A criterion on weighted \(L^{p}\) boundedness for rough multilinear oscillatory singular integrals (Q2701590)

Ricci and Stein proved that a family of oscillatory singular integral operators with polynomial phases and smooth kernel is bounded on \(L^p\). In 1992, \textit{S. Lu} and \textit{Y. Zhang} [Rev. Mat. Iberoam. 8, No. 2, 201--219 (1992; Zbl 0786.42007)] extended their result to the case of rough kernel. They also set up a criterion on \(L^p\)-mapping properties of the rough oscillatory singular integral operator. In 1996, \textit{Y. Ding} and \textit{S. Lu} [Tôhoku Math. J., II. Ser. 48, No.3, 437--449 (1996; Zbl 0873.42009)] obtained a criterion on \(L^p\)-mapping properties for the higher order commutator of the oscillatory singular integral operator with rough kernel. In 1998, \textit{W. Chen, G. Hu} and \textit{S. Lu} [Nagoya Math. J. 149, 33--51 (1998; Zbl 0929.42008)] gave a criterion on \(L^p\)-mapping properties for a family of the multilinear oscillatory singular integral operator with rough kernel. In this paper, they extend their result to the weighted case given on Muckenhoupt's class \(A_p\).