## 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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Roughly speaking, this includes: the theory of the

**Lebesgue**integral (in 1 and d dimensions), Lp spaces in Rd (1 ≤ p ≤ ∞) and their duals, and some functional analysis. We also assume that the reader knows a little about the Fourier ...

The only half-space we ever look at is Rd+1+, which equals Rd ×(0,∞). We denote the space of locally integrable functions defined on Rd by L1loc (Rd). If E is a measurable subset of Rd, we denote E's

**Lebesgue**measure by |E|.

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 ...

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). This proves completeness. Elementary functional analysis now ...

These include Lp (p = 2) and so-called weighted spaces, in which the underlying measure is no longer the familiar

**Lebesgue**one. For example, it turns out that, if 1 <p< ∞, there are constants cp and Cp so that, for all f∈ Lp(R), ...

### 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 |