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Functional calculus for the Ornstein-Uhlenbeck operator. (English) Zbl 0995.47010
Let \(\gamma\) be the Gauss probability measure on \(R^d\) with density \(\gamma_0 (x)=\pi^{-d/2} \exp(-|x |^2)\). Then the Ornstein-Uhlenbeck operator \(-1/2 \Delta +x\cdot \nabla\) is essentially selfadjoint in \(L^2 (\gamma)\). Denote by \(L\) its selfadjoint extension. It is well-known that the spectrum of \(L\) is \(\{0, 1, \dots \}\). Let \(\{ P_n\}\) be the spectral resolution of the identity for which \(Lf=\sum_{n=0}^{\infty} nP_n f, f \in \text{Dom}(L)\). The purpose of this paper is to develop a functional calculus for \(L\), i.e., to find sufficient conditions on the spectral multiplier \(M=\{ M(n): n=0, 1, \dots, \}\) for the spectral operator \(M(L)=\sum_{n=0}^{\infty} M(n)P_n\), initially defined in \(L^2 (\gamma) \cap L^p (\gamma)\), to extend to a bounded operator on \(L^p (\gamma)\), for some \(p \in (1, \infty)\).

47A60 Functional calculus for linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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