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

The subset [math]\displaystyle{ B }[/math] is relatively compact if and only if the following properties hold:

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

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

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

Special case

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

[math]\displaystyle{ \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]\displaystyle{ u_0 }[/math] smooth enough. If the solutions [math]\displaystyle{ (u_\epsilon)_\epsilon }[/math] enjoy the [math]\displaystyle{ L^1 }[/math]-contraction and [math]\displaystyle{ L^\infty }[/math]-bound properties, we will show existence of solutions of the inviscid Burgers equation

[math]\displaystyle{ \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]\displaystyle{ u,v }[/math] are solutions of the Burgers equation with [math]\displaystyle{ u_0,v_0 }[/math] as initial data, then

[math]\displaystyle{ \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]\displaystyle{ \Vert u(\cdot,t)\Vert_{L^\infty(\mathbb{R})}\leq \Vert u_0\Vert_{L^\infty(\mathbb{R})} }[/math].

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

[math]\displaystyle{ w_\epsilon(x,t):=u_\epsilon(x,t)\mathbf{1}_K(x,t), }[/math]

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

[math]\displaystyle{ \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|\lt \infty. }[/math]

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

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ L^1 }[/math]-contraction. The second term satisfies

[math]\displaystyle{ \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]\displaystyle{ L^\infty }[/math]-bound. Moreover,

[math]\displaystyle{ \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]\displaystyle{ L^1 }[/math]-contraction. The continuity of the translation mapping in [math]\displaystyle{ L^1 }[/math] then gives equicontinuity uniformly on [math]\displaystyle{ B }[/math].

Equitightness holds by definition of [math]\displaystyle{ (w_\epsilon)_\epsilon }[/math] by taking [math]\displaystyle{ r }[/math] big enough.

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

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

[math]\displaystyle{ \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

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.