## 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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Each function b(Q) has its

**support contained**in Q and satisfies ∫ b(Q) dx = 0 |b(Q) |dx ≤ 2dλ|Q|. Moreover, the family F can be chosen so that ∑ F|Q|≤ 2λ ∫ |f|>λ/2|f|dx. (1.5) and ∫ Before proving the theorem, we should explain ...

Without loss of generality, we may assume that I C J. But then, since I # J, the

**support**of h(r) is entirely

**contained**in J or J.-across which h(y) is constant. But f h(r) = 0, and so /* (a) h(J) (a) da: = 0 in this case, too.

Suppose that f's

**support**is

**contained**inside [0, 1) and that f f da = 0. The reader should satisfy himself of the truth of the following statement: If I e D and I Z [0,1), then X1(f) = 0. This says that the sum XE, AI(f)h(1) is, ...

However, if f's

**support**is

**contained**in a dyadic interval I, we have two substitute results. The first OIle IS //ølo" fol/l)" /M (nar (2.1% I I The second result is # | (Ma(f))” dr & Co (# | |f| *). (2.16) valid for 0 < 3 < 1.

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### 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 |