Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let ¿ be the Gauss measure on Image and Image the Ornstein¿Uhlenbeck operator. For every p in [1,¿)-45 degree rule{2}, set ¿p*=arcsin|2/p¿1|, and consider the sector Image . The main results of this paper are the following. If p is in (1,¿)-45 degree rule{2}, and Image , i.e., if M is an Lp(¿) uniform spectral multiplier of Image in our terminology, and M is continuous on Image , then M extends to a bounded holomorphic function on the sector S¿p*. Furthermore, if p=1 a spectral multiplier M, continuous on Image , satisfies the condition Image 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 Image belonging to a wide class, which contains Image . From these results we deduce that operators in this class do not admit an H¿ functional calculus in sectors smaller than S¿p.