A fourier-type transform on translation-invariant valuations on convex sets

Let V be a finite-dimensional real vector space. Let V al ^sm (V) be the space of translation-invariant smooth valuations on convex compact subsets of V. Let Dens(V) be the space of Lebesgue measures on V. The goal of the article is to construct and study an isomorphism $$ \mathbb{F}_V :Val^{sm} (V)\tilde \to Val^{sm} (V^* ) \otimes Dens(V) $$ such that $ \mathbb{F}_V $ commutes with the natural action of the full linear group on both spaces, sends the product on the source (introduced in 5]) to the convolution on the target (introduced in 16]), and satisfies a Planchereltype formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.