Inhomogeneous oscillatory integrals and global smoothing effects for dispersive equations

Abstract

We study oscillatory integrals of the type F−1(eita(⋅)ψ(⋅)) where a is a general function satisfying some elliptic type and non-degenerate conditions at both the origin and the infinity, and ψ belongs to some symbol class. Point-wise estimates in space–time are gained with partial sharpness. As applications, global smoothing effects of Lp–Lq as well as Strichartz type for dispersive equations are studied. An application to fractional Schrödinger equations is also given.