Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

摘要

Let gamma be the Gauss treasure on R-d and L the Ornstein-Uhlenbeck operator. For every p in [1, infinity){2}, set phi(p)(*)= arcsin2/p-1, and consider the sector S-phip(1) = {Z is an element of C :rg 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)(Lr(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-phip. Furthermore, if p = 1 a spectral multiplier M, continuous on R+, satisfies the condition sup(t>0) M(tL)(L1(gamma)) < infinity 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 R-d belonging to a wide class, which contains L. From these results we deduce that operators in this class do not admit an N-z functional calculus in sectors smaller than S-phip. (C) 2003 Els Elsevier Inc. All rights reserved.