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 need both—but we don't prove Ap factorization or the Rubio de Francia extrapolation theorem, excellent treatments of which can be found in many books (e.g.,  and ). The book is laid out this way. Chapter 1 covers Preface IX.
Chapter 1 covers some basic facts from harmonic analysis. Most of the material there will be review for many people, but we have tried to present it so as not to intimidate the non-experts. Chapter 2 introduces the one-dimensional ...
In chapter 14 we prove one theorem (Khinchin's Inequalities), but our discussion there is mainly philosophical. We look at what Littlewood-Paley theory can tell us about pointwise summation errors of Haar function expansions.
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 ...
Readers wishing to see the quick and dirty proof first will find it sketched in exercises at the end of the next chapter. We feel it fair to warn the reader that this “simpler” proof has complications of its own, including discussions ...
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A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals