Abstract. We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator L = − + |x| 2 and of the OrnsteinUhlenbeck operator L = − + 2x · ∇. The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.2000 Mathematics Subject Classification. 42C10, 42B20, 42B25.

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.

Abstract. We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form L = −∆ + V , where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H 1 , L p and BM O of classical L-square functions.

The weighted Lebesgue spaces of initial data for which almost everywhere convergence of the heat equation holds was only very recently characterized. In this note we show that the same weighted space of initial data is optimal for the heat-diffusion parabolic equations involving the harmonic oscillator and the Ornstein-Uhlenbeck operator.2010 Mathematics Subject Classification. Primary: 35K15, 35K05. Secondary: 42B37, 42B25, 42B35.

We consider degenerate identification problems with smoothing overdetermination in abstract spaces. We establish an identifiability result using a projection method and suitable hypotheses on the operators involved and develop an identification method by reformulating the problem into a nondegenerate problem. Then we use perturbation results for linear operators to solve the regular problem. The introduced identification method permits one to solve the problems under the minimum restrictions on the input data. Finally, we provide applications to degenerate differential equations that appear in mathematical physics to support the theoretical results.