Inhomogeneous oscillatory integrals and global smoothing effects for dispersive equations

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.