Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups

http://dx.doi.org/10.4153/CJM-2005-047-1
Canad. J. Math. 57(2005), 1193-1214

  Published:2005-12-01
 Printed: Dec 2005

Let $K$ be a function on a unimodular locally compact group $G$, and denote by $K_n = K*K* \cdots * K$ the $n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in $L^2(G)$, we give estimates of the norms $\|K_n\|_2$ and $\|K_n\|_\infty$ for large $n$. In contrast to previous methods for estimating $\|K_n\|_\infty$, we do not need to assume that the function $K$ is a probability density or non-negative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.