Product hardy, BMO spaces and iterated commutators associated with Bessel Schrödinger operators

In this paper we establish the product Hardy spaces associated with the Bessel Schrödinger operator introduced by Muckenhoupt and Stein, and provide equivalent characterizations in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions, and Littlewood-Paley theory, which are consistent with the classical product Hardy space theory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide another characterization via the Telyakovski’ı transform, which further implies that the product Hardy space associated with this Bessel Schrödinger operator is isomorphic to the subspace of suitable “odd functions” in the standard Chang-Fefferman product Hardy space. Based on the characterizations of these product Hardy spaces, we study the boundedness of the iterated commutator of the Bessel Riesz transforms and functions in the product BMO space associated with the Bessel Schrödinger operator. We show that this iterated commutator is bounded above, but does not have a lower bound.