Semiclassical Gevrey operators in the complex domain
Annales de l'Institut Fourier, Online first, 50 p.

We study semiclassical Gevrey pseudodifferential operators, acting on exponentially weighted spaces of entire holomorphic functions. The symbols of such operators are Gevrey functions defined on suitable I-Lagrangian submanifolds of the complexified phase space, which are extended almost holomorphically in the same Gevrey class, or in some larger space, to complex neighborhoods of these submanifolds. Using almost holomorphic extensions, we obtain uniformly bounded realizations of such operators on a natural scale of exponentially weighted spaces of holomorphic functions for all Gevrey indices, with remainders that are optimally small, provided that the Gevrey index is less than or equal to 2.

Nous étudions les opérateurs pseudo-différentiels de classe Gevrey dans le cadre semi-classique, agissant sur des sous-espaces L 2 à poids exponentiels de fonctions holomorphes. Les symboles de tels opérateurs sont des fonctions de classe Gevrey définies sur des sous-variétés I-lagrangiennes appropriées du complexifié de l’espace des phases, qui sont prolongés par des extensions presque-analytiques dans la même classe de Gevrey, ou dans un espace fonctionnel plus large, définies sur des voisinages complexes de ces sous-variétés. En utilisant les extensions presque-analytiques et la méthode de déformation de contours dans le plan complexe, nous obtenons – avec des choix de contours adéquats « bon contours » – des réalisations uniformément bornées de ces opérateurs sur une famille « naturelle » de sous-espaces L 2 à poids exponentiels de fonctions holomorphes pour tous les indices de Gevrey, avec des estimées optimales des restes sous la condition que l’indice de Gevrey est inférieur ou égal à 2.

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DOI: 10.5802/aif.3546
Classification: 30D60,  30E05,  32W05,  32W25,  35S99
Keywords: Semiclassical Gevrey pseudodifferential operator, almost holomorphic extension, FBI transform, Bargmann space, strictly plurisubharmonic weight function.
Hitrik, Michael 1; Lascar, Richard 2; Sjöstrand, Johannes 3, 4; Zerzeri, Maher 5

1 Department of Mathematics University of California Los Angeles CA 90095-1555 (USA)
2 JAD - UMR 7351 Université Côte d’Azur Parc Valrose 06108 Nice Cedex 02 (France)
3 IMB Université de Bourgogne 9, Av. A. Savary, BP 47870 21078 Dijon (France)
4 UMR 5584 CNRS
5 LAGA – UMR7539 CNRS Université Sorbonne Paris-Nord 99, avenue J.-B. Clément 93430 Villetaneuse (France)
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Hitrik, Michael; Lascar, Richard; Sjöstrand, Johannes; Zerzeri, Maher. Semiclassical Gevrey operators in the complex domain. Annales de l'Institut Fourier, Online first, 50 p.

[1] Carleson, Lennart On universal moment problems, Math. Scand., Volume 9 (1961), pp. 197-206 | DOI | MR | Zbl

[2] Dencker, Nils; Sjöstrand, Johannes; Zworski, Maciej Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math., Volume 57 (2004) no. 3, pp. 384-415 | DOI | MR | Zbl

[3] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, Cambridge, 1999, xii+227 pages | DOI | MR | Zbl

[4] Dynʼkin, E. M. An operator calculus based on the Cauchy–Green formula, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 30 (1972), pp. 33-39 | MR

[5] Dynʼkin, E. M. The pseudoanalytic extension, J. Anal. Math., Volume 60 (1993), pp. 45-70 | DOI | MR | Zbl

[6] Fürdös, Stefan; Nenning, David Nicolas; Rainer, Armin; Schindl, Gerhard Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis, J. Math. Anal. Appl., Volume 481 (2020) no. 1, 123451, 51 pages | DOI | MR | Zbl

[7] Galkowski, Jeffrey; Zworski, Maciej Outgoing solutions via Gevrey-2 properties, Ann. PDE, Volume 7 (2021) no. 1, 5, 13 pages | DOI | MR | Zbl

[8] Grigis, Alain; Sjöstrand, Johannes Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, 196, Cambridge University Press, Cambridge, 1994, iv+151 pages (An introduction) | DOI | MR | Zbl

[9] Guedes Bonthonneau, Yannick; Jézéquel, Malo FBI transform in Gevrey classes and Anosov flows (2020) (https://arxiv.org/abs/2001.03610)

[10] Hérau, Frédéric; Sjöstrand, Johannes; Stolk, Christiaan C. Semiclassical analysis for the Kramers–Fokker–Planck equation, Comm. Partial Differential Equations, Volume 30 (2005) no. 4-6, pp. 689-760 | DOI | MR | Zbl

[11] Hitrik, Michael; Lascar, Richard; Sjöstrand, Johannes; Zerzeri, Maher Semiclassical Gevrey operators and magnetic translations, J. Spectr. Theory, Volume 12 (2022) no. 1, pp. 53-82 | DOI | MR | Zbl

[12] Hitrik, Michael; Sjöstrand, Johannes Two minicourses on analytic microlocal analysis, Algebraic and analytic microlocal analysis (Springer Proc. Math. Stat.), Volume 269, Springer, Cham, 2018, pp. 483-540 | DOI | MR | Zbl

[13] Hörmander, Lars The analysis of linear partial differential operators. IV, I-IV, Springer, Berlin, 1985

[14] Hörmander, Lars Notions of convexity, Progress in Mathematics, 127, Birkhäuser Boston, Inc., Boston, MA, 1994, viii+414 pages | MR

[15] Lascar, B.; Lascar, R. Propagation des singularités Gevrey pour la diffraction, Comm. Partial Differential Equations, Volume 16 (1991) no. 4-5, pp. 547-584 | DOI | MR | Zbl

[16] Lascar, Bernard Propagation des singularités Gevrey pour des opérateurs hyperboliques, Amer. J. Math., Volume 110 (1988) no. 3, pp. 413-449 | DOI | MR | Zbl

[17] Lascar, Bernard; Lascar, Richard FBI transforms in Gevrey classes, J. Anal. Math., Volume 72 (1997), pp. 105-125 | DOI | MR | Zbl

[18] Lebeau, G. Régularité Gevrey 3 pour la diffraction, Comm. Partial Differential Equations, Volume 9 (1984) no. 15, pp. 1437-1494 | DOI | MR | Zbl

[19] Lerner, Nicolas Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, 3, Birkhäuser Verlag, Basel, 2010, xii+397 pages | DOI | MR | Zbl

[20] Martinez, André An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, New York, 2002, viii+190 pages | DOI | MR | Zbl

[21] Mather, J. N., Proceedings of Liverpool Singularities—Symposium, I (1969/70) (Lecture Notes in Mathematics), Volume 192 (1971), pp. 116-120 | MR | Zbl

[22] Melin, Anders; Sjöstrand, Johannes Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974) (Lecture Notes in Math., Vol. 459), Springer, Berlin, 1975, pp. 120-223 | MR | Zbl

[23] Melin, Anders; Sjöstrand, Johannes Bohr–Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Autour de l’analyse microlocale (Astérisque), Volume 284, Société Mathématique de France, Paris, 2003, pp. 181-244 | MR

[24] Boutet de Monvel, Louis; Krée, Paul Pseudo-differential operators and Gevrey classes, Ann. Inst. Fourier (Grenoble), Volume 17 (1967) no. 1, pp. 295-323 | DOI | MR | Zbl

[25] Rodino, Luigi Linear partial differential operators in Gevrey spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1993, x+251 pages | DOI | MR

[26] Rouleux, Michel Absence of resonances for semiclassical Schrödinger operators with Gevrey coefficients, Hokkaido Math. J., Volume 30 (2001) no. 3, pp. 475-517 | DOI | MR | Zbl

[27] Segal, I. E. Transforms for operators and symplectic automorphisms over a locally compact abelian group, Math. Scand., Volume 13 (1963), pp. 31-43 | DOI | MR | Zbl

[28] Sjöstrand, Johannes Propagation of analytic singularities for second order Dirichlet problems, Comm. Partial Differential Equations, Volume 5 (1980) no. 1, pp. 41-93 | DOI | MR | Zbl

[29] Sjöstrand, Johannes Propagation of analytic singularities for second order Dirichlet problems. II, Comm. Partial Differential Equations, Volume 5 (1980) no. 2, pp. 187-207 | DOI | MR | Zbl

[30] Sjöstrand, Johannes Propagation of analytic singularities for second order Dirichlet problems. III, Comm. Partial Differential Equations, Volume 6 (1981) no. 5, pp. 499-567 | DOI | MR | Zbl

[31] Sjöstrand, Johannes Singularités analytiques microlocales, Astérisque, 95, Société Mathématique de France, Paris, 1982, pp. 1-166 | MR

[32] Sjöstrand, Johannes Asymptotique des résonances pour des obstacles, Séminaire Bourbaki, Vol. 1989/90 (Astérisque), Volume 189-190, Société Mathématique de France, Paris, 1990, pp. 259-283 | MR

[33] Sjöstrand, Johannes Function spaces associated to global I-Lagrangian manifolds, Structure of solutions of differential equations (Katata/Kyoto, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 369-423 | MR | Zbl

[34] Zworski, Maciej Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, Providence, RI, 2012, xii+431 pages | DOI | MR | Zbl

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