We define the Hermite-Sobolev spaces naturally associated to the harmonic oscillatorH = −δ+|x|². Structural properties, relations with the classical Sobolev spaces, boundedness of operators and almost everywhere convergence of solutions of the Schrodinger equation are also considered.

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted $L^p$ of the heat and Poisson semigroups, Riesz transforms and 𝑔-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted $L^p$ boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials.