Riesz Potentials, Bessel Potentials, and Fractional Derivatives on Besov-Lipschitz Spaces for the Gaussian Measure

Abstract

In [5] Gaussian Lipschitz spaces Lip α (γ d ) were considered and then the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail. In this paper we will study the boundedness of those operators on Gaussian Besov-Lipschitz spaces B α p,q (γ d ). Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.

On the Global Gaussian Lipschitz Space

“…Consider the Euclidean space R n endowed with the Gaussian measure γ, given by dγ(x) = π −n/2 e −|x| 2 .…”

“…Bessel and Triebel-Lizorkin spaces related to GLip α were introduced and studied in [5] and [2]. These spaces must have global versions related to GGLip α and corresponding to homogeneous spaces in the Euclidean setting, which it would be interesting to explore.…”

Boundedness of Gaussian Bessel Potentials and Bessel Fractional Derivatives on variable Gaussian Besov-Lipschitz spaces

“…One of the main results in [11] was the definition of the variable Gaussian Besov-Lipschitz spaces B α p(•),q(•) (γ d ), following [13] and [7]. They were defined as follows:…”

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

“…Finally, in [6] Gaussian Lipchitz spaces Lip α (γ d ) were considered and the boundedness of Riesz Potentials, Bessel potentials and Fractional Derivatives on them and in [7] the boundedness of those operators were studied on Gaussian Besov-Lipschitz spaces B α p,q (γ d ). In the next section we are going to study the boundedness properties of those operators for Gaussian Triebel-Lizorkin spaces F α p,q (γ d ).…”

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In [7] the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces B α p,q (γ d ). In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces F α p,q (γ d ).

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