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 on , (i.e., for ) and elliptic Harnack inequality for on .
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 , (i.e., pour ) et l’inégalité de Harnack elliptique pour sur .
Keywords: Laplace equation, heat equation, Harnack inequality, Dirichlet spaces, two-sided Gaussian bounds
Mot clés : équation de Laplace, équation de la chaleur, inégalité de Harnack, espaces de Dirichlet, bornes gaussiennes
Hebisch, Waldemar 1; Saloff-Coste, Laurent 2
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TY - JOUR AU - Hebisch, Waldemar AU - Saloff-Coste, Laurent TI - On the relation between elliptic and parabolic Harnack inequalities JO - Annales de l'Institut Fourier PY - 2001 SP - 1437 EP - 1481 VL - 51 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1861/ DO - 10.5802/aif.1861 LA - en ID - AIF_2001__51_5_1437_0 ER -
%0 Journal Article %A Hebisch, Waldemar %A Saloff-Coste, Laurent %T On the relation between elliptic and parabolic Harnack inequalities %J Annales de l'Institut Fourier %D 2001 %P 1437-1481 %V 51 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1861/ %R 10.5802/aif.1861 %G en %F 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, Volume 51 (2001) no. 5, pp. 1437-1481. doi : 10.5802/aif.1861. https://aif.centre-mersenne.org/articles/10.5802/aif.1861/
[1] Invariant varieties through singularities of holomorphic vector fields, Annals of Math., Volume 115 (1982) | MR | Zbl
[1] 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] Sobolev Inequalities in Disguise, Indiana Univ. Math. J., Volume 44 (1995), pp. 1033-1073 | MR | Zbl
[3] 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] Transition densities for Brownian motion on the Sierpinski carpet, Probab. Th. Rel. Fields, Volume 91 (1992), pp. 307-330 | DOI | MR | Zbl
[5] Random walks on graphical Sierpinski carpets, Symposia Mathematica, 39, Cambridge University Press, 1999 | MR | Zbl
[6] 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] Markov Processes and Potential Theory, Academic Press, New York and London, 1968 | MR | Zbl
[8] 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] 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] On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J., Volume 89 (1997), pp. 133-199 | DOI | MR | Zbl
[11] Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana, Volume 11 (1995), pp. 687-726 | DOI | MR | Zbl
[12] Heat kernels and spectral theory, Cambridge University Press, 1989 | MR | Zbl
[13] Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math, Volume 58 (1992), pp. 99-119 | DOI | MR | Zbl
[14] Non-Gaussian aspects of Heat kernel behaviour, J. London Math. Soc., Volume 55 (1997), pp. 105-125 | DOI | MR | Zbl
[15] Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, Volume 15 (1999), pp. 181-232 | DOI | MR | Zbl
[16] Elliptic and parabolic Harnack inequalities (Potential Analysis, to appear) | MR | Zbl
[17] 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] Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. in Partial Differential Equations, Volume 19 (1994), pp. 523-604 | DOI | MR | Zbl
[19] Dirichlet forms and Symmetric Markov processes, W. de Gruyter, 1994 | MR | Zbl
[20] The heat equation on non-compact Riemannian manifolds (Matem. Sbornik), Volume 182 (1991), pp. 55-87 | Zbl
[20] The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. (Engl. Transl.), Volume 72 (1992), pp. 47-77 | DOI | MR | Zbl
[21] Heat kernel upper bounds on a complete non-compact Riemannian manifold, Revista Mat. Iberoamericana, Volume 10 (1994), pp. 395-452 | DOI | MR | Zbl
[22] Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geometry, Volume 45 (1997), pp. 33-52 | MR | Zbl
[23] 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] Estimates of heat kernels on Riemannian manifolds, Spectral Theory and Geometry (London Math. Soc. Lecture Note Series), Volume 273 (1999) | Zbl
[25] Heat kernel on connected sums of Riemannian manifolds, Mathematical Research Letters, Volume 6 (1999), pp. 1-14 | MR | Zbl
[26] Sub-Gaussian estimates of heat kernels on infinite graphs (2000) (Preprint) | MR | Zbl
[27] Metric structures for Riemannian and non-Riemannian spaces, Birkhäuser, 1998 | MR | Zbl
[28] The Poincaré inequality for vector fields satisfying the Hörmander's condition, Duke Math. J., Volume 53 (1986), pp. 503-523 | MR | Zbl
[29] A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izs, Volume 16 (1981), pp. 151-164 | DOI | Zbl
[30] Applications of Malliavin Calculus, Part 3, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., Volume 34 (1987), pp. 391-442 | MR | Zbl
[31] Counterexamples to Liouville-type theorems (Vestnik Moskov. Univ., Ser. I Mat. Mekh.), Volume 6 (1976), pp. 39-43 | Zbl
[31] Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull. (Engl. Transl.), Volume 34 (1979), pp. 35-39 | Zbl
[32] On the parabolic kernel of Schrödinger operator, Acta Math., Volume 156 (1986), pp. 153-201 | DOI | MR | Zbl
[33] On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., Volume 14 (1961), pp. 577-591 | DOI | MR | Zbl
[34] 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] On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math., Volume 24 (1971), pp. 727-740 | DOI | MR | Zbl
[36] 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] Analyse sur les groupes à croissance polynomiale, Ark. för Mat., Volume 28 (1990), pp. 315-331 | DOI | MR | Zbl
[38] Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal., Volume 98 (1991), pp. 97-121 | DOI | MR | Zbl
[39] Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Volume 36 (1992), pp. 417-450 | MR | Zbl
[40] A note on Poincaré, Sobolev and Harnack inequalities, Duke Math. J., IMRN, Volume 2 (1992), pp. 27-38 | MR | Zbl
[41] Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Volume 4 (1995), pp. 429-467 | DOI | MR | Zbl
[42] Aspects of Sobolev type inequalities (2001) (To appear in London Math. Soc. Lecture Notes Series, Cambridge University Press) | MR | Zbl
[43] 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] Analysis on local Dirichlet spaces I: Recurrence, conservativeness and -Liouville properties, J. Reine Angew. Math., Volume 456 (1994), pp. 173-196 | DOI | MR | Zbl
[45] 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] Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl., Volume 75 (1996), pp. 273-297 | MR | Zbl
[47] Local sub-Gaussian estimates of heat kernels on graphs, the strongly recurrent cases (2000) (Preprint)
[48] Fonctions harmoniques sur les groupes de Lie, CR. Acad. Sci. Paris, Sér. I Math., Volume 304 (1987), pp. 519-521 | MR | Zbl
[49] 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] Analysis and geometry on groups, Cambridge University Press, 1993 | MR | Zbl
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