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 assume that the reader has had a graduate course in measure theory, at least to the level of chapters 5 and 6 in . Roughly speaking, this includes: the theory of the Lebesgue integral (in 1 and d dimensions), Lp spaces in Rd (1 ...
A measurable function f is said to be locally integrable if ∫ K |f|dx < ∞ for every compact subset of f's domain. This domain will always be Rd or some nice subset of it (such as an interval, ball, rectangle, or half-space).
general fact about Hilbert spaces, and so hides the numerical nitty-gritty in this special case. ... These include Lp (p = 2) and so-called weighted spaces, in which the underlying measure is no longer the familiar Lebesgue one.
The proof of Theorem 2.2 is based on the following well-known equation: For any 0 < p < oo, / |f(x)" du = p / who fool-Apax (210 X O which is valid for any measure space (X, M, pl). (Equation 2.14 is frequently stated to hold for ...
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