In an article I am reading, the author proves that a function sequence $g_{n}$ is bounded in $H^{1}([0,1];\mathbb{R}^{k})$, then he uses Rellich-Kondrachov Theorem to obtain the relative compactness of $g_{n}$ in holder space $C^{0,r}([0,1];\mathbb{R}^{k})$ for any $r \in (0,1/2)$.
My question is why $g_{n}$ is relative compact in $C^{0,r}([0,1];\mathbb{R}^{k})$ for any $r \in (0,1/2)$? As I know, the Rellich-Kondrachov Theorem says the compact embedding of Sobolev space to $L^{p}$ space, not the holder space. Did I miss something? Is there another theorem that can explain this relationship?
To avoid problems caused by ignoring conditions, I put the definition of $g_{n}$ below： $$g_n=J_{\varepsilon_n} * \tilde{f}_n,$$ where $J$ is a standard mollifier with $\|J\|_{L^{\infty}} \leq \beta$, and $\varepsilon_n=\frac{1}{2 n}$. Here $\tilde{f}_n$ is the continuum extension of $f_n$ defined by $$ \tilde{f}_n(t)= \begin{cases}f_n(0) & \text { if } t<0 \\ f_n\left(t_i^{(n)}\right) & \text { if } t \in\left[t_i^{(n)}, t_{i+1}^{(n)}\right) \text { for some } i=0, \ldots, n-1 \\ f_n\left(t_{n-1}^{(n)}\right) & \text { if } t \geq 1\end{cases} $$ with $f_n:\left\{t_i^{(n)}\right\}_{i=0}^{n-1} \rightarrow \mathbb{R}^\kappa$ satisfies $$ \left\|f_n(0)\right\|^2+n \sum_{j=1}^{n-1}\left\|f_n\left(t_j^{(n)}\right)-f_n\left(t_{j-1}^{(n)}\right)\right\|^2 \leq M, \quad\left\|f_n\right\|_{L^{\infty}\left(\mu_n\right)} \leq M. $$
Thank you for your guidance.