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 will soon show that, if f is a bounded function defined on [0,1), there is a positive α such that exp(α(S(f))2) is integrable on [0,1)—and vice versa. (This is not quite like saying that |f| and S(f) are pointwise comparable, ...
Usually we are trying to show that some operator T is bounded on some Lp; i.e., we wish to show that there is a constant A such that, for all f ∈ Lp, T(f)p ≤ Afp. This is often accomplished by splitting the function f into finitely ...
The function g is good because it is bounded. It is bad because it might have unbounded support. The function b is bad because it is in general unbounded. However, it is good because it is a sum of non-interfering pieces (disjoint ...
This might seem pointless, since Ma is obviously not bounded on L'(R): if f = x|_1,1], then Ma(f) - |x|_" when |a| is large. However, if f's support is contained in a dyadic interval I, we have two substitute results.
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Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals