View source for Fréchet–Kolmogorov theorem

{{short description|Gives condition for a set of functions to be relatively compact in an Lp space}}
{{no footnotes|date=November 2011}}
In [[functional analysis]], the '''Fréchet–Kolmogorov theorem''' (the names of [[Marcel Riesz|Riesz]] or [[André Weil|Weil]] are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be [[relatively compact]] in an [[Lp space|''L''<sup>''p''</sup> space]]. It can be thought of as an ''L''<sup>''p''</sup> version of the [[Arzelà–Ascoli theorem]], from which it can be deduced.  The theorem is named after [[Maurice René Fréchet]] and [[Andrey Kolmogorov]].

== Statement ==
Let <math>B</math> be a subset of <math>L^p(\mathbb{R}^n)</math> with <math>p\in[1,\infty)</math>, and let <math>\tau_h f</math> denote the translation of <math>f</math> by <math>h</math>, that is, <math>\tau_h f(x)=f(x-h) .</math>

The subset <math>B</math> is [[Relatively compact subspace|relatively compact]] if and only if the following properties hold:

#(Equicontinuous) <math>\lim_{|h|\to 0}\Vert\tau_h f-f\Vert_{L^p(\mathbb{R}^n)} = 0</math> uniformly on <math>B</math>.
#(Equitight) <math>\lim_{r\to\infty}\int_{|x|>r}\left|f\right|^p=0</math> uniformly on <math>B</math>.

The first property can be stated as <math>\forall \varepsilon >0 \, \, \exists \delta >0  </math> such that <math>\Vert\tau_h f-f\Vert_{L^p(\mathbb{R}^n)} < \varepsilon \, \, \forall f \in B, \forall h</math> with <math>|h|<\delta .</math>

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that <math>B</math> is bounded (i.e., <math>\Vert f\Vert_{L^p(\mathbb{R}^n)}<\infty</math> uniformly on <math>B</math>). However, it has been shown that equitightness and equicontinuity imply this property.<ref>{{cite journal |last1= Sudakov |first1= V. N.|date= 1957 |title= Criteria of compactness in function spaces|language= Russian|journal= Upsekhi Math. Nauk. |volume=12 |pages=221–224}}</ref>

===Special case===

For a subset <math>B</math> of <math>L^p(\Omega)</math>, where <math>\Omega</math> is a bounded subset of <math>\mathbb{R}^n</math>, the condition of equitightness is not needed. Hence, a necessary and sufficient condition for <math>B</math> to be [[Relatively compact subspace|relatively compact]] is that the property of equicontinuity holds. However, this property must be interpreted with care as the below example shows.

==Examples==

===Existence of solutions of a PDE===
Let <math>(u_\epsilon)_\epsilon</math> be a [[sequence]] of solutions of the viscous [[Burgers' equation|Burgers equation]] posed in <math>\mathbb{R}\times(0,T)</math>:

:<math>\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial u^2}{\partial x} = \epsilon\Delta u, \quad u(x,0) = u_0(x),</math>

with <math>u_0</math> smooth enough. If the solutions <math>(u_\epsilon)_\epsilon</math> enjoy the <math>L^1</math>-contraction and <math>L^\infty</math>-bound properties,<ref>{{cite book|last1=Necas|first1=J.|last2=Malek|first2=J.|last3=Rokyta|first3=M.|last4=Ruzicka|first4=M.|title=Weak and Measure-Valued Solutions to Evolutionary PDEs|publisher=Chapman and Hall/CRC|series=Applied Mathematics and Mathematical Computation 13|year=1996|isbn=978-0412577505}}</ref> we will show existence of solutions of the inviscid [[Burgers' equation|Burgers equation]]

:<math>\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial u^2}{\partial x} = 0, \quad u(x,0) = u_0(x).</math>

The first property can be stated as follows: If <math>u,v</math> are solutions of the Burgers equation with <math>u_0,v_0</math> as initial data, then

:<math>\int_{\mathbb{R}}|u(x,t)-v(x,t)|dx\leq \int_{\mathbb{R}}|u_0(x)-v_0(x)|dx.</math>

The second property simply means that <math>\Vert u(\cdot,t)\Vert_{L^\infty(\mathbb{R})}\leq \Vert u_0\Vert_{L^\infty(\mathbb{R})}</math>.

Now, let <math>K\subset\mathbb{R}\times(0,T)</math> be any [[Compact space|compact set]], and define 

:<math>w_\epsilon(x,t):=u_\epsilon(x,t)\mathbf{1}_K(x,t),</math> 

where <math>\mathbf{1}_K</math> is <math>1</math> on the set <math>K</math> and 0 otherwise. Automatically, <math>B:=\{(w_\epsilon)_\epsilon\}\subset L^1(\mathbb{R}^2)</math> since 

:<math>\int_{\mathbb{R}^2}|w_\epsilon(x,t)|dx dt= \int_{\mathbb{R}^2}|u_\epsilon(x,t)\mathbf{1}_K(x,t)|dx dt\leq \Vert u_0\Vert_{L^\infty(\mathbb{R})}|K|<\infty.</math>

Equicontinuity is a consequence of the <math>L^1</math>-contraction since <math>u_\epsilon(x-h,t)</math> is a solution of the Burgers equation with <math>u_0(x-h)</math> as initial data and since the <math>L^\infty</math>-bound holds: We have that

:<math>\Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon\Vert_{L^1(\mathbb{R}^2)}\leq \Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon(\cdot,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}+\Vert w_\epsilon(\cdot,\cdot-h)-w_\epsilon\Vert_{L^1(\mathbb{R}^2)}.</math>

We continue by considering

:<math>\begin{align}
&\Vert w_\epsilon(\cdot-h,\cdot-h)-w_\epsilon(\cdot,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}\\
&\leq \Vert (u_\epsilon(\cdot-h,\cdot-h)-u_\epsilon(\cdot,\cdot-h))\mathbf{1}_K(\cdot-h,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}+\Vert u_\epsilon(\cdot,\cdot-h)(\mathbf{1}_K(\cdot-h,\cdot-h)-\mathbf{1}_K(\cdot,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}.
\end{align}
</math>

The first term on the right-hand side satisfies 

:<math>\Vert (u_\epsilon(\cdot-h,\cdot-h)-u_\epsilon(\cdot,\cdot-h))\mathbf{1}_K(\cdot-h,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}\leq T\Vert u_0(\cdot-h)-u_0\Vert_{L^1(\mathbb{R})}</math> 

by a change of variable and the <math>L^1</math>-contraction. The second term satisfies 

:<math>\Vert u_\epsilon(\cdot,\cdot-h)(\mathbf{1}_K(\cdot-h,\cdot-h)-\mathbf{1}_K(\cdot,\cdot-h))\Vert_{L^1(\mathbb{R}^2)}\leq \Vert u_0\Vert_{L^\infty(\mathbb{R})}\Vert \mathbf{1}_K(\cdot-h,\cdot)-\mathbf{1}_K\Vert_{L^1(\mathbb{R}^2)}</math> 
by a change of variable and the <math>L^\infty</math>-bound. Moreover,

:<math>\Vert w_\epsilon(\cdot,\cdot-h)-w_\epsilon\Vert_{L^1(\mathbb{R}^2)}\leq \Vert (u_\epsilon(\cdot,\cdot-h)-u_\epsilon)\mathbf{1}_K(\cdot,\cdot-h)\Vert_{L^1(\mathbb{R}^2)}+\Vert u_\epsilon(\mathbf{1}_K(\cdot,\cdot-h)-\mathbf{1}_K)\Vert_{L^1(\mathbb{R}^2)}.</math>

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the <math>L^1</math>-contraction.<ref>{{cite journal |last=Kruzhkov |first=S. N. |date=1970 |title=First order quasi-linear equations in several independent variables |url=https://iopscience.iop.org/article/10.1070/SM1970v010n02ABEH002156/meta |journal=Math. USSR Sbornik |volume=10 |issue=2 |pages=217–243|doi=10.1070/SM1970v010n02ABEH002156 }}</ref> The continuity of the translation mapping in <math>L^1</math> then gives equicontinuity uniformly on <math>B</math>.

Equitightness holds by definition of <math>(w_\epsilon)_\epsilon</math> by taking <math>r</math> big enough.

Hence, <math>B</math> is [[Relatively compact subspace|relatively compact]] in <math>L^1(\mathbb{R}^2)</math>, and then there is a convergent subsequence of <math>(u_\epsilon)_\epsilon</math> in <math>L^1(K)</math>. By a covering argument, the last convergence is in <math>L_{loc}^1(\mathbb{R}\times(0,T))</math>. 

To conclude existence, it remains to check that the limit function, as <math>\epsilon\to0^+</math>, of a subsequence of <math>(u_\epsilon)_\epsilon</math> satisfies 

:<math>\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial u^2}{\partial x} = 0, \quad u(x,0) = u_0(x).</math>

==See also==
*[[Arzelà–Ascoli theorem]]
*[[Helly's selection theorem]]
*[[Rellich–Kondrachov theorem]]

==References==
{{reflist}}

==Literature==
* {{cite book
  | last = Brezis
  | first = Haïm
  | authorlink = Haïm Brezis
  | title = Functional analysis, Sobolev spaces, and partial differential equations
  | publisher = [[Springer-Verlag|Springer]]
  | series = Universitext
  | year = 2010
  | isbn = 978-0-387-70913-0
  | page = 111
}}
* {{cite journal 
|last=Riesz 
|first=Marcel
|authorlink=Marcel Riesz 
|date=1933 
|title=Sur les ensembles compacts de fonctions sommables
|url=http://acta.bibl.u-szeged.hu/13418/1/math_006_136-142.pdf 
|journal=[[Acta Scientiarum Mathematicarum|Acta Sci. Math.]]
|volume=6 
|pages=136–142
}}
* {{cite book
  | last = Precup
  | first = Radu
  | title = Methods in nonlinear integral equations
  | url = https://archive.org/details/methodsnonlinear00prec
  | url-access = limited
  | publisher = [[Springer-Verlag|Springer]]
  | year = 2002
  | isbn = 978-1-4020-0844-3
  | page = [https://archive.org/details/methodsnonlinear00prec/page/n28 21]
}}

{{Functional analysis}}

{{DEFAULTSORT:Frechet-Kolmogorov Theorem}}
[[Category:Theorems in functional analysis]]
[[Category:Compactness theorems]]