The commutators of Bochner-Riesz Operators for Hermite operator

TL;DR

This paper investigates the boundedness of commutators involving Bochner-Riesz operators for the Hermite operator on L^p spaces, establishing conditions under which these commutators are bounded for certain p and q ranges.

Contribution

It provides new boundedness results for commutators of Bochner-Riesz means associated with the Hermite operator, extending understanding of their behavior on L^p spaces.

Findings

Boundedness of commutators for δ > δ(q) under specified p and q ranges.

Conditions on δ ensuring L^p-boundedness of the commutator.

Extension of boundedness results to Hermite operator context.

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 $\mathbb{R}^n$, $n\geq2$. 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 $1\leq q\leq 2n/(n+2)$.