## model theory – Roelcke precompactness and Ramsey property

A survey by Nguyen Van Thé (2014) has Conjecture 1,
which is that
“every closed oligomorphic
subgroup of $$S_∞$$ should have a metrizable universal minimal flow with a generic
orbit.” Later, it goes on to say that “it is even possible that this should be
true for a larger class of groups, called Roelcke precompact.” (Let me call this Conjecture 1′.) Now, Kwiatkowska (2018) exhibited a group without a metrizable universal minimal flow that is not Roelcke precompact, so we need to stick to Conjecture 1.

How about the converse of Conjecture 1′, i.e., the statement that a closed subgroup of $$S_∞$$ is Roelcke precompact if it has a metrizable universal minimal flow? Is there a proof or a counterexample? In absence of either of the two, do people believe it?

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## real analysis – Generalization of Kolmogorov precompactness criterion

In the book Elements in functional analysis from Hirsch and Lacombe, the Kolmogorov precompactness criterion for families of $$L^p$$ functions is stated as follows:

Theorem: Let $$H subseteq L^p(mathbb{R}^d)$$, with $$p in (1,infty)$$. Then $$H$$ is precompact if and only if the following conditions hold:

1. $$H$$ is bounded w.r.t. the norm $$||cdot||_{L^p(mathbb{R}^d)}$$;
2. $$lim_{R to +infty} sup_{f in H} int_{B_R^c} |f(x)|^p dx = 0$$, where $$B_R^c$$ denotes the complement in $$mathbb{R}^d$$ of the ball of radius $$R$$;
3. $$lim_{|a| to 0} sup_{f in H} ||tau_af-f||_{L^p(mathbb{R}^d)}=0$$, where $$tau_a$$ denotes the translation operator.

Now, I think that one can modify a little bit this statement in order to make it valid also when we consider families of functions in $$L^p(Omega)$$, where $$Omega$$ is an open set in $$mathbb{R}^d$$:

Claim: Let $$Omega$$ be an open set in $$mathbb{R}^d$$, let $$p in (1,infty)$$, and let $$H subseteq L^p(Omega)$$. For all $$R > 0$$, define
$$Omega_R = {x in Omega: text{dist}(x, {Omega}^c) > frac{1}{R}} cap B(0,R).$$
Then $$H$$ is precompact if and only id the following holds

1. $$H$$ is bounded w.r.t. the norm $$||cdot||_{L^p(Omega)}$$;
2. $$lim_{R to +infty} sup_{f in H} int_{Omega setminus Omega_R} |f(x)|^p dx = 0$$;
3. $$lim_{|a| to 0} sup_{f in H} ||tau_af-f||_{L^p(Omega_R)}=0$$ for all $$R >0$$.

I’m trying to prove this last equivalence using the theorem I’ve written above. So far this is my strategy: First, notice that we can look at $$L^p(Omega)$$ as a subset of $$L^p(mathbb{R}^d)$$ extending every function $$f$$ in $$L^p(Omega)$$ to a function $$tilde{f}$$ in $$L^p(mathbb{R}^d)$$ in this way:
$$tilde{f}(x) =begin{cases} f(x) & x in Omega \ 0 & text{otherwise} end{cases}$$
Then $$H$$ is precompact in $$L^p(Omega)$$ if and only if it’s precompact in $$L^p(mathbb{R}^d)$$.

If I prove that conditions 1,2,3 of the claim are equivalent to conditions 1,2,3 of the theorem, I’m done. Condition 1 is easy, but I can’t really see how conditions 2 and 3 of the theorem should be equivalent to condition 2 and 3 of the claim.

Any help, remark or suggestion is appreciated, thank you.

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