Introduction
Object and main results of the paper. Let us consider the heat-type
operator in
Rn+1
(0.1) H = ∂t L = ∂t
q
i,j=1
aij (t, x) XiXj
q
k=1
ak (t, x) Xk a0 (t, x)
where:
(H1) X1,X2,...,Xq is a system of real smooth vector fields which are de-
fined in some bounded domain
Rn
and satisfy ormander’s condition in Ω:
rank Lie{Xi,i = 1, 2, ..., q} = n at any point of (more precise definitions will be
given later);
(H2) A = {aij (t,
x)}q
i,j=1
is a real symmetric uniformly positive definite
matrix satisfying, for some positive constant λ,
λ−1
|ξ|2

q
i,j=1
aij (t, x) ξiξj λ
|ξ|2
for every ξ Rq, x Ω,t (T1,T2) for some T1 T2.
If d (x, y) denotes the Carnot-Carath´ eodory metric generated in by the Xi’s
and
dP ((t, x) , (s, y)) = d (x,
y)2
+ |t s|
is its “parabolic” counterpart in R × Ω, we will assume that:
(H3) aij,ak,a0 are older continuos on C = (T1,T2) × with respect to
the distance dP .
Under assumptions (H1),(H2),(H3), we shall prove the existence and basic prop-
erties of a fundamental solution h for the operator H, including a representation
formula for solutions to the Cauchy problem, a “reproduction property” for h, and
regularity results: namely, we will show that h is locally older continuous, far off
the pole, together with its derivatives Xjh, XiXjh, ∂th. To be more precise, an
explanation is in order here. The operator H is defined only on the cylinder C.
On the other hand, dealing with fundamental solutions, it is convenient to work
with an operator defined on the whole space. For this reason we will extend the
operator H to the whole Rn+1, in such a way that, outside a compact set in the
space variables, it coincides with the classical heat operator, and henceforth we will
study the fundamental solution for this extended operator.
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