Editing Lp space (section)

===Weighted {{math|''L<sup>p</sup>''}} spaces===

As before, consider a [[measure space]] <math>(S, \Sigma, \mu).</math> Let <math>w : S \to [a, \infty), a > 0</math> be a measurable function. The <math>w</math>-'''weighted <math>L^p</math> space''' is defined as <math>L^p(S, w \, \mathrm{d} \mu),</math> where <math>w \, \mathrm{d} \mu</math> means the measure <math>\nu</math> defined by
<math display="block">\nu(A) \equiv \int_A w(x) \, \mathrm{d} \mu (x), \qquad A \in \Sigma,</math>

or, in terms of the [[Radon–Nikodym theorem|Radon–Nikodym derivative]], <math>w = \tfrac{\mathrm{d} \nu}{\mathrm{d} \mu}</math> the [[Norm (mathematics)|norm]] for <math>L^p(S, w \, \mathrm{d} \mu)</math> is explicitly
<math display="block">\|u\|_{L^p(S, w \, \mathrm{d} \mu)} \equiv \left(\int_S w(x) |u(x)|^p \, \mathrm{d} \mu(x)\right)^{1/p}</math>

As <math>L^p</math>-spaces, the weighted spaces have nothing special, since <math>L^p(S, w \, \mathrm{d} \mu)</math> is equal to <math>L^p(S, \mathrm{d} \nu).</math> But they are the natural framework for several results in harmonic analysis {{harv|Grafakos|2004}}<!--Please check this reference. Appears in Grafakos "Modern Fourier analysis", Chapter 9.-->; they appear for example in the [[Muckenhoupt weights|Muckenhoupt theorem]]: for <math>1 < p < \infty,</math> the classical [[Hilbert transform]] is defined on <math>L^p(\mathbf{T}, \lambda)</math> where <math>\mathbf{T}</math> denotes the [[unit circle]] and <math>\lambda</math> the Lebesgue measure; the (nonlinear) [[Hardy–Littlewood maximal operator]] is bounded on <math>L^p(\Reals^n, \lambda).</math> Muckenhoupt's theorem describes weights <math>w</math> such that the Hilbert transform remains bounded on <math>L^p(\mathbf{T}, w \, \mathrm{d} \lambda)</math> and the maximal operator on <math>L^p(\Reals^n, w \, \mathrm{d} \lambda).</math>