I start with short introduction. Let $\Omega$ be an open, connected and bounded subset of $\mathbb{R}^d$. Let $\Gamma$ denote the boundary of $\Omega$.
1) I cover $\Omega$ with a collection $\{U_1,…,U_M\}$ of open subsets of $\mathbb{R}^d$, $\Gamma\subset \bigcup_{r=1}^MU_r$ such that
$$\Gamma_r=U_r\cap\Gamma\neq\emptyset, \qquad r=1,…,M.$$
2) For each $U_r$, I introduce an orthogonal transformation $A_r$ of coorditate system $x=(x_1,…,x_d)$ into local coordinate system $y_r=(y_{r1},…,y_{rd}): y_r=A_r(x)$.
3) I assume that there exist $\alpha, \beta>0$ and functions $f_r$ (let's say Lipschitz functions for now) defined on sets $S_r$, where
$$S_r=\{y_r^{'}=(y_{r1},…,y_{rd-1})\,|\,|y_{ri}|<\alpha\}.$$
Now I demand the following construction: $$\Gamma_{r}=U_r\cap \Gamma=\{(y_r^{'},f_{r}(y_{r}^{'}))\,|\,y_{r}^{'}\in S_{r}\},$$
$$U_r^{+}=U_{r}\cap \Omega=\{y_r\, |\,y_{r}^{'}\in S_{r},\, f_r(y_{r}^{'})<y_{rd}<f_{r}(y^{'}_{r})+\beta\},$$
$$U_r^{-}=U_{r}-cl(\Omega)=\{y_r\, |\,y_{r}^{'}\in S_{r},\, f_r(y_{r}^{'})-\beta<y_{rd}<f_{r}(y^{'}_{r})\}.$$
It follows from the construction, that $\Gamma$ can be described in the terms of hyperspaces (locally). The regularity of $\Gamma_r$ is assumed to be determined by choosing $k$ and $\lambda$ such that $f_r\in C^{k,\lambda}(cl(S_{r}))$.
Theorem. Indentyfying $H^{0}(\Gamma)$ with its dual, we have
$$(H^s(\Gamma))^*=H^{-s}(\Gamma).$$
What would be the proof?
My attempt is as follows: let $L$ be a continuous linear form on $H^{s}(\Gamma)$. Can it be written this way: $$L(u)=\sum_{r=1}^{m} \langle g_j,u_r \rangle ,$$ $$g_j\in (H^{s}(S_{r}))^*=H^{-s}(S_r),\qquad (1)$$ where $u_{r}(y_{r}^{'})=u(y^{'}_{r}, f_{r}(y_{r}^{'}))$? Does it follow from Hahn-Banach theorem? Suppose I can do that. Now, I pick a function $\psi_{r}$ such that it is positive, has a compact support in $|y_{ri}|<\alpha$ for $i=1,…,d$ and $\psi_{r}=1$ in the neighbourhood of the suppoert of $u_{r}$. So
$$L(u)=\sum_{r=1}^{M}\langle\psi_rg_r, u_r\rangle$$ and I'm stuck. What should be done next?