While solving a particular problem about composition of tempered distributions and an affine transformation, I ended up having to prove the following for $u\in\mathscr{S}'$ and a linear transformation $L:\mathbb{R}^d\rightarrow \mathbb{R}^d$: $$u(\hat{\phi}\circ L^{-1}) = |\det L| u(\widehat{\phi\circ L^t)}$$
But I am stuck from here.