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 exponential-square results (and the corresponding weighted norm inequalities) imply that this connection is pretty ... result—because we need both—but we don't prove Ap factorization or the Rubio de Francia extrapolation theorem, ...
In chapter 4 we extend the results of the preceding chapters to d dimensions and to continuous analogues of the dyadic square function. Chapters 5, 6, and 7 are devoted to the Calderón reproducing formula. The Calderón formula provides ...
Every area of mathematics—and, indeed, of learning—is a minefield of presuppositions and “well-known” results. In this section I will try to acquaint the reader with what things are taken for granted in weighted LittlewoodPaley theory.
(2.8) We have approached this formula through an L2 result 2.2, which says that f2 = S(f)2 (2.9) for f ∈ L2. Before going one step further, it will be profitable to reflect on the meaning of 2.9. It is a peculiar equation.
However, if f's support is contained in a dyadic interval I, we have two substitute results. The first OIle IS //ølo" fol/l)" /M (nar (2.1% I I The second result is # | (Ma(f))” dr & Co (# | |f| *). (2.16) valid for 0 < 3 < 1.
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