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Waldemar Hebisch; Laurent Saloff-Coste
On the relation between elliptic and parabolic Harnack inequalities
(Sur les liens entre inégalités de Harnack elliptiques et paraboliques)
Annales de l'institut Fourier, 51 no. 5 (2001), p. 1437-1481, doi: 10.5802/aif.1861
Article PDF | Reviews MR 1860672 | Zbl 0988.58007 | 2 citations in Cedram
Class. Math.: 58J05, 58J35, 31C25, 58J65, 60J65
Keywords: Laplace equation, heat equation, Harnack inequality, Dirichlet spaces, two-sided Gaussian bounds

Résumé - 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 ${\Bbb R}\times M$.

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