Resumen de Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow

Françoise Lust-Piquard

• Let 1 < q < ∞. We prove that the Riesz transforms Rk = XkL − 1 2 on a generalized Heisenberg group G satisfy PK k=1 |Rk(f)|2 1 2 Lq (G) ≤ C(q, J) kfkLq(G) where K, J are respectively the dimensions of the first and second layer of the Lie algebra of G. We prove similar inequalities on Schatten spaces Sq(H), with dimension free constants, for Riesz transforms associated to commuting inner ∗-derivations Dk and a suitable substitute of the square function. An example is given by the derivations associated to n commuting pairs of operators (Pj , Qj ) on a Hilbert space H satisfying the canonical commutation relations [Pj , Qj ] = iIH.