Since Varchenko's seminal work, the asymptotics of oscillatory integrals and related problems have been effectively analyzed using the Newton polyhedra associated with the phase function $P$. These analyses have traditionally focused on integrals supported in sufficiently small neighborhoods.
The purpose of this talk is to introduce the estimates of sub-level sets and oscillatory integrals over global domain $D$ with a primary focus on $D=\mathbb{R}^d$.
In particular, we shall discuss
(i) Piuseux series on $\mathbb{R}^2$ and resolutions of singularities,
(ii) Analogue of Varchenko's theorem on the global domain
(iii) Strichartz type inequality for dispersive partial differential equations (PDEs) as a key application.
(iv) Phong and Stein's theorem regarding the global oscillatory integral operator (a joint work with Minbeom Kang)