Abstract
If \({\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0}\) is a Hörmander partial differential operator in \({\mathbb{R}^N}\), we give sufficient conditions on the vector fields X j ’s for the existence of a Lie group structure \({\mathbb{G} = (\mathbb{R}^N, *)}\) (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that \({\mathcal{L}}\) is left invariant on \({\mathbb{G}}\). The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector fields {X 0, . . . , X m }. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff type. Examples of operators to which our results apply are also furnished.
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References
Bonfiglioli A.: Homogeneous Carnot groups related to sets of vector fields. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7, 79–107 (2004)
A. Bonfiglioli, E. Lanconelli, Lie Groups Constructed from Hörmander Operators. Fundamental Solutions and Applications to Kolmogorov-Fokker-Planck Equations, preprint (2008).
A. Bonfiglioli, E. Lanconelli, On left invariant Hörmander operators in \({\mathbb{R}^N}\). Applications to Kolmogorov-Fokker-Planck equations, Proceedings of the Fifth International Conference on Differential and Functional Differential Equations, Moscow, August 17-24, 2008 (to appear).
A. Bonfiglioli, E. Lanconelli., F. Uguzzoni, Stratified Lie Groups and Potential Theory for their sub-Laplacians, Springer Monographs in Mathematics 26, New York, NY, Springer (2007).
Bramanti M., Brandolini L.: L p-estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Rend. Sem. Mat. Univ. Politec. Torino. 58, 389–433 (2000)
Djoković D.: An elementary proof of the Baker-Campbell-Hausdorff-Dynkin formula. Math. Z. 143, 209–211 (1975)
Eichler M.: A new proof of the Baker-Campbell-Hausdorff formula. J. Math. Soc. Japan 20, 23–25 (1968)
Fefferman C.L., Sánchez-Calle A.: Fundamental solutions for second order subelliptic operators. Ann. Math. 124, 247–272 (1986)
Folland G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13, 161–207 (1975)
G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups. Mathematical Notes, 28, Princeton University Press, Princeton, N.J. (1982).
R. Godement, Introduction à la théorie des groupes de Lie. Tome 2. Publications Mathématiques de l’Université Paris VII, Université de Paris VII, U.E.R. de Mathématiques, Paris (1982).
Hartman P.: Ordinary Differential Equations. Second edition. Birkhäuser, Boston-Basel-Stuttgart (1982)
M. Hausner, J.T. Scwartz, Lie Groups. Lie Algebras. Gordon and Breach, New York-London-Paris, Notes on Mathematics and Its Applications (1968).
G. Hochschild, La structure des groupes de Lie. Monographies universitaires de mathmatiques, 27. Paris, Dunod (1968).
Hörmander L.: Hypoelliptic second order differential equations. ActaMath. 119, 147–171 (1967)
N. Jacobson, Lie algebras. Intersci. Tracts in Pure and Appl. Math. 10. N.Y. and Lond., Intersci. Publishers, John Wiley and Sons (1962).
Jerison D., Sánchez-Calle A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35, 835–854 (1986)
Kogoj A.E., Lanconelli E.: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterr. J. Math. 1, 51–80 (2004)
Kusuoka S., Stroock D.: The partial Malliavin calculus and its application to nonlinear filtering. Stochastics 12, 83–142 (1984)
Kusuoka S., Stroock D.: Applications of the Malliavin calculus. II. J. Fac. Sci., Univ. Tokyo, Sect. I A. 32, 1–76 (1985)
Kusuoka S., Stroock D.: Applications of the Malliavin calculus III. J. Fac. Sci. Univ. Tokyo, Sect. I A. 34, 391–442 (1987)
Kusuoka S., Stroock D.: Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator. Ann. of Math. 127, 165–189 (1988)
Lanconelli E., Polidoro S.: On a class of hypoelliptic evolution operators. Rend. Semin. Mat. Torino 52, 29–63 (1994)
Morbidelli D.: Fractional Sobolev norms and structure of Carnot-Carath‘eodory balls for Hörmander vector fields. Studia Math. 139, 213–242 (2000)
D. Mumford, Elastica and computer vision. Bajaj, Chandrajit L. (ed.), Algebraic geometry and its applications. Purdue University, West Lafayette, IN, U SA, June 1-4, 1990. New York, Springer-Verlag. 491–506 (1994).
Nagel A., Ricci F., Stein E.M.: Fundamental solutions and harmonic analysis on nilpotent groups. Bull. Am. Math. Soc., New Ser. 23, 139–144 (1990)
Nagel A., Ricci F., Stein E.M.: Harmonic analysis and fundamental solutions on nilpotent Lie groups. Analysis and partial differential equations, Coll. Pap. dedic. Mischa Cotlar, Lect. Notes Pure Appl. Math. 122, 249–275 (1990)
Nagel A., Stein E.M., Wainger S.: Balls and metrics defined by vector fields I, basic properties. Acta Math. 155, 103–147 (1985)
Rothschild L.P., Stein E.M.: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(1976), 247-320
A.A. Sagle, R.E. Walde, Introduction to Lie groups and Lie algebras. Pure and Applied Mathematics, 51. New York-London, Academic Press (1973)
Sánchez-Calle A.: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78, 143–160 (1984)
J.-P. Serre, Lie algebras and Lie groups. 1964 lectures, given at Harvard University. Lecture Notes in Mathematics. Berlin, Springer-Verlag (1992).
Varadarajan V.S.: Lie groups, Lie algebras and their representations. Graduate Texts in Mathematics, Springer-Verlag, New York (1984)
Varopoulos N.T., Saloff-Coste L., Coulhon T.: Analysis and geometry on groups Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge (1992)
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Bonfiglioli, A. An ODE’s Version of the Formula of Baker, Campbell, Dynkin and Hausdorff and the Construction of Lie Groups with Prescribed Lie Algebra. Mediterr. J. Math. 7, 387–414 (2010). https://doi.org/10.1007/s00009-010-0064-x
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DOI: https://doi.org/10.1007/s00009-010-0064-x