Lemma: Finite norm and shell descent

The map $F\mapsto k_F$ is continuous from each fixed-support $C^{\infty }$ test space into $\mathfrak{h}$. Also

Proof. Integrating by parts spatially gives, for every integer $n\ge 1$,

Since $m>0$, already a suitable fixed $n$ gives an integrable square on ${\mathbb{R}}^3$. On a fixed compact support, the $L^1$ norm is controlled by finitely many $C^{\infty }$ seminorms. This proves the norm estimate and continuity.

The exponential in (eq:k) solves $P_{m}v=0$. Integration by parts against a compact $v$ proves $k_{P_{m}v}=0$.