Boundedness of the maximal operator and its commutators on vanishing generalized Orlicz-morrey spaces

We prove the boundedness of the Hardy-Littlewood maximal operator and their commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces VM Phi,phi(R-n) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young function Phi(u) and the function phi(x, r) defining the space. No kind of monotonicity condition on phi(x, r) in r is imposed.Ahi Evran University [PYO.FEN.4003.13.003, PYO.FEN.4001.14.017]; Science Development Foundation under Republic of Azerbaijan [EIF-2013-9(15)-46/10/1]; Russian Fund of Basic Research [15-01-02732