On the relation between elliptic and parabolic Harnack inequalities

Waldemar Hebisch[1]; Laurent Saloff-Coste[2]

  • [1] Wroclaw University, Institute of Mathematics, Wroclaw (Pologne)
  • [2] Cornell University, Department of Mathematics, Ithaca NY (USA)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 5, page 1437-1481
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Hebisch, Waldemar, and Saloff-Coste, Laurent. "On the relation between elliptic and parabolic Harnack inequalities." Annales de l’institut Fourier 51.5 (2001): 1437-1481. <http://eudml.org/doc/115954>.

@article{Hebisch2001,
abstract = {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 ^2_t+\Delta $ on $\{\mathbb \{R\}\}\times M$.},
affiliation = {Wroclaw University, Institute of Mathematics, Wroclaw (Pologne); Cornell University, Department of Mathematics, Ithaca NY (USA)},
author = {Hebisch, Waldemar, Saloff-Coste, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Laplace equation; heat equation; Harnack inequality; Dirichlet spaces; two-sided Gaussian bounds; elliptic and parabolic Harnack inequalities; Riemannian manifold},
language = {eng},
number = {5},
pages = {1437-1481},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the relation between elliptic and parabolic Harnack inequalities},
url = {http://eudml.org/doc/115954},
volume = {51},
year = {2001},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 5
SP - 1437
EP - 1481
AB - 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 ^2_t+\Delta $ on ${\mathbb {R}}\times M$.
LA - eng
KW - Laplace equation; heat equation; Harnack inequality; Dirichlet spaces; two-sided Gaussian bounds; elliptic and parabolic Harnack inequalities; Riemannian manifold
UR - http://eudml.org/doc/115954
ER -

References

top
  1. C. Camacho, P. Sad, Invariant varieties through singularities of holomorphic vector fields, Annals of Math. 115 (1982) Zbl0503.32007MR657239
  2. P. Auscher, T. Coulhon, Gaussian bounds for random walks from elliptic regularity, Ann. Inst. Henri Poincaré, Prob. Stat. 35 (1999), 605-630 Zbl0933.60047MR1705682
  3. D. Bakry, T. Coulhon, M. Ledoux, L. Saloff-Coste, Sobolev Inequalities in Disguise, Indiana Univ. Math. J. 44 (1995), 1033-1073 Zbl0857.26006MR1386760
  4. M. Barlow, Diffusions on fractals, Lectures in Probability Theory and Statistics Ecole d'été de Probabilités de Saint Flour XXV-- 1995 1690 (1998), 1-121, Springer Zbl0916.60069
  5. M. Barlow, R. Bass, Transition densities for Brownian motion on the Sierpinski carpet, Probab. Th. Rel. Fields 91 (1992), 307-330 Zbl0739.60071MR1151799
  6. M. Barlow, R. Bass, Random walks on graphical Sierpinski carpets, 39 (1999), Cambridge University Press Zbl0958.60045MR1802425
  7. A. Bendikov, L. Saloff-Coste, On and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces, American J. Math. 122 (2000), 1205-1263 Zbl0969.31008MR1797661
  8. R. Blumental, R. Getoor, Markov Processes and Potential Theory, (1968), Academic Press, New York and London Zbl0169.49204MR264757
  9. G. Carron, Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger 1 (1996), 205-232, Soc. Math. France, Séminaires et Congrés Zbl0884.58088
  10. 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. 17 (1982), 15-53 Zbl0493.53035MR658471
  11. T. Coulhon, A. Grigor'yan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997), 133-199 Zbl0920.58064MR1458975
  12. T. Coulhon, L. Saloff-Coste, Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamericana 11 (1995), 687-726 Zbl0845.58054MR1363211
  13. E.B. Davies, Heat kernels and spectral theory, (1989), Cambridge University Press Zbl0699.35006MR990239
  14. E.B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math 58 (1992), 99-119 Zbl0808.58041MR1226938
  15. E.B. Davies, Non-Gaussian aspects of Heat kernel behaviour, J. London Math. Soc. 55 (1997), 105-125 Zbl0879.35064MR1423289
  16. T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana 15 (1999), 181-232 Zbl0922.60060MR1681641
  17. T. Delmotte, Elliptic and parabolic Harnack inequalities Zbl1081.39012MR1881595
  18. E. Fabes, D. Stroock, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rat, Mech. Anal. 96 (1986), 327-338 Zbl0652.35052MR855753
  19. B. Franchi, C. Gutiérrez, R. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. in Partial Differential Equations 19 (1994), 523-604 Zbl0822.46032MR1265808
  20. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and Symmetric Markov processes, (1994), W. de Gruyter Zbl0838.31001MR1303354
  21. A. Grigor'yan, The heat equation on non-compact Riemannian manifolds, 182 (1991), 55-87 Zbl0743.58031
  22. A. Grigor'yan, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geometry 45 (1997), 33-52 Zbl0865.58042MR1443330
  23. A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. A.M.S 36 (1999), 135-249 Zbl0927.58019MR1659871
  24. A. Grigor'yan, Estimates of heat kernels on Riemannian manifolds, Spectral Theory and Geometry 273 (1999), Cambridge University Press Zbl0985.58007
  25. A. Grigor'yan, L. Saloff-Coste, Heat kernel on connected sums of Riemannian manifolds, Mathematical Research Letters 6 (1999), 1-14 Zbl0957.58023MR1713132
  26. A. Grigor'yan, A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, (2000) Zbl1010.35016MR1853353
  27. M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, (1998), Birkhäuser Zbl1113.53001MR1699320
  28. D. Jerison, The Poincaré inequality for vector fields satisfying the Hörmander's condition, Duke Math. J. 53 (1986), 503-523 Zbl0614.35066MR850547
  29. N. Krylov, M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR-Izs 16 (1981), 151-164 Zbl0464.35035
  30. S. Kusuoka, D. Stroock, Applications of Malliavin Calculus, Part 3, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 34 (1987), 391-442 Zbl0633.60078MR914028
  31. Y.T. Kuzmenko, S.A. Molchanov, Counterexamples to Liouville-type theorems, 6 (1976), 39-43 Zbl0416.35033
  32. P. Li, S-T Yau, On the parabolic kernel of Schrödinger operator, Acta Math. 156 (1986), 153-201 Zbl0611.58045MR834612
  33. J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591 Zbl0111.09302MR159138
  34. J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 16 ; 20 (1964 ; 1967), 101-134 ; 231--236 Zbl0149.06902MR159139
  35. J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727-740 Zbl0227.35016MR288405
  36. M. Safonov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math. 21 (1983), 851-863 Zbl0511.35029
  37. L. Saloff-Coste, Analyse sur les groupes à croissance polynomiale, Ark. för Mat. 28 (1990), 315-331 Zbl0715.43009MR1084020
  38. L. Saloff-Coste, D. Stroock, Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal. 98 (1991), 97-121 Zbl0734.58041MR1111195
  39. L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom. 36 (1992), 417-450 Zbl0735.58032MR1180389
  40. L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Duke Math. J., IMRN 2 (1992), 27-38 Zbl0769.58054MR1150597
  41. L. Saloff-Coste, Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis 4 (1995), 429-467 Zbl0840.31006MR1354894
  42. L. Saloff-Coste, Aspects of Sobolev type inequalities, (2001) Zbl0991.35002MR1872526
  43. K-T. Sturm, On the geometry defined by Dirichlet forms, Seminar on Stochastic Processes, Random Fields and Applications, Ascona vol. 36 (1995), 231-242, Birkhäuser Zbl0834.58039
  44. K-T. Sturm, Analysis on local Dirichlet spaces I: Recurrence, conservativeness and L p -Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196 Zbl0806.53041MR1301456
  45. K-T. Sturm, Analysis on local Dirichlet spaces II. Upper Gaussian estimates for fundamental solutions of parabolic equations, Osaka J. Math. 32 (1995), 275-312 Zbl0854.35015MR1355744
  46. K-T. Sturm, Analysis on local Dirichlet spaces III. The parabolic Harnack inequality, J. Math. Pures Appl. 75 (1996), 273-297 Zbl0854.35016MR1387522
  47. A. Telcs, Local sub-Gaussian estimates of heat kernels on graphs, the strongly recurrent cases, (2000) 
  48. N. Varopoulos, Fonctions harmoniques sur les groupes de Lie, CR. Acad. Sci. Paris, Sér. I Math. 304 (1987), 519-521 Zbl0614.22002MR892879
  49. N. Varopoulos, Small time Gaussian estimates of the heat diffusion kernel, Part 1: the semigroup technique, Bull. Sci. Math. 113 (1989), 253-277 Zbl0703.58052MR1016211
  50. N. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and geometry on groups, (1993), Cambridge University Press Zbl0813.22003MR1218884
  51. A. Grigor'yan, Heat kernel upper bounds on a complete non-compact Riemannian manifold, Revista Mat. Iberoamericana 10 (1994), 395-452 Zbl0810.58040MR1286481
  52. A. Grigor'yan, The heat equation on non-compact Riemannian manifolds, Math. USSR Sb. (Engl. Transl.) 72 (1992), 47-77 Zbl0776.58035MR1098839
  53. Y.T. Kuzmenko, S.A. Molchanov, Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull. (Engl. Transl.) 34 (1979), 35-39 Zbl0442.35038

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