Let
- $\Omega\subset\mathbb{R}^2$ be an open bounded set,
- $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$,
- $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$,
- $u\in L^{\infty}\big(\Omega(x, \rho);\mathbb{R}^N\big)$ and
- $O(u, x, \rho)=\max_{1\leq k\leq N} \text{osc}_{\Omega(x, \rho)}u^k$, where $\text{osc}_{S} v\equiv\text{ess max}_{S}v-\text{ess min}_{S}v$.
In Wiegner's 1981 paper, "On Two-Dimensional Elliptic Systems with a One-sided Condition", he defines the following norm for the Hölder space $C^{0, \alpha}(\overline{\Omega};\mathbb{R}^N)$: \begin{equation} \|u\|_{C^{0, \alpha}(\Omega;\mathbb{R}^N)}\equiv \sup_{\Omega}|u(x)|+\sup_{\substack{x\in \Omega\\ \rho\leq\rho_0}}\{\rho^{-\alpha}\cdot O(u, x, \rho)\} \end{equation}
Does anyone understand what $\rho_0$ is? He certainly doesn't define it before that line in his paper.
I am assuming that if $u\in C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)$ then \begin{equation} \|u\|_{C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)}\equiv \sum_{k=1}^N\|u^k\|_{L^{\infty}(\Omega)}+[u^k]_{C^{0, \alpha}(\Omega)} \end{equation} would be the typical Holder norm we would assign to the space $C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)$. Wiegner's norm is as in the question but I will denote it as $\|\cdot\|_W$ to distinguish it from the one above. If Ray Yang's interpretation (see comments) is true then I should be able to show an equivalence between the two norms.
To show $\|\cdot\|_{C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)}\leq K\|\cdot\|_W$ I argued as follows:
Firstly, we see that: \begin{equation} \sup_{\Omega} |u|\geq \|u^k\|_{L^{\infty}(\Omega)}\quad\forall \ k \end{equation}which implies \begin{equation}\tag{1} N\sup_{\Omega} |u|\geq \sum_{i=1}^N\|u^k\|_{L^{\infty}(\Omega)}. \end{equation}
Now if we let $x\in\Omega$ and $\rho_o=\text{dist}(x, \partial\Omega)$ then for all $0<\rho\leq\rho_0$ and $y: |x-y|=\rho$ we deduce: \begin{align} \frac{|u^k(x)-u^k(y)|}{|x-y|^{\alpha}}&\leq\frac{\max_{B(x, \rho)}u^k(z)-\min_{B(x, \rho)}u^k(z)}{\rho^{\alpha}}\\ &\leq \rho^{-\alpha}O(u, x, \rho)\quad\ \forall\ k. \end{align} In this way as we vary $x$, we vary $\rho_0$ and $y$ , so for each $k$ we have \begin{equation} [u^k]_{C^{0, \alpha}(\Omega)}\leq \sup_{\substack{x\in\Omega\\ \rho\leq\rho_0}}\{\rho^{-\alpha}O(u, x, \rho)\} \end{equation} and therefore \begin{equation}\tag{2} \sum_{k=1}^{N}[u^k]_{C^{0, \alpha}(\Omega)}\leq N\sup_{\substack{x\in\Omega\\ \rho\leq\rho_0}}\{\rho^{-\alpha}O(u, x, \rho)\}. \end{equation} Putting (1) and (2) together we have \begin{equation} \|u\|_{C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)}\leq N\|u\|_W \end{equation} I don't know how to get \begin{equation} \|\cdot\|_W\leq K\|\cdot\|_{C^{0, \alpha}(\overline{\Omega}, \mathbb{R}^N)}. \end{equation}