Fréchet–Kolmogorov theorem

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

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

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

Special case

For a subset $B$ of $L^{p}(\Omega )$, where $\Omega$ 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 $(u_{\epsilon })_{\epsilon }$ be a sequence of solutions of the viscous Burgers equation posed in $\mathbb {R} \times (0,T)$:

${\frac {\partial u}{\partial t}}+{\frac {1}{2}}{\frac {\partial u^{2}}{\partial x}}=\epsilon \Delta u,\quad u(x,0)=u_{0}(x),$

with $u_{0}$ smooth enough. If the solutions $(u_{\epsilon })_{\epsilon }$ enjoy the $L^{1}$-contraction and $L^{\infty }$-bound properties, we will show existence of solutions of the inviscid Burgers equation

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

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

$\int _{\mathbb {R} }|u(x,t)-v(x,t)|dx\leq \int _{\mathbb {R} }|u_{0}(x)-v_{0}(x)|dx.$

The second property simply means that $\Vert u(\cdot ,t)\Vert _{L^{\infty }(\mathbb {R} )}\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}$.

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

$w_{\epsilon }(x,t):=u_{\epsilon }(x,t)\mathbf {1} _{K}(x,t),$

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

$\int _{\mathbb {R} ^{2}}|w_{\epsilon }(x,t)|dxdt=\int _{\mathbb {R} ^{2}}|u_{\epsilon }(x,t)\mathbf {1} _{K}(x,t)|dxdt\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}|K|<\infty .$

Equicontinuity is a consequence of the $L^{1}$-contraction since $u_{\epsilon }(x-h,t)$ is a solution of the Burgers equation with $u_{0}(x-h)$ as initial data and since the $L^{\infty }$-bound holds: We have that

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

We continue by considering

${\begin{aligned}&\Vert w_{\epsilon }(\cdot -h,\cdot -h)-w_{\epsilon }(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}\\&\leq \Vert (u_{\epsilon }(\cdot -h,\cdot -h)-u_{\epsilon }(\cdot ,\cdot -h))\mathbf {1} _{K}(\cdot -h,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}+\Vert u_{\epsilon }(\cdot ,\cdot -h)(\mathbf {1} _{K}(\cdot -h,\cdot -h)-\mathbf {1} _{K}(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}.\end{aligned}}$

The first term on the right-hand side satisfies

$\Vert (u_{\epsilon }(\cdot -h,\cdot -h)-u_{\epsilon }(\cdot ,\cdot -h))\mathbf {1} _{K}(\cdot -h,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}\leq T\Vert u_{0}(\cdot -h)-u_{0}\Vert _{L^{1}(\mathbb {R} )}$

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

$\Vert u_{\epsilon }(\cdot ,\cdot -h)(\mathbf {1} _{K}(\cdot -h,\cdot -h)-\mathbf {1} _{K}(\cdot ,\cdot -h))\Vert _{L^{1}(\mathbb {R} ^{2})}\leq \Vert u_{0}\Vert _{L^{\infty }(\mathbb {R} )}\Vert \mathbf {1} _{K}(\cdot -h,\cdot )-\mathbf {1} _{K}\Vert _{L^{1}(\mathbb {R} ^{2})}$

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

$\Vert w_{\epsilon }(\cdot ,\cdot -h)-w_{\epsilon }\Vert _{L^{1}(\mathbb {R} ^{2})}\leq \Vert (u_{\epsilon }(\cdot ,\cdot -h)-u_{\epsilon })\mathbf {1} _{K}(\cdot ,\cdot -h)\Vert _{L^{1}(\mathbb {R} ^{2})}+\Vert u_{\epsilon }(\mathbf {1} _{K}(\cdot ,\cdot -h)-\mathbf {1} _{K})\Vert _{L^{1}(\mathbb {R} ^{2})}.$

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 $(w_{\epsilon })_{\epsilon }$ by taking $r$ big enough.

Hence, $B$ is relatively compact in $L^{1}(\mathbb {R} ^{2})$, and then there is a convergent subsequence of $(u_{\epsilon })_{\epsilon }$ 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 $\epsilon \to 0^{+}$, of a subsequence of $(u_{\epsilon })_{\epsilon }$ satisfies

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

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" (PDF). Acta Sci. Math. 6: 136–142.
• Precup, Radu (2002). Methods in nonlinear integral equations. Springer. p. 21. ISBN 978-1-4020-0844-3.