We consider a class of oscillatory integrals with polynomial phase functions $P$ over global domains $D$ in $\mathbb{R}^2$. As an analogue of Varchenko’s theorem in a global domain, we investigate the two main issues (i) whether the integral converges or not and (ii) how fast it decays. They are described in terms of a generalized notion of Newton polyhedra associated with $(P,D)$. Finally, we discuss its applications to the Strichartz Estimates associated with the general class of dispersive equations.

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Last updated: 11 Dec 2023