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Newton polyhedra and weighted oscillatory integrals with smooth phases


Authors: Joe Kamimoto and Toshihiro Nose
Journal: Trans. Amer. Math. Soc. 368 (2016), 5301-5361
MSC (2010): Primary 58K55; Secondary 42B20, 14M25
DOI: https://doi.org/10.1090/tran/6528
Published electronically: December 2, 2015
MathSciNet review: 3458382
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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.


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Additional Information

Joe Kamimoto
Affiliation: Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan
Email: joe@math.kyushu-u.ac.jp

Toshihiro Nose
Affiliation: Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan
Address at time of publication: Faculty of Engineering, Kyushu Sangyo University, Matsukadai 2-3-1, Higashi-ku, Fukuoka, 813-8503, Japan
Email: t-nose@ip.kyusan-u.ac.jp

DOI: https://doi.org/10.1090/tran/6528
Keywords: Weighted oscillatory integrals, oscillation index and its multiplicity, resolution of singularities, Newton polyhedron, Newton distance, Newton multiplicity, principal faces
Received by editor(s): August 26, 2013
Received by editor(s) in revised form: June 9, 2014
Published electronically: December 2, 2015
Additional Notes: The first author was supported by Grant-in-Aid for Scientific Research (C) (No. 22540199), Japan Society for the Promotion of Science
Article copyright: © Copyright 2015 American Mathematical Society