## 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 6 - 10 din 91

But f h(r) = 0, and so /* (a) h(J) (a) da: = 0 in this case, too. For any fe L. (R) and I e D, we

**define**loc MU) – f found)" = (f, h(r)). where we are using (, ) to denote the usual An Elementary Introduction.

Now split f into f1 + f2, where J f(a) – fj, if a € Jr.; no-' J £, (2.3) This

**definition**forces f2 to equal JJ, if a € Jk: f2(a) = (#) if a £ U.J. This splitting has the consequence that, if I is any dyadic interval not properly ...

Because Md(f) has the absolute value inside its

**defining**integral, we always have f∗(x) ≤ Md(f)(x). What makes Md(f) really useful is the following: Theorem 2.2. For all 1 < p ≤ ∞ there is a constant Cp, depending only p, such that, ...

But, if I is big enough, g| < X (because g e L'). So, we can replace the union

**defining**SX by a union Over a subcollection; namely, those I € TP Such that |g| > X and are maximal (in the sense of set inclusion).

... we get /MAndrs/ |{a e I : Mg(f)(a) > X}| dA ~ / C +/ (#/...") dX 2|f(t) C. < c//0 (/ £). < 1 + | |f(t) log" (f(t)) dt < C / |f(a) log(e.--|f(a)) day, I proving One direction of 2.15. To show the other direction in 2.15,

**define**{I}}, ...

### Ce spun oamenii - Scrie 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 - Afișează-le pe toate

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