Inhomogeneous Oscillatory Integrals and Global Smoothing Effects for Dispersive Equations

TL;DR

This paper analyzes inhomogeneous oscillatory integrals with general phase functions, establishing sharp space-time estimates and applying these results to demonstrate global smoothing effects for dispersive equations, including fractional Schrödinger equations.

Contribution

It introduces new point-wise estimates for oscillatory integrals with general phases and applies these to obtain global smoothing effects for a broad class of dispersive equations.

Findings

Established sharp space-time estimates for oscillatory integrals.

Proved global smoothing effects for dispersive equations.

Extended results to fractional Schrödinger equations.

Abstract

We study oscillatory integrals of the type ${\mathcal F}^{-1}(e^{ita(\cdot)}\psi(\cdot))$ where $a$ is a general function satisfying some elliptic type and non-degenerate conditions at both the origin and infinity, and $\psi$ belongs to some symbol class. Point-wise estimates in space-time are gained with partial sharpness. As applications, global smoothing effects of $L^p-L^q$ as well as Strichartz type for dispersive equations are studied. An application to fractional Schr\"odinger equations is also given.