Admissible decomposition for spectral multipliers on Gaussian $$L^p$$ L p

Abstract

This paper concerns harmonic analysis of the Ornstein-Uhlenbeck operator L on the Euclidean space. We examine the method of decomposing a spectral multiplier φ(L) into three parts according to the notion of admissibility, which quantifies the doubling behaviour of the underlying Gaussian measure γ. We prove that the abovementioned admissible decomposition is bounded in L p (γ) for 1 < p ≤ 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.2010 Mathematics Subject Classification. 42B25 (Primary); 42B15, 42B30 (Secondary).