Accessible Unlicensed Requires Authentication Published by De Gruyter September 7, 2017

A groupoid approach to pseudodifferential calculi

Erik van Erp and Robert Yuncken ORCID logo

Abstract

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural +×-action. Specifically, a properly supported semiregular distribution on M×M is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the +×-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: ANR-14-CE25-0012-01

Funding statement: Robert Yuncken was supported by the project SINGSTAR of the Agence Nationale de la Recherche, ANR-14-CE25-0012-01.

Acknowledgements

The present article was inspired by an observation of Debord and Skandalis in their paper [11] that provides the first abstract characterization of the classical pseudodifferential operators in terms of the +×-action on the tangent groupoid. We wish to thank them for many discussions, particularly during the time that the first author was Professor Invité at the Université Blaise Pascal, Clermont-Ferrand II. Sincere thanks also go to Jean-Marie Lescure and Nigel Higson.

References

[1] B. Ammann, R. Lauter and V. Nistor, Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math. (2) 165 (2007), no. 3, 717–747. Search in Google Scholar

[2] R. Beals and P. Greiner, Calculus on Heisenberg manifolds, Ann. of Math. Stud. 119, Princeton University Press, Princeton 1988. Search in Google Scholar

[3] P. Carrillo Rouse, A Schwartz type algebra for the tangent groupoid, K-theory and noncommutative geometry, EMS Ser. Congr. Rep., European Mathematical Society, Zürich (2008), 181–199. Search in Google Scholar

[4] W. Choi and R. Ponge, Privileged coordinates and tangent groupoid for Carnot manifolds, preprint (2015), . Search in Google Scholar

[5] M. Christ, D. Geller, P. Głowacki and L. Polin, Pseudodifferential operators on groups with dilations, Duke Math. J. 68 (1992), no. 1, 31–65. Search in Google Scholar

[6] A. Connes, Sur la théorie non commutative de l’intégration, Algèbres d’opérateurs (Les Plans-sur-Bex 1978), Lecture Notes in Math. 725, Springer, Berlin (1979), 19–143. Search in Google Scholar

[7] A. Connes, Noncommutative geometry, Academic Press, San Diego 1994. Search in Google Scholar

[8] T. E. Cummins, A pseudodifferential calculus associated to 3-step nilpotent groups, Comm. Partial Differential Equations 14 (1989), no. 1, 129–171. Search in Google Scholar

[9] S. Dave and S. Haller, Graded hypoellipticity of BGG sequences, preprint (2017), . Search in Google Scholar

[10] C. Debord, J.-M. Lescure and F. Rochon, Pseudodifferential operators on manifolds with fibred corners, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 4, 1799–1880. Search in Google Scholar

[11] C. Debord and G. Skandalis, Adiabatic groupoid, crossed product by + and pseudodifferential calculus, Adv. Math. 257 (2014), 66–91. Search in Google Scholar

[12] C. Epstein and R. Melrose, The Heisenberg algebra, index theory and homology, unpublished book, . Search in Google Scholar

[13] L. Hörmander, The analysis of linear partial differential operators. III: Pseudodifferential operators, Grundlehren Math. Wiss. 274, Springer, Berlin 1985. Search in Google Scholar

[14] J.-M. Lescure, D. Manchon and S. Vassout, About the convolution of distributions on groupoids, J. Noncommut. Geom. 11 (2017), no. 2, 757–789. Search in Google Scholar

[15] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Ser. 124, Cambridge University Press, Cambridge 1987. Search in Google Scholar

[16] A. Melin, Lie filtrations and pseudo-differential operators, preprint (1982). Search in Google Scholar

[17] R. B. Melrose, Transformation of boundary problems, Acta Math. 147 (1981), no. 3–4, 149–236. Search in Google Scholar

[18] R. B. Melrose, The Atiyah–Patodi–Singer index theorem, Research Notes in Math. 4, A K Peters, Wellesley 1993. Search in Google Scholar

[19] R. B. Melrose and P. Piazza, Analytic K-theory on manifolds with corners, Adv. Math. 92 (1992), no. 1, 1–26. Search in Google Scholar

[20] I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Stud. Adv. Math. 91, Cambridge University Press, Cambridge 2003. Search in Google Scholar

[21] B. Monthubert, Pseudodifferential calculus on manifolds with corners and groupoids, Proc. Amer. Math. Soc. 127 (1999), no. 10, 2871–2881. Search in Google Scholar

[22] B. Monthubert and F. Pierrot, Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 2, 193–198. Search in Google Scholar

[23] A. Nagel and E. M. Stein, Lectures on pseudodifferential operators: Regularity theorems and applications to nonelliptic problems, Math. Notes 24, Princeton University Press, Princeton 1979. Search in Google Scholar

[24] V. Nistor, A. Weinstein and P. Xu, Pseudodifferential operators on differential groupoids, Pacific J. Math. 189 (1999), no. 1, 117–152. Search in Google Scholar

[25] R. Ponge, The tangent groupoid of a Heisenberg manifold, Pacific J. Math. 227 (2006), no. 1, 151–175. Search in Google Scholar

[26] R. S. Ponge, Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds, Mem. Amer. Math. Soc. 194 (2008), no. 906. Search in Google Scholar

[27] A. R. H. S. Sadegh and N. Higson, Euler-like vector fields, deformation spaces and manifolds with filtered structure, preprint (2016), . Search in Google Scholar

[28] M. Taylor, Noncommutative microlocal analysis. Part I (revised version), preprint, . Search in Google Scholar

[29] F. Trèves, Introduction to pseudodifferential and Fourier integral operators. Vol. 1, Plenum Press, New York 1980. Search in Google Scholar

[30] E. van Erp, The Atiyah–Singer index formula for subelliptic operators on contact manifolds, ProQuest LLC, Ann Arbor 2005; Ph.D. thesis, The Pennsylvania State University. Search in Google Scholar

[31] E. van Erp, The Atiyah–Singer index formula for subelliptic operators on contact manifolds. Part I, Ann. of Math. (2) 171 (2010), no. 3, 1647–1681. Search in Google Scholar

[32] E. van Erp and R. Yuncken, On the tangent groupoid of a filtered manifold, preprint (2016), . Search in Google Scholar

Received: 2016-05-25
Revised: 2017-07-26
Published Online: 2017-09-07
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston