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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... T, S(T(f)), the square function of T(f), is bounded pointwise by a function ̃S(f), where ̃S(·) is an operator similar to—and satisfying estimates similar to—S(·). This makes it possible to understand the behavior of T, because one can say: |T ...
... t under consideration, A(t) ≤ CB(t). (1.2) We often want to prove such inequalities because they help us prove equations. For example, we might have two complicated but continuous functions f(t) and g(t), and want to show f(0) = g(0) ...
... t, C1 B(t) < A(t) < C2 B(t). (1.3) The inequalities 1.3 say that A and B are roughly the same size. An example of such a pair of functions is A(t)=t(log(e+t)) and B(t)=t(log(534-H t”)), where the range of admissible t's is [0, oo); the ...
... T is bounded on some Lp; i.e., we wish to show that there is a constant A such that, for all f ∈ Lp, T(f)p ≤ Afp. This is often accomplished by splitting the function f into finitely many pieces, f = N∑ 1 fi, and showing that each T ...
... (t) = 0 and |h(r)|2 = 1. These functions h(r) are known as the Haar functions. We claim that {h(t)}rep is an orthonormal system for L*(R). We've just seen that /* h(1)(a) da: = 1. Suppose I and J belong to D and I # J. If I O J = 0 it is ...
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 |