Smoothing properties of inhomogeneous equations via canonical transforms

TL;DR

This paper introduces a novel method using canonical transforms to establish global smoothing estimates for inhomogeneous dispersive equations, extending previous techniques from homogeneous cases to more complex inhomogeneous scenarios.

Contribution

The paper develops a new approach based on canonical transformations to analyze inhomogeneous dispersive equations, enabling reduction to simpler models and deriving smoothing estimates.

Findings

Global smoothing estimates for inhomogeneous equations with lower order terms

Reduction of complex equations to low-dimensional models

Extension of previous homogeneous smoothing techniques

Abstract

The paper describes a new approach to global smoothing problems for inhomogeneous dispersive evolution equations based on an idea of canonical transformation. In our previous papers, we introduced such a method to show global smoothing estimates for homogeneous dispersive equations. It is remarkable that this method allows us to carry out a global microlocal reduction of equations to some low dimensional model cases. The purpose of this paper is to pursue the same treatment for inhomogeneous equations. Especially, time-global smoothing estimates for the operator $a(D_x)$ with lower order terms are the benefit of our new method.