Positivity 2: 123–151, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Local Lower Bounds on Heat Kernels
A.F.M. TER ELST
and DEREK W. ROBINSON
Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O.
Box 513, 5600 MB Eindhoven, The Netherlands;
Centre for Mathematics and its Applications,
School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia
(Received: 18 September 1997; accepted in revised form: 28 January 1998)
Abstract. We analyze convolution semigroups on a regular measure space which satisﬁes the local
doubling property. We assume the kernels are bounded and symmetric with the characteristic small-
time, volume-dependent, singularity. Then, using a weak conservation property, we deduce local
lower bounds with a comparable singularity.
Applications are given to a wide range of subelliptic and strongly elliptic self-adjoint, or near
self-adjoint, operators on Lie groups.
Mathematics Subject Classiﬁcation (1991): 47B38, 22E30
Key words: heat kernels, lower bounds
Davies  recently raised the question of obtaining local lower bounds on the
semigroup kernels associated with m-th order strongly elliptic operators acting on
. He developed a method which gives bounds for self-adjoint operators with
measurable coefﬁcients, in divergence form, whenever m>d. In addition he
indicated how the bounds could be derived for a class of self-adjoint operators
with regular coefﬁcients if d ≥ m. In this note we describe an alternative method
of obtaining local lower bounds which extends Davies’ results in several directions.
In particular it works with R
replaced by a general d-dimensional Lie group G.
It then gives bounds for m-th order self-adjoint operators with measurable coefﬁ-
cients in divergence form which are strongly elliptic, uniformly on G, whenever
m ≥ d. In fact the method only requires the principal part of the operator to be
self-adjoint, or close to self-adjoint, and does not impose symmetry restrictions
on the lower order terms. If d>mthen it also applies to almost self-adjoint
subelliptic operators whenever the principal coefﬁcients are m-times differentiable
in the L
Our method was used earlier (see , Chapters 3, 4 and 5) for second-order
operators with real coefﬁcients. It is based on an idea of Varopoulos  for the
sublaplacian together with perturbation arguments. The method relies on three
properties of the semigroup kernel:
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