Riesz transforms and spectral multipliers of the Hodge–Laguerre operator.

On R + d , endowed with the Laguerre probability measure μ α , we define a Hodge–Laguerre operator L α = δ δ ⁎ + 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 ∂ x i are replaced by the ‘‘Laguerre derivatives’’ x i ∂ x i , and δ ⁎ is the adjoint of δ with respect to inner product on forms defined by the Euclidean structure and the Laguerre measure μ α . We prove dimension-free bounds on L p , 1 < p < ∞ , for the Riesz transforms δ L α − 1 / 2 and δ ⁎ L α − 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 α ) is bounded on L p , 1 < p < ∞ . [ABSTRACT FROM AUTHOR]