The commutators of Bochner-Riesz Operators for Hermite operator

Abstract

In this paper, we study the $L^p$-boundedness of the commutator $[b,S_R^{\hat{I}}´(H)](f)=b{S}_R^{\hat{I}}´(H)f-S_R^{\hat{I}}´(H)(bf)$ of a BMO function $b$ and the Bochner-Riesz means $S_R^{\hat{I}}´(H)$ for Hermite operator $H=-\hat{I}”+|x{|}^2$ on ${\mathbb{R}}^n$, $n\ge 2$. We show that if $\hat{I}´>\hat{I}´(q)=n(1/q-1/2)-1/2$, the commutator $[b,S_R^{\hat{I}}´(H)]$ is bounded on $L^p({\mathbb{R}}^n)$ whenever $q<p<q^{\prime }$ and $1\le q\le 2n/(n+2)$.

15 pages