Holomorphy of spectral multipliers of the Ornstein–Uhlenbeck operator

Abstract

Let γ be the Gauss measure on R d and L the Ornstein–Uhlenbeck operator. For every p in [1,∞)⧹{2}, set φ p ∗= arcsin|2/p−1| , and consider the sector S φ p ∗ ={z∈ C : | arg z|<φ p ∗} . The main results of this paper are the following. If p is in (1,∞)⧹{2}, and sup t>0 ⦀M(t L)⦀ 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 S φ p ∗ . Furthermore, if p=1 a spectral multiplier M, continuous on R + , satisfies the condition sup t>0 ⦀M(t L)⦀ L 1(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 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 ∞ functional calculus in sectors smaller than S φ p ∗ .