Half-space weak integration by parts for boundary-compatible tests

Let $H=\{(x',t):t>0\}\subset\mathbb R^{m+1}$. Suppose $f,g$ are locally integrable for the restricted measure $\mathbf1_H\,dx$, including in neighborhoods of the boundary, and $g=\partial_i f$ weakly in the open half-space. Let $\psi\in C_c^\infty(\mathbb R^{m+1})$. If $i$ is the normal coordinate, assume $\psi(x',0)=0$; for tangential coordinates impose no boundary condition. Then

Thus weak integration by parts extends from tests supported strictly inside $H$ to smooth ambient tests compatible with the direction of differentiation. A boundary cutoff has no tangential derivative. In the normal case, the zero boundary value gives $|\psi(x',t)|\le Ct$ on the relevant compact set, controlling the cutoff derivative term by an integrable multiple of $|f|$. This is an auxiliary distributional lemma derived from the reflection construction; it does not assume that an extension theorem has already been proved.