Assume $u \in \mathscr{D}(\mathbb{R}^n)$ is a distribution of order $k$, with compact support on a smooth manifold $\Gamma \subset \mathbb{R}^n$. We know that we can write this distribution (Thm 2.3.5, Hörmander, Vol. I) as $$ \langle u,\varphi\rangle_{\mathscr{D}'\times\mathscr{D}} = \sum_{|\alpha|\leq k} u_\alpha\ \big(\partial^\alpha\varphi\big|_\Gamma\big) = \sum_{|\alpha|\leq k} u_\alpha\ \gamma_\Gamma\big(\partial^\alpha\varphi\big) $$ where $\alpha$ is a multiindex, $u_\alpha$ are distributions of order $k-|\alpha|$ with compact support on $\Gamma$, and $\gamma_\Gamma$ is the trace operator. Now, I assume it is possible to write $$ \langle u,\varphi\rangle_{\mathscr{D}'\times\mathscr{D}} = \sum_{|\alpha|\leq k} \Big( -|1|^\alpha \partial^\alpha\big(\gamma^*_\Gamma u_\alpha\big)\Big)\ \varphi $$ and thus \begin{gather} \tag{*} u = \sum_{|a|\leq k} -|1|^\alpha \partial^\alpha\big(\gamma^*_\Gamma u_\alpha\big) \end{gather} Defining the adjoint operator of the trace as $$ \langle \gamma^*_\Gamma u_\alpha,\varphi\rangle_{\mathscr{D}'(\mathbb{R}^n)\times\mathscr{D}(\mathbb{R}^n)} := \langle u_\alpha,\gamma_\Gamma\varphi\rangle_{\mathscr{D}'(\Gamma)\times\mathscr{D}(\Gamma)} $$ Question I: Is $(*)$ reasonable?
However, we could have also written $$ \tag{**} u = \sum_{|a|\leq k} -|1|^\alpha u_\alpha \partial^\alpha\delta_\Gamma $$ with $\delta_\Gamma$ the Dirac delta function on the manifold $\Gamma$.
Question II: Are $(*)$ and $(**)$ equivalent?