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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has the same domination property possessed by the Euclidean norm on Rd: if g = ∑ γnφn and |γn |≤|λ n | for all n, then g2 ≤ f2. ... also has the property that, if 1 <p< ∞, and f ∈ Lp, the Lp norms of S(f) and f are comparable.
The exponential-square results (and the corresponding weighted norm inequalities) imply that this connection is pretty ... We prove the bounded- ness of the Hardy-Littlewood operator on Lp(w) for w ∈ Ap and we prove an extrapolation ...
The scale of Orlicz spaces (which includes that of Lp spaces for 1 ≤ p ≤ ∞) provides a flexible way of keeping track of the integrability properties of functions. It is very useful in the study of weighted norm inequalities.
These include Lp (p = 2) and so-called weighted spaces, in which the underlying measure is no longer the familiar Lebesgue one. ... The reason is that having a “small” Lp norm means different things for different p.
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