## 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

The reader will learn some sufficient (and not terribly restrictive) conditions on pairs of weights which ensure that ∫ |f(x)|pv dx ≤ ∫ (S(f)(x))p w dx (0.2) or ∫ (S(f)(x))pv dx ≤ ∫ |f(x)|pw dx (0.3)

**holds**for all f in suitable ...

... which

**holds**pointwise for appropriate f, and extends, by beginning functional analysis, to all f∈ L2. Our definition of the Fourier transform satisfies f2 = ˆf2 and ̂ (f ∗ g)(ξ) = ˆf(ξ)ˆg(ξ), where f ∗ g is the usual convolution, ...

... and let FA be the family of dyadic cubes Q such that 1 - d X. :// QC = Our hypothesis on f implies that every Q € JFX is contained in some maximal Q'e F.A. (This, by the way,

**holds**even if FA is empty: check the logic!)

(Equation 2.14 is frequently stated to

**hold**for G-finite measure spaces, and proved by Fubini–Tonelli. If the reader tries to do this, he will find that the trickiest step comes in proving the measurability of certain sets in X x (0, ...

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### 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 |