On the relation between elliptic and parabolic Harnack inequalities  [ Sur les liens entre inégalités de Harnack elliptiques et paraboliques ]
Annales de l'Institut Fourier, Tome 51 (2001) no. 5, p. 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 t +Δ) et l’inégalité de Harnack elliptique pour - t 2 +Δ sur ×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 Δ on M, (i.e., for t +Δ) and elliptic Harnack inequality for - t 2 +Δ on ×M.
DOI : https://doi.org/10.5802/aif.1861
Classification:  58J05,  58J35,  31C25,  58J65,  60J65
Mots clés: équation de Laplace, équation de la chaleur, inégalité de Harnack, espaces de Dirichlet, bornes gaussiennes
@article{AIF_2001__51_5_1437_0,
     author = {Hebisch, Waldemar and Saloff-Coste, Laurent},
     title = {On the relation between elliptic and parabolic Harnack inequalities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {5},
     year = {2001},
     pages = {1437-1481},
     doi = {10.5802/aif.1861},
     zbl = {0988.58007},
     mrnumber = {1860672},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_5_1437_0}
}
Hebisch, Waldemar; Saloff-Coste, Laurent. On the relation between elliptic and parabolic Harnack inequalities. Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1437-1481. doi : 10.5802/aif.1861. http://www.numdam.org/item/AIF_2001__51_5_1437_0/

[1] C. Camacho; P. Sad Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Tome 115 (1982) | MR 657239 | Zbl 0503.32007

[1] P. Auscher; T. Coulhon Gaussian bounds for random walks from elliptic regularity, Ann. Inst. Henri Poincaré, Prob. Stat., Tome 35 (1999), pp. 605-630 | Article | Numdam | MR 1705682 | Zbl 0933.60047

[2] D. Bakry; T. Coulhon; M. Ledoux; L. Saloff-Coste Sobolev Inequalities in Disguise, Indiana Univ. Math. J., Tome 44 (1995), pp. 1033-1073 | MR 1386760 | Zbl 0857.26006

[3] M. Barlow Diffusions on fractals, Lectures in Probability Theory and Statistics Ecole d'été de Probabilités de Saint Flour XXV-- 1995, Springer (Lecture Notes in Math.) Tome 1690 (1998), pp. 1-121 | Zbl 0916.60069

[4] M. Barlow; R. Bass Transition densities for Brownian motion on the Sierpinski carpet, Probab. Th. Rel. Fields, Tome 91 (1992), pp. 307-330 | Article | MR 1151799 | Zbl 0739.60071

[5] M. Barlow; R. Bass Random walks on graphical Sierpinski carpets, Cambridge University Press, Symposia Mathematica, Tome 39 (1999) | MR 1802425 | Zbl 0958.60045

[6] A. Bendikov; L. Saloff-Coste On and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces, American J. Math., Tome 122 (2000), pp. 1205-1263 | Article | MR 1797661 | Zbl 0969.31008

[7] R. Blumental; R. Getoor Markov Processes and Potential Theory, Academic Press, New York and London (1968) | MR 264757 | Zbl 0169.49204

[8] G. Carron Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Soc. Math. France, Séminaires et Congrés, Tome 1 (1996), pp. 205-232 | Zbl 0884.58088

[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., Tome 17 (1982), pp. 15-53 | MR 658471 | Zbl 0493.53035

[10] T. Coulhon; A. Grigor'Yan On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., Tome 89 (1997), pp. 133-199 | Article | MR 1458975 | Zbl 0920.58064

[11] T. Coulhon; L. Saloff-Coste Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana, Tome 11 (1995), pp. 687-726 | Article | MR 1363211 | Zbl 0845.58054

[12] E.B. Davies Heat kernels and spectral theory, Cambridge University Press (1989) | MR 990239 | Zbl 0699.35006

[13] E.B. Davies Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math, Tome 58 (1992), pp. 99-119 | Article | MR 1226938 | Zbl 0808.58041

[14] E.B. Davies Non-Gaussian aspects of Heat kernel behaviour, J. London Math. Soc., Tome 55 (1997), pp. 105-125 | Article | MR 1423289 | Zbl 0879.35064

[15] T. Delmotte Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, Tome 15 (1999), pp. 181-232 | Article | MR 1681641 | Zbl 0922.60060

[16] T. Delmotte Elliptic and parabolic Harnack inequalities (Potential Analysis, to appear) | MR 1881595 | Zbl 1081.39012

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

[18] B. Franchi; C. Gutiérrez; R. Wheeden Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. in Partial Differential Equations, Tome 19 (1994), pp. 523-604 | Article | MR 1265808 | Zbl 0822.46032

[19] M. Fukushima; Y. Oshima; M. Takeda Dirichlet forms and Symmetric Markov processes, W. de Gruyter (1994) | MR 1303354 | Zbl 0838.31001

[20] A. Grigor'Yan The heat equation on non-compact Riemannian manifolds (Matem. Sbornik) Tome 182 (1991), pp. 55-87 | Zbl 0743.58031

[20] A. Grigor'Yan The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. (Engl. Transl.), Tome 72 (1992), pp. 47-77 | Article | MR 1098839 | Zbl 0776.58035

[21] A. Grigor'Yan Heat kernel upper bounds on a complete non-compact Riemannian manifold, Revista Mat. Iberoamericana, Tome 10 (1994), pp. 395-452 | Article | MR 1286481 | Zbl 0810.58040

[22] A. Grigor'Yan Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geometry, Tome 45 (1997), pp. 33-52 | MR 1443330 | Zbl 0865.58042

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

[24] A. Grigor'Yan; E.B. Davies And Y. Sasarov, Eds Estimates of heat kernels on Riemannian manifolds, Spectral Theory and Geometry, Cambridge University Press (London Math. Soc. Lecture Note Series) Tome 273 (1999) | Zbl 0985.58007

[25] A. Grigor'Yan; L. Saloff-Coste Heat kernel on connected sums of Riemannian manifolds, Mathematical Research Letters, Tome 6 (1999), pp. 1-14 | MR 1713132 | Zbl 0957.58023

[26] A. Grigor'Yan; A. Telcs Sub-Gaussian estimates of heat kernels on infinite graphs (2000) (Preprint) | MR 1853353 | Zbl 1010.35016

[27] M. Gromov Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser (1998) | MR 1699320 | Zbl 05114904

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

[29] N. Krylov; M. Safonov A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izs, Tome 16 (1981), pp. 151-164 | Article | Zbl 0464.35035

[30] S. Kusuoka; D. Stroock Applications of Malliavin Calculus, Part 3, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Tome 34 (1987), pp. 391-442 | MR 914028 | Zbl 0633.60078

[31] Y.T. Kuzmenko; S.A. Molchanov Counterexamples to Liouville-type theorems (Vestnik Moskov. Univ., Ser. I Mat. Mekh.) Tome 6 (1976), pp. 39-43 | Zbl 0416.35033

[31] Y.T. Kuzmenko; S.A. Molchanov Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull. (Engl. Transl.), Tome 34 (1979), pp. 35-39 | Zbl 0442.35038

[32] P. Li; S-T Yau On the parabolic kernel of Schrödinger operator, Acta Math., Tome 156 (1986), pp. 153-201 | Article | MR 834612 | Zbl 0611.58045

[33] 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

[34] J. Moser A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., Tome 16 ; 20 (1964 ; 1967), p. 101-134 ; 231--236 | Article | MR 159139 | Zbl 0149.06902

[35] J. Moser On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math., Tome 24 (1971), pp. 727-740 | Article | MR 288405 | Zbl 0227.35016

[36] M. Safonov Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math., Tome 21 (1983), pp. 851-863 | Article | Zbl 0511.35029

[37] L. Saloff-Coste Analyse sur les groupes à croissance polynomiale, Ark. för Mat., Tome 28 (1990), pp. 315-331 | Article | MR 1084020 | Zbl 0715.43009

[38] L. Saloff-Coste; D. Stroock Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal., Tome 98 (1991), pp. 97-121 | Article | MR 1111195 | Zbl 0734.58041

[39] L. Saloff-Coste Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Tome 36 (1992), pp. 417-450 | MR 1180389 | Zbl 0735.58032

[40] L. Saloff-Coste A note on Poincaré, Sobolev and Harnack inequalities, Duke Math. J., IMRN, Tome 2 (1992), pp. 27-38 | MR 1150597 | Zbl 0769.58054

[41] 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

[42] L. Saloff-Coste Aspects of Sobolev type inequalities (2001) (To appear in London Math. Soc. Lecture Notes Series, Cambridge University Press) | MR 1872526 | Zbl 0991.35002

[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, Birkhäuser (Progress in Probability) Tome vol. 36 (1995), pp. 231-242 | Zbl 0834.58039

[44] K-T. Sturm Analysis on local Dirichlet spaces I: Recurrence, conservativeness and L p -Liouville properties, J. Reine Angew. Math., Tome 456 (1994), pp. 173-196 | Article | MR 1301456 | Zbl 0806.53041

[45] K-T. Sturm Analysis on local Dirichlet spaces II. Upper Gaussian estimates for fundamental solutions of parabolic equations, Osaka J. Math., Tome 32 (1995), pp. 275-312 | MR 1355744 | Zbl 0854.35015

[46] K-T. Sturm Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl., Tome 75 (1996), pp. 273-297 | MR 1387522 | Zbl 0854.35016

[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., Tome 304 (1987), pp. 519-521 | MR 892879 | Zbl 0614.22002

[49] N. Varopoulos Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semigroup technique, Bull. Sci. Math., Tome 113 (1989), pp. 253-277 | MR 1016211 | Zbl 0703.58052

[50] N. Varopoulos; L. Saloff-Coste; T. Coulhon Analysis and geometry on groups, Cambridge University Press (1993) | MR 1218884 | Zbl 0813.22003