Editing Lp space (section)

==Generalizations and extensions==

===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]].

===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> 

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

One may also define spaces <math>L^p(M)</math> on a manifold, called the '''intrinsic <math>L^p</math> spaces''' of the manifold, using [[Density on a manifold|densities]].

===Vector-valued {{math|''L<sup>p</sup>''}} spaces===

Given a measure space <math>(\Omega, \Sigma, \mu)</math> and a [[Locally convex topological vector space|locally convex space]] <math>E</math> (here assumed to be [[Complete topological vector space|complete]]), it is possible to define spaces of <math>p</math>-integrable <math>E</math>-valued functions on <math>\Omega</math> in a number of ways. One way is to define the spaces of [[Bochner integral|Bochner integrable]] and [[Pettis integral|Pettis integrable]] functions, and then endow them with [[Locally convex topological vector space|locally convex]] [[Vector topology|TVS-topologies]] that are (each in their own way) a natural generalization of the usual <math>L^p</math> topology. Another way involves [[topological tensor product]]s of <math>L^p(\Omega, \Sigma, \mu)</math> with <math>E.</math> Element of the vector space <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> are finite sums of simple tensors <math>f_1 \otimes e_1 + \cdots + f_n \otimes e_n,</math> where each simple tensor <math>f \times e</math> may be identified with the function <math>\Omega \to E</math> that sends <math>x \mapsto e f(x).</math> This [[tensor product]] <math>L^p(\Omega, \Sigma, \mu) \otimes E</math> is then endowed with a locally convex topology that turns it into a [[topological tensor product]], the most common of which are the [[projective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\pi E,</math> and the [[injective tensor product]], denoted by <math>L^p(\Omega, \Sigma, \mu) \otimes_\varepsilon E.</math> In general, neither of these space are complete so their [[Complete topological vector space|completions]] are constructed, which are respectively denoted by <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\pi E</math> and <math>L^p(\Omega, \Sigma, \mu) \widehat{\otimes}_\varepsilon E</math> (this is analogous to how the space of scalar-valued [[simple function]]s on <math>\Omega,</math> when seminormed by any <math>\|\cdot\|_p,</math> is not complete so a completion is constructed which, after being quotiented by <math>\ker \|\cdot\|_p,</math> is isometrically isomorphic to the Banach space <math>L^p(\Omega, \mu)</math>). [[Alexander Grothendieck]] showed that when <math>E</math> is a [[nuclear space]] (a concept he introduced), then these two constructions are, respectively, canonically TVS-isomorphic with the spaces of Bochner and Pettis integral functions mentioned earlier; in short, they are indistinguishable.

==={{math|''L''<sup>0</sup>}} space of measurable functions===

The vector space of ([[equivalence class]]es of) measurable functions on <math>(S, \Sigma, \mu)</math> is denoted <math>L^0(S, \Sigma, \mu)</math> {{harv|Kalton|Peck|Roberts|1984}}. By definition, it contains all the <math>L^p,</math> and is equipped with the topology of ''[[convergence in measure]]''. When <math>\mu</math> is a probability measure (i.e., <math>\mu(S) = 1</math>), this mode of convergence is named ''[[convergence in probability]]''. The space <math>L^0</math> is always a [[topological abelian group]] but  is only a [[topological vector space]] if <math>\mu(S)<\infty.</math> This is because scalar multiplication is continuous if and only if <math>\mu(S)<\infty.</math> If <math>(S,\Sigma,\mu)</math> is <math>\sigma</math>-finite then the [[weaker topology]] of [[local convergence in measure]] is an [[F-space]], i.e. a [[Complete topological vector space|completely]] [[metrizable topological vector space]]. Moreover, this topology is isometric to global convergence in measure <math>(S,\Sigma,\nu)</math> for a suitable choice of [[probability measure]] <math>\nu.</math>

The description is easier when <math>\mu</math> is finite. If <math>\mu</math> is a [[finite measure]] on <math>(S, \Sigma),</math> the <math>0</math> function admits for the convergence in measure the following [[fundamental system of neighborhoods]]
<math display="block">V_\varepsilon = \Bigl\{f : \mu \bigl(\{x : |f(x)| > \varepsilon\} \bigr) < \varepsilon \Bigr\}, \qquad \varepsilon > 0.</math>

The topology can be defined by any metric <math>d</math> of the form
<math display="block">d(f, g) = \int_S \varphi \bigl(|f(x) - g(x)|\bigr)\, \mathrm{d}\mu(x)</math>
where <math>\varphi</math> is bounded continuous concave and non-decreasing on <math>[0, \infty),</math> with <math>\varphi(0) = 0</math> and <math>\varphi(t) > 0</math> when <math>t > 0</math> (for example, <math>\varphi(t) = \min(t, 1).</math> Such a metric is called [[Paul Lévy (mathematician)|Lévy]]-metric for <math>L^0.</math> Under this metric the space <math>L^0</math> is complete. However, as mentioned above, scalar multiplication is continuous with respect to this metric only if <math>\mu(S)<\infty</math>. To see this, consider the Lebesgue measurable function <math>f:\mathbb R\rightarrow \mathbb R</math> defined by <math>f(x)=x</math>. Then clearly <math>\lim_{c\rightarrow 0}d(cf,0)=\infty</math>. The space <math>L^0</math> is in general not locally bounded, and not locally convex.

For the infinite Lebesgue measure <math>\lambda</math> on <math>\Reals^n,</math> the definition of the fundamental system of neighborhoods could be modified as follows
<math display="block">W_\varepsilon = \left\{f : \lambda \left(\left\{x : |f(x)| > \varepsilon \text{ and } |x| < \tfrac{1}{\varepsilon}\right\}\right) < \varepsilon\right\}</math>

The resulting space <math>L^0(\Reals^n, \lambda)</math>, with the topology of local convergence in measure, is isomorphic to the space <math>L^0(\Reals^n, g \, \lambda),</math> for any positive <math>\lambda</math>&ndash;integrable density <math>g.</math>