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We obtain heat kernel estimates for a class of fourth order non-uniformlyelliptic operators in two dimensions. Contrary to existing results, theoperators considered have symbols that are not strongly convex. This risescertain difficulties as it is known that, as opposed to the strongly convexcase, there is no absolute exponential constant. Our estimates involve sharpconstants and Finsler-type distances that are induced by the operator symbol.The main result is based on two general hypotheses, a weighted Sobolevinequalitry and an interpolation inequality, which are related to thesingularity or degeneracy of the coefficients.

Heat kernel estimates for fourth order non-uniformly elliptic operators with non strongly convex symbols