×

On the relation between elliptic and parabolic Harnack inequalities. (English) Zbl 0988.58007

The present work deals with Harnack inequalities on manifolds. The authors show that if \((M,g)\) is a complete Riemannian manifold satisfying the scale-invariant local Sobolev inequality and the scale-invariant elliptic Harnack inequality, then \((M,g)\) satisfies the scale-invariant parabolic Harnack inequality. Besides that the authors consider non-classical parabolic Harnack inequalities where the time-space scaling \((t^2,t)\) is replaced by a more general one including scaling of the type \((t^\omega,t)\) for large \(t\) with \(\omega>2.\) They also establish an equivalence between such parabolic Harnack inequalities and certain non-classical two-sided Gaussian estimates of the heat kernel.

MSC:

58J05 Elliptic equations on manifolds, general theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds
31C25 Dirichlet forms
58J65 Diffusion processes and stochastic analysis on manifolds
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI Numdam Numdam EuDML

References:

[1] Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115 (1982) · Zbl 0503.32007
[2] Gaussian bounds for random walks from elliptic regularity, Ann. Inst. Henri Poincaré, Prob. Stat., 35, 605-630 (1999) · Zbl 0933.60047 · doi:10.1016/S0246-0203(99)00109-0
[3] Sobolev Inequalities in Disguise, Indiana Univ. Math. J., 44, 1033-1073 (1995) · Zbl 0857.26006
[4] Diffusions on fractals, Lectures in Probability Theory and Statistics Ecole d’été de Probabilités de Saint Flour XXV– 1995, 1690, 1-121 (1998) · Zbl 0916.60069
[5] Transition densities for Brownian motion on the Sierpinski carpet, Probab. Th. Rel. Fields, 91, 307-330 (1992) · Zbl 0739.60071 · doi:10.1007/BF01192060
[6] Random walks on graphical Sierpinski carpets, 39 (1999) · Zbl 0958.60045
[7] On and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces, American J. Math., 122, 1205-1263 (2000) · Zbl 0969.31008 · doi:10.1353/ajm.2000.0043
[8] Markov Processes and Potential Theory (1968) · Zbl 0169.49204
[9] Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, 1, 205-232 (1996) · Zbl 0884.58088
[10] Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., 17, 15-53 (1982) · Zbl 0493.53035
[11] On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., 89, 133-199 (1997) · Zbl 0920.58064 · doi:10.1215/S0012-7094-97-08908-0
[12] Variétés riemanniennes isométriques à l’infini, Rev. Mat. Iberoamericana, 11, 687-726 (1995) · Zbl 0845.58054 · doi:10.4171/RMI/190
[13] Heat kernels and spectral theory (1989) · Zbl 0699.35006
[14] Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math, 58, 99-119 (1992) · Zbl 0808.58041 · doi:10.1007/BF02790359
[15] Non-Gaussian aspects of Heat kernel behaviour, J. London Math. Soc., 55, 105-125 (1997) · Zbl 0879.35064 · doi:10.1112/S0024610796004607
[16] Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, 15, 181-232 (1999) · Zbl 0922.60060 · doi:10.4171/RMI/254
[17] Elliptic and parabolic Harnack inequalities · Zbl 1081.39012
[18] A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash, Arch. Rat, Mech. Anal., 96, 327-338 (1986) · Zbl 0652.35052
[19] Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. in Partial Differential Equations, 19, 523-604 (1994) · Zbl 0822.46032 · doi:10.1080/03605309408821025
[20] Dirichlet forms and Symmetric Markov processes (1994) · Zbl 0838.31001
[21] The heat equation on non-compact Riemannian manifolds, 182, 55-87 (1991) · Zbl 0743.58031
[22] The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. (Engl. Transl.), 72, 47-77 (1992) · Zbl 0776.58035 · doi:10.1070/SM1992v072n01ABEH001410
[23] Heat kernel upper bounds on a complete non-compact Riemannian manifold, Revista Mat. Iberoamericana, 10, 395-452 (1994) · Zbl 0810.58040 · doi:10.4171/RMI/157
[24] Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geometry, 45, 33-52 (1997) · Zbl 0865.58042
[25] Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. A.M.S, 36, 135-249 (1999) · Zbl 0927.58019 · doi:10.1090/S0273-0979-99-00776-4
[26] Estimates of heat kernels on Riemannian manifolds, Spectral Theory and Geometry, 273 (1999) · Zbl 0985.58007
[27] Heat kernel on connected sums of Riemannian manifolds, Mathematical Research Letters, 6, 1-14 (1999) · Zbl 0957.58023
[28] Sub-Gaussian estimates of heat kernels on infinite graphs (2000) · Zbl 1010.35016
[29] Metric structures for Riemannian and non-Riemannian spaces (1998) · Zbl 1113.53001
[30] The Poincaré inequality for vector fields satisfying the Hörmander’s condition, Duke Math. J., 53, 503-523 (1986) · Zbl 0614.35066
[31] A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izs, 16, 151-164 (1981) · Zbl 0464.35035 · doi:10.1070/IM1981v016n01ABEH001283
[32] Applications of Malliavin Calculus, Part 3, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 34, 391-442 (1987) · Zbl 0633.60078
[33] Counterexamples to Liouville-type theorems, 6, 39-43 (1976) · Zbl 0416.35033
[34] Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull. (Engl. Transl.), 34, 35-39 (1979) · Zbl 0442.35038
[35] On the parabolic kernel of Schrödinger operator, Acta Math., 156, 153-201 (1986) · Zbl 0611.58045 · doi:10.1007/BF02399203
[36] On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, 577-591 (1961) · Zbl 0111.09302 · doi:10.1002/cpa.3160140329
[37] A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 16 ; 20, 101-134 ; 231-236 (19641967) · Zbl 0149.06902 · doi:10.1002/cpa.3160170106
[38] On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math., 24, 727-740 (1971) · Zbl 0227.35016 · doi:10.1002/cpa.3160240507
[39] Harnack’s inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., 21, 851-863 (1983) · Zbl 0511.35029 · doi:10.1007/BF01094448
[40] Analyse sur les groupes à croissance polynomiale, Ark. för Mat., 28, 315-331 (1990) · Zbl 0715.43009 · doi:10.1007/BF02387385
[41] Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal., 98, 97-121 (1991) · Zbl 0734.58041 · doi:10.1016/0022-1236(91)90092-J
[42] Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., 36, 417-450 (1992) · Zbl 0735.58032
[43] A note on Poincaré, Sobolev and Harnack inequalities, Duke Math. J., IMRN, 2, 27-38 (1992) · Zbl 0769.58054
[44] Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, 4, 429-467 (1995) · Zbl 0840.31006 · doi:10.1007/BF01053457
[45] Aspects of Sobolev type inequalities (2001) · Zbl 0991.35002
[46] On the geometry defined by Dirichlet forms, Seminar on Stochastic Processes, Random Fields and Applications, Ascona, vol. 36, 231-242 (1995) · Zbl 0834.58039
[47] Analysis on local Dirichlet spaces I: Recurrence, conservativeness and \(L^p\)-Liouville properties, J. Reine Angew. Math., 456, 173-196 (1994) · Zbl 0806.53041 · doi:10.1515/crll.1994.456.173
[48] Analysis on local Dirichlet spaces II. Upper Gaussian estimates for fundamental solutions of parabolic equations, Osaka J. Math., 32, 275-312 (1995) · Zbl 0854.35015
[49] Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl., 75, 273-297 (1996) · Zbl 0854.35016
[50] Local sub-Gaussian estimates of heat kernels on graphs, the strongly recurrent cases (2000)
[51] Fonctions harmoniques sur les groupes de Lie, CR. Acad. Sci. Paris, Sér. I Math., 304, 519-521 (1987) · Zbl 0614.22002
[52] Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semigroup technique, Bull. Sci. Math., 113, 253-277 (1989) · Zbl 0703.58052
[53] Analysis and geometry on groups (1993) · Zbl 0813.22003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.