Characterizations for multi-sublinear operators and their commutators on three kinds of generalized weighted Morrey spaces and applications

The main questions raised in this paper are to find the sufficient conditions that make multi-sublinear operators $T$ and their commutators ${T_{\prod \vec b }}$, ${T_{\sum {\vec b} }}$ to be bounded on three kinds of generalized weighted Morrey spaces. We give the main theorems of this paper to solve the above related questions. As corollaries of the main theorems, we give sufficient and necessary conditions for a class of multi-sublinear operators which are bounded on three kinds of generalized weighted Morrey spaces. As some inportant applications, we apply the main results to the multilinear vector-valued Calder\'on-Zygmund operators, multilinear Littlewood-Paley square operators, multilinear pseudo-differential operators and multilinear paraproducts.