Functional Calculus for the Ornstein–Uhlenbeck Operator

Let γ be the Gauss measure on Rd and L the Ornstein–Uhlenbeck operator, which is self adjoint in L2(γ). For every p in (1, ∞), p≠2, set φ*p=arc sin |2/p−1| and consider the sector Sφ*p={z∈C : |arg z|<φ*p}. The main result of this paper is that if M is a bounded holomorphic function on Sφ*p whose boundary values on ∂Sφ*p satisfy suitable Hörmander type conditions, then the spectral operator M(L) extends to a bounded operator on Lp(γ) and hence on Lq(γ) for all q such that |1/q−1/2|⩽|1/p−1/2|. The result is sharp, in the sense that L does not admit a bounded holomorphic functional calculus in a sector smaller than Sφ*p.