Classical and Multilinear Harmonic Analysis

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Cambridge University Press, Jan 31, 2013 - Mathematics - 387 pages
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This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón-Zygmund and Littlewood-Paley 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; Coifman-Meyer 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.
 

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Contents

Harmonic functions Poisson kernel
28
Conjugate harmonic functions Hilbert transform
52
The Fourier transform on Rd and on LCA groups
73
groups
90
Introduction to probability theory
106
Fourier series and randomness
136
CalderonZygmund theory of singular integrals
166
LittleWoodPaley theory
196
Almost orthogonality
235
The uncertainty principle
268
Fourier restriction and applications
287
Introduction to the Weyl calculus
315
References
360
Index
367
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About the author (2013)

Camil Muscalu is Associate Professor of Mathematics at Cornell University, New York.

W. Schlag is Professor in the Department of Mathematics at the University of Chicago.

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