Weighted Littlewood-Paley Theory and Exponential-Square IntegrabilitySpringer, 31 dec. 2007 - 227 pagini Littlewood-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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... dyadic interval is one of the form [j/2k,(j + 1)/2k), where j and k are integers. A dyadic cube Q ⊂ Rd is a cube whose component intervals are all dyadic. The family of all dyadic cubes in Rd is denoted by Dd. Strictly speaking, the family ...
... 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!) Call this family of maximal cubes F. If Q e ...
... dyadic cubes. It lets us write any function f satisfying 1.4 as the sum of two functions (usually called g and b, for “good” and “bad”). As we shall see, “good” and “bad” must be used advisedly, because the functions g and b are both ...
... dyadic cubes satisfying #/ - f| da = \. K.I./." By Our Observation, 1 #/. |f|da s 2"X for every Qe FA. Define - (a) = f(a) if a £ UF, Q, QC ) = E.J., f dt if t e Q e FA. Then g is clearly bounded by 2"A almost everywhere. Set b = f – g ...
... dyadic cubes J'A such that b = XX, bø). where the functions bø satisfy the support and cancelation conditions, and also have | Wolars 2"Q/2)Q-2'No. Summing up the measures of the Q's, we get 2 2 |Q| < # / Alts #/ |f|dt, 2. 2.5 .." A J|f ...
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 Previzualizare limitată - 2008 |