Fundamental solutions for sum of squares of vector fields operators

Forum Math. 24 (2012), 973 – 1011
DOI 10.1515 / FORM.2011.093
Forum Mathematicum
© de Gruyter 2012
Fundamental solutions for sum of squares of
vector fields operators with C 1;˛ coefficients
Maria Manfredini
Communicated by Christopher D. Sogge
Abstract. We adapt Levi’sP
parametrix method to construct local fundamental solutions
2
for operators of the form m
i D1 Xi , where X1 ; : : : ; Xm are Hörmander vector fields of
step 2 having non-smooth coefficients. We also provide estimates of and of its derivatives.
Keywords. Sub-elliptic operators, fundamental solutions.
2010 Mathematics Subject Classification. Primary 35A08, 35H20, 43A80; secondary
35A17, 35J70.
1 Introduction
We consider operators on Rn given by
LD
m
X
Xi2 ;
(1.1)
i D1
where 1 m n, and X1 ; : : : ; Xm are locally Euclidean Lipschitz continuous
vector fields in Rn of the form
Xj D @j C
n
X
aj;k .x/@k
.j D 1; : : : ; m/:
(1.2)
.j D m C 1; : : : ; n/;
(1.3)
kDmC1
Suppose also that
@j D
X
ji;k .x/ŒXi ; Xk .x/
1i<km
n
for every x 2 Rn , where ji;k 2 L1
loc .R / for j D m C 1; : : : ; n, 1 i < k m.
Hörmander’s result in [18] was the starting point of an extensive research aiming to investigate the regularity properties of operators which are sum of squares
of C 1 vector fields. The existence of a fundamental solution and of a control
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M. Manfredini
distance made it possible to develop a general theory of the regularity in Sobolev
and Hölder spaces (related to the vector fields). However, for the application to
the quasilinear case we need to consider operators with less regular coefficients.
We stress that a general theory for operators with non-smooth coefficients is not
available. In [17], the authors prove a first regularity result for solutions of a linear equation with continuous vector fields. We also quote [6], where Kolmogorov
type operator, with coefficients that may be discontinuous, is studied. The paper [4] considers operators which are linear combination of smooth vector fields
and non-smooth functions. In [23], the authors study Hörmander Lipschitz continuous vector fields of step 2, plus some other regularity conditions on the commutators.
In the context of non-linear equations, which can be written as sum of squares
of non-linear vector fields, we refer to the Levi type equations and the papers [9]
and [11], and references therein. In [9], a modification of the freezing method of
[26] is developed and is based on the notion of an intrinsic Taylor expansion of the
coefficients of the vector fields, which depend on the first-order derivatives of the
solutions. In [14], a non-linear Kolmogorov type equation, which arises in some
recent problems of mathematical finance, is studied. In [8] and [7], the authors
consider the regularity of the Lipschitz intrinsic minimal graphs in the Heisenberg
groups Hn , with n > 1 and n D 1 respectively. In [7, 8, 14], a technique similar
to the one in [9] is used. The regularity of solution is obtained through a boostrap
argument in suitable intrinsic spaces of Hölder continuous functions.
Based on these considerations, if we study operators sum of squares of nonsmooth vector fields, then it seems natural to suppose that the coefficients belong
to a class of Hölder continuous functions (intrinsically defined), modeled on the
fields.
In this paper, we contribute a study of operators (1.1) when the coefficients aj;k
are intrinsic Cd1;˛
.Rn / continuous, according to Definition 2.3. This means that
c ;loc
the derivatives of aj;k along the vector fields are Hölder continuous with respect
to a distance dc , defined in (2.10).
We construct a local fundamental solution for L by adapting the classical
Levi parametrix method, see [21]. Levi’s method allows us to construct a fundamental solution for an elliptic operator starting from a fundamental solution of the
operator which results from freezing all coefficients at some point.
For any 2 Rn we consider the operator L whose coefficients are the first
order Taylor expansions of coefficients aj;k of L in the directions X1 ; : : : ; Xm with
initial point . Our freezing method has been first introduced in [9]. The frozen
vector fields obtained in this way are smooth and generate a Lie algebra which
has the same structure at every point. Besides, there exists a canonical change of
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Fundamental solutions
variables which transforms the family of frozen vector fields into a nilpotent family
of vector fields independent of , whose corresponding sub-Laplacian admits a
fundamental solution. We use this fundamental solution to construct a parametrix
for the original operator L.
For more details about Levi’s parametrix method we refer to [22] and [20],
where fundamental solutions are constructed for classical elliptic operators.
In [2], the authors consider the
P smooth case and construct a fundamental solution for operator of the form m
i;j D1 ai;j Xi Xj and for its parabolic version,
where X1 ; : : : ; Xm are smooth Hörmander vector fields generating a stratified Lie
algebra and .ai;j /i;j m is a positive definite matrix having Hölder continuous entries. See also [5], where more general smooth vector fields have been considered.
We also quote [25], where Levi’s method is applied to ultraparabolic operators of
Kolmogorov type.
Note that, in the smooth case, hypotheses (1.2) and (1.3) ensure Hörmander’s
condition of step 2. Let us remark that hypothesis (1.2) is not restrictive. Inn
deed, if a family X1 ; : : : ; Xm of Lipschitz continuous vector fields
Pm in R is linearly independent at a point x, we can find a new family Yj D kD1 cj;k Xk for
j D 1; : : : ; m of the form (1.2), where .cj;k /j;k is a non-singular matrix having
Lipschitz
P continuous entries in a neighborhood of x. In other words, suppose that
Xj D nkD1 bj;k @k for j D 1; : : : ; m and consider the matrix of the vector fields
.X1 ; : : : ; Xm /. Without loss of generality, we can suppose thatPthe determinant
of the first m lines and m columns is non-singular. Let Xj0 WD m
kD1 bj;k @k for
0 ;e
j D 1; : : : ; m and consider the family X10 ; : : : ; Xm
;
:
:
:
;
e
,
where
e1 ; : : : ; e n
mC1
n
n
is the standard basis in R . There is a change of variables ' such that '.Xj0 / D ej
for j D 1; : : : ; m, '.ej / D ej for j D m C 1; : : : ; n, and the matrix associated to
' has Lipschitz continuous entries in a neighborhood of x. Finally
'.Xj / D
'.Xj0 /
C
n
X
bj;k '.ek / D @j C
kDmC1
n
X
bj;k @k
.j D 1; : : : ; m/:
kDmC1
In our case, X1 ; : : : ; Xm are of step 2, in the sense of (1.3). This ensures that
the total number m.m 1/=2 of commutators ŒXi ; Xj for 1 i < j satisfies
m.m 1/=2 n m. We set
Q D m C 2.n
m/:
(1.4)
Let 0 < ˛ 1 and assume the following intrinsic condition:
aj;k 2 Cd1;˛
.Rn / \ Liploc .Rn /
c ;loc
.j D 1; : : : ; m; k D m C 1; : : : ; n/:
(1.5)
Our main result is the following theorem.
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M. Manfredini
Theorem 1.1. Let 2 Rn . There exist a subset T of Rn which contains and a
function D .z; /, defined for z 2 T n ¹º, such that
Lz .z; / D
ı .z/:
Here ı is the Dirac measure supported at ¹º. has the properties listed below:
2;ˇ
(i) . ; / 2 Cdc ;loc .T n ¹º/ for every 0 < ˇ < ˛ (see Definition 2.3) and
Lz .z; / D 0 for z ¤ , in classical sense;
(ii) for every compact subset K of T there exists a positive constant c such that
j.z; /j c d2
Q
.z; /;
and
jXiz .z; /j c d1
Q
.z; /
.i D 1; : : : ; m/;
(1.6)
for every z 2 K n ¹º. (The superscript z means that Xi is acting on that
variable).
We infer that there exists a local fundamental solution satisfying Lz .z; / D
ı .z/ in a neighborhood of the pole . In the setting of Carnot groups, global
result can be inferred from the local result by using the homogeneity of . Since
we are interested in local regularity results, global fundamental solutions are out
of our scope.
The paper is organized as follows. In Section 2, we introduce a control distance
associated to X1 ; : : : ; Xm and define the Hölder classes associated to this distance
and prove some properties of their elements. In Section 3, we define the frozen
vector fields and describe properties of relevant distances. Section 4 is devoted to
the construction of a fundamental solution for L and to the proof of Theorem 1.1.
In the Appendix, we prove some lemmas which are used throughout.
2
Connectivity and Hölder spaces
We recall the definition of derivative along an Euclidean Lipschitz continuous vector field. Let i 2 ¹1; : : : ; mº, and let be the integral curve of Xi such that
.0/ D x. Let u W Rn ! R. If
d
.u ı / jhD0
dh
exists, it is called the derivative of u in the direction Xi at the point x, and is
denoted by Xi u.x/. We will denote by CX1 .Rn / the set of functions u such that
X1 u; : : : ; Xm u exist and are continuous functions.
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Fundamental solutions
In the following,
we will adopt the notation: I will be the identity map on Rn ;
Pn
if X equals j D1 bj @j ; then XI will be the column vector of the components of
X and X 2 I will be the column vector of components Xb1 ; : : : ; Xbn .
The following theorem holds:
Theorem 2.1. Suppose that aj;k 2 CX1 .Rn / for j D 1; : : : ; m, k D m C 1; : : : ; n.
For every x; x0 2 Rn there exists a function W Œ0; T ! Rn , connecting x0 and x,
which is a piecewise continuous integral curve of X1 ; : : : ; Xm .
Proof. We denote by exp.tXi /.y/ the integral curve of Xi which starts from y at
t D 0. We set
x1 D exp..x
x0 /1 X1 /.x0 /;
x2 D exp..x
::
:
x0 /2 X2 /.x1 /;
xm D exp..x
x0 /m Xm /.xm
(2.1)
1 /;
where .x x0 /k denotes the k-th component of x x0 . The point xm has the first
m components respectively equal to the first m components of x.
Let j D m C 1, 1 i < k m, let ji;k be the function in (1.3) and d 2 R.
We set
!
m
X
i;k
xmC1 D exp d
j .xm /Xk .xm /;
kD1
xmC2 D exp d
m
X
!
ji;k .xm /Xi .xmC1 /;
i D1
xmC3 D exp
d
m
X
(2.2)
!
ji;k .xm /Xk
.xmC2 /;
kD1
xmC4 D exp
d
m
X
!
ji;k .xm /Xi
.xmC3 /:
iD1
We claim that
xmC4 D xm C d 2
ji;k .xm /ŒXi ; Xk I.xm / C o.d 2 /:
X
(2.3)
1i<km
If we define
.t/ WD exp t d
m
X
!
ji;k .xm /Xk
.xm /;
.0/ D xm ;
.1/ D xmC1 ;
kD1
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M. Manfredini
then, by the definition of the exponential map,
P .t/ D d
m
X
ji;k .xm /.Xk I /..t //;
.t/
R
Dd
kD1
2
m
X
.ji;k .xm //2 .Xk2 I /..t//:
kD1
By the Euclidean Taylor formula,
xmC1 D xm C d
m
X
ji;k .xm /.Xk I /.xm /
kD1
m
1 X i;k
C d2
.j .xm //2 .Xk2 I /.xm / C o.d 2 /;
2
(2.4)
kD1
xmC2 D xmC1 C d
m
X
ji;k .xm /.Xi I /.xmC1 /
i D1
m
1 X i;k
.j .xm //2 .Xi2 I /.xmC1 / C o.d 2 /:
C d2
2
(2.5)
i D1
By the mean value theorem, we have
.Xi I /.xmC1 / D .Xi I /..1// D .Xi I /..0// C .Xi I ı /0 .c/
D .Xi I /.xm / C d
m
X
ji;k Xi Xk I..c//
i;kD1
D .Xi I /.xm / C d
m
X
(2.6)
ji;k Xi Xk I.xm / C o.d /;
i;kD1
for suitable c 2 0; 1Œ. Inserting (2.4) in (2.5) and using (2.6), one has
X i;k
xmC2 D xm C d
j .xm /.Xk I /.xm /
k
X i;k
1 X i;k
.j .xm //2 .Xk2 I /.xm / C d
j .xm /.Xi I /.xm /
C d2
2
i
k
X i;k
C d2
.j .xm //2 .Xi Xk I /.xm /
i;k
1 X i;k
C d2
.j .xm //2 .Xi2 I /.xmC1 / C o.d 2 /
2
i
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Fundamental solutions
D xm C d
X
ji;k .xm /.Xk I /.xm /
k
X i;k
1 X i;k
.j .xm //2 .Xk2 I /.xm / C d
C d2
j .xm /.Xi I /.xm /
2
i
k
X i;k
2
2
Cd
.j .xm // .Xi Xk I /.xm /
i;k
1 X i;k
.j .xm //2 .Xi2 I /.xm / C o.d 2 /:
C d2
2
(2.7)
i
Arguing as before and using (2.7), we obtain
xmC3 D xmC2
d
X
ji;k .xm /.Xk I /.xmC2 /
i
1 X i;k
C d2
.j .xm //2 .Xk2 I /.xmC2 / C o.d 2 /
2
i
X i;k
j .xm /.Xk I /.xm /
D xmC2 d
i
d
2
X
.ji;k .xm //2 .Xk Xi I /.xm /
k
X i;k
d2
.j .xm //2 .Xk2 I /.xm /
k
1 X i;k
.j .xm //2 .Xk2 I /.xmC2 / C o.d 2 /
C d2
2
i
1 X i;k
ji;k .xm /.Xi I /.xm / C d 2
.j .xm //2 .Xi2 I /.xm /
2
k
k
X i;k
C d2
.j .xm //2 .Xi Xk I Xk Xi I /.xm / C o.d 2 /:
(2.8)
D xm C d
X
i;k
Hence
xmC4 D xmC3
d
X
ji;k .xm /.Xi I /.xmC3 /
i
1 X i;k
C d2
.j .xm //2 .Xi2 I /.xmC3 / C o.d 2 /
2
i
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M. Manfredini
(by the mean value theorem and (2.8))
X i;k
D xmC3 d
j .xm /.Xi I /.xm /
i
1 2 X i;k
d
.j .xm //2 .Xi2 I /.xm / C o.d 2 /
2
i
X i;k
D xm C d 2
.j .xm //2 ŒXi ; Xk I.xm / C o.d 2 /:
(2.9)
i;k
This proves (2.3). Next, by (1.3),
xmC4 D xm C d 2 .@mC1 I /.xm / C o.d 2 /:
In particular, the points xmC4 and xm differ in the j -th component.
Analogously, we can reach the points of type
xmC4 D xm
d 2 .@mC1 I /.xm / C o.d 2 /
by replacing Xk with Xk and Xi with Xi .
The connectivity of R allows us to apply (2.2) a finite number of times, and
we can assume that the .m C 1/-th component of the new xmC4 , obtained by this
iteration, equals the .m C 1/-th component of x.
Finally, applying this procedure for every j D m C 2; : : : ; n, we can reach the
final point x and find the required path.
The previous result allows us to define a control distance associated to the vector
fields (see [24] for the definition in the smooth case).
Definition 2.2. For all x; x0 2 Rn we call P .x; x0 / the set of the piecewise continuous integral curves of X1 ; : : : ; Xm connecting x0 and x. We define
®
dc .x0 ; x/ D inf T > 0 j 9 W Œ0; T ! Rn ; 2 P .x; x0 /;
¯
(2.10)
.0/ D x0 ; .T / D x :
We define spaces of Hölder continuous functions related to the distance dc :
Definition 2.3. Let 0 < ˛ 1 and let Rn . We say that u W ! R is of class
Cd˛c ./ if there exists a positive constant M such that
ju.x/
u.x0 /j Mdc˛ .x; x0 /;
for every x; x0 2 .
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Fundamental solutions
We say that u 2 Cd1;˛
./ if u 2 CX1 ./ and Xi u 2 Cd˛c ./ for i D 1; : : : ; m
c
2;˛
and that u 2 Cdc ./ if Xi Xj u 2 Cd˛c ./ for i; j D 1; : : : ; m.
Finally, u 2 Cd˛c ;loc ./ if u 2 Cd˛c .0 / for every compact subset 0 of .
Several versions of the Taylor formula for smooth functions on homogeneous
Carnot groups have been proved in Chapter 20 of [3] (see also [1]). In our case,
a function belonging to Cd1;˛
has the following natural Taylor expansion, which is
c
deeply related to Theorem 2.1.
Theorem 2.4. Let be a open subset of Rn . For any u 2 Cd1;˛
./ and x0 2
c
we have
u.x/ D u.x0 / C
m
X
Xi u.x0 /.x
x0 /i C O dc .x0 ; x/1C˛
(2.11)
i D1
as x ! x0 .
Hence, we will call the function
Px10 u.x/
WD u.x0 / C
m
X
Xi u.x0 /.x
x0 /i
(2.12)
i D1
the Taylor polynomial of order one of u with initial point x0 .
Proof. Let x; x0 be fixed. We choose x1 ; x2 ; : : : ; xm ; xmC1 ; : : : ; xmC4 as in Theorem 2.1 and assume, for simplicity, that x D xmC4 . Furthermore, suppose that
.t/ D exp.t.x x0 /1 X1 /.x0 / is the integral curve of .x x0 /1 X1 connecting
x0 and x1 given by (2.1). Then u ı is of class C 1;˛ in the Euclidean sense, and
by the classical Taylor formula, we have
u.x1 / D u.x0 / C X1 u.x0 /.x x0 /1 C O dc .x0 ; x/1C˛
as x ! x0 : (2.13)
Analogously,
u.x2 / D u.x1 / C X2 u.x1 /.x
D u.x0 / C X1 u.x0 /.x
x0 /2 C O dc .x0 ; x/1C˛
x0 /1 C X2 u.x0 /.x
x0 /2 C O dc1C˛ .x0 ; x/ :
Using repeatedly that u 2 Cd1;˛
./ and the definition of x3 ; : : : ; xm in (2.1), we
c
get
u.xm / D u.x0 / C
m
X
i D1
Xi u.x0 /.x
x0 /i C O dc1C˛ .x0 ; x/
as x ! x0 :
(2.14)
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Note that we have dc .x0 ; xm / dc .x0 ; x/ and dc .xm ; xmC4 / D dc .xm ; x/ c.dc .xm ; x0 / C dc .x0 ; x// 2c dc .x0 ; x/. Set d D dc .xm ; xmC4 /. Now, by the
definition of xmC4 in (2.2) and by the classical Taylor formula, we obtain
X i;k
u.x/ D u.xmC4 / D u.xmC3 / d
mC1 .xm /Xi u.xmC3 / C O.d 1C˛ /
i
(by the definitions of xmC3 , xmC2 and xmC1 in (2.2))
X i;k
D u.xmC2 / d
mC1 .xm /Xk u.xmC2 /
k
X
d
i;k
mC1 .xm /Xi u.xmC3 /
C O.d 1C˛ /
i
D u.xm / C d
X
i;k
mC1 .xm /Xk u.xm /
k
Cd
X
X
i;k
.x
/X
u.x
/
mC2
k
mC1 m
k
i;k
mC1 .xm /Xi u.xmC1 /
i
X
i;k
.x
/X
u.x
/
C O.d 1C˛ /:
i
mC3
mC1 m
i
The conclusion results from inserting in the last identity the expression of u.xm /
given by (2.14), and using the regularity of u.
3
Frozen vector fields
Assume the intrinsic condition (1.5). Because of Theorem 2.4, it seems natural to
call
n
X
Xj; WD @j C
P1 aj;k .x/ @k .j D 1; : : : ; m/
(3.1)
kDmC1
the freezing of X1 ; : : : ; Xm at 2 Rn .
Remark 3.1. We have
Xj; D Xj C
n
X
.P1 aj;k .x/
aj;k .x//@k
.j D 1; : : : ; m/:
(3.2)
kDmC1
Since the derivatives @k , for k D m C 1; : : : ; n, act as second-order derivative,
the operator .P1 aj;k .x/ aj;k .x//@k is a differential operator of degree 1 ˛,
according to a definition in [26], while Xi and Xi; have degree 1. This means
that the intrinsic derivative equals the frozen derivative, plus a lower order term.
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Fundamental solutions
Note that, for every 2 Rn the following relations hold:
ŒXl; ; Xj; D ŒXl ; Xj ./
D
n
X
.Xl aj;k ./
Xj alk .//@k
.l; j D 1; : : : ; m/:
kDmC1
Then the Lie algebra generated by X1; ; : : : ; Xm; has the same structure as the
Lie algebra generated by X1 ; : : : ; Xm , at least up to the step 2.
3.1
Distances
As in [24], we consider two natural distances associated to the given vector fields
X1 ; : : : ; Xm and X1; ; : : : ; Xm; .
Remark 3.2. If x1 2 Rn , then the map
Ex1 W u ! exp
m
X
ui Xi C
i D1
n
X
!
ui @i .x1 /
(3.3)
i DmC1
is a diffeomorphism of a neighborhood of the origin of Rn into a neighborhood
Ux1 of x1 in Rn . Its inverse defines the canonical change of variables associated
to the vector fields X1 ; : : : ; Xm , @mC1 ; : : : ; @n and center x1 . By the local invertibility theorem, the open set Ux1 continuously depends on x1 . Hence, in the sequel
we will assume that U b Rn is a fixed open set such that Ex11 .x2 / exists for any
x1 ; x2 2 U .
We are now in a position to give the following definition.
Definition 3.3. Let U be as above and let x1 ; x2 2 U . Let us denote the canonical
coordinates of x2 around x1 , with respect to X1 ; : : : ; Xm , by u1 ; : : : ; un , i.e.
!
m
n
X
X
ui Xi C
ui @i .x1 /:
x2 D exp
i D1
i DmC1
We define Q by (1.4) and
d.x1 ; x2 / m
X
i D1
Q
jui j C
n
X
jui j
Q
2
!1=Q
:
i DmC1
It has been proved in [23] that d is locally equivalent to the control distance dc
defined in (2.10).
Analogously, we can define a distance with respect to the frozen vector fields.
In this case, it is well known that the exponential map is a global diffeomorphism.
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M. Manfredini
Definition 3.4. Let 2 Rn . For every x1 ; x2 2 Rn we denote the canonical
coordinates of x2 around x1 , with respect to X1; ; : : : ; Xm; , by u1 ; : : : ; un , i.e.
!
m
n
X
X
x2 D exp
ui Xi; C
ui @i .x1 /:
i D1
i DmC1
We define
d .x1 ; x2 / m
X
n
X
Q
jui j C
i D1
jui j
Q
2
!1=Q
:
i DmC1
We generalize a result from [12] and prove that the distances in Definition 3.3
and in Definition 3.4 are locally equivalent.
Lemma 3.5. Let U be as in Remark 3.2 and let K be a compact subset of U . There
are positive constants c1 and c2 such that
c1 d.x; x0 / dx0 .x; x0 / c2 d.x; x0 /;
for every x; x0 2 K.
Proof. Let vi be the canonical coordinates
of x around xP
0 w.r.t. X1;x0 ; : : : ; Xm;x0 .
P
n
If is the integral curve .t
P /D m
v
X
..t//
C
i D1 i i;x0
iDmC1 vi @i ..t// such
that .0/ D x0 , we have
001
001
0
1
0
1
0
0
1
0
Px1 a1;n
Px1 am;n
B
C
B
C
B 00 C
B 00 C
1
0
C
B
C
B:C
B
C
B
::
::
B
C
B
C
B:C
B C
.t/
P
D v1 B 1 :
CC Cvm B 1 :
CCvmC1 B :: CC Cvn B :: C :
B Px0 a1;mC1 C
B Px0 am;mC1 C
B1C
B0C
::
::
@
A
@
A
@ :: A
@ :: A
:
:
:
:
0
0
0
1
If i , x0i and xi are the components of , x0 and x respectively, then
vi D xi
x0i
.i D 1; : : : ; m/:
Let us consider the last components. We have
Pi .t/ D
m
X
vk Px10 ak;i ..t // C vi
.i D m C 1; : : : ; n/:
kD1
Integrating between 0 and 1, the above identity gives
Z 1
m
X
xi x0i D
vk
Px10 ak;i ..//d C vi .i D m C 1; : : : ; n/:
kD1
(3.4)
0
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985
Fundamental solutions
Analogously, let ui be the canonical coordinates of the point xP
around the point x0
P .t/ D m
with
respect
to
X
;
:
:
:
;
X
.
If
e
is
the
integral
curve
e
.t// C
1
m
i D1 ui Xi .e
Pn
u
@
..t
//
such
that
e
.0/
D
x
,
then
0
i DmC1 i i
vi D ui
.i D 1; : : : ; m/:
Arguing as before yields
x0i D
xi
m
X
1
Z
ak;i .e
.//d C ui
uk
0
kD1
.i D m C 1; : : : ; n/:
(3.5)
Subtracting (3.4) and (3.5), and using the regularity of ak;i , we get
Z 1
m
X
1
vi ui D
uk
ak;i .e
.// Px0 ak;i ..// d 0
kD1
m
X
(3.6)
juk j.dx0 .x; x0 / C d.x; x0 //
.i D m C 1; : : : ; n/
kD1
by the definitions of Px10 ak;i , dx0 .x; x0 / and d.x; x0 /.
From the definition of d.x; x0 / we have
0
!1=2 1
m
n
X
X
2
A
d Q .x; x0 / c @
jui j C
jui j
i D1
2
Q
x0
cd
i DmC1
.x; x0 / C
n
X
(3.7)
!1=2
juk
vk j
:
kDmC1
Then, by (3.6), we have
n
X
2
Q
x0
jui vi j c d
.x; x0 /Cc
i DmC1
m
X
!1=2
juk j.dx0 .x; x0 /Cd.x; x0 //
; (3.8)
kD1
hence
d
2
Q
2
Q
x0
.x; x0 / c d
.x; x0 / C c
!1=2
m
1X
2
2
jui j C "d .x; x0 /
"
i D1
for every " > 0.
Then, for d.x; x0 / and " sufficiently small, we obtain
d.x; x0 / C dx0 .x; x0 /:
The inequality dx0 .x; x0 / c2 d.x; x0 / follows analogously.
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986
3.2
M. Manfredini
The frozen fundamental solution
Let 2 Rn . Consider the frozen vector fields X1; ; : : : ; Xm; defined in (3.1).
We call
m
X
2
(3.9)
L WD
Xj;
j D1
the freezing of L at . L is a sub-Laplacian on a homogeneous Carnot group up
to a change of variables, in particular it admits a fundamental solution. Indeed,
the family X1; ; : : : ; Xm; has the same structure at every point, then it can be
transformed, by the canonical change of variables
W Rn ! Rn
associated to it, into a family Y1 ; : : : ; Ym which does not depend on , see Section 5.3.1 in [3]. More precisely, the exponential map E;
E; W u ! exp
m
X
i D1
ui Xi; C
n
X
!
ui @i ./
i DmC1
is a diffeomorphism. Its inverse is called the canonical change of variables associated to Xi; and center . We denote by the canonical change of variables
with D . In our case, the exponential map E; can be explicitly written by
solving the Cauchy problem
8
m
n
X
X
ˆ
ˆ
< .t
P /D
uj Xi; ..t // C
uj @j ..t//;
j D1
ˆ
ˆ
:
j DmC1
.0/ D ;
and letting E; .u/ D .1/. That is
!
8
m
n
n
X
X
X
ˆ
ˆ
1
< .t/
P
D
uj ej C
P aj;k ..t //ek C
uj ej ..t//;
j D1
ˆ
ˆ
:
.0/ D ;
kDmC1
j DmC1
(3.10)
where e1 ; : : : ; en is the standard basis in Rn . Now, Pk .t/ D uk ek for k D 1; : : : ; m,
so that
k .t/ D uk t C k .k D 1; : : : ; m/:
(3.11)
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987
Fundamental solutions
If k D m C 1; : : : ; n, by definition of and (3.11), we have
m
X
Pk .t/ D
uj P1 aj;k ..t // C uk
j D1
m
X
D
uj aj;k ./ C
m
X
j D1
sD1
m
X
m
X
D
uj aj;k ./ C
!
/s C uk
Xs aj;k ./..t/
!
Xs aj;k ./us t
C uk :
sD1
j D1
Thus integrating over the interval Œ0; 1, the above identity gives
k .1/ D k C
m
X
aj;k ./uj C uk C
m
1 X
Xs aj;k ./uj us :
2
j;sD1
j D1
Then
E; .u/ D .1/
m
n
X
X
1
D
i C Ai ./u C ui C XAi ./u u ei ;
.ui C i /ei C
2
i D1
i DmC1
where
m
X
Ai ./u D
ak;i ./uk ;
XAi ./u u D
m
X
Xs aj;i ./uj us ;
j;sD1
kD1
for i D m C 1; : : : ; n. Finally
m
X
.zi
.z/ D
i D1
i /ei C
n
X
i DmC1
zi
i
Ai ./ .z
1
XAi ./.z
2
/
/ .z
/ :
Notice that since the family X1; ; : : : ; Xm; has the form (3.1), the determinant
of the Jacobian matrix of , det J , depends only on . Besides, it is locally
bounded by the regularity assumption on coefficients ai;j , i.e., for every compact
set K Rn there are positive constants c1 ; c2 such that c1 det J c2 for
every 2 K, and the function ! det J is Cd˛c ;loc continuous.
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988
M. Manfredini
We indicate by
Y1 ; : : : ; Ym
the vector fields which are the image of X1; ; : : : ; Xm; with respect to the canonical change of variables, that is, for every sufficiently smooth function u
Xi; u D Yi uY
.i D 1; : : : ; m/;
(3.12)
where uY D u ı 1 .
We note that the family Y1 ; : : : ; Ym generates a Lie algebra whose Lie group
G D .Rn ; ˚; ı / is a homogeneous Carnot group of step two, with group law ˚
and dilation group ı .x/ D .x1 ; : : : ; xm ; 2 xmC1 ; : : : ; 2 xn /, > 0. A homogeneous norm with respect to the dilation ı is given by
m
X
kykY n
X
Q
jyi j C
i D1
jyi j
Q
2
!1=Q
;
i DmC1
and the associated distance is
dY .x; y/ D ky
1
˚ xkY :
Then the following equation holds:
d .z; / D dY .
.z/;
.//:
Besides,
L u D LY uY ;
Pm
where LY D i D1 Yi2 is a sub-Laplacian on G, according to a definition in [3].
Then there exists a fundamental solution Y for LY (Theorem 5.3.2 in [3]), which
is invariant with respect to the left ˚-translation and is homogeneous of degree
2 Q. Hence
.z; / D .det J / Y . .z/; .//
(3.13)
is a fundamental solution for L . The function Y and its derivatives are homogeneous with respect to the dilations, and satisfy the following standard inequalities:
2 Q
.x; y/;
1 Q
.x; y/;
jY .x; y/j c dY
jYix .x; y/j c dY
Q
jYjx Yix .x; y/j c dY .x; y/
.x ¤ y; i; j D 1; : : : ; m/:
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989
Fundamental solutions
Hence
j .z; /j cjY .
.z/;
.//j
2 Q
c dY
.
.z/;
.//
D c d2
Q
.z; /
and
jXi; .z; /j D jYi Y .
.z/;
.//j
c d1
Q
.z; /:
Analogous inequalities hold also for the second-order derivatives. We can estimate
.z; / .z 0 ; / and the difference of the frozen derivatives similarly. Then
the following result holds.
e be a compact subset of Rn and let 2 K.
e There exists a
Theorem 3.6. Let K
fundamental solution D .z; / of L such that is smooth away from the
diagonal of Rn Rn and
Lz .z; / D
ı .z/:
For every compact subset K of Rn Rn and p 2 ¹0; 1; 2º there exists c > 0 such
that
jXiz1 ; Xizp ; .z; /j c d2
p Q
.z; /
.i1 ; ip 2 ¹1; : : : ; mº/;
(3.14)
for every z; 2 K satisfying z ¤ . If p D 0, no derivative is applied on . For
i; j 2 ¹1; : : : ; mº there exist positive constants c and C such that
1 Q
1 Q 0
.z; / C d
.z ; / ;
(3.15)
j .z; / .z 0 ; /j c d .z; z 0 / d
Q
Q
z
z
jXi;
.z; / Xi;
.z 0 ; /j c d .z; z 0 / d .z; / C d .z 0 ; / (3.16)
and
z
z
jXi;
Xi;
.z; /
z
z
Xi;
Xi;
.z 0 ; /j
1 Q
c d .z; z 0 / d
.z; / C d
1 Q
.z 0 ; / ;
(3.17)
for every z; ; z 0 such that d .z; z 0 / C d .z; /, z; z 0 ¤ . The constants c; C
e
are independent of 2 K.
We close this section by estimating the derivatives of with respect to the
initial vector fields X1 ; : : : ; Xm .
For convenience, in the following proofs we shall denote by c and C constants
that will not be always the same.
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990
M. Manfredini
e be compact subsets of Rn and let 2 K.
e Let p 2 ¹1; 2º.
Proposition 3.7. Let K,K
Then there exists a positive constant c such that
jXiz1 Xizp .z; /j c d2
p Q
.z; /
.i1 ; ip 2 ¹1; : : : ; mº/;
(3.18)
e
for every z 2 K, z ¤ . The constant c is independent of 2 K.
Proof. Let p D 1 and i 2 ¹1; : : : ; mº. By (3.2), Taylor’s formula (2.11) and (3.14),
we obtain
z
z
jXiz .z; /j D j.Xiz Xi;
/ .z; / C Xi;
.z; /j
ˇ m
ˇ
ˇX
ˇ
1
z
z
ˇ
Dˇ
.ai;k .z/ P ai;k .z//@k .z; / C Xi; .z; /ˇˇ
kD1
c d1C˛ .z; /d Q .z; / C d1
Q
.z; / c d1
Q
.z; /:
Let p D 2 and i; j 2 ¹1; : : : ; mº. We write
Xiz Xjz .z; / D Xiz .Xjz
D .Xiz
z
z
Xj;
/ .z; / C Xiz .Xj;
/ .z; /
z
Xi;
/..Xjz
z
C Xi;
.Xjz
C .Xiz
z
Xj;
/ .z; //
(3.19)
z
Xj;
/ .z; /
z
z
z
z
Xi;
/Xj;
.z; / C Xi;
Xj;
.z; /
and evaluate each term separately.
By (3.2) and arguing as before, one has
j.Xiz
z
z
Xi;
/..Xjz Xj;
/ .z; //j
ˇX
X
m
ˇ m
.ai;k .z/ P1 ai;k .z//@zk
.aj;l .z/
D ˇˇ
kD1
lD1
ˇ m
ˇX
.ai;k .z/
D ˇˇ
P1 ai;k .z//
kD1
C
m
X
.ai;k .z/
m
X
ˇ
ˇ
P1 aj;l .z//@zl .z; / ˇˇ
@k aj;l .z/@zk .z; /
lD1
P1 ai;k .z//.aj;l .z/
k;lD1
c.d1C˛ .z; /d Q .z; / C d2C2˛ .z; /d 2
ˇ
ˇ
P1 aj;l .z//@2kl .z; /ˇˇ
Q
.z; //
(3.20)
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991
Fundamental solutions
since @2kl acts as a derivative of order 4. Similarly,
z
jXi;
.Xjz
z
Xj;
/ .z; /j
ˇ
X
m
ˇ z
.aj;k .z/
D ˇˇXi;
ˇ
ˇ
ˇ
ˇ
P1 aj;k .z//@zk .z; /
kD1
ˇX
ˇ m z
D ˇˇ
Xi; .aj;k .z/
(3.21)
P1 aj;k .z//@zk .z; /
kD1
C
m
X
ˇ
ˇ
z z
P1 aj;k .z//Xi;
@k .z; /ˇˇ:
.aj;k .z/
kD1
z
We first evaluate the term Xi;
.aj;k .z/
we have
P1 aj;k .z//. By the definition of P1 aj;k ,
z
Xi;
.aj;k .z/
P1 aj;k .z//
z
aj;k .z/ aj;k ./
D Xi;
m
X
Xl aj;k ./.z
/l
lD1
D Xi; aj;k .z/
m
X
Xl aj;k ./Xi; .z
/l
lD1
D Xi; aj;k .z/
Xi aj;k ./
D Xi; aj;k .z/
Xi; aj;k ./ C Xi; aj;k ./
Xi aj;k ./:
Hence, by Remark 3.1, we get
z
jXi;
.aj;k .z/
P1 aj;k .z//j c d˛ .z; /:
By (3.21), (3.22), (3.2), (2.11) and (3.14), we obtain
z
z
jXi;
.Xjz Xj;
/ .z; /j c d˛ Q .z; / C d1C˛
1 Q
(3.22)
.z; / :
(3.23)
Analogously,
j.Xiz
z
z
Xi;
/Xj;
.z; /j c d1C˛
Q
.z; /:
(3.24)
Finally, from (3.19), (3.20), (3.23), (3.24) and (3.14), we get the desired estimates.
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992
M. Manfredini
Under the same assumptions as in the previous proposition we have:
Proposition 3.8. Let i; j 2 ¹1; : : : ; mº. There exist positive constants c and C
such that
jXiz .z; / Xiz .z 0 ; /j c d .z; z 0 / d Q .z; / C d Q .z 0 ; /
(3.25)
and
jXiz Xjz .z; /
Xiz Xjz .z 0 ; /j
c d .z; z 0 / d 1 Q .z; / C d 1
Q
(3.26)
.z 0 ; / ;
for every z; z 0 2 K such that d .z; z 0 / C d .z; /, z; z 0 ¤ . The constants c
e
and C are independent of 2 K.
Proof. We write Xiz .z; /
Using (3.2), we have
Xiz .z 0 ; / in terms of the frozen vector fields.
jXiz .z; / Xiz .z 0 ; /j
ˇX
ˇ m
D ˇˇ
.ai;k .z/ P1 ai;k .z//@zk .z; /
kD1
m
X
.ai;k .z 0 /
P1 ai;k .z 0 //@zk .z 0 ; /
kD1
z
C Xi;
.z; /
m
X
ˇ
ˇ
z
Xi;
.z 0 ; /ˇˇ
P1 ai;k .z/
jai;k .z/
ai;k .z 0 / C P1 ai;k .z 0 /jj@zk .z; /j
kD1
C
C
m
X
jai;k .z 0 /
kD1
z
jXi;
.z; /
P1 ai;k .z 0 /jj@zk .z; /
@zk .z 0 ; /j
z
Xi;
.z 0 ; /j:
(3.27)
Theorem 2.4 gives
P1 ai;k .z/
jai;k .z/
ai;k .z 0 / C P1 ai;k .z 0 /j
D jai;k .z/ ˙ Pz10 ai;k .z/
jai;k .z/
Pz10 ai;k .z/j C
ai;k .z 0 /
m
X
P1 ai;k .z/ C P1 ai;k .z 0 /j
jXs ai;k .z 0 /
Xs ai;k .//.z
z 0 /i j
sD1
c.d1C˛ .z; z 0 /
C
d˛ .z 0 ; /d .z; z 0 //:
(3.28)
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993
Fundamental solutions
Then
m
X
jai;k .z/
P1 ai;k .z/
ai;k .z 0 / C P1 ai;k .z 0 /jj@zk .z; /j
kD1
c.d1C˛ .z; z 0 / C d˛ .z 0 ; /d .z; z 0 //d Q .z; /
(3.29)
c d .z; z 0 /.d Q .z; / C d Q .z 0 ; //;
provided d .z; z 0 / C d .z; /.
Since @k acts as a second-order derivative, the mean value theorem (see for
example [3]) yields
j@zk .z; / @zk .z 0 ; /j c d .z; z 0 / d 1 Q .z; / C d 1 Q .z 0 ; / ;
for suitable constants c; C and for every z; ; z 0 such that d .z; z 0 / C d .z; /,
z; z 0 ¤ . Hence, by Theorem 2.4, we have
m
X
jai;k .z 0 /
P1 ai;k .z 0 /jj@zk .z; /
@zk .z 0 ; /j
kD1
c d1C˛ .z; z 0 /d .z; z 0 / d 1
Q
.z; / C d 1
Q
.z 0 ; /
(3.30)
c d .z; z 0 / d Q .z; / C d Q .z 0 ; / ;
provided d .z; z 0 / C d .z; /. Finally, by (3.27), (3.29), (3.30), and (3.16), we
obtain (3.25).
The proof of (3.26) is essentially analogous and is omitted.
Remark 3.9. Thanks to Proposition 3.7, the inequalities in Proposition 3.8 hold
also for any z; z 0 such that d .z; z 0 / C d .z; /. We will use this fact systematically.
4
Parametrix method
In this section, we describe the Levi parametrix method to construct a fundamental
solution for the operator L. We recall that denotes a fundamental solution for
the frozen operator L . According to Levi’s method, we look for a fundamental
solution in the form
.z; / D .z; / C J.z; /:
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994
M. Manfredini
The function J.z; / is unknown and supposed to have the form
Z
J.z; / D .z; /'.; /d ;
where ' has to be determined via successive approximations.
Let be fixed in Rn , U be as in Remark 3.2 and contain . Let T be a bounded
subset of U which contains and will be chosen later.
We set
Z1 .z; / D L.z ! .z; // .z 2 T; z ¤ /;
(4.1)
and, for every j 2 N, put
Z
Zj C1 .z; / D
Z1 .z; /Zj .; /d
T
.z 2 T; z ¤ /:
(4.2)
Remark 4.1. We have
Z1 .z; / D .L L /.z ! .z; //
D
m
X
n
X
P1 ai;k .z//2 @2kk .z; /
.ai;k .z/
i D1 kDmC1
C2
n
X
.ai;k .z/
kDmC1
n
X
C
(4.3)
z
P1 ai;k .z//@zk Xi;
.z; /
!
Xi; ai;k .//@zk .z; /
.Xi; ai;k .z/
;
kDmC1
for every z 2 T n ¹º. Then, by (2.11) and the equivalence of distances, we get
jZ1 .z; /j c d2C2˛ .z; /d
2 Q
C c d˛ .z; /d
c d
QC˛
.z; / C c d1C˛ .z; /d
Q
.z; /
1 Q
.z; /
(4.4)
.z; /;
for every z 2 T n ¹º, where c depends on the intrinsic Hölder norm of coefficients ai;k .
Proposition 4.2. There exists a subset T of Rn which contains and such that
equation (4.2) makes sense. There are j0 2 N, positive constants cT ; c1 , with
c1 < 1, such that, for any z 2 T , z ¤ , one has
jZj .z; /j cT d .z; /
and
QCj˛
j j0 1
jZj .z; /j cT c1
.j j0 /
.j > j0 /:
(4.5)
(4.6)
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995
Fundamental solutions
Proof. Suppose 2˛ < Q. Let be fixed in Rn and let T Rn contain . The
definition of Z2 , estimate (4.4) and the equivalence of distances give
ˇZ
ˇ
ˇ
ˇ
ˇ
jZ2 .z; /j D ˇ Z1 .z; /Z1 .; /dˇˇ
T
Z
˛ Q
c
d˛ Q .z; /d
.; /d
T
Z
cT
d ˛ Q .z; /d ˛ Q .; /d
T
2˛ Q
cT d
.z; /;
by Lemma A.2 in the Appendix. We indicate by cT a positive constant which
depends on T .
Iterating these estimates we reach some j0 such that .j0 C 1/˛ > Q, and
jZj0 .z; /j cT d
QCj0 ˛
.z; / and
jZj0 C1 .z; /j cT :
Then, by the last inequality and (4.4),
Z
Z
ˇ
ˇ
ˇZj C1 .z; /Z1 .; /dˇ cT
jZ1 .; /jd cT c1 ;
jZj0 C2 .z; /j 0
T
T
where c1 is smaller than 1, if the Lebesgue measure of T is sufficiently small. We
have
Z
Z
ˇ
ˇ
ˇ
ˇ
Zj0 C2 .z; /Z1 .; /d cT
jZ1 .; /jd cT c12 ;
jZj0 C3 .z; /j T
T
therefore an induction argument proves (4.6).
We define
'.z; / D
1
X
Zj .z; /;
(4.7)
j D1
for every z 2 T , z ¤ .
As a consequence of Proposition 4.2, we have
Proposition 4.3. The series in (4.7) uniformly converges, and for every compact
subset K of T there exists c > 0 such that
j'.z; /j c d
QC˛
.z; /;
(4.8)
for every z 2 T , z ¤ .
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Remark 4.4. In consequence of Proposition 4.3 we get
Z
1 Z
X
Z1 .z; /'.; /d D
Z1 .z; /Zj .; /d T
j D1 T
D
1
X
Zj C1 .z; /
j D1
D '.z; /
Hence
Z1 .z; /:
Z
'.z; / D Z1 .z; / C
Z1 .z; /'.; /d ;
(4.9)
T
for every z 2 T , z ¤ .
Let us now estimate Z1 .z; / Z1 .z 0 ; /, and consequently deduce an estimate
of '.z; / '.z 0 ; /.
In what follows, will be fixed in Rn and T will be the set as in Proposition 4.2.
Lemma 4.5. For every ˇ, 0 < ˇ < ˛, there exist positive constants c and C such
that
ˇ
˛ ˇ Q
˛ ˇ Q 0
.z; / C d
.z ; / ;
jZ1 .z; / Z1 .z 0 ; /j c d .z; z 0 / d
for every z; ; z 0 such that d .z; z 0 / C d .z; /, z; z 0 2 T; z; z 0 ¤ .
Proof. Using expression (4.3) of Z1 , we write
Z1 .z; /
Z1 .z 0 ; / D .f1 .z; /
f1 .z 0 ; // C .f2 .z; /
C .f3 .z; /
0
f2 .z 0 ; //
f3 .z 0 ; //
0
(4.10)
0
I1 .z; ; z / C I2 .z; ; z / C I3 .z; ; z /;
where
f1 .z; / D
m
n
X
X
.ai;k .z/
P1 ai;k .z//2 @2kk .z; /;
(4.11)
i D1 kDmC1
f2 .z; / D 2
m
n
X
X
.ai;k .z/
z
P1 ai;k .z//@zk Xi;
.z; /;
(4.12)
i D1 kDmC1
f3 .z; / D
m
n
X
X
.Xi; ai;k .z/
Xi; ai;k .//@zk .z; /:
(4.13)
i D1 kDmC1
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Fundamental solutions
We estimate each term in (4.10) separately. Let us start with I1 and write
I1 .z; ; z 0 / D
X
.ai;k .z/
P1 ai;k .z//2
.ai;k .z 0 /
i;k
C
X
P1 ai;k .z 0 //2
.ai;k .z 0 /
i;k
P1 ai;k .z 0 //2 @2kk .z; /
.@2kk .z; /
(4.14)
@2kk .z 0 ; //;
where
j.ai;k .z/
P1 ai;k .z//2
D jai;k .z/
.ai;k .z 0 /
ai;k .z 0 /
P1 ai;k .z 0 //2 j
P1 ai;k .z/ C P1 ai;k .z 0 /j
jai;k .z/ C ai;k .z 0 /
P1 ai;k .z 0 /
(4.15)
P1 ai;k .z/j:
Using (2.11), we have
jai;k .z/ C ai;k .z 0 /
P1 ai;k .z 0 /
P1 ai;k .z/j
c d1C˛ .z; / C d1C˛ .z 0 ; /:
(4.16)
If C is sufficiently small, (4.15), (4.16) and (3.28) tell us that the first term in
(4.14) obeys
P1 ai;k .z//2 .ai;k .z 0 / P1 ai;k .z 0 //2 j j@2kk .z; /j
c d1C˛ .z; z 0 / C d˛ .z 0 ; /d .z; z 0 /
2 Q
d1C˛ .z; / C d1C˛ .z 0 ; / d
.z; /
˛ 1 Q
˛ 1 Q 0
c d .z; z 0 / d
.z; / C d
.z ; / ;
j.ai;k .z/
(4.17)
provided d .z; z 0 / C d .z; /.
Moreover, by the mean value theorem and (2.11), there are c and C such that
P1 ai;k .z//2 .@2kk .z; / @2kk .z 0 ; //j
3 Q
3 Q 0
c d2C2˛ .z; /d .z; z 0 / d
.z; / C d
.z ; /
˛ 1 Q
˛ 1 Q 0
c d .z; z 0 / d
.z; / C d
.z ; / ;
j.ai;k .z/
(4.18)
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provided d .z; z 0 / C d .z; /. Then it follows that
jI1 .z; ; z 0 /j D jf1 .z; /
f1 .z 0 ; /j
˛ 1 Q
˛ 1 Q 0
c d .z; z 0 / d
.z; / C d
.z ; /
ˇ
˛ ˇ Q
˛ ˇ Q 0
c d .z; z 0 / d
.z; / C d
.z ; / ;
(4.19)
for every ˇ, provided C is sufficiently small and d .z; z 0 / C d .z; /.
Let us estimate I2 :
X
I2 .z; ; z 0 / D 2
ai;k .z/ P1 ai;k .z/
z
i;k
ai;k .z 0 / C P1 ai;k .z 0 / @zk Xi;
.z; /
X
.ai;k .z 0 / P1 ai;k .z 0 //
C2
i;k
z
.@zk Xi;
.z; /
z
@zk Xi;
.z 0 ; //:
Hence, by (3.28) and the mean value theorem again,
1 Q
jI2 .z; ; z 0 /j c d1C˛ .z; z 0 / C d˛ .z 0 ; /d .z; z 0 / d
.z; /
2 Q
2
.z; / C d
C c d1C˛ .z; z 0 /d .z; z 0 / d
ˇ
˛ ˇ Q
˛ ˇ Q 0
c d .z; z 0 / d
.z; / C d
.z ; / ;
Q
.z 0 ; /
(4.20)
provided C is sufficiently small and d .z; z 0 / C d .z; /.
We have
X
I3 .z; ; z 0 / D
Xi; ai;k .z/ Xi; ai;k .z 0 / @zk .z; /
i;k
C
X
Xi; ai;k .z 0 /
Xi; ai;k ./ @zk .z; /
@zk .z 0 ; / :
i;k
Then
jI3 .z; ; z 0 /j c d˛ .z; z 0 /d
Q
.z; /
1 Q
1
C c d˛ .z; /d .z; z 0 / d
.z; / C d
ˇ
˛ ˇ Q
˛ ˇ Q 0
c d .z; z 0 / d
.z; / C d
.z ; /
Q
.z 0 ; /
(4.21)
if ˇ < ˛, C is sufficiently small and d .z; z 0 / C d .z; /.
By (4.10), (4.19), (4.20) and (4.21), we conclude the proof.
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999
Fundamental solutions
Proposition 4.6. For every ˇ, 0 < ˇ < ˛, there exist c; C > 0 such that
ˇ
˛ ˇ Q
˛ ˇ Q 0
j'.z; / '.z 0 ; /j c d .z; z 0 / d
.z; / C d
.z ; / ;
(4.22)
for every z; ; z 0 such that d .z; z 0 / C d .z; /, z; z 0 2 T , z; z 0 ¤ .
Proof. The assertion can be demonstrated by arguing as in the proof of Proposition 4.2, using Lemma 4.5, the definition of ' and Lemma A.2 in the Appendix.
Let be fixed. We set
Z
J.z; / D
.z; /'.; /d (4.23)
T
and define
.z; / D J.z; / C .z; /;
(4.24)
for every z 2 T , z ¤ .
Proposition 4.7. Let K be a compact subset of T . There exist c; C > 0 such that
2C˛ Q
jJ.z; /j c d
.z; /
(4.25)
and
jJ.z; /
˛C1
J.z 0 ; /j c d .z; z 0 / d
Q
˛C1 Q
.z; / C d
.z 0 ; / ;
(4.26)
for every z; ; z 0 such that d .z; z 0 / C d .z; /, z; z 0 2 K, z; z 0 ¤ .
Proof. By (4.8) and Theorem 3.6, we have
ˇZ
ˇ
Z
ˇ
ˇ
2
ˇ
ˇ
d
jJ.z; /j D ˇ .z; /'.; /d ˇ c
T
T
Q
˛ Q
.z; /d
.; /d (by the locally equivalence of distances and Lemma A.2 in the Appendix)
2C˛ Q
c d
.z; /:
By (3.15) and Lemma A.2 in the Appendix,
Z
0
jJ.z; / J.z ; /j j .z; / .z 0 ; /j j'.; /jd T
Z
1 Q
1 Q 0
˛
c
d .z; z 0 / d
.z; / C d
.z ; / d
T
˛C1 Q
˛C1 Q 0
d .z; z 0 / d
.z; / C d
.z ; / :
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1000
M. Manfredini
As a consequence of Propositions 4.7 and 3.7 we have
Proposition 4.8. Let K be a compact subset of T . There exists a positive constant
c such that
QC2
j.z; /j c d
.z; /;
(4.27)
for every z 2 K, z ¤ .
In the following, we estimate the first and the second-order derivatives of J. ; /
and Lz J.z; /.
Lemma 4.9. The function J. ; / is differentiable along X1 ; : : : ; Xm away from and
Z
z
(4.28)
Xjz .z; /'.; /d .j D 1; : : : ; m/;
Xj J.z; / D
T
for every z 2 T , z ¤ . For every compact subset K of T there exists c > 0 such
that
1C˛ Q
.z; / .j D 1; : : : ; m/;
(4.29)
jXjz J.z; /j c d
for every z 2 K, z ¤ .
Proof. We write
Xjz .z; /'.; / D Xjz .z; /
D
n
X
z
z
Xj;
.z; / '.; / C Xj;
.z; /'.; /
P1 aj;k .z/ @zk .z; /'.; /
aj;k .z/
kDmC1
z
C Xj;
.z; /'.; /:
By (2.11), (4.8), Theorem 3.6, and the equivalence of distances, we have
jXjz .z; /'.; /j c d1C˛ .z; /d
1 Q
C c d
˛ Q
c d
Q
˛ Q
.z; /d
˛ Q
.z; /d
1 Q
.z; /d
.; /
.; /
.; /:
By Lemma A.2 in the Appendix, for every compact subset K of T , we have
Z
1C˛ Q
jXjz .z; /'.; /jd c d
.z; / .z 2 K; z ¤ /:
T
This proves that
R
Xjz .z; /'.; /d is well defined for z ¤ .
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1001
Fundamental solutions
Next we prove (4.28). Let us fix a function 2 C 1 .R/ satisfying 0 1,
.t/ D 0 for t 1 and .t/ D 1 for t 2. Now, for every " > 0, we define
Z
J" .z; / D
.z; /'.; / " .z; /d ;
T
d .z;/ "
is such that jXiz " .z; /j c" for i D 1; : : : ; m,
for k D m C 1; : : : ; n. Observe that
Z
Xjz J" .z; / D
Xjz . .z; / " .z; //'.; /d where " .z; / WD
and j@zk " .z; /j c
"2
T
and
Z
Xjz .z; /'.; /d Xjz J" .z; /
T
Z
D
Xjz . .z; /. " .z; / 1//'.; /d T
Z
z
D .Xjz Xj;
/. .z; /. " .z; / 1//'.; /d T
Z
z
. .z; /. " .z; / 1//'.; /d :
Xj;
C
(4.30)
T
We estimate the first integral on the right-hand side of (4.30):
ˇ
ˇZ
ˇ
ˇ
ˇ .X z X z /. .z; /. " .z; / 1//'.; /d ˇ
j
j;
ˇ
ˇ
T
ˇZ
n
X
ˇ
D ˇˇ
.aj;k .z/ P1 aj;k .z//@zk . .z; /. " .z; /
T
kDmC1
Z
¹Wd .z;/2"º
ˇ X
ˇ n
ˇ
.aj;k .z/
ˇ
kDmC1
Z
C
¹W"d .z;/2"º
Z
c
¹Wd .z;/2"º
kDmC1
Z
Cc
2˛ Q
c " d
P1 aj;k .z//@zk .z; /
ˇ
ˇ
. " .z; / 1/'.; /ˇˇd ˇ X
ˇ n
ˇ
.aj;k .z/
ˇ
d1C˛ .z; /d
Q
P1 aj;k .z// .z; /
@zk . " .z; /
˛ Q
.z; /d
2 Q
¹W"d .z;/2"º
ˇ
ˇ
1//'.; /d ˇˇ
d1C˛ .z; /d
ˇ
ˇ
1/'.; /ˇˇd .; /d .z; /
1 ˛
d
"2 Q
.; /d .z; /;
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M. Manfredini
by Lemma A.2 in the Appendix. This proves that the first integral on the right-hand
side of (4.30) converges uniformly to zero on every compact subset of T n ¹º.
Similarly, the second integral on the right-hand side of (4.30) converges uniformly to zero. Lemma 4.9 is proved.
Lemma 4.10. The function J. ; / satisfies
Z
z
Lz .z; /'.; /d L J.z; / D
'.z; /;
(4.31)
T
for every z 2 T , z ¤ .
Proof. We first note that the integral on the right-hand side is convergent. Indeed,
Lz .z; / D Z1 .z; / and Z1 .z; / satisfies (4.4).
Let j 2 ¹1; : : : ; mº. We compute the derivative Xjz of Xjz J.z; / in the distributional sense. We denote by Xj the adjoint operator of Xj . If h 2 C01 .Rn /, we
have
ZZ
Xjz .z; /'.; /d Xj h.z/ dz
T
Z Z
D
T
Xjz .z; /Xj h.z/dz '.; /d Z Z
D lim
"!0 T ¹zW .z;/"2
Qº
Xjz .z; /Xj h.z/dz '.; /d Z Z
D
lim
"!0 T ¹zW .z;/"2
Qº
.Xjz /2 .z; /h.z/dz '.; /d Xjz .z; /Xjz .z; /
Z Z
C lim
"!0 T ¹zW .z;/D"2
Qº
jr z .z; /j
h.z/ dz '.; /d ;
(4.32)
where r denotes the Euclidean gradient and d the surface measure.
Our first goal is to prove that
Xjz .z; /Xjz .z; /
Z Z
T ¹zW .z;/D"2
jr z .z; /j
Qº
z
z
Xj;
.z; /Xj;
.z; /
Z Z
D
T ¹zW .z;/D"2
h.z/ dz '.; /d Qº
jr z .z; /j
dz h./'.; /d C O."˛ /
(4.33)
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1003
Fundamental solutions
as " ! 0. Now,
Xjz .z; /Xjz .z; /
jr z .z; /j
D
h.z/
z
z
Xj;
.z; /Xj;
.z; /
jr z .z; /j
h./
z
z
.z; /
.z; /Xj;
Xj;
Xjz .z; /Xjz .z; /
jr z .z; /j
C
z
z
Xj;
.z; /Xj;
.z; /
jr z .z; /j
.h.z/
h.z/
h.//;
where
ˇ z
ˇ
ˇ X .z; /X z .z; / X z .z; /X z .z; / ˇ
j
j
ˇ
ˇ
j;
j;
ˇ
ˇ
ˇ
ˇ
jr z .z; /j
ˇ
ˇ Pn
ˇ.
P1 aj;k .z//@zk .z; //2 ˇˇ
kDmC1 .aj;k .z/
ˇ
Dˇ
ˇ
ˇ
ˇ
jr z .z; /j
c
d .z; /2C2˛ 2Q
:
jr z .z; /j
It follows that
Z
ˇ z
ˇ
ˇ X .z; /X z .z; / X z .z; /X z .z; / ˇ
j
ˇ j
ˇ
j;
j;
ˇ
ˇ h.z/dz
z .z; /j
ˇ
ˇ
2
Q
jr
¹zW .z;/D"
º
Z
1
1
c "1C2˛ Q Q 1
dz ;
(4.34)
z .z; /j
2
Q
jr
"
¹ .z;/D"
º
and, by means of Lemma A.3 in the Appendix, we conclude that the integral on
the right-hand side of (4.34) equals O."2˛ / as " ! 0.
Analogously, we obtain
ˇ z
ˇ
2 2QC˛
ˇ X .z; /X z .z; /
ˇ
d
.z; /
ˇ
ˇ j;
j;
.h.z/ h.//ˇ c
:
ˇ
z
z
ˇ
ˇ
jr .z; /j
jr .z; /j
By Lemma A.3 in the Appendix, we have
ˇ z
Z Z
ˇ X .z; /X z .z; /
ˇ j;
j;
.h.z/
ˇ
jr z .z; /j
T ¹zW .z;/D"2 Q º ˇ
ˇ
ˇ
ˇ
h.//ˇ dz D O."˛ /:
ˇ
(4.35)
Then inequality (4.33) follows from (4.34) and (4.35).
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1004
M. Manfredini
By identity (4.32) and inequality (4.33), one has
Z Z
Xjz .z; /'.; /d Xj h.z/ dz
T
Z Z
D lim
.Xjz /2 .z; /h.z/ dz'.; /d "!0 T ¹zW .z;/"2
Qº
z
z
.z; /Xj;
.z; /
Xj;
Z Z
C lim
"!0 T ¹zW .z;/D"2
ZZ
D
T
jr z .z; /j
Qº
dz h./'.; /d .Xjz /2 .z; /'.; /d h.z/ dz
z
z
Xj;
.z; /Xj;
.z; /
Z Z
C lim
"!0 T ¹zW .z;/D"2
Qº
jr z .z; /j
dz h./'.; /d :
(4.36)
Since is a fundamental solution of L , we have (see [10])
lim
"!0
z
z
Xj;
.z; /Xj;
.z; /
m Z
X
2
j D1 ¹zW .z;/D"
Qº
jr z .z; /j
dz D 1:
(4.37)
Then (4.36) and (4.37) complete the proof of the lemma.
4.1
Properties of
Let T be the set as in Proposition 4.2.
Proposition 4.11. We have
Lz .z; / D
ı .z/
on T;
where ı is the Dirac measure supported at ¹º.
Proof. By Lemma 4.10 and identity (4.9), we have
Z
ZZ
Z
z
J.z; /L h.z/ dz D
L .z; /'.; /d h.z/ dz
'.z; /h.z/ dz
T
ZZ
Z
D
Z1 .z; /'.; /d h.z/ dz
'.z; /h.z/ dz
T
Z
D
Z1 .z; / h.z/ dz;
(4.38)
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1005
Fundamental solutions
for every h 2 C01 .T /. Moreover,
Z
Z
.z; /L h.z/ dz D
Lz .z; /h.z/ dz
h./
p:v:
(4.39)
Z
D
Z1 .z; /h.z/ dz
h./:
Hence, by (4.38) and (4.39), the conclusion follows.
Proposition 4.12. The function J. ; / is twice differentiable along X1 ; : : : ; Xm
away from , and for every i; j D 1; : : : ; m
Xiz Xjz J.z; / D
Z
p:v:
Xjz Xiz .z; /'.; /d 1
C cij
.det J
(4.40)
2
/
'.z; /
.z 2 T; z ¤ /;
for a suitable constant cij which does not depend on . (The integral on the righthand side is a principal value integral.)
Proof. We argue as in Lemma 4.10. For every i; j D 1; : : : ; m we compute the
derivative Xiz of Xjz J.z; / in (4.28) in the distributional sense: if h 2 C01 .Rn /,
we have
ZZ
Xjz .z; /'.; /d Xi h.z/ dz
T
ZZ
D
T
Xiz Xjz .z; /'.; /d h.z/ dz
z
z
Xi;
.z; /Xj;
.z; /
Z Z
C lim
"!0 T ¹zW .z;/D"2
Qº
jr z .z; /j
dz h./'.; /d :
In order to evaluate the term
z
z
Xi;
.z; /Xj;
.z; /
Z
¹zW .z;/D"2
Qº
jr z .z; /j
dz ;
we argue as in Lemma A.2 in the Appendix. We recall that is defined as
in (3.13). By Federer’s coarea formula (see Theorem 3.2.12 in [15]) and using
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1006
M. Manfredini
the canonical change of variables, we have
"
Z
1
tQ
0
z
z
.z; /
.z; /Xj;
Xi;
Z
1
¹zW .z;/Dt 2
jr z .z; /j
Qº
Z
D .Q
2/
¹zW
.z;/>"2 Q º
z
z
Xi;
.z; /Xj;
.z; / dz
Z
D .Q
2/
¹zW.det J
2
.z/;0/>"
/Y .
Z
D .Q
2/
¹W.det J
dz dt
/Y .;0/>"
2 Qº
Qº
Yiz Y .
z
.z/; 0/Yj Y . .z/; 0/ dz
Yiz Y .; 0/Yjz Y .; 0/
1
det J
d:
(4.41)
Using the homogeneity of Y and of its derivatives Yi Y , i D 1; : : : ; m, and
1
letting e
D ıY .." det J / Q 2 /, we have
Z
Yi Y .; 0/Yj Y .; 0/ d
¹W.det J
2 Qº
/Y .;0/>"
1
D
det J
"
2 Q
(4.42)
Z
Yi Y .e
; 0/Yj Y .e
; 0/ de
:
¹e
WY .e
;0/>1º
Differentiating (4.41) with respect to " and using (4.42), we get
1
"Q
z
z
Xi;
.z; /Xj;
.z; /
Z
1
¹zW .z;/D"2
D .Q
2/.2
Qº
Q/"1
dz
jr z .z; /j
Z
1
Q
Yi Y .e
; 0/Yj Y .e
; 0/ de
;
.det J /2 ¹e
WY .e
;0/>1º
which ends the proof if we put
Z
cij D .Q
2/.2
Q/
Yi Y .e
; 0/Yj Y .e
; 0/ de
:
¹e
WY .e
;0/>1º
Proposition 4.13. For every 0 < ˇ < ˛ the function J. ; / belongs to the space
2;ˇ
Cdc ;loc .T n ¹º/.
Proof. By representation formula (4.28) of the first-order derivatives of J and by
results on fractional integrals (see [4]), we obtain J. ; / 2 Cd˛c ;loc .T n ¹º/. By
representation formula (4.40) of the second-order derivatives and the Schauder
estimates proved in [4], we get the desired result.
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Fundamental solutions
1007
Proposition 4.14. We have
Lz .z; / D 0;
for every z 2 T , z ¤ .
Proof. By (4.24) and Lemma 4.10, we have
Lz .z; / D Lz .J.z; / C .z; //
Z
D
Lz .z; /'.; / d '.z; / C Z1 .z; /
T
Z
Z1 .z; /'.; / d '.z; / C Z1 .z; /;
D
T
and the last term equals 0 according to (4.9).
Proof of Theorem 1.1. By Proposition 4.11, we have Lz .z; / D ı .z/ on T ,
and by Proposition 4.14, we get Lz .z; / D 0 for z ¤ .
QC2
.z; / locally, and (4.29), (3.18) give
Estimate (4.27) gives j.z; /j c d
1
Q
z
.z; / locally. Finally, by Propositions 4.13 and 3.8, we have
jXj .z; /j c d
2;ˇ
. ; / 2 Cdc ;loc .T n ¹º/, for every ˇ < ˛.
A
Appendix
Lemma A.1. Let ˛ 2 RC and let K be a compact subset of Rn . There exists a
positive constant c such that
Z
˛ Q
d
.z; / d c r ˛ ;
¹Wd .z;/rº
for every r > 0 and for all z 2 K.
Proof. The proof is a standard computation, see for example Proposition A.3
in [10].
Lemma A.2. Let ˛; ˇ 2 RC . Given a bounded subset T of Rn , there exists a
positive constant cT such that
Z
d ˛ Q .z; /d ˇ Q .; / d cT d ˛Cˇ Q .z; /
T
if ˛ C ˇ < Q, and for every z; 2 T , z ¤ . If ˛ C ˇ > Q, then the integral on
the left-hand side is bounded.
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1008
M. Manfredini
Proof. Suppose that ˛ C ˇ < Q. The proof is similar to that of Proposition A.5
in [10] and is obtained by breaking the set T into
T1 D ¹ W d.z; / " d.; /º;
T2 D ¹ W d.; / " d.z; /º;
T3 D ¹ W d.z; / " d.z; /º n .T1 [ T2 /;
T4 D T n .T1 [ T2 [ T3 /;
where " > 0 will be chosen later. We estimate the corresponding integral separately. Let us first consider the integral over T1 . If 2 T1 , then
d.z; / c.d.z; / C d.; // c.1 C "/d.; /;
d.; / c d.; z/ C c "d.; /:
If " is small enough, there are c1 ; c2 > 0 such that
c2 d.; / d.; z/ c1 d.; /:
Then
Z
d˛
Q
.z; /d ˇ
Q
.; / d
T1
cd
ˇ Q
c dˇ
Z
.z; /
d˛
Q
.z; / d
T1
Q
Z
d˛
.z; /
c d ˇ C˛
Q
.z; / d
¹W d.z;/c d.z;/º
Q
.z; /;
by means of Lemma A.1. The estimates of the integrals over T2 and T3 are similar.
In T4 , the distance d.; / is equivalent to d.z; /. Then
Z
d ˛ Q .z; /d ˇ Q .; / d
T4
Z
c
d ˛Cˇ
Q
.z; / d
T4
1 Z
X
kD1
c
d ˛Cˇ
Q
.z; / d
¹2k d.z;/d.z;/2kC1 d.z;/º
Z
1
X
d ˛Cˇ .z; /2.˛Cˇ /.kC1/
d
d 2Q .z; /22kQ
¹d.z;/2kC1 d.z;/º
kD1
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1009
Fundamental solutions
cd
˛Cˇ 2Q
1
X
2Q.kC1/ d Q .z; / .˛Cˇ /.kC1/
2
.z; /
22kQ
kD1
c d ˛Cˇ
Q
1
X
.z; /
2k.˛Cˇ
Q/
;
kD1
where the series converges since ˛ C ˇ < Q.
If ˛ C ˇ > Q, the integral over T4 is bounded.
Lemma A.3. Let 2 Rn . Then for any " > 0
Z
1
1
dz D c "Q
"Q 1 ¹zW .z;/D"2 Q º jr z .z; /j
R
where c D Q.Q 2/ .det J /1=.Q 2/ ¹WY .;0/>1º d.
1
;
Proof. By Federer’s coarea formula (Theorem 3.2.12 in [15]), we get
Z "
Z
1
1
dz dt
z
Q
1
¹zW .z;/Dt 2 Q º jr .z; /j
0 t
Z "Z
1
dz dt
D .Q 2/
1
z
0
¹zW. .z;// 2 Q Dt º jr .z; /j
Z
dz:
D .Q 2/
¹zW .z;/> "2
(A.1)
Qº
We recall that .z; / D .det J /Y . .z/; .// D .det J /Y . .z/; 0/.
Hence, by the change of variables D .z/ defined in Section 3.2, we have
Z
Z
dz D
dz
¹zW .z;/> "2
Qº
¹zW.det J
D
1
det J
D
1
det J
/ Y .
2 Qº
.z/;0/> "
Z
d
¹W.det J
"Q
(A.2)
2 Qº
/ Y .;0/> "
1
.det J /Q=.2
Z
Q/
d;
¹WY .;0/>1º
by means of the homogeneity of Y . Hence, differentiating (A.1) with respect to
" and using (A.2), we get the desired result.
Acknowledgments. I would like to thank Professor G. Citti for many encouragements and useful conversations during the preparation of this work.
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1010
M. Manfredini
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Received September 15, 2008; revised September 27, 2010.
Author information
Maria Manfredini, Dipartimento di Matematica, Universitá di Bologna,
Piazza Porta S. Donato 5, 40126 Bologna, Italy.
E-mail: manfredi@dm.unibo.it
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