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 under divergence form on Rn. Ellipticity is in the sense of Gårding inequality. We show that for homogeneous operators Gaussian upper bounds and Hölder regularity of the heat kernel is equivalent to local regularity of weak solutions. We also show stability of such bounds under L∞-perturbations of the coefficients or under perturbations with bounded coefficients lower order terms. Such a criterion allows us to obtain heat kernel bounds for operators or systems with uniformly continuous or vmo coefficients.