On anomalous asymptotics of heat kernels (Q2708555)

Assume \(A_1,A_2,A_3\) is a vector space basis, formed by right invariant vector fields, of the Lie algebra \({\mathcal G}\) of the three-dimensional Lie group \(G\) of Euclidean motions of the plane.NEWLINENEWLINENEWLINEThe authors prove that for \(m\geq 4\) the semigroup kernel \(K_t\) associated with the strongly elliptic operator NEWLINE\[NEWLINEH=(-1)^{m/2} \sum^3_{i=1}A_i^mNEWLINE\]NEWLINE satisfies \(m\)-th order Gaussian bounds for all \(t\geq 1\) if, and only if, two of the \(A_i\) generate the nilradical of \({\mathcal G}\).NEWLINENEWLINENEWLINEIt is also studied what does happen if this condition is not satisfied.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].