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

The subset ${\displaystyle B}$ is relatively compact if and only if the following properties hold:

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

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

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that ${\displaystyle B}$ is bounded (i.e., ${\displaystyle \Vert f\Vert _{L^{p}(\mathbb {R} ^{n})}<\infty }$ uniformly on ${\displaystyle B}$). However, it has been recently shown that equitightness and equicontinuity imply this property.[1]

Special case

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

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

with ${\displaystyle u_{0}}$ smooth enough. If the solutions ${\displaystyle (u_{\epsilon })_{\epsilon }}$ enjoy the ${\displaystyle L^{1}}$-contraction and ${\displaystyle L^{\infty }}$-bound properties,[2] we will show existence of solutions of the inviscid Burgers equation

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

The first property can be stated as follows: If ${\displaystyle u,v}$ are solutions of the Burgers equation with ${\displaystyle u_{0},v_{0}}$ as initial data, then

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

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

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

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

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

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

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

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

We continue by considering

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

The first term on the right-hand side satisfies

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

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

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

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

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

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the ${\displaystyle L^{1}}$-contraction.[3] The continuity of the translation mapping in ${\displaystyle L^{1}}$ then gives equicontinuity uniformly on ${\displaystyle B}$.

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

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

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

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

See also

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

References

1. Hanche-Olsen, H.; Holden, H.; Malinnikova, E. (2019). "An improvement of the Kolmogorov–Riesz compactness theorem". Expo. Math. 37 (1): 84–91.
2. Necas, J.; Malek, J.; Rokyta, M.; Ruzicka, M. (1996). Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation 13. Chapman and Hall/CRC. ISBN 978-0412577505.
3. Kruzhkov, S. N. (1970). "First order quasi-linear equations in several independent variables". Math. USSR Sbornik. 10: 217–243.

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.