Some Conditions for Decay of Convolution Powers and Heat Kernels on Groups
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.
22E30 - Analysis on real and complex Lie groups [See also 33C80, 43-XX]
35B40 - Asymptotic behavior of solutions
43A99 - None of the above, but in this section