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Fundamental solutions for non-divergence form operators on stratified groups


Authors: Andrea Bonfiglioli, Ermanno Lanconelli and Francesco Uguzzoni
Journal: Trans. Amer. Math. Soc. 356 (2004), 2709-2737
MSC (2000): Primary 35A08, 35H20, 43A80; Secondary 35A17, 35J70
DOI: https://doi.org/10.1090/S0002-9947-03-03332-4
Published electronically: October 21, 2003
MathSciNet review: 2052194
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Abstract: We construct the fundamental solutions $\Gamma$ and $\gamma$for the non-divergence form operators $\,{\textstyle\sum_{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial_t\,$ and ${\,\textstyle\sum_{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G} $ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments.


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  • 1. G. K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc., 155 (2002), no. 739. MR 2003c:22015
  • 2. A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), 1153-1192.
  • 3. A. Bonfiglioli and F. Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, to appear in Forum Math.
  • 4. A. Bonfiglioli and F. Uguzzoni, A note on lifting of Carnot groups, preprint.
  • 5. A. Bonfiglioli and F. Uguzzoni, Harnack inequality for non-divergence form operators on stratified groups, preprint.
  • 6. J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), 277-304. MR 41:7486
  • 7. M. Bramanti and L. Brandolini, $L^p$ estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781-822. MR 2000c:35026
  • 8. M. Bramanti and L. Brandolini, $L^p$ estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, to appear in Rend. Sem. Mat. Univ. Politec. Torino.
  • 9. L. Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867-889. MR 98k:22037
  • 10. L. Capogna, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263-295. MR 2000a:35027
  • 11. L. Capogna, D. Danielli, and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765-1794. MR 94j:35038
  • 12. L. Capogna, D. Danielli, and N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153-1196. MR 97k:35033
  • 13. L. Capogna and Q. Han, Pointwise Schauder estimates for second order linear equations in Carnot groups, preprint.
  • 14. G. Citti, N. Garofalo, and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), 699-734. MR 94m:35069
  • 15. G. Citti, E. Lanconelli, and A. Montanari, Smoothness of Lipschitz continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), 87-128.
  • 16. G. Citti, M. Manfredini, and A. Sarti, A note on the Mumford-Shah functional in Heisenberg space, preprint.
  • 17. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. MR 58:13215
  • 18. B. Franchi, G. Lu, and R. L. Wheeden, Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations, Potential Anal. 4 (1995), 361-375. MR 97e:35018
  • 19. A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964. MR 31:6062
  • 20. P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. MR 2000j:46063
  • 21. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 36:5526
  • 22. G. Huisken and W. Klingenberg, Flow of real hypersurfaces by the trace of the Levi form, Math. Res. Lett. 6 (1999), 645-661. MR 2001f:53141
  • 23. D. Jerison and J. M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167-197. MR 88i:58162
  • 24. D. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835-854. MR 88c:58064
  • 25. H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo (2) 41 (1992), no. 2, 251-294. MR 93m:35054
  • 26. S. Kusuoka and D. Stroock, Applications of the Malliavin calculus III, J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 34 (1987), 391-442. MR 89c:60093
  • 27. S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), 165-189. MR 89b:35022
  • 28. E. Lanconelli, Non-linear equations on Carnot groups and CR-curvature problems, Proceeding of the Conference ``Renato Caccioppoli and Modern Analysis", to appear in Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
  • 29. E. Lanconelli, A. Pascucci, and S. Polidoro, Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, Nonlinear Problems in Mathematical Physics and Related Topics, II, in Honor of Professor O.A. Ladyzhenskaya, International Mathematical Series, 2, to appear.
  • 30. E. Lanconelli and A.E. Kogoj, $X$-elliptic operators and $X$-control distances, Contributions in honor of the memory of Ennio De Giorgi, Ricerche Mat. 49 (2000), suppl., 223-243. MR 2002c:35121
  • 31. G. Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Rev. Mat. Iberoamericana 8 (1992), 367-439. MR 94c:35061
  • 32. G. Lu, Existence and size estimates for the Green's functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), 1213-1251. MR 93i:35030
  • 33. G. Lu, On Harnack's inequality for a class of strongly degenerate Schrödinger operators formed by vector fields, Differential Integral Equations 7 (1994), 73-100. MR 95f:35032
  • 34. A. Montanari, Real hypersurfaces evolving by Levi curvature: Smooth regularity of solutions to the parabolic Levi equation, Comm. Partial Differential Equations 26 (2001), 1633-1664. MR 2002i:35110
  • 35. C. Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. MR 44:1924
  • 36. R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI, 2002. MR 2002m:53045
  • 37. J. Petitot and Y. Tondut, Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Math. Inform. Sci. Humaines 145 (1999), 5-101. MR 2000j:92007
  • 38. S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania) 49 (1994), no. 1, 53-105. MR 97a:35133
  • 39. L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. MR 55:9171
  • 40. Z. Slodkowski and G. Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), 392-407. MR 93c:32018
  • 41. E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002
  • 42. F. Uguzzoni, A note on a generalized form of the Laplacian and of sub-Laplacians, to appear in Arch. Math. (Basel).
  • 43. N.T. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992. MR 95f:43008
  • 44. C.J. Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), 77-96. MR 93b:35042

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Additional Information

Andrea Bonfiglioli
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy
Email: bonfigli@dm.unibo.it

Ermanno Lanconelli
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy
Email: lanconel@dm.unibo.it

Francesco Uguzzoni
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy
Email: uguzzoni@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9947-03-03332-4
Keywords: Non-divergence sub-elliptic operators, stratified groups, fundamental solutions
Received by editor(s): November 21, 2002
Published electronically: October 21, 2003
Additional Notes: Investigation supported by University of Bologna, Funds for selected research topics
Article copyright: © Copyright 2003 American Mathematical Society

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