The Properties Of Commutators Generated By Lipschitz Functions And O-U Operators

In this paper, we study the boundedness of the commutator [b,T] which are generated by Lipschitz functions and the local singular integral operator associated to Ornstein-Uhlenbeck semigroup under the Gaussian measure. Furthermore, we also research the boundedness of the commutator [b,Iαβ] generated by Lipschitz functions and local frac-tional integral operator.The main purpose of this paper is to develop a theory of singular operators acting on function spaces over the measured metric space (Rd,ρ,γ), whereρdenotes the Euclidean distance andγthe Gauss measure. Our theory plays for the Ornstein-Uhlenbeck operator the same role that the classical Calderon-Zygmund theory plays for the Laplacian on (Rd,ρ,λ).Firstly, we define the Lipschitz functions in both forms of average and step in the condition of Gauss measure. And also we discuss the relation between the two forms of Lipschitz functions.Next, we discuss the boundedness of [b,T] on the Triebel-Lizorkin space which is defined in the way of average. And we get the Triebel estimate for the commutator.Finally, we research the boundedness of commutator [b, Iαβ] generated by Lipschitz functions and local fractional integral opetators Iαβon the Triebel-Lizorkin space. In order to get this result, we define the local fractional integral opetator Iαβin the condition of Gauss measure, and we use the boundedness of Iαβ.This paper, we discuss the boundedness of commutators generated by some local operators related to Ornstein-Uhlenbeck semigroup in Gauss measure. Our work develop the singular integrals theory on the Gauss measure and we get some valuable results.