Localization of trace norms in two and three dimensions

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