Local Operators in k- forms An operator $L : Ω^*(M) → Ω^*(M)$ is support - decreasing if $suppL(w) \subset supp (w)$ for every k-form $w \in Ω^*(M)$ for all $0≤k$. Show that an operator on $Ω^*(M)$ is local if and only if it is support-decreasing.
Some help please.
An operator D : Ω^(M) → Ω^(M) is said to be local if for all $k ≥ 0$, whenever a k-form ω ∈ Ω^k(M) restricts to 0 on an open set U, then $Dω ≡ 0$ on U.

Suppose $L:\Omega^\ast (M)\to\Omega^\ast(M)$ is support-decreasing. Let $k\geq 0$ and $U$ be an open set such that $L(\omega)=0$ for all $\omega\in\Omega^k(M)$ such that $\omega|_U=0.$ Then $\operatorname{supp}\omega\subset M\smallsetminus U$, so that since $L$ is support-decreasing, $\operatorname{supp}L(\omega)\subset\operatorname{supp}\omega\subset M\smallsetminus U$, and in particular $L(\omega)|_U=0.$ So $L$ is local. (I suppose technically this was only shown for $L$ restricted to pure forms, but it isn't hard to extend to mixed forms.)
The other direction is similar.
