Author | Rosenblum, Marvin. author |
---|---|
Title | Topics in Hardy Classes and Univalent Functions [electronic resource] / by Marvin Rosenblum, James Rovnyak |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1994 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8520-1 |
Descript | XII, 250 p. online resource |
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 Harnackโs 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 Fatouโs 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 Jensenโs 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 Vallรฉe Poussin and Nagumo -- 3.11 Singular component of associated measures -- 3.12 Sufficient conditions for absolute continuity -- 3.13 Theorem of Szegรถ-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 Szegรถโs 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 Beurlingโs theorem -- 4.31 Theorem of Szegรถ, 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 Fatouโs 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 Szegรถโs 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? fโi f on Lp(-?, ?) to HP(R) -- 5.34 M Riesz theorem -- 5.35 Algebraic properties of Hilbert transforms -- Examples and addenda -- 6 Phragmรฉn-Lindelรถf Principle -- 6.1 Introduction -- 6.2 Phragmรฉn-Lindelรถf 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?nโs 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 Bieberbachโs theorem -- 7.7 Size of the image domain -- 7.8 Distortion theorem -- 7.9 Carathรฉodory convergence theorem -- 7.10 Subordination -- 7.11 Technical lemmas -- 7.12 Parametric representation of Loewner families -- 8 Loewnerโs Differential Equation -- 8.1 Loewner families and associated semigroups -- 8.2 Estimates derived from Schwarzโs lemma -- 8.3 Absolute continuity -- 8.4 Herglotz functions -- 8.5 Loewnerโs differential equation -- 8.6 Solution of the nonlinear equation -- 8.7 Solution of Loewnerโs differential equation -- 9 Coefficient Inequalities -- 9.1 Three famous problems -- 9.2 de Brangesโ method -- 9.3 Construction of the weight functions -- 9.4 Askey-Gasper inequality -- Notes