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Produktbild: Function Classes on the Unit Disc
Band 52

Function Classes on the Unit Disc An Introduction

Aus der Reihe De Gruyter Studies in Mathematics

173,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

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Beschreibung

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

12.12.2013

Verlag

De Gruyter

Seitenzahl

449

Maße (L/B/H)

24,6/17,5/3,8 cm

Gewicht

906 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-3-11-028123-1

Beschreibung

Rezension

"In this ambitious book, the author treats a number of topics from the theory of functions and function spaces on the unit disc in the complex plane. The selection of topics is far ranging, and includes both classical and modern ideas. Many of his proofs are new or unusual, and many of his ideas and presentations appear here in book form for the first time." - Steven G. Krantz, Mathematical Reviews

"[...] this is a well-written and detailed text with concise proofs. Graduate students and researchers who are pursuing research in harmonic or holomorphic function theory of one or several variables will find this book to be an excellent addition to their personal library." - Manfred Stoll, Zentralblatt für Mathematik

Produktdetails

Einband

Gebundene Ausgabe

Erscheinungsdatum

12.12.2013

Verlag

De Gruyter

Seitenzahl

449

Maße (L/B/H)

24,6/17,5/3,8 cm

Gewicht

906 g

Auflage

1. Auflage

Sprache

Englisch

ISBN

978-3-11-028123-1

Herstelleradresse

Walter de Gruyter
Genthiner Straße 13
10785 Berlin
DE

Email: productsafety@degruyterbrill.com

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  • Produktbild: Function Classes on the Unit Disc
  • Preface

    1 Quasi-Banach spaces

    1.1 Quasinorm and p-norm

    1.2 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 theorem

    2.3 The maximal function and Lebesgue points

    2.4 The Rademacher functions and Khintchine's inequality

    2.5 Nikishin's theorem

    2.6 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 h1

    3.3 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 lemma

    4 Subharmonic functions

    4.1 Basic properties

    4.1a The maximum principle

    4.1b Approximation by smooth functions

    4.2 Properties of the mean values

    4.3 Integral means of univalent functions

    Prawitz' theorem

    Distortion theorems

    4.4 The subordination principle

    4.5 The Riesz measure

    Green's formula

    The Riesz measure of | f |p (f ¿ H(D)) and | u |p (u ¿ hp)

    5 Classical Hardy spaces

    5.1 Basic properties

    The decomposition lemma of Hardy and Littlewood

    5.1a Radial limits

    The Poisson integral of log | f* |

    5.2 The space H1

    5.3 Blaschke products

    Riesz' factorization theorem

    5.4 Some inequalities

    5.5 Inner and outer functions

    5.5a Beurling's approximation theorem

    5.6 Composition with inner functions. Stephenson's theorems

    5.6a Approximation by inner functions

    6 Conjugate functions

    6.1 Harmonic conjugates

    6.1a The Privalov/Plessner theorem and the Hilbert operator

    6.2 Riesz projection theorem

    6.2a The Hardy/Stein identity

    6.2b Proof of Riesz' theorems

    6.3 Applications of the projection theorem

    6.4 Aleksandrov's theorem: Lp(T) = Hp(T) + \overline{Hp}(T)

    6.5 Strong convergence in H1

    6.6 Quasiconformal harmonic homeomorphisms and the Hilbert transformation

    7 Maximal functions, interpolation, and coefficients

    7.1 Maximal theorems

    7.1a Hardy/Littlewood/Sobolev theorem

    7.2 Maximal characterization of Hp (Burkholder, Gundy and Silverstein)

    7.3 "Smooth" Cesàro means

    ¿¿-maximal theorem

    The "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 coefficients

    7.5 Lacunary series

    7.6 A proof of the ¿¿-maximal theorem

    8 Bergman spaces: Atomic decomposition

    8.1 Bergman spaces

    8.2 Reproducing kernels

    8.3 The Coifman/Rochberg theorem

    q-envelops of Hardy spaces

    8.4 Coefficients of vector-valued functions. Kalton's theorems

    8.4a Inequalities for a Hadamard product

    8.4b Applications to spaces of scalar valued functions

    9 Lipschitz spaces

    9.1 Lipschitz spaces of first order

    9.2 Conjugate functions

    9.3 Lipschitz condition for the modulus. Dyakonov's theorems with simple proofs by Pavlovic

    9.4 Lipschitz spaces of higher order

    9.5 Lipschitz spaces as duals of Hp, p < 1

    10 Generalized Bergman spaces and Besov spaces

    10.1 Decomposition of mixed norm spaces: case 1 < p < ¿

    10.1a Besov spaces

    10.2 Decomposition of mixed norm spaces: case 0 < p ¿ ¿

    10.2a Radial limits of Hardy/Bloch functions

    10.2b Fractional integration and differentiation

    10.3 Möbius invariant Besov spaces

    10.4 Mean Lipschitz spaces

    10.4a Lacunary series in mixed norm spaces

    10.5 Duality in the case 0 < p ¿ ¿

    10.6 Appendix: Characterizations of Besov spaces

    11 BMOA, Bloch space

    11.1 The dual of H1 and the Carleson measures

    Proof of Fefferman's theorem

    11.2 Vanishing mean osillation

    11.3 BMOA and mean Lipschitz spaces

    11.4 Coefficients of BMOA-functions

    11.4a Lacunary series

    11.5 The Bloch space

    11.5a On the predual of B

    Functions with decreasing coefficients

    12 Subharmonic behavior

    12.1 Subharmonic behavior and Bergman spaces

    Two simple proofs of Hardy/Littlewood/Fefferman/Stein theorem

    12.2 The space hp, p < 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 functions

    12.4 A generalization of the Littlewood/Paley theorem

    12.4a Invariant Besov spaces and the derivatives of the integral means

    12.4b 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 functions

    13.2 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 means

    13.4 The Littlewood/Paley g-theorem in a generalized form

    An improvement

    13.5 Proof of Calderon's theorem

    14 Tauberian theorems and lacunary series on the interval (0,1)

    14.1 Karamata's theorem and Littlewood's theorem

    14.1a Tauberian nature of ¿p1/p

    14.2 Lacunary series in C[0,1]

    14.2a Lacunary series on weighted L¿-spaces

    14.3 Lp-integrability of lacunary series on (0,1)

    14.3a Some consequences

    Bibliography