## Compactness between holder space

Let $\Omega$ is a domain in $\mathbb{R}^n$, $u$ is function on $\Omega$, $\alpha\in[0,1]$

$\displaystyle [u]_{\alpha;\Omega}=\sup\limits_{x,y}\frac{u(x)-u(y)}{|x-y|^\alpha}$ if $\alpha>0$

$\displaystyle [u]_{\alpha;\Omega}=\sup\limits_{x}|u(x)|$ if $\alpha=0$

And for $k=0,1,2,\cdots$

$[u]_{0,\alpha;\Omega}=[u]_{\alpha;\Omega}$,

$[u]_{k,\alpha;\Omega}=\sup\limits_{|\beta|=k}\sup\limits_{\Omega}[D^\beta u]_{\alpha;\Omega}$

Then we can define a norm

$\displaystyle ||u||_{k,\alpha;\Omega}=\sum\limits_{j=0}^{k}[u]_{k,0;\Omega}+[ u]_{k,\alpha;\Omega}$

So we have generalized holder space $C^{k,\alpha}(\overline{\Omega})=\{u|||u||_{k,\alpha;\Omega}<\infty\}$. In arbitrary domain $\Omega$, we don’t have the inclusion $C^{k,\alpha}(\overline{\Omega})\subset C^{j,\beta}(\overline{\Omega})$ when $j+\beta. But for $C^{0,1}$ domain, we have

$\mathbf{Thm:}$ Suppose $j+\beta, where $j=0,1,2,\cdots,$, $k=1,2,\cdots$ and $0\leq \alpha,\beta\leq 1$. Let $\Omega$ be a $C^{0,1}$ domain and $u\in C^{k,\alpha}(\overline{\Omega})$. Then for any $\epsilon >0$ and some constant $C=C(\epsilon,j,k,\Omega)$, we have

$\displaystyle |u|_{j,\beta;\Omega}\leq C|u|_{0;\Omega}+\epsilon |u|_{k,\alpha;\Omega}$

From this interpolation result, we can prove the compactness between holder space

$\mathbf{Thm:}$ Suppose $j+\beta with $k\geq 1$. $\Omega$ as before. The  inclusion $C^{k,\alpha}(\overline{\Omega})\hookrightarrow C^{j,\beta}(\overline{\Omega})$ is compact

$\mathbf{Remark:}$GT p137