Abstract
In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels TΩ,αA1,A2,…,Ak,$T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces M p,ϕ ( w ). We find the sufficient conditions on the pair ( ϕ 1 , ϕ 2 ) with w ∈ A p,q which ensures the boundedness of the operators TΩ,αA1,A2,…,Ak,$T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ from Mp,φ1wptoMp,φ2wq${M_{p,{\varphi _1}}}\left( {{w^p}} \right)\,{\rm{to}}\,{M_{p,{\varphi _2}}}\left( {{w^q}} \right)$ for 1 < p < q < ∞. In all cases the conditions for the boundedness of the operator TΩ,αA1,A2,…,Ak,$T_{\Omega ,\alpha }^{{A_1},{A_2}, \ldots ,{A_k}},$ are given in terms of Zygmund-type integral inequalities on ( ϕ 1 , ϕ 2 ) and w , which do not assume any assumption on monotonicity of ϕ 1 ( x,r ) , ϕ 2 ( x, r ) in r .