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Holomorphy of spectral multipliers of the Ornstein–Uhlenbeck operator. (English) Zbl 1069.47017
The closure $$\mathcal L$$ of the Ornstein–Uhlenbeck operator has spectral resolution ${\mathcal L}f=\sum_{n=0}^\infty n P_n f,$ where $$P_n$$ is the orthogonal projection onto the linear span of Hermite polynomials of degree $$n$$ in $$d$$ variables. For a given function $$M$$ defined in $$\mathbb R^+$$, define the operator $$M({\mathcal L})f=\sum_{n=0}^\infty M(n) P_n f$$. Then $$M$$ is said to be an $$L^p(\gamma)$$ spectral multiplier if $$M({\mathcal L})$$ extends to a bounded operator on $$L^p(\gamma)$$ ($$\gamma$$ being the Gauss measure on $$\mathbb R^d$$).
In this paper, the authors study the problem of finding conditions on the $$L^p(\gamma)$$ spectral multipliers $$M$$ that force $$M$$ to extend to a holomorphic function in some sector containing the positive real line. In fact, they prove that if $$1<p<\infty$$, $$p\neq 2$$, and $$M$$ is a continuous function on $$\mathbb R^+$$ such that $$\sup_{t>0} | | M(t{\mathcal L})| | _{L^p(\gamma)}<\infty,$$ then $$M$$ extends to a holomorphic function in the sector $$\{z\in{\mathbb C}: | \arg\;z| <\arcsin | 2/p-1| \}$$. This type of result is a continuation of some previous results in [J. Funct. Anal. 183, 413–450 (2001; Zbl 0995.47010)]. Related results for the case $$p=1$$ are also presented.

MSC:
 47A60 Functional calculus for linear operators 42B15 Multipliers for harmonic analysis in several variables 47D03 Groups and semigroups of linear operators 60G15 Gaussian processes
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