= Support law for distributional dilation
For $t>0$, the <support of a distribution> satisfies
$$
\operatorname{supp}(\delta_tu)=t^{-1}\operatorname{supp}u.
$$
The duality formula $\langle\delta_tu,\varphi\rangle=t^{-n}\langle u,\varphi(\mathord\cdot/t)\rangle$ proves one inclusion by transporting <test functions>; applying the inverse dilation proves the other. For a compact support, the <Paley–Wiener–Schwartz theorem> gives another proof: $\widehat{\delta_tu}(\zeta)=t^{-n}\widehat u(\zeta/t)$ rescales the exponential growth indicator by $1/t$.
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