On the relation between elliptic and parabolic Harnack inequalities

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 ${\partial }_t+\Delta$) et l’inégalité de Harnack elliptique pour $-{\partial }_t^2+\Delta$ sur $ℝ\times M$.

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 $-{\partial }_t^2+\Delta$ on $ℝ\times M$.

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