Pointwise lower bounds on the heat kernels of higher order elliptic operators

Abstract

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(R^N, 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.