Inégalités de Carleman sur les cônes symétriques. (Carleman inequalities on symmetric cones) (Q908415)

We extend to arbitrary symmetric cones a family of Carleman estimates for partial differential equations with constant coefficients introduced by A. Unterberger; these estimates generalize at once Treves' inequalities on \({\mathbb{R}}^ n\) and the \(L^ 2\)-type Hardy inequalities on the half- line. The method is based on a coherent state transformation and on the description of symmetric cones by means of Jordan algebras. As a corollary we give a solution, on such a cone, of an equation \(P(D)u=f\in C_ 0^{\infty}\), depending in a covariant way on the pair (P,f).