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 [CM1], which extends the
bilinear estimates of Germain, Hani and Walsh’s in [PZS] to
multilinear case for all dimensions. As applications, we give the
estimate of Leibniz’s 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 $d=1,2$.