Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let gamma be the Gauss measure on R^d and L the Ornstein-Uhlenbeck operator. For every p in [1, infinity)\{2}, set phi*_p= arcsin|2/p-1|, and consider the sector S_phi*_p = {Z in C :|arg z| < phi*_p)}. The main results of this paper are the following. If p is in (1, infinity)\{2}, and sup(t>0) |||M(tL)|||(L^r(gamma)) < infinity, i.e., if M is an L^p(gamma) 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 S_phi*_p. Furthermore, if p = 1 a spectral multiplier M, continuous on R+, satisfies the condition sup(t>0) |||M(tL)|||(L^1(gamma)) < infinity if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value 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 R^d belonging to a wide class, which contains L. From these results we deduce that operators in this class do not admit an H^\infty functional calculus in sectors smaller than S_phi*_p.