Let $K \in \mathbb{R}^d$ be a compact set and consider the space of Hölder continuous functions $C^{0,\gamma}(K)$ with norm $||f||_{C^{0,\gamma}}:=||f||_{\infty}+\sup_{x,y \in K,x \neq y}\frac{|f(x)-f(y)|}{|x-y|^{\gamma}}$. Assume we have a bounded sequence $\{f_n\} \subset C^{0,\gamma}(K)$, i.e. $\exists C>0$ s.t. $\sup_{n}||f_n||_{C^{0,\gamma}} \leq C$. Under what conditions can we say the sequence $\{f_n\}$ is pre-compact in $C^{0,\gamma}(K)$? In other words, what are the sufficient conditions that guarantee there exist a subsequence $f_{n_k}\subset f_n$ and a $f \in C^{0,\gamma}(K)$ such that $f_{n_k} \xrightarrow{C^{0,\gamma}}f$ ?
Thank you very much!
A sufficient condition is: the sequence $f_n$ is bounded in $C^{0,\gamma+\epsilon}$ for some $\epsilon>0$. For a proof, see Is there a reference for compact imbedding of Hölder space?
A slightly weaker sufficient condition: there exists a modulus of continuity $\omega:(0,\infty)\to(0,\infty)$ such that $\omega(\delta)=o(\delta^\gamma) $ as $\delta\to 0$ and $$|f_n(a)-f_n(b)|\le \omega(|a-b|) \tag{1}$$ for all $a,b$ in $K$, and for all $n$.
The proof is the same as in thread linked above. Namely, extract a uniformly convergent subsequence by Ascoli-Arzelà and use (1) in the form $$\frac{|f_n(a)-f_n(b)|}{|a-b|^\gamma} \le \tau(|f_n(a) -f_n(b) |) $$ where $\tau$ is such that $\tau(t)\to 0$ as $t\to 0$. The existence of $\tau$ follows by rewriting (1) as $$\omega^{-1}(|f_n(a)-f_n(b)|)\le |a-b| $$ where $\omega^{-1}$ is the inverse of $\omega$, hence satisfies $\omega^{-1}(t)/t^{1/\gamma}\to\infty$ as $t\to0$.