Product Hardy, BMO spaces and iterated commutators associated with Bessel Schr\"odinger operators

TL;DR

This paper develops a theory of product Hardy spaces linked to Bessel Schr"odinger operators, providing multiple characterizations and analyzing the boundedness of associated iterated commutators.

Contribution

It introduces new characterizations of product Hardy spaces for Bessel Schr"odinger operators and studies the boundedness properties of their iterated commutators.

Findings

Product Hardy spaces are characterized via Riesz transforms, maximal functions, and Littlewood--Paley theory.

The Hardy space is isomorphic to a subspace of standard product Hardy space with odd functions.

Iterated commutators are bounded above but lack a lower bound.

Abstract

In this paper we establish the product Hardy spaces associated with the Bessel Schr\"odinger 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\'i transform, which further implies that the product Hardy space associated with this Bessel Schr\"odinger 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 Bessel Schr\"odinger operator. We show that this iterated commutator is bounded above, but does not have a lower bound.