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**Real-variable methods in harmonic analysis.**
*(English)*
Zbl 0621.42001

Pure and Applied Mathematics, Vol. 123. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers) XII, 462 p. (1986).

The classical books by A. Zygmund [Trigonometric Series (1959; Zbl 0085.05601)] and E. M. Stein [Singular Integrals and Differentiability Properties of functions (1970; Zbl 0207.13501)] have been a basic source of reference for analysts working in classical harmonic analysis.

The last 20 years or so have witnessed a number of major developments in the area. Let us mention, for example the duality theorem of C. Fefferman stating that the dual of \(H^ 1\) is BMO, the real variable theory of \(H^ p\) spaces by C. Fefferman and E. M. Stein, the atomic theory of \(H^ p\) spaces, Muckenhoupt \(A_ p\) theory of weights, the boundedness of Calderón commutators and their application to boundary value problems.

One of the purposes of the book under review is to present an introduction to the subject that leads to the recent developments in the field. The book starts with the classical question of convergence of Fourier series (Chapter 1 deals with pointwise convergence, Chapter 2 with Cesàro (C,1) convergence, Chapters 3 and 5 deal with norm convergence while Chapter 7 is concerned with Abel convergence). In Chapter 4 the author deals with the Calderón-Zygmund decomposition. Chapter 8 studies the class BMO of John-Nirenberg. This first part of the book could be used for an introductory 1 semester course in Fourier series for students with an elementary knowledge of the Lebesgue integral.

From this point the setting is \({\mathbb R}^ n\) and the material more advanced. In Chapter 9 the theory of \(A_ p\) weights is presented while Chapter 13 covers the good \(\lambda\) inequalities of Burkholder-Gundy. Chapters 14 and 15 develop the modern theories of \(H^ p\) spaces and Carleson measures. Chapters 16 and 17 present Calderón’s Cauchy integrals on Lipschitz curves and their applications to boundary value problems on \(C^ 1\) domains. In the remaining 4 chapters the author presents more classical material. Chapter 10 covers Riesz transforms and Sobolev inequalities. Chapters 11, 12 and 13 deal with the essentials of the Calderón Zygmund theory of singular integral operators including vector valued singular integrals and Littlewood-Paley theory. The second half of the book could be easily adapted for an advanced course or seminar in harmonic analysis.

The book has a very good bibliography of over 200 items. Notes at the end of each chapter will enable readers to track down the references easily. Professor Torchinsky’s book is an important expository contribution and is likely to become a standard reference on the subject.

The last 20 years or so have witnessed a number of major developments in the area. Let us mention, for example the duality theorem of C. Fefferman stating that the dual of \(H^ 1\) is BMO, the real variable theory of \(H^ p\) spaces by C. Fefferman and E. M. Stein, the atomic theory of \(H^ p\) spaces, Muckenhoupt \(A_ p\) theory of weights, the boundedness of Calderón commutators and their application to boundary value problems.

One of the purposes of the book under review is to present an introduction to the subject that leads to the recent developments in the field. The book starts with the classical question of convergence of Fourier series (Chapter 1 deals with pointwise convergence, Chapter 2 with Cesàro (C,1) convergence, Chapters 3 and 5 deal with norm convergence while Chapter 7 is concerned with Abel convergence). In Chapter 4 the author deals with the Calderón-Zygmund decomposition. Chapter 8 studies the class BMO of John-Nirenberg. This first part of the book could be used for an introductory 1 semester course in Fourier series for students with an elementary knowledge of the Lebesgue integral.

From this point the setting is \({\mathbb R}^ n\) and the material more advanced. In Chapter 9 the theory of \(A_ p\) weights is presented while Chapter 13 covers the good \(\lambda\) inequalities of Burkholder-Gundy. Chapters 14 and 15 develop the modern theories of \(H^ p\) spaces and Carleson measures. Chapters 16 and 17 present Calderón’s Cauchy integrals on Lipschitz curves and their applications to boundary value problems on \(C^ 1\) domains. In the remaining 4 chapters the author presents more classical material. Chapter 10 covers Riesz transforms and Sobolev inequalities. Chapters 11, 12 and 13 deal with the essentials of the Calderón Zygmund theory of singular integral operators including vector valued singular integrals and Littlewood-Paley theory. The second half of the book could be easily adapted for an advanced course or seminar in harmonic analysis.

The book has a very good bibliography of over 200 items. Notes at the end of each chapter will enable readers to track down the references easily. Professor Torchinsky’s book is an important expository contribution and is likely to become a standard reference on the subject.

Reviewer: M.Milman

### MSC:

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

42A20 | Convergence and absolute convergence of Fourier and trigonometric series |

40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |

42B30 | \(H^p\)-spaces |

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |