Newton polyhedra and weighted oscillatory integrals with smooth phases

Kamimoto, Joe; Nose, Toshihiro

doi:10.1090/tran/6528

Newton polyhedra and weighted oscillatory integrals with smooth phases
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by Joe Kamimoto and Toshihiro Nose PDF

Trans. Amer. Math. Soc. 368 (2016), 5301-5361 Request permission

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.

References

V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191, DOI 10.1007/978-1-4612-3940-6
M. F. Atiyah, Resolution of singularities and division of distributions, Comm. Pure Appl. Math. 23 (1970), 145–150. MR 256156, DOI 10.1002/cpa.3160230202
I. N. Bernšteĭn and S. I. Gel′fand, Meromorphy of the function $P^{\lambda }$, Funkcional. Anal. i Priložen. 3 (1969), no. 1, 84–85 (Russian). MR 0247457
Edward Bierstone and Pierre D. Milman, Resolution of singularities in Denjoy-Carleman classes, Selecta Math. (N.S.) 10 (2004), no. 1, 1–28. MR 2061220, DOI 10.1007/s00029-004-0327-0
Koji Cho, Joe Kamimoto, and Toshihiro Nose, 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. MR 3055595
Tristan C. Collins, Allan Greenleaf, and Malabika Pramanik, A multi-dimensional resolution of singularities with applications to analysis, Amer. J. Math. 135 (2013), no. 5, 1179–1252. MR 3117305, DOI 10.1353/ajm.2013.0042
Jan Denef, Johannes Nicaise, and Patrick Sargos, Oscillating integrals and Newton polyhedra, J. Anal. Math. 95 (2005), 147–172. MR 2145563, DOI 10.1007/BF02791501
William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
Michael Greenblatt, The asymptotic behavior of degenerate oscillatory integrals in two dimensions, J. Funct. Anal. 257 (2009), no. 6, 1759–1798. MR 2540991, DOI 10.1016/j.jfa.2009.06.015
Michael Greenblatt, Oscillatory integral decay, sublevel set growth, and the Newton polyhedron, Math. Ann. 346 (2010), no. 4, 857–895. MR 2587095, DOI 10.1007/s00208-009-0424-7
Michael Greenblatt, Resolution of singularities, asymptotic expansions of integrals and related phenomena, J. Anal. Math. 111 (2010), 221–245. MR 2747065, DOI 10.1007/s11854-010-0016-1
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Isroil A. Ikromov, Michael Kempe, and Detlef Müller, Estimates for maximal functions associated with hypersurfaces in $\Bbb R^3$ and related problems of harmonic analysis, Acta Math. 204 (2010), no. 2, 151–271. MR 2653054, DOI 10.1007/s11511-010-0047-6
Isroil A. Ikromov and Detlef Müller, On adapted coordinate systems, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2821–2848. MR 2775788, DOI 10.1090/S0002-9947-2011-04951-2
Isroil A. Ikromov and Detlef Müller, Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction, J. Fourier Anal. Appl. 17 (2011), no. 6, 1292–1332. MR 2854839, DOI 10.1007/s00041-011-9191-4
A. Iosevich and E. Sawyer, Maximal averages over surfaces, Adv. Math. 132 (1997), no. 1, 46–119. MR 1488239, DOI 10.1006/aima.1997.1678
Jun-ichi Igusa, Forms of higher degree, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 59, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546292
Pierre Jeanquartier, Développement asymptotique de la distribution de Dirac attachée à une fonction analytique, C. R. Acad. Sci. Paris Sér. A-B 201 (1970), A1159–A1161 (French). MR 420695
Joe Kamimoto and Toshihiro Nose, Toric resolution of singularities in a certain class of $C^{\infty }$ functions and asymptotic analysis of oscillatory integrals, J. Math. Sci. Univ. Tokyo, to appear, arXiv:1208.3924.
Joe Kamimoto and Toshihiro Nose, Asymptotic analysis of weighted oscillatory integrals via Newton polyhedra, Topics in finite or infinite dimensional complex analysis, Tohoku University Press, Sendai, 2013, pp. 3–12. MR 3074732
Bernard Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. École Norm. Sup. (4) 7 (1974), 405–430 (1975) (French). MR 372243, DOI 10.24033/asens.1274
Toshihiro Nose, Asymptotic analysis of oscillatory integrals and local zeta functions via Newton polyhedra, Ph.D. Thesis, Kyushu University, 2012.
Toshihisa Okada and Kiyoshi Takeuchi, Meromorphic continuations of local zeta functions and their applications to oscillating integrals, Tohoku Math. J. (2) 65 (2013), no. 2, 159–178. MR 3079283, DOI 10.2748/tmj/1372182720
D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152. MR 1484770, DOI 10.1007/BF02392721
Malabika Pramanik and Chan Woo Yang, Decay estimates for weighted oscillatory integrals in ${\Bbb R}^2$, Indiana Univ. Math. J. 53 (2004), no. 2, 613–645. MR 2060047, DOI 10.1512/iumj.2004.53.2388
Helmut Schulz, Convex hypersurfaces of finite type and the asymptotics of their Fourier transforms, Indiana Univ. Math. J. 40 (1991), no. 4, 1267–1275. MR 1142714, DOI 10.1512/iumj.1991.40.40056
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Vincent Thilliez, On quasianalytic local rings, Expo. Math. 26 (2008), no. 1, 1–23. MR 2384272, DOI 10.1016/j.exmath.2007.04.001
A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Functional Anal. Appl. 10 (1976), no. 3, 175–196.
V. A. Vassiliev, Asymptotic behavior of exponential integrals in the complex domain, Functional Anal. Appl. 13 (1979), no. 4, 239–247.
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1