## Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilityLittlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense. It does so by letting us control certain oscillatory infinite series of functions in terms of infinite series of non-negative functions. Beginning in the 1980s, it was discovered that this control could be made much sharper than was previously suspected. The present book tries to give a gentle, well-motivated introduction to those discoveries, the methods behind them, their consequences, and some of their applications. |

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We have put

**exercises**at the end of almost every chapter. Some of them expand on topics treated in the text; some tie up loose ends in proofs; some are referred to later in the book. We encourage the reader to understand all of them and ...

The family of all dyadic cubes in Rd is denoted by Dd. Strictly speaking, the family of dyadic intervals should be D1, but we will usually refer to it by D. The reader's first

**exercise**is to show that, if Q and Q are two dyadic cubes in ...

... if a € Jk: f2(a) = (#) if a £ U.J. This splitting has the consequence that, if I is any dyadic interval not properly contained in some Jk, then //ar = // da!. (2.4) I I Establishing 2.4 is an eacellent

**exercise**for the reader.

Readers wishing to see the quick and dirty proof first will find it sketched in

**exercises**at the end of the next chapter. We feel it fair to warn the reader that this “simpler” proof has complications of its own, including discussions ...

We will prove these inequalities when f > 0 and |I| = |f| = 1, leaving the general cases as

**exercises**. The splitting argument we used to prove Theorem 2.2 implies that, for every X > 1, |{a e I : Mg(f)(a) > X}| < # / |f(t) dt, ...

### Ce spun oamenii - Scrieți o recenzie

### Cuprins

1 | |

9 | |

Exponential Square 39 | 38 |

Many Dimensions Smoothing | 69 |

The Calderón Reproducing Formula I | 85 |

The Calderón Reproducing Formula II | 101 |

The Calderón Reproducing Formula III | 129 |

Schrödinger Operators 145 | 144 |

Orlicz Spaces | 161 |

Goodbye to Goodλ | 189 |

A Fourier Multiplier Theorem | 197 |

VectorValued Inequalities | 203 |

Random Pointwise Errors | 213 |

References | 219 |

Index 223 | 222 |

Some Singular Integrals | 151 |

### Alte ediții - Afișați-le pe toate

Weighted Littlewood-Paley Theory and Exponential-Square ..., Ediția 1924 Michael Wilson,Professor Michael Wilson Previzualizare limitată - 2008 |