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

Authors: HUIXIA MO, DONGYAN YU , HUIPING ZHOU

Abstract: 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_l\in L_c^{\infty}(R^n). Let x\notin\cap_{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{\prod\limits_{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_1\dots dy_l. When D^{\gamma}A_i\in \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.

Keywords: Multilinear fractional integral, commutator, Triebel--Lizorkin space, Lipschitz function space

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