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
@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}, pages = {1437--1481}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {51}, number = {5}, year = {2001}, doi = {10.5802/aif.1861}, mrnumber = {1860672}, zbl = {0988.58007}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1861/} }
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 - http://archive.numdam.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 http://archive.numdam.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. http://archive.numdam.org/articles/10.5802/aif.1861/
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