On the relation between elliptic and parabolic Harnack inequalities

Hebisch, Waldemar; Saloff-Coste, Laurent

doi:10.5802/aif.1861

Hebisch, Waldemar; Saloff-Coste, Laurent

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

Abstract
Résumé

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 $\partial_{t} + Δ$ ) and elliptic Harnack inequality for $- \partial_{t}^{2} + Δ$ on $ℝ \times M$ .

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 $\partial_{t} + Δ$ ) et l’inégalité de Harnack elliptique pour $- \partial_{t}^{2} + Δ$ sur $ℝ \times M$ .

MR 1860672 | Zbl 0988.58007 | 3 citations in Numdam

DOI : https://doi.org/10.5802/aif.1861
Classification: 58J05, 58J35, 31C25, 58J65, 60J65
Keywords: Laplace equation, heat equation, Harnack inequality, Dirichlet spaces, two-sided Gaussian bounds

@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, Volume 51 (2001) no. 5, pp. 1437-1481. doi : 10.5802/aif.1861. http://www.numdam.org/item/AIF_2001__51_5_1437_0/

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