Harnack inequality for non-divergence form operators on stratified groups. (English) Zbl 1113.35032
The authors consider the following operators in non-divergence form
\[ \sum_{i,j}a_{i,j}(x,t) X_iX_j - \partial_t \text{ and } \sum_{i,j}a_{i,j}(x) X_iX_j, \]
where the \(X_i\)’s are Hörmander vector fields generating a stratified group G and \((a_{i,j})\) is a positive defined matrix with Hölder continuous entries. A lower bound estimate is proved for the fundamental solution as well as an invariant Harnack inequality. Some relevant properties of the Green function are also studied.
\[ \sum_{i,j}a_{i,j}(x,t) X_iX_j - \partial_t \text{ and } \sum_{i,j}a_{i,j}(x) X_iX_j, \]
where the \(X_i\)’s are Hörmander vector fields generating a stratified group G and \((a_{i,j})\) is a positive defined matrix with Hölder continuous entries. A lower bound estimate is proved for the fundamental solution as well as an invariant Harnack inequality. Some relevant properties of the Green function are also studied.
Reviewer: Marco Biroli (Milano)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35A08 | Fundamental solutions to PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 43A80 | Analysis on other specific Lie groups |
| 35H20 | Subelliptic equations |
Keywords:
Fundamental solution; analysis on Carnot groups; Harnack inequality; Hörmander vector fields; Green functionReferences:
| [1] | Georgios K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc. 155 (2002), no. 739, x+101. · Zbl 0994.22006 · doi:10.1090/memo/0739 |
| [2] | A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian estimates for the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), no. 10, 1153 – 1192. · Zbl 1036.35061 |
| [3] | Andrea Bonfiglioli, Ermanno Lanconelli, and Francesco Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2709 – 2737. · Zbl 1072.35011 |
| [4] | Andrea Bonfiglioli and Francesco Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, Forum Math. 16 (2004), no. 3, 403 – 415. · Zbl 1065.35102 · doi:10.1515/form.2004.018 |
| [5] | A. Bonfiglioli, F. Uguzzoni, A note on lifting of Carnot groups, Rev. Mat. Iberoamericana 21 (2005), 1013-1035. · Zbl 1100.35029 |
| [6] | Marco Bramanti and Luca Brandolini, \?^{\?} estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781 – 822. · Zbl 0935.35037 |
| [7] | M. Bramanti and L. Brandolini, \?^{\?} estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, Rend. Sem. Mat. Univ. Politec. Torino 58 (2000), no. 4, 389 – 433 (2003). · Zbl 1072.35058 |
| [8] | Luca Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867 – 889. , https://doi.org/10.1002/(SICI)1097-0312(199709)50:93.0.CO;2-3 · Zbl 0886.22006 |
| [9] | Luca Capogna, Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263 – 295. · Zbl 0927.35024 · doi:10.1007/s002080050261 |
| [10] | Luca Capogna, Donatella Danielli, and Nicola Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765 – 1794. · Zbl 0802.35024 · doi:10.1080/03605309308820992 |
| [11] | Luca Capogna, Donatella Danielli, and Nicola Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153 – 1196. · Zbl 0878.35020 |
| [12] | L. Capogna, Q. Han, Pointwise Schauder estimates for second order linear equations in Carnot groups, Proceedings for the AMS-SIAM Harmonic Analysis conference in Mt. Holyhoke, 2001. · Zbl 1330.35056 |
| [13] | Giovanna Citti, Nicola Garofalo, and Ermanno Lanconelli, Harnack’s inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), no. 3, 699 – 734. · Zbl 0795.35018 · doi:10.2307/2375077 |
| [14] | G. Citti, E. Lanconelli, and A. Montanari, Smoothness of Lipchitz-continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), no. 1, 87 – 128. · Zbl 1030.35084 · doi:10.1007/BF02392796 |
| [15] | G. Citti, M. Manfredini, A. Sarti, A note on the Mumford-Shah functional in Heisenberg space, preprint. · Zbl 1198.49043 |
| [16] | E. B. Fabes and D. W. Stroock, A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal. 96 (1986), no. 4, 327 – 338. · Zbl 0652.35052 · doi:10.1007/BF00251802 |
| [17] | G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161 – 207. · Zbl 0312.35026 · doi:10.1007/BF02386204 |
| [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), no. 4, 361 – 375. Potential theory and degenerate partial differential operators (Parma). · Zbl 0841.46018 · doi:10.1007/BF01053453 |
| [19] | Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. · Zbl 0954.46022 · doi:10.1090/memo/0688 |
| [20] | Gerhard Huisken and Wilhelm Klingenberg, Flow of real hypersurfaces by the trace of the Levi form, Math. Res. Lett. 6 (1999), no. 5-6, 645 – 661. · Zbl 0956.32034 · doi:10.4310/MRL.1999.v6.n6.a5 |
| [21] | David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167 – 197. · Zbl 0661.32026 |
| [22] | S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391 – 442. · Zbl 0633.60078 |
| [23] | 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), no. 1, 165 – 189. · Zbl 0699.35025 · doi:10.2307/1971418 |
| [24] | Ermanno Lanconelli, Nonlinear equations on Carnot groups and curvature problems for CR manifolds, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 14 (2003), no. 3, 227 – 238 (2004). Renato Caccioppoli and modern analysis. · Zbl 1225.35059 |
| [25] | Ermanno Lanconelli and Alessia Elisabetta Kogoj, \?-elliptic operators and \?-control distances, Ricerche Mat. 49 (2000), no. suppl., 223 – 243. Contributions in honor of the memory of Ennio De Giorgi (Italian). · Zbl 1029.35102 |
| [26] | Ermanno Lanconelli, Andrea Pascucci, and Sergio 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, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 243 – 265. · Zbl 1032.35114 · doi:10.1007/978-1-4615-0701-7_14 |
| [27] | E. Lanconelli, F. Uguzzoni, Cone criterion for non-divergence equations modeled on Hörmander vector fields, preprint. · Zbl 1165.35007 |
| [28] | Guozhen Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Rev. Mat. Iberoamericana 8 (1992), no. 3, 367 – 439. · Zbl 0804.35015 · doi:10.4171/RMI/129 |
| [29] | Guozhen Lu, Existence and size estimates for the Green’s functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), no. 7-8, 1213 – 1251. · Zbl 0798.35002 · doi:10.1080/03605309208820883 |
| [30] | Guozhen Lu, On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields, Differential Integral Equations 7 (1994), no. 1, 73 – 100. · Zbl 0827.35032 |
| [31] | A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, Comm. Partial Differential Equations 26 (2001), no. 9-10, 1633 – 1664. · Zbl 1019.35056 · doi:10.1081/PDE-100107454 |
| [32] | Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. · Zbl 1044.53022 |
| [33] | Jean Petitot and Yannick Tondut, Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Math. Inform. Sci. Humaines 145 (1999), 5 – 101 (French, with English and French summaries). |
| [34] | Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247 – 320. · Zbl 0346.35030 · doi:10.1007/BF02392419 |
| [35] | L. Saloff-Coste and D. W. Stroock, Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal. 98 (1991), no. 1, 97 – 121 (French, with English summary). · Zbl 0734.58041 · doi:10.1016/0022-1236(91)90092-J |
| [36] | Zbigniew Slodkowski and Giuseppe Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), no. 2, 392 – 407. · Zbl 0744.35015 · doi:10.1016/0022-1236(91)90164-Z |
| [37] | Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001 |
| [38] | N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. · Zbl 0813.22003 |
| [39] | Chao Jiang Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), no. 1, 77 – 96. · Zbl 0827.35023 · doi:10.1002/cpa.3160450104 |
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