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 ℝ × 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 -

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