Weighted Lp Spaces
As before, consider a measure space (S, Σ, μ). Let be a measurable function. The w-weighted Lp space is defined as Lp(S, w dμ), where w dμ means the measure ν defined by
or, in terms of the Radon–Nikodym derivative,
The norm for Lp(S, w dμ) is explicitly
As Lp-spaces, the weighted spaces have nothing special, since Lp(S, w dμ) is equal to Lp(S, dν). But they are the natural framework for several results in harmonic analysis (Grafakos 2004); they appear for example in the Muckenhoupt theorem: for 1 < p < ∞, the classical Hilbert transform is defined on Lp(T, λ) where T denotes the unit circle and λ the Lebesgue measure; the (nonlinear) Hardy–Littlewood maximal operator is bounded on Lp(Rn, λ). Muckenhoupt's theorem describes weights w such that the Hilbert transform remains bounded on Lp(T, w dλ) and the maximal operator on Lp(Rn, w dλ).
Read more about this topic: Lp Space
Famous quotes containing the word spaces:
“In any case, raw aggression is thought to be the peculiar province of men, as nurturing is the peculiar province of women.... The psychologist Erik Erikson discovered that, while little girls playing with blocks generally create pleasant interior spaces and attractive entrances, little boys are inclined to pile up the blocks as high as they can and then watch them fall down: “the contemplation of ruins,” Erikson observes, “is a masculine specialty.””
—Joyce Carol Oates (b. 1938)