Aspects of Bounded Integral Operators in L^p Spaces

Title Page......Page 1
Preface......Page 2
Contents......Page 4
1.1 Normed linear spaces......Page 8
1.2 Examples of Banach spaces......Page 10
1.3 Topology in normed linear spaces......Page 16
1.4 Separable normed linear spaces......Page 19
1.5 Compact sets: the Heine-Borel theorem......Page 20
1.6 Baire\'s category theorem......Page 22
1.7 Continuous mappings between normed linear spaces......Page 23
1.8 Banach\'s fixed point theorem......Page 29
1.9 Linear transformations in normed linear spaces......Page 30
1.10 Extensions of linear mappings: the Hanh-Banach theorem......Page 36
1.11 Adjoint mappings......Page 40
1.12 Banach algebras......Page 44
1.13 Inner product spaces and Hilbert spaces......Page 51
1.14 Exercises to Chapter 1......Page 65
2.1 Measures and measure spaces......Page 76
2.2 Properties of outer measures, and the Lebesgue measure......Page 78
2.3 Limit theorems for measures and outer measures......Page 84
2.4 Measurable functions......Page 86
2.5 Simple functions......Page 91
2.6 Convergence theorems......Page 93
2.7 The Lebesgue integral of simple functions......Page 95
2.8 The Lebesgue integral of a bounded measurable function.......Page 97
2.9 The Riemann and Lebesgue integrals on R^1......Page 105
2.10 The bounded convergence theorem......Page 107
2.11 Integrable functions......Page 109
2.12 Distribution functions......Page 111
2.13 Convergence theorems......Page 113
2.14 Product spaces, product measures: Fubini\'s theorem......Page 116
2.15 Radon-Nikodym theorem: change of variables, derivatives......Page 122
2.16 Derivatives and integrals on the real line......Page 132
2.17 Appendix to Chapter 2: change of variables in integrals over R^n......Page 140
2.18 Exercises to Chapter 2......Page 145
3.1 Introduction......Page 159
3.2 Holder\'s inequality......Page 160
3.3 Minkowski\'s inequality and Jensen\'s inequality......Page 164
3.4 Completeness of L^p (X, \\textcal{X}, m), p geq 1 trigonometric series......Page 168
3.5 Approximations in L^p(X)......Page 170
3.6 Continuity of norm in L^p(R^n)......Page 175
3.7 Representation of linear functionais in L^p(X): the adjoint operator......Page 177
3.8 Isometries in L^p(X)......Page 180
3.9 The pointwise convergence of classes of bounded operators on L^p(R^n)......Page 186
3.10 Exercises to Chapter 3......Page 189
4.1 A general inequality: Young\'s inequality for convolutions......Page 197
4.2 Inequalities for homogeneous kernels......Page 200
4.3 Convolutions and approximations to the identity......Page 204
4.4 Operator approximations......Page 210
4.5 Special integral operators......Page 212
4.6 Exercises to Chapter 4......Page 224
5.1 The Riesz-Thorin convexity theorem and extensions......Page 237
5.2 Distributions functions and the Marcinkiewicz-Zygmund interpolation theorem......Page 241
5.3 An extension of Young\'s inequality for generalized convolution......Page 252
5.4 Non-increasing rearrangements......Page 255
5.5 Lorentz spaces......Page 261
5.6 Further results involving convolutions......Page 265
5.7 Maximal functions......Page 271
5.8 Weighted norms for operators mapping into L^infty......Page 279
5.9 Exercises to Chapter 5......Page 281
6.1 Fourier transforms of functions in L^1(R^n)......Page 299
6.2 Some special theorems: special functions and radial functions......Page 303
6.3 Fourier transforms of functions in L^2(R^n)......Page 311
6.4 Fourier transforms in L^p(R^n), 1 geq p geq 2......Page 317
6.5 Fourier transforms with weight functions......Page 320
6.6 Fourier transforms, fractional integrals and Bessel potentials......Page 337
6.7 Operators with Fourier-type kernels: generalized transforms......Page 341
6.8 Laplace transforms and Mellin transforms......Page 351
6.9 Exercises to Chapter 6......Page 355
7.1 Introduction to the Hilbert transform......Page 368
7.2 The Hilbert transform in L^2(R)......Page 372
7.3 Hilbert transforms and conjugate functions in L^p(R)......Page 378
7.4 Fourier transform multipliers: Dirichlet projections......Page 393
7.5 Estimates for singular integrals: An estimate for differentiable multipliers......Page 407
7.6 Maximal functions for convolution operators......Page 429
7.7 Exercises to Chapter 7......Page 436
8.2 Equivalent functional relations: translation invariant operators......Page 444
8.3 Vector-valued functions on measure spaces......Page 451
8.4 Estimates for vector-valued singular integrals......Page 462
8.5 Mixed-norm estimates for singular integrals......Page 464
8.6 Semi-groups of operators......Page 474
8.7 Examples of special semi-groups of operators on L^p(R^n)......Page 490
8.8 Remarks on semi-group potential theory......Page 494
8.9 Exercises to Chapter 8......Page 495
Notation not defined in Text......Page 509
Bibliography and References......Page 511
Author Index......Page 518
Subject Index......Page 521