Fréchet–Kolmogorov theorem

Short description: Gives condition for a set of functions to be relatively compact in an Lp space

In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an L^p space. It can be thought of as an L^p 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 $B$ be a subset of $L^p({\mathbb{ℝ}}^n)$ with $p\in [1,\mathrm{\infty })$, and let ${\tau }_{h}f$ denote the translation of $f$ by $h$, that is, ${\tau }_{h}f(x)=f(x-h).$

The subset $B$ is relatively compact if and only if the following properties hold:

1. (Equicontinuous) $\underset{|h|\to 0}{\lim }\Vert {\tau }_{h}f-f{\Vert }_{L^p({\mathbb{ℝ}}^n)}=0$ uniformly on $B$.
2. (Equitight) $\underset{r\to \mathrm{\infty }}{\lim }\underset{|x|>r}{\int }{\left|f\right|}^p=0$ uniformly on $B$.

The first property can be stated as $\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0$ such that $\Vert {\tau }_{h}f-f{\Vert }_{L^p({\mathbb{ℝ}}^n)}<\epsilon \mathrm{\forall }f\in B,\mathrm{\forall }h$ with $|h|<\delta .$

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that $B$ is bounded (i.e., $\Vert f{\Vert }_{L^p({\mathbb{ℝ}}^n)}<\mathrm{\infty }$ uniformly on $B$). However, it has been shown that equitightness and equicontinuity imply this property.

Special case

For a subset $B$ of $L^p(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is a bounded subset of ${\mathbb{ℝ}}^n$, the condition of equitightness is not needed. Hence, a necessary and sufficient condition for $B$ to be 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 $(u_{ϵ}{)}_{ϵ}$ be a sequence of solutions of the viscous Burgers equation posed in $\mathbb{ℝ}\times (0,T)$:

$\frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial u^2}{\partial x}=ϵ\mathrm{\Delta }u,u(x,0)=u_0(x),$

with $u_0$ smooth enough. If the solutions $(u_{ϵ}{)}_{ϵ}$ enjoy the $L^1$-contraction and $L^{\mathrm{\infty }}$-bound properties, we will show existence of solutions of the inviscid Burgers equation

$\frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial u^2}{\partial x}=0,u(x,0)=u_0(x).$

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

$\underset{\mathbb{ℝ}}{\int }|u(x,t)-v(x,t)|dx\le \underset{\mathbb{ℝ}}{\int }|u_0(x)-v_0(x)|dx.$

The second property simply means that $\Vert u(\cdot ,t){\Vert }_{L^{\mathrm{\infty }}(\mathbb{ℝ})}\le \Vert u_0{\Vert }_{L^{\mathrm{\infty }}(\mathbb{ℝ})}$.

Now, let $K\subset \mathbb{ℝ}\times (0,T)$ be any compact set, and define

$w_{ϵ}(x,t):=u_{ϵ}(x,t){\mathbf{𝟏}}_K(x,t),$

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

$\underset{{\mathbb{ℝ}}^2}{\int }|w_{ϵ}(x,t)|dxdt=\underset{{\mathbb{ℝ}}^2}{\int }|u_{ϵ}(x,t){\mathbf{𝟏}}_K(x,t)|dxdt\le \Vert u_0{\Vert }_{L^{\mathrm{\infty }}(\mathbb{ℝ})}|K|<\mathrm{\infty }.$

Equicontinuity is a consequence of the $L^1$-contraction since $u_{ϵ}(x-h,t)$ is a solution of the Burgers equation with $u_0(x-h)$ as initial data and since the $L^{\mathrm{\infty }}$-bound holds: We have that

$\Vert w_{ϵ}(\cdot -h,\cdot -h)-w_{ϵ}{\Vert }_{L^1({\mathbb{ℝ}}^2)}\le \Vert w_{ϵ}(\cdot -h,\cdot -h)-w_{ϵ}(\cdot ,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}+\Vert w_{ϵ}(\cdot ,\cdot -h)-w_{ϵ}{\Vert }_{L^1({\mathbb{ℝ}}^2)}.$

We continue by considering

$\begin{array}{rl}& \Vert w_{ϵ}(\cdot -h,\cdot -h)-w_{ϵ}(\cdot ,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}\\ & \le \Vert (u_{ϵ}(\cdot -h,\cdot -h)-u_{ϵ}(\cdot ,\cdot -h)){\mathbf{𝟏}}_K(\cdot -h,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}+\Vert u_{ϵ}(\cdot ,\cdot -h)({\mathbf{𝟏}}_K(\cdot -h,\cdot -h)-{\mathbf{𝟏}}_K(\cdot ,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}.\end{array}$

The first term on the right-hand side satisfies

$\Vert (u_{ϵ}(\cdot -h,\cdot -h)-u_{ϵ}(\cdot ,\cdot -h)){\mathbf{𝟏}}_K(\cdot -h,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}\le T\Vert u_0(\cdot -h)-u_0{\Vert }_{L^1(\mathbb{ℝ})}$

by a change of variable and the $L^1$-contraction. The second term satisfies

$\Vert u_{ϵ}(\cdot ,\cdot -h)({\mathbf{𝟏}}_K(\cdot -h,\cdot -h)-{\mathbf{𝟏}}_K(\cdot ,\cdot -h)){\Vert }_{L^1({\mathbb{ℝ}}^2)}\le \Vert u_0{\Vert }_{L^{\mathrm{\infty }}(\mathbb{ℝ})}\Vert {\mathbf{𝟏}}_K(\cdot -h,\cdot )-{\mathbf{𝟏}}_K{\Vert }_{L^1({\mathbb{ℝ}}^2)}$

by a change of variable and the $L^{\mathrm{\infty }}$-bound. Moreover,

$\Vert w_{ϵ}(\cdot ,\cdot -h)-w_{ϵ}{\Vert }_{L^1({\mathbb{ℝ}}^2)}\le \Vert (u_{ϵ}(\cdot ,\cdot -h)-u_{ϵ}){\mathbf{𝟏}}_K(\cdot ,\cdot -h){\Vert }_{L^1({\mathbb{ℝ}}^2)}+\Vert u_{ϵ}({\mathbf{𝟏}}_K(\cdot ,\cdot -h)-{\mathbf{𝟏}}_K){\Vert }_{L^1({\mathbb{ℝ}}^2)}.$

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the $L^1$-contraction. The continuity of the translation mapping in $L^1$ then gives equicontinuity uniformly on $B$.

Equitightness holds by definition of $(w_{ϵ}{)}_{ϵ}$ by taking $r$ big enough.

Hence, $B$ is relatively compact in $L^1({\mathbb{ℝ}}^2)$, and then there is a convergent subsequence of $(u_{ϵ}{)}_{ϵ}$ in $L^1(K)$. By a covering argument, the last convergence is in $L_{loc}^1(\mathbb{ℝ}\times (0,T))$.

To conclude existence, it remains to check that the limit function, as $ϵ\to 0^{+}$, of a subsequence of $(u_{ϵ}{)}_{ϵ}$ satisfies

$\frac{\partial u}{\partial t}+\frac{1}{2}\frac{\partial u^2}{\partial x}=0,u(x,0)=u_0(x).$

See also

• Arzelà–Ascoli theorem
• Helly's selection theorem
• Rellich–Kondrachov theorem

Literature

• Brezis, Haïm (2010). Functional analysis, Sobolev spaces, and partial differential equations. Universitext. Springer. p. 111. ISBN 978-0-387-70913-0. 
• Riesz, Marcel (1933). "Sur les ensembles compacts de fonctions sommables". Acta Sci. Math. 6: 136–142. http://acta.bibl.u-szeged.hu/13418/1/math_006_136-142.pdf. 
• Precup, Radu (2002). Methods in nonlinear integral equations. Springer. p. 21. ISBN 978-1-4020-0844-3. https://archive.org/details/methodsnonlinear00prec.