## Modern Fourier AnalysisThe great response to the publication of the book Classical and Modern Fourier Analysishasbeenverygratifying.IamdelightedthatSpringerhasofferedtopublish the second edition of this book in two volumes: Classical Fourier Analysis, 2nd Edition, and Modern Fourier Analysis, 2nd Edition. These volumes are mainly addressed to graduate students who wish to study Fourier analysis. This second volume is intended to serve as a text for a seco- semester course in the subject. It is designed to be a continuation of the rst v- ume. Chapters 1–5 in the rst volume contain Lebesgue spaces, Lorentz spaces and interpolation, maximal functions, Fourier transforms and distributions, an introd- tion to Fourier analysis on the n-torus, singular integrals of convolution type, and Littlewood–Paley theory. Armed with the knowledgeof this material, in this volume,the reader encounters more advanced topics in Fourier analysis whose development has led to important theorems. These theorems are proved in great detail and their proofs are organized to present the ow of ideas. The exercises at the end of each section enrich the material of the corresponding section and provide an opportunity to develop ad- tional intuition and deeper comprehension. The historical notes in each chapter are intended to provide an account of past research but also to suggest directions for further investigation. The auxiliary results referred to the appendix can be located in the rst volume. |

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### Contents

LXXXVII | 238 |

LXXXVIII | 239 |

LXXXIX | 242 |

XC | 246 |

XCI | 250 |

XCII | 253 |

XCIII | 256 |

XCV | 257 |

IX | 22 |

X | 24 |

XI | 25 |

XII | 27 |

XIII | 31 |

XIV | 34 |

XV | 37 |

XVII | 40 |

XVIII | 53 |

XIX | 56 |

XX | 58 |

XXI | 63 |

XXII | 66 |

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XXVI | 76 |

XXVII | 78 |

XXX | 82 |

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XXXII | 90 |

XXXIII | 93 |

XXXV | 96 |

XXXVII | 104 |

XXXVIII | 111 |

XXXIX | 116 |

XL | 118 |

XLI | 124 |

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LXX | 191 |

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LXXX | 222 |

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XCVI | 260 |

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C | 275 |

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CIII | 283 |

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CXXXVI | 368 |

CXXXVIII | 370 |

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CXL | 387 |

CXLI | 388 |

CXLII | 390 |

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CXLV | 402 |

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CL | 419 |

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CLIX | 452 |

CLX | 456 |

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### Common terms and phrases

annulus Ap weights BMO function bounded function boundedness Carleson measure centered characterization compact support concludes the proof condition consequence constant Cn constant multiple contained converges Corollary cube Q deduce defined Deﬁnition denote disjoint dyadic cube equal equivalent estimate Exercise exists a constant expression fact finite ﬁrst follows Fourier transform functions f given by convolution Hardy spaces Hardy-Littlewood maximal operator hence Hint holds identity implies Lebesgue Lebesgue measure Lemma linear operator locally integrable function LP norm Lp(Rn maps maximal function multi-indices nonnegative norm normalized bumps observe obtain pointwise proof of Theorem Proposition prove rectangles required conclusion result satisfies Schwartz function sequence Show side length singular integrals smooth function Sobolev space standard kernel subset sup sup Suppose supremum tempered distribution Theory tiles top(T unit ball valid weak type 1,1 write yields zero