Harnack inequality for non-divergence form operators on stratified groups

Bonfiglioli, Andrea; Uguzzoni, Francesco

doi:10.1090/S0002-9947-07-04273-0

Harnack inequality for non-divergence form operators on stratified groups
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by Andrea Bonfiglioli and Francesco Uguzzoni PDF

Trans. Amer. Math. Soc. 359 (2007), 2463-2481 Request permission

Abstract:

We prove lower bounds for the fundamental solutions of the non-divergence form operators \[ {\textstyle \sum _{i,j}} a_{i,j}(x,t) X_iX_j-\partial _t \quad \text {and}\quad {\textstyle \sum _{i,j}}a_{i,j}(x) X_iX_j,\] 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 then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.

References

Georgios K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc. 155 (2002), no. 739, x+101. MR 1878341, DOI 10.1090/memo/0739
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. MR 1919700
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. MR 2052194, DOI 10.1090/S0002-9947-03-03332-4
Andrea Bonfiglioli and Francesco Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, Forum Math. 16 (2004), no. 3, 403–415. MR 2050190, DOI 10.1515/form.2004.018
A. Bonfiglioli, F. Uguzzoni, A note on lifting of Carnot groups, Rev. Mat. Iberoamericana 21 (2005), 1013–1035.
Marco Bramanti and Luca Brandolini, $L^p$ estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781–822. MR 1608289, DOI 10.1090/S0002-9947-99-02318-1
M. Bramanti and L. Brandolini, $L^p$ estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, Rend. Sem. Mat. Univ. Politec. Torino 58 (2000), no. 4, 389–433 (2003). MR 1962808
Luca Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867–889. MR 1459590, DOI 10.1002/(SICI)1097-0312(199709)50:9<867::AID-CPA3>3.0.CO;2-3
Luca Capogna, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263–295. MR 1679786, DOI 10.1007/s002080050261
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. MR 1239930, DOI 10.1080/03605309308820992
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. MR 1420920, DOI 10.1353/ajm.1996.0046
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.
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. MR 1221840, DOI 10.2307/2375077
G. Citti, E. Lanconelli, and A. Montanari, Smoothness of Lipchitz-continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), no. 1, 87–128. MR 1947459, DOI 10.1007/BF02392796
G. Citti, M. Manfredini, A. Sarti, A note on the Mumford-Shah functional in Heisenberg space, preprint.
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. MR 855753, DOI 10.1007/BF00251802
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
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). MR 1354890, DOI 10.1007/BF01053453
Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
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. MR 1739222, DOI 10.4310/MRL.1999.v6.n6.a5
David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
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. MR 914028
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. MR 924675, DOI 10.2307/1971418
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. MR 2064269
Ermanno Lanconelli and Alessia Elisabetta Kogoj, $X$-elliptic operators and $X$-control distances, Ricerche Mat. 49 (2000), no. suppl., 223–243. Contributions in honor of the memory of Ennio De Giorgi (Italian). MR 1826225
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. MR 1972000, DOI 10.1007/978-1-4615-0701-7_{1}4
E. Lanconelli, F. Uguzzoni, Cone criterion for non-divergence equations modeled on Hörmander vector fields, preprint.
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. MR 1202416, DOI 10.4171/RMI/129
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. MR 1179284, DOI 10.1080/03605309208820883
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. MR 1250940
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. MR 1865940, DOI 10.1081/PDE-100107454
Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
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). MR 1697185
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
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). MR 1111195, DOI 10.1016/0022-1236(91)90092-J
Zbigniew Slodkowski and Giuseppe Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), no. 2, 392–407. MR 1136942, DOI 10.1016/0022-1236(91)90164-Z
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. MR 1232192
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. MR 1218884
Chao Jiang Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), no. 1, 77–96. MR 1135924, DOI 10.1002/cpa.3160450104