## 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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The function S(f)(x) varies from point to point, but, if f and g are two functions such that |〈g,φ〉| ≤ |〈f,φ〉| for all φ, then S(g)(x) ≤ S(f)(x)

**everywhere**.

Notice that, by the Lebesgue differentiation theorem, |f| ≤ λalmost

**everywhere**off the set ∪ Fλ Q. Now, what is this good for? Harmonic analysis is about the action of linear operators on functions. Usually we are trying to show that ...

... 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 FA. Then g is clearly bounded by 2"A almost

**everywhere**.

Notice that |fo| < \/2

**everywhere**. Now we apply the previous splitting argument to fi, but use A/2 as our cut-off height, instead of A. We obtain two functions j and b. and a disjoint family of dyadic cubes J'A such that b = XX, ...

...

**everywhere**, while the second sum on the left converges to Zero

**everywhere**; so, the left-hand side of 2.7 converges to f almost

**everywhere**. Meanwhile, the right-hand side of 2.7 will converge to f in L*. THEREFORE, the left-hand side ...

### 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 |