Holomorphy of spectral multipliers of the Ornstein–Uhlenbeck operator

Let γ be the Gauss measure on Rd and L the Ornstein–Uhlenbeck operator. For every p in [1,∞)⧹{2}, set φp∗=arcsin|2/p−1|, and consider the sector Sφp∗={z∈C : |arg z|<φp∗}. The main results of this paper are the following. If p is in (1,∞)⧹{2}, and supt>0 ⦀M(tL)⦀L p(γ)<∞, i.e., if M is an L p(γ) uniform spectral multiplier of L in our terminology, and M is continuous on R+, then M extends to a bounded holomorphic function on the sector Sφp∗. Furthermore, if p=1 a spectral multiplier M, continuous on R+, satisfies the condition supt>0 ⦀M(tL)⦀L1(γ)<∞ if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i•) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on Rd belonging to a wide class, which contains L. From these results we deduce that operators in this class do not admit an H∞ functional calculus in sectors smaller than Sφp∗.