BOCHNER-RIESZ MEANS ON EUCLIDEAN SPACES

by Dunyan Yan, Shanzhen Lu

Overview

This book mainly deals with the Bochner-Riesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the Bochner-Riesz means and important achievements attained in the last 50 years. For the Bochner-Riesz means of multiple Fourier integral, it includes the Fefferman theorem which negates the Disc multiplier conjecture, the famous Carleson-Sjölin theorem, and Carbery-Rubio de Francia-Vega's work on almost everywhere convergence of the Bochner-Riesz means below the critical index. For the Bochner-Riesz means of multiple Fourier series, it includes the theory and application of a class of function space generated by blocks, which is closely related to almost everywhere convergence of the Bochner-Riesz means. In addition, the book also introduce some research results on approximation of functions by the Bochner-Riesz means.

Product Details

ISBN-13:	9789814458788
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/28/2013
Sold by:	Barnes & Noble
Format:	eBook
Pages:	388
File size:	32 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface vii

1 An introduction to multiple Fourier series 1

1.1 Basic properties of multiple Fourier series 3

1.2 Poisson summation formula 10

1.3 Convergence and the opposite results 15

1.4 Linear summation 34

2 Bochner-Riesz means of multiple Fourier integral 41

2.1 Localization principle and classic results on fixed-point convergence 41

2.2 L^p-convergence 45

2.3 Some basic facts on multipliers 48

2.4 The disc conjecture and Fefferman theorem 51

2.5 The L^p-boundedness of Bochner-Riesz operator T_α with α > 0 59

2.6 Oscillatory integral and proof of Carleson-Sjölin theorem 61

2.7 Kakeya maximal function 78

2.8 The restriction theorem of the Fourier transform 89

2.9 The case of radial functions 97

2.10 Almost everywhere convergence 105

2.11 Commutator of Bochner-Riesz operator 129

3 Bochner-Riesz means of multiple Fourier series 141

3.1 The case of being over the critical index 141

3.2 The case of the critical index (general discussion) 146

3.3 The convergence at fixed point 167

3.4 LP approximation 177

3.5 Almost everywhere convergence (the critical index) 194

3.6 Spaces related to the a.e. convergence of the Fourier series 208

3.7 The uniform convergence and approximation 244

3.8 (C, 1) means 251

3.9 The saturation problem of the uniform approximation 259

3.10 Strong summation 280

4 The conjugate Fourier integral and series 293

4.1 The conjugate integral and the estimate of the kernel 293

4.2 Convergence of Bochner-Riesz means for conjugate Fourier integral 303

4.3 The conjugate Fourier series 309

4.4 Kernel of Bochner-Riesz means of conjugate Fourier series 316

4.5 The maximal operator of the conjugate partial sum 319

4.6 The relations between the conjugate series and integral 324

4.7 Convergence of Bochner-Riesz means of conjugate Fourier series 332

4.8 (C, 1) means in the conjugate case 334

4.9 The strong summation of the conjugate Fourier series 337

4.10 Approximation of continuous functions 347

Bibliography 367

Index 375