Journal of the Mathematical Society of Japan

Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude

Koji CHO, Joe KAMIMOTO, and Toshihiro NOSE

Full-text: Open access

Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 2 (2013), 521-562.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1366896643

Digital Object Identifier
doi:10.2969/jmsj/06520521

Mathematical Reviews number (MathSciNet)
MR3055595

Zentralblatt MATH identifier
1271.32035

Subjects
Primary: 58K55: Asymptotic behavior
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Keywords
oscillatory integrals oscillation index and its multiplicity local zeta function asymptotic expansion Newton polyhedra of the phase and the amplitude essential set

Citation

CHO, Koji; KAMIMOTO, Joe; NOSE, Toshihiro. Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude. J. Math. Soc. Japan 65 (2013), no. 2, 521--562. doi:10.2969/jmsj/06520521. https://projecteuclid.org/euclid.jmsj/1366896643


Export citation

References

  • V. I. Arnol'd, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of Differentiable Maps. II, Monogr. Math., 83, Birkhäuser, 1988.
  • B.-Y. Chen, J. Kamimoto and T. Ohsawa, Behavior of the Bergman kernel at infinity, Math. Z., 248 (2004), 695–708.
  • K. Cho, J. Kamimoto and T. Nose, Asymptotics of the Bergman function for semipositive holomorphic line bundles, Kyushu J. Math., 65 (2011), 349–382.
  • J. Denef, A. Laeremans and P. Sargos, On the largest nontrivial pole of the distribution $|f|^s$, Sûrikaisekikenkyûsho Kôkyûroku, 999 (1997), 1–9.
  • J. Denef, J. Nicaise and P. Sargos, Oscillating integrals and Newton polyhedra, J. Anal. Math., 95 (2005), 147–172.
  • J. Denef and P. Sargos, Polyèdre de Newton et distribution $f_+^s$. I, J. Analyse Math., 53 (1989), 201–218.
  • J. Denef and P. Sargos, Polyèdre de Newton et distribution $f_+^s$. II, Math. Ann., 293 (1992), 193–211.
  • M. V. Fedorjuk, Non-homogeneous generalized functions of two variables. (Russian), Mat. Sb. (N.S.), 49 (1959), 431–446.
  • W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud., 131, Princeton University Press, Princeton, NJ, 1993.
  • I. M. Gel'fand and G. E. Shilov, Generalized Functions. I, Academic Press, New York, 1964.
  • M. Greenblatt, A direct resolution of singularities for functions of two variables with applications to analysis, J. Anal. Math., 92 (2004), 233–257.
  • M. Greenblatt, An elementary coordinate-dependent local resolution of singularities and applications, J. Funct. Anal., 255 (2008), 1957–1994.
  • M. Greenblatt, The asymptotic behavior of degenerate oscillatory integrals in two dimensions, J. Funct. Anal., 257 (2009), 1759–1798.
  • M. Greenblatt, Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Math. Ann., 346 (2010), 857–895.
  • M. Greenblatt, Resolution of singularities, asymptotic expansions of oscillatory integrals, and related phenomena, J. Anal. Math., 111 (2010), 221–245.
  • H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2), 79 (1964), 109–326.
  • J. Igusa, Forms of Higher Degree. Tata Inst. Fund. Res. Lectures on Math. and Phys., 59, New Delhi, 1978.
  • J. Kamimoto, Newton polyhedra and the Bergman kernel, Math. Z., 246 (2004), 405–440.
  • A. Kaneko, Newton diagrams, Singular points and Oscillatory integrals, Lecture Note at Sophia University, 11 (in Japanese), 1981.
  • G. Kempf, F. F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Math., 339, Springer-Verlag, 1973.
  • B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. (4), 7 (1974), 405–430.
  • T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, 1988.
  • M. Oka, Non-Degenerate Complete Intersection Singularity, Actualites Math., Hermann, Paris, 1997.
  • M. Pramanik and C. W. Yang, Decay estimates for weighted oscillatory integrals in ${\mathbb R}^2$, Indiana Univ. Math. J., 53 (2004), 613–645.
  • H. Schulz, Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms, Indiana Univ. Math. J., 40 (1991), 1267–1275.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton University Press, Princeton, NJ, 1993.
  • V. A. Vasilíev, Asymptotic behavior of exponential integrals in the complex domain, Funct. Anal. Appl., 13 (1979), 239–247.
  • A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funct. Anal. Appl., 10 (1976), 175–196.
  • G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math., 152, Springer-Verlag, New York, 1995.