Admissible decomposition for spectral multipliers on Gaussian L^p

TL;DR

This paper studies a new way to decompose spectral multipliers of the Ornstein-Uhlenbeck operator on Gaussian space, proving boundedness in L^p spaces for 1<p≤2 using a novel connection with hypercontractivity.

Contribution

It introduces an admissible decomposition method for spectral multipliers and establishes its boundedness in Gaussian L^p spaces, linking it with hypercontractivity.

Findings

Admissible decomposition is bounded in L^p(\gamma) for 1<p≤2.

The proof connects admissibility with E. Nelson's hypercontractivity theorem.

Provides a new harmonic analysis approach for Gaussian measures.

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 \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.