A Fourier type transform on translation invariant valuations on convex sets

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.