## Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilitySpringer, 31 dec. 2007 - 227 pagini Littlewood-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. |

### Din interiorul cărții

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...

**let**l(I) denote I's length (which is the same as |I|). A cube

**Q**⊂ Rd is a cartesian product of d intervals all having the same length. We refer to this common length by l(

**Q**), and we call it

**Q's**sidelength. Notice that |

**Q**|= l(

**Q**)d ...

...

**Let**X be a positive number, and

**let**FA be the family of dyadic cubes

**Q**such that 1 - d X. :// QC = Our hypothesis on f implies that every

**Q**€ JFX is contained in some maximal

**Q**'e F.A. (This, by the way ...

**̃Q**||

**Q**|λ, from 4 1 Some Assumptions.

Michael Wilson. But 1|

**Q**| ∫

**Q**|f|dx≤ 1|

**Q**| ∫

**̃Q**|f|dx≤ |

**̃Q**||

**Q**|λ, from which the inequality follows. Notice that, by ...

**Let**f satisfy 1.4. For every λ > 0, there is a (possibly empty) family F of pairwise disjoint dyadic cubes such that ...

...

**let**FA (note that we have dropped the prime') be the family of maximal dyadic cubes satisfying #/ - f| da = \. K.I./." By Our Observation, 1 #/. |f|da s 2"X for every Qe FA. Define - (a) = f(a) if a £ UF,

**Q**, QC ) = E.J., f dt if t e

**Q**e ...

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### 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 - Afișează-le pe toate

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