Riesz transforms and spectral multipliers of the Hodge-Laguerre operator

On R-+(d), endowed with the Laguerre probability measure mu(alpha), we define a Hodge-Laguerre operator Le, = delta delta* + delta*delta acting on differential forms. Here delta is the Laguerre exterior differentiation operator, defined as the classical exterior differential, except that the partial derivatives partial derivative(xi) are replaced by the "Laguerre derivatives" root xi partial derivative(xi) and delta* is the adjoint of delta with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure mu(alpha). We prove dimension-free bounds on L-P, 1 < p < infinity, for the Riesz transforms delta L-alpha(-1/2) and delta*L-alpha(-1/2) As applications we prove the strong Hodge-de Rahm-Kodaira decomposition for forms in L-p and deduce existence and regularity results for the solutions of the Hodge and de Rham equations in L-p. We also prove that for suitable functions m the operator m(L-alpha) is bounded on L-P, 1 <p < infinity. (C) 2015 Elsevier Inc. All rights reserved.