On multilinear distorted multiplier estimate and its applications

TL;DR

This paper extends multilinear multiplier estimates to Schrödinger operators with potential, providing new tools for analyzing nonlinear PDEs and establishing scattering results in low dimensions.

Contribution

It introduces a distorted multilinear Coifman-Meyer multiplier theorem for Schrödinger operators, extending previous bilinear results to multilinear cases across all dimensions.

Findings

Established multilinear distorted multiplier estimates.

Derived Leibniz rule estimates for distorted multipliers.

Proved small data scattering for generalized NLS with potential in low dimensions.

Abstract

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 Walsh's in \cite{PZS} to the 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 $d=1,2$.