We extend a localization result for the $H^{1/2}$ norm by B. Faermann to a wider class of subspaces of $H^{1/2}(\Gamma)$, and we prove an analogous result for the $H^{-1/2}(\Gamma)$ norm, $\Gamma$ being the boundary of a bounded polytopal domain $\Omega$ in $\mathbb{R}^n$, $n=2,3$. As a corollary, we obtain equivalent, better localized, norms for both $H^{1/2}(\Gamma)$ and $H^{-1/2}(\Gamma)$, which can be exploited, for instance, in the design of preconditioners or of stabilized methods.

翻译：我们将B. Faermann关于$H^{1/2}$范数的局部化结果推广到$H^{1/2}(\Gamma)$的更广泛子空间类，并证明了$H^{-1/2}(\Gamma)$范数的类似结果，其中$\Gamma$为$\mathbb{R}^n$（$n=2,3$）中有界多面体区域$\Omega$的边界。作为推论，我们获得了$H^{1/2}(\Gamma)$和$H^{-1/2}(\Gamma)$的等价且具有更强局部性的范数，这些范数可应用于例如预处理器或稳定化方法的设计中。