## 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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Rezultatele 1 - 5 din 92

We might want to work in L4, with its norm defined by f4≡ (∫ |

**f**(x)|4dx ) 1/4. To make things specific,

**let's suppose**that our functions are defined on [0,1). The collection {exp(2πinx)}∞−∞ defines a complete orthonormal family in ...

where

**f**∗ g is the usual convolution,

**f**∗g(x)= ∫ Rd

**f**(x−y)g(y)dy= ∫ Rd

**f**(y)g(x−y)dy defined for appropriate pairs of ... We will also use |·| to denote the norm of a vector in Rd. If I ⊂ R is an interval, we

**let**l(I) denote I's ...

**Suppose**that

**f**is a locally integrable function with the property that, for every e > 0, there exists an R = 0 such that, if Q is any cube with ((Q) > R, then 1 -

**f**|da: • 1.4 This hypothesis is not very restrictive: it is satisfied by ...

**Let f**satisfy 1.4. 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 ...

To see this, take f e L”, and

**suppose**that (f h(t) = 0 for all I e D. The claim will be proved if we can show that

**f**= 0. For this it is sufficient to prove that

**f**is (a.e.) constant on (–oo,0) and (0, oo). A little computation shows ...

### 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 |