Optimal trace norms for Helmholtz problems

The natural $H^1(\mathrm{\Omega })$ energy norm for Helmholtz problems is weighted with the wavenumber modulus $\sigma$ and induces natural weighted norms on the trace spaces $H^{\pm 1\mathrm{/}2}(\mathrm{\Gamma })$ by minimial extension to $\mathrm{\Omega }\subset {\mathbb{R}}^n$. This paper presents a rigorous analysis for these trace norms with an explicit characterisation by weighted Sobolev-Slobodeckij norms and scaling estimates, highlighting their dependence on the geometry of the extension set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and the weight $\sigma$. The analysis identifies conditions under which these trace norms are intrinsic to the isolated boundary component $\mathrm{\Gamma }\subset \mathrm{\partial }\mathrm{\Omega }$ and provides $\sigma$-explicit estimates for trace inequalities in weighted spaces. In these natural wavenumber-weighted norms, the boundary integral operators allow improved continuity estimates that do \emph{not} deterioriate as $\sigma \to 0$.