Introduction to the Besov Spaces and Triebel-Lizorkin Spaces for Hermite and Laguerre expansions and some applications

We introduced new definitions of Besov spaces and Triebel-Lizorkin spaces associated with multidimensional Hermite expansions and multidimensional Laguerre expansions. We showed that the set of p-integrable functions is a Triebel-Lizorkin space with respect to the Gaussian measure and similarly, with respect to the probabilistic Gamma measure. Also, we showed that the Gaussian Sobolev spaces and Laguerre Sobolev spaces are Triebel-Lizorkin spaces, associated with Hermite and Laguerre expansions respectively. We defined Carleson measures with respect to the Gaussian measure and probabilistic Gamma measure. By using maximal functions, related to the Ornstein Uhlenbeck semigroup and Laguerre semigroup, we studied these measures, giving a version of Fefferman