Abstract
We consider a class of operators of the type sum of squares of real analytic vector fields satisfying the Hörmander bracket condition. The Poisson-Treves stratification is associated to the vector fields. We show that if the deepest stratum in the stratification, i.e., the stratum associated to the longest commutators, is symplectic, then the Gevrey regularity of the solution is better than the minimal Gevrey regularity given by the Derridj-Zuily theorem.
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References
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 G. Chinni, Minimal microlocal Gevrey regularity for “sums of squares”, Int. Math. Res. Not. IMRN, 2009, 2275–2302.
P. Albano, A. Bove, and M. Mughetti, Analytic hypoellipticity for sums of squares and the Treves conjecture, preprint, 2016, http://arxiv.org/abs/1605.03801.
A. Bove, Gevrey hypo-ellipticity for sums of squares of vector fields: some examples, in Geometric Analysis of PDE and Several Complex Variables, Contemp. Math. 368 (2005), 41–68.
A. Bove and M. Mughetti, Analytic hypoellipticity for sums of squares and the Treves Conjecture, II, Analysis and PDE 10 (2017), 1613–1635.
A. Bove and D. S. Tartakoff, A class of sums of squares with a given Poisson-Treves stratification, J. Geom. Anal. 13 (2003), 391–420.
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.
P. Cordaro and N. Hanges, Hypoellipticity in spaces of ultradistributions–study of a model case, Israel J. Math. 191 (2012), 771–789.
M. Derridj and C. Zuily, Régularité analytique et Gevrey d’opérateurs elliptiques dégénérés, J. Math. Pures Appl. (9) 52 (1973), 65–80.
M. Derridj and C. Zuily, Sur la régularité Gevrey des opérateurs de Hörmander, J. Math. Pures Appl. (9) 52 (1973), 309–336.
D. Gilbarg and N. S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
A. Grigis and J. Sjöstrand, Front d’onde analytique et somme de carrés de champs de vecteurs, Duke Math. J. 52 (1985), 35–51.
L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
L. Hórmander, Fourier integral operators. I, Acta Math. 127 (1971), 79–183.
B. Simon, Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Physics 146 (1983), 209–220.
B. Simon, Nonclassical eigenvalue asymptotics, J. Funct. Anal. 53 (1983), 84–98.
J. Sjöstrand, Singularités analytiques microlocales, Astérisque 95 (1982).
J. Sjöstrand, Analytic wavefront set and operators with multiple characteristics, Hokkaido Math. J. 12 (1983), 392–433.
F. Treves, Symplectic geometry and analytic hypo-ellipticity, in Differential Equations: La Pietra 1996, Amer. Math. Soc., Providence, RI, 1999, pp. 201–219.
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Albano, P., Bove, A. The presence of symplectic strata improves the Gevrey regularity for sums of squares. JAMA 134, 139–155 (2018). https://doi.org/10.1007/s11854-018-0005-3
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DOI: https://doi.org/10.1007/s11854-018-0005-3