Lipschitz Estimates for Multilinear Commutator of Pseudo-differential Operators

As the development of singular integral operators, their commutators and multilinear operators have been well studied (see [4, 5, 6, 7, 8, 9, 10]). In [4, 5, 6, 7, 8, 9, 10], the authors prove that the commutators and multilinear operators generated by the singular integral operators and BMO functions are bounded on L(R) for 1 < p <∞; Chanillo (see [2]) proves a similar result when singular integral operators are replaced by the fractional integral operators. In [4, 11, 12, 13, 16], the boundedness for the commutators and multilinear operators generated by the singular integral operators and Lipschitz functions on Triebel– Lizorkin and L(R)(1 < p < ∞) spaces are obtained. In [14], the weighted boundedness for the commutators generated by the singular integral operators and BMO and Lipschitz functions on L(R)(1 < p < ∞) spaces are obtained. The purpose of this paper is to prove the weighted boundedness on Lebesgue spaces for some multilinear operators associated to the pseudo-differential operators and the weighted Lipschitz functions. To do this, we first prove a sharp function estimate for the multilinear operators. Our results are new, even in the unweighted cases.