Estimates on the boundary for differential operators with constant coefficients in a half-space

For differential operators $A(D)$, $P_j(D)$ ($j=1,\dots,N$, $D=(\partial/i\partial x_1,\dots,\partial/i\partial x_{n-1};\partial/i\partial t)$) with constant complex coefficients in the half-space $\mathbf R^n_+=\{(x;t),x\in\mathbf R^{n-1},t\geqslant0\}$ we present a precise description of the “space of traces” $A(D)u|_{t=0}$ of elements $u$ in the completion of the space $C^\infty_0(\mathbf R^n_+)$ with respect to the metric $\sum_{j=1}^N\|P_j(D)u\|^2$ ($\|\cdot\|$ is the norm in $L_2(\mathbf R^n_+)$). We consider the case of the metric $\|P(D)u\|^2+\|u\|^2$ in detail.
We establish necessary and sufficient conditions for validity of the inequality
$$ \bigl\langle A(D)u\bigr\rangle_{s_0}^2\leqslant C\biggl(\sum_{j=1}^N\|P_j(D)u\|^2+\sum_{k=1}^r\langle B_k(D)u\rangle_{s_k}^2\biggr) $$
for all $u(x;t)\in C^\infty_0(\mathbf R^n_+)$ ($\langle\cdot\rangle$ is the norm in $\mathscr H_s(\partial\mathbf R^n_+)$).