## 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 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 |

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absolute constant analytic apply approximate identity arbitrary assume Banach basic boundedness Calderon—Zygmund Carleson Carleson measure Chapter claim coefﬁcients compact compactly supported conclude condition convolution Corollary cubes decomposition deﬁned Deﬁnition 7.1 denote Dirichlet kernel duality equation example Exercise exists f G L1(T fact Fejér kernel ﬁnal ﬁnd ﬁnite ﬁrst ﬁx ﬁxed formula Fourier series Fourier transform function f G M(T Haar functions harmonic analysis harmonic functions Hilbert transform implies inequality inﬁnite interpolation Lebesgue measure Lemma Let f linear Littlewood—Paley martingale means norm Note obtain orthogonality paraproducts Plancherel Problem proof of Theorem Proposition prove Rademacher random variables reader result Riesz right-hand side satisﬁes Schwartz sense sequence Show Sidon sets singular integral operator smooth Sobolev space speciﬁc subharmonic sufﬁces sufﬁcient supp(f Suppose symbol classes theory trigonometric polynomial uniformly verify weak-L1 bound Weyl calculus whence