2 INTRODUCTION
Strictly related to the proof of the existence of h, and of independent interest,
are several sharp Gaussian bounds for h that we will establish:
e−cd(x,y)2/(t−s)
c B
(
x,

t s
) h (t, x; s, y) c
e−d(x,y)2/c(t−s)
B
(
x,

t s
)
|Xih (t, x; s, y)|
c

t s
e−d(x,y)2/c(t−s)
B
(
x,

t s
)
(0.2)
|XiXjh (t, x; s, y)| + |∂th (t, x; s, y)|
c
t s
e−d(x,y)2/c(t−s)
B
(
x,

t s
)
where x, y Rn, 0 t−s T and |B (x, r)| denotes Lebesgue measure of the d-ball
B (x, r). The constant c in these estimates depends on the coefficients aij,ak,a0
only through their older moduli of continuity and the ellipticity constant λ.
A precise list of the results we prove about h is contained in Theorem 10.7,
stated at the beginning of Part II (see also Remark 10.9).
A remarkable consequence of these bounds is a scaling invariant Harnack in-
equality for H, and for its stationary counterpart L in (0.1), which will be proved
throughout Part III. In that part we will assume a0, the zero order term of H, to
be identically zero. Precise results are stated in Theorems 15.1 and 15.3 at the
beginning of Part III.
As we mentioned before, all the results we have described so far are proved
for an operator defined on the whole space, which extends H, initially defined only
locally. At the end of this work (see Section 19) we will also show how to come back
to the original operator, deducing local results from the above global theorems (see
Theorems 19.1 and 19.2). We could also say that the final goal of all our theory is
to prove local properties of our operators, so that the theory itself is local, in spirit,
although it exploits, for technical convenience, objects that are defined globally.
An announcement of the results contained in this paper has appeared in [13].
Previous results and bibliographic remarks. Gaussian estimates for the
fundamental solution of second order partial differential operators of parabolic type,
or, somehow more generally, for the density function of heat diffusion semigroups,
have a long history, starting with Aronson’s work [1]. The relevance of two-sided
Gaussian estimates to get scaling invariant Harnack inequalities for positive solu-
tions was firstly pointed out by Nash in the Appendix of his celebrated paper [46].
However, a complete implementation of the method outlined by Nash was given
much later by Fabes and Stroock in [22], also inspired by some ideas of Krylov
and Safonov (see [32], [33], [50]). Since then, the full strength of Gaussian esti-
mates has been enlightened by several authors, showing their deep relationship not
only with the scaling invariant Harnack inequality, but also with the ultracontrac-
tivity property of heat diffusion semigroups, with inequalities of Nash, Sobolev or
Poincar´ e type, and with the doubling property of the measure of “intrinsic” balls.
We directly refer to the recent monograph by Saloff-Coste [51] for a beautiful ex-
position of this circle of ideas, and for an exhaustive list of references on these
subjects. Here we explicitly recall just the results in literature strictly close to the
core of our work.
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