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**Modern Fourier analysis.
2nd ed.**
*(English)*
Zbl 1158.42001

Graduate Texts in Mathematics 250. New York, NY: Springer (ISBN 978-0-387-09433-5/hbk). xv, 504 p. (2009).

The second part of the two volume treatise in harmonic analysis entitled “Modern Fourier Analysis” is designed to be a continuation of the first volume “Classical Fourier Analysis” [Graduate Texts in Mathematics 249. New York, NY: Springer (2008; Zbl 1220.42001)], where the basics of the subject such as Lebesgue and Lorentz spaces, interpolation, Fourier transforms and distributions, singular integrals of convolution type are treated. In this volume some more advanced topics in Fourier analysis are predented.

The book opens with a large chapter 6 on differentiability and smoothness of functions. A fact of paramount importance is that smoothness can be measured using the Fourier transform. One of the major accomplishments of this chapter is the characterization of the basic spaces (Lipschitz, Sobolev, Hardy) via Littlewood–Paley theory, and their atomic characterization. The author also investigates the action of singular integrals on these spaces.

The space of functions of bounded mean oscillation BMO appears in chapter 7. This space shares similar properties with the space \(L^\infty\) and often serves as a substitute for the latter. Along with the standard properties (the range of classical singular integrals on \(L^\infty\), interpolation) this space arises for instance in the characterization of the \(L^2\) boundedness of nonconvolution singular integral operators with standard kernels, examined in detail in the next chapter 8.

The theory of weighted inequalities in chapter 9 is a natural development of the principles and methods displayed in earlier chapters. A better understanding of the subject was spurred by Mackenhoupt’s characterization of positive functions \(w\) for which the Hardy–Littlewood maximal operator acts in \(L^p(\mathbb{R}^n, wdx)\), and the celebrated \(A_p\) condition.

In chapter 10, 11 the author returns to fundamental questions in Fourier analysis related to convergence of Fourier series and integrals, in particular, the almost everywhere convergence of the partial Fourier integrals of \(L^p\) functions on the line.

The exercises at the end of each section supplement the material of the section nicely and provide a good chance to develop additional intuition and deeper comprehension. 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.

The book opens with a large chapter 6 on differentiability and smoothness of functions. A fact of paramount importance is that smoothness can be measured using the Fourier transform. One of the major accomplishments of this chapter is the characterization of the basic spaces (Lipschitz, Sobolev, Hardy) via Littlewood–Paley theory, and their atomic characterization. The author also investigates the action of singular integrals on these spaces.

The space of functions of bounded mean oscillation BMO appears in chapter 7. This space shares similar properties with the space \(L^\infty\) and often serves as a substitute for the latter. Along with the standard properties (the range of classical singular integrals on \(L^\infty\), interpolation) this space arises for instance in the characterization of the \(L^2\) boundedness of nonconvolution singular integral operators with standard kernels, examined in detail in the next chapter 8.

The theory of weighted inequalities in chapter 9 is a natural development of the principles and methods displayed in earlier chapters. A better understanding of the subject was spurred by Mackenhoupt’s characterization of positive functions \(w\) for which the Hardy–Littlewood maximal operator acts in \(L^p(\mathbb{R}^n, wdx)\), and the celebrated \(A_p\) condition.

In chapter 10, 11 the author returns to fundamental questions in Fourier analysis related to convergence of Fourier series and integrals, in particular, the almost everywhere convergence of the partial Fourier integrals of \(L^p\) functions on the line.

The exercises at the end of each section supplement the material of the section nicely and provide a good chance to develop additional intuition and deeper comprehension. 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.

Reviewer: Leonid Golinskii (Kharkov)

### MSC:

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

45E99 | Singular integral equations |

42B25 | Maximal functions, Littlewood-Paley theory |