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.
Unfortunately, L2 is not always the most useful function space for a given problem. ... also has the property that, if 1 <p< ∞, and f ∈ Lp, the Lp norms of S(f) and f are comparable. The combination of these two facts—domination plus ...
The scale of Orlicz spaces (which includes that of Lp spaces for 1 ≤ p ≤ ∞) provides a flexible way of keeping track of the integrability properties of functions. It is very useful in the study of weighted norm inequalities.
Roughly speaking, this includes: the theory of the Lebesgue integral (in 1 and d dimensions), Lp spaces in Rd (1 ≤ p ≤ ∞) and their duals, and some functional analysis. We also assume that the reader knows a little about the Fourier ...
Here is a question: To what extent does this equivalence extend to other L” spaces, and even to weighted LP spaces? (We'll explain what weighted spaces are in a bit.) To put it more generally: When can we use a vector-space ...
These include Lp (p = 2) and so-called weighted spaces, in which the underlying measure is no longer the familiar Lebesgue one. For example, it turns out that, if 1 <p< ∞, there are constants cp and Cp so that, for all f∈ Lp(R), ...
Ce spun oamenii - Scrie o recenzie
Goodbye to Goodλ
A Fourier Multiplier Theorem
Random Pointwise Errors
Some Singular Integrals