Extrapolation on product Morrey-Herz spaces and applications

Recently, the extrapolation theory has become a mainsteam method to investigate some integral type operators, since it does not depend on the density of spaces. The purpose of this paper is threefold. The first is to establish product Morrey-Herz spaces and product block-Herz spaces, study the duality between them, and prove the boundedness of strong maximal operators product block-Herz spaces, which is ready for extrapolation. Secondly, we set up extrapolation on product Morrey-Herz spaces by the methods of Rubio de Francia. Finally, as applications, we prove some completely new boundedness and estimates for some operators on product Morrey-Herz spaces, such as Fefferman-Stein vector-valued strong maximal inequalities, John-Nirenberg inequality and a new characterization of bmo in terms of product Morrey-Herz spaces, and the boundedness of bi-parameter Calder\'on-Zygmund operators and their commutators.