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ANALYTIC HYPOELLIPTICITY FOR SUMS OF SQUARES IN THE PRESENCE OF SYMPLECTIC NON TREVES STRATA

Part of: Close-to-elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  07 January 2019

Antonio Bove  and
Marco Mughetti
Antonio Bove
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, Italy (bove@bo.infn.it; marco.mughetti@unibo.it)
Marco Mughetti
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna, Italy (bove@bo.infn.it; marco.mughetti@unibo.it)
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Abstract

In Albano, Bove and Mughetti [J. Funct. Anal. 274(10) (2018), 2725–2753]; Bove and Mughetti [Anal. PDE 10(7) (2017), 1613–1635] it was shown that Treves conjecture for the real analytic hypoellipticity of sums of squares operators does not hold. Models were proposed where the critical points causing a non-analytic regularity might be interpreted as strata. We stress that up to now there is no notion of stratum which could replace the original Treves stratum. In the proposed models such ‘strata’ were non-symplectic analytic submanifolds of the characteristic variety. In this note we modify one of those models in such a way that the critical points are a symplectic submanifold of the characteristic variety while still not being a Treves stratum. We show that the operator is analytic hypoelliptic.

Keywords

sums of squares of vector fieldsanalytic hypoellipticityTreves conjecture

MSC classification

Primary: 35H10: Hypoelliptic equations 35H20: Subelliptic equations
Secondary: 35B65: Smoothness and regularity of solutions 35A20: Analytic methods, singularities 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 6 , November 2020 , pp. 1877 - 1888
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Copyright
© Cambridge University Press 2019

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References

Albano, P., Bove, A. and Mughetti, M., Analytic hypoellipticity for sums of squares and the Treves conjecture, J. Funct. Anal. 274(10) (2018), 2725–2753.Google Scholar
Bove, A. and Mughetti, M., Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable, J. Differential Equations 255(4) (2013), 728–758.Google Scholar
Bove, A. and Mughetti, M., Analytic hypoellipticity for sums of squares and the Treves conjecture, II, Anal. PDE 10(7) (2017), 1613–1635.Google Scholar
Bove, A. and Tartakoff, D. S., Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm. Partial Differential Equations 22(7–8) (1997), 1263–1282.Google Scholar
Bove, A. and Treves, F., On the Gevrey hypo-ellipticity of sums of squares of vector fields, Ann. Inst. Fourier (Grenoble) 54 (2004), 1443–1475.Google Scholar
Calderón, A. P. and Vaillancourt, R., A class of bounded pseudo-differential operators, Proc. Natl Acad. Sci. USA 69(5) (1972), 1185–1187.Google Scholar
Ehrenpreis, L., Solutions of some problems of division IV, Amer. J. Math. 82 (1960), 522–588.Google Scholar
Hörmander, L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.Google Scholar
Hörmander, L., Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math. 24 (1971), 671–704.Google Scholar
Hörmander, L., The Analysis of Partial Differential Operators, I (Springer, Berlin, 1985).Google Scholar
Hörmander, L., The Analysis of Partial Differential Operators, III (Springer, Berlin, 1985).Google Scholar
Kumano-Go, H., Pseudo-Differential Operators (MIT press, London, 1982).Google Scholar
Mughetti, M., Regularity properties of a double characteristics differential operator with complex lower order terms, J. Pseudo-Differ. Oper. Appl. 5(3) (2014), 343–358.Google Scholar
Mughetti, M., On the spectrum of an anharmonic oscillator, Trans. Amer. Math. Soc. 367 (2015), 835–865.Google Scholar
Trèves, F., Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ∂̄-Neumann problem, Comm. Partial Differential Equations 3(6–7) (1978), 475–642.Google Scholar
Treves, F., Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1 (Plenum Press, New York-London, 1980).Google Scholar
Treves, F., Symplectic geometry and analytic hypo-ellipticity, in Differential Equations, La Pietra 1996 (Florence), Proceedings of Symposia Pure Mathematics, Volume 65, pp. 201–219 (American Mathematical Society, Providence, RI, 1999).Google Scholar
Treves, F., On the analyticity of solutions of sums of squares of vector fields, in Phase Space Analysis of Partial Differential Equations, Bove, Colombini (ed. Del Santo), Progress in Nonlinear Differential Equations and their Applications, Volume 69, pp. 315–329 (Birkhäuser Boston, Boston, MA, 2006).Google Scholar
Treves, F., Aspects of Analytic PDE, book in preparation.Google Scholar