Stability results for Harnack inequalities
Annales de l'Institut Fourier, Volume 55 (2005) no. 3, p. 825-890
We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain non-uniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete non-compact manifolds having non-negative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically non-negative sectional curvature.
Nous développons de nouvelles techniques pour obtenir des inégalités de Harnack uniformes elliptiques et paraboliques sur les variétés riemanniennes à poids. Nous démontrons en particulier la stabilité de ces inégalités pour certains changements de poids. Nous donnons une condition nécessaire et suffisante pour ces inégalités dans le cas des variétés riemanniennes complètes à courbure de Ricci positive ou nulle en dehors d'un compact et dont le premier nombre de Betti est fini, ou sous la condition de courbure sectionnelle asymptotiquement positive ou nulle.
DOI : https://doi.org/10.5802/aif.2116
Classification:  58J35,  31C12
Keywords: Harnack inequality, Riemannian manifold, heat equation
@article{AIF_2005__55_3_825_0,
     author = {Grigor'yan, Alexander and Saloff-Coste, Laurent},
     title = {Stability results for Harnack inequalities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     pages = {825-890},
     doi = {10.5802/aif.2116},
     mrnumber = {2149405},
     zbl = {02171527},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_3_825_0}
}
Stability results for Harnack inequalities. Annales de l'Institut Fourier, Volume 55 (2005) no. 3, pp. 825-890. doi : 10.5802/aif.2116. https://aif.centre-mersenne.org/item/AIF_2005__55_3_825_0/

[1] D.G. Aronson Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., Tome 73 (1967), pp. 890-896 | Article | MR 217444 | Zbl 0153.42002

[2] M.T. Barlow; R.F. Bass Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., Tome 54 (1999), pp. 673-744 | MR 1701339 | Zbl 0945.60071

[3] M.T. Barlow; R.F. Bass; M. Picardello, W.Woess Random walks on graphical Sierpinski carpets, Random walks and discrete potential theory (Cortona, Italy, 1997), Cambridge Univ. Press, Cambridge (Symposia Math.) Tome 39 (1999), pp. 26-55 | Zbl 0958.60045

[4] P. Buser A note on the isoperimetric constant, Ann. Sci. École Norm. Sup., Tome 15 (1982), pp. 213-230 | Numdam | MR 683635 | Zbl 0501.53030

[5] M. Cai Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc., Tome 24 (1991), pp. 371-377 | Article | MR 1071028 | Zbl 0728.53026

[6] I. Chavel Eigenvalues in Riemannian geometry, Academic Press, New York (1984) | MR 768584 | Zbl 0551.53001

[7] J. Cheeger; M. Gromov; M. Taylor Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., Tome 17 (1982), pp. 15-53 | MR 658471 | Zbl 0493.53035

[8] J. Cheeger; S.-T. Yau A lower bound for the heat kernel, Comm. Pure Appl. Math., Tome 34 (1981), pp. 465-480 | Article | MR 615626 | Zbl 0481.35003

[9] S.Y. Cheng; S.-T. Yau Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., Tome 28 (1975), pp. 333-354 | Article | MR 385749 | Zbl 0312.53031

[10] F.R.K. Chung Spectral Graph Theory, Amer. Math. Soc. Publications, CBMS Regional Conference Series in Mathematics, Tome 92 (1996) | Zbl 0867.05046

[11] T. Coulhon; L. Saloff-Coste Variétés riemanniennes isométriques à l'infini, Revista Matematica Iberoamericana, Tome 11 (1995) no. 3, pp. 687-726 | MR 1363211 | Zbl 0845.58054

[12] T. Delmotte Graphs between elliptic and parabolic Harnack inequalities, Potential Analysis, Tome 16 (2000) no. 2, pp. 151-168 | MR 1881595 | Zbl 01723365

[13] P. Diaconis; L. Saloff-Coste What do we know about the Metropolis Algorithm?, J. Computer and System Sciences, Tome 57 (1998), pp. 20-36 | Article | MR 1649805 | Zbl 0920.68054

[14] J. Dodziuk Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., Tome 32 (1983) no. 5, pp. 703-716 | Article | MR 711862 | Zbl 0526.58047

[15] E.B. Fabes; D.W. Stroock A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash, Arch. Rat. Mech. Anal., Tome 96 (1986), pp. 327-338 | MR 855753 | Zbl 0652.35052

[16] R. Greene; W. Wu Function theory of manifolds which possess a pole, Springer, Lecture Notes Math., Tome 699 (1979) | MR 521983 | Zbl 0414.53043

[17] A. Grigor'Yan The heat equation on non-compact Riemannian manifolds (Russian), Mat. Sbornik, Tome 182 (1991) no. 1, pp. 55-87 | Zbl 0743.58031

[18] A. Grigor'Yan Heat kernel upper bounds on a complete non-compact manifold, Revista Matematica Iberoamericana, Tome 10 (1994) no. 2, pp. 395-452 | MR 1286481 | Zbl 0810.58040

[19] A. Grigor'Yan Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., Tome 36 (1999), pp. 135-249 | Article | MR 1659871 | Zbl 0927.58019

[20] A. Grigor'Yan ; L. Saloff-Coste Surgery of Faber-Krahn inequalities and applications to heat kernel upper bounds on manifolds with ends (2000) (in preparation)

[21] A. Grigor'Yan; L. Saloff-Coste Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math., Tome 55 (2002), pp. 93-133 | Article | MR 1857881 | Zbl 1037.58018

[22] A. Grigor'Yan; L. Saloff-Coste Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures et Appl., Tome 81 (2002), pp. 115-142 | Article | MR 1994606 | Zbl 1042.58022

[23] M. Gromov Structures métriques pour les variétés Riemannienes, Cedic/Ferdnand Nathan, Paris (1981) | MR 682063 | Zbl 0509.53034

[24] P. Hajlasz; P. Koskela Sobolev Met Poincaré, Memoirs of the AMS Tome 688 (2000) | MR 1683160 | Zbl 0954.46022

[25] W. Hebisch; L. Saloff-Coste On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier, Tome 51 (2001) no. 5, pp. 1437-1481 | Article | Numdam | MR 1860672 | Zbl 0988.58007

[26] D. Jerison The Poincaré inequality for vector fields satisfying Hörmander condition, Duke Math. J., Tome 53 (1986), pp. 503-523 | MR 850547 | Zbl 0614.35066

[27] M. Kanai Analytic inequalities, and rough isometries between non-compact Riemannian manifolds, Springer (Lecture Notes Math.) Tome 1201 (1986), pp. 122-137 | Zbl 0593.53026

[28] A. Kasue Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I., Geometry and Analysis on Manifolds (Katata/Kyoto, 1987), Springer (Lecture Notes Math.) (1988), pp. 158-181 | Zbl 0685.31004

[29] J.L. Kazdan; F.W. Warner Prescribing curvatures, Proceedings of Symposia in Pure Mathematics, Tome 27 (1975) no. 2, pp. 309-319 | MR 394505 | Zbl 0313.53017

[30] S. Kusuoka; D. Stroock Application of Malliavin calculus, III, J. Fac. Sci. Tokyo Univ., Sect. 1A, Math., Tome 34 (1987), pp. 391-442 | MR 914028 | Zbl 0633.60078

[31] E.M. Landis The second order equations of elliptic and parabolic type (Russian), Nauka, Moscow (1971) | Zbl 0226.35001

[32] P. Li; L.F. Tam Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. Math., Tome 125 (1987), pp. 171-207 | Article | MR 873381 | Zbl 0622.58001

[33] P. Li; L.F. Tam Green's function, harmonic functions and volume comparison, J. Diff. Geom., Tome 41 (1995), pp. 277-318 | Zbl 0827.53033

[34] P. Li; S.-T. Yau On the parabolic kernel of the Schrödinger operator, Acta Math., Tome 156 (1986) no. 3,4, pp. 153-201 | MR 834612 | Zbl 0611.58045

[35] Z.-D. Liu Ball covering property and nonnegative Ricci curvature outside a compact set, Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Amer. Math. Soc., Providence, RI (Proceedings of Symposia in Pure Mathematics) Tome 54, Part 3 (1993), pp. 459–464 | Zbl 0788.53028

[36] A.G. Losev Some Liouville theorems on Riemannian manifolds of a special type (Russian), Izv. Vyssh. Uchebn. Zaved. Matematika, Tome 12 (1991), pp. 15-24 | MR 1205018 | Zbl 0764.58035

[37] G. Lu Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Revista Matematica Iberoamericana, Tome 8 (1992) no. 3, pp. 367-439 | MR 1202416 | Zbl 0804.35015

[38] P. March Brownian motion and harmonic functions on rotationally symmetric manifolds, Ann. Prob., Tome 14 (1986) no. 3, pp. 793-801 | Article | MR 841584 | Zbl 0593.60078

[39] J. Moser On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., Tome 14 (1961), pp. 577-591 | Article | MR 159138 | Zbl 0111.09302

[40] J. Moser A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., Tome 17 (1964), pp. 101-134 | Article | MR 159139 | Zbl 0149.06902

[41] M. Murata; M. Kishi Positive harmonic functions on rotationary symmetric Riemannian manifolds, Potential Theory, Walter de Gruyter, Berlin (1992), pp. 251-259 | Zbl 0777.53039

[42] F.O. Porper; S.D. Eidel'Man Two-side estimates of fundamental solutions of second-order parabolic equations and some applications (Russian), Uspekhi Matem. Nauk, Tome 39 (1984) no. 3, pp. 101-156 | MR 747792 | Zbl 0582.35052

[43] L. Saloff-Coste A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, Tome 2 (1992), pp. 27-38 | MR 1150597 | Zbl 0769.58054

[44] L. Saloff-Coste Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Tome 4 (1995), pp. 429-467 | Article | MR 1354894 | Zbl 0840.31006

[45] L. Saloff-Coste Lectures on finite Markov chains,, Springer, Lecture Notes Math. (1997) | MR 1490046 | Zbl 0885.60061

[46] L. Saloff-Coste Aspects of Sobolev inequalities, Cambridge Univ. Press (London Math. Soc. Lecture Notes Series) Tome 289 (2002) | Zbl 0991.35002

[47] K.-Th. Sturm Sharp estimates for capacities and applications to symmetrical diffusions, Probability theory and related fields, Tome 103 (1995) no. 1, pp. 73-89 | Article | MR 1347171 | Zbl 0828.60062

[48] C.-J. Sung; L.-F. Tam; J. Wang Spaces of harmonic functions, J. London Math. Soc., Tome 2 (2000) no. 3, pp. 789-806 | MR 1766105 | Zbl 0963.31004

[49] S.-T. Yau Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., Tome 28 (1975), pp. 201-228 | Article | MR 431040 | Zbl 0291.31002