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Function Classes on the Unit Disc

An Introduction

Miroslav Pavlović (Autor)

Buch | Hardcover

XIII, 449 Seiten

2013
De Gruyter (Verlag)
978-3-11-028123-1 (ISBN)

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The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

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.

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 l^p

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 L^p-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 h^p and L^p(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 h^p)

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: L^p(T) = H^p(T) + /overline{H^p}(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 H^p (Burkholder, Gundy and Silverstein)

7.3 "Smooth" Cesàro means

s-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 s-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 H^p, 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 h^p, 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 ^p_1/p

14.2 Lacunary series in C[0,1]

14.2a Lacunary series on weighted L-spaces

14.3 L^p-integrability of lacunary series on (0,1)

14.3a Some consequences

Bibliography

"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

Erscheint lt. Verlag	12.12.2013
Reihe/Serie	De Gruyter Studies in Mathematics ; 52
Verlagsort	Berlin/Boston
Sprache	englisch
Maße	170 x 240 mm
Gewicht	908 g
Themenwelt	Schulbuch / Wörterbuch ► Lexikon / Chroniken
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Schlagworte	Bergman space • Bergman spaces • Besov-Lipschitz Space • bounded mean oscillation • Funktion (Mathematik) • Hardy space • Hardy Space; Bergman Space; Bounded Mean Oscillation; Besov-Lipschitz Space; Littlewood-Paley g-Function • Hardy Spaces • Harmonic Functions • H^p spaces • Littlewood-Paley g-Function • L^p spaces • Quasinormed spaces
ISBN-10	3-11-028123-6 / 3110281236
ISBN-13	978-3-11-028123-1 / 9783110281231
Zustand	Neuware