On multilinear distorted multiplier estimate and its applications

In this article, we investigate the multilinear distorted multiplier estimate (Coifman-Meyer type theorem) associated with the Schr{o}dinger operator $H=-Delta + V$ in the framework of the corresponding distorted Fourier transform. Our result is the distorted analog of the multilinear Coifman-Meyer multiplier operator theorem in cite{CM1}, which extends the bilinear estimates of Germain, Hani and Walshs in cite{PZS} to the multilinear case for all dimensions. As applications, we give the estimate of Leibnizs law of integer order derivations for the multilinear distorted multiplier for the first time and we obtain small data scattering for a kind of generalized mass-critical NLS with good potential in low dimensions $d=1,2$.