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 close. We have tried to make this book self-contained, not too long, and accessible to non-experts.
Let X be a positive number, and let FA be the family of dyadic cubes Q such that 1 - d X. :// QC = Our hypothesis on f implies that every Q € JFX is contained in some maximal Q'e F.A. (This, by the way, holds even if FA is empty: check ...
Since Jo and K0 are arbitrarily small, the Lebesgue differentiation theorem implies that f is a.e. constant on (0, oo). Obviously, the same argument works as well on (—oo,0). This proves completeness. Elementary functional analysis now ...
... if a £ U.J. This splitting has the consequence that, if I is any dyadic interval not properly contained in some Jk, then //ar = // da!. (2.4) I I Establishing 2.4 is an eacellent exercise for the reader. Equation 2.4 implies ...
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Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals