Sublevel-Set Estimates Over Global Domains.

Since Varchenko’s seminal paper, the asymptotics of oscillatory integrals and related problems have been elucidated through the Newton polyhedra associated with the phase P. The supports of those integrals are concentrated on sufficiently small neighborhoods. The aim of this paper is to investigate the estimates of sublevel sets and oscillatory integrals whose supports are global domains D. A basic model of D is R d . For this purpose, we define the Newton polyhedra associated with (P, D) and establish analogues of Varchenko’s theorem in global domains D, under nondegeneracy conditions of P. [ABSTRACT FROM AUTHOR]