  This paper concerns harmonic analysis of the Ornstein--Uhlenbeck operator L
on the Euclidean space. We examine the method of decomposing a spectral
multiplier \phi(L) into three parts according to the notion of admissibility,
which quantifies the doubling behaviour of the underlying Gaussian measure
\gamma. We prove that the above-mentioned admissible decomposition is bounded
in L^p(\gamma) for 1 < p \leq 2 in a certain sense involving the Gaussian
conical square function. The proof relates admissibility with E. Nelson's
hypercontractivity theorem in a novel way.
