrie de ϕ(D). Pour la raison donn´ee ci-dessus. Soit Ω d´esigne un sous-domaine simplement connexe de D tel que ∂Ω ∩ ∂D = {1}. Nous allons ´egalement supposer que ∂Ω au voisinage de +1, a une ´equation polaire 1 − r = γ(|θ|), O`u γ : [0, δ] → [0, 1] est une fonction croissante continue avec γ(0) = 0. Soit ϕ une fonction holomorphe univalent de D dans Ω avec ϕ(1) = 1.
Par le th´eor`eme de Tsuji-Warschawski, ϕ admet une d´eriv´e angulaire en +1 si et seulement si Z δ
0
γ(t)
t2 dt < +∞.
Cela signifie, par le th´eor`eme et la compacit´e et le crit`ere de Julia-Carath´eodory, que
Cϕ est compact sur Hα⇐⇒
Z δ
0
γ(t)
t2 dt = +∞.
Si T est un op´erateur compact agissant sur un espace de Hilbert s´eparable H, (T∗T )1/2 est un op´erateur compact positif, o`u T∗d´esigne l’adjoint de T . Il en r´esulte que le spectre de (T∗T )1/2se compose de plusieurs valeurs propres d´enombrable (sn) sur [0, +∞[. Si la suite des valeurs propres de (T∗T )1/2r´eside dans `palors
T appartient `a la classe se Schatten, not´ee Sp(H). Pour plus de d´etails, voir la section 1.3. Notez que les
classes de Schatten de Hα v´erifions le principe de comparaison. Par cons´equent, sous quelle condition sur γ
nous avons l’appartenance `a Sp(Hα).
Th´eor`eme .1. Soit Ω, γ et ϕ comme ci-dessus. Pour α > 0 et p > 0, on a Cϕ∈ Sp(Hα) si et seulement si
Z δ 0 e−pα2 Γ(t) γ(t) dt converge. Avec Γ(t) = 2 π Z δ t γ(s) s2 ds.
Les mesures de Carleson sur les espaces de Bergman ont ´et´e ´etudi´es par Hasting, puis par Oleinik, Luecking et autres ([7], [16], [29], [30], [32] et [34]). Le cas des espaces Fock a ´egalement ´et´e examin´e par plusieurs auteurs (voir [18], [37] · · · ).
Soit Ω un domaine dans le plan complexe C et soit dA repr´esentent la mesure de Lebesgue sur C. ´etant donn´e une fonction non n´egative continue ω : Ω → (0, ∞), L’espace de Bergman `a poids A2ω associ´ee `a ω est d´efini par A2ω =f ∈ Hol(Ω) : kf kω= Z Ω |f (z)|2ω2(z)dA(z) 1/2 < ∞ .
´etant donn´e une mesure de Borel positive µ sur Ω, l’op´erateur de Toeplitz Tµassoci´ee `a µ sur A2ω, est la
transformation lin´eaire d´efinie par
Tµf (z) :=
Z
Ω
K(z, ζ)f (ζ)ω2(ζ)dµ(ζ), z ∈ Ω.
O`u K est le noyau reproduisant de A2ω. D. Luecking dans [26] est le premier `a ´etudier l’op´erateur de Toeplitz sur l’espace de Bergman avec des mesures comme un symbole ϕ. L’appartenance des op´erateurs de Toeplitz dans les classes de Schatten sur les espace de Bergman `a poids a ´et´e ´egalement ´etudier (voir par exemple [25], [26], [31], [33], [44] et [41])).
Notre objectif est de donner une caract´erisation g´eom´etrique des op´erateurs de Toeplitz compacts dont les valeurs propres ont une d´ecroissance lente vers z´ero. Nos r´esultats concernent une grande classe de poids qui seront d´esign´es par W. Cette classe de poids comprennent tous les exemples classiques. Pour simplifier l’expos´e de nos r´esultats, nous allons exprimer notre th´eor`eme principal seulement dans les espaces de Bergman standard sur le disque unit´e.
Soit ωαun poids sur D donn´e par ωα(z) =
q
1+α
π (1 − |z|2)α/2, α > −1. La d´ecomposition dyadique du
disque D, ∆n,j, n ≥ 0 et 0 ≤ j ≤ 2n− 1, est donn´e par
∆n,j = reit : 1 2n+1 < 1 − r ≤ 1 2n and 2πj 2n ≤ t < 2π(j + 1) 2n .
Soit (Rn) une num´erotation de (∆n,j). Il est connu, dans ce cas, que Tµest born´e (resp. compact) si et
seule-ment si µ(Rn) = O(|Rn|) (resp. o(|Rn|)), avec |Rn| d´esigne l’aire euclidienne de Rn. La suite d´ecroissante
des valeurs propres de l’op´erateur compact T sera not´ee λn(T ). Nous avons
Th´eor`eme .2. Soit η : [1, +∞) → (0, ∞) une fonction convexe d´ecroissante satisfaisant η(∞) = 0, et η(x2) η(x). Soit µ une mesure de Borel positive sur D telle que Tµest compact surA2ωα. Alorsλn(Tµ) = O(η(n)) si
et seulement s’il existe une constantec such that
X hη cµ(Rn) |Rn| < ∞,
Introduction
This thesis entitled Asymptotic Behavior of Toeplitz and composition operators is located at the intersection of functional analysis, operator theory and complex analysis. It collects various works bearing on classes of composition operators and Toeplitz operators on several analytic function spaces: Hardy space, weighted Bergman spaces, weighted Dirichlet spaces, Fock space, · · ·
The study of the composition of operators has been the subject of many mathematicians see for example the books of C. Cowen and B. MacCluer [8] and J. H. Shapiro [39], and it lies at the interface of operator theory and holomorphic function theory. E. Nordgren began the study of the composition operator with a fixed function ϕ acting on a space of holomorphic functions in the mid 1960’s and gave their name ”Composition operator”. During the years following, the bibliography has been extended in a way where it would be difficult to read all the papers on the subject.
To each holomorphic function ϕ that takes D (The unit disk of the complex plane) into itself, we associate the composition operator Cϕdefined by
Cϕf = f ◦ ϕ , f holomorphic on D.
We study the relationship between properties of Cϕand properties of the symbol map ϕ: the goal is to see the
boundedness, compactness, membership the the Schatten p-ideals, · · · , of Cϕ as consequences of particular
holomorphic and geometric feature of the symbol ϕ.
The boundedness of the composition operator is a basic issue, it happens that some composition operators are unbounded because the space is too small or too big. We will see that in the most important classical spaces, the composition operator induced by a holomorphic map of the unit disk into itself is bounded operator. If all compositions operators whose symbol is an automorphism is bounded in a space of holomorphic functions, then the boundedness of many other composition operators follows immediately. The Littlewood subordination principle ( Theorem 4.6 ), gives boundedness of composition operators on many spaces under the condition ϕ(0) = 0.
Denote by H(D) the space of all holomorphic functions on D. For α ≥ 0, the weighted analytic spaces Hα
is defined by Hα:= f ∈ H(D) : Z D |f0(z)|2dAα(z) < ∞ , where dAα(z) = (1 + α)(1 − |z|2)αdA(z).
For α ∈ [0, 1), Hα := Dα are the weighted Dirichlet spaces (the classical Dirichlet space corresponds to
weighted Bergman space defined by A2β := f ∈ H(D) : Z D |f (z)|2dA β(z) < ∞ , (β > −1),
then Hα= A2α−2. By Littlewood subordination principles, every composition operator is bounded on Hα.
Since Cϕfixes constants, kCϕkHα ≥ 1. It follows that kCϕkHα = 1 for holomorphic function satisfying
ϕ(0) = 0.
The compactness of composition operators means that for which inducing map ϕ the operator Cϕ
com-presses bounded subsets into relatively compact ones. The study of compactness was initiated by H. J. Schwarts in [36], who obtained that if kϕk∞< 1 then Cϕis a compact operator on Hα. On the other hand, if the symbol
ϕ touches the unit circle T in a part of Lebesgue measure not null, that is m({eit, |ϕ(eit)| = 1}) > 0,
then Cϕ is not compact on Hα. The question arises, what happens when the symbol ϕ touches the unit circle
at one point.
It is the intermediate cases that make the problem interesting. For example, suppose ϕ takes D into a polygon inscribed in the unit circle. Then the operator Cϕ will be compact on Hα, even if kϕk∞ = 1.
Addi-tional subtlety arises from the fact that this result persists even when the corners of the inscribed polygon are rounded a bit, but the corners cannot be rounded too much. H. J. Schwartz in [36] observed that the mapping ϕ(z) = (1 + z)/2, wich takes the unit disk conformally onto a disk touching the unit circle only at point 1, induces on Hardy space a noncompact composition operator.
Figure 1: Maps inducing noncompact and compact operators
Note that if ϕ, ψ are two univalent holomorphic functions such that ϕ(D) = ψ(D) then Cϕ= ChCψwith h
an automorphism of the disk, and therefore Cϕand Cψare equal up to multiplicative invertible operator. Then
Cϕ is compact if and only if Cψ satisfies the same property. This is called Comparison Principle. So, it is
natural to ask how to characterize these properties in terms of the geometry of ϕ(D).
For the reason given above. Suppose Ω denotes a simply connected sub-domain of D such that ∂Ω ∩ ∂D = {1}. We will also suppose that ∂Ω has, in a neighborhood of +1, a polar equation 1 − r = γ(|θ|), where γ : [0, δ] → [0, 1] is a continuous, increasing function with γ(0) = 0. Let ϕ be a univalent map of D onto Ω with ϕ(1) = 1.
Tsuji-Warschawski’s theorem ([39]), asserts that ϕ has an angular derivative at +1 if and only if Z δ
0
γ(t)
t2 dt < +∞.
This means, by Julia-Caratheodory’s theorem and compactness criterion, that we have
Cϕ is compact on Hα⇐⇒
Z δ 0
γ(t)
Figure 2: Domain Ω
If T is a compact operator acting on a separable Hilbert space H, then (T∗T )1/2 is a positive compact op-erator, where T∗denotes the adjoint of T . It follows that the spectrum of (T∗T )1/2consists of countably many eigenvalues (sn) on [0, +∞[. If the sequence of the eigenvalues of (T∗T )1/2lies in `p then T is said to belong
to the Schatten p-class, denoted by Sp(H) . For more details see Section 1.3. Note that the Schatten p-class
of Hαsatisfying the comparison principle. Therefore, under which condition on γ we have the membership to
Sp(Hα).
Theorem 0.1. Let Ω, γ and ϕ as above. Suppose that
γ(t) = Ot/ logβ(1/t) for someβ > 1/2.
Forα > 0 and p > 0, we have Cϕ ∈ Sp(Hα) if and only if
Z δ 0 e−pα2 Γ(t) γ(t) dt converges. Where Γ(t) = 2 π Z δ t γ(s) s2 ds.
The Carleson measures on Bergman spaces were first studied by Hasting and thereafter by Oleinik, Lueck-ing and others ( [7], [16], [29], [30], [32] and [34]). The case of Fock spaces was also considered by several authors (see [18], [37] and references therein).
Let Ω be a domain in the complex plane C and let dA denote the Lebesgue measure on C. Given a nonnegative continuous function ω : Ω → (0, ∞), the weighted Bergman space A2ω associated with ω is defined by A2ω =f ∈ Hol(Ω) : kf kω= Z Ω |f (z)|2ω2(z)dA(z) 1/2 < ∞ .
transformation defined by
Tµf (z) :=
Z
Ω
K(z, ζ)f (ζ)ω2(ζ)dµ(ζ), z ∈ Ω.
Where K is the reproducing Kernel of A2ω. D. Luecking in [26] is the first to study Toeplitz operator on the Bergman space with measures as symbols. The membership of Toeplitz operators to the Schatten Ideals was also considered for some weighted analytic spaces (see for instance [25], [26], [31], [33], [44] and [41]).
Our aim is to give a geometrical characterization of compact Toeplitz operators whose eigenvalues have a prescribed slow decay to zero. Our results concern a large class of weights which will be denoted by W. This class of weights include all classical examples. To simplify the exposition of our results we will state our main theorem only in the standard Bergman spaces on the unit disc.
Let ωαbe the weight on D given by ωα(z) =
q
1+α
π (1 − |z|2)α/2, α > −1. The dyadic discs ∆n,j, n ≥ 0
and 0 ≤ j ≤ 2n− 1, are given by
∆n,j = reit : 1 2n+1 < 1 − r ≤ 1 2n and 2πj 2n ≤ t < 2π(j + 1) 2n .
Let (Rn) be a numeration of (∆n,j). It is known, in this case, that Tµis bounded (resp. compact) if and only if
µ(Rn) = O(|Rn|) (resp. o(|Rn|)), where |Rn| denotes the area of Rn. The decreasing sequence of eigenvalues
of compact operator T will be denoted by λn(T ). We have
Theorem 0.2. Let η : [1, +∞) → (0, ∞) be a decreasing convex function satisfying η(∞) = 0, and η(x2)
η(x). Let µ be a positive Borel measure on D such that Tµis compact onA2ωα. Thenλn(Tµ) = O(η(n)) if and
only if there is a positive constantc such that
X hη cµ(Rn) |Rn| < ∞, wherehη is defined byhη(η(x)) = 1/x.
This work is organized as follows:
The first chapter is devoted to the theory of bounded operators. We will introduce a new class of operators, which generalizes the famous Schatten p-class. Thereafter we give some interesting properties of these classes.
In the second chapter we give definitions and properties of reproducing kernel Hilbert spaces with some theorems. Thereafter we will show that every operator T on these spaces gives rise to a smooth function ˜T in a natural way. We call ˜T the Berezin transform of T . It is a useful tool to study operators on any reproducing kernel Hilbert space.
The third chapter is devoted to Ahlfors distortion theorem and Warschawski theorem. for that we may need to study the extremal distance and harmonic measure.
The fourth chapter is concerned with the spaces of holomorphic functions, such that Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and Fock space.
In the fifth chapter, we will study the composition operator, we give the Littlewood subordination principle. And we will discuss the compactness of this operator. The relationship with the contact.
In the sixth chapter, we study composition operators, induced by a sub-domain of the unit disc whose boundary intersects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation 1 − r = γ(|θ|). We obtain an explicit characterization for the membership in Schatten p- classes, in terms of γ.
In the seventh chapter, we consider Toeplitz operators on a class of analytic function spaces, including stan-dard Bergman spaces on the unit disc and Fock spaces. We give a complete characterization of those compact operators whose eigenvalues have a prescribed slow decay to zero.
1
Bounded Operators
1.1
Definitions
In this part, we give an overview on the theory of bounded linear operators on Banach spaces in general and Hilbert spaces in particular. Throughout this section H is a Hilbert space. Let X be a Banach space, we define X∗ to be the space of all bounded linear functional on X. X∗is a Banach space with norm:
kx∗kX∗ := sup{|x∗(x)| : kxkX = 1}.
Suppose X and Y are two Banach spaces and T : X → Y be an operator. We say that T is bounded from X into Y if there exist a strictly positive constant C such that:
kT xkY ≤ CkxkX, ∀x ∈ X,
we denote by L(X, Y ) the set of all bounded operators from X into Y , and by L(X) the set L(X, X). Given two bounded operators T and S in L(H), we define T S as the composition of T with S:
(T S)(x) := T (Sx) , ∀x ∈ H,
and similarly to define T + S and αT for α is a complex number. Let T ∈ L(X, Y ) then T induces an operator T∗ : Y∗ → X∗defined by:
T∗(y)(x) = y(T x), ∀x ∈ X , ∀y ∈ Y∗ .
It is easy to see that T∗ ∈ L(Y∗, X∗) and kT k = kT∗k. The operator T∗is called the adjoint of T .
Theorem 1.1 (Riesz Representation Theorem). Let H be a Hilbert space and y ∈ H. The operator Ty : H → C
defined by:
Ty(x) = hx, yi, ∀x ∈ H,
is bounded. Conversely ifT : H → C be a bounded operator, then there exist a unique vector y ∈ H such that T = Ty.
By Riesz representation theorem, H is isomorphic to H∗. In fact, H and H∗are related by the equation: hT x, yi = hx, T∗yi, ∀x, y ∈ H .
Definition 1.1. Let T and S are two bounded operators on H. We say that 1. T is self-adjoint if T∗ = T .
2. T is positive if hT x, xi ≥ 0 , ∀x ∈ H.
3. T is unitary if T∗T = T T∗ = I (where I is the identity operator).
4. T and S are unitarily equivalent if there is a unitary operator U on H such that T = U∗SU . 5. T is normal if T∗T = T T∗.
It is clear that self-adjoint, positive and unitary operators are normal. Suppose that T is self-adjoint, then for all x ∈ H, we have
hT x, xi = hx, T xi = hT x, xi,
so, T is self-adjoint if and only if hT x, xi ∈ R. In particular, every positive operator is self-adjoint. An example of positive operator is T∗T ( hT∗T x, xi = kT xk2 ).
Suppose T is a positive operator on H and n ∈ N, there is a unique positive operator denoted by T1n such
that T = (Tn1)n. Because T∗T is a positive operator, then the square root of T∗T is well defined, denoted by
|T |. The following result, called the polar decomposition, shows the important role played by |T | in operator theory.
Theorem 1.2 (Polar decomposition). If T is a bounded operator on H, then, there is a unique bounded operator V on H such that T = V |T |.
Proof. Let x ∈ H, by a simple calculation, we have k|T |xk = kT xk. Then, there exists an operator V on Im(|T |), such that
V |T |x = T x , ∀x ∈ H.
We extend V in a unique way on Im(T ). Let V the operator defined on H by: V x = V x if x ∈ Im(T ) and V x = 0 if x ∈ Ker(T ). Therefore, T = V |T |.
1.2
Compact Operators on Hilbert Spaces
Let H be a separable Hilbert space. We denote by B1(H) the closed unite ball of H,
B1(H) := {x ∈ H : kxk ≤ 1} .
Definition 1.2. T is compact operator on H if T (B1(H)) is relatively compact.
It is easy to see that a compact operator is bounded. We denote by K(H) the set of compact operators on H.
Proposition 1.1. Let H be separable Hilbert space. An operator T on H is compact if and only if for any bounded sequence(xn)n in H there exists a sub-sequence (xnk)k such that (T xnk)k converge in the norm
topology ofH.
Proof. I Suppose T is compact, since (xn)nis bounded, then (T xn)n is contained in a compact set. Thus
(T xn)ncontains a convergent sub-sequence (T xnk)k.
I Conversely, the non-compactness of T will imply that there exists a sequence (xn)nin B1(H) such that
(T xn)ncontains no convergent sub-sequence.
Theorem 1.3. An operator T on H is compact if and only if for every xn→ 0 weakly in H, (T xn)nconverge
to zero in the norm topology.
Proof. I Suppose that T is compact and xn→ 0 weakly in H. If kT xnk 9 0, there exists A > 0 such that
kT xnk ≥ A , ∀n ∈ N.
Since xn→ 0, then (xn)nis bounded, by Proposition 1.1, there exists a sub-sequence (xnk)ksuch that (T xnk)k
converge to y in H. Thus, T xnk → y weakly in H. The operator T is bounded, then y = 0, and we have
kT xnkk → 0. Contradiction with the fact that kT xnk ≥ A.
I Conversely, if T is not compact, by proposition 1.1, there exists a sequence (xn)n ⊂ B1(H) such that
(T xnk)kdoes not converge for every sub-sequence (xnk)k. By Alaoglu’s theorem, B1(H) is compact for the
weak topology. Then there exists a convergent sub-sequence (xnk)k. Let yk = xnk, so ykis weakly convergent,
but kT ykk has no convergent sub-sequence.
Theorem 1.4. Suppose T is compact operator on H, for every bounded operator S on H, the operators T S andST are compact. Furthermore, if S is compact then T + S is also compact.
Proof. Let (xn)n → 0 weakly in H, thus kT xnk → 0, and by boundedness of S, we have kST xnk → 0, and
Sxn→ 0 weakly, then kT Sxnk → 0. Therefore, T S and ST are both compact. If S is compact, we know that
k(T + S)xnk ≤ kT xnk + kSxnk, so T + S is compact.
Theorem 1.5. A bounded operator T on H is compact if and only if T∗is compact.
Proof. Since (T∗)∗= T , then, we need only to prove that T is compact, which would imply that T∗is compact. Suppose T is compact, let xn→ 0 weakly in H, there exists C > 0 such that kxnk ≤ C, for all n. The operator
T∗ is bounded, then T∗xn→ 0 weakly in H, so by using Cauchy-Schwarz inequality, we have
kT∗xnk2= hT T∗xn, xni
≤ kT T∗xnkkxnk
≤ CkT T∗xnk.
T∗xn→ 0 weakly in H, and T is compact, so kT T∗xnk → 0. Thus T∗is compact.
Theorem 1.6. Let T be a bounded operator on H. The following assertions are equivalent 1. T is compact.
2. T∗T is compact. 3. |T | is compact.
Proof. If T is compact, by Theorem 1.4, the operator T∗T is compact. If |T | is compact, by Theorems 1.4 and 1.2, T is compact. Suppose T∗T is compact. Let S = |T |, we have S2 = T∗T , let xn → 0 weakly in H, by
compactness of S2, we have kS2xnk → 0, and
kSxnk = hS2xn, xni ≤ kS2xnkkxnk.
Where the last inequality is obtained by Cauchy-Schwarz. Therefore, S = |T | is compact.
Theorem 1.7. If (Tn)nis a sequence of compact operators onH such that kTn− T k → 0 for some operator
T on H, then T is compact.
Proof. Let xn→ 0 weakly in H, then for all k ∈ N we have
kTkxnk → 0 , n → 0 .
Suppose that kxnk ≤ 1, and let > 0, there is k > 0 such that kTk− T k < . Let N ∈ N such that
kTkxnk < , ∀n ≥ N .
Therefore,
kT xnk = kT xn− Tkxn+ Tkxnk ≤ kT xn− Tkxnkk + Tkxnk ≤ 2 ,
therefore, T is compact.
Proposition 1.2. Suppose (en)nand(σn)nare two orthonormal sets inH, and (sn)na sequence of complex
numbers tending to0. The operator T defined on H by
T : H → H x → ∞ X n=1 snhx, eniσn is compact.
Proof. Let xk→ 0 weakly in H, for all n ∈ N, we have hxk, eni → 0, as n → ∞, and there exists c > 0, such
that kxkk ≤ c, for all k ∈ N. Let > 0, then there exists N ∈ N such that |sn| < , for all n ≥ N . Thus
kT xkk2 = ∞ X n=1 |sn|2|hxk, eni|2 = N X n=1 |sn|2|hxk, eni|2+ ∞ X n=N +1 |sn|2|hxk, eni|2 ≤ N X n=1 |sn|2|hxk, eni|2+ ∞ X n=N +1 2|hxk, eni|2. We have N X n=1 |sn|2|hx k, eni|2 → 0 , k → ∞, and ∞ X n=N +1 |hxk, eni|2 ≤ c2. Then T is compact.
The converse of Proposition 1.2 is true, the following result will show that.
Theorem 1.8. Suppose T is a compact self-adjoint operator on H, then there exists a sequence (sn)nof real
numbers tending to0, and orthonormal set (en)ninH such that
T x =
∞
X
n=1
snhx, enien, x ∈ H .
Proof. The operator T is self-adjoint, then we have
kT k = sup {|hT x, xi| : kxk = 1} . Assume that there exists a sequence (xn)nin H of norm equal to 0 such that
lim
n→∞hT xn, xni = kT k,
since hT xn, xni ≤ kT xnkkxnk ≤ kT k , for all n ∈ N. Then we have
lim
n→∞kT xnk = kT k.
Because kT k = kkT kxnk, we have
lim
n→∞kT xn− kT kxnk = 0.
Since T is compact and the sequence (xn)nis bounded, by Proposition 1.1 there is a sub-sequence (xnk)ksuch
that
lim
for some y ∈ H, then
kkT kxn− yk = kkT kxn− T xn+ T xn− yk
≤ kkT kxn− T xnk + kT xn− yk.
Therefore, (xn)nconverge to x = y/kT k in H, with kxk = 1, hence T x = kT kx. Thus, we have the existence
of nonzero eigenvalue of T . The compactness of T implies that either T has a finite number of distinct eigen-values or the eigeneigen-values of T form a sequence tending to 0.
T is compact, the eigenspace corresponding to each of its nonzero eigenvalues is finite dimensional. Oth-erwise, let (en)nbe an orthonormal basis of the eigenspace corresponding to s 6= 0, then en → 0 weakly in H
and the compactness of T implies that lim
n→∞|s| = limn→∞kT enk = 0 ,
contradiction to s 6= 0. Let (λn)nbe the sequence of nonzero eigenvalues of T , arranged so that |λn| ≥ |λn+1|
for n ∈ N∗. and let αnbe the dimension of the eigenspace corresponding to λn. Let (sn)nthe sequence defined
by (sn)n= (s1, s2, s3, . . .) := (λ1, λ1, . . . , λ1 | {z } α1-times , λ2, λ2, . . . , λ2 | {z } α2-times , . . . , λn, λn, . . . , λn | {z } αn-times , . . .)
Let (en)nbe a sequence of unit vectors consisting of an orthonormal basis of eigenspace of s1, followed by an
orthonormal basis of the eigenspace of s1, and so on. The set (en)nis orthonormal in H, and T en= λnen, for
all n ∈ N. Let T0the operator defined by
T0x =
∞
X
n=1
snhx, enien, x ∈ H ,
by Proposition 1.2, the operator T0 is compact. Let M be the closed subspace of H spanned by (en)n, we have
T0M ⊂ M and T0M⊥= {0}. It is clear that T M ⊂ M and T M⊥⊂ M⊥.
Let S = T − T0, then S has no nonzero eigenvalues, in fact, if λ is a nonzero eigenvalue of S and x its corresponding eigenvector, then
T (x −
∞
X
n=1
hx, enien) = T x − T0x = λx.
Since T en= λnenfor all n ∈ N, then x ∈ M⊥, because x −Pn=1∞ hx, enien ∈ M⊥and T M⊥ ⊂ M⊥. This
implies that T0x = 0. Thus T x = λx and λ = sn for some n ∈ N, then x ∈ M ∩ M⊥ = {0}. And we
have a contradiction. The operator S = T − T0 is self-adjoint compact operator with no nonzero eigenvalues, therefore, T = T0, and the proof is complete.
If the operator T is compact but not self-adjoint, by the polar decomposition (see Theorem 1.2), there exists a unitary operator V such that T = V |T |. The operator |T | is positive, then self-adjoint and by theorem 1.6 is also compact. By using Theorem 1.8, we have
|T |x =
∞
X
n=1
snhx, enien, x ∈ H. (1.1)
Where (sn)nis a sequence of real numbers tending to 0 and (en)nis orthonormal set in H. Therefore
T x =
∞
X
n=1
snhx, eniσn, x ∈ H. (1.2)
Where σn= V en, then σnstill an orthonormal set. The equation 1.2 will be called the canonical
1.3
S
h(H) Class Operators
Recall that if T is compact operator on separable Hilbert space H, then |T | is positive compact operator. The eigenvalues of |T | arranged in decreasing order and repeated according to multiplicity form a sequence of positive numbers tending to 0, these numbers are called the singular values of the operator T , we write sn(T )
(or simply sn) for the nth singular value of T . And there exist orthonormal sets enand σnin H such that
T x =
∞
X
n=1
snhx, eniσn, x ∈ H.
Proposition 1.3. The singular values sn(T ) of a compact operator T are given by the following formula
sn+1(T ) = min y1,...,yn∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n h|T |x, xi (1.3)
Proof. The spectrum of |T | consists of a sequence (sn)nof positive numbers, it is seen that the set of
eigen-vectors belonging to any nonzero eigenvalue form a finite dimensional space. The dimension of this spaces is known as multiplicity of the corresponding eigenvalue. Let us suppose the eigenvalues enumerated in decreas-ing order, each eigenvalue bedecreas-ing repeated a number of times equal to its multiplicity, then
h|T |x, xi = hX
n
snhx, enien, xi ≤ s1kxk2,
so that h|T |x, xi ≤ s1, for all kxk = 1. By the proof of Theorem 1.8, s1 = k|T |k = kT k and |T | is positive,
then
s1= kT k = k|T |k = max
kxk=1h|T |x, xi. (1.4)
Let y ∈ H, then for all α1e1+ α2e2⊥ y we have
h|T |(α1e1+ α2e2), α1e1+ α2e2i = |α1|2s1ke1k2+ |α2|2s2ke2k2
≥ s2(|α1|2ke1k2+ |α2|2ke2k2)
≥ s2kα1e1+ α2e2k,
where the last inequality by Cauchy-Schwarz inequality. Therefore for all y in H, we have
s2 ≤ sup kxk = 1 hx, yi = 0 h|T |x, xi (1.5) and s2= h|T |e2, e2i . thus s2= min y∈H kxk = 1max hx, yi = 0 h|T |x, xi .
Hence, by the same calculation, we have sn+1(T ) = min y1,...,yn∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n h|T |x, xi
Since kT xk = hT x, T xi = hT∗T x, xi and the eigenvalues of T∗T are the square of the singular values of T then we have sn+1(T ) = min y1,...,yn∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n kT xk (1.6)
Proposition 1.4. The singular values of a compact operator satisfy the inequalities sn+m+1(T + S) ≤ sn+1(T ) + sm+1(S) sn+m+1(T S) ≤ sn+1(T )sm+1(S) Proof. We have sn+m+1(T + S) = min y1,...,yn+m∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n + m k(T + S)xk ≤ min y1,...,yn+m∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n + m kT xk + kSxk ≤ min y1,...,yn∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n kT xk + min yn+1,...,yn+m∈H max kxk = 1 hx, yii = 0 i = n + 1, . . . , n + m kSxk = sn+1(T ) + sm+1(S) .
And similarly sn+m+1(T S) = min y1,...,yn+m∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n + m k(T S)xk ≤ min y1,...,yn+m∈H max kxk = 1 hx, S∗y ii = 0 , i = 1, . . . , n hx, yii = 0 , i = n + 1, . . . , n + m k(T S)xk ≤ min y1,...,yn+m∈H max kxk = 1 hx, S∗y ii = 0 , i = 1, . . . , n hx, yii = 0 , i = n + 1, . . . , n + m kT (Sx)k kSxk kSxk ≤ min y1,...,yn∈H max kxk = 1 hx, yii = 0 i = 1, . . . , n kT xk + min yn+1,...,yn+m∈H max kxk = 1 hx, yii = 0 i = n + 1, . . . , n + m kSxk ≤ sn+1(T )sm+1(S)
The Schatten Class Sp(H) ( 0 < p < ∞ ) is defined by
Sp(H) := {T ∈ K(H) , (sn(T ))n∈ lp} .
In this part we give definition for generalized classes of operators and we will prove that several properties from the classes Sp(H) can be generalized to them.
Definition 1.3. Let h : R+ → R+ be a continuous increasing function such that h(0) = 0. We say that
T ∈ Sh(H) if there exists c > 0 such that ∞
X
n=1
h(csn(T )) < ∞. (1.7)
Note that for the function h(t) = tp, 0 < p < ∞, the classes Sh(H) coincides with the Schatten classes.
These classes are introduced by the team consisting of O. El Fallah, H. Mahzouli, I. Marrhich and H. Naqos, aims to estimate the eigenvalues of Toeplitz operators. In this part we will give some properties of these classes.
Lemma 1.1. Let h : R+ → R+ be a continuous increasing function such thath(0) = 0. Then S
h(H) is a
Proof. Let T, S ∈ Sh(H), there exist strictly positive constants c1and c2such that ∞ X n=1 h (c1sn(T )) < ∞ and ∞ X n=1 h (c2sn(S)) < ∞ . (1.8)
Let c = min(c1, c2), then we have ∞ X n=1 hc 2sn+1(T + S) = ∞ X n=1 hc 2s2n+1(T + S) + ∞ X n=1 hc 2s2n+2(T + S) , by Proposition 1.4, we have s2n+1(T + S) ≤ sn+1(T ) + Sn+1(T ) and s2n+2(T + S) ≤ sn+1(T ) + Sn+2(T ) . Thus ∞ X n=1 hc 2s2n+1(T + S) ≤ ∞ X n=1 hc 2(sn+1(T ) + Sn+1(S)) ≤ ∞ X n=1 h (c max{sn+1(T ), sn+1(S)}) ≤ 2 ∞ X n=1 h (csn+1(T )) + h (csn+1(S))
by the same calculation
∞ X n=1 h c 2s2n+2(T + S) ≤ ∞ X n=1 h c 2(sn+1(T ) + Sn+2(S)) ≤ ∞ X n=1 h (c max{sn+1(T ), sn+2(S)}) ≤ 2 ∞ X n=1 h (csn+1(T )) + h (csn+2(S)) Thus ∞ X n=1 hc 2sn+1(T + S) ≤ 4 ∞ X n=1 h (c1sn+1(T )) + 4 ∞ X n=1 h (c1sn+1(S)) .
Hense T + S ∈ Sh(H). Therefore, Sh(H) is an additive group. Now, let T ∈ Sh(H) and S ∈ L(H), we have
T S and ST are both compact. Then by Theorem 1.4
sn+2(T S) ≤ sn+1(T )s1(S) and sn+2(ST ) ≤ sn+1(T )s1(S).
Because s1(S) = kSk (see proof of Theorem 1.8). Thus ∞ X n=1 h c1 kSksn+2(ST ) ≤ ∞ X n=1 h c1 kSkkSksn+1(T ) = ∞ X n=1 h (c1sn+1(T )) < ∞.
By the same calculation, ∞ X n=1 h c1 kSksn+2(T S) ≤ ∞ X n=1 h (c1sn+1(T )) < ∞.
Thus, Sh(H) is two-sided ideal in L(H).
As a consequence of the previous theorem, Sh(H) is a linear vector space.
Theorem 1.9. Suppose T is a compact operator on H, and h be a continuous increasing convex function such thath(0) = 0. Then T is in Sh(H) if and only if
∞
X
n=1
h (c|hT en, eni|) < ∞. (1.9)
For all orthonormal sets(en)n.
Proof. Suppose T is a compact operator on H and h is convex function. Assume that T is self-adjoint. Then there exist orthonormal sets (σn)nsuch that
T x =X n snhx, σniσn, x ∈ H, since sn= hT σn, σni, we have X n h (Csn) = X n h (ChT σn, σni) . (1.10)
On the other hand, if (en)nis an orthonormal set in H, then for any m ≥ 0, we have
hT em, emi = h X n snhem, σniσn, emi =X n sn|hσn, emi|2.
Thus, by Jensen’s inequality, we have
h (ChT em, emi) ≤
X
n
|hσn, emi|2h(Csn).
Applying Fubini’s theorem, we obtain
X m h (ChT em, emi) ≤ X n h(Csn) X m |hσn, emi|2 ! . Therefore X m h (ChT em, emi) ≤ X n h(Csn). (1.11)
Proposition 1.5. Let h : R+ → R+be a continuous increasing function such thath(0) = 0. Suppose S is a
bounded surjective operator onH and T is any bounded operator on H. Then, we have S∗T S ∈ Sh(H) ⇔ T ∈ Sh(H).
Proof. If T ∈ Sh(H), then T 0
= S∗T S ∈ Sh(H), because Sh(H) is a two sided ideal. See Lemma 1.1. Let
X = H ker(S). Then S : X → H is bounded, one-to-one, and onto, so it has a bounded inverse A. Thus A is a bounded linear operator on H with SA = I, the identity operator on H. Such an operator A is called a right inverse of A. Now if T0is in Sh(H), then A∗T
0
A is also in Sh(H). Since
A∗T0A = A∗S∗T SA = (SA)∗T SA = T.
2
Reproducing Kernel Hilbert Spaces
2.1
Definitions and Properties
Definition 2.1. Let H be a Hilbert space of functions on a set Ω. The complex valued function K(z, w) defined onΩ × Ω is called a reproducing kernel of H if the following are satisfied:
1: For everyz ∈ Ω, the function Kw(z) = K(z, w) belongs to H.
2: For everyz ∈ Ω, and for all f ∈ H
f (z) = hf, Kzi . (2.1)
Applying (2.1) to the function Kw, we find
K(w, z) = hKz, Kwi , ∀z, w ∈ Ω.
In addition, we have kKzk2 = hKz, Kzi = K(z, z), for all z ∈ Ω.
Definition 2.2. A Hilbert space H of functions on a set Ω is called a Reproducing kernel Hilbert space if there exists a reproducing kernelK of H.
Theorem 2.1. Let H be a Hilbert space which admits a reproducing kernel K. Then this kernel is unique, and determined only by the Hilbert spaceH.
Proof. Let K(z, w) be a reproducing kernel of H. Suppose that H admits another reproducing kernel ˜K(z, w). Then for every z, w ∈ Ω, we have
kKz− ˜Kzk2= hKz− ˜Kz, Kz− ˜Kzi
= hKz− ˜Kz, Kzi − hKz− ˜Kz, ˜Kzi
= Kz(z) − ˜Kz(z) − Kz(z) + ˜Kz(z)
Theorem 2.2. Let H be a Hilbert space of functions on a set Ω. Then H admits a reproducing kernel if and only if for everyz ∈ Ω, the linear functional f 7→ f (z) is bounded.
Proof. Let K be the reproducing kernel of H. Using Schwartz’s inequality, for every z ∈ Ω, we have |f (z)| = |hf, Kzi| ≤ kf kkKzk,
therefore, the evaluation at z is bounded on H. Conversely, suppose that for every z ∈ Ω, the mapping f 7→ f (z) is bounded. Then, by Riesz Representation Theorem, for every z ∈ Ω there exists a function hz ∈ H, such that
f (z) = hf, hzi.
We put Kzinstead of hz. Then, H admits a reproducing kernel.
Definition 2.3. Let Ω be a set and K : Ω × Ω → C be a function. We say that K is hermitian, if for any finite set of points(zn)N1 ⊂ Ω, and any (cn)N1 ⊂ C, we have
N X i=1 N X j=1 cjciK(zj, zi) ∈ R .
K is called positive definite if
N X i=1 N X j=1 cjciK(zj, zi) ≥ 0 .
Theorem 2.3. Let H be a reproducing kernel Hilbert space, and K be the reproducing kernel of H. Then K is positive definite. Proof. N X i=1 N X j=1 cjciK(zj, zi) = N X i=1 N X j=1 cjcihKzi, Kzji = N X i=1 N X j=1 hciKzi, cjKzji = N X i=1 ciKzi 2 ≥ 0.
Theorem 2.4. Suppose K : Ω × Ω → C is positive definite. Then, there is a unique Hilbert space H of functions onΩ admitting K as a reproducing kernel.
Proof. Let ˜ H = {f = n X i=1 aiK(., zi) : n ∈ N , (ai)N1 ⊂ C , (zi)N1 ⊂ Ω} . For f =Pn
i=1aiK(., zi) and g =Pmj=1bjK(., wj), we define the inner product on ˜H by
hf, giH˜ = n X i=1 m X j=1 aibjK(zi, wj) . (2.2) Thus, we have hf, KziH˜ = n X i=1 aiK(z, zi) = f (z),
and since K is positive definite, then for every f ∈ ˜H, we have hf, f iH˜ ≥ 0. And by Schwartz inequality, if
kf kH˜ = 0 then f = 0. Thus ( ˜H, h·, ·iH˜) is a pre-Hilbert space. Let H = ˜H. We need to prove that H is the unique Hilbert space admitting K as a reproducing kernel. Let (fn)n∈Nbe a Cauchy sequence in ˜H. Then, for
all z ∈ Ω, we have
|fm(z) − fn(z)| = |hfm, KziH˜− hfn, KziH˜|
= |hfm− fn, KziH˜|
≤ kfm− fnkH˜kKzkH˜ .
Therefore, there exists a function f : Ω → C, such that for any z ∈ Ω lim
n→+∞fn(z) = f (z) ,
and further
lim
n→+∞kfnkH˜ = kf kH.
Let f ∈ H, there exists (fn)n∈N⊂ ˜H such that fn→ f , thus
f (z) = lim
n→Nfn(z) = limn→Nhfn, KziH0 = hf, KziH.
Then, H admits a reproducing kernel K. Now suppose that this space is not unique, let H1 another Hilbert
space that admits K as reproducing kernel. By construction of H, we have H1 ⊂ H. We will show that
H ⊂ H1. By definition, for all z ∈ Ω, Kz ∈ H1, thus ˜H ⊂ H1. Furthermore we have for all f, g ∈ ˜H
hf, giH˜ = hf, giH1 . (2.3)
Let f ∈ H1, such that hf, KziH1 = 0, for all z ∈ Ω. Then f = 0 on Ω. Therefore the family {Kz : z ∈ Ω}
is total in H1. Then, for all f ∈ H1, There exists a Cauchy sequence in ˜H, such that
lim
Therefore H ⊂ H1.
Let f, g ∈ H, there exist (fn)n∈N, (gn)n∈Ntwo Cauchy sequences in ˜H which converge to f and g
respec-tively. Then
hf, giH1 = lim
n→+∞hfn, gniH˜ = hf, giH.
This completes the proof.
Proposition 2.1. Given a reproducing kernel Hilbert space H and its kernel K on Ω. Let z0 ∈ Ω. Then the
following are equivalent 1. kKz0k = 0.
2. Kw(z0) = 0 , (∀w ∈ Ω).
3. f (z0) = 0 , (∀f ∈ H).
Proof. The proof is obvious
2.2
Berezin Transform
Let H be a reproducing kernel Hilbert space on an open set Ω of C. If H has the property that for any z ∈ Ω there exists f ∈ H such that f (z) 6= 0, then by Proposition 2.1, kKzk > 0, ∀z ∈ Ω. In this case we can
normalize the reproducing kernel to obtain a family of unit vectors kz, as follows
kz(w) :=
K(w, z)
kKzk , ∀z, w ∈ Ω. (2.4)
kzis called the normalized reproducing kernel of H.
Definition 2.4. Let H be a reproducing kernel Hilbert space on an open set Ω of C. If T is a bounded linear operator onH. The Berezin transform of T ; denoted by ˜T , is the complex valued function on Ω defined by
˜
T (z) := hT kz, kzi.
The operator T is bounded, then
| ˜T (z)| = |hT kz, kzi|
≤ kT kzk ≤ kT k.
Where the last inequality is by Cauchy-Schwarz. Thus, the mapping T → ˜T is a contractive linear operator from L(H) into L∞(Ω).
Proposition 2.2. Let H be a reproducing kernel Hilbert space on an open set Ω of C. If T is a bounded linear operator onH, then the Berezin transform of T has the following properties
1. IfT is self-adjoint, then ˜T is real-valued. 2. IfT is positive, then ˜T is nonnegative
3. The Berezin transform of the adjointT∗ of T is the complex conjugate of ˜T
Proof. The proof is obvious.
Proposition 2.3. Let H be a reproducing kernel Hilbert space of holomorphic functions on a open set Ω of C. The mappingT → ˜T is one-to-one from L(H) into L∞(Ω).
Proof. Suppose ˜T = 0. Since
˜ T (z) = hT Kz, Kzi kKzk2 , z ∈ Ω. We have hT Kz, Kzi = 0, z ∈ Ω.
Let’s consider the function f (z, w) = hT Kw, Kzi, z, w ∈ Ω. It is clear that f (z, w) is holomorphic in z, and
antiholomorphic in w, and vanishes on the diagonal:
f (z, z) = 0, ∀z ∈ Ω,
then, f = 0, and we have
f (z, w) = (T kw)(z), z, w ∈ Ω.
It follows that T Kw = 0, for all w ∈ Ω. As the reproducing kernels span the whole space H, we must have
T = 0.
Proposition 2.4. Let H be a reproducing kernel Hilbert space on an open set Ω of C. Suppose kz → 0 weakly
inH, as z → ∂Ω. If T is compact operator on H, then ˜
T (z) → 0, asz → ∂Ω.
Proof. By Cauchy-Schwarz inequality, we have
| ˜T (z)| ≤ kT kzk.
3
Ahlfors Distortion Theorem
3.1
Harmonic Measure
Let H = {z ∈ C : Im (z) > 0}. For a < b the function
θ(z) = arg z − b z − a (3.1) is harmonic on H, in fact θ(z) = Im log z − a z − b
with θ(z) = π on ]a, b[ and θ(z) = 0 on R\[a, b].
Figure 3.1: The harmonic function θ
Let E = ∪ni=1]ai, bi[ be a finite union of open intervals such that bj−1 < aj < bj. And let
θj(z) = arg
z − bj
z − aj
,
then, we define the harmonic measure of E at z by :
ω(z, E, H) = n X j=1 θj(z) π . (3.2)
We have the following properties: 1. 0 < ω(z, E, H) < 1 : ∀z ∈ H
2. ω(z, E, H) → 1 : z → E 3. ω(z, E, H) → 0 : z → R\E
The function ω(z, E, H) is the unique harmonic function on H that satisfies the properties (1), (2) and (3). The uniqueness of ω(z, E, H) is a consequence of the following lemma.
Lemma 3.1 (Lindel¨of maximum principle). Let f (z) be a harmonic bounded function on Ω such that Ω ( C. And letF be a finite subset of ∂Ω, suppose that
lim sup
z→ξ
f (z) ≤ 0 : ∀ξ ∈ ∂Ω\F.
Thenf (z) ≤ 0 on Ω
Proof. Fix w /∈ Ω. The application 1/(z − w) transforms Ω into a bounded region. Then without loss of generality we may assume that Ω is bounded. Let F = {ξ1, ξ2, ..., ξn}, and let > 0, we define
f(z) = f (z) − n X j=1 log diam(Ω) |z − ξj|
f(z) is harmonic on Ω, and lim supz→ξu(z) ≤ 0, ∀ξ ∈ ∂Ω and ∀ > 0, Thus
f (z) ≤ lim →0 n X j=1 log diam(Ω) |z − ξj| = 0
Throughout, we will define the harmonic measure on the Borel set ∂H = R in term of the Poisson kernel of H, then, we need the following proposition
Proposition 3.1. Soient a, b ∈ R, (a < b). Then
ω(z, ]a, b[, H) = Z b a y (t − x)2+ y2 dt π (3.3) for allz = x + iy ∈ H.
Proof. Let z = x + iy ∈ H
ω(z, ]a, b[, H) = arg (x − b) + iy (x − a) + iy
= arg((x − b) + iy) − arg((x − a) + iy) = 1 πarctan x − a y − 1 π arctan x − b y = Z a b x−t y 0 1 + x−t y 2 dt π = Z b a y (t − x)2+ y2 dt π
Definition 3.1. Let E ⊂ R is measurable set, we define the harmonic measure of E at z ∈ H by
ω(z, E, H) = Z E y (t − x)2+ y2 dt π (3.4)
When E is a finite union of open intervals, the definition 3.4 is equivalent to 3.1. For z = x + iy ∈ H, the function Pz(t) = 1 π y (t − x)2+ y2
is called the Poisson kernel of H.
Note that the harmonic measure ω(z, E, H) is a harmonic function in its first variable z and a probability measure in its second variable E.
Definition 3.2. Lat Ω ( C, for z ∈ Ω and E ⊂ ∂Ω we define the harmonic measure on Ω by
ω(z, E, Ω) = ω(ϕ(z), ϕ(E), H) (3.5)
whereϕ : Ω → H is a conformal map of Ω onto H.
The harmonic function ω(z, E, Ω) satisfies the properties (1), (2) and (3), and by lemma 3.1, the definition (3.5) is independent of the choice of ϕ.
Proposition 3.2. Let z ∈ D and E ⊂ ∂D measurable, then
ω(z, E, D) = Z E 1 − |z|2 |eiθ− z|2 dθ 2π (3.6)
The function Pz(θ) = 1 2π 1 − |z|2 |eiθ− z|2 (3.7)
is called the Poisson kernel of D.
Figure 3.2: The harmonic measure of an arc of T
Note that for z = 0, the harmonic measure ω(0, E, D) = A(E), where A is the Lebesgue measure on T.
3.2
Extremal Distance
Let Ω be domain. By definition, a Path family Γ in Ω is a non empty set of countable unions of rectifiable arcs in Ω. Each element of γ ∈ Γ is called curve, the curve γ is not necessarily connected and may have many self intersections. The euclidean length of the path family Γ is given by
inf
γ∈Γ
Z
γ
ds (3.8)
where ds is the arc length. The quantity (3.8) is not invariant by conformal mapping. For this reason we will define a distance invariant under conformal transformations, and therefore we need more definitions.
Definition 3.3. Let ρ : Ω → R+. We say thatρ is a metric if the quantity
A(Ω, ρ) = Z
Ω
ρ2(x + iy)dxdy (3.9)
satisfies0 < A(Ω, ρ) < +∞
Definition 3.4. Let ρ be a metric and Γ a path family. We define the ρ-length of Γ by
L(Γ, ρ) = inf
γ∈Γ
Z
γ
ρ(z)|dz|
and theextremal length of Γ by
λΩ(Γ) = sup ρ
L(Γ, ρ)2
whereA(Ω, ρ) is the area given in (3.9).
For ϕ is a conformal mapping defined on Ω, by change of variable we find that ϕ transforms the metric ρ on Ω into the metric ρ(ϕ−1(z))|(ϕ−1)0(z)| on ϕ(Ω). Then the extremal length is invariant under conformal mapping in the sense
λϕ(Ω)(ϕ(Γ)) = λΩ(Γ)
Definition 3.5 (Extremal Distance). Let Ω be a domain E ⊂ Ω and F ⊂ Ω. The extremal distance dΩ(E, F )
betweenE and F is defined by
dΩ(E, F ) = λΩ(Γ)
whereΓ is the family of connected arcs in Ω that join E and F .
Figure 3.3: Connecting curves γ
Proposition 3.3 (Rectangle). Let R = {z = x + iy ∈ C : 0 < x < l et 0 < y < h} be a rectangle of length l and heighth. The extremal distance between the vertical sides E and F is given by
dR(E, F ) =
l
h. (3.11)
Proof. Let Γ be the path family of connected arcs in R joining E and F , and let ρ be a metric in R. Then by Cauchy-Schwarz inequality L2(Γ, ρ) = Z l 0 ρ(x + iy)dx 2 ≤ l Z l 0 ρ2(x + iy)dx. (3.12)
Integrating (3.12) with respect to y, we have
Figure 3.4 therefore dR(E, F ) = sup ρ L2(Γ, ρ) A(R, ρ) ≤ l h.
For equality, we use the metric ρ(z) = 1, and then we have L(Γ, ρ) = l and A(R, ρ) = hl.
The metric ρ that verify the equality in (3.10) is called the extremal metric. More generally, let Ω be a Jordan domain, and let E = [ξ1, ξ2] ⊂ ∂Ω and F = [ξ3, ξ4] ⊂ ∂Ω (E ∩ F = ∅). Then there exists a unique
l > 0 and a conformal maps ϕ from Ω into R = {z = x + iy ∈ C : 0 < x < l et 0 < y < 1} such that
ϕ(ξ1) = 0 , ϕ(ξ2) = i , ϕ(ξ3) = l + i et ϕ(ξ4) = l.
By the invariance of the extremal distance by conformal transformations, we have
dΩ(E, F ) = dR(ϕ(E), ϕ(F )) = l,
and the extremal metric is |ϕ0|.
Proposition 3.4 (The annulus). Let A = {z ∈ C : r < |z| < R}, Cr = {z ∈ C : |z| = r} and
CR= {z ∈ C : |z| = R}. Then the extremal distance dA(Cr, CR) between CrandCRinA is
dA(Cr, CR) = 1 2πlog R r . (3.13)
Proof. Without loss of generality assume that 0 < r < R < +∞, let Γ path family of connected arcs A joining E and F , and let ρ be a metric on A. So, by the Cauchy-Schwarz inequality, we have
L2(Γ, ρ) ≤ Z R r ρ(teiθ)dt 2 ≤ log R r Z R r ρ(teiθ)2tdt. (3.14)
Integrating (3.14) with respect to θ, we have
2πL2(Γ, ρ) ≤ log R r A(A, ρ). Thus dA(Cr, CR) ≤ 1 2πlog R r .
By using the metric ρ(z) =|z|1 . Then L(Γ, ρ) = Z R r dt t = log R r et A(A, ρ) = Z 2π 0 Z R r 1 t2tdtdθ = 2π log R r . Therefore dA(Cr, CR) = 1 2πlog R r .
More generally, if Ω is a doubly connected domain. By the theorem of Tsuji, there exists a conformal map ϕ : Ω → A. So, the extremal distance between E and F of Ω is
dΩ(E, F ) = 1 2πlog R r ,
and the extremal metric is ρ(z) = |ϕ|ϕ(z)|0(z)| .
Proposition 3.5. Let Ω ⊂ Ω0two domains, and letΓ a path family in Ω. Then
λΩ0(Γ) = λΩ(Γ) . (3.15)
Furthermore, ifΓ0is a path family inΩ0 such that for everyγ0 ∈ Γ0, there existsγ ∈ Γ such that γ ⊂ γ0. Then
λΩ0(Γ0) ≥ λΩ(Γ) (3.16)
Proof. Let ρ0 be a metric on Ω0, and ρ0 = ρ/Ω, then L(Γ, ρ) = L(Γ, ρ0) and A(Ω, ρ) ≤ A(Ω0, ρ). Taking the
supremum on ρ0, we find
λΩ0(Γ) ≤ λΩ(Γ) .
Let ρ be a metric on Ω, we define the metric ρ0 on Ω0 by ρ0 = ρχΩ. Then L(Γ, ρ) = L(Γ, ρ0) and A(Ω, ρ) =
A(Ω0, ρ0). Taking the supremum on ρ, we have
λΩ0(Γ) ≥ λΩ(Γ) .
Thus, it has (3.15). To prove (3.16), we use (3.15) and the fact that λΩ0(Γ0) ≥ λΩ0(Γ).
The equality (3.15) indicates that the extremal length depends of the path family Γ and not the domain Ω, for this reason we write often λ(Γ) for λΩ(Γ). The Proposition 3.5 show that the extremal distance decreases
if one of the sets E, F and Ω decreases, and when E and F are close, we have dΩ(E, F ) = dΩ\E∪F(E, F ).
Proposition 3.6. Let Γ1 andΓ2 two paths family in two domains Ω1 andΩ2 respectively. Let Γ be a path
family in domainΩ which contains Ω1∪ Ω2. If for everyγ ∈ Γ, there exist γ1 ∈ Γ1 andγ2 ∈ Γ2 such that
γ1∪ γ2⊂ γ. Then
λ(Γ) ≥ λ(Γ1) + λ(Γ2). (3.17)
Proof. If λ(Γ1) or λ(Γ2) equal to 0 or +∞, then the result comes from the Proposition 3.5. Let ρ1, ρ2be two
metrics on Ω1, Ω2respectively, normalized by
L(Γ1, ρ1)
A(Ω1, ρ1)
= L(Γ2, ρ2) A(Ω2, ρ2)
= 1 .
Then, we define the metric ρ3on Ω by ρ3= ρ1χΩ1 + ρ2χΩ2. Thus
L(Γ, ρ) ≥ L(Γ1, ρ1) + L(Γ2, ρ2) ,
and
A(Ω, ρ) = A(Ω1, ρ1) + A(Ω2, ρ2),
therefore
L2(Γ, ρ)
A(Ω, ρ) ≥ L(Γ1, ρ1) + L(Γ2, ρ2) . (3.18)
Taking the supremum on ρ1and ρ2in (3.18), and on ρ we find (3.17).
Proposition 3.7. Let Γ1, Γ2be path families contained in disjoint open setsΩ1andΩ2respectively. LetΓ be a
path family inΩ which contains Ω1∪ Ω2such that for everyγ ∈ Γ1∪ Γ2, there existsγ0 ∈ Γ such that γ0 ⊂ γ.
Then 1 λ(Γ) ≥ 1 λ(Γ1) + 1 λ(Γ2) . (3.19)
Proof. Let ρ be a metric on Ω normalized by L(Γ, ρ) = 1. Then L(Γ1, ρ) ≥ 1 and L(Γ2, ρ) ≥ 1. So
A(Ω, ρ) ≥ A(Ω1, ρ) + A(Ω2, ρ) .
By passing to supremum, we obtain (3.19).
3.3
Ahlfors Distortion Theorem
Proposition 3.8. Let RL= {z ∈ C ; |Re(z)| < L and |Im(z)| < 1}. If EL := {z ∈ ∂RL ; |Re(z)| < L} is
the union of the vertical edges ofRL, then
e−π2L≤ ω(0, EL, RL) ≤ 8
πe
−π
Proof. Let
SL= {z ∈ ∂RL; |Re(z)| > −L and |Im(z)| < 1}
and let
ωL(z) = ω(z, ∂SL∩ {Re(z) = −L}, SL).
Using the map z → eπ2z, we obtain
ωL(z) = 2 πarg eπ2z+ ie− π 2L eπ2z− ie− π 2L ! . (3.21) In particular ωL(0) = π4tan−1(e− π 2L). By equation 3.21, we have ωL(z) ≤ ωL(L), Re(z) = L. Thus ω(z, EL, RL) ≤ ωL(z) + ωL(−z) ≤ (1 + ωL(L))ω(z, EL, RL) For z = 0, we have 2 ωL(0) 1 + ωL(L) ≤ ω(z, EL, RL) ≤ 2ωL(0). (3.22)
We have, ωL(L) = ω2L(0) = π4 tan−1(e−πL). Therefore, 8 πtan −1(e−π 2L 1 +π4tan−1(e−πL ≤ ω(0, EL, RL) ≤ 8 πtan −1(e−π 2L (3.23)
Now, from the inequality
π 4t ≤ tan −1(t) ≤ min(t,π 4), ∀t ∈ (0, 1), we have e−π2L≤ ω(0, EL, RL) ≤ 8 πe −π 2L
The Proposition 3.8 connects the harmonic measure of the ends of RL at its center point to the extremal
distance L between the two ends.
Let Ω be a Jordan domain, and let E be an arc on ∂Ω and let z0 ∈ Ω. Consider all Jordan arcs σ ⊂ Ω that
join z0to ∂Ω \ E, and we define
λ(z, E) = sup
σ
dΩ\σ(σ, E).
Where the supremum is taken over all Jordan arcs.
Theorem 3.1. Let Ω be a Jordan domain, let E be an arc of ∂Ω and let z0 ∈ Ω. Then
e−πλ(z0,E) ≤ ω(z
0, E, Ω) ≤
8 πe
Proof. By conformal invariance we may suppose that Ω = D, and z0 = 0 and E is an arc of T. Let E1, E2be
two disjoint arcs on ∂D given by
E1∪ E2 = {eiθ , e2iθ∈ E}.
The arcs E1 and E2 are symmetric about 0, there exists a conformal map ϕ of D onto a rectangle R such that
R has center 0 and sides parallel to the axes, such that ϕ(0) = 0 and
b
E1:= ϕ(E1) , Eb2:= ϕ(E2),
are the vertical sides of R. Let σ be any arc in D connecting 0 to ∂D \ E and letσ = {z ∈ R , (ϕb
−1(z))2 ∈ σ}.
The map√· is conformal on D \ σ, and ϕ ◦√· map D \ σ conformally onto R \bσ. By Proposition 3.6, we have
dR( bE1, bE2) ≥ d( bE1,bσ) + d(σ, bb E2) = 2dD(σ, E). (3.24)
Because ω(z2, E, D) = ω(z, E1∪ E2, D), we have
ω(0, E, D) = ω(0, E1∪ E2, D) = ω(0, bE1∪ bE2, R).
Then, by Proposition 3.8, we obtain the result.
Theorem 3.2. Let Ω be a Jordan domain, and let E be a finite union of arcs contained in ∂Ω and z0∈ Ω. Then
ω(z0, E, Ω) ≤
8 πe
−πλ(z0,E).
Proof. By conformal invariance we may suppose that Ω = D and z0 = 0. Let σ be an arc from ∂D \ E and set
α = dΩ\E(σ, E). Let σ1 = {z ∈ C , z2 ∈ σ} and E1∪ E2 = {eiθ , e2iθ∈ E}. There exists a conformal map
ϕ of D onto a rectangle R with horizontal slits removed so that
b
E1:= ϕ(E1) , Eb2:= ϕ(E2),
are the vertical ends of R. Then by Theorem 3.1, we obtain
dR( bE1, bE2) ≥ 2dD(σ, E).
Therefore
ω(0, E, D) ≤ 8 πe
−πλ(0,E).
Now, let us consider the strip domain
Ω := z = x + iy ∈ C , |y − m(x)| < θ(x) 2 and a < x < b .
Let a = x0 < x1 < · · · < xn= b and σj = Ω ∩ {z ∈ C , Re(z) = xj, where j = 1, · · · , n. By Theorem
3.6, we have dΩ(σ0, σn) ≥ n X j=1 d(σj−2, σj).
Figure 3.5: Strip domain Ω
If xj − xj−1is small, the region between σj−1and σj is approximately a rectangle having xj− xj−1as base,
θ(xj) as height, and xj + im(xj) as the midpoint of its tight vertical side. The map
z → z − Im(xj) θ(xj)
send this rectangle to a rectangle centered on R with height 1 and width (xj−xj−1)/θ(xj). Then by Proposition
3.3 d(σj−1, σj) xj− xj−1 θ(xj) . The map ψ : z = x + iy → Z x a 1 θ(t)dt + i y − m(x) θ(x) , defined from Ω onto a rectangle R of height 1 and length
Z b
a
1 θ(t)dt. The extremal metric for dΩ(σ0, σn) is given by ρ(z) = |ϕ 0
(z)|, where ϕ is a conformal map of Ω onto a rectangle. The idea is to replace the holomorphic function ϕ by ψ and using as metric
ρ(z) = |∇Re(ψ(z))| = 1
θ(x) , z = x + iy. Let γ is a curve connecting σ0and σnin Ω, then
Z γ ρ(z)|dz| ≥ Z b a 1 θ(x)dx. Furthermore Z γ ρ2(z)dydx ≥ Z b a 1 θ(x)dx. Thus dΩ(σ0, σn) ≥ Z b a 1 θ(x)dx. (3.25)
Theorem 3.3. Let Ω be a Jordan domain and z0 ∈ Ω. Let Re(z0) = x0 < b and suppose F ⊂ {z ∈
∂Ω , Re(z) ≥ b}. Assume that for x0 < x < b, there exists Ix ⊂ Ω ∩ {z , Re(z) = x} separating z0fromF
and setθ(x) := `(Ix). Then
ω(z0, F, Ω) ≤ 8 πe −πRb x0 dx θ(x).
Where`(Ix) is the Euclidean length of Ix.
Proof. Let ˜Ω denote the points of Ω which are not separated from z0by some Ixand ˜F denote the point of ∂ ˜Ω
which are separated from z0by all Ix, where x0< x < b. We can suppose that ˜F is finite union of arcs on ∂ ˜Ω.
By replacing Ω with ϕ−1(|z| < r) where ϕ is a conformal map of Ω onto D. Let σ ⊂ {Re(z) = x0} be a curve
connecting z0to ∂ ˜Ω. By Proposition 3.5, we have
ω(z0, F, Ω) ≤ ω(z0, ˜F , ˜Ω),
then the result is by inequality 3.25 and Theorem 3.2
The converse of Theorem 3.3 is true under conditions. It is proved by constructing a new metric. And in this case we need to define the conjugate extremal distance d∗Ω(E, F ) to be the extremal length of the family Γ∗of curves that separate E from F . For the proof of the converse theorem see [15] page 160.
Theorem 3.4. Suppose Ω := z = x + iy ∈ C , |y − m(x)| < θ(x) 2 andx > a ,
is a Jordan domain and Letz0= x0+ iy0∈ Ω. Suppose
{z = x + iy ∈ C , |x − x0| < δ and |y − y0| < δ} ⊂ Ω. If Z +∞ a m0(x)2− θ0(x)2 θ(x) dx = A < ∞. Then wheneverx0 < b and F = {z ∈ Ω , Re(z) ≥ b},
ω(z0, F ; Ω) ≥ Ce −πRb
x0 dx θ(x).
WhereC is a constant depending only on A and θ.
For x ∈ (x0− δ, x0+ δ). Theorem 3.4 implies that Theorem 3.3 is the best result possible for domains with
4
Spaces Of Holomorphic Functions
We denote by D the open unit disc of the complex plane C
D = {z ∈ C : |z| < 1} and by T the unit circle of C
T = {z ∈ C : |z| = 1} .
The set of holomorphic functions on open set Ω of C is denoted by H(Ω). We denote by C2(Ω) the set of functions of class C2on Ω. If w ∈ Ω, The laplacian of f in w is defined by
∆f (w) = ∂ 2f ∂x2(w) + ∂2f ∂y2(w). (4.1) We find easily ∆f (w) = 4 ∂ 2f ∂z∂z(w) . The Lebesgue measure normalized on D will be noted by dA
dA(z) = 1
πdxdy = 1
πrdrdθ, z = x + iy = re
iθ
Theorem 4.1 (H¨older inequality). Let X be a measure space, of positive measure µ. Let f, g two measurable functions onX with values in R+. Then :
Z X f (x)g(x)dµ(x) ≤ Z X f (x)pdµ(x) 1pZ X f (x)qdµ(x) 1q (4.2) ∀p, q ∈ R+such that 1 p + 1 q = 1.
Theorem 4.2 (Minkowski inequality). Let X be a measure space, of positive measure µ. Let f, g two measur-able functions onX with values in R+. Then :
Z X (f (x) + g(x))pdµ(x) 1p ≤ Z X f (x)pdµ(x) 1p + Z X f (x)pdµ(x) 1p (4.3)
