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 ...
... which holds pointwise for appropriate f, and extends, by beginning functional analysis, to all f∈ L2. Our definition of the Fourier transform satisfies f2 = ˆf2 and ̂ (f ∗ g)(ξ) = ˆf(ξ)ˆg(ξ), where f ∗ g is the usual convolution, ...
... and let FA be the family of 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!)
(Equation 2.14 is frequently stated to hold for G-finite measure spaces, and proved by Fubini–Tonelli. If the reader tries to do this, he will find that the trickiest step comes in proving the measurability of certain sets in X x (0, ...
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