## Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory IntegralsThis book contains an exposition of some of the main developments of the last twenty years in the following areas of harmonic analysis: singular integral and pseudo-differential operators, the theory of Hardy spaces, L\sup\ estimates involving oscillatory integrals and Fourier integral operators, relations of curvature to maximal inequalities, and connections with analysis on the Heisenberg group. |

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### Contents

PROLOGUE | 3 |

Examples | 9 |

Generalization of the CalderonZygmund decomposition | 16 |

Examples of the general theory | 23 |

Truncation of singular integrals | 30 |

Further results | 37 |

MORE ABOUT MAXIMAL FUNCTIONS | 49 |

HARDY SPACES | 87 |

Averages over a fcdimensional submanifold of finite type | 476 |

Averages on variable hypersurfaces | 493 |

Further results | 511 |

INTRODUCTION TO THE HEISENBERG GROUP | 527 |

Geometry of the complex ball and the Heisenberg group | 528 |

The CauchySzego integral | 532 |

Formalism of quantum mechanics and the Heisenberg group | 547 |

Weyl correspondence and pseudodifferential operators | 553 |

Hl AND BMO | 139 |

PseudoDifferential and Singular Integral | 228 |

An L2 theorem | 234 |

Singular integral realization of pseudodifferential operators | 241 |

Estimates in Lp Sobolev and Lipschitz spaces | 250 |

Compound symbols | 258 |

PseudoDifferential and Singular Integral | 269 |

Almost orthogonality | 278 |

L2 theory of operators with CalderonZygmund kernels | 289 |

The Cauchy integral | 310 |

Further results | 317 |

Oscillatory Integrals of the First Kind | 329 |

Oscillatory Integrals of the Second Kind | 375 |

Some Examples | 433 |

MAXIMAL AVERAGES AND OSCILLATORY INTEGRALS | 467 |

Maximal averages and square functions | 469 |

Twisted convolution and singular integrals on Hn | 557 |

Representations of the Heisenberg group | 568 |

Further results | 574 |

MORE ABOUT THE HEISENBERG GROUP | 587 |

The CauchyRiemann complex and its boundary analogue | 589 |

The operators Bb and Db on the Heisenberg group | 594 |

Applications of the fundamental solution | 605 |

The Lewy operator | 611 |

Homogeneous groups | 618 |

The dNeumann problem | 627 |

Further results | 632 |

BIBLIOGRAPHY | 645 |

AUTHOR INDEX | 679 |

685 | |

### Common terms and phrases

analogue apply argument assertion assume assumption asymptotic atomic decomposition bounded function bounded operator boundedness bump function Cauchy-Szego compact support condition consider constant converges convolution corollary corresponding curvature defined denote differential differential inequalities dilations distribution equation equivalent estimate fact Fefferman finite fixed follows formula Fourier integral operators Fourier transform Gaussian curvature given gives Hardy space Heisenberg group Holder inequality holds holomorphic homogeneous of degree identity inequality kernel L2 norm lemma linear LP(R Lp(Rn mapping maximal function maximal operator measure Moreover multiple nonnegative norm Note observe origin orthogonal oscillatory integrals Plancherel's theorem Poisson integral polynomial previous chapter proof properties proposition prove pseudo-differential operators radius Re(s rectangles restriction result satisfies shows singular integrals space square functions Stein sufficiently Suppose symbol theorem theory tion unit ball vanishes variables variant vector fields weak-type write