The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.2000 Mathematics Subject Classification . 58K55 (14B05, 14M25). Key words and phrases. oscillatory integrals, oscillation index and its multiplicity, local zeta function, asymptotic expansion, Newton polyhedra of the phase and the amplitude, essential set.

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in C n+1 , the Bergman kernel B(z) of takes the form near a boundary point p:where (w, ρ) is some polar coordinates on a nontangential cone with apex at p and ρ means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ρ 1/m and log(1/ρ) as ρ → 0 on . The values of d F > 0, m F ∈ Z + and m ∈ N are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as ρ → 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel.

Abstract. In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals, whose phases and weights are contained in a certain class of smooth functions including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.

We give a precise decay rate of the Bergman kernel and metric at infinity on model domains, characterized in terms of certain convex polyhedron.