Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators

Pascucci, Andrea; Polidoro, Sergio

doi:10.1090/S0002-9947-06-04050-5

Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
HTML articles powered by AMS MathViewer

by Andrea Pascucci and Sergio Polidoro PDF

Trans. Amer. Math. Soc. 358 (2006), 4873-4893 Request permission

Abstract:

We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.

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
D. G. Aronson and James Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal. 25 (1967), 81–122. MR 244638, DOI 10.1007/BF00281291
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
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
David S. Jerison and Antonio 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 865430, DOI 10.1512/iumj.1986.35.35043
Velimir Jurdjevic, Geometric control theory, Cambridge Studies in Advanced Mathematics, vol. 52, Cambridge University Press, Cambridge, 1997. MR 1425878
Alessia Elisabetta Kogoj and Ermanno Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math. 1 (2004), no. 1, 51–80. MR 2088032, DOI 10.1007/s00009-004-0004-8
Alessia Elisabetta Kogoj and Ermanno Lanconelli, One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations, Geometric analysis of PDE and several complex variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 305–312. MR 2126477, DOI 10.1090/conm/368/06786
L. P. Kupcov, The fundamental solutions of a certain class of elliptic-parabolic second order equations, Differencial′nye Uravnenija 8 (1972), 1649–1660, 1716 (Russian). MR 0315290
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
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29–63. Partial differential equations, II (Turin, 1993). MR 1289901
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
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Andrea Pascucci and Sergio Polidoro, On the Harnack inequality for a class of hypoelliptic evolution equations, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4383–4394. MR 2067125, DOI 10.1090/S0002-9947-04-03407-5
Andrea Pascucci and Sergio Polidoro, A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations, J. Math. Anal. Appl. 282 (2003), no. 1, 396–409. MR 2000352, DOI 10.1016/S0022-247X(03)00159-8
Sergio Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania) 49 (1994), no. 1, 53–105 (1995). MR 1386366
Sergio Polidoro, A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations, Arch. Rational Mech. Anal. 137 (1997), no. 4, 321–340. MR 1463798, DOI 10.1007/s002050050031
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
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