  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.
