This book mainly concerns with the Bochner-Riesz means of multiple Fourier integral and series, which shall be simply termed the Bochner-Riesz means for short. Bochner-Riesz means is an important branch of multiple Fourier analysis initiated in the 1930s by Bochner. At that time, Bochner and his fellows restricted their studies to the cases when its degree is greater than the critical index. In the late of 1940s, Minde Cheng, Bochner's student, carried out a systematic research about the Bochner-Riesz means. In the late 50s and early 60s, Stein contributed significantly to the research where the degree is at and below the critical index. In the same period, Minde Cheng and his students also started the research on approximation of functions by the Bochner-Riesz means. Since the 1970s, inspired by Stein's work, many researchers in Europe, America, Soviet Union and China have joined the study. A number of important or valuable results are achieved in this field. Although the development in this field took nearly half of a century with many achievements, there still remain many fundamental problems waiting to be resolved. To introduce these results and some remaining problems to Chinese scholars, Shanzhen Lu and Kunyang Wang published a monograph "Bochner-Riesz Means" in Chinese in 1988. We note that there have been some important progresses in this field since then. To supplement the previous book with these important results, we wish to publish a new monograph in English with the present title.
The book is aimed at giving a systematical introduction to fundamental theories of the Bochner-Riesz means and important achievements attained through the latest 50 years. Numerous results illustrate that the Bochner- Riesz means offer the most natural way for the extension of Fourier series from the case of single variable to that of several variables. This book consists of four chapters. The first chapter is a brief introduction to the theory of multiple Fourier series. Chapter 2 and 3 contain main topics of this book which are closely related to each other. Chapter 2 introduces the Bochner-Riesz means of multiple Fourier integral, including Fefferman theorem which negates the Disc multiplier's conjecture and the famous Carleson-Sjölin theorem. For almost everywhere convergence of the Bochner-Riesz means below the critical index, we introduce Carbery-Rubio de Francia-Vega's work. Some recent results on commutators of the Bochner-Riesz means are also included in the last section of this chapter. Chapter 3 concerns with the Bochner-Riesz means of multiple Fourier series, including the theory and application of a class of function space (block space), developed by Taibleson, Weiss and other mathematicians, which is closely related to almost everywhere convergence of the Bochner-Riesz means. In addition, a class of function space named spaces generated by smooth blocks is also introduced in this chapter. Such spaces are closely related to the rate of convergence of the Bochner-Riesz means. Chapter 4 discusses the Bochner-Riesz means of conjugate Fourier integral and conjugate Fourier series, where the concepts of conjugate integral and conjugate series are based on the theory of singular integral by Calderón-Zygmund.
We would like to thank Kunyang Wang whose research work contributes nicely to this book. We have to give our thanks to Bolin Ma who provides some useful materials related to new progress in this field since 1988. In addition, Shanzhen Lu wishes to express his gratitude to Guido Weiss for their collaboration in the early 1980s. This book is dedicated to Guido Weiss on the occasion of his 85th birthday. The book is also in memory of Minde Cheng, Yongsheng Sun and Mitchell H. Taibleson.

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 Lp-convergence 45

2.3 Some basic facts on multipliers 48

2.4 The disc conjecture and Fefferman theorem 51

2.5 The Lp-boundedness of Bochner-Riesz operator Tα 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

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