Non-divergence equations structured on Hörmander vector fields : heat kernels and Harnack inequalities

136 Pages·2010·0.961 MB·English

by  Marco Bramanti, Luca Brandolini, Ermanno Lanconelli, Francesco Uguzzoni

MEMOIRS
of the
American Mathematical Society
Number 961
Non-Divergence Equations Structured
on Ho¨rmander Vector Fields:
Heat Kernels and
Harnack Inequalities
Marco Bramanti
Luca Brandolini
Ermanno Lanconelli
Francesco Uguzzoni
March 2010 • Volume 204 • Number 961 (end of volume) • ISSN 0065-9266
American Mathematical Society
Number 961
Non-Divergence Equations Structured
on Ho¨rmander Vector Fields:
Heat Kernels and
Harnack Inequalities
Marco Bramanti
Luca Brandolini
Ermanno Lanconelli
Francesco Uguzzoni
March2010 • Volume204 • Number961(endofvolume) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Non-divergence equations structured on H¨ormander vector fields: Heat kernels and Harnack in-
equalities/MarcoBramanti... [etal.].
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 961)
“Volume204,number961(endofvolume).”
Includesbibliographicalreferences.
ISBN978-0-8218-4903-3(alk.paper)
1.Vectorfields. 2.Differentialinequalities. 3.Heatequation. 4.Partialdifferentialopera-
tors. I.Bramanti,Marco,1963-.
QA613.619.N66 2010
515(cid:1).353—dc22 2009050034
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Contents
Introduction 1
Part I: Operators with constant coefficients 7
1. Overview of Part I 7
2. Global extension of H¨ormander’s vector fields and geometric
properties of the CC-distance 9
2.1. Some global geometric properties of CC-distances 10
2.2. Global extension of H¨ormander’s vector fields 13
3. Global extension of the operator H and existence of a fundamental
A
solution 15
4. Uniform Gevray estimates and upper bounds of fundamental
solutions for large d(x,y) 18
5. Fractional integrals and uniform L2 bounds of fundamental solutions
for large d(x,y) 25
6. Uniform global upper bounds for fundamental solutions 30
Homogeneous groups 31
6.2. Upper bounds on fundamental solutions 37
7. Uniform lower bounds for fundamental solutions 54
8. Uniform upper bounds for the derivatives of the fundamental
solutions 57
9. Uniform upper bounds on the difference of the fundamental solutions
of two operators 60
Part II: Fundamental solution for operators with Ho¨lder continuous
coefficients 67
10. Assumptions, main results and overview of Part II 67
11. Fundamental solution for H: the Levi method 74
12. The Cauchy problem 86
13. Lower bounds for fundamental solutions 89
14. Regularity results 93
Part III: Harnack inequality for operators with H¨older continuous
coefficients 99
15. Overview of Part III 99
16. Green function for operators with smooth coefficients on regular
domains 101
iii
iv CONTENTS
17. Harnack inequality for operators with smooth coefficients 108
18. Harnack inequality in the non-smooth case 111
Epilogue 115
19. Applications to operators which are defined only locally 115
20. Further developments and open problems 117
References 121
Abstract
In this work we deal with linear second order partial differential operators of
the following type:
(cid:1)q (cid:1)q
H =∂ −L=∂ − a (t,x)X X − a (t,x)X −a (t,x)
t t ij i j k k 0
i,j=1 k=1
whereX ,X ,...,X isasystemofrealH¨ormander’svectorfieldsinsomebounded
1 2 q
domainΩ⊆Rn,A={a (t,x)}q isarealsymmetricuniformlypositivedefinite
ij i,j=1
matrix such that:
(cid:1)q
λ−1|ξ|2 ≤ a (t,x)ξ ξ ≤λ|ξ|2 ∀ξ ∈Rq,x∈Ω,t∈(T ,T )
ij i j 1 2
i,j=1
for a suitable constant λ > 0 a for some real numbers T < T . The coefficients
1 2
a ,a ,a are Ho¨lder continuous on (T ,T )×Ω with respect to the parabolic CC-
ij k 0 1 2
metric (cid:2)
d ((t,x),(s,y))= d(x,y)2+|t−s|
P
(where d is the Carnot-Carath´eodory distance induced by the vector fields X ’s).
i
We prove the existence of a fundamental solution h(t,x;s,y) for H, satisfying
natural properties and sharp Gaussian bounds of the kind:
e−cd(x,y)2/(t−s) e−d(x,y)2/c(t−s)
c(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3) ≤h(t,x;s,y)≤c(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
c e−d(x,y)2/c(t−s)
|Xih(t,x;s,y)|≤ √t−s(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
c e−d(x,y)2/c(t−s)
|XiXjh(t,x;s,y)|+|∂th(t,x;s,y)|≤ t−s(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
where |B(x,r)| denotes the Lebesgue measure of the d-ball B(x,r). We then
use these properties of h as a starting point to prove a scaling invariant Harnack
inequality for positive solutions to Hu =0, when a ≡0. All the constants in our
0
ReceivedbytheeditorOctober27,2006.
ArticleelectronicallypublishedonNovember9,2009;S0065-9266(09)00605-X.
2000MathematicsSubjectClassification. Primary35H20,35A08,35K65;Secondary35H10,
35A17.
Key words and phrases. H¨ormander’svectorfields,heatkernels,Gaussianbounds,Harnack
inequalities.
(cid:1)c2009 American Mathematical Society
v
vi ABSTRACT
estimates and inequalities will depend on the coefficients a ,a ,a only through
ij k 0
their Ho¨lder norms and the number λ.
Introduction
Object and main results of the paper. Let us consider the heat-type
operator in Rn+1
(cid:1)q (cid:1)q
(0.1) H =∂ −L=∂ − a (t,x)X X − a (t,x)X −a (t,x)
t t ij i j k k 0
i,j=1 k=1
where:
(H1) X ,X ,...,X is a system of real smooth vector fields which are de-
1 2 q
fined in some bounded domain Ω ⊆ Rn and satisfy Ho¨rmander’s condition in Ω:
rank Lie{X ,i=1,2,...,q} = n at any point of Ω (more precise definitions will be
i
given later);
(H2) A = {a (t,x)}q is a real symmetric uniformly positive definite
ij i,j=1
matrix satisfying, for some positive constant λ,
(cid:1)q
λ−1|ξ|2 ≤ a (t,x)ξ ξ ≤λ|ξ|2
ij i j
i,j=1
for every ξ ∈Rq, x∈Ω,t∈(T ,T ) for some T <T .
1 2 1 2
If d(x,y) denotes the Carnot-Carath´eodory metric generated in Ω by the X ’s
i
and
(cid:2)
d ((t,x),(s,y))= d(x,y)2+|t−s|
P
is its “parabolic” counterpart in R×Ω, we will assume that:
(H3) a ,a ,a are Ho¨lder continuos on C = (T ,T )×Ω with respect to
ij k 0 1 2
the distance d .
P
Underassumptions(H1),(H2),(H3),weshallprovetheexistenceandbasicprop-
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 Ho¨lder continuous, far off
the pole, together with its derivatives X h,X X h, ∂ h. To be more precise, an
j i j t
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
spacevariables,itcoincideswiththeclassicalheatoperator,andhenceforthwewill
study the fundamental solution for this extended operator.
1
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) e−d(x,y)2/c(t−s)
c(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3) ≤h(t,x;s,y)≤c(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
c e−d(x,y)2/c(t−s)
(0.2) |Xih(t,x;s,y)|≤ √t−s(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
c e−d(x,y)2/c(t−s)
|XiXjh(t,x;s,y)|+|∂th(t,x;s,y)|≤ t−s(cid:3)(cid:3)B(cid:4)x,√t−s(cid:5)(cid:3)(cid:3)
wherex,y ∈Rn,0<t−s<T and|B(x,r)|denotesLebesguemeasureofthed-ball
B(x,r). The constant c in these estimates depends on the coefficients a ,a ,a
ij k 0
only through their Ho¨lder 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 a , the zero order term of H, to
0
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. Attheendofthiswork(seeSection19)wewillalsoshowhowtocomeback
totheoriginaloperator, deducinglocalresultsfromtheaboveglobaltheorems(see
Theorems 19.1 and 19.2). We could also say that the final goal of all our theory is
toprovelocalpropertiesofouroperators,sothatthetheoryitselfislocal,inspirit,
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
fundamentalsolutionofsecondorderpartialdifferentialoperatorsofparabolictype,
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-
mateshas beenenlightened by several authors, showing their deeprelationship 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.
INTRODUCTION 3
For heat operators of the kind
(cid:1)q
(0.3) H =∂ − X2
t i
i=1
withX leftinvarianthomogeneousvectorfieldsonaCarnotgroupinRn,Gaussian
i
bounds have been proved by Varopoulos ([57], [58], see also [59]):
(0.4) 1 e−c(cid:7)y−1◦x(cid:7)2/t ≤h(t,x,y)≤ c e−(cid:7)y−1◦x(cid:7)2/ct
ctQ/2 tQ/2
for any x,y ∈ Rn,t > 0, where Q is the homogeneous dimension of the group,
and (cid:7)·(cid:7) any homogeneous norm of the group. Two-sided Gaussian estimates and a
scaling invariant Harnack inequality for the operator
(cid:1)q
H =∂ − X (a X )
t i ij j
i,j=1
have been proved by Saloff-Coste and Stroock in [52], where {a } is a uniformly
ij
positive matrix with measurable entries, and the vector fields X are left invariant
i
with respect to a connected unimodular Lie group with polynomial growth.
Inabsenceofagroupstructure,Gaussianboundsforoperators(0.3)havebeen
proved, on a compact manifold and for finite time, by Jerison-Sanchez-Calle [30],
with an analytic approach (see also the previous partial result in [53]), and, on
the whole Rn+1, by Kusuoka-Stroock, [35], [36], using the Malliavin stochastic
calculus.
Unlike the study of “sum of squares” H¨ormander’s operators, the investiga-
tion of non-divergence operators of H¨ormander type has a relatively recent history.
Stationary operators of kind
(cid:1)q
(0.5) L= a (x)X X
ij i j
i,j=1
with X ,...,X system of H¨ormander’s vector fields have been studied by Xu [60],
1 q
Bramanti, Brandolini [10], [11], Capogna, Han [15]. A first attempt to study
Cordesand/orAlexandrov-Bakelman-Pucciestimatesforoperators(0.5)withmea-
surable coefficients a and particular classes of vector fields X are contained in
ij i
[19], [20], [21].
Evolution operators of kind (0.1) have been considered by Bonfiglioli, Lan-
conelli, Uguzzoni [3], [4], [6], Bramanti, Brandolini [12]. In [9] also more general
operators of kind
(cid:1)q
(0.6) L= a (x)X X +a (x)X
ij i j 0 0
i,j=1
with X ,X ,...,X system of H¨ormander’s vector fields have been studied.
0 1 q
In these papers, the matrix {a } is assumed symmetric and uniformly elliptic,
ij
and the entries a typically belong to some function space defined in terms of the
ij
vector fields X and the metric they induce. In particular, these operators do not
i
have smooth coefficients, so they are no longer hypoelliptic. Therefore the mere
existenceofafundamentalsolutionistroublesome. Fortheoperators(0.1)(without
lowerorderterms)withX leftinvarianthomogeneousHo¨rmander’svectorfieldson
i
a stratified Lie group and under assumptions (H1), (H2), (H3), it has been proved