  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 sub-level-sets and
oscillatory integrals whose supports are global domains $D$. A basic model of
$D$ is $ \mathbb{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 non-degeneracy conditions of $P$.
