## 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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This ingenious

**argument**, due to Fedor Nazarov, completely avoids the use of good-λ inequalities (which we introduce in chapter 2). These have been a mainstay of analysis since the early 1970s. In the opinion of some researchers, ...

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, bø). where the functions bø satisfy the ...

But a similar

**argument**proves that fix, = flo. Thus fj, = fix. Since Jo and K0 are arbitrarily small, the Lebesgue differentiation theorem implies that f is a.e. constant on (0, oo). Obviously, the same

**argument**works as well on (—oo,0) ...

<e (I) <2\ e(Io) There is clearly nothing special about 10, IN, Or a in this

**argument**. We are allowed to say the following: If I and J are dyadic intervals, I C J, and a € I, then fi – f.J = X Ak(f)h(x)(x). (2.6) K e D: I C K C J £(I) ...

This will also provide a chance to introduce the sorts of

**arguments**we will frequently see in this book, and to review some of the fundamentals from chapter 1. By the Lebesgue differentiation theorem, |f(x)| ≤ f∗(x) almost everywhere.

### 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 |