Function Classes on the Unit Disc: An Introduction

Description

This monograph contains a study on various function classes, a number of new results and new or easy proofs of old results (Fefferman-Stein theorem on subharmonic behavior, theorems on conjugate functions and fractional integration on Bergman spaces, Fefferman's duality theorem), which are interesting for specialists; applications of the Hardy-Littlewood inequalities on Taylor coefficients to (C, α)-maximal theorems and (C, α)-convergence; a study of BMOA, due to Knese, based only on Green's formula; the problem of membership of singular inner functions in Besov and Hardy-Sobolev spaces; a full discussion of g-function (all p > 0) and Calderón's area theorem; a new proof, due to Astala and Koskela, of the Littlewood-Paley inequality for univalent functions; and new results and proofs on Lipschitz spaces, coefficient multipliers and duality, including compact multipliers and multipliers on spaces with non-normal weights. It also contains a discussion of analytic functions and lacunary series with values in quasi-Banach spaces with applications to function spaces and composition operators. Sixteen open questions are posed. The reader is assumed to have a good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series. Further information can be found at the author's website at http://poincare.matf.bg.ac.rs/~pavlovic. Preface 1 The Poisson integral and Hardy spaces 1.1 The Poisson integral 1.1.1 Borel measures and the space h1 1.2 Spaces hp and Lp(T) (p > 1) 1.3 Space hp (p < 1) 1.4 Harmonic conjugates 1.4.1 Privalov–Plessner’s theorem and the Hilbert operator 1.5 Hardy spaces: basic properties 1.5.1 Radial limits and mean convergence 1.5.2 Space H1 1.6 Riesz projection theorem 1.6.1 Aleksandrov’s theorem Further notes and results 2 Subharmonic functions and Hardy spaces 2.1 Basic properties of subharmonic functions 2.1.1 Maximum principle 2.2 Properties of the mean values 2.3 Riesz measure 2.3.1 Riesz’ representation formula 2.4 Factorization theorems 2.4.1 Inner–outer factorization 2.5 Some sharp inequalities 2.6 Hardy–Stein identities 2.6.1 Lacunary series 2.7 Subordination principle 2.7.1 Composition with inner functions 2.7.2 Approximation with inner functions Further notes and results 3 Subharmonic behavior and mixed norm spaces 3.1 Quasi-nearly subharmonic functions 3.2 Regularly oscillating functions 3.3 Mixed norm spaces: definition and basic properties 3.4 Embedding theorems 3.5 Fractional integration 3.6 Weighted mixed norm spaces 3.6.1 Lacunary series in mixed norm spaces 3.6.2 Bergman spaces with rapidly decreasing weights 3.6.3 Mixed norm spaces with subnormal weights 3.7 Lq-integrability of lacunary power series 3.7.1 Lacunary series in C[0, 1] Further notes and results 4 Taylor coefficients with applications 4.1 Using interpolation of operators on Hp 4.1.1 An embedding theorem 4.1.2 The case of monotone coefficients 4.2 Strong convergence in H1 4.2.1 Generalization to (C, a)-convergence 4.3 A (C, a)-maximal theorem Further notes and results 5 Besov spaces 5.1 Decomposition of Besov spaces: case 1 < p < ⋄ 5.2 Maximal function 5.3 Decomposition of Besov spaces: case 0 < p =8 5.3.1 Radial limits of Hardy–Bloch functions 5.4 Duality in the case 0 < p ≤∞ 5.5 Embeddings between Hardy and Besov spaces 5.6 Best approximation by polynomials 5.7 Normal Besov spaces 5.8 Inner functions in Besov and Hardy–Sobolev spaces 5.8.1 Approximation of a singular inner function 5.8.2 Hardy–Sobolev space Sp 1/p 5.8.3 f-property and K-property Further notes and results 6 The dual of H1 and some related spaces 6.1 Norms on BMOA 6.2 Garsia’s and Fefferman’s theorems 6.2.1 Fefferman’s duality theorem 6.3 Vanishing mean oscillation 6.4 BMOA and Bp 1/p 6.4.1 Tauberian nature of Bp 1/p 6.5 Coefficients of BMOA functions 6.6 Bloch space 6.7 Mean growth of Hp-Bloch functions 6.8 Composition operators on B and BMOA 6.8.1 Weighted Bloch spaces 6.9 Proof of the bi-Bloch lemma Further notes and results 7 Littlewood–Paley theory 7.1 Vector maximal theorems and Calderon’s area theorem 7.2 Littlewood–Paley g-theorem 7.3 Applications of the (C,m)-maximal theorem 7.4 Generalization of the 𝑔-theorem 7.5 Proof of Calderón’s theorem 7.6 Littlewood–Paley inequalities 7.7 Hyperbolic Hardy classes Further notes and results 8 Lipschitz spaces of first order 8.1 Definitions and basic properties 8.1.1 Lipschitz spaces of analytic functions 8.1.2 Mean Lipschitz spaces 8.2 Lipschitz condition for the modulus 8.3 Composition operators 8.4 Composition operators into HΛ pω 8.5 Inner functions Further notes and results 9 Lipschitz spaces of higher order 9.1 Moduli of smoothness and related spaces 9.2 Lipschitz spaces and spaces of harmonic functions 9.3 Conjugate functions 9.4 Integrated mean Lipschitz spaces 9.4.1 Generalized Lipschitz spaces 9.5 Invariant Besov spaces 9.6 BMO-type characterizations of Lipschitz spaces 9.6.1 Division and multiplication by inner functions Further notes and results 10 One-to-one mappings 10.1 Integral means of univalent functions 10.1.1 Distortion theorems 10.2 Membership of univalent functions in some function classes 10.3 Quasiconformal harmonic mappings 10.3.1 Boundary behavior of QCH homeomorphisms of the disk 10.4 Hp-classes of quasiconformal mappings Further notes and results 11 Coefficients multipliers 11.1 Multipliers on abstract spaces 11.1.1 Compact multipliers 11.2 Multipliers for Hardy and Bergman spaces 11.2.1 Multipliers from H1 to BMOA 11.3 Solid spaces 11.3.1 Solid hull of Hardy spaces (0 < p < 1) 11.4 Multipliers between Besov spaces 11.4.1 Monotone multipliers 11.5 Multipliers of spaces with subnormal weights 11.6 Some applications to composition operators Further notes and results 12 Toward a theory of vector-valued spaces 12.1 Some properties of admissible spaces 12.2 Subharmonic behavior of ||F(z)||x 12.2.1 Banach envelope of Hp(X), 0 < p < 1 12.3 Linear operators on Hardy and Bergman spaces 12.4 Proof of the Coifman–Rochberg theorem Further notes and results A Quasi-Banach spaces A.1 Quasi-Banach spaces A.2 q-Banach envelopes A.3 Closed graph theorem A.4 F-spaces A.4.1 Nevanlinna class A.5 Spaces lp A.6 Lacunary series in quasi-Banach spaces A.6.1 Lp-integrability of lacunary series on (0, 1) Further notes and results B Interpolation and maximal functions B.1 Riesz–Thorin theorem B.2 Weak Lp-spaces and Marcinkiewicz’s theorem B.3 Classical maximal functions B.4 Rademacher functions and Khintchin’s inequality B.5 Nikishin’s theorem B.6 Nikishin–Stein’s theorem B.7 Banach’s principle and the theorem on a.e. convergence B.8 Vector-valued maximal theorem Further notes and results Bibliography Index