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

Localization of trace norms in two and three dimensions