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

Abstract

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 L p α (γ d ) 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.

A characterization of the Gaussian Lipschitz space and sharp estimates for the Ornstein–Uhlenbeck Poisson kernel

“…Since now x 1 > 2y 1 , we see that log x 1 y 1 −log 5 4 1, which implies that s ≃ log It follows that…”

“…Gatto and Urbina [3] introduced the Gaussian Lipschitz spaces; see also [2] and [4]. Let α ∈ (0, 1), which will be fixed throughout the paper.…”

Riesz potentials, Bessel potentials and fractional derivatives on Triebel–Lizorkin spaces for the Gaussian measure

“…Next we want to study the boundedness properties of the Bessel potentials on Triebel-Lizorkin spaces. In [10], Theorem 2.4, the following result was proved, for completeness the proof will be given here too.…”

“…That is to say, we can defined in analogous manner Laguerre-Triebel-Lizorkin spaces, and Jacobi-Triebel-Lizorkin spaces then prove that the corresponding notions of Fractional Integrals and Fractional Derivatives behave similarly. In order to see this it is more convenient to use the representation (1.7) of P t in terms of the one-sided stable measure µ (1/2) t (ds), see [10].…”

“…The definition of F α p,q (γ d ) does not depend on which k > α is chosen and the resulting norms are equivalent, for the proof of this result and other properties of these spaces see [10].…”