Editing Lp space (section)

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

Let <math>(S, \Sigma, \mu)</math> be a measure space, and <math>f</math> a [[measurable function]] with real or complex values on <math>S.</math> The [[cumulative distribution function|distribution function]] of <math>f</math> is defined for <math>t \geq 0</math> by
<math display="block">\lambda_f(t) = \mu\{x \in S : |f(x)| > t\}.</math>

If <math>f</math> is in <math>L^p(S, \mu)</math> for some <math>p</math> with <math>1 \leq p < \infty,</math> then by [[Markov's inequality]],
<math display="block">\lambda_f(t) \leq \frac{\|f\|_p^p}{t^p}</math>

A function <math>f</math> is said to be in the space '''weak <math>L^p(S, \mu)</math>''', or <math>L^{p,w}(S, \mu),</math> if there is a constant <math>C > 0</math> such that, for all <math>t > 0,</math>
<math display="block">\lambda_f(t) \leq \frac{C^p}{t^p}</math>

The best constant <math>C</math> for this inequality is the <math>L^{p,w}</math>-norm of <math>f,</math> and is denoted by
<math display="block">\|f\|_{p,w} = \sup_{t > 0} ~ t \lambda_f^{1/p}(t).</math>

The weak <math>L^p</math> coincide with the [[Lorentz space]]s <math>L^{p,\infty},</math> so this notation is also used to denote them.

The <math>L^{p,w}</math>-norm is not a true norm, since the [[triangle inequality]] fails to hold. Nevertheless, for <math>f</math> in <math>L^p(S, \mu),</math>
<math display="block">\|f\|_{p,w} \leq \|f\|_p</math>
and in particular <math>L^p(S, \mu) \subset L^{p,w}(S, \mu).</math> 

In fact, one has
<math display="block">\|f\|^p_{L^p} = \int |f(x)|^p d\mu(x) \geq \int_{\{|f(x)| > t \}} t^p + \int_{\{|f(x)| \leq t \}} |f|^p \geq t^p \mu(\{|f| > t \}),</math>
and raising to power <math>1/p</math> and taking the supremum in <math>t</math> one has
<math display="block">\|f\|_{L^p} \geq \sup_{t > 0} t \; \mu(\{|f| > t \})^{1/p} = \|f\|_{L^{p,w}}.</math>

Under the convention that two functions are equal if they are equal <math>\mu</math> almost everywhere, then the spaces <math>L^{p,w}</math> are complete {{harv|Grafakos|2004}}.

For any <math>0 < r < p</math> the expression
<math display="block">\|| f |\|_{L^{p,\infty}} = \sup_{0<\mu(E)<\infty} \mu(E)^{-1/r + 1/p} \left(\int_E |f|^r\, d\mu\right)^{1/r}</math>
is comparable to the <math>L^{p,w}</math>-norm. Further in the case <math>p > 1,</math> this expression defines a norm if <math>r = 1.</math> Hence for <math>p > 1</math> the weak <math>L^p</math> spaces are [[Banach space]]s {{harv|Grafakos|2004}}.

A major result that uses the <math>L^{p,w}</math>-spaces is the [[Marcinkiewicz interpolation|Marcinkiewicz interpolation theorem]], which has broad applications to [[harmonic analysis]] and the study of [[singular integrals]].