Riesz transforms and spectral multipliers of the Hodge-Laguerre operator

On R+d, endowed with the Laguerre probability measure μα, we define a Hodge-Laguerre operator Lα=δδ*+δ*δ acting on differential forms. Here δ is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives ∂xi are replaced by the ''Laguerre derivatives'' xi∂xi, and δ* is the adjoint of δ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure μα. We prove dimension-free bounds on Lp, 1<p<∞, for the Riesz transforms δLα-1/2 and δ*Lα-1/2. As applications we prove the strong Hodge-de Rahm-Kodaira decomposition for forms in Lp and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in Lp. We also prove that for suitable functions m the operator m(Lα) is bounded on Lp, 1<p<∞.