On the relation between elliptic and parabolic Harnack inequalities

Hebisch, Waldemar; Saloff-Coste, Laurent

Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1437-1481.

Sous l’hypothèse qu’une certaine inégalité de Sobolev est satisfaite, nous montrons qu’une inégalité de Harnack elliptique uniforme implique sa version parabolique. Ni l’inégalité de Sobolev ni l’inégalité de Harnack elliptique, n’implique à elle seule l’inégalité de Harnack parabolique en question; chacune est une condition nécessaire. En conséquence, nous obtenons l’équivalence entre l’inégalité de Harnack parabolique pour le laplacien sur une variété riemannienne $M$, (i.e., pour ${\partial }_t+\Delta$) et l’inégalité de Harnack elliptique pour $-{\partial }_t^2+\Delta$ sur $ℝ\times M$.

We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suffices to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suffices to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for $\Delta$ on $M$, (i.e., for ${\partial }_t+\Delta$) and elliptic Harnack inequality for $-{\partial }_t^2+\Delta$ on $ℝ\times M$.

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Volume 115 (1982) | MR | Zbl

[1] P. Auscher; T. Coulhon Gaussian bounds for random walks from elliptic regularity, Ann. Inst. Henri Poincaré, Prob. Stat., Volume 35 (1999), pp. 605-630 | DOI | Numdam | MR | Zbl

[2] D. Bakry; T. Coulhon; M. Ledoux; L. Saloff-Coste Sobolev Inequalities in Disguise, Indiana Univ. Math. J., Volume 44 (1995), pp. 1033-1073 | MR | Zbl

[3] M. Barlow Diffusions on fractals, Lectures in Probability Theory and Statistics Ecole d'été de Probabilités de Saint Flour XXV-- 1995 (Lecture Notes in Math.), Volume 1690 (1998), pp. 1-121 | Zbl

[4] M. Barlow; R. Bass Transition densities for Brownian motion on the Sierpinski carpet, Probab. Th. Rel. Fields, Volume 91 (1992), pp. 307-330 | DOI | MR | Zbl

[5] M. Barlow; R. Bass Random walks on graphical Sierpinski carpets, Symposia Mathematica, 39, Cambridge University Press, 1999 | MR | Zbl

[6] A. Bendikov; L. Saloff-Coste On and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces, American J. Math., Volume 122 (2000), pp. 1205-1263 | DOI | MR | Zbl

[7] R. Blumental; R. Getoor Markov Processes and Potential Theory, Academic Press, New York and London, 1968 | MR | Zbl

[8] G. Carron Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Volume 1 (1996), pp. 205-232 | Zbl

[9] 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., Volume 17 (1982), pp. 15-53 | MR | Zbl

[10] T. Coulhon; A. Grigor'yan On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., Volume 89 (1997), pp. 133-199 | DOI | MR | Zbl

[11] T. Coulhon; L. Saloff-Coste Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana, Volume 11 (1995), pp. 687-726 | DOI | MR | Zbl

[12] E.B. Davies Heat kernels and spectral theory, Cambridge University Press, 1989 | MR | Zbl

[13] E.B. Davies Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math, Volume 58 (1992), pp. 99-119 | DOI | MR | Zbl

[14] E.B. Davies Non-Gaussian aspects of Heat kernel behaviour, J. London Math. Soc., Volume 55 (1997), pp. 105-125 | DOI | MR | Zbl

[15] T. Delmotte Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, Volume 15 (1999), pp. 181-232 | DOI | MR | Zbl

[16] T. Delmotte Elliptic and parabolic Harnack inequalities (Potential Analysis, to appear) | MR | Zbl

[17] E. Fabes; D. Stroock A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rat, Mech. Anal., Volume 96 (1986), pp. 327-338 | MR | Zbl

[18] B. Franchi; C. Gutiérrez; R. Wheeden Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. in Partial Differential Equations, Volume 19 (1994), pp. 523-604 | DOI | MR | Zbl

[19] M. Fukushima; Y. Oshima; M. Takeda Dirichlet forms and Symmetric Markov processes, W. de Gruyter, 1994 | MR | Zbl

[20] A. Grigor'yan The heat equation on non-compact Riemannian manifolds (Matem. Sbornik), Volume 182 (1991), pp. 55-87 | Zbl

[20] A. Grigor'yan The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. (Engl. Transl.), Volume 72 (1992), pp. 47-77 | DOI | MR | Zbl

[21] A. Grigor'yan Heat kernel upper bounds on a complete non-compact Riemannian manifold, Revista Mat. Iberoamericana, Volume 10 (1994), pp. 395-452 | DOI | MR | Zbl

[22] A. Grigor'yan Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geometry, Volume 45 (1997), pp. 33-52 | MR | Zbl

[23] A. Grigor'yan Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. A.M.S, Volume 36 (1999), pp. 135-249 | DOI | MR | Zbl

[24] A. Grigor'yan; E.B. Davies and Y. Sasarov, Eds Estimates of heat kernels on Riemannian manifolds, Spectral Theory and Geometry (London Math. Soc. Lecture Note Series), Volume 273 (1999) | Zbl

[25] A. Grigor'yan; L. Saloff-Coste Heat kernel on connected sums of Riemannian manifolds, Mathematical Research Letters, Volume 6 (1999), pp. 1-14 | MR | Zbl

[26] A. Grigor'yan; A. Telcs Sub-Gaussian estimates of heat kernels on infinite graphs (2000) (Preprint) | MR | Zbl

[27] M. Gromov Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, 1998 | MR | Zbl

[28] D. Jerison The Poincaré inequality for vector fields satisfying the Hörmander's condition, Duke Math. J., Volume 53 (1986), pp. 503-523 | MR | Zbl

[29] N. Krylov; M. Safonov A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izs, Volume 16 (1981), pp. 151-164 | DOI | Zbl

[30] S. Kusuoka; D. Stroock Applications of Malliavin Calculus, Part 3, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Volume 34 (1987), pp. 391-442 | MR | Zbl

[31] Y.T. Kuzmenko; S.A. Molchanov Counterexamples to Liouville-type theorems (Vestnik Moskov. Univ., Ser. I Mat. Mekh.), Volume 6 (1976), pp. 39-43 | Zbl

[31] Y.T. Kuzmenko; S.A. Molchanov Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull. (Engl. Transl.), Volume 34 (1979), pp. 35-39 | Zbl

[32] P. Li; S-T Yau On the parabolic kernel of Schrödinger operator, Acta Math., Volume 156 (1986), pp. 153-201 | DOI | MR | Zbl

[33] J. Moser On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., Volume 14 (1961), pp. 577-591 | DOI | MR | Zbl

[34] J. Moser A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., Volume 16 ; 20 (1964 ; 1967), p. 101-134 ; 231--236 | DOI | MR | Zbl

[35] J. Moser On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math., Volume 24 (1971), pp. 727-740 | DOI | MR | Zbl

[36] M. Safonov Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., Volume 21 (1983), pp. 851-863 | DOI | Zbl

[37] L. Saloff-Coste Analyse sur les groupes à croissance polynomiale, Ark. för Mat., Volume 28 (1990), pp. 315-331 | DOI | MR | Zbl

[38] L. Saloff-Coste; D. Stroock Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal., Volume 98 (1991), pp. 97-121 | DOI | MR | Zbl

[39] L. Saloff-Coste Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Volume 36 (1992), pp. 417-450 | MR | Zbl

[40] L. Saloff-Coste A note on Poincaré, Sobolev and Harnack inequalities, Duke Math. J., IMRN, Volume 2 (1992), pp. 27-38 | MR | Zbl

[41] L. Saloff-Coste Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Volume 4 (1995), pp. 429-467 | DOI | MR | Zbl

[42] L. Saloff-Coste Aspects of Sobolev type inequalities (2001) (To appear in London Math. Soc. Lecture Notes Series, Cambridge University Press) | MR | Zbl

[43] K-T. Sturm; E. Bolthausen et al. Ed. On the geometry defined by Dirichlet forms, Seminar on Stochastic Processes, Random Fields and Applications, Ascona (Progress in Probability), Volume vol. 36 (1995), pp. 231-242 | Zbl

[44] K-T. Sturm Analysis on local Dirichlet spaces I: Recurrence, conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math., Volume 456 (1994), pp. 173-196 | DOI | MR | Zbl

[45] K-T. Sturm Analysis on local Dirichlet spaces II. Upper Gaussian estimates for fundamental solutions of parabolic equations, Osaka J. Math., Volume 32 (1995), pp. 275-312 | MR | Zbl

[46] K-T. Sturm Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl., Volume 75 (1996), pp. 273-297 | MR | Zbl

[47] A. Telcs Local sub-Gaussian estimates of heat kernels on graphs, the strongly recurrent cases (2000) (Preprint)

[48] N. Varopoulos Fonctions harmoniques sur les groupes de Lie, CR. Acad. Sci. Paris, Sér. I Math., Volume 304 (1987), pp. 519-521 | MR | Zbl

[49] N. Varopoulos Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semigroup technique, Bull. Sci. Math., Volume 113 (1989), pp. 253-277 | MR | Zbl

[50] N. Varopoulos; L. Saloff-Coste; T. Coulhon Analysis and geometry on groups, Cambridge University Press, 1993 | MR | Zbl