Sharp estimates for the Ornstein-Uhlenbeck operator.

Abstract

Let L be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure γ on Rd. We prove a sharp estimate of the operator norm of the imaginary powers of L on Lp(γ), 1 < p < ∞. Then we use this estimate to prove that if b is in [0,∞) and M is a bounded holomorphic function in the sector {z ∈ C : |arg(z − b)| < arcsin |2/p−1|} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on Lp(γ). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea.