Functional calculus for the Ornstein-Uhlenbeck operator

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 \ < <phi>(p)*}. The main result of this paper is t hat 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 holomorp hic functional calculus in a sector smaller than S-phip*. (C) 2001 Academic Press.