On anomalous asymptotics of heat kernels

Research Report MRR99-042

On anomalous asymptotics of heat kernels

A.F.M. ter Elst and Derek W.Robinson

Abstract: Let $A_1,A_2, A_3$ denote a vector space basis, formed by right invariant vector fields, of the Lie algebra $\gotg$ of the three-dimensional Lie group $G$ of Euclidean motions of the plane. We demonstrate that for $m\geq 4$ the semigroup kernel $K_t$ associated with the strongly elliptic operator $H=(-1)^{m/2}\sum^3_{i=1}A_i^m$ satisfies $m$-th order Gaussian bounds for all $t\geq1$ if, and only if, two of the $A_i$ span the nilradical of $\gotg$. If this condition is not satisfied the kernel has an anomalous asymptotics. It behaves like an $m$-th order kernel in one direction and like a second-order kernel in the other two directions. No such anomaly occurs for the kernels associated with the operators $H=(-\sum^3_{i=1}A_i^2)^{m/2}$.

Download paper:	PDF file (206K)
	gzipped DVI file (27K)

This service is maintained by the Mathematical Sciences Institute (MSI)
Comments to webmaster@maths.anu.edu.au URL: http://wwwmaths.anu.edu.au/