The commutators of Bochner-Riesz Operators for Hermite operator

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