Product BMO, little BMO and Riesz Commutators in the Bessel setting

TL;DR

This paper explores the properties of product BMO and little BMO spaces in the Bessel setting, establishing their connections with commutators and extending classical results to this context.

Contribution

It introduces the boundedness of iterated commutators with Bessel Riesz transforms and characterizes the little BMO space via commutators in the Bessel setting.

Findings

Product BMO space ensures boundedness of iterated commutators.

Little BMO space characterized by commutators in Bessel setting.

Little BMO is a proper subspace of product BMO, extending classical results.

Abstract

In this paper, we study the product BMO space, little bmo space and their connections with the corresponding commutators associated with Bessel operators studied by Weinstein, Huber, and by Muckenhoupt-Stein. We first prove that the product BMO space in the Bessel setting can be used to prove the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little $\rm bmo$ space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators. We further show that in analogy with the classical setting, the little $\rm bmo$ space is a proper subspace of the product $\rm BMO$ space. These extend the previous related results studied by Cotlar-Sadosky and Ferguson-Sadosky on the bidisc to the Bessel setting.