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, a)-maximal theorems and (C, a)-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.

Autorentext

Miroslav Pavlovi****, University of Belgrade, Serbia.

Inhalt

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/Ste...