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 Lp space. It can be thought of as an Lp 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 be a subset of with , and let denote the translation of by , that is,

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

  1. (Equicontinuous) uniformly on .
  2. (Equitight) uniformly on .

The first property can be stated as such that with

Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that is bounded (i.e., uniformly on ). However, it has been recently shown that equitightness and equicontinuity imply this property.[1]

Special case

For a subset of , where is a bounded subset of , the condition of equitightness is not needed. Hence, a necessary and sufficient condition for 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 be a sequence of solutions of the viscous Burgers equation posed in :

with smooth enough. If the solutions enjoy the -contraction and -bound properties,[2] we will show existence of solutions of the inviscid Burgers equation

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

The second property simply means that .

Now, let be any compact set, and define

where is on the set and 0 otherwise. Automatically, since

Equicontinuity is a consequence of the -contraction since is a solution of the Burgers equation with as initial data and since the -bound holds: We have that

We continue by considering

The first term on the right-hand side satisfies

by a change of variable and the -contraction. The second term satisfies

by a change of variable and the -bound. Moreover,

Both terms can be estimated as before when noticing that the time equicontinuity follows again by the -contraction.[3] The continuity of the translation mapping in then gives equicontinuity uniformly on .

Equitightness holds by definition of by taking big enough.

Hence, is relatively compact in , and then there is a convergent subsequence of in . By a covering argument, the last convergence is in .

To conclude existence, it remains to check that the limit function, as , of a subsequence of satisfies

See also

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

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