Muscalu C., Schlag W. Classical and multilinear harmonic analysis. Volume 1

Cambridge University Press, 2013. — 390 p. — (Cambridge Studies in Advanced Mathematics 137) — ISBN: 0521882451This 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.Fourier series: convergence and summability
Harmonic functions; Poisson kernel
Conjugate harmonic functions; Hilbert transform
The Fourier transform on Rd and on LCA groups
Introduction to probability theory
Fourier series and randomness
Calderón–Zygmund theory of singular integrals
Littlewood–Paley theory
Almost orthogonality
The uncertainty principle
Fourier restriction and applications
Introduction to the Weyl calculus