## 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

... is bounded pointwise by a function ̃S(f), where ̃S(·) is an operator similar to—and satisfying

**estimates**similar to—S(·). This makes it possible to understand the behavior of T, because one can say: |T(f)| is controlled by S(T(f)), ...

In practice, |f(t) − g(t)| is hard to

**estimate**directly, but B(t) is easy (or easier) to handle. An inequality like 1.2 usually follows from a chain of inequalities, like so, A(t) ≤ C1A1(t) ≤ C2A2(t) ≤ ··· ≤ C129 B(t), ...

I λI(f)h(I) can represent f pretty well, and that good

**estimates**of cp and Cp tell us how careful we have to be in computing the λI(f)'s, if we want this representation to be faithful. Our first major goal is a proof of Theorem 2.1.

{t: |f(t)|>X/2} We plug in our

**estimate**for |{x : Mg(f)(a) > X}| and apply 2.14: /(MAD) as of x- |: | no-yo" * d}\ 0 2|f(t) = / |f(t) |/ 2pxP-2 e dt 0 – or "- p–1 -2';*)//0/0." -2';*)/U() at p – 1 We've used Fubini–Tonelli in the ...

If we multiply both sides of these

**estimates**by pXP-' (i.e., 1) and integrate from 0 to oo, 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.

### 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 |