On the relation between elliptic and parabolic Harnack inequalities

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 .