This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and useful to graduates and researchers in pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. The first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional CalderónZygmund and LittlewoodPaley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; CoifmanMeyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and the Cauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.
Recenzijos
'The two-volume set under review is a worthy addition to this tradition from two of the younger generation of researchers. It is remarkable that the authors have managed to fit all of this into [ this number of] smaller-than-average pages without omitting to provide motivation and helpful intuitive remarks. Altogether, these books are a most welcome addition to the literature of harmonic analysis.' Gerald B. Folland, Mathematical Reviews
Daugiau informacijos
This two-volume, contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
Volume 1: Preface; Acknowledgements;
1. Fourier series: convergence and
summability;
2. Harmonic functions, Poisson kernel;
3. Conjugate harmonic
functions, Hilbert transform;
4. The Fourier Transform on Rd and on LCA
groups;
5. Introduction to probability theory;
6. Fourier series and
randomness;
7. CalderónZygmund theory of singular integrals;
8.
LittlewoodPaley theory;
9. Almost orthogonality;
10. The uncertainty
principle;
11. Fourier restriction and applications;
12. Introduction to the
Weyl calculus; References; Index. Volume 2: Preface; Acknowledgements;
1.
Leibniz rules and gKdV equations;
2. Classical paraproducts;
3. Paraproducts
on polydiscs;
4. Calderón commutators and the Cauchy integral;
5. Iterated
Fourier series and physical reality;
6. The bilinear Hilbert transform;
7.
Almost everywhere convergence of Fourier series;
8. Flag paraproducts;
9.
Appendix: multilinear interpolation; Bibliography; Index.
Camil Muscalu is Associate Professor of Mathematics at Cornell University, New York. Wilhelm Schlag is Professor in the Department of Mathematics at the University of Chicago.
