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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Because of 2.1, we must have fjo = fj = = fj, = flo. But a similar argument proves that fix, = flo. Thus fj, = fix. Since Jo and K0 are arbitrarily small, the Lebesgue differentiation theorem implies that f is a.e. constant on (0, oo).
Theorem 2.1. For all 1 <p< ∞, there are constants cp and Cp, depending only on p, so that, for all f∈ Lp(R), cpfp ≤ S(f)p ≤ Cpfp. (2.12) We will prove Theorem 2.1 shortly. However, before doing so, we wish to describe one possible ...
Our first major goal is a proof of Theorem 2.1. There are at least two approaches we could take here. We could give a quick-and-dirty, relatively dir- ect proof. Unfortunately, this proof does not generalize to small p (0 < p ≤ 1) or ...
The kind of argument used to prove Theorem 2.2 is called interpolation. We showed—or could see directly—that Ma was ... The first OIle IS //ølo" fol/l)" /M (nar (2.1% I I The second result is # | (Ma(f))” dr & Co (# | |f| *).
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Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals