An L^1-estimate for certain spectral multipliers associated with the Ornstein--Uhlenbeck operator

TL;DR

This paper establishes an L^1 estimate for spectral multipliers related to the Ornstein-Uhlenbeck operator, connecting their integrability to conical square functions and maximal functions within Gaussian measure spaces.

Contribution

It provides a new sufficient condition for the integrability of spectral multipliers of the Ornstein-Uhlenbeck operator based on conical square and maximal functions.

Findings

Derived a sufficient condition for spectral multiplier integrability

Connected spectral multiplier estimates with Gaussian measure tools

Enhanced understanding of spectral multipliers in harmonic analysis

Abstract

We study a class of spectral multipliers \phi(L) for the Ornstein--Uhlenbeck operator L arising from the Gaussian measure on R^n and find a sufficient condition for integrability of \phi(L)f in terms of the admissible conical square function and a maximal function.