## 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

Rezultatele 1 - 5 din 39

... 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, f∗g(x)= ∫ Some Assumptions.

Each function b(Q) has its support contained in Q and

**satisfies**∫ b(Q) dx = 0 |b(Q) |dx ≤ 2dλ|Q|. Moreover, the family F can be chosen so that ∑ F|Q|≤ 2λ ∫ |f|>λ/2|f|dx. (1.5) and ∫ Before proving the theorem, we should explain ...

Every Qe FA

**satisfies**1 o's ' | Ula. osy /, / But these cubes are also disjoint. Therefore 1 1 XD|Q|<XD # / flats # / flat, Q 3FA JFX which is nearly right. To get our final decomposition, we first split f in 6 1 Some Assumptions.

... we let I, and I, denote (respectively) the left and right halves of I. For each I e D, Set |I| "/* if c e II: h(1)(a) = ( – IT'/* if t e I, 0 if a # I. Notice that each h(r)

**satisfies**J h(t) = 0 and |h(r)|2 = 1.

Each I

**satisfies**1 H/ |g| dt = A; 1 Wils : / old. X I; implying Therefore, |SA|= XII. 1. dt 'X' | |g| :/old A / 10 at This is the celebrated weak-type inequality for the Hardy–Littlewood maximal function. Now take fe LP, ...

### 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 |