Suppose that H=H*[ges ]0 on
L2(X, dx) and that
e−Ht has an integral kernel
K(t, x, y) which is a continuous
function of all three variables. It follows from the
fact that e−Ht is a non-negative self-adjoint
operator
that K(t, x, x)[ges ]0 for all t>0
and
x∈X. Our main abstract results, Theorems 2 and 3,
provide a positive lower
bound on K(t, x, x) under
suitable general hypotheses. As an application we obtain
a explicit positive lower bound on K(t, x, y)
when x is close enough to y and H is a
higher order uniformly elliptic operator in divergence form acting in
L2(RN, dx); see
Theorem 6.We emphasize that our results are not applicable to second order elliptic
operators
(except in one space dimension). For such operators much stronger lower
bounds can
be obtained by an application of the Harnack inequality. For higher order
operators,
however, we believe that our result is the first of its type which does
not impose any
continuity conditions on the highest order coefficients of the operators.
Heat Kernels of Second Order Complex Elliptic Operators and Applications