A Fourier type transform on translation invariant valuations on convex sets

Let V be a finite dimensional real vector space. Let V alsm(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 FV : V alsm(V )−̃→V alsm(V ∗)⊗Dens(V ) such that FV 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 [15]), and satisfies a Plancherel type formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.