Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

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 Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,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 (ϕ1, ϕ2) and w, which do not assume any assumption on monotonicity of ϕ1 (x,r), ϕ2(x, r) in r.

Keywords

• fractional multilinear integral
• rough kernel
• bmo
• generalized weighted morrey space
• 42b20
• 42b35
• 47g10