Is there a reference for compact imbedding theory of Hölder space?

Question

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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.

Answer 1

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.

Answer 2

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.