Marcel Dekker, Inc., 1979

Preface (excerpts)

This book is an introduction to harmonic analysis on Euclidean spaces, aiming at the study of singular integrals. Thus it provides a basis for the study of topics such as differentiability properties of functions of several variables and the applications to partial differential equations, and for some recent developments in classical harmonic analysis. In particular, it leads to the more advanced treatises of Stein, and Stein and Weiss. While certain topics had to be excluded, some of those which are presented here are not found in the existing introductory texts on harmonic analysis, as the Hardy-Littlewood theory of maximal functions and some of its modern applications, the Marcinkiewicz interpolation theorem, the class of functions of bounded mean oscillations, and ergodic theorems.

I hope that the book will be accessible to a wide audience that includes graduate students first approaching the subject. For this purpose, I tried to make the text as self-contained as possible, and most proofs are given in great detail, thereby making the development understandable for a beginner. The paragraphs marked with an asterisk are more technical and its reading can be omitted without altering the comprehension of the remaining text.

...

This book is the outgrowth of a volume of Lecture Notes written at the Universidad Central de Venezuela (Publ. Mat., U.C.V., Segunda Serie, Fasc. 1) in 1976. They correspond to material presented in courses taught at the Universidad Central de Venezuela in 1975, 1977 and 1978. The origin of those notes is much older, starting with a course taught a the Universidad de le Republica, Montevideo, Uruguay, in 1970, and a three months lecture series given in 1973 at the Universidad del Sur, Bahia Blanca, Argentina.

The initial inspiration on the treatment of this subject comes from magnificent courses given by E. M. Stein, G. Weiss, and A. P. Calderón, which I attended as a graduate student a the University of Chicago and the University of Buenos Aires. The overall influence is that of Professor A. Zygmund who taught me how beautiful singular integrals are and induced the will to try to share with others the pleasure of their beauty.

**Contents**

- Preliminaries
- Some definitions from measure theory
- Polar coordinates in R
^{n} - The spaces C and L and their duals
- Hilbert and Banach spaces
- The Three lines theorem

References

- Convolution Units and the Group Algebra
- Convolution of functions
- Pointwise convergence
- Convolution of finite measures
- The group algebra of R
^{n}and its characters - Remarks on the periodic case

References

- Fourier Transforms of Integrable Functions and Finite Measures
- Fourier transforms in L
^{1}(R^{n}) - Fourier transforms of finite measures
- Positive functionals on L
^{1}and*A* - The Bochner theorem
- An application to convergence theorems

References

- Fourier transforms in L
- Inversion Theory and Harmonic Functions
- Summation of Fourier Integrals
- Fourier transforms in L
^{2}and the Plancherel theorem - Harmonic functions
- Poisson integrals

References

- Interpolation of Operators in L
^{p}Spaces- The M. Riesz-Thorin convexity theorem
- Proof of the M. Riesz-Thorin theorem by the complex method
- Distribution functions and weak type operators
- The Marcinkiewicz interpolation theorem: diagonal case
- The Marcinkiewicz interpolation theorem: general case (*)
- Kolmogoroff and Zygmund conditions

References

- Maximal Theory and the Space BMO
- The Hardy-Littlewood maximal theorem
- Applications to Poisson integrals
- Maximal operators and the space BMO
- The method of maximal functions
- Ergodic theorems

References

- Singular Integrals
- The Hilbert transform in L
^{2} - Singular integrals: the L
^{2}theory - General theorems in L
^{p}and BMO - The Calderón-Zygmund singular integrals
- Pointwise convergence of singular integrals
- Extensions to Lebesgue spaces with weighted measures (*)

References

- The Hilbert transform in L

Appendix A: Singular Integrals and Partial Differential Equations

Appendix B: The Complex Method of Interpolation