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[其它题目] 如何定义曲面上的Holder 空间

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发表于 2021-9-10 20:29:10 | 显示全部楼层 |阅读模式
We know the definition of H\"{o}lder space $C^{0,\gamma}$ on the domain $U$ (an open and connect domain in $\mathbb{R}^n$):
$$C^{0,\gamma}(U)=\left\{\,u\,\Big|\,\sup\limits_{x,y\in U,x\neq y}\frac{|u(x)-u(y)|}{|x-y|^\gamma}<+\infty\,\right\}.$$
But how can we define the H\"{o}lder space on unit sphere
$$B_1=\left\{\,x\in\mathbb{R}^n\,|\,|x|=1\,\right\}.$$

For example, let $n>3$ and $\phi(x)=\ln |x-\xi|$ with $\xi\in \text{int}\, B_1$, then
$$\|\phi\|_{C^{2,\gamma}(\partial B_1)}= \sum_{k=0}^2\sup |D^k\phi|+\sup_{x,y\in\partial B_1,x\neq y}\frac{|D^2\phi(x)-D^2\phi(y)|}{d^\gamma(x,y)}$$
where $d(x,y)$ is the distance between $x$ and $y$. But what's the meaning of $D^k\phi$ on $\partial B_1$?

I would appreciate it if someone could give me a definition or a reference!

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