On Gaussian Lipchitz spaces and the boundedness of fractional integrals and fractional derivatives on them

Abstract

The main purpose of this paper is to study the boundedness of Gaussian fractional integrals and derivatives associated to Hermite polynomial expansions on Gaussian Lipschitz spaces Lipα(γ). To get these results we introduce formulas for these operators in terms of the Hermite-Poisson semigroup as well as the Gaussian Lipschitz spaces. This approach was originally developed for the classical Poisson integral. These proofs can also be extended to the case of Laguerre and Jacobi expansions. In subsequent papers we will study the same operators on Gaussian Besov-Lipschitz and Triebel-Lizorkin spaces.

Keywords: Fractional integration, fractional differentiation, Lipschitz spaces, Gaussian measure.