L^p Space - Weighted Lp Spaces

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).