Extrapolation for Weighted Product Morrey Spaces and Some Applications

This paper builds the extrapolation theory on weighted product Morrey spaces. To prove the main result, we introduce the weighted product block spaces which are the pre-duals of weighted product Morrey spaces and show the boundedness of the strong maximal operator on weighted product block spaces. By using this extrapolation theory, we first obtain the John-Nirenberg inequality on weighted product Morrey spaces, and then give a new characterization of little bmo in terms of weighted product Morrey spaces, which has its own interest. As applications of our extrapolation theory, we also give the Fefferman-Stein vector-valued inequalities and the mapping properties of the bi-parameter singlular integral operator and its commutator on weighted product Morrey spaces. Even in the unweighted setting, our results are new.

Automated summary

This paper establishes the extrapolation theory on weighted product Morrey spaces and introduces the weighted product block spaces to prove the main results. The results include the John-Nirenberg inequality, a new characterization of little bmo, and the mapping properties of certain operators on weighted product Morrey spaces. The results are novel even in the unweighted setting.