Proposition: Support of a finite normal-jet distribution

For

one has

Consequently, if $C\subset U$ is relatively closed, a supported normal-order-preserving change $\Delta t=\sum {\mathsf{E}}_{\gamma }{d}_{\gamma }$ is supported in $C$ if and only if every $d_{\gamma }$ is supported in

Proof. The inclusion from right to left as an upper bound on the support follows from the insertion definition.

For the reverse inclusion, suppose $t$ vanishes in a product neighborhood of $(x,0)$. Choose the cutoff in ${\mathsf{J}}_{\gamma }$ inside that normal neighborhood. Testing there extracts every coefficient, so all $b_{\gamma }$ vanish near $x$. This proves the support equality and then the stated equivalence.