Strong type inequalities for sublinear operator on weighted modular Banach function spaces

In this paper, we prove the boundedness of sublinear operator on weighted modular Banach function space (BFS) under a certain size condition. We establish sufficient conditions on weight functions and on the geometry of modular BFS for the validity of the strong inequality for sublinear operator on weighted modular BFS under certain size condition. We will assume that the BFS is p-convex and the modular defining the BFS satisfies some growth condition. In particular, we obtain the boundedness of the sublinear operator on weighted Musielak-Orlicz spaces. The size condition is satisfied by most of the operators in harmonic analysis, such as the Calder?n-Zygmund singular integral operator, Hardy-Littlewood maximal operator, Bochner-Riesz means at the critical index, Carleson maximal operator, Ricci-Stein?s oscillatory singular integrals, C. Fefferman?s singular multiplier operator, R. Fefferman?s singular integral operator and so on. The main result is new in the case of an unweighted setting.