## 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. |

### Din interiorul cărții

Rezultatele 1 - 5 din 47

A

**cube Q**⊂ Rd is a cartesian product of d intervals all having the same length. ... 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 ...

The collection of all such

**Q's**is F. If f is locally integrable and E is an appropriate measurable subset of f's domain (in practice, E is almost always a

**cube**), then fE is f's average value over E, defined by fE ≡ 1|E| ∫ E f dx.

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, holds even if FA is empty: check ...

But 1|

**Q**| ∫

**Q**|f|dx≤ 1|

**Q**| ∫

**̃Q**|f|dx≤ |

**̃Q**||

**Q**|λ, from which the inequality follows. Notice that, by the Lebesgue differentiation ... is based on the foregoing observation about dyadic

**cubes**. It lets us write any function f satisfying ...

To begin: 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., ...

### 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 |