Duke Mathematical Journal

Integration of oscillatory and subanalytic functions

Raf Cluckers, Georges Comte, Daniel J. Miller, Jean-Philippe Rolin, and Tamara Servi

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We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This article extends the investigation started by Lion and Rolin and Cluckers and Miller to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.

Article information

Duke Math. J., Volume 167, Number 7 (2018), 1239-1309.

Received: 20 July 2016
Revised: 6 November 2017
First available in Project Euclid: 14 March 2018

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Zentralblatt MATH identifier

Primary: 26B15: Integration: length, area, volume [See also 28A75, 51M25]
Secondary: 03C64: Model theory of ordered structures; o-minimality 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05] 32B20: Semi-analytic sets and subanalytic sets [See also 14P15] 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 14P10: Semialgebraic sets and related spaces 33B10: Exponential and trigonometric functions

stability under integration oscillatory integrals Fourier transforms globally subanalytic functions constructible functions preparation theorems uniformly distributed functions oscillation index families of exponential periods o-minimality


Cluckers, Raf; Comte, Georges; Miller, Daniel J.; Rolin, Jean-Philippe; Servi, Tamara. Integration of oscillatory and subanalytic functions. Duke Math. J. 167 (2018), no. 7, 1239--1309. doi:10.1215/00127094-2017-0056. https://projecteuclid.org/euclid.dmj/1521014410

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  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.
  • [2] V. I. Arnol’d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of Differentiable Maps, Vol. II, Monogr. Math. 83, Birkhäuser, Boston, 1988.
  • [3] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5–42.
  • [4] S. Bloch and H. Esnault, Gauß-Manin determinant connections and periods for irregular connections, Geom. Funct. Anal. 2000, special volume, 1–31.
  • [5] R. Cluckers, J. Gordon, and I. Halupczok, Local integrability results in harmonic analysis on reductive groups in large positive characteristic, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 1163–1195.
  • [6] R. Cluckers, T. Hales, and F. Loeser, “Transfer principle for the fundamental lemma” in On the Stabilization of the Trace Formula, Stab. Trace Formula Shimura Var. Arith. Appl. 1, Int. Press, Somerville, Mass., 2011, 309–347.
  • [7] R. Cluckers and F. Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2) 171 (2010), 1011–1065.
  • [8] R. Cluckers and D. J. Miller, Stability under integration of sums of products of real globally subanalytic functions and their logarithms, Duke Math. J. 156 (2011), 311–348.
  • [9] R. Cluckers and D. J. Miller, Loci of integrability, zero loci, and stability under integration for constructible functions on Euclidean space with Lebesgue measure, Int. Math. Res. Not. IMRN 2012, no. 14, 3182–3191.
  • [10] G. Comte, J.-M. Lion, and J.-P. Rolin, Nature log-analytique du volume des sous-analytiques, Illinois J. Math. 44 (2000), 884–888.
  • [11] S. Denkowska and J. Stasica, Ensembles sous-analytiques à la polonaise, Hermann, Paris, 2008.
  • [12] A. Gabrièlov, Projections of semianalytic sets (in Russian), Funkcional. Anal. i Priložen. 2 (1968), 18–30; English translation in Functional Anal. Appl. 2 (1968), 282–291.
  • [13] C. Gasquet and P. Witomski, Fourier Analysis and Applications, Texts Appl. Math. 30, Springer, New York, 1999.
  • [14] L. Hörmander, The Analysis of Linear Partial Differential Operators, IV, Classics Math., Springer, Berlin, 2009.
  • [15] E. Hrushovski and D. Kazhdan, “Integration in valued fields” in Algebraic Geometry and Number Theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 261–405.
  • [16] T. Kaiser, Integration of semialgebraic functions and integrated Nash functions, Math. Z. 275 (2013), 349–366.
  • [17] M. Kontsevich and D. Zagier, “Periods” in Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001, 771–808.
  • [18] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Pure Appl. Math., Wiley, New York, 1974.
  • [19] J.-M. Lion and J.-P. Rolin, Théorème de Gabrielov et fonctions log-exp-algébriques, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 1027–1030.
  • [20] J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble) 47 (1997), 859–884.
  • [21] J.-M. Lion and J.-P. Rolin, Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier (Grenoble) 48 (1998), 755–767.
  • [22] S. Łojasiewicz, “Sur les ensembles semi-analytiques” in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 1971, 237–241.
  • [23] B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 405–430.
  • [24] D. J. Miller, A preparation theorem for Weierstrass systems, Trans. Amer. Math. Soc. 358 (2006), no. 10, 4395–4439.
  • [25] A. Parusiński, “On the preparation theorem for subanalytic functions” in New Developments in Singularity Theory (Cambridge, 2000), NATO Sci. Ser. II Math. Phys. Chem. 21, Kluwer, Dordrecht, 2001, 193–215.
  • [26] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
  • [27] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • [28] L. van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), 189–193.
  • [29] L. van den Dries, On the elementary theory of restricted elementary functions, J. Symbolic Logic 53 (1988), 796–808.
  • [30] L. van den Dries, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, Cambridge, 1998.
  • [31] L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), 183–205.
  • [32] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497–540.
  • [33] A. N. Varchenko, Newton polyhedra and estimates of oscillatory integrals (in Russian), Funkcional. Anal. i Priložen. 10 (1976), no. 3, 13–38; English translation in Funct. Anal. Appl. 18 (1976), 175–196.
  • [34] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.