Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$
$$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp \,(u)\subset H\} $$ Where, $H$ is a Half Space defined by a fixed vector $N\neq 0$ in $\mathbb R^n$ $$ H=\{y\in \mathbb R^n \,|\, (y,N)\geq 0\} $$
$(,)$ is inner product in $\mathbb R^n$.
I wish to show that for every $y \in H$, there exist a compact set $K\subset \mathbb R^n$ and constants $C $ and $k$ (all depending on y) such that
$$ |u(y)|\leq C \sum_{|\alpha |\leq k} \sup _{K}|D^{\alpha} \,T(u)| $$
For all $u \in X$
PS: In my context the operator $T$ is a differential operator, if someone thinks the question in under-informed, I can provide extra details.
PPS: It seems to me that the structure preserving Automorphisms preserves the semi-norms, the the result should follow. But I cannot figure out whey the constants and compact set depend on the point we take.