A weighted L^2 estimate of Commutators of Bochner-Riesz Operators for Hermite operator

Abstract

Let H be the Hermite operator -\Delta +|x|^2 on \mathbb{R}^n. We prove a weighted L^2 estimate of the maximal commutator operator \sup_{R>0}|[b, S_R^\lambda(H)](f)|, where [b, S_R^\lambda(H)](f) = bS_R^\lambda(H) f - S_R^\lambda(H)(bf) is the commutator of a BMO function b and the Bochner-Riesz means S_R^\lambda(H) for the Hermite operator H. As an application, we obtain the almost everywhere convergence of [b, S_R^\lambda(H)](f) for large \lambda and f\in L^p(\mathbb{R}^n).