Weighted Littlewood-Paley Theory and Exponential-Square Integrability
Springer, 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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The reader will learn some sufficient (and not terribly restrictive) conditions on pairs of weights which ensure that ∫ |f(x)|pv dx ≤ ∫ (S(f)(x))p w dx (0.2) or ∫ (S(f)(x))pv dx ≤ ∫ |f(x)|pw dx (0.3) holds for all f in suitable ...
We have devoted three chapters to it because we believe the reader will gain more by seeing essentially the same problem (the convergence of the Calderón integral formula) treated in increasing levels of generality, than in having one ...
We encourage the reader to understand all of them and to attempt at least a few. (We have supplied hints for the more difficult ones.) The author wishes to thank the many colleagues who have offered suggestions, helped him track down ...
In this section I will try to acquaint the reader with what things are taken for granted in weighted LittlewoodPaley theory. Some of these things are definitions and notations, and some of them are theorems. We assume that the reader ...
The family of all dyadic cubes in Rd is denoted by Dd. 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 ...
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