Classical and Multilinear Harmonic Analysis This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and is intended for 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 transforms. 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. This 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 been collected previously in book form.

We prove L p L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer (1991), as well as the bilinear Hilbert transform and other operators with large groups of modulation symmetries.

In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As an example of an application of our main theorem, we generalize the Kato-Ponce inequality in nonlinear PDE. Then, we show that the double bilinear Hilbert transform does not satisfy any L p estimates.

Abstract. We prove L p estimates (Theorem 1.2) for the "biest", a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [16] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.

We develop a new method of proving vector-valued estimates in harmonic analysis, which we like to call "the helicoidal method". As a consequence of it, we are able to give affirmative answers to several questions that have been circulating for some time. In particular, we show that the tensor product BHT ⊗ Π between the bilinear Hilbert transform BHT and a paraproduct Π satisfies the same L p estimates as the BHT itself, solving completely a problem introduced in [MPTT04]. Then, we prove that for "locally L 2 exponents" the corresponding vector-valued − −− → BHT satisfies (again) the same L p estimates as the BHT itself. Before the present work there was not even a single example of such exponents.Finally, we prove a bi-parameter Leibniz rule in mixed norm L p spaces, answering a question of Kenig in nonlinear dispersive PDE.

Abstract. The theory of Carleson measures, stopping time arguments, and atomic decompositions has been well-established in harmonic analysis. More recent is the theory of phase space analysis from the point of view of wave packets on tiles, tree selection algorithms, and tree size estimates. The purpose of this paper is to demonstrate that the two theories are in fact closely related, by taking existing results and reproving them in a unified setting. In particular we give a dyadic version of extrapolation for Carleson measures, as well as a two-sided local dyadic T(b) theorem which generalizes earlier T(b) theorems of David, Journe, Semmes, and Christ.

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.

We prove L p estimates (Theorem 1.8) for the Walsh model of the "biest", a trilinear multiplier with singular symbol. The corresponding estimates for the Fourier model will be obtained in the sequel [15] of this paper.