Generalized higher commutators generated by the multilinear fractional integrals and Lipschitz functions

Let l in N and vec{A}=(A_1,dots,A_l) and vec{f}=(f_1,dots,f_l) be 2 finite collections of functions, where every function A_i has derivatives of order m_i and f_1,dots,f_lin L_c^{infty}(R^n). Let xnotincap_{i=1}^lSupp f_i. The generalized higher commutator generated by the multilinear fractional integral is then given by I_{alpha,m}^{vec{A}}(vec{f})(x) =dint_{(R^n)^m} frac{prodlimits_{i=1}^lR_{m_i+1}(A_i;x,y_i)f_{i}(y_i)}{|(x-y_1,dots ,x-y_m)|^{ln+(m_1+m_2+dots+m_l)-alpha}} dy_1dots dy_l. When D^{gamma}A_iin dot{Lambda}_{beta_i}(0 beta_i 1, |gamma|=m_i), i=1,cdots,m, the authors establish the boundedness of I_{alpha,m}^{vec{A}} on the product Lebesgue space, Triebel--Lizorkin space, and Lipschitz space.