8
$\begingroup$

This question is posted and unanswered from math.stackexchange.

Suppose $0 < \alpha < \beta$ and $\Omega$ is bounded. Then, the Hölder space $C^\beta(\Omega)$ is compactly imbedded to $C^\alpha(\Omega)$. See the wikipedia page:

http://en.wikipedia.org/wiki/H%C3%B6lder_condition

More precisely, I want to know the exact reference of the theory related to the following statement: [Claim] Given $\{f_n\}$ is a sequence of functions with $\|f_n\|_\beta <1$ for all $n$. Then, there exists a subsequence $\{n_k\}$ and $f\in C^\alpha$ such that, $\lim_{k\to \infty} \|f_{n_k} - f\|_\alpha = 0$. In the above, $\|\cdot\|_\alpha$ is Hölder-$\alpha$ norm.

However, I could not find a precise reference from some books on functional analysis.

1) Can anybody indicate a precise reference for this theorem?

2) If possible, I would like to know a reference on the similar result on parabolic Hölder space.

Thanks.

$\endgroup$
4
  • 2
    $\begingroup$ The proof for (1) is already given in the Wiki article you linked to. $\endgroup$ Jul 24, 2013 at 11:52
  • 1
    $\begingroup$ @WillieWong Yes, I guess for (2), the proof is similar, but more complicated. I just want to save my work by by citing a reference, but wiki is not acceptable formally. Nevertheless, it's not for parabolic H\"older space. $\endgroup$
    – kenneth
    Jul 24, 2013 at 14:21
  • 2
    $\begingroup$ I think that the Arzela-Ascoli theorem does the trick. $\endgroup$ Jul 24, 2013 at 15:12
  • 2
    $\begingroup$ Actually it was me who wrote that section on compactness in the wiki article. Since the proof is one line, and can be checked immediately, I didn't bother to look for a reference. $\endgroup$ Feb 28, 2015 at 19:35

3 Answers 3

2
$\begingroup$

For non-integer values of $\alpha$, the space $C^\alpha$ has a nice characterisation in terms of wavelet coefficients, see "Wavelets and Operators" by Yves Meyer. With that characterisation (essentially a weighted $\ell^\infty$ bound on the wavelet coefficients), the compactness statement boils down to the (trivial) statement that the set of sequences bounded by some fixed sequence $\{a_n\}$ converging to $0$ is compact in $\ell^\infty$. For parabolic Hölder spaces (or any non-Euclidean scaling for that matter), Meyer's characterisation and therefore the compactness of the embedding still works, provided that one considers a suitably scaled tensor product wavelet basis.

$\endgroup$
1
  • $\begingroup$ But Mayer's book (in that Chapter 6) only consider the case where $\Omega=\mathbb{R}^n$; but do Meyer's characterizations hold for bounded open domains with Lipschitz boundaries (like in OP's question)? $\endgroup$
    – ABIM
    Jul 24, 2022 at 11:15
1
$\begingroup$

Watch this paper. Proposition 24.23

http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Holder-spaces.pdf

$\endgroup$
3
  • $\begingroup$ Did you mean 24.13? $\endgroup$ Feb 28, 2015 at 19:40
  • $\begingroup$ Yes. It is clear. $\endgroup$ May 28, 2015 at 17:11
  • $\begingroup$ If you want 24.14 please ask. $\endgroup$ May 28, 2015 at 17:18
0
$\begingroup$

I may be wrong, but I think this fits the requirements of what you are asking. Consider Holder continuous functions defined on $\mathbb{R}$. Specifically, $f_n:\mathbb{R}\to\mathbb{R}$ given by $$ f_n(x) = \frac{1}{2}(\max\{ x-n,0\})^\beta \ \ \ \forall x\in\mathbb{R},\ n\in\mathbb{N} .$$ It follows that $f_n\in C^\beta (\mathbb{R})$ and $||f_n||_\beta = \tfrac{1}{2}<1$ for all $n\in\mathbb{N}$. It also follows that $f_n(x)\to 0$ for all $x\in\mathbb{R}$ as $n\to\infty$ (albeit not uniformly). However, on any compact interval (where $f_n$ does converges uniformly to 0 as $n\to\infty$) any subsequence of $f_n$ must also converge to 0 as $n\to\infty$. Therefore, there is only one function corresponding to $f$ in this scenario, namely $f\equiv 0$. It also follows that for any $\alpha \in (0,\beta )$, we have $f_n\not\in C^\alpha (\mathbb{R})$ (as it is only locally Holder continuous of degree $\alpha$, not globally). This eliminates the possibilty of taking the difference $||f_n-f||_\alpha$ completely.

I also feel I should ask why you have supposed the condition $||f_n||_\beta <1$. It doesn't seem that there is too much special about 1. It does seem that the intention here is to consider $f_n$ defined on a compact/bounded set, however in your claim, it was not explicitly stated.

$\endgroup$
4
  • $\begingroup$ perhaps one should take $||f_{n_k}- f||_\alpha$ with the Holder $\alpha$ norm on an arbitrary compact subset of the original set. $\endgroup$
    – JCM
    Jul 24, 2013 at 11:46
  • 2
    $\begingroup$ Compact embedding is only true on bounded domains. In the Wiki article that the OP referred to the assumption is stated, but the OP failed to copy it into the question. $\endgroup$ Jul 24, 2013 at 11:54
  • $\begingroup$ @JCM Thanks, your answer is correct. But, as you mentioned, I forgot to mention its domain is a bounded set. This is already added in the above question now. By the way, one can just put $\|f_n\|_\beta < K$ instead of $<1$. $\endgroup$
    – kenneth
    Jul 24, 2013 at 14:42
  • $\begingroup$ On second look, it seems that I was using the Holder semi-norm and not the Holder norm. The argument I employed was not quite correct. $\endgroup$
    – JCM
    Jul 24, 2013 at 14:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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