Functional calculus for the Ornstein-Uhlenbeck operator

摘要

Let gamma be the Gauss measure on R-d and L the Ornstein-Uhlenbeck operator, which is self adjoint in L-2(gamma). For every p in (1, infinity), p not equal 2, set phi (p)* = arc sin 2/p - 1 and consider the sector S-phip* = {z epsilon C : arg z < hi>(p)*}. The main result of this paper is that if M is a bounded holomorphic function on S-phip*whose boundary values on aS(phip)*. satisfy suitable Hormander type conditions. then the spectral operator M(L) extends to a bounded operator on L-p(gamma) and hence on L-q(gamma) for all q such that 1/q - 1/2 less than or equal to 1p- 1/2 . The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than S-phip*. (C) 2001 Academic Press. [References: 16]