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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... cubes in Rd is denoted by Da . Strictly speaking , the family of dyadic intervals should be D1 , but we will usually refer to it by D. The reader's first exercise is to show that , if Q and Q ' are two dyadic cubes in Rd , then either Q CQ ...
... Q is any cube with l ( Q ) > R , then 121 Q 1 / 11 de < e . E. ( 1.4 ) This hypothesis is not very restrictive : it ... cubes Q such that 1 dx > X. Our hypothesis on ƒ implies that every Q € F 、 is contained in some maximal Q ' F ...
... cubes such that f = g + b , where || g || ≤ 2d and b EQEFbQ ) . Each function b ( q ) has its support contained in Q and satisfies = [ b ( Q ) dx = 0 and [ \ b ( Q ) dx ≤ 2a XQ ] . < Moreover , the family F can be chosen so that 2 The ...
... cubes satisfying 1 Tal / If dx > X. By our observation , 1 Q for every QE Fx . Define Then g g ( x ) = = { / f ( x ) ... cubes are also disjoint . Therefore < Σas nas fina Fx Fx 1 To get our final decomposition , we first split ƒ.
... cubes F such that b = Σb ( Q ) , where the functions ( q ) satisfy the support and cancelation conditions , and also have [ [ b ( q ) | dx ≤ 2d + 1 ( \ / 2 ) | Q | = 2a \ | Q | . Summing up the measures of the Q's , we get Σια < Σ | f1 ...
Cuprins
1 | |
9 | |
Exponential Square 39 | 40 |
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 GoodX | 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 |