Product BMO, Little BMO, and Riesz Commutators in the Bessel Setting

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 Muckenhoupt-Stein. We first prove that the product BMO space in the Bessel setting can be used to deduce the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators, where the main tool that we develop is the characterization of the predual of little bmo and its weak factorizations. We further show that in analogy with the classical setting the little space is a proper subspace of the product space. These extend the previous related results studied by Cotlar-Sadosky and Ferguson-Sadosky on the bidisc to the Bessel setting, where the usual analyticity and Fourier transform do not apply.