L I P S C H I T Z C O N T I N U I T Y ,

G L O B A L S M O O T H A P P R O X I M A T I O N S A N D

E X T E N S I O N T H E O R E M S F O R S O B O L E V F U N C T I O N S

I N C A R N O T - - C A R A T H I E O D O R Y S P A C E S

NICOLA GAROFALO* AND DuY-MINH NHIEU t

1 I n t r o d u c t i o n and s t a t e m e n t o f the resul ts

The study of the local properties of solutions to linear and nonlinear pde's

arising from non-commuting vector fields has received considerable attention over

the last fifteen years. On the other hand, little is known concerning boundary value

problems for such classes of equations and the corresponding analytic and geo-

metric properties of solutions. In this paper we address some questions which are

of interest in this context: approximation of Sobolev functions by functions which

are smooth up to the boundary of a domain, global Morrey type embeddings, ex-

tension properties of Sobolev spaces. Our main results are Theorem 1.7, Theorem

1.8 and Theorem 1.10. To develop our study we also establish a characterization

of those functions which are Lipschitz continuous with respect to a given Carnot-

Carath6odory metric. An interesting consequence of such a characterization is the

existence of appropriate cut-off functions supported in metric balls.

We work with systems of vector fields satisfying minimal smoothness require-

ments. No explicit geometric assumption, such as the H6rmander finite rank

condition, is made. The interest of such a general setting stems from the following

considerations. On the one hand, it includes the important case of C ~ systems

of H6rmander type [28]; on the other, it also incorporates the general subelliptic

operators studied in [38], [15], since by the results in [39] the factorization matrix

of a smooth positive semi-definite matrix has in general at most Lipschitz contin-

uous entries. A further motivation comes from the fact that there are interesting

*First author supported by NSF Grant No. DMS-9404358. t Second author supported by a grant of the Purdue Research Foundation and also by the first author's

NSF Grant no. DMS-9404358.

67 JOURNAL D'ANALYSE MATHEMATIQUE, X.bl. 74 (1998)

68 N. G A R O F A L O AND D. M. NHIEU

classes of operators (such as, e.g., those of Baouendi--Grushin type) which arise

from systems of non-smooth vector fields. Our main point here is to prove that,

remarkably, even in such a general context the above-mentioned properties can be

deduced from three basic assumptions, listed as (1.1), (1.4) and (1.5) below. In a

different direction, the results in this paper apply to the setting of a connected Rie-

mannian manifold with non-negative Ricci tensor, provided that one replaces the

gradient along the system of vector fields with the Riemannian one, and Lebesgue

measure with the Riemannian volume. We emphasize that in the first part of the

paper, where we study only local results, assumptions (1.4) and (1.5) are not used.

They only enter later in Section 3, where the global theory is developed. In a

different (yet related) context, Saloff-Coste [40] first succeeded in proving that the

Harnack inequality for the heat flow associated to a smooth subelliptic operator can

be deduced from (and it is in fact equivalent to) assumptions (1.1), (1.4) and (1.5).

Subsequently, many authors have obtained results in the same direction. In [24] we

proved that (I.1), (1.4) and (1.5) suffice to develop a complete theory of isoperi-

metric and Sobolev inequalities associated to a system of vector fields with merely

locally Lipschitz coefficients for a class of domains called Poincarr-Sobolev, or

(PS)-domains, which is essentially as large as possible.

Before we can state our results, we need to introduce the main assumptions and

briefly discuss some relevant consequences of them. Given in I~ n a system of locally

Lipschitz real-valued vector fields X = {X1 ..... Am}, following C. Fefferman and

Phong [14] we say that a piecewise C 1 curve 7 : [0, T] ---* ~'~ is a sub-unit if

whenever 3"(t) exists one has for ~ �9 1~"

< 3"(t),~ >2<_ ~ < xj(-y(t)),~ >2. j = l

The sub-unit length of 3' is by definition ls(7) = T. We make the following

basic hypothesis: For any x, y �9 ~,'~ there exists a sub-unit curve 3' : [0, T] --* Rn,

such that 3'(0) = 23, 3"(T) = y. Denote by S(x, y) the collection of all sub-unit

curves joining x to y. It is then clear that

d(x, y) = inf{l,(3') 13, �9 8(x, y)}

defines a distance on IR '~, usually called the Carnot-Carathrodory distance

associated to X. Metric and Euclidean balls will be denoted respectively by

B(xo,R) = {x �9 IR'* Id(x, xo) < R}, B~(xo, R) = {z �9 ~" l l z - zo l < R}. Through-

out the paper we make use of the openness of the balls B(xo, R) in the (pre-existing)

Euclidean topology. Since this property is not guaranteed in the generality within

LIPSCHITZ CONTINUITY 69

which we work, see [ 1, p. 18], we introduce it as an assumption:

(1.1) i : (~'~,[. [) ~ (llT*,d) is continuous.

When the vector fields Xj are C ~ and satisfy Hrrmander ' s finite rank condition

[28], (1.1) can be deduced from the following estimate proved in [36]: For every

connected ~ c c it~ '~ there exist C, e > 0 such that

(1.2) C t x - yl <_ dn(z,y) < C - Z I x - yl ~, x, y E ~.

Here, dn is the distance defined by subunit curves whose trace lies in 12. (Note that

Chow's accessibility theorem [5] guarantees that for any x, y E f't there exists such

a curve joining x to y.) Since, obviously, d(x, y) <_ dn(x, y) for x, y E fl, from (1.2)

wc obtain (l . 1 ).

It is interesting to note that (1.1) suffices to guarantee that the metric and the

Euclidean topology are equivalent; see [24]. An important consequence of (1.1) is

the following

Proposition l.1. Assume (1.1). Then, (~n,d) is locally compact.

Furthermore, for any bounded set U c Rn there exists Ro = Ro(U) > 0 such

that the closed balls B(xo, R), with xo E U and 0 < R < Ro, are compact.

R e m a r k 1.2. The reader should be aware that even for C ~ systems X =

{X1, ..., Xm} of Hrrmander type, compactness of balls o f large radii may fail in

general; see [24]. However, there are important cases in which Proposition 1.1

holds globally, in the sense that one can take U to coincide with the whole ambient

space and Ro = oo. One example is that ofni lpotent Lie groups whose Lie algebra

admits a stratification; see Proposition 2.8 below. Another interesting case is that

in which the vector fields Xi have coefficients which are globally Lipschitz; see

Proposition 2.11.

We next introduce the relevant Sobolev spaces. Given an open set f~ c R", the

weak Sobolev space s 1 < p < oc, is the Banach space

s = { f e LP(f~)lXjf E LP(f~), j = 1,. . ,m}

endowed with the norm m

Ilfllcl.~(~) = I{fl}L~(~) + ~ IlXjfllL~(n). j = l

In the above definition, X j f denotes the distributional derivative o f f E L~oc(12)

defined by the identity

< xj.f,r > = f i x ; e r i c , r .In

70 N. G A R O F A L O AND D. M. NHIEU

where

is the formal adjoint of

X i = _ k = l

n

x j = F_,bJ o Ox~ "

k = l

We also need the space of functions having "strong derivatives" along the vector

fields

Sl'~(~) = C~r n s ll~,,.~.~, 1 < p < ~ .

In [24] we proved the following result of Meyers---Serrin type [34] (see also

[12] for the Beppo Levi spaces):

(1.3) s = Sl,p(ft), 1 _< p < ~ .

This remarkable identity plays an important role in this paper. In its local form,

s ~'p = S~o~(fl),- it was already discovered filly years ago by Friedrichs [22], a fact which appears not to be well known among workers in pde. The identity (1.3) has

also been found independently in [20]. Our first result brings up a useful property of functions which are Lipschitz

continuous with respect to the distance d(x, y).

T h e o r e m 1.3. Let ft C R '~ be a bounded, open set with da = sup=,uea d( x, y) <

oo. I f f o r a given funct ion u : f~ --* R we have for some constant C > 0

Cd(=,y) f o r x, y e a,

then u E s

We remark explicitly that a hypothesis that guarantees dn < c~ is (1.1). We

also have the following converse to Theorem 1.3.

lto0 T h e o r e m 1.4. Supposethat(1.1) holds. Letf~ C R" b e o p e n a n d u e s (f2).

Then for every w CC ~ CC ft there exists C = C(w, Co) > 0 such that, after

redefinition on a set o f measure zero in w, we have f o r x, y E

- (y)l -< Cll ll ,, ( ld( =, Y). /,1,oo Theorems 1.3 and 1.4 characterize elements of "-.zo~ as those functions which

are locally Lipschitz continuous with respect to the metric d(x, y).

An important consequence of Theorem 1.3 is the following existence result for

cut-off functions which are tailored on the Carnot-Carath6odory balls.

LIPSCHITZ CONTINUITY 71

T h e o r e m 1.5. Assume (1.1). Let B( Xo, R) be a bounded metric ball. Then,for

every 0 < s < t < R, there exists a d-Lipschitz continuous function r : ]R" ~ [0, co)

such that r E s every 1 < p < ~ . Furthermore, we have

(i) r _-__ 1 on B(zo, s) and ~ = 0 outside B(Xo, t),

(ii) IXr ___ c3 / ( t - s ) fo r a.e. x E R n.

It is worthwhile stressing that the usefulness of Theorem 1.5 goes beyond the

scope of this paper. One important application of it, for instance, is to the study

0fregularity properties of weak solutions of linear and nonlinear subelliptic pde's;

see [23]. Theorems 1.3 and 1.5 have recently been established independently by

Franchi, Serapioni and Serra Cassano in [21 ].

We now turn to some results of a global nature. To achieve them, however, we

need to impose geometric conditions on the ground domain and also introduce the

two basic hypotheses (1.4) and (1.5), which have been earlier referred to.

In what follows, if B = B(xo ,R) , we write for brevity a B = B ( x o , a R ) .

Furthermore, if E C IR '~ denotes a Lebesgue measurable set with Lebesgue

measure IEI, we let u s = [El -1 f s u d , x denote the average of a function u on

E. The notation X u = (X~u, ..., X , , u ) will indicate the horizontal gradient of u,

whereas IXul = ( ~ = l ( x J u ) = ) x/= will denote its length.

(1.4) For every bounded set U c IR '~ there exist constants C1,Ro > 0 such that

for Xo E U and 0 < R < Ro one has

IB(zo, 2R)I < C~lB(xo, R)I.

Property (1.4) is the familiar "doubling condition" which is assumed to hold

m any space of homogeneous type [8]. In addition, we will need the following

Poincar~ type inequality.

(1.5) Given U as in (1.4), there exist constants Cz, Ro > 0 and ~ > 1 such that

[Or Zo E U and O < R < Ro one has f o r u E C I ( B ( x o , a R ) )

l u - uBIdx < C 2 R f IXuldz . (xo,R) J B(zo,aR)

We emphasize that the above hypotheses are satisfied in a wide variety

of situations. For instance, if X is a system of C ~ vector fields satisfying

HSrmander's finite rank condition [28], then (1.4) and (1.5) respectively follow

from the fundamental works of Nagel, Stein and Wainger [36], and of D. Jerison

[29]. For non-smooth vector fields of Baouendi--Grushin type, we refer the reader

to the works of Franchi and Lanconelli [17, 18, 19].

In view of the results in [24] the assumption (1.5) can be replaced by the weaker

72 N. GAROFALO AND D. M. NHIEU

f ( l .5) ' sup[.Xl{= e B(xo,R) llu(x) - uBI > ,X}l] _< C2n/ IX l dx.

)~>0 JB(xo,aR)

This would not affect the validity of the results in this paper. However, to avoid

a certain amount o f additional technical work, we have preferred to employ (1.5),

rather than (1.5)'.

R e m a r k 1.6. The number Ro = Ro(U) in (1.4) and (1.5) will always he

chosen to accommodate Proposition 1.1. B y this we mean that for those balls

involved in assumptions (1.4) and (1.5) we can (and will) always assume in view

of Proposition 1.1 that they have compact closure.

The class of domains which we consider is a generalization of those introduced

by P. Jones in his famous extension paper [31]. They are the so-called (E,6)-

domains, also known as uniform domains in the case 5 = oo. A well known

classical result states that if ft C L~ '~ is a bounded, Lipschitz domain, then C~((~)

is dense in the Sobolev space WI,p(Ft); see, e.g., [13], Theorem 3, p.127. This

approximation theorem was generalized in [31 ] to (e, 5)-domains.

When f~ is an (e, 5)-domain in a Camot-Carath6odory space, we have the

following delicate global result.

T h e o r e m 1.7. Suppose that (1.1) and (1.4) and (1.5) hold. Given a bounded

(~, 5)-domain f~ c R ~ with rad(f~) > O, one has f o r 1 < p < oo

Oo, o(h)ll I,"Lp(n) = ~X,p(~.~).

Theorem 1.7 plays an important role in the study o f boundary value problems.

It is also instrumental in the proof o f the following extension result.

T h e o r e m 1.8. Assume (1.1), (1.4) and (1.5). Let 1 < p < oz. l f f~ C ~,'~

is a bounded (~, 6)-domain with rad(ft) > 0, then there exists a l inear operator

g : t~l'p(f~) ~ s such that f o r some C > 0 one has f o r f 6 s

(i) E l ( z ) = f ( z ) f o r a.e. z E f L

(i i) IIEfllz:,,.(a,,) < Cllf l l , : , , , ,(.). l f p = oo, then under assumptions (1.1) and (1.4) exists an extension operator

E : s ~ s satisfying (i) and (ii) with p = oo.

R e m a r k 1.9. For the definition of an (e, 6)-domain in a metric space, and that

o f rad(f~) (radius of f~), we refer the reader to Section 3. The operator norm o f

s depends only on c, 5,p, rad(s as well as on the constants in (1.4) and (1.5).

Concerning the part relative to p = oo in Theorem 1.8, one should see Theorem

2.7 below.

LIPSCHITZ CONTINUITY 73

For the Heisenberg group iE n, with non-commuting vector fields

0 0 0 2xj O j = l , . , n , x j = oz~ + 2y j-b-i, ~ = Oy--j - ' "

Theorem 1.8 is a special case of the results established by one of us in [37]. The

latter work also deals with extension results for higher order Sobolev spaces, but

in the general context o f this paper such a problem appears very difficult and we

presently have nothing to say about it. When the system X equals the standard

basis o f T ~ " , i.e.,X = {O/Ox~ . . . . . O/Ox,,}, then d(x, y) = Ix - y}, and Theorem 1.8

is nothing but the case k = 1 o f R Jones' cited extension theorem for the ordinary

Sobolev spaces W k'p [31 ].

In the Euclidean setting, the class of (e, 5)-domains is very wide; see [31],

[30], [33]. For a general Carnot-Carath6odory metric, finding examples o f (e, 5)-

domains is a question which presents serious difficulties. For instance, using the

results in [25] one can find examples of C ~''~ domains in the Heisenberg group H '~

which are not (e, 5). In [42] it is proved that each Carnot-Carath6odory ball in N'~

is an (e, 6)-domain. This fact also follows from the results in [4]. We mention

that, contrary to common belief, metric balls in N'~ are not NTA (non-tangentially

accessible) domains, according to [30]. This negative result is proved in [4]. The

latter paper also contains a detailed study of significant examples o f NTA domains.

Since every NTA domain is also a (e, 5)-domain, [4] provides various interesting

classes of domains to which the results of the present paper apply. A classical

theorem of Morrey [35], which plays an important role in the regularity theory o f

linear and nonlinear pde's, states that i f ~ C R 'L is a bounded Lipschitz domain,

and i fp > n, then W~,P(ft) ~ Ca(ft) , with ~ = 1 - n/p. Our next result extends

Morrey's theorem. To state it we need to introduce the notion of local homogeneous

dimension. Let U C N" be a bounded open set. With Ct as in (1.4) we set

Q = log 2 C~

and call this number the local homogeneous dimension relative to U (and to the

system X = {Xz . . . . ,Xm}. Note that when X = {O/Oxl,... ,O/Ox,~}, then

d(x, 9) = Ix - Yl, and therefore Q = n. Another case of geometric relevance

is that of a stratified, nilpotent Lie group G, with Lie algebra g. Suppose that

g = V~ ~ V2 �9 .-. | V~ is a stratification o f the Lie algebra, with [Vt, Vj-] C V~.+~,

j = 1 , . . . ; r - 1, [V1,V~] = {0}; then any basis X of the first layer V~ in

the stratification of g satisfies HSrmander's finite rank condition. In this case

the number Q is constant throughout the group G and equals ~2~=1 j dim(V~);

see, e.g., [16].

74 N. GAROFALO AND D. M. NHIEU

T h e o r e m 1.10. A s s u m e (1.1), (1.4) a n d (1.5). Le t f2 C (2 C U be a (e,6)-

domain with dn = supx,u~ ~ d(x, y) < Ro(U), u E s with p > Q. Then u can

be modi f ied on a set o f measure zero in such a way that u E F~ with 0 = 1 - Q/p.

By this we mean that u has a cont inuous representat ive in (2 (also deno t ed by u)

sat is fying

r (x) - (y)r s u p < oc.

�9 ,u~f~,zev d(x, y)O

We emphasize the global nature of the above results. We mention that a

local version of Theorem 1.10 was found in [24]. We have recently received

an interesting preprint by Lu [32], dated 1994, in which, among other results,

the author independently establishes the local result in the context of H6rmander

vector fields, but with techniques different from those employed in [24].

A remarkable consequence of Theorem 1.10 is the d-H61der regularity up to

the boundary of weak solutions to nonlinear equations which arise from the system

X = {X1, ..., Xm}. We confine ourselves to discussing the relevant model equation.

Consider the functional

f )=falXul pdz, Jp( ;

Its Euler-Lagrange equation is given by

m

(1.6) f-.pU = _,Xff(IXul -2Xju) = 0 in j = l

For a study of the local properties of solutions to a general class o f nonlinear

equations modeled on (1.6) in the framework of C ~ vector fields o f H6rmander

type, we refer to [2], [3], [9]. A characteristic aspect of the results in these

papers is that the coefficients of the lower order terms are allowed to belong to

functional classes which are optimal within the scale of Lebesgue spaces. For

some nontrivial improvements of the results in [2] within the scale o f subelliptic

Morrey spaces, the reader should consult the papers [ 10], [ 11 ], [32]. Here, we note

that as a consequence of Theorem 1.10 it is not necessary to study the regularity

of (weak) solutions u r Z:I'P(f~) to (1.6) when f~ C R '~ is a bounded (e,f)-domain

and Q < p < oc. In such a situation, in fact, the mere membership o f u in

s guarantees that (up to modification on a set of measure zero) u is d-HSlder

continuous up to the boundary, with HSlder exponent 0 = 1 - Q/p . Concerning

the HSlder continuity up to the boundary of weak solutions to (1.6) in the range

1 < p < Q, we refer the reader to [9].

L I P S C H I T Z C O N T I N U I T Y 75

2 L o c a l r e s u l t s

This section is mainly devoted to proving Theorems 1.3, 1.4 and 1.5.

preliminary step we note the following elementary

P r o p o s i t i o n 2.1. The inclusion i : (~'~, d) --, (I~ '~, 1.1) is continuous.

As a

Proof . Given Xo E ~'~ and r > 0 we let

m(xo,r ) : max IXCz)!, zEB~(Zo.~)

where i f X = {X1, ..., Xm} and Xj = ~ = 1 bjkO/Oz~, then

We note that, since X3 has locally Lipschitz continuous coefficients, then M(xo, r) <

~. To prove the proposition we argue by contradiction and suppose that there exist

Zo, {zk}~E.~ and eo > 0 such that d(xk,Xo) --, 0 as k ~ ~ , but Ixk - Xol >_ eo. For

each k E N we can find a sub-unit curve ~'k E S(Xo,Xk) such that 18(7k) = Tk <_

2d(zo,xk). By connectedness, there exists t~ �9 [O,T~) such that 7k(t) �9 Be(xo, eo) forO <_ t < tk, andpk = ";'k(tk) E OB~(zo, eo). This yields

0 < Co = Izo - pkl = 7~(t ) dt < IX(Tk( t ) ) l dt

< M(xo, r <_ 2M(xo, eo)d(xo, Xk).

Since d(xo,xk) ~ 0 as k ~ cr we have reached a contradiction. []

We now give the

P r o o f o f P r o p o s i t i o n 1.1. To begin we note that, thanks to Proposition

2.1 and to the assumption (1.1), the metric topology is equivalent to the Euclidean

topology of IR '~. In particular, compact sets coincide in either topology. In order

to prove the (locally) uniform compactness o f small metric balls, we are going to

show that for any xo E l~" and r > 0 one has

7" (2.1) B(zo, M(xo, r)) C B~(zo,r) ,

where M(xo, r) is as in the proof of Proposition 2.1. To see this, we let y E

B(Zo, r/M(xo, r)). For any e > 0 with e < r/M(xo, r) - d(xo, y), there exists a 7 �9

S(zo, y) such that d(xo, y) <_ I~(7) = T < d(xo, y) + 4. By a connectivity argument

76 N. GAROFALO AND D. M. NHIEU

similar to that in the proof o f Proposition 2.1, one sees that {7} C B~(zo,r).

Therefore,

/o' /0 Ixo - Yl = "y'(t) dt <_ IX('~(t))l dt <_ M(xo, r ) T < r. i

This proves (2.1). To complete the proof consider a bounded set U c R '~ and

define

Ro = inf �9 eC' M(x , 3)'

where 6 = sup~,,yeu I x - Yl < oo. I f Ro = 0, by the compactness o f / ) we can find

a sequence {Xk}k~,,~ and xo in (l such that Ixk - xo[ --* 0 and M(xk ,6 ) ---+ oc. But

this is impossible since M(Xk,6) < M(xo,26) < oc. Therefore Ro > 0, and (2.1)

implies for x E U and 0 < R < Ro

6 D(zo, R) c [~(Xo, M(xo,~----)) c [~e(Xo,~),

which proves that [7(Xo, R) is compact. []

We now turn our attention to the proof o f Theorem 1.3. We need to recall a

few basic facts about the local one-parameter group action O(t, x) generated by a

vector field Y = ~2=1 bkO/Oxk. Y is assumed to have locally Lipschitz continuous

coefficients. Then O(t, x) solves the Cauchy problem

{ ~o(t, x) = y(o(t, x)), 0(o, ~) = z.

In the next lemma we collect some known properties o f 0(t, x) which will prove

useful in the sequel. We refer the reader to [26], [27].

L e m m a 2.2. Let ,J c C ft c c II~ '~ be open sets. There exist positive numbers

M and T (depending on w, f~ and on the Lipschitz constant o f Y on 12) such that

for x, x' E o; and It[ _< T,

(2.2) t 0 ( t , x ) - 0 ( t , x ' ) l <_ M i x - x' l .

Furthermore, i f JO = (OOi/Ozj) denotes the Jacobian o f O, then for a.e. z E w

and for it[ <_ T one has

(2.3) JO(t, x) = I + ~(t, x),

LIPSCHITZ CONTINUITY 77

where I = identity matrix in R '~, and rl(t, x) = (rhd (t, z ) ) is a matrix-valued function such that for a.e. x E w and for It[ _< T,

1

IIw(t,x)ll = ,T i , j ( t , z ) z < Cltl, i , j - - : l

for some constant C > 0 depending only on w, f~ and on the Lipschitz constant o f Y in fL In particular, one has

(2.4) det JO(t, x) = 1 + A(t, x)

with I,~(t, z)] _< Cltlfor(t,:c) e [-T,T] • w.

We are now ready to give the proof o f Theorem 1.3.

P r o o f o f T h e o r e m 1.3. Fix Xo E Ft. Then we have for every x E f~

lu(x)l _< lu(xo)l + lu(x) - U(zo)l _< lu(xo)l + Cdn.

This shows u E L~ In order to prove u E s we need to show Xju E L:c(f~), j = 1, ...,m. Let then Y = Xj. Consider r E C~( f l ) and choose an open

set w c ~ c Ft such that supp r c w. By Lemma 2.2 we can find a number T > 0

such that the flow O(t, x) associated to Y satisfies (2.2)--(2.4) on [ -T , T] x w for

suitable constants M and C. Since

(2.5) < Y u , r >= s u Y ' r f a u Y C d x - f n u d i v Y C d x ,

in order to show that Yu E L ~ (gt) it will be enough to prove that each o f the integrals

on the right-hand side o f (2.5) is bounded in absolute value by const .11r (n). To

this end, we first consider fn u divYr dx. By the assumption on the coefficients o f

the vector fields in X, we see that IldivYIIL~(n ) < co, and therefore

f ud ivYCdx < IldivYllLoo(a)llUllL~C(n)llr ).

The integral on the left-hand side o f the latter inequality thus defines an element

of Ll(Ft) *. We show next that

s uYCdx _< ClI r (2.6)

for some C > 0 independent o f r We have

/o' r x)) - r = r x)) ds = Y(r x)) ds = tg(t, x),

78 N. GAROFALO AND D. M. NHIEU

where g(O, x) = Y4)(x). Since (t, x) ~ O(t, x) is continuous in I -T , T] x w, we see

that g E L~176 T] x w). Therefore, by the Lebesgue dominated convergence one

has

(2.7) s 1 6 3 t dx.

Now we recall that the map x ~ O(t, x) is invertible, its inverse being given by

O(-t, .). This observation and the change o f variable y = O(t, x) allow us to infer

that for each fixed Itl < T

~ u(x)o(O(t,x))dx = ~ u(O(-t,y))i)(y) IdetJO(-t,y)Idy.

We use this equality to obtain

/a u(x) r - r dx = In u(O(-t,Y))t - u(y) r [detJO(-t,y),dy

/~ ]detJO(-t, y ) ] - 1 dy. + u(y)e(y) t

If we now let 7(t) = O(t,x), then for any 0 < t _< T, 7 : [0,t] --, IR '~ is a sub-unit

curve joining x to yt = O(t, x). Therefore

d(x,O(t,x)) <_ Is(y) = t

for every x E w and 0 < t < T. As a consequence o f the assumption on u,

u(O(-t,y)) - u(y)[ < cd(O(-t ,y) ,y) < C i t ] - - t - -

for y E w and Itl < T. Then (2.4) allows us to conclude that

lim fa ~(0(-t, y)) - ~(y) r I det JO(-t, Y)I dy < cllr t~O t

Analogously, one has

lira f u" "r ' IdetJO(-t'Y)l - 1 t__.0]a ty) ty) t dy < Cl[r ).

The latter two inequalities and (2.7) imply (2.6), and this completes the proof. []

R e m a r k 2.3. We note that iff~ is as in Theorem 1.3, then a sufficient condition

for dn < c~ is assumption (1.1). The latter, in fact, implies that f~ is d-compact. It is

also worthwhile to observe that if (1. I) holds, then a function u as in the statement

o f Theorem 1.3 is also continuous in f~ with respect to the Euclidean topology.

L I P S C H I T Z C O N T I N U I T Y 79

An important consequence o f Theorem 1.3 is the existence o f suitable cut-off

functions (Theorem 1.5), which are tailored on the Carnot-Carath6odory balls.

Before proving such a result we recall the following lemma whose proof can be

found in [24].

L e m m a 2 . 4 . Let f E C 1 ( ' ~ ) with If'l ~ M. If1 ~ p < ~ , then f o r any u E Z21'P(f2) we have f o u E Z;I'P(f2). Furthermore, f o r j = 1, . . . ,m

X j ( f o u) = ( f ' o u ) X j u in D'(I2).

We now turn to the proof o f Theorem 1.5.

P r o o f o f T h e o r e m 1.5. Let h E C~([0, 2 ) ) be such that 0 < h < 1, h - 1

on [0, s], h - 0 on [t, oe), and Ih'l _< C / ( t - s). Consider the function r =

h(d(x, xo)). By Theorem 1.3 the function :c ~ d(x, Xo) belongs to s176162 R))

and, in fact, the p roof shows that [Xjd(., Zo)[ <_ 1, j = 1, ..., rn. In view o f Lemma

2.4, we conclude that O E s for any 1 _< p < cr and moreover

X j r = h'(d(., Xo))Xjd(. , , xo) in D'(f~), j = 1,..., rn. This proves (ii) o f the theorem.

Since (i) is obvious, we have reached the conclusion. []

R e m a r k 2.5. Concerning the assumptions in Theorem 1.5 we recall that, by

Proposition 1.1, for every bounded set U c ll~ '~ there exists Ro = Ro(U) > 0 such

that the balls B(xo, R), with :co E U and 0 < R < Ro, are compact. This property,

however, fails in general for large radii. An interesting situation in which the

boundedness of all metric balls is guaranteed is that in which the vector fields have

globally Lipschitz continuous coefficients; see Proposition 2.11 below.

As the reader may surmise, Theorem 1.5 plays a basic role in the study o f partial

differential equations arising from a system X = {X1, ..., Xm} of vector fields. As

we mentioned earlier, when (1.1) holds the Lipschitz continuity with respect to 1 , ~ d(x, y) of a function u is necessary and sufficient for its membership to Eqo c . To

prove this fact we will need a localized version which has independent interest and

will prove useful in what follows as well. We first recall an approximation result

which is the key to the proof o f (1.3).

Let K E Co~(l~ n) with suppK C {x E R'~[[x[ < 1} and fR, K ( h ) d h = 1. 1 n Given a function u E Lbc(IR ) we let J,u = K, �9 u where K, = E-nK(e -1.). Let

n Y = ~ k ~ l bkO/Oxk be a vector field on R n with locally Lipschitz coefficients.

L e m m a 2.6. Let u E Z2loc (~ ). I f w CC we have

Y ( J , u ) = J , (Yu) + L u in D'(a:),

80 N. GAROFALO AND D. M. NHIEU

where

and

J~u(x) = fn- u(x + eh)K~(x, h) dh,

R~(z,h)= 1~ O e -~k [(bk(X + e.) -- bk(X))g] (h). k = t

Lemma 2.6 is due to Friedrichs [22]. We mention the two notable properties o f

the ke rne l /~ (x , h) which will be needed in the sequel:

(2.8) f [s d h = O f o r x � 9 and e > 0 , J~

t ' (2.9) ] sup IRa(z, h)l dh < C3,

J.~ XC~

where C3 > 0 is independent o fe > 0.

T h e o r e m 2.7. Suppose (1.1) holds. Given a bounded open set U c R '~, there

exist Ro = Ro(U) > 0 and C = C(U) > 0 such that i f u E El '~(B(xo ,3R)) , with Xo E U and 0 < R < Ro/3, then u can be modified on a set o f measure zero in

[3 = B(xo, R) so as to satisfy

I~(x) - ~ (u) l < C d ( x , u ) l l ~ l l L , , = ( 3 ~ l

for every x, y �9 B(xo, R). In particular, one has

[lu - UBIIL•(B) <_ CRIlullc',•(sB).

Proof . We fix Ro = Ro(U) > 0 as in Proposition 1.1, and let 0 < R < Ro/3. Then, for each Xo �9 U the closed ball B(xo, 3R) is compact. At first we consider

u �9 C~(B(xo ,3R) ) ,q s176176 Given x ,y E B = B(xo,R) , then for every

0 < e < 1 there exists a sub-unit curve -y : [0, T] ---, ~'~ such that 7(0) = x, -y(T) = y

and d(x, y) _< 18(7) = T < (1 + e)d(x, y). We claim that {"/} c (2 + e)B. To see this

let z �9 {7}. Suppose that

d(x, z) >_ �89 + e)d(x, y) and d(z, y) >_ �89 + e)d(x, y).

Denoting by t~(-y; x, z) and/s(7; z, y) the sub-unit lengths o f those portions o f ' r

which, respectively, join x to z and z to y, we find

l~('~) =/8(-~; x, z) + ls('r; z, y) >_ d(x, z) + d(z, y) >_ (1 + e)d(x, y),

LIPSCHITZ CONTINUITY 81

which contradicts our choice o f 7. Therefore, we must have

d(x, z) < �89 + e)d(x, V) or d(z, y) < �89 + e)d(x, y).

Assuming that the former possibility occurs, one finds

d(z,Xo) <_ d(x,z) + d(x,Xo) < 1 ( 1 + e)d(x,y) + R <_ (2 + e)R.

Since an analogous conclusion is reached in the second case, the claim follows.

By the assumption -'/E S(x, y), and by the claim, we have

lu(x) - u(y)l = u(,~(s)) <_ IXu('Y(s))l ds

<_ IlXullLo~<r < (1 + ~)d(x,Y)llXullLoo<<2.~,~Bi.

Letting e --, 0 we conclude

(2.10) t u (x ) - u(y)l < d(x,y)llXu[IL~(2m, z , y �9 B.

We next remove the assumption u �9 C ~ ( 3 B ) . Let then u �9 Z:I,~(3B), extend

it to all o f R" by setting u = 0 in IR"\(3B), and consider the Sobolev-Fr iedr ichs

mollifier J,u introduced above. Recall now that by Proposit ion 1.1 the ball 3B is

compact . Since u �9 / :1,~(3B), we then have u �9 EI,P(3B) for any 1 < p < cx~.

Thus, i f we fix 1 < p < co, by (1.3) we can find a subsequence c3 ~ 0 such that

vj(x) = J~ju(x) ~ u(x) as j --, ~ for a.e. x �9 3B, and moreover

Ilvj - UllL,(am ~ 0, IlXvj - XullL,<aB) ~ O.

L e m m a 2.6, with a~ = ~B, gives for j >_ jo (with jo �9 N large enough) and

k = 1, . . . ,m,

Xkvj(z) = J,,(Xku)(x) + Zju(z) for a .e .x �9 ~B.

From this we obtain

(2.11) IlXkvjllL~c2m <_ IIJ,,(Xku)llL~(2B) + IIJ~ullL~(2m.

The first term on the right-hand side is classically estimated as follows:

IIJ,~(Xku)llL=<2m <_ IlXkullL~(3m <--IIX~llLooC3m.

For the second term we use (2.9) in L e m m a 2.6, which gives for x �9 2B

IL~u(x)l <- JR[- [u(x + ejh)llg,, (x, h)l ds < C311UllLOO(3B).

82 N. GAROFALO AND D. M. NHIEU

From these inequalities and (2.1 1) we conclude for j > jo

(2.12) IlXvjllL~(2m < CIl,~llL,,~(3m.

I f we now apply (2.10) with u = v~ we obtain for x, y E B

(2.13) Ivj(x) - vj(y)l < d(z,Y)llXvjllLo~(2m <_ Cd(z,Y)llullc.,,~(3m,

where, in the last inequality, (2.12) has been used. Since we also have

(2.14) IlvjllL=(=m = IIJ, ju l lL~(2m _< IlullL~(am,

inequalities (2.13) and (2.14) allow us to conclude that the sequence o f C ~

functions {vj }~>_~o is equicontinuous and equibounded in/3. By Proposition 1.1,

(/3, d) is a compact metric space, therefore the theorem of Ascol i -Arzelh guaran-

tees the existence of a subsequence, still denoted {Vj}j>_jo, and o f a continuous

function v on/3, such that vj ---, v as j ~ ~ uniformly in/3. Since vj(x) ---, u(x) as

j ---, oc in a.e. x E/3, we see that by modifying u on a set o f measure zero we can

assume that u is continuous in/3. Passing to the limit as j ---, oo in the left-hand

side o f (2.1 3) we obtain

(2.15) lu(x) - u ( y ) l < d(x,Y)llull~.~,~(3B), x , y ~ [~,

which shows that u is, in fact, d-Lipschitz continuous in B. Finally, one has

from (2.1 5)

Ilu - uBIIL=(m _< 2RIlul lL, ,~(3m,

so the proof is complete. []

We are now ready to give the

P r o o f o f T h e o r e m 1.4. Thanks to (1. I) and Proposition 2.1 the d-topology

coincides with the Euclidean one. Therefore & and a5; are d-compact sets and

5 = ldist(~, 003) > 0, where the distance is taken with respect to d(x, y). By

Theorem 2.7 there exists Ro = Ro(&) > 0 such that for u E s162162 n C~(&) and

B(xo, 3R) C &, with 3R < Ro, one has

(2.16) lu(x) - u(y)l ___ Cllull~.,,~(~)d(x,y)

for x, y E B(xo, R). Let 8o = min(Ro/6, 6/3). I f x, y E w and d(x, y) < 50, then

(2.16) holds. If, instead, x , y E w, but d(x ,y) > 50, then we have trivially

2 lu(x)- u(y)l < 21lullL~(~)___ ~llullL~(o~)d(z,Y).

L I P S C H I T Z C O N T I N U I T Y 83

In all cases one concludes that

I~,(z) - u(y)l <_ Cl lul lL, ,~(~/d(z , y)

for every x, y E ~. To remove the assumption u E C ~ ( ~ ) , one now proceeds

exactly as in the proof o f Theorem 2.7. The almost obvious modifications o f the

argument there are left to the reader. []

As we mentioned in the introduction, Proposition 1.1 holds globally in the

case o f a stratified, nilpotent Lie group or in the case where the coefficients o f

the vector field Xj are globally Lipschitz. Before stating the relevant result, we

recall some elementary facts about stratified groups. Let G be a Lie group with

Lie algebra g. The lower central series of g (and G) are defined inductively

by g(1) = g (G(~) = G ) , g ( j ) = [ g , i ~ ( j _ x ) ] ( G ( j + l ) = [G,G(j_I)]) . 9 (or G ) i s

called ni lpotent of step r E 1~" i f g(~+l) = {0} (G(~+I) = {e}) but g(r) r {0}

(G(~) r {e}). A Lie algebra g is strat i f ied i f it admits a vector space decomposit ion

g = V1 r V2 ~ . . . �9 V~ with [V~, V3. ] C Vj+I, j = 1, . . . , r - 1, [V1, V~] = {0}, and

if, furthermore, V1 generates g as an algebra. A stratified, n i lpotent Lie group G

is a connected, simply connected Lie group whose Lie algebra g is nilpotent and

stratified. A natural family o f dilations on a stratified nilpotent Lie algebra g is

given as follows: For A > 0 and X �9 g with X = ~ = 1 Xj, Xj �9 V3, let

A~(X) = AX1 +A~X2 + - . - +A'X~.

Since exp: g ---, G is a diffeomorphism, we can use it to define a family o f

dilations on G by letting, for g �9 G,

(2.17) 5x(g) = exp oA~ o exp -1 (g).

In what follows we identify the elements o f X E g with the corresponding

lei~-invariant vector fields on G. It easy to recognize that X �9 Vj i f and only i f X

is homogeneous o f degree j with respect to (2.17). Next, we choose a Euclidean

norm II II on ~ with respect to which the V~'s are mutually orthogonal and define

for X = X l + .. . + X j �9 g

j = l

A homogeneous norm on G is then given by

Iglc = tXl~,

i f g = exp X. Since V1 generates g as an algebra, i f we choose a basis X =

{X1, ..., Arm} of V1, then X satisfies Hf rmander ' s finite rank condition. Denote by

84 N. GAROFALO AND D. M. NHIEU

d the corresponding Camot--Carath6odory distance defined in Section 1. By the left-invariance of X, d is also left-invariant, i.e., we have

(2.18) d(g' g,g'h) = d(g, h)

for any g, h, g' 6 G.

Proposition 2.8, Let G be a stratified, nilpotent Lie group o f step r with a

Carnot--Carath~odory distance d. Then for each xo 6 G and R > 0 the closed ball

B(xo, R) is compact

The proof of Proposition 2.8 is based on (2.18) and on two interesting properties

of the distance d(z, y) which we collect below as Propositions 2.9 and 2.10. We

omit their elementary proofs, referring the reader, e.g., to [23].

Proposition 2,9. For any 9, h 6 G and A > 0

= ha(g, h).

The next proposition shows, in particular, that the topology generated by I I G is

compatible with the metric one.

Proposition 2.10. There exist constants C', C" > 0 such that

C'lg- lhlc < d(g,h) < C"lg-lhlc

for every g, h E G.

For g �9 G and R > 0, we let Ba(g, R) = {h �9 G l ig- lh lc < R} denote the

ball in the homogeneous norm I Ic centered at g with radius R. Proposition 2.10

implies

(2.19) Bc(g ,C" - lR) C B(g,R) C BG(g,C'-*R), g 6 a, n > O,

hence the metric topology and that generated by I 1G coincide. We are now in a

position to give the

Proof of Propos i t ion 2.8. By lemma 1.4 in [ 16] each closed homogeneous

ball [3G(g, R) = {h �9 G I Jg-lhlc <_ R} is compact in the topology of G. Thanks

to the estimate

d(x, y) < Cda(x,y) 1/~,

where d•(x,y) is the Riemannian distance on G (see, e.g., [41], p.40), we have

i : (G, dR) ---, (G,d) is continuous, so /3c(g,R) is also compact in the metric

topology. By (2.19), we infer the compactness of/}(g, R). []

LIPSCHITZ CONTINUITY 85

We now turn to the case of vector fields having globally Lipschitz coefficients.

P r o p o s i t i o n 2.11. Suppose that X = {X1, ..., X,n } have coefficients in Lip(R '~).

If(1.1) holds, then B(xo, R) is compact for every Xo E ~,'~ and 1t > O.

Proof . By the hypothesis on the Xj ' s there exists a constant M > 0 such that

(2.20) IX(x)[ = IXj(x)l ~ _< M(1 + ]xl) j = l

for any x E R '~ �9 Fix Xo, y E R '~ and let 7 : [0, T] ~ R '~, 3' E S(Xo, y) be a sub-unit

curve. Letting y(t) = 13'(t)l 2 we obtain

y'(t) = 2 < 3"(t),3''(t) > < 21"~(t)ll3''(t)l <_ 213"(t)llX(3"(t))l.

Using (2.20) we infer that

y'(t) <_ C(1 + y(t))

for some C > 0 depending only on M. Integrating the latter inequality one has

(2.21) I'Y(t)l _< x/X + Ixol2~ c~, t ~ [0,T].

The estimate (2.21) shows, in particular, that

(2.22) B(xo, R) C Be(O, X/1 + ]Zol2eCR).

We conclude that B(Xo, R) is Euclidean compact. By (1.1) it is also

d-compact. []

3 G l o b a l resul ts

This section is devoted to Theorems 1.7 and 1.8. The proof o f these results relies

on what has been established in Section 2, as well as on geometric properties o f

(E, 6)-domains. Such properties have, in one form or another, already been studied

extensively in previous works, especially in [31]. For the reader's convenience we

present below the main steps in the proof of Theorem 1.7, whereas we will omit

altogether the similar constructions in the proof o f Theorem 1,8. For this and other

proofs that will be omitted the reader is referred to the following sources: [31],

[6], [7], [37].

86 N. GAROFALO AND D. M. NHIEU

D e f i n i t i o n 3.1. An open set f~ C/ t~ '~ is called an (e, 6)-domain if there exist

0 < 6 < oo, 0 < e < 1 such that for any pair o f points p, q �9 f~, ifd(p, q) < (5, then one

can find a continuous, rectifiable curve 7 : [0, T] ~ fl, for which 7(0) = p, -y(T) = q,

and

1 (3.1) l(-~) < ed(p,q) ,

(3.2) d(z,Ogl) >_ e m i n ( d ( p , z ) , d ( z , q ) ) for all z E {'y}.

We recall that i f 7 : [a, b] ~ R '~, then one defines the metric length l(-y) as I(q) =

sup ~-~,~=1 d(?(t i ) , "~(ti,-1)), the supremum being taken on all finite partitions a =

tl < t2 < . . . < t v < tv+l = b o f the interval [a,b].

One should notice that there exists a close connection between the class o f

NTA domains studied in [4] and that o f (e, ~o)-domains (or uniform domains). In

particular, every NTA domain is an (e, oo)-domain. In fact, i f from the definition

o f NTA domain one removes the requirement about the exterior corkscrew, then

one has an (e, oo)-domain.

D e f i n i t i o n 3.2. Let fl C R '~ be an open set. Then the radius o f f~ is defined to

be the quantity

rad(F/) =

sup{r > 0 [ for every 0 <_ s < r, and p E fl, there exists q E f~ with d(p, q) = s}.

It is reasonable to assume that rad(f~) > 0. Note also that i f 6 = oo, or fl is

connected, then rad(f~) > 0 automatically.

Let ~ be a bounded (e, 6)-domain and U C iR 7' be a bounded set such that ~ / c U.

Let Ro = Ro(U), and C1 and C2 be the constants in (1.4) and (1.5) (recall Remark

1.6). Thanks to (1.4), for any 0 < t < Ro(U) we can find a covering .Tt o f f l with

1 B(bj , t /6 ) are pairwise disjoint. balls Bj = B(bj , t /2) such that 5B# =

I f B is a ball, then r ( B ) denotes the radius o f B. We define

n t = {B# e ~ t I B j C fl},

n ' t = {By �9 Tit Id(B#,Ofl ) >_ (20/e)h}.

Throughout the end o f this section we fix

1602. e2 Ro t = - - ~ - h , with 0 < h < 160---2"

For B# �9 R'~ we write

and B;* = B(b , 1602 --fi-- ] .

LIPSCHITZ CONTINUITY 87

L e m m a 3.3. Assume (1.1) and (1.4). I f h is su:ff~ciently small one has

ft c U~je~ B~.

Proof . Given z Ef] , define

az = inf{d(z, B) I B E TO't}.

Observe that i faz < (400/e2)h, then z E B o for some Bo E 7"r Next, we show that if h is small enough, then for all z E ft, we have a~ <_ (400/e2)h. We simply

choose h so that 8-50 h

e2 < min(6, rad(f~)).

Fix z E ft. The assumption rad(ft) > 0 implies the existence o f a point x = x(z) E ~Q

such that

d(x, z) = �89 min(5, a~, rad(ft)) > 0.

Let 7 be the curve given by Definition 3.1 joining x to z. Let Xo E {7} be such

that d(z, Xo) = �89 = �88 min(6,~r~, rad(f~)). (The existence of such an Xo is

guaranteed by the intermediate value theorem.) Now this choice of Xo gives

d(x, Xo) > d(x, z) - d(xo, z) = d(x, z) - �89 z) = �89 z). (3.3)

Also

Now if

(3.4)

then

s d(xo,Oft) >_ emin(d(xo,x) ,d(xo, Z)) > -~d(x,z) (by (3.3))

= ~ min(min(5, rad(f~)), cry)

> _e8 min ~""-7- (8" 50 h a~) (by the choice of h).

and we are done. If, instead,

min ( ~ h , a ~ ) = a s ,

40Oh, ~ <_ - j -

( 8 . 5 0 h , 8 .50 h m i n \ ~ as) = e2 ,

let B = B(b, h) be a ball containing the point xo. Letp E/} be such that d(B, Of~) = d(p, Of~). Such a point exists since, by Proposition 1.1,/3 is compact. Then

d(B, Oft) = d(p, Oft) > d(xo, Of~) - d(p, Xo)

(by (3.4)) >_ 5Oh - 2h _> 2Oh. E

88 N. G A R O F A L O A N D D. M. N H I E U

The above calculation shows that B E R~. But then

1 crz <_ d(z,B) <_ d(z, Xo) = �88 min(6, rad(~),az) < ~a~.

We have reached a contradiction. This completes the proof. []

L e m m a 3.4. Under the same assumptions of Lemma 3.3, the number h can be chosen such that i f Bj, Bk E ~ , and B~* N B~* # 0, then there exists a chain

Gj,k = {Bt, ..., Bm E R[ Bt = B3, B m = Bk and Bz n Bz+l # O for 1 < l < m - 1},

with m <_ c = c(n, 6).

Proof . First, observe that if aBj n aBk # O, then we have

(3.5) d(Bj,Bk) <_ d(Bj,q) + d(Bk,q) <_ 2(h + a h ) = 2(1 + a)h

where q E aBj N aBk. With this observation, we set

and choose

1602 h = 62

5 h < (loo2

2\ e2 +

so that ifB~* M B~* # 0, then d(Bj,Bk) <_ 6. Let -y be the curve given by Definition 3.1. Consider Bj and Bk E R.~. By an

easy compactness argument, we have

(3.6) l('7) <_ l d(Bj,Bk),

d(z,O~) > emin(d(z, Bj),d(z, Bk)) for all z E {'7}.

We assume without loss of generality that d(z, Bj) <_ d(z, Bk), and distinguish

two cases.

Case 1: d(z,Bj) < S-h - - 6 "

Then

d(z, Off) > d(Bj, OfZ)- d(z, Bj) >_ d(Bj, Of f ) - 5h E

> 2 0 h - 5h > 14h. s s

Case 2: d(z, Bj) > 5-h

LIPSCHITZ CONTINUITY 89

Then

d(z,Of2) > e min(d(z,Bj),d(z, Bk)) = Ed(z, Bj) > r = 5h. s

In both cases, we have d(z, 012) > 5h. Now if B(bj, h/2) tq {3'} ~ 0, then let z

be in this intersection. We have

d(B(bj, h), Oft) > d(z, 012) - diam(B(bj, h)) > 5h - 2h = 3h > O.

Hence, B(bj,h) C 12. This implies B(bj,h) E 7~. Next, we consider G =

{B(b~, h) [ B(bj, h/2) n {'y} r 0}. Using (3.6) and (I .4) it is now easy to see that G

is finite and its cardinality is bounded by a universal constant depending only on e.

Since the balls B(bj, hi2) cover 12, and therefore cover {7}, a suitable subset o f G

provides the sought for chain. []

L e m m a 3.5. We assume the hypothesis o f Lemma 3.4. Then

BoER~t BjC=~(Bo)

where for each Bo E 7~ we have let ~'(Bo) = {Bj E 7~ I B~* n Bo* ~ •} and

G(Bo) = U Co,j.

Proof . For each Bo E 7~' t the inner sum is finite. Fix x E U Go,j; then there

exists a ball B(bt, h) in God for which x 6 B(bt, h). Observe that for all z E 13o we

have

d(z,x) < d(z, bo) + d(bo,bl) + "" + d(bl-l,bt) + d(bt,x) < (m + 2)h,

where m is the cardinality o f the chain Go,j. Therefore Bo C B(x, (m + 2)h). This

inclusion, together with (1.4), allows us to reach the conclusion. []

In what follows we let f2 c /R '~ and set f~s = {x 6 9t I d(x, 012) > s}.

L e m m a 3.6. Suppose that (1.1) and (1.4) hold. l f s > 0 is given, then there

exists h > 0 (small enough) such that i f Bo, B~ E ~ and Bo r (f~ \ 08) r 0,

Bo* n B~* r 0, then

U G0j c (f~ \ f~28).

P r o o f . It suffices to show that for h sufficiently small and B(bl, h) in G0,s,

we have: z E B(b ,h) implies d(z,Oft) < 28. First, Bo N (12 \ 12~) # 0 implies

(3.7) d(Bo, Of~) <_ s.

90 N. GAROFALO AND D. M. NHIEU

Also, we have from (3.5)

(3.8) /(7) _< !d (Bo ,B j ) < c(e)h.

Take q e B(bL, h) cl {3'} (by the construction of the chain such a q exists). Let

z e B(bl; h). Then

d(z, Oft) < d(z, q) + d(q, 0~2) < d(z, q) + d(q, Bo) + d(Bo, O~) + diam(Bo)

< 2h + l(-~) + d(Bo, 0~) + diam(Bo)

(by (3.7),(3.8)) 5 2h + c(e)h + s + 2h = (4 + c(e))h + s.

Thus, once s is given, it suffices to choose h < s/(4 + c(e)). []

L e m m a 3.7. Assume (1.1) and (1.4), I f s > 0 is given, h can be chosen so

that fftsi2 \ 9t, C Uno~'< Bo.

Proof . Note that i f z 6 Os/z then d(z, 0~) >_ s/2. Let B = B(b, h/2) be a bali

containing z, and q 6 B be such that d(B, Off) = d(q, 0~). We show that if h is

chosen small enough, then B E 7~. One simply chooses

8 h < - 2 ( ~ + 1)'

Then

s ~h. d(B, Of~) =- d(q,O~) > d(z,O~) - d(q,z) > ~ - h > e

This implies that B E 7Z~. Since such B's cover fl, we conclude

(as/2\a )c ,nc U B. [] Senl

With the geometric Lemmas 3.3--3.7 in hand, we now turn our attention to the

proof of Theorem 1.7. This will be done in several steps.

P r o o f o f T h e o r e m 1.7. For a fixed 1 < p < c~, let f E s and r /> 0.

We want to produce a function ~ E C~(f ) ) such that

I t f - ~.11~,,.<~) -< c n ,

where C is a constant depending on various parameters, but not on ~/.

(1) We fix h > 0 such that Lemmas 3.3--3.7 hold, and with t = 1602/e 2, we

consider the relative covering ~ . For each B~. E 7~ let

Pj = J

LIPSCHITZ CONTINUITY 91

(2) Let {r be a partition of unity subordinated to the covering {B~* [B~ E R't}.

Theorem 1.5 allows one to choose the Cj's as follows:

(i) (~j E Lipd(B;*), supp Cj C B;*,

(ii) 0 < r 1 6 2 U B~*Df~, j Bj E ~

(iii) [XCj[ < c/h.

We set

go= Z P *J. 8j~7r

The function go will approximate f near 012.

(3) Fix s E (0,1) so that [lf[l~l.p(~\a2o) _< 7/. Using Theorem 1.5 and a

standard partition of unity argument, we construct a function V) E Lipa(~ '~),

0 _< r _< 1, ~ - 1 on f2~ and ~b = 0 on 1R n \ f~/2, with [Xr _< c/s.

(4) Let f r = J~f be the standard Sobolev-Friedrichs mollification o f f . We fix

7- E (0, s/2) such that [[f - fr[IL',,(~,/2) <- rls.

(5) Finally, let gl = (1 -~b)go = (1 - ~ ) ~ B j ~ T z ~ P~r g2 = e f t , and set

gr/ = ffl + g 2 .

We have g~ E Lipd(IR '~). Our main goal is to show that

(3.9) Ilg,7 - f]]~l,p(a) < Cr].

Taking (3.9) for granted for a moment we can complete the proof of Theorem

1.7 as follows. Let ~ be a bounded open set containing ~. By Theorem 1.3, we

know that g,7 is in s hence in s for any 1 < p < ~o. With ~2 playing

the role off~ in (1.3), we can find go E C~ such that

(3.10) ][gv- g,~llL~.,(a)<-II~,- gollc~.,(~)-< ~/2.

The conclusion of Theorem 1.7 now follows by combining (3.9) and (3.10). In

order to complete the proof we are thus left with proving (3.9)

We start with three lemmas whose proofs are standard adaptations o f similar

Euclidean results. Therefore the proofs are omitted.

L e m m a 3.8. Suppose (1.1) holds. Let U C R ~ be a bounded set and

f~ c c U. There exists a positive number Ro = Ro(U) such that for any balls

Bx,Bz, with r(Bi) < Ro and [Bx O B2[ ~> N max(lBl[, [B21), then one has for

u E El'V(2B1 U 2B2):

92 N. GAROFALO AND D. M. NHIEU

(i) If(1.4) and (1.5) hold, 1 < p < c~, and f2B,u2B2 u = O, then

IlulIL,(2B,o2B~) _< Cr(Bx)llXullL,cmB,u2B=).

(ii) Assuming (1.4) , / fp = c~ and fB,oB~ u = O, then

L e m m a 3.9. Let (1.1), (1.4) and (1.5) be true. I l l < p <_ oo, Bj �9 7"r then

there exists a constant C independent o f h such that

IIPjlIL,(%-) -< C(IIZIIL,r + hllX fllr, cB~)).

L e m m a 3.10. Under the same hypothesis o f Lemma 3.9, i f Bo, Bj �9 7Z~ and

Bo* Cl B~* r 0, then

IlPo - PjIIL,~(Bo) -< C h l l X f i l , - , . , (UCo. j ) .

We are now in a position to complete the proof o f Theorem 1.7. We recall

that r -- 1 on f/ , , f~ c fl~/2. By (4) above, and the fact that s < 1, we have

Ilf - gollz',~(n,) < r]. Thus, to complete the proof it suffices to show

I I f - (g~ + g2)llL,,,(a\a~ -< c v .

We let X o F denote F and for 0 < k < m write

X k [ f -- (gX "q-g2)] = X k [ f -- (1 -- ff3)g o -- ~P f r "~- ~.3 f -- e l ]

= Xk[ ( f - fr)~b] + Xk[( f -- go)(1 - r

= A1 + A2.

Now by the choice of r in (3), by (4) above, and the fact that s < 1, we have

IIA1 [[LP(akao) <_ Cr/. To estimate IIA2[Iz,,(a\a,) we consider two cases.

C a s e I: A2 = (1 - r go).

By (2) and Lemma 3.10 one infers

(3.11) Z IlXk[(Po - Pj)r162 <-- Chl-min(k'l) Bj ~ i

(3.12)

Using (3.11) and Lemma 3.9 one obtains

IIXIIIL'r Bj 6 7~' t

B;'nB:'#O

IIXk(Y~ PjCj)IIL,(B:) J

C(IIXdlIL,r + Chl-min(k'l) Z IIXIIIL,(UGo.~)). B, e ~',

B;'nB2"eO

LIPSCHITZ CONTINUITY 93

Now, observe that the sum ~ B~e~', in (3.11) and (3.12) is a finite sum. By

the choice of s in (3), and taking h < 1, we estimate

I1(1 - ~b)Xk(f P P �9 - - g o ) l l L , ( m a ~ g o ) l l L , ( m a . ) < I [ X k ( f -

(by Lemma 3.3)<_C([[Xkfl[~,(a\a~ ~ ]lXk ( ~ Pj4~j) [I~,(B; 1) B o E "t~ ' t j

B'n(a\n,)#0

(by (3.12)) < C[IXkf[[~,(aXa~

+ C Z ([IXkf[lPLn(Bo) -t- h p(1-min(k'l))

BoE~'t B 2 n(n\a~162

I]Xf]l~'(U ~o.j))

(by Lemma 3.6) , Z [[Xkf[[L,(a\a.) + BoE'R' t

B2n(f2\~.)r

t l x s . x k f l l ~ , ( m , ~ . )

nt_ hP(1-min(k,1)) E Z B~q(fl\fl.)#O B;*nB;'#r

][XO co,jXf[[~,(m~=,))

(by Lemma 3.5) _< ClIXkfll[~(n\a~~ + CIIXdIlPL~(ma~,)

+ ChP(1-mln(kJ))ChPO-min(k,1))IIXf [[~,(n\ll2.)

<_ Crl p -t- Crl p + Ch2P(1-min(k'l))UP <_ CU p.

Taking the pth root we finally conclude that Ilf - gollr' p(a\a.) -< Cr~ with C independent of 77.

C a s e II : A2 = ( f - g o ) X k ( 1 - ~P), k # O.

Then, Xk(1 - ~P) is supported in fls/2 \ fls. Recalling that by the choices we made, 0 < h < s < 1, we have

n(f go) \k(1 P - - r : I [ ( f - go) \ k (1 - r 1 7 6

94 N. GAROFALO AND D. M. NHIEU

C o P ~ l l ( f - g )llL,(a,,=\a.)

(by Lemma 3.7) <

(by Lemma 3.8) <

C -~,~ go)IIL,~Bo) I1(/- P

Bo67~' t Bon(a./2\n,)r

C ( ~ hPlIXIII~,(Bo)

BoETg~t Bon(n,/2\n,)r

+ E Z B o e r ; B, en;

Bon(n0/2\x2,)#r B;*AB2" r

II(eo - Pj)r If~,(.o))

c( (by Lemma 3.6) < ~ hPllXfrl~,(a\a=,)

+

Bo n(n./2 \n0 )r B;" nBO" #0

rl(Po - P~)r II~,(Bo))

(by Lemma 3.10) C (h Lfllo,.(akn=./ _<~ P p

+ ~ ~ ChPlIXYlI[.(Uao,~)) Boer; s~en',

B0nCn./=Xn,)Ce BI'nB2* #e

C (hPllfll~,.,,(n\a=,)+ hPllfll~,,(a\~,)) (by Lemmas 3.6, 3.5) <

_< Cry.

Combining Cases I and II with the estimates for A1, we conclude that (3.9) holds

and this completes the proof. []

Finally, we present the proof of Theorem 1.10. The latter is based on its local

version proved in [24] and on Theorem 1.8 above. Before we turn to the proof,

LIPSCHITZ CONTINUITY 95

we take the occasion to clarify some o f the assumptions made in the paper [24].

It was assumed there that besides (1.1), the metric space (R'~,d) is a complete length-space. We have later discovered that this assumption is not necessary for

the results in [24] to hold. For a detailed discussion of this and related matters, we

refer the reader to the book [23].

P r o o f o f T h e o r e m 1.10. Let B be a ball with radius r(B) < Ro(U) and containing ~. Given u E E l'p(f~) with p > Q, we extend u to ,~u E s Thanks

to Proposition 1.1, ~ is compact, since such i s /3 D ~. Therefore, by the local

version of Theorem 1.10 in [24] (see Theorem 1.11 in [24]) we infer the existence

of C > 0 such that, after modification on a set o f measure zero in ~, we have

lu(x) - u(y)l < Cd(x, y)o,

for x, y E ~. This proves the first part o f Theorem 1.10.

As for the second part, we know from Propositions 2.8 and 2.11 that in the

situations described there any closed ball is compact. Let then ft be a (e, 6)-domain

with unrestricted diameter either in R '~ or in a stratified, nilpotent Lie group G. Let

B be a ball containing (L From the compactness o r b it follows that ~ is compact.

Repeating the above arguments we reach the conclusion. []

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Nicola Garq/alo DEPARTMENT OF MATHEMATICS

PURDUE UNIVERSITY WEST LAFAYETTE, IN 47907-1968, USA

DIPARTIMENTO DI ,~E'I'ODI E MODELL] MATEMATICI VIA BELZONI, 7

35100 PADOVA, ITALY email: garofaloOmat h.purdue.edu

Duy-Minh Nhieu DEPARTMENT OF MATHEMATICS

PURDUE UNIVERSITY WEST LAFAYETTE, IN 47907-1968, USA

Current address." INSTITUTE OF MATHEMATICS

ACADEMIA SINICA NANKANG, TAIPEI 11529

TAIWAN R.O.C. email: d rnn hieu~gate,sinica.edu.tw

(Received November 6, 1996 and in revised form October 20, 1997)