We analyze boundedness properties of some operators related to the heat-diffusion semigroup associated to Laguerre functions systems. In particular, for any α > −1, we introduce appropriate Laguerre Riesz Transforms and we obtain power-weighted L p inequalities, 1 < p < ∞. We achieve this result by taking advantage of the existing classical relationship between n-variable Hermite polynomials and Laguerre polynomials on the half line of type α = n/2 − 1. Such connection allows us to transfer known boundedness properties for Hermite operators to Laguerre operators corresponding to those specific values of α. To extend the results to any α > −1, we make use of transplantation and some weighted inequalities we obtain in the Hermite setting (which we beleive of independent interest).

Let L be either the Hermite or the Ornstein-Uhlenbeck operator on R d. We find optimal integrability conditions on a function f for the existence of its heat and Poisson integrals, e −tL f (x) and e −t √ L f (x), solutions respectively of Ut = −LU and Utt = LU on R d+1 + with initial datum f. As a consequence we identify the most general class of weights v(x) for which such solutions converge a.e. to f for all f ∈ L p (v), and each p ∈ [1, ∞). Moreover, if 1 < p < ∞ we additionally show that for such weights the associated local maximal operators are strongly bounded from L p (v) → L p (u) for some other weight u(x).

We characterize the weighted Lebesgue spaces, L p (R n , v(x)dx), for which the solutions of the Heat and Poisson problems have limits a.e. when the time t tends to zero.