Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.

The following is from the book "Sobolev spaces" by Adams and Fournier:

Theorem. Let $1\leq p < \infty$. A bounded subset $K \subseteq L^p(\Omega)$ with $\Omega \subseteq \mathbb R^n$ a domain is precompact in $L^p(\Omega)$ if and only if for every number $\varepsilon >0$ there exists a number $\delta >0$ and a subset $G \subset \subset \Omega$ such that for every $u\in K$ and $h \in \mathbb R^n$ with $|h| < \delta$ both of the following inequalities hold: $$\int_\Omega |\tilde u(x+h) -\tilde u(x)|^p\,dx <\varepsilon^p \quad \text{and} \quad \int_{\Omega \setminus \overline G} |u(x)|^p\,dx < \varepsilon^p.$$

Here, $\tilde u$ is defined by $\tilde u = u$ on $\Omega$ and $\tilde u = 0$ on $\mathbb R^n\setminus \Omega$. The theorem is obtained by proving the version for $\mathbb R^n$ and applying this to $\tilde K = \{\tilde u : u \in K\}$.

In a book by Brezis ("Analisis functional, Theoria y aplicaciones"), the following theorem is stated:

Corollary. Let $\Omega \subseteq \mathbb R^n$ be open and let $\mathscr F$ be a bounded subset of $L^p(\Omega)$ with $1\leq p <\infty$. Suppose that

  • $\forall \varepsilon >0 \; \forall \omega \subset \subset \Omega,\;\exists \delta >0,\;\delta < \mathrm{dist}(\omega,\Omega^c) \text{ such that } \|\tau_h f -f\|_{L^p(\omega)} < \varepsilon \quad \forall |h|<\delta \text{ and } f\in\mathscr{F}.$
  • $\forall \varepsilon >0 \; \exists \omega \subset \subset \Omega \text{ such that } \|f\|_{L^p(\Omega\setminus\omega)} < \varepsilon \quad \forall f \in \mathscr F$

Then $\mathscr F$ is relatively compact in $L^p(\Omega)$.

Here, $\tau_h$ denotes the operator $\tau_h f(x) = f(x+h)$.

(Disclaimer: I translated this from the Spanish version of the book and I really don't know any Spanish (the original version is in French; unfortunately there is no English version).)

The crucial part here is that the first property need only be tested on $\omega \subset \subset \Omega$.

Using the Theorem from Adams and Fournier, the statement of Brezis should also characterize relative compactness, since $\|\tau_h f - f\|_{L^p(\omega)} \leq \|\tau_h \tilde f - \tilde f\|_{L^p(\Omega)}$ for every $\omega \subset \subset \Omega$.

My question is this:

  • Does anybody know if there are other references (preferably in English) where relative compactness is characterized by the properties in the Corollary of Brezis?
  • As a follow-up, are there references which deal with relative compactness in the case of weighted $L^p$-spaces? Specifically, I'm interested in the case where the weight takes the form $e^{-\varphi}$ for some $C^2$ function $\varphi$.

One probably could take the proofs for non-weighted spaces and adapt them, but I guess one would have to deal with adjusting the necessary tools (e.g. smoothing by convolution) to the weighted case, hence a reference would be nice.

Thanks in advance!

share|improve this question
1  
There is an English version of The Book by Brezis. –  user53153 Jan 26 '13 at 20:11
    
Thanks, 5PM! I probably missed it since I was looking for and English version of the second book on the list at the bottom of this page. Strangely, this book only deals with $L^p(\mathbb R^n)$. –  fbg Jan 26 '13 at 20:15
2  
You might be interested in having a look at this nice article by Harald Hanche-Olsen and Helge Holden for an exposition of the Kolmogorov-Riesz compactness theorem. –  Martin Jan 26 '13 at 20:30
    
Martin: Thanks, but I already knew of this article and while it is very nicely written, it doesn't really address my questions. –  fbg Jan 27 '13 at 14:59
1  
Okay, in that case this wasn't helpful at all... I wasn't sure and didn't check whether the article is really relevant. // If you include "@Martin" or "@5PM" in your comments then we are notified of your reactions. –  Martin Jan 27 '13 at 15:11

1 Answer 1

For a weighted version of the Kolmogorov-Riesz theorem you can have a look over this article (see Thm. 5). The authors used some results proved by Hanche-Olsen and Holden (cited as Lemma 6 in the same work).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.