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. |
Din interiorul cărții
... defined by fE ≡ 1|E| ∫ E f dx. Very early in the book, the reader will encounter two conventions, endemic in Fourier analysis, which he might find a little disturbing. They are “the constantly changing constant”1 and the use of ...
... 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. A little computation shows 0 a # UF, Q, b(a) = to- if a € Qe FA. We set b(Q)(a) = (f(a) – fo)xo(a) ...
... definition forces f2 to equal JJ, if a € Jk: f2(a) = (#) if a £ U.J. This splitting has the consequence that, if I is any dyadic interval not properly contained in some Jk, then //ar = // da!. (2.4) I I Establishing 2.4 is an eacellent ...
... Definition 2.1. Let f : R→ R be locally integrable. The dyadic maximal function of f, f∗, is given by: f∗(x) ≡ sup |fI|. I:x∈I∈D The dyadic Hardy-Littlewood maximal function of f, Md(f), is defined by: Md(f)(x)≡ supI:x∈I∈D|f|I ...
... defining SX by a union Over a subcollection; namely, those I € TP Such that |g| > X and are maximal (in the sense of set inclusion). Call this set of maximal intervals {I}}. Because these intervals are dyadic, they are pairwise disjoint ...
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 |