Regularity for Quasilinear
Second-Order Subelliptic Equations
CHAO-JIANG XU
Courant Institute
and
Wuhan University
Abstract
In this paper, we study the regularity of solutions of the quasilinear equation
where X = ( X , ; . . , X , , , ) is a system of real smooth vector fields, A i j , B E Cw(Q x R m + l ) .
Assume that X satisfies the Hormander condition and ( A , , ( x , z , c ) )is positive definite. We
prove that if u E S2@(Q)(see Section 2) is a solution of the above equation, then u E Cw(Q).
Introduction
In this work, we study the regularity of solutions of the following quasilinear second-order degenerate elliptic equation:
is a system of real smooth vector fields defined
where X = (X,,...,X,)
in an open domain M of R", n 2 3, and Ajj, B E C"(R x Rrn+'), i, j =
1,. - . ,rn. We assume X satisfies the Hormander condition (see Section 1)
and ( A j j ( X , z , < ) )is positive definite on M x BBrn+'. Equation (*) is degenerate elliptic in general. But this degeneracy is described by a system of vector
fields and it is independent of solutions. So we call equation (*) subelliptic
as in the linear case. The Hormander condition permits us to define a metric p ( x , y ) associated with X on SZ cc M (see Section 1). In the induced
geometry, the Hormander operator H = Cy==,
X2 + c(x) possesses properties
similar to those of the Laplacian. For examp[e, one has the precise estimation of Green's kernel, the Poincare and Harnack inequalities, etc. (See
[l], [2], [7], [8], [ 101.) We use these properties to construct a Schauder type
estimate for the operator H in the associate function space Sk@(SZ)(see Section 2). As for the elliptic equations, we prove that the Schauder estimate implies the regularity of solutions for quasilinear equations, i.e., if u E S2*a(SZ),
Communications on Pure and Applied Mathematics, Vol. XLV, 77-96 (1992)
CCC 00 10-3640/92/010077-20$04.00
0 1992 John Wiley & Sons, Inc.
78
C.-J. XU
> 0, is a solution of equation (*), then u E Cm(sZ)(see Theorem 4.2). The
existence of solutions for the related variational problems is studied in [ 131.
cr
1. Metrics and Geometry Defined by a System of Vector Fields
Suppose that X I Xz,.
, . X m are a system of smooth real vector fields defined on an open set M of R“, n 2 3. Our results will be restricted to the
case where X = (XI,
. . ,X m )satisfies the following two conditions:
(HI
X together with their commutators X , = [X,, , . [X,,-, ,XaS]. . .]]
up to some fixed length r span the tangent space at each point of M.
+ ,
+
+
For each j 2 r, the dimension of the space spanned by the
commutators of length 2 j at each point is constant in a
neighborhood (by convention, we define X k to be of length one).
By conditions (H) and (M), we can choose, in a neighborhood at each point,
a system of vector fields
klk,
{A>k};lr ,
-
where
X;k
is a commutator of length
j , such that for any r’ f r the subspace spanned by { X ; k , j
5
r’} is the
same as that spanned by all commutators of length 2 r’. If the Xi’s are
not linearly independent, we may replace them by a linearly independent
subset which still satisfies the conditions (H) and (M). Hence we will assume
Xlk = Xk, k 2 kl = rn. For fixed x E M , the choice of { X j k } will be
defined as a (canonical) coordinate system around x . More precisely, for a
small neighborhood V, of x , if y E V,, and y = exp{x U ; k X ; k } X , we define
a mapping Ox by
dx(y) = u = (Ujk).
(1.1)
Thus we obtain a C” diffeomorphism from V, c M to a neighborhood of 0
in W”.
We define now the metric on M associated with X as in [7]. Let us briefly
recall the properties of this metric; for the details of this aspect see [2], [7], [9].
DEFINITION
1.1. Let Cl(6) be a class of absolutely continuous mappings
4 : [0,1] -, M which almost everywhere satisfy the differential equation
( 1*2)
4’(4
=
c
a J ( t )XJ(4(t)1
lJl$
with
( t )I < dJ1,then we define
(1.3)
p l ( x , y )= inf{d > 0
I 34 E Cl(6) with $(O)
=x
$(I) = Y ) .
QUASILINEAR SUBELLIPTIC EQUATIONS
79
Then p1 is a local metric on M, and for any small compact subset K c M,
there exists a constant C > 0 such that
(1.4)
for any x , y
(1.5)
- yl
E K.
2 p1(x,y) 2 C l x - y p r
We can define a family of balls by this metric.
B I ( X , E ) = {Y E M ; P l ( X , Y ) < €1
for x E M, and e > 0 small enough. Denote by B E.( x , E
) Euclidean ball.
. the
Then the property (1.4) implies that, for all compact K
constants C1 > 0, C2 > 0, and €0 > 0 such that:
for all x E K and 0 < E 5 €0.
If in Definition 1.1 we replace Cl(6)by the following:
j=l
then we can define another metric p2(x,y) and balls & ( x , e ) as above. Observe that the curves in C2(6) point only in the directions of the original
. . ,X m , it is not obvious a priori that p2(x, y ) is finite for
vector fields XI,.
every ( x , y ) E M x M. But from [7] we know that p1 is equivalent to p2, i.e.,
for every compact K c M, there exists a constant C such that
From (1.4) and (1.7), p1 and p2 are only Holder continuous. So it is difficult
to study the associated geometry. For this reason, we introduce the following
quasi-distance function defined in terms of the exponential map.
For xo E M, Vxoa neighborhood of xo in which the exponential map is
well defined, and for x , y E Vxo,we define
Then p3 is equivalent to p1 on Vxo. Since we are interested in the local
properties, in later discussions we shall use metric p3 on Go. Denote by
{ X j k } (1 5 j 5 r, k 2 k j ) the base of tangent space on KO,
then n = Cis, kj.
80
C.-J. XU
We have
PROPOSITION
1.2.
(1.9)
Let x E K cc V, SO > 0 so that B3(x,60) c KO,
then
I B ~ ( x ,=
~ )~(l A ( x ) (,6 ~v o < 6
5 60
where ( B ( denotes the Lebesgue measure of B c R", N = Cis,j k j ,
and A(x) = det(Xjk)(x). C depends only on K and SO.
where aj = (jl,-.-,jt)
with 1 5 j t
Proof: Denote J = (al,...,a,)
and la,( = l 6 r , d ( J ) = C,"=,
Iajl. We define
2m
2 . d ~ )= det(X,,,...,&,)(x)
From Theorem 7 of [7], it sufficesto prove that there exists a constant q 2 > 0
such that if x E K and 0 < 6 j60 we have
(1.10)
(A(x)lP'2 q 2 mJax I A . , ( x ) I ~ ~ ( ~ )
First, we suppose that in &(x) there exists only one vector different from that
in A(x), for example X j o b is substituted by X,, hence d ( J ) = N - j o + ( a ( .
Using the hypothesis (M), we have on b,,
x a
=
fik(x)Xjk
jzlal
By repeating the above arguments, we have ( 1.10) and so the proposition.
From this proposition, we deduce immediately the following
LEMMA1.3.
(i) For every compact K cc
< $60, we have
KO,there exists a
x E K and 6
(1.11)
I&(x,26)1
5 CI&(X,6)1 .
(ii) For x E K cc V,,, 0 < 6 5 60,we have
constant C such that i j
QUASILINEAR SUBELLIPTIC EQUATIONS
81
where (I: > 0, C depends on K only.
Proof: As in Theorem 7 of [7], we have
where JOY1 is the Jacobian of
e;',
and
with Q(6) = { u E R"; llull < 6). Hence
c
J
d"pla-n
d t = C;P ,
BE(O,1)
which proves (ii) of the lemma, and (i) is a direct consequence of Proposition 1.2.
Let u = ( u j k ) be a coordinate system on R". We introduce a family of dilatations d , ( ~=
) (tju,k) for t > 0. A function f(u) is called homogeneous of
degree s if f(dtu) = t ' f ( u ) . A differential operator of the form f ( u ) ( a / a u j k )
is called homogeneous of degree j - s if f(u) is homogeneous of degree s.
A differential operator D has local degree less than or equal to j on R" if
its Taylor expansion at 0 is a sum of homogeneous differential operators of
degree less than or equal to j . Define X,k,x = e:(X,k). We have (see [6])
h
Xj,x = Xj,x
+Rj,x ,
j = 1,.
..,m
zj,x
where
is homogeneous of degree one and the degree of Rj,x 5 0. Now we
can differentiate the quasi-distance p3(x,y ) and construct the cutoff function
associated to distance.
LEMMA 1.4.
(i) For x , y E Vxo,we have
(1.13)
IX"P3(X,Y)l I CoP3(x,Y)1-lal
82
C.-J. XU
where X" =
. . Xat, and the diferentiation is taken in x or y .
(ii) For all K cc V,,, and K3€ = { x E R"; p 3 ( x , y ) < 3~ ,y E K } c V,
there exists (v E C?(K3€), 0 5
- (v 5 1, ~ ( x=) 1 on K such that for all a,
for some constant C,.
Proof To show (i), we note that llull is a function of homogeneous degree one and X a p 3 ( x , y )= X,"(llull).
To prove (ii), we construct the function (v as follows: Let
for x E V, and
E
> 0 small, we define
for { X j k }is a base of tangent space on V,,, and IA(x)l > 0 for x E Vxo.Using
(i), we have
Ixjac(X)I = I€-"
XjX$(P3(X,Y)/E)dYI
Similarly, we have IX"a,(x)1 5 Ca~-lal.
We define now,
Then ( v ( x )satisfies the properties of (ii).
83
QUASILINEAR SUBELLIPTIC EQUATIONS
2. Function Space Associated with a System of Vector Fields
Let us introduce now a class of Holder continuous functions with respect
to the metric p3, and denote V,, by R. For 1 > a > 0, we define (So(R)=
CO(fiZ>>
and for k E N, 1 > a
(2.2)
2 0, we define
Sk,a(Q)= { U E S"(R) ; X J u E S"(R), V IJI
2k} .
and
The norms on Sk@(R)are given by
Using the condition (H), we have the following lemma.
LEMMA2.1.
(i) For 0 < a
5 1,
SyR) c cqn)
is a continuous embedding where Cp(R) is the usual Holder space.
(ii) For k E N, we have
Sk'9O
(a)c CLi,( R) ,
where CLiPis the Lipschitz space.
(iii) IfF(x, z) E Cw(Q x W), u E Sk*"(Q),is real, then
F ( x ,u(x) ) E S k @ ( Q ) .
(iv) The space S k @ ( Q )is a Banach space.
Proof: (i), (ii), and (iii) can be found in [7]. To prove (iv), we assume
that {f;.}
c Skia(Q)is a Cauchy sequence. Thus Ilfjllsk..(n) 5 M < +m. Using
84
C.-J. XU
the results of (i) and (ii), { A } is equicontinuous, so there exists fo E C o ( i f )
such that fi + fo in C o ( i f ) .For 0 < a < 1, x # y , x , y E i f , we have
That proves fo E S " ( i f ) . For k = 1, we have similar X j f m + in C ( i f ) ,
and E P ( i f )So
. we have to prove A ( x ) = X,fo(x)
for all x E i f only. If
X,(x) = 0, then X,fo(x) = Xjf,(x) = & ( x ) = 0. Assume now X j ( X ) # 0,
and denote by $(t) the integral curve of Xj with $(O) = x, then for small I t \ ,
A
t
fm($(t) )
- fm(+(o) ) =
J Xjfm($(s) d s .
0
Because f, and Xjf, are all uniformly convergent, we have
As for the classical Holder space, we also have the following interpolation
inequalities in the space S","(n).
PROPOSITION
2.2, Suppose j + p < k + a , J ,k E N, 0 a,/? 5 1, and
u E Sksa(if).Then for any E > 0 there exists a constant C = C(E,j , k , i f , r )
such that
Proof: It is sufficient to prove the following interpolation inequality for
seminorms:
We prove (2.5) by induction, and suppress the index
enough, such that
ifd
if,
= { x E i f ; p ( x , a i f )> d } # 0 .
take d > 0 small
QUASILINEAR SUBELLIPTIC EQUATIONS
85
(a) Let j = 1, k = 2, ct = j3 = 0, we need to prove
By definition [u]f = supj supxERl X j U ( X ) l . For u E Sz(R) fixed, there exists
such that [u]f = IXjoU(Xo)l. Let p E (0, to be chosen; we
first consider the case B(x0,p d ) c a.
For [u]f # 0, we have X j o ( X o ) # 0. Let # ( t ) be the integral curve of Xjo
with #(O) = XO, take pd 2 6 2 ( p / 2 ) d , such that # ( 6 ) = x2 E B(x0,pd).
Then
4 x 0 ) - 4 x 2 ) = u(#(O)1 - u(#(a)1 = X j o u ( # ( e ) ) 6 *
j o , and xo E
a
Let d(0) = x
E
i)
B ( x 0 , d ) . Then
On the other hand, there exists $ 1
E
C2(pd) such that
x, hence
so
41 (0) = x g and 41 (1) =
m
Take p > 0 small enough such that p d m 5 E , we have proved (2.6) in the
case B(x0,pd) c R.
For the case p(x0,aR) < p d , we consider B(x1,pd) c R, where x1 E
sZ,d n B(x0,pd). If X j o ( x l ) = 0, we have
.I
hence
m
m
If X j o ( X 1 ) # 0, as above, there exists x E B ( x l , p d ) such that I X j O u ( ~ ) II
SIulo and p(X,xo) 5 2pd, then we can obtain (2.6) as above.
86
C.-J. XU
(b) Let j = k = 2, /3 = 0, a > 0, and u E S2sU(Q),by definition we have
[u]; = SUP,,SUP,~R IXiXjU(X)J= lXioXjoU(Xo)l.AS in point (a), we consider
only the case xo E R,,d. Assume that X,,(xo) # 0 and XjoU(Xo)- Xj0u(X2)=
Xi0X,,u(x)6 with x0,X E B(xl,pd) and pd 2 6 2 (p/2)d. Then
IJsing (a) we have proved [u]; 5 e[u]fa
The other cases are similar.
+ Clulo with e = 2(pd)".
As an application of Proposition 2.2, we have the following compactness
results.
j
PROPOSITION
2.3. Let K be a bounded subset of Sk,"(Q),k
<
k
+
a
,
then
K is precompact in Sj.p(Q).
+
+
(1:
> 0. If
3. Schauder Estimate for the Hiirmander Operators
We study now the following so-called Hormander operator
m
H = E X ; + C(x)
(3.1)
j =1
with C(x) 5 co < 0. By [I], there exists a Green's kernel G(x,y) for H, and
from [ 101 we have the following
LEMMA3.1.
For n 2 3,
where diferentiations are taken in x or y. Also
(3.3)
-G(X,Y)
near the diagonal of R x Q.
I
cP(x,Y)21B(x9P ( x , Y ) ) l - '
87
QUASILINEAR SUBELLIPTIC EQUATIONS
Using this estimate, we will prove a Schauder type estimate for operator H
in the function space Sk?.(Q)as in the function space
for Laplacian
Ax. Let xo E R, so that the exponential map and the metric p = p3 are welldefined on Q. Let R > 0 be so small that B(xo,2R) c Q, and B1 = B(x0, R).
THEOREM
3.2.
Let f E P(Q),with suppf c B I , Q > 0, and
v ( x ) = JB, G ( x , y )f ( y )d y . Then v E S21a(B1)and
(3.4)
11u11S2.n(B,)
<= c l l f l l S n ( B , )
9
where C is independent of u and R.
First let us show
LEMMA
3.3.
(3.5)
Let f be bounded and integrable in Q, then v
Xjv(x)=
LI
E S'(B1) and
X f G ( X , y ) f ( y ) d y , j = l , * * * , m ,x
E B1
,
where superscript x means X; is acting on that variable, and
(3.6)
Proof:
I X ~ ~ I O , B I CRlflO,B,
For x E B I ,by Lemma 1.3 and 3.1, we have
which proves the function J', X T G ( x , y )f ( y )d y is well defined. Hence it
remains to prove X ; v ( x ) = JBI X f G ( x , y ) f ( y ) d y .Let us fix a function q E
C'(W1)
satisfying, 0 q 1, 0 <= q' 5 2, q ( t ) = 0 for t 5 1 and q ( t ) = 1 for
t 2 2. Now, for E > 0, we define
88
C.-J. XU
Using Lemma 1.3, we have IXjqfI 2 Iqi$Xjp(x,y)l 2 C/E,and
Let E -+ 0, then V, and Xj V, converge uniformly to w and XjV in B1, respectively. This proves Lemma 3.3.
LEMMA3.4.
that
Let 1 2 j 2 rn, then there exists R i , k = O , l , . . . ,rn such
(3.7)
k= 1
with R i are the diferential operator of local degree
<= 0.
This is just a theorem of [9].
LEMMA
3.5. Let f E P ( n )with supp f
and for x E B I we have
c B1 and cr > 0, then
2)
E
S2(Bl)
for i , j = l,...,rn, where G Y ( x , y ) satisfies the estimate (3.2).
Using the uniqueness theorem for the operator H , we have supp w
c B1, take 4 E C r ( B 2 ) with + ( x )= 1 on B I and IXJ41 5 C J / R I ~ I .
Set z ‘ ( x , y ) = 4 ( x )G ( x , y )+ ( y ) , and denote b y w ( x ) the right side function
of (3.8), then, for x E B I ,
Proof:
= supp f
QUASILINEAR SUBELLIPTIC EQUATIONS
and
so that
which proves X,X,vc
-+
w ( x ) in CO(B1) and so the lemma.
89
90
C.-J. XU
Now, for x,X E Bl, set 6 = p ( x , x), take
6/2, then for i, j = l,...,rn, we have
We have, via the above, that
In order to estimate Z3,we consider first
< E B1 such that p ( x , < ) ,p ( ( , X ) 5
QUASILINEAR SUBELLIPTIC EQUATIONS
91
Since
k=l
for some iE B(<,( 3 / 4 ) 6 ) ,hence
1-l
If (Y
Now, for y $Z B(<,d), we have
where we have used 0 < a < 1 . Summing the above terms and the easy
estimates of 1s and 1 6 , we finally prove the following
for x,x E B1, with C depending only on (Y and n. This completes the proof
of the theorem.
Similarly, we have
THEOREM
3.6.
Let f E Sk?"(n)with k E N,
v E Sk+2@(B1)
and
where C is independent of v and R.
(Y
> 0, supp f c B 1 , then
92
C.-J. XU
Now let u E Co(Q)be a solution of the following equation,
m
(3.10)
Hu
= C X j u + c ( x ) u= f ,
j= 1
then u = ul
+ u2 such that
(3.1 1)
Hul=O
on0
and
The subelliptic condition (H) implies that U I E Cm(Q). Then for any K cc
Q, and k E N, there exists a constant D which depends on K , k , X, and
Iu1 Jtmp)
only such that
(3.13)
We therefore obtain
THEOREM
3.7. Let f E Sksa(Q),with supp f
solution of equation (3.10) then
c B1, and
u E C O ( Q be
) a
where Dk , C are the constants independent off.
In the above theorem, we have imposed the compact support condition on
the function f.We study now the general case. We need a technical lemma.
LEMMA
3.8. Let +(t)be a non-negative bounded function on [To,T I ]with
5 TI we have
0 5 TO< T I .Assume that for any TO5 t < s
(3.15)
with 1 > 8 > 0, A , B , p 2 0; then we have
for all To I t < s 2 T I ,where C depends on p and 8 only.
QUASILINEAR SUBELLIPTIC EQUATIONS
Using ( 1.4) we have, for 0 < t < s
93
1,
where Bt = B(x0,tR) and dE the Euclidean distance, thus there exists a
function I( E Cr(B,)such that C(x) = 1 on Bt and
(3.16)
[XkC]O4-( ( S - t)R)'a[XkC]:5
c k ( (S -
t)R)-'k
for all k E N, where [XkC]= & , k [ X J [ ] .
Let f E S k , a ( R )and u E SO(R)be a weak solution of equation (3.10),
then
L(CU) = C f
m
m
j= 1
j= 1
- C 2 ( X j O x j u - C(x,?C).-
Using Theorem 3.7 and the interpolation inequality, we have
THEOREM
3.9. Let f E Sk,u(Q)for some k E N, a > 0, and u E Co(Q)
be a weak solution of the problem H u = f. Then for all xo E R, there exists
R > 0 such that
for all o < t < s
1, where Dk,
ck
and
Z;k
are independent on f.
4. Regularity of the Solutions
We first study the following linear equation
where aij, bj, and c are defined on R c W" and the matrix (ajj(x))is symmetric for all x E Q. Then we have
94
C.-J. XU
THEOREM
4.1. Let u E
be a solution of equation Lu = f.Assume
that a,,, bj, c, and f E Sk+(i2)for some k E N and a > 0, and (ai,(x)) is
positive definite for all x E a. Then we have u E Sk+2,a(i2).
S210(Q)
Proof: Given xo E R, then there exists Ro > 0 such that B(xo,2Ro) c a.
We will prove that u E Skf2>"(B(xo,
$Ro)) for Ro > 0 small enough.
We first study the operator of frozen coefficients Lo = C5=1
AijXiXj + Co
with CO< 0 and A = (Ai,) the constant positive definite matrix. Then there
exists a matrix Q of dimension m such that Q'AQ = E with Q = PD,P is
orthogonal and D = (AT"2Sij) is diagonal, Al,...,Am are the eigenvalues of
A. Set 2 = Q X , then LO= fi with H = Cj"==,
2; + COand 2 satisfies the
same condition of X. Since A is positive definite, we also have
x.
It follows that S$0(!2) = S$"(n) for all
for all x , y , where P is defined by
k E fU, a > 0 and two norms are equivalent. Using Theorem 3.9, if LOW= f
with f E Skxa(Q),we have
with Bk, C1, and C2 depending in addition on A = min(,ll), and A = max(Aj).
We now rewrite Lu = f in the following way.
ij=l
Using (4.2), we have by interpolation inequality
take R > 0 small enough. Using Lemma 3.8, we have
QUASILINEAR SUBELLIPTIC EQUATIONS
95
We have therefore proved u E Sk+2@(Bt)
for 0 < t < 1, which proves Theorem 4.1.
The linear regularity theorem implies immediately the following nonlinear
regularity results.
THEOREM
4.2.
eauation.
Let u E S2>@(R)
be a solution of the following quasilinear
ij=1
where A j j , B E C“(R x W x R m ) . Assume that A = (Aj,(X,C,())
definite on R x R x Wm. Then u E Cw(R).
is positive
Proof:
Set ajj(X) = A j j ( X 7 u ( x ) ,X U ( X ) )f(x)
.
= B ( x , u ( x ) ,X U ( X ) )then
,
S1@(R)by Lemma 2.1. So Theorem 4.1 implies that u E S3ya(R).
Now a j j , f are the function of class S2,@(R),
so u will be the class S43a(i2)
and so on, we can prove that u E Sk@(R)for all k E N; Lemma 2.1 implies
that u E Cw(R).
aij,f E
Acknowledgments. The main part of this research was done while the author was visiting the Courant Institute of Mathematical Sciences. He wishes
to thank Professor Louis Nirenberg for his invitation and the Courant Institute for its warm hospitality. The work on this paper was supported in part
by the Fok Ying Tung Education Foundation.
Bibliography
[ I ] Bony, 3 . M., Principe du maximum, inegalite de Harnack et unicite du probkme de Cauchy
pour les operateurs elliptiques degenerees, Ann. Inst. Fourier 19, 1969, pp. 227-304.
[2] Fefferman, C., and Phong, D. H., Subelliptic eigenvalue problems, Proceedings of the Conference on Harmonic Analysis, in honor of A. Zygmund, Wadsworth Math. Series, 1981,
pp. 590-606.
[3] Gilbarg, D., and Trudinger, N. S., Elliptic partial diferential equations of second order,
Grundlehren der Mathematischen Wissenschaften 224, Springer-Verlag, Berlin, 1983.
[4] Hormander, L., Hypoelliptic second order diferential equations, Acta Math. 119, 1967, pp.
141-171.
[ 5 ] Ladyzhenskaya, 0. A., and Ural’tseva, N. N., Linear and Quasilinear Elliptic Equations,
Academic Press, New York, 1968.
[6] Metivier, G., Fonctions spectrales et valeurs propres dttne class d’kquations non elliptiques,
Comm. P.D.E. 1, 1976,pp. 467-519.
[7] Nagel, A., Stein, E. M., and Wainger, S., Balls and metrics defined by vectorfields I, basic
properties, Acta Math. 155, 1985, pp. 103-147.
[8] Oleinik, O., and Radkevitch, E., Second Order Equations with Nonnegative Characteristic
Form, American Math. SOC.,New York, 1973.
[9] Rothschild, L., and Stein, E. M., Hypoelliptic diferential operators and nilpotent Lie groups,
Ada Math. 137, 1977, pp. 247-320.
96
C.-J. XU
[ 101 Sanchez-Calle, A., Fundamental solutions and geometry of the sum of squares of vectorJelds,
Invent. Math. 78, 1984,pp. 143-160.
[ 1 11 Xu, C. J., R&gularitPdes solutions d’equations aux ddrivees partielles non lindaires assocides
~3un systkme de champs de vecteurs, Ann. Inst. Fourier 31, 1987,pp. 105-1 13.
[ 121 Xu, C. J., Operateurs sous-elliptiques et r&gularitksdes solutions d’equations aux dkrivees
partielles non lineaires du second ordre en deux variables, Comm. P.D.E. 11, 1986, pp.
1575-1603.
[13] Xu,C. J., Subelliptic variational problems, Bull. SOC.Math. France 118, 1990,pp. 147159.
Received May 1990.