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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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