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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We use C∞ 0 (Rd) to denote the family of infinitely differentiable functions with compact supports. ... some Q∈ F. The proof is: For every Q∈ F, let Q be the maximal element (in the sense of set inclusion) of F that 2 1 Some Assumptions.
element (in the sense of set inclusion) of F that contains Q; such a maximal element must exist because of 1.1. ... We have two positive quantities—call them A and B—that depend on something else: a variable, a vector, a function, ...
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 ... 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!)
The function g is good because it is bounded. ... The function b is bad because it is in general unbounded. ... To begin: let FA (note that we have dropped the prime') be the family of maximal dyadic cubes satisfying #/ - f| da = \.
The dyadic maximal function of f, f∗, is given by: f∗(x) ≡ sup |fI|. I:x∈I∈D The dyadic Hardy-Littlewood maximal function of f, Md(f), is defined by: Md(f)(x)≡ supI:x∈I∈D|f|I. We will be seeing a lot of maximal functions like ...
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