Lipschitz estimates for rough fractional multilinear integral operators on local generalized Morrey spaces

We obtain the Lipschitz boundedness for a class of fractional multilinear operators I-Omega,alpha(A,m) with rough kernels Omega is an element of L-s(Sn-1), s > n/(n-alpha) on the local generalized Morrey spaces LMp,phi{x(0)}, generalized Morrey spaces M-p,M-phi and vanishing generalized Morrey spaces VMp,phi, where the functions A belong to homogeneous Lipschitz space Lambda(beta), 0 < beta < 1. We find the sufficient conditions on the pair (phi(1),phi(2)) which ensures the boundedness of the operators I-Omega,alpha(A,m) from LMp,phi 1{x(0)} to LMq,phi 2{x(0)}, from M-p,M-phi 1 to M-q,M-phi 2 and from VMp,phi 1 to VMq,phi 2 for 1 < p < q < infinity and 1/p-1/q = (alpha + beta)/n. In all cases the conditions for the boundedness of the operator I-Omega,alpha(A,m) is given in terms of Zygmund-type integral inequalities on (phi(1), phi(2)), which do not assume any assumption on monotonicity of phi(1)(x, r), phi(2)(x, r) in r.