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LIPSCHITZ CONTINUITY, GLOBAL SMOOTH APPROXIMATIONS AND EXTENSION THEOREMS FOR SOBOLEV FUNCTIONS IN CARNOT--CARATHIEODORY SPACES

NICOLA GAROFALO* AND D u Y - M I N H NHIEU t

1

I n t r o d u c t i o n a n d s t a t e m e n t o f the r e s u l t s

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 geometric 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, extension 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 CarnotCarath6odory 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 requirements. 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 continuous entries. A further motivation comes from the fact that there are interesting *First author supported by NSF Grant No. DMS-9404358. t Secondauthor supportedby a grant of the PurdueResearchFoundation and also by the firstauthor's NSF Grant no. DMS-9404358. 67 JOURNALD'ANALYSEMATHEMATIQUE,X.bl.74 (1998)

68

N. G A R O F A L O A N D 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 Riemannian 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 isoperimetric 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),~ >22. 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 - z o l < R}. Throughout 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

69

LIPSCHITZ CONTINUITY

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

C t x - yl 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 (xo,R)

IXuldz.

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

(l.5)' sup[.Xl{= e B(xo,R) llu(x) )~>0

f

-

uBI > ,X}l] _< C2n/

IX l

dx.

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. Theorem

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. Theorem

1.8. A s s u m e (1.1), (1.4) and (1.5). L e t 1 < p < oz. l f f~ C ~,'~

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

(ii) IIEfllz:,,.(a,,) < Cllfll,:,,,,(.). 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 of 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.


73

For the Heisenberg group iEn, with non-commuting vector fields 0 x j=

0

oz~ + 2y j-b-i,

0 ~

= Oy--j -

2xj O

'

j=l,

"

.,n,

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) = I x - 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 if ~ C R 'L is a bounded Lipschitz domain, and ifp > 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,...;r1, [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


T h e o r e m 1.10. A s s u m e (1.1), (1.4) a n d (1.5). L e t f2 C (2 C U be a (e,6)d o m a i n with dn = supx,u~ ~ d(x, y) < Ro(U), u E s

with p > Q. Then u can

be m o d i f i e d on a set o f m e a s u r e zero in such a w a y that u E F~

with 0 = 1 - Q/p.

B y this w e m e a n that u has a continuous representative in (2 (also d e n o t e d by u) satisfying r (x) sup

9 ,u~f~,zev

(y)r

d(x, y)O

< oc.

We emphasize the global nature of the above results. We mention that a local version o f 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

Jp( ; f

)=falXul pdz,

Its Euler-Lagrange equation is given by m

(1.6)

f-.pU =

_,Xff(IXul

-2Xju)

= 0 in

j=l

For a study o f 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 o f ( w e a k ) 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].

75


2

Local results

This section is mainly devoted to proving Theorems 1.3, 1.4 and 1.5. As a 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. 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 0 one has 7"

(2.1)

B(zo, M(xo, r)) C B~(zo,r),

where M(xo, r) is as in the p r o o f o f 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) 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 p r o o f 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]. Lemma

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 f o r x, x' E o; and It[ _< T, (2.2)

t0(t,x) - 0(t,x')l

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

< Y u , r >= s u Y ' r

(2.5)

fauYCdx-

fnudivYCdx,

in order to show that Y u 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. B y the assumption on the coefficients o f the vector fields in X, we see that IldivYIIL~(n ) < co, and therefore

f u d i v Y C d x < 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 (2.6)

s

uYCdx _< ClIr

for some C > 0 independent o f r We have r

x)) - r

=

r

x)) ds =

/o'

Y(r

x)) ds = tg(t, x),

78


where g(O, x) = that g E L~176

Y4)(x). Since (t, x) ~ O(t, x) is continuous in I - T , T] x w, we see T] x w). Therefore, by the Lebesgue dominated convergence one

has

s163

(2.7)

t

Now we recall that the map

O(-t, .).

x ~ O(t, x)

is invertible, its inverse being given b y

This observation and the change o f variable y =

that for each fixed

Itl

dx. O(t, x)

allow us to infer

< 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, u(y)e(y) t

= O(t,x), then for any yt = O(t, x). Therefore

If we now let 7(t) curve joining x to

[detJO(-t,y),dy y ) ] - 1 dy.

0 < t _< T, 7 : [0,t] --, IR'~ is a sub-unit

d(x,O(t,x)) 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 o f 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} o f vector fields. As we mentioned earlier, when (1.1) holds the Lipschitz continuity with respect to 1,~ d(x, y) o f 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 p r o o f 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 ) we let J,u = K, 9 u where K , = E-nK(e -1.). Let Given a function u E Lbc(IR

Y = ~ k n~ 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 C C

Y(J,u) = J,(Yu) + L u

we have in D'(a:),

80

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

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

R~(z,h)=e1~ -~kO [(bk(X + e.) -- bk(X))g] (h). k=t

L e m m a 2.6 is due to Friedrichs [22]. We mention the two notable properties o f the k e r n e l / ~ ( x , h) which will be needed in the sequel: (2.8)

f [s J~

dh=O

forx 9

and

e>0,

t'

] sup IRa(z, h)l dh < C3, J.~ XC~

(2.9)

where C3 > 0 is independent o f e > 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 if 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

f o r every x, y 9 B(xo, R). In particular, one has [lu - UBIIL•(B) 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 if 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~,

if g = exp X. Since V1 generates g as an algebra, if we choose a basis X = {X1, ..., Arm} o f V1, then X satisfies H f r m a n d e r ' s finite rank condition. Denote b y

84


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 IG 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 lig-lhlc < R} denote the ball in the homogeneous norm I Ic centered at g with radius R. Proposition 2.10 implies (2.19)

B c ( g , C " - l R ) 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 o f P r o p o s i t i o n 2.8. By lemma 1.4 in [ 16] each closed homogeneous ball [3G(g, R) = {h 9 G I Jg-lhlc O. P r o o f . 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 _ 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 ~-~,~=1d(?(ti), "~(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, if 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 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 iR7' 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 R e m a r k 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. t = --~-h,

e2 Ro with 0 < h < 160---2"

For B# 9 R'~ we write 1602

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

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 ifaz < (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~ 0 implies the existence o f a point x = x(z) E ~Q such that

d(x, z) = 89min(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 = 88min(6,~r~, rad(f~)). (The existence of such an Xo is guaranteed by the intermediate value theorem.) Now this choice of Xo gives

(3.3)

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

z) = 89

z).

Also s

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

(3.4)

(by (3.3))

= ~ min(min(5, rad(f~)), cry) > _e8min ~""-7(8" 50 h a~)

(by the choice of h).

Now if min ( ~ h ,

a~)=as,

then

~ 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