Modern Fourier Analysis

1. Front Matter

   Pages 1-13

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2. Smoothness and Function Spaces

   • Loukas Grafakos

   Pages 1-115

3. BMO and Carleson Measures

   • Loukas Grafakos

   Pages 1-51

4. Singular Integrals of Nonconvolution Type

   • Loukas Grafakos

   Pages 1-110

5. Weighted Inequalities

   • Loukas Grafakos

   Pages 1-60

6. Boundedness and Convergence of Fourier Integrals

   • Loukas Grafakos

   Pages 1-84

7. Time Frequency Analysis and the Carleson–Hunt Theorem

   • Loukas Grafakos

   Pages 1-63

8. Back Matter

   Pages 1-24

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The great response to the publication of the book Classical and Modern Fourier Analysishasbeenverygratifying.IamdelightedthatSpringerhasofferedtopublish the second edition of this book in two volumes: Classical Fourier Analysis, 2nd Edition, and Modern Fourier Analysis, 2nd Edition. These volumes are mainly addressed to graduate students who wish to study Fourier analysis. This second volume is intended to serve as a text for a seco- semester course in the subject. It is designed to be a continuation of the rst v- ume. Chapters 1–5 in the rst volume contain Lebesgue spaces, Lorentz spaces and interpolation, maximal functions, Fourier transforms and distributions, an introd- tion to Fourier analysis on the n-torus, singular integrals of convolution type, and Littlewood–Paley theory. Armed with the knowledgeof this material, in this volume,the reader encounters more advanced topics in Fourier analysis whose development has led to important theorems. These theorems are proved in great detail and their proofs are organized to present the ow of ideas. The exercises at the end of each section enrich the material of the corresponding section and provide an opportunity to develop ad- tional intuition and deeper comprehension. The historical notes in each chapter are intended to provide an account of past research but also to suggest directions for further investigation. The auxiliary results referred to the appendix can be located in the rst volume.

• Singular integral
• Smooth function
• bounded mean oscillation
• calculus
• convolution
• differential equation
• functional analysis
• measure

From a reviews:

“Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Moreover, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.” - Ken Ross, MAA Online

From the reviews of the second edition:

"The author … has produced a very well-written, polished, and exciting graduate textbook which easily doubles as a reference book in a number of areas belonging to or touching on Fourier analysis. … Modern Fourier Analysis sports a huge number off well-designed problems and exercises, and every chapter ends with an exceedingly informative set of ‘Historical Notes’. … I think it’s nigh-on indispensable for the aspiring Fourier analyst." (Michael Berg, MAA Online, January, 2009)

"The second part of the two volumetreatise in harmonic analysis entitled ‘Modern Fourier Analysis’ is designed to be a continuation of the first volume … . The historical notes in each chapter are intended to provide an account of past research as well as to suggest directions for further investigation. The volume is mainly addressed to graduate students who wish to study harmonic analysis." (Leonid Golinskii, Zentralblatt MATH, Vol. 1158, 2009)

“Intended for graduate students who wish to study Fourier analysis. … also suitable for self-study. Proofs are provided in great detail. Each chapter is followed by historical notes with references, often including a discussion of further results. There are numerous exercises of varying difficulty, with hints and references provided for the harder ones. … certainly a valuable and useful addition to the existing literature and can serve as textbooks or as reference books. Students will especially appreciate the extensive collection of exercises.”­­­ (Andreas Seeger, Mathematical Reviews, Issue 2011 d)

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