Gaussian Fractional Integrals and Fractional Derivatives, and Their Boundedness on Gaussian Function Spaces

In this chapter, we study several important operators in Gaussian harmonic analysis. First, we consider Riesz and Bessel potentials with respect to the Ornstein–Uhlenbeck operator L, and then, Riesz and Bessel fractional derivatives. We study their regularity on Gaussian Lipschitz spaces, on Gaussian Besov–Lipschitz spaces, and on Gaussian Triebel–Lizorkin spaces. The results obtained are essentially similar to the classical results, as mentioned before, the methods of proofs are completely different. The boundedness results for Gaussian Besov–Lipschitz and Triebel–Lizorkin spaces were obtained by A. E. Gatto, E. Pineda, and W. Urbina, and appeared initially in [110] and [111]. These results can be extended to the case of Laguerre and Jacobi expansions by analogous arguments.