Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces Lαp(γd) are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.