The Commutators of Bochner–Riesz Operators for Hermite Operator

Abstract

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