O N J O R D A N A R C S A N D L I P S C H I T Z C L A S S E S OF F U N C T I O N S D E F I N E D ON T H E M
B Y
A. S. BESICOVITCH and I. J. S C H O E N B E R G
Philadelphia, Pa, U.S.A. (1)
Introduction L e t the equations
x=/(t), y=g(t)
(0~<t~< 1),define a continuous are in the plane E 2 and let us assume t h a t the derivative of g (t) with respect to /(t) vanishes everywhere. According to Lebesgue ([4], p. 296) this means t h a t
g (t + h) - g (t)
lim 0 (0~<t~<l),
h-~0 / (t + h) - / (t)
where we ignore as h-->0 those values of h which produce simultaneously vanishing increments A / and A g and where the above limit relation is assumed to hold, b y definition, in the interior of a n y common interval of constancy f o r / a n d g. Lebesgue showed t h a t g (t) is necessarily constant provided t h a t we assume / (t) to be of bounded variation. R. Caccioppoli [2] and J. Petrovski [6] showed t h a t g (t) is constant even without the last additional assumption concerning /(t).
H. W h i t n e y [8] showed t h a t the situation is different for skew arcs: W h i t n e y constructs in the complex x-plane a J o r d a n arc
J:x=/(t)
(0 ~t~< 1), (1)(1) T h e m a i n r e s u l t s of t h e p r e s e n t p a p e r were a n n o u n c e d in t h e n o t e : S u r les arcs ascendants pente partout nulle et des probl~mes qui s'y rattachent, C. R . Acad. Paris, 249 (1959), 1079-1080.
S u b s e q u e n t l y M. G. Glaeser k i n d l y b r o u g h t to o u r a t t e n t i o n t h e r e f e r e n c e s [3] a n d [8] w h i c h h e l p e d u s to s h o r t e n a n d i m p r o v e o u r p a p e r .
8 - 61173055. Aeta mathematica. 106. Imprlra4 le 28 septembre I961.
114 A, S. B E S I C O V I T C H A N D I . J , S C I t O E N B E R G
a n d also a real-valued, non-decreasing, n o n - c o n s t a n t continuous function g ( t ) i n [0, 1]
such t h a t
lim g (t + h) - g (t)
h-~o I / (t + h) - / (t) l = 0 for all t in [0, 1]. (2) I t is clear t h a t the point (/(t), g (t)) describes a J o r d a n arc, in the 3-dimensional space, which is rising while having, in view of (2), everywhere vanishing slopes with respect t o the complex x-plane which is t h o u g h t of as horizontal. F o r a p a r t i c u l a r l y simple example of such a skew arc (whose projection J is the arc of I t . y o n Koch) see G.
Glaeser ([3], 57-58).
Our first result is
T H ] ~ O R ~ t 1. There exists in the complex x-plane a Jordan arc J, having the/ol- lowing properties: Let v and v' be distinct points o/ J and let J (v, v') be the subarc o / J having v, v' as end points while m 2 J (v, v') denotes its 2-dimensional Lebesgue measure.
To every positive e there corresponds a constant C~ such that /or all subarcs
O < m ~ J (v, v ' ) < C ~ [ v - v ' [ ~-~.
(3)
A n arc enjoying these properties will be c o n s t r u c t e d in w 1 below. Before we discuss t h e significance of T h e o r e m 1 let us first show how it furnishes one more example of an arc of the k i n d first c o n s t r u c t e d b y W h i t n e y . To o b t a i n it we erect at each point v, of J , an ordinate y = G ( v ) = m S J (0, v). This is a continuous p o i n t - f u n c t i o n on J which increases strictly in view of the first inequality (3): If J (0, v) is a proper subarc of J(O,v') t h e n G ( v ' ) - G ( v ) = m 2 J ( v , v ' ) > O . B y (3)
G (v') - G (v) = m2 J (v, v')
]v'-v] !lv-v']
< C ~ l v - v ' l 1-~I f we select ~ < 1 a n d let v'-->v we see t h a t the skew arc described b y (v, G(v)), v E J , has e v e r y w h e r e a vanishing slope.
Observe t h a t the e appearing in Theorem 1 is required to be positive. This is n o t an accident because of
TI~EOREY~ 2. Let J be a plane Jordan arc such that m ~ J > O . Then
lim m2 J (v, v') _ + oo (4)
Iv-v'l 2
holds at almost all points v, o/ J, in the sense o] the ms-measure.
O N J O R D A N A R C S A N D L I P S C I t I T Z C L A S S E S O F F U N C T I O N S 115 T h i s r e s u l t allows a n a p p l i c a t i o n t o t h e n o t i o n of lower q u a d r a t i c l e n g t h of arcs.
W e use t h e following
D E F I N I T I O N 1. Let the complex-valued /unction x = / ( t ) , ( 0 ~ t ~ < l ) , describe a continuous arc B in the plane. I / t O = 0 < t 1 < t 2 < " " < t n = 1 , w e de/ine the lower qua- dratic length o/ B by
5 (2) B = lim ~ ]/(t~) - / (t~-l)[2, (5)
- - i = l
where the limes in/erior is taken as m a x ] t i - t ~ _ l l - > 0 .
I t was s h o w n b y A. Ville [7] t h a t L a) B = 0 p r o v i d e d t h a t m 2 B = 0. I t n o w t u r n s o u t t h a t t h e a d d i t i o n a l c o n d i t i o n m a y b e i g n o r e d since we h a v e t h e following
T H E O R E M 3. The lower quadratic length o/ any plane continuous arc vanishes.
U s i n g T h e o r e m 2 we first p r o v e T h e o r e m 3 for t h e case of a J o r d a n a r c (Sec- t i o n 2.2). A l e m m a t o t h e effect t h a t a n y c o n t i n u o u s arc m a y b e r e d u c e d t o a J o r d a n a r c b y r e m o v i n g a p p r o p r i a t e loops e a s i l y allows t o c o m p l e t e a general p r o o f of T h e o - r e m 3 (Section 2.3).
I n c o n t r a s t to T h e o r e m 2 we h a v e a d i f f e r e n t s i t u a t i o n for J o r d a n arcs of f i n i t e
~ - d i m e n s i o n a l H a u s d o r f f m e a s u r e ; we s t a t e t h i s as T H E O R E M 4. Let 1 < ~ < 2 .
A~-measure such that
/or all subarcs J (v, v').
There are plane Jordan arcs o/ /inite and positive
A~ J (v' v') < K (6)
Iv-v,I
H o w e v e r , a w e a k e r a n a l o g u e of T h e o r e m 2 still h o l d s w h i c h shows t h a t t h e ex- p o n e n t ~ of I v - v' ]~ in (6) can n o t be i n c r e a s e d . I n d e e d , we h a v e
T H E O R E M 5 . I / J is a plane Jordan arc such that O < A ~ J < ~ , 1 < o~ < 2 then - - A ~ J ( v , v')>~
l i m [ v - v ' 1 (7)
Vt~O'Y [ o~
at almost all points v in the sense o/ the A~-measure.
W e t u r n n o w t o a discussion of L i p s c h i t z classes of p o i n t - f u n c t i o n s G ( v ) d e f i n e d on an arc J . W e shall use t h e following
D E F I N I T I O N 2. Let ~ ( x ) be de/ined /or x > 0 and be positive, continuous, non- decreasing and such that ~ ( + O)= O. Let J be a Jordan arc and G (v) be de/ined on, J.
116 A . S. B E S I C O V I T C H A N D I . J . S C H O E N B E R G
We write G (v) E Lipj ~ (x) provided that there is a /inite-valued positive ]unction A (v) such that
I G ( v ) - G ( v ' ) l < A ( v ) r (v, v ' E J , v=~v'), (8) and we say that G (v) is o/ Lipschitz class r (x) along J. I / A (v) is bounded we write
G (v) E U Lipj r (x)
and say that G (v) is uni/ormly o/ Lipschitz class ~ (x) along J.
I t is well k n o w n t h a t if J is the segment [0, 1] a n d r ( x ) = o ( x ) , as x->0, t h e n constants are t h e only elements of the class Lipj r (x). T h e situation is different for plane arcs J : F o r the are J of T h e o r e m 1 a n d the function G ( v ) - m~J (0, v) we see f r o m (3) t h a t
G(v) e U L i p j x 2 ~ ( ~ > 0 ) , while G (v) is certainly n o t constant.
W h a t a b o u t t h e class U Lipj x ~ obtained b y letting here ~ become zero? The answer becomes obvious if we a p p l y our T h e o r e m 3. Indeed, let G (v) satisfy t h e inequality
] G ( v ) - G ( v ' ) [ < A [ v - v ' [ 2 (v, v ' e J ; A const.).
I f J is t r a c e d o u t b y x = / ( t ) , 0~<t~< 1, a n d if a a n d fl are t h e endpoints of J t h e n [G (fl) - G (cr ~< Y. I G (/(t~)) - G (] (t~-l)) I < A ~ 1 ] (t~) - / (t,_l)12.
1
However, we k n o w t h a t the last-written s u m will converge to zero for a n appro- priate sequence of divisions b y v i r t u e of T h e o r e m 3. T h u s G ( ~ ) = G (fl). Since this a r g u m e n t m a y be applied to a n y subarc we h a v e established
T ~ E O R E M 6. I] J is a plane Jordan arc and the ]unction G (v) is uni/ormly o/
the Lipschitz class x 2 along J, then G (v) is necessarily a constant.
L e t n o w J be a J o r d a n arc in the plane such t h a t A ~ J < ~ ( 1 < ~ < 2 ) .
B y T h e o r e m 4 we see t h a t T h e o r e m 6 does n o t generalize to such arcs, for if J is an arc as described b y T h e o r e m 4 a n d G (v) = A ~ J (0, v) then (6) implies t h a t G (v) E U Lipj x ~ while G (v) is n o t constant. However, a slightly weaker analogue of T h e o r e m 6 holds which we state as
ON J O R D A ~ ARCS AI~D LIPSCI-IITZ CLASSES OF F U N C T I O N S 117 T~V, OREM 7. I] J is a plane Jordan arc o/ finite A%measure, 1 < a < 2 , and G (v) is o/ Lipschitz class r (x) along J, then
( x ) = o ( x ~) as x->O, (9)
implies that G (v) is a constant.
W e conclude our I n t r o d u c t i o n w i t h a few r e s u l t s w h e n t h e J o r d a n a r c J is in a space of d i m e n s i o n h i g h e r t h a n t w o . T h e r e is a n a t u r a l g e n e r a l i z a t i o n of T h e o r e m 7:
I / J c E ~ , A ~ J < ~ , l < ~ < ~ n and G(v) e L i p ~ r then
r (z) = o (x ~) (10)
implies that G (v) in a constant.
T h e o r e m 7 a n d its g e n e r a l i z a t i o n j u s t s t a t e d suggest t h a t if J is a n a r c of t h e real H i l b e r t space H , a g a i n t h e class Lip~ ~ (x) will c o n t a i n o n l y c o n s t a n t s p r o v i d e d t h a t t h e s c a l e - f u n c t i o n r (x) t e n d s t o zero s u f f i c i e n t l y f a s t as x--> + 0. H o w e v e r , i t is a curious f a c t t h a t such is n o t t h e case a n d we s t a t e t h i s as our l a s t
T ~ E O R E M 8. Let r (x) be a given scale-/unction subject to the conditions o/ De- finition 2. There are in the Hilbert space H Jordan arcs J such that the class U L i p j • (x) contains /unctions which are not constants.
O b s e r v e t h a t t h e s c a l e - f u n c t i o n r (x) m a y t e n d t o zero as f a s t as we wish.
w 1. Proof of Theorem l
1.1. T H E CONSTRUCTION OF THE ARC J . L e t S O be t h e u n i t - s q u a r e , one side of w h i c h connects x = 0 t o x = 1. T h i s a n d all following s q u a r e s will b e a s s u m e d t o b e closed. W e shall n o w c o n s t r u c t a c o n t i n u u m J1 as follows: L e t
1 1
0 ~ - 2 16n2 ( n = l , 2, ...). (1.1)
I n S O we c o n s t r u c t f o u r c o r n e r s q u a r e s s~, s~, Sl ~, s~ of sides = 01. W e n o w c o n n e c t t h e s e s q u a r e s b y t h r e e s e g m e n t s (or links) as shown in fig. 1, o b s e r w i n g t h a t t w o of t h e s e links lie along t h e t w o v e r t i c a l sides of S o while t h e t h i r d l i n k a b lies on t h e line w h i c h carries t h e t w o lower sides of s~ a n d s~.
On t h e l i n k ab we consider i t s C a n t o r m i d d l e - t h i r d set ~ a n d in p a r t i c u l a r its c o m p l e m e n t a r y set of i n t e r v a l s . O n each of t h e s e i n t e r v a l s a s side we c o n s t r u c t a square, l y i n g a b o v e a b, a n d d e n o t e b y a t h e set of s q u a r e s so o b t a i n e d . W e n o w f o r m t h e u n i o n [a, b/U a w h i c h is e v i d e n t l y a c o n t i n u u m j o i n i n g a t o b. W e r e p e a t
118 A. S. B E S I C O V I T O H A N D I. J . S C t t 0 E N B E R G
D a b E
-] E
sl
0 I
Fig. I.
the same construction on each of the remaining two links placing the sets of squares as indicated in fig. 1. This completes the construction of the continuum Jl. Observe t h a t J1 is c o m p o s e d of f o u r corner squares, e n u m e r a b l y m a n y intermediate s q u a r e s a n d f i n a l l y t h r e e C a n t o r sets. L e a v i n g o u t t h e C a n t o r sets we h a v e a collection of s q u a r e s s~ w h i c h we d e n o t e b y S 1. W e e s t a b l i s h a n o r d e r r e l a t i o n a m o n g t h e e l e m e n t s of S 1 = {s~} o b t a i n e d b y t r a v e r s i n g J1 f r o m x = 0 t o x = 1. E a c h s q u a r e s 1 h a s a n entry point a n d a n exit point d e f i n e d in a n o b v i o u s w a y .
T h e s e c o n d s t e p of our c o n s t r u c t i o n is as follows: I n e a c h s q u a r e sl(slES1)we j o i n its e n t r y p o i n t t o its e x i t p o i n t b y a c o n t i n u u m s i m i l a r in s t r u c t u r e t o J1, t h e o n l y difference b e i n g t h a t t h e sides of its four corner s q u a r e s a r e n o w = 0 3 9 side s 1.
R e p l a c i n g in J1 e a c h s q u a r e Sl b y its s u b - c o n t i n u u m so c o n s t r u c t e d we o b t a i n o u r s e c o n d c o n t i n u u m J~. I t is c o m p o s e d of a set S 2 of s q u a r e s s2=s~ a n d e n u m e r - a b l y m a n y C a n t o r sets.
T h i s c o n s t r u c t i o n is n o w r e p e a t e d i n d e f i n i t e l y b y o b t a i n i n g Jn f r o m J n - 1 b y r e p l a c i n g e a c h sn-1 (ESn_I) b y a c o n t i n u u m s i m i l a r in s t r u c t u r e t o J1, h a v i n g 4 c o r n e r s q u a r e s of sides = 0n" side sn 1. S, = {sn} will d e n o t e t h e set of s q u a r e s of Jn.
E v i d e n t l y J1 D J~ ~ ...
a n d J = N J r (1.2)
is e a s i l y s h o w n t o b e a J o r d a n arc j o i n i n g t h e p o i n t x = 0 t o x = 1.
L e t us s h o w t h a t
m S J > 0. (1.3)
T o see t h i s l e t ~ n ( n = l , 2, ...) d e n o t e t h e set of t h o s e 4 ~ e l e m e n t s of Sn w h i c h a r e o b t a i n e d b y c o n s t r u c t i n g , s t a r t i n g f r o m S o , o n l y c o r n e r s q u a r e s while o m i t t i n g t h e
O N J O R D A N A R C S A N D L I P S C H I T Z C L A S S E S O F F U N C T I O N S 119 i n t e r m e d i a t e s q u a r e s a l t o g e t h e r . These 4 ~ e l e m e n t s of ~ , a r e s q u a r e s of sides = 0102... 0~
a n d J n ~ = . B y (1.1) a n d (1.2) we t h e r e f o r e f i n d
m 2 J = l i r a m ~ J ~ >~ lim m 2 ~,~ = l i m 4 ~ ( 0 1 0 2 . . . 0 ~ ) 2 = 1 - > 0
a n d (1.3) is e s t a b l i s h e d .
A s i m i l a r discussion shows e a s i l y t h a t e v e r y s u b a r c J (v, v') of J has p o s i t i v e m~-measure a n d t h i s a l r e a d y e s t a b l i s h e s t h e first i n e q u a l i t y (3). W e n o w t u r n t o a p r o o f of t h e s e c o n d i n e q u a l i t y (3).
1.2. P R O O F O r T H E O R E M 1. A p r o o f of t h e second i n e q u a l i t y ( 3 ) w i l l r e q u i r e a closer discussion of t h e r e l a t i o n b e t w e e n a s u b a r c J (v, v') a n d t h e s q u a r e s s~ of t h e c o n t i n u u m Jn" T h e inclusion r e l a t i o n J ( v , v ' ) c s ~ r e q u i r e s no e x p l a n a t i o n ; if s~f) J c J ( v , v') t h e n we shall s a y t h a t J ( v , v') c o n t a i n s t h e s q u a r e s,, or t h a t s~ is c o n t a i n e d in J ( v , v'). T h e s y m b o l Sn will also b e u s e d t o d e n o t e t h e a r e a of t h e s q u a r e sn. T h e s q u a r e s. c o n t a i n s f o u r c o r n e r s q u a r e s s=+l; t h e l e a s t d i s t a n c e or t h e w i d t h of t h e c o r r i d o r b e t w e e n t w o of t h e s e will be d e n o t e d b y c o r r s = , i t s v a l u e b e i n g c o r r s~ = (1 - 2 0~+1) side sn 8 (n + 1) 2 side s~. 1 (1.4)
O u r p r o o f is b a s e d o n t h e following p r e l i m i n a r y r e m a r k s :
1. The distance between two complementary intervals o/ the Cantor set is at least equal to the length o/ the smaller interval. The distance between a complementary inter- val and an endpoint o/ the Cantor set is never less than the length o/ the interval.
2. Given e >O there is a constant Be such that
S n
(rs~)'cor - - "2-" < B , (I .5) /or all n and all squares s~.
O m i t t i n g t h e simple p r o o f of t h e first r e m a r k , we t u r n t o t h e second.
of (1.4) a n d t h e e v i d e n t i n e q u a l i t y side s ~ < 2 -~, we o b t a i n
I n v i e w
S n Sn,2) ~ < 8 2-~ (n + 1) 2(2-~) (side s~) ~ < 8 2 (n + 1) 4 2 -~n,
(corr
w h i c h is a b o u n d e d sequence a n d (1.5) is e s t a b l i s h e d .
G i v e n t h e a r c J (v, v') we define t h e i n t e g e r n such t h a t J (v, v') is c o n t a i n e d i n a s q u a r e sn b u t n o t in a n y s~+l. W e n o w d i s t i n g u i s h t h r e e cases d e p e n d i n g on t h e r e l a t i o n of J (v, v') t o t h e four c o r n e r s q u a r e s of s~.
120 A. S. B E S I C O V I T C I t A N D I . J . S C H O E N B E R G
1. J (v, v') contains points o/ at least two corner squares o/ s n. F r o m t h e defini- tion of corr sn a n d the i n e q u a l i t y (1.5) we o b t a i n t h a t
m 2 J (v, v ' ) i s ~
Iv - v' 12-~ ~ icorr sn) 2-~ < B,. (1.6)
2. J (v, v') ]ully contains a corner square sn+l, o/ sn, but does not contain points o] any o/ the other corner squares o/ sn. A glance at fig. 1 (where t h e large square n o w represents sn) shows t h a t
1 1
I v - v' I > 12 side 8 n +1 = ~ On +1 side Sn > ~ side sn.
B u t t h e n m2 J (v, v') s~
I v - v ' 12-~ < (8 1 side sn) 2-* < 82 (side sn) ~ ~< 8 2. (1.7) 3. I n the remaining cases (see fig. 1) all the squares sn+l containing points of J (v, v') are based on one a n d the same straight line. This will i m p l y t h a t t h e are J (v, v') is fairly stretched, in fact we shall prove the following: I[
d = diam J (v, v') (1.8)
then [v - v ' [ ~ ~ d. (1.9)
i s j a n d to fix the ideas we shall assume t h a t the square Indeed, let vEs,+l, v'E ~+1
8 i n+l does n o t exceed s~+l in size. L e t v 0 be the o r t h o g o n a l projection of v on to the c o m m o n base line of our squares. L e t v~ be the exit p o i n t of s t ~+1 a n d v~ the e n t r y point of 8~+ 1.
We distinguish t w o cases depending on w h e t h e r I v 2 - v, I is ~> d / 1 3 or < d/13.
I n the first case when ] v 2 - v l l ~>d/13 it is evident t h a t also
iv _ v, I >~ _1 d (1.10)
13 "
L e t us n o w assume The opening r e m a r k /ortiori
IV-Vol< d,
as well as
d i a m J (v, v2) • diam J (v, vl) 9 diam J (vl, v2) < ~ d 9 13 ~ d.
]v2-vl] < ~d.
1 of Section 1.2 implies t h a t side sn+1<~lv2-vll<d/13 a n d a
(1.11)
O N J O R D A N A R C S A N D L I T S C H I T Z C L A S S E S O F F U N C T I O N S 121
B y (1.12) and therefore
We now conclude from (1.8) t h a t
10 (1.12)
diam J (v~, v') > i 3 d.
Consider now the sequence of corner squares sn+~c s~+l which have the common entry point v 2 (~=1, 2 . . . . ;sn+l=S~+l) and let p be such t h a t
V I ~8n+p~ V p ~ 8 n + p + l .
diam Sn+~>~ diam J (v~, v') >~i~d 10
1 10 2
Iv'-v I> side
diam3V~ ]
1 3 d > ~ d- a /ortiori iv ' _ vo [ > 2B u t then d.
This and (1.11) imply (1.9) which has now been shown to hold in a n y ease.
Returning to our proof of (3) we observe (1.8) implies t h a t m,~J(v, v ' ) < d ~ and now b y (1.9)
m 2 J (v, v')
I v _ v , 12_ ~ < 13~ d~< 132. (1.1a)
The estimates (1.6), (1.7) and (1.13) establish (3) and our proof is completed.
w 2. The lower quadratic length of plane ares
2.1. P g o o F oF T ~ o R ~ 2. The k e y to our discussion of quadratic length is Theorem 2 which we are now going to establish. Let m 2 J > 0. Denote by E the set of points v of J to which corresponds some v ' ( 4 v ) such t h a t
m 2 J ( v , v'
[ v _ v , 12 )> A > 8, (2.1)
where A is a certain constant. E is open. Let E 1 be the complement of E oll J . We shall show that, for a n y A, m2E1=O.
Suppose t h a t for a certain A this is not true, hence m 2 E 1 > 0 , and let an in- terior point v 0 of J be a density point of E 1. Then to a n y ~ > 0 corresponds an r 0 > 0 such t h a t
m s (C (Vo, r) -- E l ) < 7 2 r ~ if r < ro, (2.2) 9 - 61173055. Acta mathematica. 106. Imprim6 le 27 septembre 1961.
122 A . S. BESICOVITCIt A:ND I. J . SCHOE:NBEI~G
where c (Vo, r) denotes t h e circle h a v i n g center v 0 a n d radius r; this circle a n d all circles of t h e present discussion will be considered to be closed. A t this point we select positive quantities ($, ~, r subject to t h e inequalities
( ~ < 2 , ~ 3 ~ < ~ , 4 A r < r o , (2.3)
a n d notice t h a t t h e first t w o i m p l y ~]2 32 A < 6 2 < ~ " 2 / A hence
16 A 2 ~]s < ~. (2.3')
We shall use the order relation a m o n g t h e points of J using t h e s y m b o l ~ , a n d shall speak of the first a n d last point of J in a given closed set, denoting b o t h as t h e extreme points of J . L e t n o w v 1 a n d v s, vl~,Vo~V s be t h e extreme points of J belonging t o the circle c (v0, r) so t h a t no v-<v 1, or v>-v 2 belongs to t h e circle. Ob- viously Iv1 - v 0 I = ]vs - v01 = r, while the arc J (v 1, v2) need n o t belong entirely to the circle c (v 0, r). I n fact t h e d i a m e t e r of the arc J (vl, vs) m a y well be large c o m p a r e d with 2 r . As v 0 does n o t belong to E we h a v e
m 2 J (v 1, v o ) < ~ A [ v l - v o ] S = A r 2, m s J (v o, vs) <~ A [ v s - v o ]2 = A r s,
a n d therefore m s J (vl, vs) ~< 2 A r ~. (2.4)
Writing U = E 1 J (v 1, vs)
we have a /ortiori m 2 U <~ 2 A r ~. (2.5)
We denote b y d (p; U) the distance from t h e p o i n t p t o t h e set U a n d b y (l, U}
t h e set of points p of the plane such t h a t d ( p , U)~<l a n d not belonging to U. W e shall n o w s t u d y t h e set
V = { ~ r , U } ' c ( v o, 4 A r )
in its relation to the arc J . First we a d d t o V such points of U which lie in c(v0, 4 A r ) to o b t a i n t h e closure V. L e t n o w v a a n d v a be t h e extreme points of J belonging to V a n d let us show t h a t
V3"~Vl, Vs'~V 4. (2.6)
To see this we h a v e to show t h a t v 1 E V a n d v s E V . Suppose t h a t v I ~ V so t h a t E 1 c (vl, (~ r) = o. B u t e v i d e n t l y
m 2 {c (vo, r ) , (~ (Vl, (~ r)} > 6 2 r 2,
oN ;IORDAI~ ARCS A~D L n ' s c l t l T Z CLASSES OF rU~CTIO~S 123 w h i l e t h e s e t of p o i n t of c ( v o , r ) w h i c h a r e n o t i n E 1 is of m e a s u r e < 9 2 r s, w h i c h is < ~s r e. A s i m i l a r a r g u m e n t s h o w s t h a t v s E V a n d t h e r e l a t i o n s (2.6) a r e e s t a b l i s h e d . W e f i n a l l y o b s e r v e t h a t v 3 a n d v 4 c a n n o t b e l o n g t o U = E 1 J (vl, vs) a n d t h e r e f o r e v 3 a n d v 4 lie in V. T h u s v a a n d v4 a r e also t h e e x t r e m e p o i n t s of J b e l o n g i n g t o
V. S i n c e v a a n d v 4 b e l o n g t o {~r, U}, t h e r e a r e p o i n t s v~, v~ of U s u c h t h a t
B y (2.2) a n d t h e l a s t c o n d i t i o n (2.3)
m s {c (v 0, 4 A r) - E l } < *]216 A s r ~ f r o m w h i c h , in v i e w of V = c (v 0, 4 A r), w e c o n c l u d e t h a t
m s ( V -
VE1)
<,}2 16 A 2 r e. (2.8)B u t t h e p a r t of E 1 t h a t b e l o n g s t o V lies o n t h e a r c J (va, va), a n d a g a i n t h e p o i n t s of E 1 J ( v l , v s ) = U d o n o t b e l o n g t o V. W e c o n c l u d e t h a t
E~ V ~ J (va, v~) + J (v2, va). (2.9)
N o w V = ( V - E 1 V) + E 1 V,
m s V = m s ( V - V E 1 ) + m e E 1 V , a n d (2.8), (2.9) i m p l y
m 2 V < *]2 16 A 2 r 2 + m 2 J (v3, Vl) + m 2 J (vs, v4) a n d a / o r t i o r i
m s J (va, v~) + m s J (v~, v4) > m s V - ~s 16 A s r 2. (2.10) T o e s t i m a t e m s V f r o m b e l o w w e s h a l l i n t r o d u c e p o l a r c o o r d i n a t e s (if, 0) w i t h t h e o r i g i n a t vo, a n d we w r i t e
l(O)= U(Q, 0), (rl, rS, 0)= U (~,0).
~>~0 rl~<q~<r ~
W e c o n s i d e r t h e set of d i r e c t i o n s
01 [0 ] u . ((t -*]) r, r, 0) = ~].
O b s e r v i n g t h a t all p o i n t s of E 1 c (v 0, r) lie o n t h e a r c J (vl, vs), we h a v e E 1 c (v o, r) -- E 1 J (v 1, vs) c (Vo, r) = U c (v o, r)
a n d t h u s c (v o, r) - E 1 = c (vo, r) - U .
124 A. S. B E S I C O V I T C I / A N D I. J. S C H O E N B E R G
B y (2.2) m s {c (v o, r) - U} < U s r ~,
f r o m which it follows at once t h a t
Consider n o w the set
m 0 1 < 2 U.
(2.11)
Os [01 m { U . (r, 4 A r, 0)} > (4 A - 1 - ~) r].
B y (2.5) a n d writing So = (r, 4 A r, O) we have 2 A r S > m a U > ~ m s U 9 (.J (r, 4 A r , O)
OeOz
=ff~176176176176176 o, aorj ao>r'r
2 A 8
Hence m 02 < 4-A - 1 - (~ < ~ " (2.12)
Consider finally t h e set @ a which is t h e c o m p l e m e n t of @1+ Os.
B y (2.11) a n d (2.12) m O a > 2 ~ - l . (2.13)
L e t C U denote t h e c o m p l e m e n t of U. F o r a n y 0 6 O a t h e segment ( ( 1 - ~ ) r , r, O) contains points of U while m {(r, 4 A r, 0) U} ~< (4 A - 1 - ~) r a n d therefore
m {(r, 4 A r , O ) . C V } >(~r. (2.14) Consider now, for a fixed 0 6 Oa, the intersections of the sets U a n d C U with t h e closed segment ( ( 1 - ~ ) r , 4 A r , 0): I t s intersection with U is a closed n o n - v o i d set while its intersection w i t h C U is a n open set, i.e. a collection of non-overlapping open intervals of total measure > ~ r b y (2.14). I f none of these intervals exceeds ~ r in length t h e n t h e y belong to {(5 r, U} b y t h e definition of this set. If one of these intervals, I say, exceeds (~ r in length t h e n obviously the t w o sub-intervMs of length (5 r, co-terminal with I , m u s t belong to {~ r, U}. I n a n y case we h a v e shown t h a t
m ( ( i l - • ) r , 4 A t , 0 ) { S r , U})>~($r.
N o w b y (2.13)
m 2 (J [ ( ( 1 - ~ ) r , 4 A r , O){(~r, U } J > ( 1 - u ) r ~ r m |
Oe@8
a n d a /ortiori, b y the definition of V,
m s V > 5 r 2 &
O:bT J O R D A N A R C S A N D L 1 T S C H I T Z C L A S S E S O F F U : N C T I O N S 125 B y (2.10) a n d (2.3')
m 2 J (Va, v~) + m 2 J (v~, v4) > 4 r 2 (~
a n d a t l e a s t one of t h e t e r m s on t h e l e f t side, s a y t h e first one, satisfies t h e i n e q u a l i t y m2 J (v3, v~) > 2 r 2 ~.
N o w b y (2.7) a n d (2.3)
2 r 2 ~ 2 m s J ( v 3 ' ) > ~2r 2 ~ > A
I v y - v; =
w h i c h is i m p o s s i b l e b e c a u s e v~ E E 1. T h i s c o n t r a d i c t i o n e s t a b l i s h e s T h e o r e m 2.
2.2. P R O O F OF T H E O R E M 3 W ~ E N B IS A J O R D A N ARC. L e t J = J ( 0 , 1) be a J o r d a n are. G i v e n ~ we a r e t o s h o w t h a t we can inscribe a p o l y g o n of v e r t i c e s
0 = Uo<Ul'<
... < u s = 1, (2.15)s ch t h a t I s < (2.16)
i = l
S u p p o s e t h i s t o be a l r e a d y e s t a b l i s h e d ; t h e a d d i t i o n a l r e q u i r e m e n t of t h e t h e o r e m , t h a t (2.16) can be a c h i e v e d while m a x l u ~ - u ~ _ l l is a s s m a l l as we please, c a n n o w be s a t i s f i e d in a n o b v i o u s w a y . I n d e e d , we can first s u b d i v i d e J i n t o a f i n i t e se- q u e n c e of a r c s of s u f f i c i e n t l y s m a l l d i a m e t e r s a n d t h e n a p p l y t h e r e s u l t (2.16) t o e a c h of these arcs.
W e m a y ignore t h e s i m p l e case w h e n
m2J=O
for t w o reasons:(1) I t is e a s i l y d i s p o s e d of b y t h e s e c o n d p a r t of o u r p r o o f w h i c h uses coverings U of s m a l l ~ d2; (2) I t is c o v e r e d b y A. Ville's t h e o r e m of 1936. W e m a y t h e r e f o r e a s s u m e t h a t m 2 J > 0.
L e t e > 0 be given. F o r ($1>0 d e n o t e b y E~,, t h e set of t h o s e p o i n t s v, of J , t o w h i c h c o r r e s p o n d p o i n t s
v"
w i t hI v - v'l> ~51
a n d s a t i s f y i n g t h e c o n d i t i o nm~J(v, v') 2 m 2 J
I v - v ' l 2 > (2.17)
lira m 2 E~, = m 2 J .
&-+0
B y T h e o r e m 2
W e a s s u m e ~1 so chosen t h a t
m2E~'
> ~3 m2 J"2
126 A . S. B E S I G O V I T C I ~ A:ND I . J . S C H O E N B E R G
D e n o t e b y E ~ a n d E~, t h e d i s j o i n t sets of t h o s e p o i n t s of E ~ t o w h i c h c o r r e s p o n d p o i n t s v'>-v or v " < v r e s p e c t i v e l y : E ~ = E ~ + E ~ . L e t E ' ~, b e one of t h e sets o n t h e r i g h t h a n d side whose m e a s u r e is > ~ m 2 J . 1 S u p p o s e i t is E~,.
W e can o b v i o u s l y select a sequence of d i s j o i n t a r c s J (vii, V ; 1 ) , J ( v 1 2 , v;2), . . . , J ( v 1 .... v;.~,),
, t p p
in n a t u r a l o r d e r a l o n g J , where vH, v12 . . . . , vl, ~, are p o i n t s of E~, = E~ +, a n d V l l , V12 . . . Vl, n~
t h e c o r r e s p o n d i n g p o i n t s s a t i s f y i n g (2.17), so t h a t t h e m e a s u r e of E;, o u t s i d e t h e s e n 1 a r c s be as s m a l l as we please. These a r c s a r e p i c k e d successively a l o n g J a n d t h e i r n u m b e r n 1 is n e c e s s a r i l y finite b e c a u s e t h e m2-measure of e a c h a r c exceeds 2 ~ m 2 J / ~ . W r i t i n g
F i : J (v11, ?)11) + J (v12, v~e) § § J (Vx . . . . Vl, n,), we m a y t h e r e f o r e a s s u m e t h a t
ms F1 > ~ ms J . 1
L e t n o w ~e > 0 a n d d e n o t e b y E ~ t h e set of t h o s e v of J - F 1 t o w h i c h corre- s p o n d p o i n t s v' s a t i s f y i n g (2.17), v' b e l o n g i n g t o t h e s a m e a r c of J - l " 1 as v, a n d such t h a t ]v - v' I > ~2. As before m 2 E,~-->m 2 ( J - F1) as ~2-->0. A s s u m e ~e so chosen t h a t
m 2 Eo, > ~ m s (J 2 -- F1).
' E '
W e n o w d e f i n e t h e set E~, a s ~, was d e f i n e d before a n d a set F 2 of d i s j o i n t a r c s in J - l " 1 , s u c h t h a t for e a c h a r e J (v, v') of F 2 (2.17) holds, while
m2 F 2 > 3 m2 1 (J - 1~1).
S i m i l a r l y sets Pa, Fa . . . Fk a r e d e f i n e d s u c c e s s i v e l y such t h a t m 2 F , > ~ m 2 ( J - F 1 1 ... F~ 1) ( i = 2 , . . . , / c ) .
Since t h e m e a s u r e of e a c h 1~1 e x c e e d s a t h i r d of t h e r e m a i n i n g m e a s u r e , we can r e a c h a v a l u e k s u c h t h a t
m 2 ( J - F 1 - . . . - Fk) < ~. (2.18)
O l q J O R D A : N A R C S A N D L I P S C H I T Z C L A S S E S O F F U N C T I O : N S
L e t J (v~, v~), B y (2.17)
127 ( i = 1 . . . N), be all t h e a r c s of F I + ... + F k is a s c e n d i n g order.
N N
" 2 ~ ~ m 2 J ( v ~ , v~)<~ ~ (2.19)
T h e d i s t a n c e b e t w e e n a n y p a i r of a r c s of J - 1~1- . . . - F~ b e i n g p o s i t i v e , l e t i t b e g r e a t e r t h a n 2 ~ ( > 0 ) . D e n o t e b y U = U ( ~ , J - F 1 - . . . - F k ) a collection of closed c o n v e x sets (e.g. s q u a r e s w i t h sides p a r a l l e l t o f i x e d directions), e a c h set of d i a m e t e r
< a a n d such t h a t e v e r y p o i n t of t h e closure J - F 1 - . . . - Fk is a n i n t e r i o r p o i n t of a t l e a s t one of t h e sets. U m a y a l w a y s be a s s u m e d t o consist of a f i n i t e n u m b e r of sets. I f we d e n o t e b y d t h e d i a m e t e r of t h e g e n e r a l set of U t h e n , b y (2.18), we can choose U so t h a t
W e m a y w r i t e
d < 2 ( 2 . 2 0 )
U
N
J - F 1 - . . . - F k = ~ J (v~, V~+l) ,
i = 0
w h e r e v 0 = 0 a n d VN+I= 1, while t h e first a n d t h e l a s t a r c of t h i s s u m m a y n o t e x i s t . A n y e l e m e n t of U can cover p o i n t s of one a r c o n l y . T h u s we c a n w r i t e
N
u.
where U~ consists of t h o s e sets of U w h i c h cover p o i n t s of t h e a r c J (v~, v~+l). Clearly, b y (2.20),
N 2 _ 2
- Z d (2.21)
~ = 0 Ui U
Take the general arc J (v;, v,+1) and define on it a finite sequence of points
v~ = w~.0, wi.1 . . . wi.~ = v~,l i (2.22)
i n t h e following w a y : L e t w~. 0 be i n t e r i o r to t h e set U(~ 1). I f also vi+l is in U(1 ~) t h e n p ~ = 1 a n d we are t h r o u g h . I f n o t , l e t w~. 1 be t h e l a s t p o i n t of t h e a r c J ( v ~ , V~+l) w h i c h belongs t o U (1)~ . C l e a r l y w~. 1 is on t h e b o u n d a r y of U (1)'~ , l e t w~. 1 be i n t e r i o r t o U~ 2). I f also V~+l belongs t o U~ 2) t h e n p ~ = 2 a n d we s t o p t h e process. I f n o t , l e t wi.z be t h e l a s t p o i n t of J ( w L 1 ,
Vi+l)
b e l o n g i n g t o U~ 2). C o n t i n u i n g in t h i s w a y , we o b t a i n t h e sequence of p o i n t s (2.22) such t h a t t h e p o i n t s w~.j 1 a n d w~. s b e l o n g t o t h e s a m e set U~ j) ( ] = 1 . . . Pi), where t h e p~ sets U~ s) a r e d i s t i n c t e l e m e n t s of t h e collection U~. W e conclude t h a t128 A . S . B E S I C O V I T C H A N D I . J . S C H O E N B E R G p~
Iw,,J
1 ~ 1 Ut
and therefore
N pi
2 e
i = 0 ] = 1
(2.23)
We have thus obtained the following monotone sequence of points along J :
p
0 ~ W 0 , 0 ~ W 0 , 1 , . . . ~ WO, p o ~ V 1, V l = W l , 0~ W1,1~ . . . ~ W l , p l ~ V 2 ~ p
V2 = W 2 , 0 , . . . , WN, P N ~ I .
Denoting them in order b y 0 = % , ul, . . . , u s = l , we have
N p~
l-u,12=Zlv,-v,t + 5
2Xlw,,
i = 0 ] = 1
b y (2.19) and (2.23), and the desired inequality (2.16) is established and therefore also the theorem for the ease when B is a J o r d a n arc.
2.3. A L E M M A O N C O N T I N U O U S A R C S A N D P R O O F O F T H E O R E M 3. L e t B be a non-closed continuous arc in the plane. B y omitting from B subarcs with coin- cident endpoints (loops) we m a y reduce B to become a J o r d a n are J joining the original endpoints of B. A precise description of this intuitive idea is given by (1)
LEMMA 1. Let x = / ( t ) be an continuous complex-valued /unction o/ t E I = [ 0 , 1]
such that / ( 0 ) # / ( 1 ) . We can / i n d i n I a per/ect set F such that the image / ( F ) is a J o r d a n arc J , having as endpoints / ( 0 ) and /(1), i n the sense that the relations
a e F , a ' e F , a < a ' / ( a ) = / ( a ' ) (2.24) hold i/ and only i/ the open interval (a, a') is contiguous to F .
R e m a r k 1. The set F is b y no means always uniquely defined. An arc B in the shape of a pretzel, with its ends slightly extended, admits three distinct sets ~' obtained b y removing from I appropriate single open intervals.
R e m a r k 2. The lemma and its proof require nothing beyond the continuity of / (t). The lemma therefore holds as stated if the values of / (t) are in a Hausdorff space.
(1) Lemma 1 is a special case of the following Arcwise Connectedness Theorem: Every two points a and b o / a locally connected continuum M can be joined in M by a simple continuous arc.
(See [9], p. 36.) However, a simple proof of Lemm~ 1 is here included for the reader's convenience.
O1~ J O R D A I ~ ARCS AI~D L I P S C H I T Z C L A S S E S O F F U N C T I O N S 129 Proo/: W e call t h e o p e n i n t e r v a l S = (t, t') a loopsegment p r o v i d e d t h a t / (t) = / (t').
L e t L d e n o t e t h e t o t a l i t y of l o o p s e g m e n t s . Since L is e v i d e n t l y c o m p a c t , t h e r e e x i s t s a l o n g e s t loolosegment w h i c h we d e n o t e b y S 1 = (tl, t~)and define F 1 = I - S 1. O b s e r v e t h a t if S E L, S c F1, t h e n S can n o t a b u t on S 1 since t h e i r u n i o n w o u l d g i v e a l o n g e r ]OOlosegment. L e t S 2 be t h e l o n g e s t a m o n g t h e S c _ ~ 1 a n d consider F 2 = I - S 1 - S 2.
W e r e p e a t t h i s o p e r a t i o n s u c c e s s i v e l y o b t a i n i n g t h e loolosegments S 1, S 2 .. . . such t h a t t h e closed s e g m e n t s $1, ~2 . . . . a r e p a i r w i s e d i s j o i n t a n d 1 ( $ 1 ) ) l ( $ 2 ) ) . . . . E i t h e r t h e process t e r m i n a t e s w h e n F n = I - S 1 - . . . - S~ c o n t a i n s no f u r t h e r l o o p s e g m e n t , or else i t c o n t i n u e s i n d e f i n i t e l y w h e n e v i d e n t l y l (S~)-+0. I n e i t h e r case l e t f 2 = ~ S ~ a n d c o n s i d e r t h e p e r f e c t set F = I - ~ .
L e t (a,a') s a t i s f y t h e c o n d i t i o n s (2.24). W e c a n n o t h a v e [ a , a ' ] c F . I n d e e d , ( a , a ' ) E L a n d s h o u l d h a v e b e e n r e m o v e d before l ( S n ) h a s b e c o m e < a ' - a . H e n c e [a, a ' ] ~: F a n d t h e r e f o r e (a, a ' ) ~ S~ = (t~, t~) for s o m e i a n d where we choose /or i the least value which will do. N o w we m u s t h a v e (a, a ' ) = St = (t~, t~) for if (a, a ' ) ~= S~ t h e n a ' - a > l (S~) a n d (a, a ' ) s h o u l d h a v e b e e n r e m o v e d be/ore S~. T h i s p r o v e s o u r l e m m a e x c e p t , p e r h a p s , t h e m a i n p o i n t t h a t J is a J o r d a n arc. To see this, l e t ~ = T (t) be a c o n t i n u o u s n o n - d e c r e a s i n g f u n c t i o n in t h e r a n g e I , ~ ( 0 ) = 0 , ~ ( 1 ) = 1 a n d such t h a t (t) = ~ (t') for t < t' if a n d o n l y if t h e i n t e r v a l (t, t') is c o n t a i n e d in f2 = ~ S~. I f we n o w i d e n t i f y t h e t w o e n d p o i n t s t~ a n d t~ of S~ for all i, we o b t a i n a set F 1 w h i c h b y T = T (t) is homeomorlohic w i t h t h e r a n g e 0 ~ ~ ~< 1. O n t h e o t h e r h a n d , we h a v e s h o w n t h a t J = / ( F ) = / ( F i ) is a h o m e o m o r p h of F 1. I t t h e r e f o r e follows t h a t J is a h o m e o m o r l o h of t h e i n t e r v a l 0 ~< T ~ < 1 a n d o u r l e m m a is e s t a b l i s h e d .
A g e n e r a l p r o o f of T h e o r e m 3 n o w b e c o m e s obvious. G i v e n s > 0 a n d a p p l y i n g T h e o r e m 3 t o t h e Jordan a r c J j u s t c o n s t r u c t e d we c a n f i n d a d i v i s i o n
O ~ t (~ ( 1 ) < . . . < t (n)=l where all t (~) E F a n d such t h a t
II - I
(t(' ,)12
<a n d t h i s a l r e a d y e s t a b l i s h e s t h e t h e o r e m .
w 3. On plane Jordan arcs of finite and positive A~-measure
3.1. P R o o F o F T H E O r e M 4. To o b t a i n a n a r c J h a v i n g t h e p r o p e r t i e s r e q u i r e d b y T h e o r e m 4 we r e p e a t w i t h some s i m p l i f i c a t i o n s t h e c o n s t r u c t i o n of w 1.1: W e n o w choose 0~ = 0, i n d e p e n d e n t of n, s a t i s f y i n g t h e e q u a t i o n
130 A . S . B E S I C O V I T C I - I A N D I . J . S C I t O E N B E R G
4 0 ~ = 1 (1 < ~ < 2 ) .
S t a r t i n g as in w 1.1 w i t h t h e u n i t s q u a r e So, l e t t h e c o n t i n u u m
J1
consist of f o u r c o r n e r s q u a r e s of sides = 0 a n d of t h r e e r e c t i l i n e a r links (fig. 1). J2 is o b t a i n e d f r o m J1, b y r e p l a c i n g e a c h s q u a r e s 1 b y a c o n t i n u u m g e o m e t r i c a l l y s i m i l a r t o J1 (because 02 = 01 = 0) w h i c h joins its e n t r y p o i n t t o its e x i t p o i n t a n d so f o r t h . N o w J = N J~is our p r e s e n t J o r d a n arc. I f we o b s e r v e t h a t J is c o v e r e d b y collection ~ n of 4 n s q u a r e s h a v i n g d i a m e t e r s O n V2, we see t h a t
A ~ J ~< (V2) ~
a n d we l e a v e i t t o t h e r e a d e r t o show t h a t A ~ J > 0. I n t e r m s of t h e n o t a t i o n s of w 1 we can s a y t h a t J consists of t h e set
E = lim E= 175
n--->cr n n
p l u s a n e n u m e r a b l e set of links whose A ~ - m e a s u r e is 0. To a n y arc J (v, v'), w h i c h is n o t a r e c t i l i n e a r s e g m e n t , c o r r e s p o n d s a v a l u e n such t h a t J (v, v ' ) ~ ~ belongs t o one s q u a r e of ~ n b u t t o m o r e t h a n one s q u a r e of ~ n + l . F r o m t h i s i t follows t h a t
A ~ J (v, v') < 0n~289
On t h e o t h e r h a n d I v - v' I is s u r e l y g r e a t e r t h a n or e q u a l t o t h e w i d t h of t h e cor- r i d o r s of Sn. Since corr s~ =O n ( 1 - 2 0) we o b t a i n
Iv-v'l>o (1-2o).
H e n c e
w h i c h p r o v e s T h e o r e m 4.
W e m i g h t r e m a r k t h a t t h e r e a r e p l a n e J o r d a n a r c s 1 < ~ < 2, such t h a t
l i ~ A:J(v' v') +
v, v I v - v ' [J of finite A=-measure,
a t a l m o s t all p o i n t s v in t h e sense of A=-measure, b u t we do n o t d w e l l on g i v i n g a n e x e m p l e here.
3.2. P ~ o o F o F T ~ O R E M 5. W e p i c k 8 > 0 a n d A such t h a t 0 < A < I a n d d e n o t e b y E t h e set of those p o i n t s v of J, t o w h i c h c o r r e s p o n d v' s a t i s f y i n g t h e i n e q u a l i t i e s
ON" J O R D A N A R C S A N D L I P S C H I T Z C L A S S E S O F F U N C T I O N S 131
A ~ J ( v ' ~ ) > A , ] v - v ' l < a .
(3.1)Iv-v'j
T h e set E is either v o i d or open; i n a n y case its c o m p l e m e n t E 1 = J - E is closed.
L e t us n o w show t h a t if
A ~ E~ = 0 (3.2)
for e v e r y f r a c t i o n a l A a n d e v e r y d t h e n our t h e o r e m follows. I n d e e d , l e t A~, 0 < A n < 1, a n d (5~ (n = 1, 2 . . . . ) be such t h a t
l i m A , = l , l i m d ~ = O ,
a n d let E (~), E(1 ~) be t h e corresponding sets defined above. W e a s s u m e t h a t A ~ E(1 ~) = 0 for e v e r y n a n d therefore F = [7 ~ E(1 ~) also has t h e p r o p e r t y A ~ F = 0. I f v E J - F t h e n
A~J(v, v'~)
Iv_vs
for a p p r o p r i a t e p o i n t s v~, a n d t h e i n e q u a l i t y (7) follows o n l e t t i n g n - - > ~ .
To e s t a b l i s h (3.2) l e t us a s s u m e t h a t A ~ E 1 > 0 a n d see t h a t we get a c o n t r a d i c t i o n . B y the d e f i n i t i o n of A~-measure, for e v e r y e > 0 we c a n f i n d a closed c o n v e x set U of d i a m e t e r
d U
as small as we please, i n p a r t i c u l a rd U
< 8, a n d such t h a tA ~ (E 1 U) > (1 - s) (d U) ~, a n d i n p a r t i c u l a r such t h a t
A ~ (E 1 U) > A (d U) ~. (3.3)
L e t vl, v~ be t h e e x t r e m e p o i n t s of J b e l o n g i n g to t h e closed set E 1 U. I n p a r t i c u l a r J (vl, v~) D E 1 U. B u t t h e n A ~ J (Vl, v2) ~> A ~ (E 1 U) a n d (3.3) implies
A ~ J ( v . v2) > A ( d U ) ~
a n d
a /ortiori
A ~ J (vl, %)iv_v l >A, IVl-V l<dU< .
However, these i n e q u a l i t i e s i m p l y t h a t v 1 a n d v 2 b e l o n g to E while t h e y a c t u a l l y be- long to E 1 b y c o n s t r u c t i o n . This c o n t r a d i c t i o n establishes t h e t h e o r e m .
132 A . S. B E S I C O V I T C H A N D I . J . S C H O E N B E R G
w 4. On Lipschitz classes o f functions defined o n Jordan arcs
4.1. P R O O F OF T H E O R E M 7. I t follows f r o m t h e a s s u m p t i o n s of T h e o r e m 7, n a m e l y G ( v ) E L i p j q~(x) a n d (9) t h a t t o a n y a > 0 , h o w e v e r small, c o r r e s p o n d s a f u n c t i o n (~ (v) > 0 such t h a t
IG(v)-G(v')l<alv-v'l if
L e t E~ be t h e set of p o i n t s v of J , for w h i c h
I G ( v ) - G ( v ' ) l < a l v - v ' l ~ if Iv-v'l< 2 -n
T h e set
E n
is closed a n d liraE,~=J.
T a k e a sequence{'~n},
e ~ > e ~ + l , en-->0, a n d such t h a t Z s~ < A ~ J . H e r e we a s s u m e t h a t A ~ J > 0, t h e p r o o f for t h e case w h e n A ~ J = 0 b e i n g a s i m p l i f i e d v e r s i o n of t h e p r e s e n t one. F o r e v e r y n we can f i n d a setPn=Pn (En-E~-I,
2 -n) of c o n v e x sets, e a c h of d i a m e t e r < 2 -n, a n d such t h a t e v e r y p o i n t ofE,~-En_I
is a n i n t e r i o r p o i n t of a t l e a s t one of t h e sets. W e shall also a s s u m e t h a t~,d~<A~(E~-E~_I)+~,
Pn
w h e r e d d e n o t e s t h e d i a m e t e r of a g e n e r a l set of t h e collection
Pn.
E v e r y p o i n t of J is a n i n t e r i o r p o i n t of a t l e a s t one c o n v e x set of t h e collection ~ P= a n d b y t h e H e i n e - B o r e l t h e o r e m t h e r e is a finite s u b c o l l e c t i o n P of ~ P= w i t h t h e s a m e p r o p e r t y . W e o b v i o u s l y h a v e~ d~<A~J + ~ ~ < 2 A ~ J .
(4.1)P
L e t p(1), p(2), ..-, p(k) d e n o t e all t h e c o n v e x sets w h i c h a r e e l e m e n t s of P . I f
p(i)EP~,
t h e n l e t v (~) b e a p o i n t of( E n - E ~ - I ) n p (i).
W r i t i n gr(i)=dp(~)<2 -~,
we con- s t r u c t t h e circle c (~)= c (v (~), r (i)) w h i c h c l e a r l y c o n t a i n s p(~).T h u s J ~ C - c (1) + c (~) + ... + e (~)
k
a n d b y (4.1) ~ ( r ( ~ ) ) ~ < 2 A ~ J . (4.2)
B y t h e d e f i n i t i o n of
En
we see t h a t for a n y v Ec(~)f3 J ]G (v) - G (v(~)) ] < a Iv - v (~) I ~ < a (r(~)) ~ a n d t h e r e f o r e for a n y p a i r v', v" of p o i n t s of c(~)f3 Jl G (v') - G (v") I < 2 a (r(~)) ~.
O N J O R D A I g A R C S A N D L I P S C I I I T Z C L A S S E S O F F U N C T I O N S 1 3 3
L e t v 0 ~ v * b y a n y pair of points of J a n d let v o be an interior point of c (~~
D e n o t e b y v 1 the last point v>-v o, v ~ v * , such t h a t v E c (~'). We h a v e {G (v0) - G (vl){ < 2 a (r(~l)) ~.
N o w v 1 is an interior point of one of the circles of C, s a y of c (~'), i1:4:io, a n d let v~
be the last p o i n t v>-vl, v < v * , such t h a t v E c (~'). As before {G (v~) - G (v~){ < 2 a (r(~')) ~
a n d so forth. After a finite n u m b e r of steps, in fact after k'~< k steps, we shall reach the p o i n t v*. B y (4.2) we find
{ G (V0) -- G (V*){ < {G (v0) -- ~ (Vl) { + { ~ (Vl) - e (v2) { + . . . + { G (Vk,) -- ~ (V*){
< 2 a ((r(~~ ~ + (r(~1>) ~ § ... § (r(i~')) ~) < 4 a A ~ J . Since a was a r b i t r a r y we conclude t h a t
G (v0) - G (v*) = 0 which was to be proved.
4.2. P ~ O O F OF T ~ E O ~ M 8. There remains as our last t a s k to furnish a proof of T h e o r e m 8. L e t the positive m o n o t o n e function ~ (x) of t h a t t h e o r e m be given.
We select a function ~ (x) which is convex a n d continuously differentiable in t h e range [0, 1] a n d such t h a t
0 < y J ( x ) < r ( 0 < x ~ < l ) ~f(0)=O. (4.3)
L e t t = A ~ (x), A =~r/~0 (1). (4.4)
This is a relation which m a p s the range O~<x~< 1 onto 0~<t~<z. W e n o w i n v e r t (4.4) o b t a i n i n g t h e concave increasing function
x = ( F ( t ) ) ~ ( o < t < ~ , F(t)~>o). (4.5)
We n o w consider t h e function
h ( t ) = l - ( F ( t ) ) 2 (0~<t<z), (4.6) which has the following properties: h ( 0 ) = 1, h (7~)=O, h (t) is convex in [0, ~] a n d c o n t i n u o u s l y differentiable in (O, 7e). Notice in particular t h a t in t h e range (0, z), h' (x) < 0 a n d non-decreasing. We n o w e x t e n d t h e definition of h (t) to the range [ - ~, ~]
134
so as to be even, a n d e x p a n d it in cosine series
oo
h (t) = ~ A~ cos v t.
o
Clearly A o > O. B u t also all A~ > O. I n d e e d
f: =-fo
A , = h(t) c o s v t d t 1
A . S. B E S I C O V I T ( J I I A N D I . J . S C H O E N B E R G
( - h ' (t)) sin v t d t > O .
(4.7)
1 = ~ 2 a ~ .
0
E 2 (t) = ~ 2 a~ (1 -- cos v t) = 4 a~ sin2 --.v t (4.8)
1 1 2
This expansion implies t h a t 2' (t) is a screw /unction in Hilbert space which corre- sponds to a closed screw line of t h a t space. We refer t o y o n N e u m a n n a n d Schoen- berg [5] for f u r t h e r i n f o r m a t i o n on this subject; we, however, need none whatever, because w h a t we need is perfectly e l e m e n t a r y a n d explained in a few words: We m e a n t h a t there is in the Hflbert space H a d o s e d curve
C : x = / (t), (0 ~< t ~< 2 z ; / (t) of period 2 z~), (4.9) such t h a t for all real t a n d t'
.F (t - t') = [I/(t) - / (t')[1- (4.10)
This curve is i m m e d i a t e l y constructed, for (4.8) gives F 2 (t - t') = ~ 4a~ sin 2 ~ v (t - t') 1
= ~ {(a~ cos v t - a ~ cos v t ' ) 2 + (a~ sin v t--a~ sin v t')2}. (4.11) N o w (4.6) gives
a n d in particular, for t = 0
t h e last integral being positive, because - h ' (t) is positive a n d decreasing. (Compare B o c h n e r [1], 76-77.)
W e m a y therefore write t h e expression (4.7) as h ( t ) = ~ 2 a ~ c o s v t , (a~>0),
0
0 N $ O t ~ D A N A R C S A N D L I P S C I t I T Z C L A S S E S O F F U N C T I O N S 135 I n the space H o~ real sequences {xn}y with ~ x~ < ~ a n d the usual n o r m tl xll = ( ~ X~n) 89 we indeed see b y (4.11) t h a t the closed curve C t r a c e d o u t b y
/ ( t ) = { a l c o s t , a l s i n t , a 2 c o s 2 t , a n s i n 2 t , . . . } ( 0 ~ < t < 2 ~ ) enjoys the p r o p e r t y (4.10).
Along the J o r d a n arc
we n o w define the f u n c t i o n
F : x = / ( t ) (O~<t~<~), (4.12)
g (t) = t. (4.13)
F o r a n y t w o values t, t' such t h a t 0 ~< t < t' ~ g (t') - g (t) r (It/(t,) -
/(t)II)
t' - - t
a n d b y (4.3) this is < A, w ( F~ it' - t)) t h e last e q u a l i t y relation
W e h a v e therefore shown t h a t
t t - - t
r (F (t' - t))
holding because the relation (4.5) is the inverse of (4.4).
g(t')-g(t)<Ar
( 0 < t < t ' < ~ ) .R e t u r n i n g to our old n o t a t i o n v =/(t), G (v)=g (t), this is precisely the relation ] G (v') - G (v)] < A r (If v' - v It)
which was to be established a n d which shows t h a t G(v)EU Lipl r G(v) =g (t)=t ist n o t a c o n s t a n t o u r T h e o r e m 8 is t h e r e b y established.
Since
136 A. S. BESICOVITCH A~D I. J. SCItOENBERG
References
[1]. BOC~NER, S., Vorlesungen is Fouriersche Integrale. Leipzig, 1932.
[2]. CACCIOI'POLI, R., Sul lemma fondamentale del calcolo integrale. Atti Mere. Accad. Sci.
Padova, 50 (1934), 93-98.
[3]. G~ESER, G., ]~tude de quelques alg~bres Tayloriennes. J . Analyse Math., 6 (1958), 1-124.
[4]. L~BESOUE, H., Legons sur l'int~gration et la recherche des ]onctions primitives. Paris, 1928.
[5]. y o n NEUMn_~, J. & SCHOENBE~G, I. J., Fourier integrals a n d metric geometry. Trans.
Amer. Math. Soe., 50 (1941), 226-251.
[6]. PETROVSKY, J., Sur l'unicit@ de la fonction primitive par rapport ~ une fonction continue arbitraire. Rec. Math. Soc. Math. Moscou, 41 (1934), 48-58.
[7]. V~LE, A., E i n Satz fiber quadratische L~nge. Ergebnisse eines math. Kolloquiums (Wien), Heft. 7 (1936), 22-23.
[8]. WHrrNEY, H., A function n o t constant on a connected set of critical points. Duke Math.
J., 1 (1935), 514-517.
[9]. W ~ Y B u ~ , G. T., Analytic Topology. New York, 1942.
Received June 11, 1960
