36 research outputs found
Asymptotic expansion of the Bergman kernel for weakly pseudoconvex tube domains in C^2
In this paper we give an asymptotic expansion of the Bergman kernel for
certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic
formula asserts that the singularity of the Bergman kernel at weakly
pseudoconvex points is essentially expressed by using two variables; moreover
certain real blowing-up is necessary to understand its singularity. The form of
the asymptotic expansion with respect to each variable is similar to that in
the strictly pseudoconvex case due to C. Fefferman. We also give an analogous
result in the case of the Szego kernel
Newton polyhedra and weighted oscillatory integrals with smooth phases
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.Comment: 67pages. arXiv admin note: text overlap with arXiv:1208.392
ON THE REGULARITY OF THE BERGMAN KERNEL ON THE BOUNDARY
This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #e
Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude
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.Comment: 36 page