**Description:**
In just over 100 pages, this book
provides basic, essential knowledge of some of the tools of real analysis: the
Hardy–Littlewood maximal operator, the Calderón–Zygmund theory, the
Littlewood–Paley theory, interpolation of spaces and operators, and the basics
of H1 and BMO spaces. This concise text offers brief proofs and exercises of
various difficulties designed to challenge and engage students.

*An Introduction to Singular
Integrals* is meant to give first-year graduate
students in Fourier analysis and partial differential equations an introduction
to harmonic analysis. While some background material is included in the
appendices, readers should have a basic knowledge of functional analysis, some
acquaintance with measure and integration theory, and familiarity with the
Fourier transform in Euclidean spaces.

**Contents:**

**Preface**

**Chapter 1:** **The Hardy–Littlewood maximal operator** • The
Hardy–Littlewood operator • The Lebesgue derivation theorem • Regular families
• Control of some convolutions • Exercises

**Chapter 2:** **Principal values, and some Fourier transforms** • Operators
commuting with translations • Principal values • Some Fourier transforms •
Homogeneous kernels • Exercises

**Chapter 3:** **The Calderón–Zygmund theory** • The dyadic cubes • The
Calderón–Zygmund decomposition • Singular integrals • Exercises

**Chapter 4:** **The Littlewood–Paley theory** • Vector-valued singular
integrals • The Littlewood–Paley inequalities • The Marcinkiewicz multiplier
theorem • Exercises

**Chapter 5:** **Higher Riesz transforms **• Spherical harmonics • Higher
Riesz transforms • Nonsmooth kernels • Exercises

**Chapter 6:** **BMO and ***H*^{1} • The BMO space • The *H*^{1}
(R^{n}) space • Duality of *H*^{1}–BMO • Exercises

**Chapter 7:** **Singular integrals on other groups** • The torus • Z • Some
totally disconnected groups • Exercises

**Chapter 8: Interpolation **• Real methods • Complex methods • Exercises

**A. Background material **• Vector-valued integrals • Convolution • Polar coordinates •
Distribution functions and weak *L*^{p} spaces • Laplace transform
• Khintchine inequalities • Exercises

**B. Notation and conventions **• Glossary of notation and symbols • Conventions

**Postface**

**Bibliography**

**Index**

**About the Author:**

**Jacques Peyrière** is an emeritus professor of mathematics at Université Paris-Sud
(Orsay). He has been head of the Equipe d’Analyse Harmonique (a CNRS team)
there for 10 years. Professor Peyrière has published two books and more than 60
articles on harmonic analysis and related topics in mathematical journals,
including *Duke Mathematical Journal, Advances in Mathematics,* and *Probability
Theory and Related Fields.* His research interests are harmonic analysis,
probability theory, and fractals.

**Target Audience:**

Useful for first-year graduate students in Fourier analysis and partial
differential equations.