Topics in Hardy Classes and Univalent Functions

These notes are based on lectures given at the University of Virginia over the past twenty years. They may be viewed as a course in function theory for nonspecialists. Chapters 1-6 give the function-theoretic background to Hardy Classes and Operator Theory, Oxford Mathematical Monographs, Oxford University Press, New York, 1985. These chapters were written first, and they were origi­ nally intended to be a part of that book. Half-plane function theory continues to be useful for applications and is a focal point in our account (Chapters 5 and 6). The theory of Hardy and Nevanlinna classes is derived from proper­ ties of harmonic majorants of subharmonic functions (Chapters 3 and 4). A selfcontained treatment of harmonic and subharmonic functions is included (Chapters 1 and 2). Chapters 7-9 present concepts from the theory of univalent functions and Loewner families leading to proofs of the Bieberbach, Robertson, and Milin conjectures. Their purpose is to make the work of de Branges accessible to students of operator theory. These chapters are by the second author. There is a high degree of independence in the chapters, allowing the material to be used in a variety of ways. For example, Chapters 5-6 can be studied alone by readers familiar with function theory on the unit disk. Chapters 7-9 have been used as the basis for a one-semester topics course.

1 Harmonic Functions.- 1.1 Introduction.- 1.2 Uniqueness principle.- 1.3 The Poisson kernel.- 1.4 Normalized Lebesgue measure.- 1.5 Dirichlet problem for the unit disk.- 1.6 Properties of harmonic functions.- 1.7 Mean value property.- 1.8 Harnack’s theorem.- 1.9 Weak compactness principle.- 1.10 Nonnegative harmonic functions.- 1.11 Herglotz and Riesz representation theorem.- 1.12 Stieltjes inversion formula.- 1.13 Integral of the Poisson kernel.- 1.14 Examples.- 1.15 Space h1(D).- 1.16 Characterization of h1(D).- 1.17 Nontangential convergence.- 1.18 Fatou’s theorem.- 1.19 Boundary functions.- Examples and addenda.- 2 Subharmonic Functions.- 2.1 Introduction.- 2.2 Upper semicontinuous functions.- 2.3 Subharmonic functions.- 2.4 Some properties of subharmonic functions.- 2.5 Maximum principle.- 2.6 Convergence of mean values.- 2.7 Convex functions.- 2.8 Structure of convex functions.- 2.9 Jensen’s inequality.- 2.10 Composition of convex and subharmonic functions.- 2.11 Vector- and operator-valued functions.- 2.12 Subharmonic functions from holomorphic functions.- 3 Part I Harmonic Majorants Part II Nevanlinna and Hardy-Orlicz Classes.- 3.1 Introduction.- 3.2 Least harmonic majorant.- 3.3 Existence of least harmonic majorants.- 3.4 Construction of harmonic majorants.- 3.5 Class shl(D).- 3.6 Characterization of sh1(D).- 3.7 Absolutely continuous component of a related measure.- 3.8 Uniformly integrable family.- 3.9 Strongly convex functions.- 3.10 Theorem of de la Vallée Poussin and Nagumo.- 3.11 Singular component of associated measures.- 3.12 Sufficient conditions for absolute continuity.- 3.13 Theorem of Szegö-Solomentsev.- 3.14 Remark.- 3.15 Hardy and Nevanlinna classes.- 3.16 Linearity of the classes.- 3.17 Properties of log+x.- 3.18 Majorants for strongly convex functions.- 3.19 Compositions and restrictions.- 3.20 Quotients of bounded functions.- Examples and addenda.- 4 Hardy Spaces on the Disk.- 4.1 Introduction.- 4.2 Inner and outer functions.- 4.3 Rational inner functions.- 4.4 Infinite products.- 4.5 An infinite product.- 4.6 Blaschke products.- 4.7 Inner functions with no zeros.- 4.8 Singular inner functions.- 4.9 Factorization of inner functions.- 4.10 Boundary functions for N(D).- 4.11 Characterization of N(D).- 4.12 Condition on zeros.- 4.13 N(D) as an algebra.- 4.14 Characterization of N+(D).- 4.15 N+(D) as an algebra.- 4.16 Estimates from boundary functions for N+(D).- 4.17 Outer functions in N+(D).- 4.18 Characterization of ??(D).- 4.19 Nevanlinna and Hardy-Orlicz classes on the boundary.- 4.20 Szegö’s problem.- 4.21 Classes HP(D) and HP(?).- 4.22 Characterization of HP(D).- 4.23 Characterization of HP(?).- 4.24 Connection between HP(D) and HP(?).- 4.25 Hp(?) as a subspace of LP(?).- 4.26 Hp(D) and HP(?) as Banach spaces.- 4.27 F and M Riesz theorem.- 4.28 H2(D) and H2(?).- 4.29 Sufficient conditions for outer functions.- 4.30 Beurling’s theorem.- 4.31 Theorem of Szegö, Kolmogorov, and Kre?n.- 4.32 Closure of trigonometric functions in Lp(?).- 5 Function Theory on a Half-Plane.- 5.1 Introduction.- 5.2 Poisson representation.- 5.3 Nevanlinna representation.- 5.4 Stieltjes inversion formula.- 5.5 Fatou’s theorem.- 5.6 Boundary functions for N(?).- 5.7 Limits of nondecreasing functions.- 5.8 Nonnegative harmonic functions.- 5.9 Theorem of Flett and Kuran.- 5.10 Nevanlinna and Hardy-Orlicz classes.- 5.11 Notation and terminology.- 5.12 Szegö’s problem on the line.- 5.13 Inner and outer functions.- 5.14 Examples and miscellaneous properties.- 5.15 Hardy classes.- 5.16 Characterization of ?P(I?).- 5.17 Inclusions among classes.- 5.18 Poisson representation for ?P(?).- 5.19 Cauchy representation for Hp(?).- 5.20 Characterization of HP(?).- 5.21 Hp(?) as a subspace of N+(?).- 5.22 Condition for mean convergence.- 5.23 Hp(?)and ?P(?) as subspaces of N+(?).- 5.24 HP(?)and ?p(?) as Banach spaces.- 5.25 Local convergence to a boundary function.- 5.26 Remark on the definition of HP(?).- 5.27 Plancherel theorem.- 5.28 Paley-Wiener representation.- 5.29 Natural isomorphisms.- 5.30 Hilbert transforms.- 5.31 Real and imaginary parts of boundary functions.- 5.32 Cauchy transform on Lp(??, ?).- 5.33 Mapping f? f—i f on Lp(-?, ?) to HP(R).- 5.34 M Riesz theorem.- 5.35 Algebraic properties of Hilbert transforms.- Examples and addenda.- 6 Phragmén-Lindelöf Principle.- 6.1 Introduction.- 6.2 Phragmén-Lindelöf principle.- 6.3 Functions on a sector.- 6.4 Estimate from behavior on the imaginary axis.- 6.5 Blaschke products on the imaginary axis.- 6.6 Equivalence of the unit disk and a half-disk.- 6.7 Function theory on a half-disk.- 6.8 Estimates on a half-disk.- 6.9 Test to belong to N(?).- 6.10 Asymptotic behavior of Poisson integrals.- 6.11 Estimate from behavior on semicircles.- 6.12 Blaschke products on semicircles.- 6.13 Factorization of bounded type functions.- 6.14 Nevanlinna factorization and mean type.- 6.15 Formulas for mean type.- 6.16 Exponential type.- 6.17 Kre?n’s theorem.- 6.18 Inequalities for mean type.- Examples and addenda.- 7 Loewner Families.- 7.1 Definitions and overview of the subject.- 7.2 Preliminary results.- 7.3 Riemann mapping theorem.- 7.4 The Dirichlet space and area theorem.- 7.5 Generalization of the Dirichlet space.- 7.6 Bieberbach’s theorem.- 7.7 Size of the image domain.- 7.8 Distortion theorem.- 7.9 Carathéodory convergence theorem.- 7.10 Subordination.- 7.11 Technical lemmas.- 7.12 Parametric representation of Loewner families.- 8 Loewner’s Differential Equation.- 8.1 Loewner families and associated semigroups.- 8.2 Estimates derived from Schwarz’s lemma.- 8.3 Absolute continuity.- 8.4 Herglotz functions.- 8.5 Loewner’s differential equation.- 8.6 Solution of the nonlinear equation.- 8.7 Solution of Loewner’s differential equation.- 9 Coefficient Inequalities.- 9.1 Three famous problems.- 9.2 de Branges’ method.- 9.3 Construction of the weight functions.- 9.4 Askey-Gasper inequality.- Notes.