## 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 reader's third exercise is to generalize the second exercise to Rd. The first exercise has this consequence: If F ⊂ Dd is any collection of dyadic cubes such that sup Q∈F l(Q) < ∞ (1.1) then there exists a

**disjoint**collection F ...

For every λ > 0, there is a (possibly empty) family F of pairwise

**disjoint**dyadic cubes such that f = g + b, where g∞ ≤ 2dλ and b = ∑ Q∈F b(Q). Each function b(Q) has its support contained in Q and satisfies ∫ b(Q) dx = 0 |b(Q) |dx ...

However, it is good because it is a sum of non-interfering pieces (

**disjoint**supports) with controlled L' norms, and which satisfy a cancelation condition, and with a total support that is also controlled, at least in terms of measure.

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 ...

Suppose that {J.), is an arbitrary

**disjoint**collection of dyadic intervals. 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 ...

### 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 |