# Analytic Hypoellipticity and the Treves Conjecture

## DOI:

https://doi.org/10.6092/issn.2240-2829/6690## Keywords:

Sums of squares of vector fields, Analytic hypoellipticity, Treves conjecture## Abstract

We are concerned with the problem of the analytic hypoellipticity; precisely, we focus on the real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We discuss a model operator,*P*, (firstly appeared and studied in [3]) having a single symplectic stratum and prove that it is not analytic hypoelliptic. This yields a counterexample to the sufficient part of Treves conjecture; the necessary part is still an open problem.

## References

P. Albano, A. Bove and G. Chinni, Minimal Microlocal Gevrey Regularity for Sums of Squares", International Mathematics Research Notices, Vol. 2009, No. 12, 2275-2302.

P. Albano and A. Bove, Wave front set of solutions to sums of squares of vector fields, Mem. Amer. Math. Soc., 221 (2013), no. 1039.

P. Albano, A. Bove and M. Mughetti, Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture, https://arxiv.org/abs/1605.03801

M. S. Baouendi and C. Goulaouic, Nonanalytic-hypoellipticity for some degenerate operators, Bull. A. M. S., 78 (1972), 483-486.

F.A. Berezin and M.A. Shubin, The Schrödinger equation, Mathematics and its Applications (Soviet Series), 66. Kluwer Academic Publishers Group, Dordrecht, 1991

A. Bove, M. Mughetti, On a new method of proving Gevrey hypoellipticity for certain sums of squares. Adv. Math. 293 (2016), 146-220.

A. Bove, M. Mughetti, Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable. J. Differential Equations 255 (2013), no. 4, 728758.

A. Bove, M. Mughetti and D. S. Tartakoff, Hypoellipticity and Non Hypoellipticity for Sums of Squares of Complex Vector Fields, Analysis & PDE 6 (2013), no. 2, 371-446.

A. Bove and D. S. Tartakoff, Optimal non-isotropic Gevrey exponents for sums of squares of vector fields, Comm. Partial Differential Equations 22 (1997), no. 7-8,1263-1282.

A. Bove and F. Treves, On the Gevrey hypo-ellipticity of sums of squares of vector fields, Ann. Inst. Fourier (Grenoble) 54(2004), 1443-1475.

M. Christ, Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 16 (1991), 1695-1707.

P. D. Cordaro and N. Hanges, A New Proof of Okaji's Theorem for a Class of Sum of Squares Operators, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 595-619.

M. Derridj, Un problème aux limites pour une classe d'opérateurs du second ordre hypoelliptiques, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 4, 99-148.

V. S. Fedii, A certain criterion for hypoellipticity, Mat. Sb. 14 (1971), 1545.

A. Grigis and J. Sjöstrand, Front d'onde analytique et sommes de carr_es de champs de vecteurs, Duke Math. J. 52 (1985), no. 1, 35-51.

V. V. Grušin, On a class of elliptic pseudodifferential operators degenerating at a submanifold, Mat. Sbornik 84 (2) (1971), 163-195.

N. Hanges and A. A. Himonas, Singular solutions for sums of squares of vector fields, Comm. In Partial Differential Equations 16 (1991), 1503-1511.

B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics, no. 1336, 1988, Springer Verlag.

B. Helffer and J. Sjöstrand, Multiple Wells in the Semi-classical Limit I , Communications in Partial Differential Equations 9(4) (1984), 337-408.

L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171.

L. Hörmander, Uniqueness Theorems and Wave Front Sets for Solutions of Linear Differential Equations with Analytic Coefficients, Communications Pure Appl. Math. 24 (1971), 671-704.

L. Hörmander, The Analysis of Partial Differential Operators, I, Springer Verlag, 1985.

L. Hörmander, The Analysis of Partial Differential Operators, III, Springer Verlag, 1985.

L. Maniccia and M. Mughetti, SAK principle for a class of Grushin-type operators. Rev. Mat. Iberoam. 22 (2006), n. 1, 259286.

A. Martinez, Estimations de l'effet tunnel pour le double puits. I, J. Math. Pures Appl. 66 (1987), no. 2, 195-215.

G. Métivier, Non-hypoellipticité Analytique pour D_x^2+(x2 + y2)D_y^2, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 7, 401-404.

Y. Morimoto, On the hypoellipticity for infinitely degenerate semi-elliptic operators, Math. Soc. Japan, 30 (1978) , 327-358.

M. Mughetti and F. Nicola, On the generalization of Hormander's inequality, Comm. in Partial Differential Equations, (2005) 30, 4-6, 509-537.

M. Mughetti and F. Nicola, Hypoellipticity for a class of operators with multiple characteristics. J. Anal. Math. 103 (2007), 377-396.

M. Mughetti, C. Parenti and A. Parmeggiani Lower bound estimates without transversal ellipticity. Comm. Partial Differential Equations 32 (2007), n. 7-9, 1399-1438.

M. Mughetti, Hypoellipticity and higher order Levi conditions. J. Differential Equations 257 (2014), n. 4, 1246-1287.

M. Mughetti, Regularity properties of a double characteristics differential operator with complex lower order terms. J. of Pseudo-Differential Operators and Applications, 5, 3,(2014) 343-358.

M. Mughetti, On the spectrum of an anharmonic oscillator , Trans. Amer. Math. Soc. 367 2 (2015), 835-865.

B. Simon, Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: asymptotic expansions, Annales de l'I. H. P., section A, 38(3) (1983), 295-308.

B. Simon, ERRATA: Semiclassical analysis of low lying eigenvalues. I. Non-degenerate minima: asymptotic expansions, Annales de l'I. H. P., section A, 40(2) (1984), 224.

J. Sjöstrand, Analytic wavefront set and operators with multiple characteristics, Hokkaido Math. J. 12(1983), 392-433.

J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982).

D.S. Tartakoff, On the Local Real Analyticity of Solutions to □_{

F. Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ∂-Neumann problem, Comm. Partial Differential Equations 3 (1978), no. 6-7, 475-642.

F. Trèves, Symplectic geometry and analytic hypo-ellipticity, in Differential equations, La Pietra 1996 (Florence), Proc. Sympos. Pure Math. 65, Amer. Math. Soc., Providence, RI, 1999, 201-219.

F. Trèves, On the analyticity of solutions of sums of squares of vector fields, Phase space analysis of partial differential equations, Bove, Colombini, Del Santo ed.'s, 315-329, Progr. Nonlinear Differential Equations Appl., 69, Birkhäuser Boston, Boston, MA, 2006.

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*Bruno Pini Mathematical Analysis Seminar*,

*7*(1), 53–68. https://doi.org/10.6092/issn.2240-2829/6690

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