Equivalence between regularity theorems and heat kernel estimates for higher order elliptic operators and systems under divergence form

We study the heat kernel of higher order elliptic operators or systems tind er divergence form on R-n. Ellipticity is in the sense of Garding inequalit y. We show that for homogeneous operators Gaussian upper bounds and Holder regularity of the heat kernel is equivalent to local regularity of weak sol utions, We also show stability of such bounds tinder L-infinity-perturbatio ns of the coefficients or under perturbations with bounded coefficients low er order terms. Such a criterion allows us to obtain heat kernel bounds for operators or systems with uniformly continuous or vmo coefficients. (C) 20 00 Academic Press.