  Using some harmonic extensions on the upper-half plane, and probabilistic
representations, and curvature-dimension inequalities with some negative
dimensions, we obtain some new opimal functional inequalities of the Beckner
type for the Cauchy type distributions on the Euclidean space. These optimal
inequalities appear to be equivalent to some non tight optimal Beckner
inequalities on the sphere, and the family appear to be a new form of the
Sobolev inequality.
