Support law for distributional dilation

For $t>0$, the support of a distribution satisfiesThe duality formula $⟨{\delta }_{t}u,\phi ⟩=t^{-n}⟨u,\phi (\cdot /t)⟩$ 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: $\widehat{{\delta }_{t}u}(\zeta )=t^{-n}\hat{u}(\zeta /t)$ rescales the exponential growth indicator by $1/t$.