CMA Research Report
MRR01-006
Nick Dungey, A.F.M. ter Elst and Derek W. Robinson
On anomalous asymptotics of heat kernels on groups of polynomial growth
Abstract:
Let K denote the kernel of the continuous semigroup
S generated by H
= (-1)m/2 ∑i=1d'
Aim where A1, ... ,
Ad' are a
generating basis of right-invariant fields acting on L2(G)
with G a Lie group of polynomial growth and m an even positive
integer. If G is connected, simply connected, and has an abelian
nilshadow we establish that |Kt(g)|
≤ c∫Na dh
Gb,t(m)
(gh-1) Gb,t(2)(h)
for all g ∈ G and
all t ≥ 1, where Na is a subgroup of the abelian
nilradical, G(m) denotes an m-th order Gaussian over
G and G(2) the second-order Gaussian over Na.
The group Na is determined by the choice of the generating
basis and in general is non-zero. Analogous estimates are derived for
various derivatives of the kernel. Further, through the use of
homogenization theory, we establish asymptotic estimates for S and K.
These estimates imply that the above kernel bounds give the correct
asymptotic behaviour of K, e.g., if m ≥ 4 and Na
≠ 0
then K decreases faster than G(m) as t
→ ∞.
AMS Classification:
22E25, 35B40, 35B27, 42B10
Date:
September 2000
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