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]_{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<k+\alpha. But for C^{0,1} domain, we have

\mathbf{Thm:} Suppose j+\beta<k+\alpha, 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<k+\alpha 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

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