  We consider a class of fourth order uniformly elliptic operators in planar
Euclidean domains and study the associated heat kernel. For operators with
$L^{\infty}$ coefficients we obtain Gaussian estimates with best constants,
while for operators with constant coefficients we obtain short time asymptotic
estimates. The novelty of this work is that we do not assume that the
associated symbol is strongly convex. The short time asymptotics reveal a
behavior which is qualitatively different from that of the strongly convex
case.
