On the Harnack inequality for a class of hypoelliptic evolution equations

Pascucci, Andrea; Polidoro, Sergio

doi:10.1090/S0002-9947-04-03407-5

On the Harnack inequality for a class of hypoelliptic evolution equations
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by Andrea Pascucci and Sergio Polidoro

Trans. Amer. Math. Soc. 356 (2004), 4383-4394

DOI: https://doi.org/10.1090/S0002-9947-04-03407-5

Published electronically: January 16, 2004

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Abstract:

We give a direct proof of the Harnack inequality for a class of degenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find explicitly the optimal constant of the inequality.

References

Giles Auchmuty and David Bao, Harnack-type inequalities for evolution equations, Proc. Amer. Math. Soc. 122 (1994), no. 1, 117–129. MR 1219716, DOI 10.1090/S0002-9939-1994-1219716-X
E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci. 11 (2001), no. 3, 475–497. MR 1830951, DOI 10.1142/S0218202501000945
Huai Dong Cao and Shing-Tung Yau, Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields, Math. Z. 211 (1992), no. 3, 485–504. MR 1190224, DOI 10.1007/BF02571441
J. J. Corliss, Upper limits to the real roots of a real algebraic equation, Amer. Math. Monthly 46 (1939), 334–338. MR 4
Sydney Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases; In co-operation with D. Burnett; With a foreword by Carlo Cercignani. MR 1148892
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
James J. Duderstadt and William R. Martin, Transport theory, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. MR 551868
Nicola Garofalo and Ermanno Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc. 321 (1990), no. 2, 775–792. MR 998126, DOI 10.1090/S0002-9947-1990-0998126-5
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
S. D. Īvasishen and O. G. Voznyak, On the fundamental solutions of the Cauchy problem for a class of degenerate parabolic equations, Mat. Metodi Fiz.-Mekh. Polya 41 (1998), no. 2, 13–19 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., J. Math. Sci. (New York) 99 (2000), no. 5, 1533–1540. MR 1710034, DOI 10.1007/BF02674176
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
L. P. Kupcov, On parabolic means, Dokl. Akad. Nauk SSSR 252 (1980), no. 2, 296–301 (Russian). MR 571952
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
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
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
A. Pascucci, Hölder regularity for a Kolmogorov equation, Trans. Amer. Math. Soc., 355 (2003), pp. 901–924.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Sergio Polidoro, Uniqueness and representation theorems for solutions of Kolmogorov-Fokker-Planck equations, Rend. Mat. Appl. (7) 15 (1995), no. 4, 535–560 (1996) (English, with English and Italian summaries). MR 1387312
I. M. Sonin, A class of degenerate diffusion processes, Teor. Verojatnost. i Primenen. 12 (1967), 540–547 (Russian, with English summary). MR 0215367