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.
Miroslav Pavlović, University of Belgrade, Serbia.
Preface1 Quasi-Banach spaces 1.1 Quasinorm and p-norm1.2 Linear operators 1.3 The closed graph theorem The open mapping theorem The uniform boundedness principle The closed graph theorem 1.4 F-spaces 1.5 The spaces lp 1.6 Spaces of analytic functions 1.7 The Abel dual of a space of analytic functions 1.7a Homogeneous spaces 2 Interpolation and maximal functions 2.1 The Riesz/Thorin theorem 2.2 Weak Lp-spaces and Marcinkiewicz’s theorem2.3 The maximal function and Lebesgue points2.4 The Rademacher functions and Khintchine’s inequality 2.5 Nikishin’s theorem2.6 Nikishin and Stein’s theorem 2.7 Banach’s principle, the theorem on a.e. convergence, and Sawier’s theorems 2.8 Addendum: Vector-valued maximal theorem 3 Poisson integral 3.1 Harmonic functions 3.1a Green’s formulas 3.1b The Poisson integral 3.2 Borel measures and the space h13.3 Positive harmonic functions 3.4 Radial and non-tangential limits of the Poisson integral 3.4a Convolution of harmonic functions 3.5 The spaces hp and Lp(T) 3.6 A theorem of Littlewood and Paley 3.7 Harmonic Schwarz lemma4 Subharmonic functions 4.1 Basic properties4.1a The maximum principle4.1b Approximation by smooth functions 4.2 Properties of the mean values 4.3 Integral means of univalent functionsPrawitz’ theorem Distortion theorems4.4 The subordination principle 4.5 The Riesz measure Green’s formulaThe Riesz measure of | f |p (f є H(D)) and | u |p (u є hp) 5 Classical Hardy spaces 5.1 Basic propertiesThe decomposition lemma of Hardy and Littlewood5.1a Radial limitsThe Poisson integral of log | f* | 5.2 The space H15.3 Blaschke products Riesz’ factorization theorem5.4 Some inequalities5.5 Inner and outer functions 5.5a Beurling’s approximation theorem5.6 Composition with inner functions. Stephenson’s theorems 5.6a Approximation by inner functions6 Conjugate functions 6.1 Harmonic conjugates6.1a The Privalov/Plessner theorem and the Hilbert operator 6.2 Riesz projection theorem6.2a The Hardy/Stein identity6.2b Proof of Riesz’ theorems6.3 Applications of the projection theorem 6.4 Aleksandrov’s theorem: Lp(T) = Hp(T) + \overline{ Hp}(T) 6.5 Strong convergence in H16.6 Quasiconformal harmonic homeomorphisms and the Hilbert transformation7 Maximal functions, interpolation, and coefficients7.1 Maximal theorems 7.1a Hardy/Littlewood/Sobolev theorem7.2 Maximal characterization of Hp (Burkholder, Gundy and Silverstein) 7.3 “Smooth” Cesàro means σα-maximal theoremThe “W-maximal” theorem 7.4 Interpolation of operators on Hardy spaces 7.4a Application to Taylor coefficients and mean growth 7.4b On the Hardy/Littlewood inequality 7.4c The case of monotone coefficients7.5 Lacunary series7.6 A proof of the σα-maximal theorem 8 Bergman spaces: Atomic decomposition8.1 Bergman spaces8.2 Reproducing kernels8.3 The Coifman/Rochberg theorem q-envelops of Hardy spaces8.4 Coefficients of vector-valued functions. Kalton’s theorems 8.4a Inequalities for a Hadamard product 8.4b Applications to spaces of scalar valued functions9 Lipschitz spaces 9.1 Lipschitz spaces of first order9.2 Conjugate functions 9.3 Lipschitz condition for the modulus. Dyakonov’s theorems with simple proofs by Pavlovic9.4 Lipschitz spaces of higher order9.5 Lipschitz spaces as duals of Hp, p < 110 Generalized Bergman spaces and Besov spaces 10.1 Decomposition of mixed norm spaces: case 1 < p < ∞10.1a Besov spaces10.2 Decomposition of mixed norm spaces: case 0 < 1 Two open problems posed by Hardy and Littlewood 12.3 Subharmonic behavior of smooth functions 12.3a Quasi-nearly subharmonic functions 12.3b Regularly oscillating functions12.4 A generalization of the Littlewood/Paley theorem 12.4a Invariant Besov spaces and the derivatives of the integral means12.4b Addendum: The case of vector valued functions 12.5 Mixed norm spaces of harmonic functions 13 Littlewood/Paley theory 13.1 Some more vector maximal functions13.2 The Littlewood/Paley g-function Calderon’s generalization of the area theorem (p > 0) A proof of a the Littlewood/Paley g-theorem (p > 0)13.3 Applications of Cesàro means13.4 The Littlewood/Paley g-theorem in a generalized form An improvement 13.5 Proof of Calderon’s theorem14 Tauberian theorems and lacunary series on the interval (0,1) 14.1 Karamata’s theorem and Littlewood’s theorem 14.1a Tauberian nature of Λp1/p14.2 Lacunary series in C[0,1]14.2a Lacunary series on weighted L∞-spaces14.3 Lp-integrability of lacunary series on (0,1) 14.3a Some consequencesBibliography
