## Classical and Multilinear Harmonic AnalysisThis 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 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

Classical paraproducts | 29 |

Paraproducts on polydisks | 79 |

Lipschitz curves | 126 |

Iterated Fourier series and physical reality | 187 |

The bilinear Hilbert transform | 208 |

Almost everywhere convergence of Fourier series | 244 |

Flag paraproducts | 275 |

Multilinear interpolation | 311 |

References | 318 |

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### Common terms and phrases

arbitrary argument assume Banach bi-tiles bilinear Hilbert transform bilinear operator biparameter boundedness bump functions Calderon commutators Calderon—Zygmund Cauchy integral Chapter classical clearly Coifman Coifman—Meyer theorem consequence consider convergence corresponding decompose decomposition deﬁned deﬁnition denote difﬁcult disjoint dyadic intervals equation expression fact ﬁrst ﬁrst term ﬁxed ﬂag paraproducts follows formula Fourier coefﬁcients Fourier series functions f given Hardy space harmonic analysis Heisenberg boxes Holder inequality inequality integral on Lipschitz interpolation L2-normalized lacunary left-hand side Leibnitz rule Lemma linear Lipschitz curves Littlewood—Paley LP space Math maximal operator multilinear Muscalu natural nonlinear observe obtain particular PDEs phase-space portrait polydisk Problem proof of Theorem prove real number recall restricted weak type rewrite satisﬁes scaling invariance Schwartz functions similar similarly smaller smooth smooth function square function sufﬁciently supp supremum symbol Theorem 2.3 Thiele trees tri-tiles trilinear tuple variables wave packets weak type write