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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!

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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
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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

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