T H E B O U N D A R Y C O R R E S P O N D E N C E U N D E R Q U A S I C O N - F O R M A L M A P P I N G S
BY
A . B E U R L I N G and L. AHLFORS in Princeton, N. .]. (A. B.) and Cambridge, Mass. (L. A.)
1. I n t r o d u c t i o n
t . 1 . T h e d i l a t a t i o n D > 1 of a d i f f e r e n t i a b l e t o p o l o g i c a l m a p p i n g / :
(x, y)--> (u,v)
be- t w e e n p l a n e regions is d e t e r m i n e d b y1 Ux ~ + ~,~ + v~ + v~
2(l)
D + D = [u~v~-n~Vxl
G e o m e t r i c a l l y , D r e p r e s e n t s t h e r a t i o b e t w e e n m a j o r a n d m i n o r axis of t h e i n f i n i t e s i m a l ellipse o b t a i n e d b y m a p p i n g a n i n f i n i t e s i m a l circle of c e n t e r (x, y). F r o m t h i s i n t e r p r e t a t i o n i t is e v i d e n t t h a t t h e d i l a t a t i o n is u n a f f e c t e d b y c o n f o r m a l m a p p i n g s of t h e planes. F u r t h e r - more, a m a p p i n g ] a n d its i n v e r s e /-1 h a v e t h e s a m e d i l a t a t i o n a t c o r r e s p o n d i n g p o i n t s . A m a p p i n g is s a i d to be
quasicon/ormal
if D is b o u n d e d . T h e least u p p e r b o u n d of D is r e f e r r e d t o as t h emaximal dilatation.
1.2, I t is k n o w n t h a t a q u a s i e o n f o r m a l m a p p i n g of x 2 § y'~ < 1 o n t o u" + ' v 2 < 1 re- m a i n s c o n t i n u o u s on t h e b o u n d a r y . 1 H e n c e i t i n d u c e s a t o p o l o g i c a l c o r r e s p o n d e n c e 1)e- t w e e n t h e t w o circumferences. W e shall be c o n c e r n e d w i t h t h e p r o b l e m of c h ~ r a c t e r i z i n g t h i s c o r r e s p o n d e n c e b y s i m p l e n e c e s s a r y a n d sufficient c o n d i t i o n s . More g e n e r a l l y , we shall look for c o n d i t i o n s which are c o m p a t i b l e w i t h a m a p p i n g whose m a x i m a l d i l a t a t i o n does n o t e x c e e d a g i v e n n u m b e r K > 1.
I n view of t h e i n v a r i a n c e w i t h r e s p e c t to c o n f o r m a l m a p p f n g s we m a y r e p l a c e t h e d i s k b y t h e u p p e r h a l f - p l a n e s y > 0 a n d v > 0, a n d we m a y a s s u m e t h a t t h e -points a t c<~
1 L. AHLFORS, Oft q u a s i c o n f o r m a l m a p p i n g s . Jourt~al ,l'A~rdyse 3Iathem(itique, vol. 7 . l
(~ :~3/a4).
1 2 6 A . B E U R L I N G A:ND L. A H L F O R S
c o r r e s p o n d t() e a c h o t h e r u n d e r t h e m a p p i n g . I n t h e s e c i r c u m s t a n c e s t h e b o u n d a r y corre- s p o n d e n c e is d e t e r m i n e d b y a s t r i c t l y m o n o t o n e c o n t i n u o u s f u n c t i o n tt (x), in t h e sense t h a t t h e p o i n t (x,0) is m a p p e d on (,u (x),0). I t is sufficient t o consider t h e case of a n i n c r e a s i n g /~ (x).
t . 3 . T h e m a i n t h e o r e m t h a t will be p r o v e d in t h i s p a p e r is t h e following:
T H E O R E M 1. There exists a quasicon/ormal m a p p i n g o/ the hal/-planes with the boundary correspondence x - > # ( x ) i/ and only i/
1 / ~ ( x + t ) - ~ ( x )
- _< ~ ~o ( 2 )
o # (x) - /t (x - t) /or some constant Q > 1 a n d / o r all real x and t.
More precisely, i/ (2) is/ul/illed there exists a m a p p i n g whose dilatation does not exceed Q2. On the other hand, every m a p p i n g with the boundary correspondence a must have a m a x i m a l dilatation ~ 1 § A log o where A is a certain numerical constant ( = .2284).
C o n d i t i o n (2), which will be r e f e r r e d to as t h e ~-condition, i n d i c a t e s t h a t # ( x ) m u s t possess a d e g r e e of a p p r o x i m a t e s y m m e t r y w h e n x a p p r o a c h e s a n y v a l u e f r o m t i l e r i g h t a n d fronl t h e left. T h e ~ - c o n d i t i o n is n o t i n v a r i a n t wi~h r e s p e c t t o a r b i t r a r y l i n e a r t r a n s - f o r m a t i o n s , n o r does t h e i n v e r s e m a p p i n g s a t i s f y t h e s a m e o - c o n d i t i o n . H o w e v e r , t h e v e r y s i m p l e f o r m t h a t t h e c o n d i t i o n t a k e s w h e n t h e p o i n t s a t ' o o are s i n g l e d o u t is sufficient r e a s o n to give p r e f e r e n c e to t h e f o r m u l a t i o n t h a t we h a v e chosen.
1.4. I n S e c t i o n 2 we use T h e o r e m 1 to d e r i v e a d i f f e r e n t c h a r a c t e r i s t i c of quasicon- f o r m a l m a p p i n g s . I n S e c t i o n 3 t h e p r o b l e m is f u r t h e r a n a l y z e d , a n d in S e c t i o n 4 we p r o v e t h e e a s y p a r t of T h e o r e n l 1. S e c t i o n 5 d e a l s w i t h a class of e x p l i c i t m a p p i n g s a n d serves t o e x h i b i t l i m i t s b e y o n d w h i c h t h e e s t i m a t e s in T h e o r e m 1 c a n n o t be i m p r o v e d . T h e c o m p l e t e p r o o f of T h e o r e m 1 follows in S e c t i o n 6, a n d in a final s e c t i o n we give t h e a n s w e r t o an open q u e s t i o n c o n c e r n i n g a b s o l u t e c o n t i n u i t y .
The i n v e s t i g a t i o n r e q u i r e s r a t h e r e x t e n s i v e c o m p u t a t i o n s . I n o r d e r to f a c i l i t a t e t h e r e a d i n g m o s t of t h e s e c o m p u t a t i o n s a r e g i v e n in c o m p l e t e d e t a i l . A r e a d e r who is i n t e r e s t e d o n l y in t h e q u a l i t a t i v e a s p e c t s m a y o m i t t h e c o m p u t a t i o n s in S e c t i o n 4 a n d all of S e c t i o n 5.
2. Compact Families of Mappings
2.1. I n t h i s s e c t i o n we show t h a t t h e 9 - c o n d i t i o n can also be i n t e r p r e t e d as a com- p a c t n e s s c o n d i t i o n . This a n a l y s i s can be c a r r i e d o u t a l t e r n a t i v e l y for t r a n s f o r m a t i o n s of t h e u n i t circle or for t r a n s f o r m a t i o n s of t h e line which l e a v e t h e p o i n t a t c~ fixed. Since t h e t r a n s i t i o n is v e r y e a s y we shall o n l y t r e a t t h e case of tile line.
T H E B O U N D A R Y C O R R E S P O N D E N C E U N D E R Q U A S I C O N F O R M A L M A P P I ~ I G S 1 2 7
As before, # (x) shall d e n o t e a n i n c r e a s i n g f u n c t i o n which sets u p a 1-1 c o r r e s p o n d e n c e of t h e r e a l line w i t h itself. T h e l i n e a r t r a n s f o r m a t i o n s x-->ax + b, a > 0 will be d e n o t e d b y S, T. W e s a y t h a t a f a m i l y M of t r a n s f o r m a t i o n s # is closed u n d e r l i n e a r t r a n s f o r m a t i o n s if all c o m p o s e d m a p p i n g s S~t T a r e c o n t a i n e d in M t o g e t h e r w i t h / z .
W e shall also s a y t h a t a m a p p i n g is normalized if # ( 0 ) - 0 , ~ ( 1 ) = 1.
2.2. L e t u s consider t h e following c o m p a c t n e s s c o n d i t i o n :
(a) Every in/inite set o/ normalized mappings # E M contains a sequence {#n}T which converges to a strictly increasing limit/unction.
T h e following t h e o r e m holds:
T H E O R E M 2. The mappings # in a/amily M, which is c/osed under linear trans/orma- tions, satis/y a e-condition, the same/or all #, i/ and only i/condition (a) is [ul/illed.
2.3. S u p p o s e f i r s t t h a t t h e ~ - c o n d i t i o n is satisfied. I f # is n o r m a l i z e d we f i n d / ~ ( 2 - ~ + 1 ) = > ( 1 + ~ ) # ( 2 -~)
for a n y n, a n d since # ( 1 ) - 1 i t follows t h a t
~ (2 -~) =< (3)
for p o s i t i v e a n d n e g a t i v e i n t e g e r s n.
F o r a n y a t h e f u n c t i o n
/z (a + x) --/z (a)
~u (a + 1) --/z (a) is n o r m a l i z e d . B y (3) we h a v e c o n s e q u e n t l y
~ (a + x / - ~ (~/< (~ (a + ] / - ~ (all o-~i (4) as soon as 0 ~ x < 2 -~. On t h e o t h e r h a n d , if a is r e s t r i c t e d t o a f i n i t e i n t e r v a l , i n e q u a l i t y (3) w i t h n e g a t i v e n y i e l d s a b o u n d for ~ ( a + l) - # ( a ) . H e n c e (4) c o n s t i t u t e s a n cquicon- t i n u i t y c o n d i t i o n on a n y c o m p a c t set.
I t follows t h a t we c a n select a c o n v e r g e n t s e q u e n c e f r o m e v e r y i n f i n i t e set of n o r m a - lized m a p p i n g s . T h e l i m i t f u n c t i o n is n o r m a l i z e d a n d satisfies t h e ~-condition. F o r t h i s r e a s o n i t c a n n o t be c o n s t a n t on a n y i n t e r v a l . I n o t h e r words, c o n d i t i o n (a) is fulfilled.
2.~. Conversely, s u p p o s e t h a t (a) is fulfilled. W e set cr ~ inf # ( - 1), /3 - sup ~t ( - 1 ),
9 - 563802. Acta rnathematica. 96. I m p r i r a 4 le 22 o c t o b r e 1956.
128
where be ranges over all n o r m a l i z e d m a p p i n g s i n M. There exists a sequence of #~ E M such t h a t #~ ( - 1)-+j3. Since a s u b s e q u e n c e converges to a s t r i c t l y m o n o t o n e f u n c t i o n it follows t h a t / 3 < 0. T h e same r e a s o n i n g yields ~ > - oo.
Consider a n y # i n M. T h e m a p p i n g
Et (y + t x) - be (y) (x) be (y + t) - # (y)
< t t ( Y - - t ) - - be(Y) <_/3
~ = ~ + t ) - - # ( y )
1
~ # ( y + t ) - - # ( y ) < _ 1.
o r
:,. = # (y) - # (y - t) = fl Clearly, this implies t h e ~ - e o n d i t i o n for ~ = m a x - e , - 9
2.5. T h e o r e m s 1 a n d 2 yield t h e following:
COIr
A boundary mapping/~ can be extended to a quasicon/ormal mapping
o/the hal/planes i / a n d only i/ the /amily o/ all mappings She T satis/ies condition
(a).I n s t e a d of r e l y i n g o n q u a n t i t a t i v e s t a t e m e n t s this criterion emphasizes a d i s t i n c t i v e q u a l i t a t i v e feature.
3. Quantities Related to Dilatation
3.1.. If / is a differentiable m a p p i n g of t h e h a l f - p l a n e on itself we shall d e n o t e its m a x i m a l d i l a t a t i o n b y K I. F o r a g i v e n b o u n d a r y correspondence ~t we set
K ( ~ ) = inf K r ,
where t h e i n f i m u m is w i t h respect to all m a p p i n g s / such t h a t / = # on t h e real axis.
T h e q u a n t i t y K (be) is i n t h e f o r e g r o u n d of our i n t e r e s t , b u t we shall also find it illu- m i n a t i n g to i n t r o d u c e some other q u a n t i t i e s of similar character.
3.2. A q u a s i c o n f o r m a l m a p p i n g with the m a x i m a l d i l a t a t i o n K has t h e following p r o p e r t y : if t h e r e a l - v a l u e d f u n c t i o n s U t a n d U2 are r e l a t e d b y U 1 (z) = U2 (/(z)), t h e n t h e Dirichlet i n t e g r a l s of U1 a n d U 2 over c o r r e s p o n d i n g d o m a i n s h a v e a r a t i o which lies be- t w e e n
1 / K
a n d K .I n p a r t i c u l a r , we m a y confine t h e a t t e n t i o n to Dirichlet i n t e g r a l s over t h e whole half-plane. L e t u 1 a n d u2 be defined on t h e real axis a n d r e l a t e d b y u 1 (x) = u 2 (# (x)). W e choose U2 as t h e solution of D i r i c h l e t ' s p r o b l e m w i t h b o u n d a r y values u 2. T h e n t h e Dirich- let i n t e g r a l ~ (U~) is g i v e n e x p l i c i t l y b y I (u~) where
A. B E U R L I N G A N D L . A I t L F O R S
is n o r m a l i z e d a n d i n M . H e n c e
T H E : B O U N D A t ~ u C O R I ~ E S t ' O : N D E ~ C E U N D E R Q U A S I C O N F O I ~ M A L M A P P I N G S 129
; ; ( : )
1 u ( x ) - (y) 2
. . . . d x d y
x (~) 2 ~ x
o r - r
is the well-known Douglas functional.
B y use of the Dirichlet principle we find at once that l (u,) <= ~ ( v l ) <_ K ~ ( U~) = K I (u2).
Since the same reasoning can be applied to the inverse mapping we have also I(uz)<=
K I (ul).
This result leads us to introduce a quantity K 1 (#), defined as the least number K 1 such t h a t
I < I (u2)
K~ = I (ul) -< K1 (5)
for all pairs of corresponding functions Ul,U2. We have just shown t h a t
- K 1 (#) :~ K (t~).
The quantity KI(#) m a y be regarded as more explicit t h a n K(/~) inasmuch as its definition does not involve two-dimensional mappings.
3.3. I n the circular representation, let ~1,/~1 be a n y two disjoint arcs, and denote their images under the mapping # by ~2,/~2. The extremal distance between :r and ill, to be denoted by d (:% ill), is a function-theoretic quantity which in this simple case can be computed explicitly. I t is equal to 1/O(U1) , where U 1 is the harmonic function with values 0,1 on :r whose normal derivative vanishes on the complementary arcs.
The function U 1 minimizes the Dirichlet integral for the prescribed boundary values.
Therefore,
1
- min ~ ( U I ) - min I (ul) d (~1, ill)
for the class of all function u 1 which are 0 and 1 on ~1, ill. But it follows from (5) that min I ( u l ) <= K 1 rain I ( u 2 ) , and hence that d(a:, fie) Z K 1 d(0~l, ill)"
I n the present case there is s y m m e t r y between # and its inverse mapping, for the complementary arcs of :r have the extrema] distance 1/d(~l,/~). Hence the definition
d ( ~ , fl~)
K 0 (#) = sup d (:r ill) (6)
implies the double inequality
K 0 (#) = d (0~1, ~1) ~
1~~
]30 A . B E U R L I I ~ G A:ND L. A H L F O R S
W e have proved that
Ko(#) g K~ (/~) =< K ( # ) .
Of these three quantities K o is t h e m o s t explicit, for it is defined as t h e m a x i m u m of a function r a t h e r t h a n a functional.
3.4. F o r a given m a p p i n g / ~ of the real axis we denote b y ~ (#) the smallest value of Q such t h a t the Q-condition (2) is fulfilled. Our efforts will be directed t o w a r d s p r o v i n g inequalities of t h e f o r m
9 (e(#)) =< K 0 ( # ) ~ K (/~) =< ~ (#(#)).
H e r e we m a y let (I) a n d q? denote t h e best possible funetions of their kind. This a m o u n t s to setting
( I ) ( o ) - i n f K 0 ( / ~ ) for Q(,u)~ e q ? ( o ) - s u p K ( / ~ ) for e(#)--<~.
The necessity in T h e o r e m 1 will be p r o v e d if we show t h a t lira(I) (~) - c ~ , a n d the sufficiency follows u p o n showing t h a t qP (~) is finite. I t t u r n s o u t t h a t (I)(~) can be deter- mined explicitly, a l t h o u g h in t r a n s c e n d e n t a l form, a n d the t r a n s c e n d e n t a l expression leads to t h e e l e m e n t a r y m i n o r a n t m e n t i o n e d in t h e theorem. As for ~ ( # ) we prove a slightly b e t t e r inequality t h a n XF (~) ~ ~2, and for comparison we shall also derive a m i n o r a n t of q?.
4 . P r o o f o f t h e N e c e s s i t y
4.1. I n order to determine a m i n o r a n t of K0(/~), defined b y (6), we observe t h a t the extremal distance d (~1,/51) is i n v a r i a n t u n d e r linear transformations, a n d hence a function of the cross-ratio of t h e end points of ~1 a n d ill. If ~1 =
(tl,
t2), fil =(t~, Q)
we set= t 3 _ _ - t 2 : - - - t 4 -- t 3 t 2 - t 1 t4-- t I a n d obtain d(~l,/31) - P ( ~ ) , where P is a k n o w n function.
The n o t a t i o n is such t h a t P ( 0 ) - 0, P (c~) = c~. Also, since t h e c o m p l e m e n t a r y inter- vals (t2, t3), (Q, tl) lead to the reciprocal cr0ss-ratio, P ( ~ ) P ( 1 / ~ ) = 1 and P ( 1 ) = 1.
4.2. A m i n o r a n t for K 0(/l) is obtained b y restricting the choice of ~1,/31 to intervals of the form (x - t, x) a n d
(x § t,c~).
I n this case ). = l, while the corresponding intervals:r determine
(x + t) - # (x)
# (x) - ~ (x - t)
T H E B O U N D A R Y C O R R E S P O N D E N C E U N D E R Q U A S I C O ~ F O R M A L M A P P I N G S 1 3 1
Since sup ~' = Q (/z) it follows b y (6) t h a t
Ko >-P(q).
4.3. On the other hand, let ~ > 1 be given. The u p p e r half-plane with vertices at - 1 , 0 , 1 , c ~ is eonformally a square, a n d t h e half-plane with vertices at - 1 1 0 , o , oo is a rectangle whose sides h a v e the ratio P (~). A n affine m a p p i n g of the square on the rectangle determines an extremal quasieonformM m a p p i n g of the half-plane with c o n s t a n t dilatation P(~). If t h e induced m a p p i n g of the real axis is d e n o t e d b y v we h a v e t h u s
K(v)
= P ( Q ) , while evidently ~ (v) ~ ~. The latter inequality implies K 0 (v) _~ P (~), a n d therefore K 0 (v) == K (v) = P (~). This proves:
The best lower bound r o/ Ko(#) is equal to P(~), the extremal distance between ( - 1,0) and (q, co).
4 . 4 . I t is worthwhile to determine e l e m e n t a r y estimates for P ( o ) . The exact expres- sion for P(~) reads
or 1
P(e)= VX(x-1)(x+e) Vx(1 x)(x+e)
1 0
On i n t r o d u c i n g the hypergeometrie function
I
F (t)= F (89 89 1, t) 1 f dx : ~ l l x ( l _ x i i l _ x t )
0
it is possible to write
F Q
P (e) =
A classical c o m p u t a t i o n (Carath6odory,
Funktionentheorie
I I , p. 169) yields 1 F ( 1 ~-2"Here F [1/(~ + 1)] varies between F(O) = 1 a n d F ( 8 9 = 1.1803. Consequently we can write
P(O) - 1 =O(q) log O, where 0(~) increases f r o m 0 (1) = .2284 to 0 (o~) - 1 / ~ = .3183.
132 A . B E U R L I N G A ~ D L , A H L F O R S
5. A Class of Explicit Mappings
5.1. I n t h i s s e c t i o n we s t u d y a class of e x p l i c i t m a p p i n g s for w h i c h K ( # ) a n d ~ ( # ) c a n b e c o m p u t e d . T h e y a r e u s e d t o d e t e r m i n e a m i n o r a n t of ~F (if) (see 3.4 for t h e d e f i n i t i o n o f
~F).
F o r a n y g i v e n at > 0 we c o n s i d e r
# (x) = s g n x . ] x L ~. (7)
A c o r r e s p o n d i n g m a p p i n g is o b t a i n e d b y s e t t i n g ]/(z)l = I z ]~, a r g / (z) - a r g z. I n t e r m s of s = log z a n d a = log /(z) t h e m a p p i n g is affine,
a = a t R e s § I m s,
a n d we see a t o n c e t h a t t h e d i l a t a t i o n is c o n s t a n t l y e q u a l t o at if at >_ 1 a n d 1/at if at g 1.
I t f o l l o w s t h a t K (/~) __< m a x (at, 1/:r
5.2. I n o r d e r t o d e t e r m i n e K 0 ( # ) we c o n s i d e r t h e i n t e r v a l s ( - r, - i / r ) a n d ( l / r , r) f o r r > 1. T h e i r e x t r e m a l d i s t a n c e is g i v e n b y P ( 4 r 2 / ( r 2 - 1)2), a n d t h e i m a g e i n t e r v a l s h a v e t h e e x t r e m a l d i s t a n c e P ( 4 r2~/(r 2~ - 1 )2). F r o m t h e a s y m p t o t i c d e v e l o p m e n t P (t) ~ ~ / l o g ( 1 / t ) f o r s m a l l t w e o b t a i n
4 r 2 ~ ~ r - - > ~
P ~(r 2 - 1 ) 2 / ~ 2 l o g r
P - - - 2 ~ r - ~ ,
2 Vog r
a n d h e n c e t h e r a t i o t e n d s t o at. W e c o n c l u d e t h a t K 0 ( # ) > m a x ( a t , l / a t ) . T o g e t h e r w i t h K0(/x ) g K ( # ) g m a x (at, 1/at) i t f o l l o w s t h a t
K 0 (#) = K (#) = m a x (at, if:c).
5.3. B e c a u s e # (kx) = sgn k" I k [~ # (x), t h e r a t i o
# (x + t) - # (x) (x) - ~ (x - t) t a k e s t h e s a m e v a l u e s as
# ( l + t ) - I
] - ~ ( 1 -t)" (8)
W e m u s t t h e r e f o r e s t u d y t h e m a x i m u m of (8) in its d e p e n d e n c e on t h e p a r a m e t e r at. T h e c a s e s at > 1 a n d at g 1 will b e t r e a t e d s e p a r a t e l y .
THE BOUNDARY CORRESPONDENCE U N D E R QUASICOS/FORMAL MAPPINGS 1 3 3
5.4. S u p p o s e f i r s t t h a t ~ > 1 . F o r 0 < t < l (8) t a k e s t h e f o r m ( l + t ) ~ - 1
1 - (1 - t ) ~" (9)
A l o o k a t t h e g r a p h of x ~ shows t h a t [(1 + t) ~ - 1]It i n c r e a s e s a n d [1 - (1 - t)~]/t d e c r e a s e s w i t h i n c r e a s i n g t. H e n c e (9) is i n c r e a s i n g a n d a t t a i n s i t s m a x i m u m for t = 1, i t s m i n i m u m 1 f o r t = 0.
F o r t > 1 (8) b e c o m e s
( t + 1) ~ - 1 q (t) -
(t-1)~+1"
A s i m p l e c a l c u l a t i o n s h o w s t h a t q' (t) h a s t h e s a m e s i g n as p ( t ) = (t + 1) 1-~ + (t - l ) 1-cr -- 2.
B u t p (t) d e c r e a s e s f r o m o0 for t = 1 t o - 2 for t = c~. T h e r e f o r e q(t) h a s a s i n g l e m a x i m u m w h i c h is a t t a i n e d a t t h e r o o t t~ of t h e e q u a t i o n
(t Jr ] ) 1 - ~ A_ (t - l ) 1-~r = 2.
T h e v a l u e of t h e m a x i m u m e q u a l s
q ( t ~ ) = \ ~ ! = 2 ( t ~ + 1 ) ~ - 1 - 1. (10)
F r o m w h a t we h a v e s a i d i t follows also t h a t t h e v a l u e s of (8) a r e > 1 for t > 0, a n d h e n c e < 1 for t < 0. W e c o n c l u d e t h a t Q (it) = q (t~).
5 . 5 . A s ~ - + 1 i t is clear t h a t t~-+]/2. T o o b t a i n a n e l e m e n t a r y b o u n d for q(t~) w e o b s e r v e t h a t p ( V 2 ) > 0 for t h e s i m p l e r e a s o n t h a t (L/2 + 1) ~-~ a n d ( L / 2 - 1) ~-~ a r e reci- p r o c a l s . T h i s i m p l i e s t~ >V~2, a n d b y (10) we f i n d t h a t
Q (/~) = q(t~) < (]/2 + l ) 2(~-1).
W e k n o w t h a t K ( # ) = ~ a n d a r e t h u s a b l e t o c o n c l u d e t h a t
~F(~) > l + log~o . (11)
= 2 1 o g ( ~ + l )
5.6. F o r large v a l u e s of ~ t h e e s t i m a t e (11) c a n b e r e p l a c e d b y a m u c h b e t t e r o n e w h i c h we f i n d b y c h o o s i n g :~ = 1 / K < 1. T h e c o r r e s p o n d i n g v a l u e of Q (#) c a n b e c a l c u l a t e d i n t h e s a m e w a y as a b o v e , w i t h t h e d i f f e r e n c e t h a t (8) is n o w < 1 for p o s i t i v e t. A c c o r d i n g l y , we f i n d t h a t 9(/~) = 1 / q ( t ~ ) w h e r e q(t) a n d t~ h a v e t h e s a m e m e a n i n g as before.
134 A. B : E U R L I N G A N D L. A H L F O R S
This t i m e we are i n t e r e s t e d in small v a l u e s of ~, a n d it is easily seen t h a t t~ ~ 1 +~. log 2 for :r W e shall show in a m o m e n t t h a t p(1 + e log 2) > 0, p r o v i d e d t h a t
is s u f f i c i e n t l y small. Since p (t) is now i n c r e a s i n g t h i s implies t~ < 1 + ~ log 2.
B y use of (10) we o b t a i n
q (G) > 2 ('2 + ~ log 2 ) ~ - 1 - - 1 2 ~ 1 + ~ log 2
1 > 1
l + : ~ l o g 2 I + ~ l o g 2
5 log "2 1 + 2 log "2
On s u b s t i t u t i n g q(G) = 1/o a n d ~ -
1/K we
are led to t h e e s t i m a t e> log 2
~ ( 0 . ) = ~ ( e 1) (12)
which is v a l i d for s u f f i c i e n t l y large e.
To p r o v e o u r c o n t e n t i o n t h a t p(1 + s l o g 2 ) > 0 we m a k e use of t h e i n e q u a l i t i e s e x ~ l ~ , x a n d ( 1 + x ) - ~ > l - ~ x . W e o b t a i n
( 2 + c ~ l o g 2 ) 1 - ~ = ( 2 + ~ t o g 2 ) 2 ~ 1 + . ~ l o g 2
"~ (2 + ~ log 2) ( 1 - ~ log 2) 1 - ~ l o g 2
> 2 - ~ l o g 2 - 0 c 2 ( 1 + log 2) l o g 2 a n d f u r t h e r
(e log 2 ) 1 e - - ~ log 2 (~ log 2) -~
> e log 2 (1 -- e log e -- ~ log log 2).
W h e n t h e s e i n e q u a l i t i e s are a d d e d we f i n d t h a t p(1 + ~ log 2) > 0 as soon as log 1 / ~ > 1 + log 2 + log log 2,
i.e. if ~ < - -1 9
= 2 e log 2 H e n c e t h e e s t i m a t e (12) is v a l i d for o > 4e + 1.
6. T h e S u f f i c i e n c y P r o o f
6.3. I t r e m a i n s to f i n d a n u p p e r b o u n d for K w h e n Q is given. To t h i s e n d we m u s t c o n s t r u c t e x p l i c i t q u a s i c o n f o r m a l m a p p i n g s w i t h a g i v e n b o u n d a r y c o r r e s p o n d e n c e #.
W e define a m a p p i n g
[(x, y) - u ( x , y)+ iv(x, y)
b yT H E B O U N D A R Y C O R R E S P O N D E N C E U N D E R Q U A S I C O N F O R M A L M A P P I N G S 135
1 K x - t / t ( t ) d t = K ~ ( t ) ~ t ( x + y t ) d t I
i ,
;l ,x t, ; j
where t h e kernels K1, K2 are g i v e n as follows I ( l ( x ) = [ 89 for - l < x < l
for Ixl l '
W e observe
K,~ (x) = K 1 (x) ~ign x.
u e s # (x),
(13)
t h a t
/(x, y)
is d e f i n e d b y a l i n e a r o p e r a t i o n on t h e b o u n d a r y val-/ X - - t
(K = K 1 +irK2).
which we shall use later to m a k e t h e d i l a t a t i o n as small
K1, K 2 t h e d e f i n i t i o n can be r e p h r a s e d as
[# (x § ty) +/~ (x - ty)] dt
(l.4) [# ( x + t y ) - / z ( x - t y ) ] dti
T h e c o n s t a n t r > 0 is a p a r a m e t e r as w e c a n .
W i t h o u t use of t h e n o t a t i o n s I u ( x , y ) = ~ .
0 1
v (x, y) = ~ . 0
6.2. T h e following properties are e v i d e n t : u a n d v are defined a n d c o n t i n u o u s in t h e closed h a l f - p l a n e y _> 0,v > 0 for y > 0, a n d u (x,0) =/~ (x), v (x,0) = 0. H e n c e / (z) has t h e r i g h t b o u n d a r y values.
As to t h e b e h a v i o r a t oo we see t h a t
u(x, y)--~
+ oo for x - >t-c<)andu(x,
y ) - - > - o o for x--> - ce, u n i f o r m l y i n y . Similarly,v(x, y)--~ + c<~
for y--> + co, a n d t h e convergence is u n i f o r m w h e n x is r e s t r i c t e d to a finite i n t e r v a l . W e conclude t h a t ] ( z ) ~ oo for z~cx~.I t will be seen t h a t t h e J a c o b i a n of t h e m a p p i n g is always positive. I n view of this fact, a n d b y v i r t u e of t h e b o u n d a r y correspondence, it is a simple m a t t e r to show t h a t t h e m a p p i n g is a u t o m a t i c a l l y topological, a n d t h a t t h e image of t h e half-plane, y > 0 is t h e whole h a l f - p l a n e v > 0.
6.3. W e shall h a v e to d e t e r m i n e t h e p a r t i a l d e r i v a t i v e s of u a n d v a t a p o i n t
(xo, Yo),
Y0 > 0. I t is easy to p r o v e t h a t t h e d e r i v a t i v e s of a c o n v o l u t i o n of t h e form used in (13) are given b y
136 A . B E U R L I N G A N D L . A H L F O R S
~x
-~- K ( t ) # ( x + y t ) d t = y K ( t ) d # ( x + y t )
9 fK(t)~(x+yt)dt=~ tK(t)d~(x+yt).
~ Y . Y
C o n s e q u e n t l y , if we i n t r o d u c e t h e n o t a t i o n s
we o b t a i n
1 0
~:yo d#(Xo ~yot), t~=yo
0 1
/ / /
, . 1 . t d # ( x o + y o t ) , fl, . . ~ 1 t d # ( x ~ 1 7 6
YO 0 1
u z = ~ + ~ ,
u~=~'-fl'
v~ = r ( ( z - fl),
vy=r(:r247
(15)
On s u b s t i t u t i n g t h e s e v a l u e s in (1) we f i n d
1 ( ~ § 2 4 7 2 4 7 2 4 7 2)
D + D - 2 r ( :c fl ' § cc' fl ) (16)
I t is t o b e n o t e d t h a t a, fi, ~', fl' a r e all p o s i t i v e . T h i s s u b s t a n t i a t e s o u r c l a i m t h a t uxvy - uyvx = 2 r ( ~ fl' § ~r fl) > 0.
6.4. S i n c e ~ (t) =/~ (x 0 § Y0 t) s a t i s f i e s t h e s a m e e - c o n d i t i o n as/~ (t) we lose n o g e n e r a l i t y if we c h o o s e x 0 - 0, Yo = 1 i n (15). I n o t h e r w o r d s , i t a m o u n t s m e r e l y t o a c h a n g e of n o t a - t i o n if we r e p l a c e (15) b y
1 0
0 - - 1
1 0
0 - - 1
B y m e a n s of t h e e - c o n d i t i o n
1 # ( x + t ) - # ( x )
- < --<e (17)
e t t ( x ) - # ( x - t )
we o b t a i n a t o n c e : r :r (18)
T H E B O U N D A R Y " C O I : ~ K E S P O N D E N C E U N D E I r Q U A S I C O N F O R M A L M A P P I N G S 137 6.5. Corresponding estimates of ~'/:~ a n d
fl'/fl
are a little less obvious. W e prove in this respect:LEMMA.
T]~e ratios ~'/~ and fi'/fi lie between
1/'(9 i l)and ~/(~ -i-
1).F o r the purpose of p r o v i n g t h e L e m m a we m a y assume t h a t # (0) = 0,/~ (1) -- 1. T h e n 1
r162
i -f # d t ,
0
a n d we h a v e to prove t h a t
l
- - - - ~ # d t S ~ 9
(19)~ o + 1 - ~ ~o§
L e t Co be t h e family of all # which are normalized b y #(0) = O, #(1) - 1 and satisfy (17) for points x - t, x, x + t contained in t h e interval (0,1). W e set
M (x) = sup /~ (x).
;~eCo
I t is clear t h a t M ( i ) ~ / ( ~ + 1). F u r t h e r m o r e , it follows from t h e definition of M (x) t h a t
M(x) <_ M( 89 0 <_ x < 89
M ( 8 9 1 8 9 O g x 2 8 9
Hence
M(x) + M ( 1- + x) ~ M(1) +
M ( 2 x )1 1/2 I
a n d
f M ( x ) d x = f [ M ( x ) + M ( 8 9 M ( 1 ) + 8 9 fM(x)dx.
0 0 0
We conclude t h a t
1 1
# d t < M ( x ) d x g M ( 8 9 + 1
0 0
The left-hand inequality in (19) follows on replacing #(t) b y 1 - / t ( 1 - t ) . The same b o u n d s for fl'/fl are f o u n d when we replace # (t) b y - # ( - t).
6.6. W e simplify (16) b y writing ~ r fl'=~//~. T h e n
D-~ D - 2 r ( ~ + ~ )
~ ( 1 + ~ 2 ) + - ( 1 + ~ 2)( l + r 2 ) + 2 ( 1 - ~ r ] ) ( 1 - r ~ ) ,
a n d t h e point (~, ~1) is restricted to t h e square1
~ + 1 ~ , ~ 9
138 A . B : E U R L I N G A : N D L . A H L F O I ~ S
B e c a u s e of t h e s y m m e t r y we m a y s u p p o s e t h a t ~ > z]. U n d e r t h i s c o n d i t i o n our e x p r e s s i o n has its l a r g e s t v a l u e w h e n ~/fl a t t a i n s its m a x i m u m 9- H e n c e we f i n d t h a t
D + ]5 --< F (~, ~) 1 (20)
w h e r e F (~, ~) - a (~, ~1) ~- b ($, ~) r,
r
( e + l ) 2 ? ( 9 ~ - ~ ] ) 2 b(~, ~ 1 ) _ ( 9 - - 1 ) ' ~ (95 ~-~) 3
a (~, ~j) - '2 (~ + ~) ' 2 (~ + V) "
T h e f u n c t i o n s a (~, ~) a n d b (~, zt) a r e seen t o be c o n v e x (from below) for ~ + ~ > 0.
H e n c e t h e m a x i m u m of F ( $ , ~1) in t h e t r i a n g l e u n d e r c o n s i d e r a t i o n can o n l y be a t t a i n e d a t one of t h e v e r t i c e s
(1 1)( o 1)
9 + 1 ' ~ + 1 ' 9 ~ - - 1 ' e ~ - - 1 or 9 + 1 ' 9 + 1
W e d e n o t e t h e v a l u e s a t t h e s e p o i n t s b y F~, F 3, F a r e s p e c t i v e l y , a n d set F~ = ai/r + b~ r.
6.7. W e h a v e a l r e a d y p r o v e d t h a t t h e d i l a t a t i o n is b o u n d e d , for i t follows f r o m (20) t h a t
D + ] 5 < 1 m a x ( F 1 , F 3, F3).
I t r e m a i n s t o d e t e r m i n e a n e x p l i c i t b o u n d b y s u i t a b l e choice of r.
I n o r d e r t o t r e a t t h i s t e c h n i c a l q u e s t i o n we c o m p u t e ( 9 - 1 ) ( 0 2 + 3 9 §
a l - a 3 = 4 ( 9 + 1 ) bl _ b3 = ( 9 - 1) ( g a - 9 - 4)
4 (9 + 1) ~
( 9 - 1) ( 9 4 + 3 9 3 + 8 9 2 + 3 9 + 1)
a l - a a = 493 (9 + 1)
( 9 - 1) ( 9 5 - 9~ + 3 9 + 1) b 1 - b 3 =
4 93 (9 + 1) 3
I t is seen t h a t a 1 - - a 3 a n d b 1 - b a are b o t h > 0 for Q > 1, a n d h e n c e F1 > F 3 for all r.
C o n s e q u e n t l y , we n e e d o n l y c o m p a r e F~ a n d F 2.
The i n e q u a l i t y F x > F 2 h o l d s if
(al--a2) ~-(bl-b2)r2~O.
THE BOUNDARY CORRESPONDENCE U57DER QUASICONFORMAL MAPPINGS 1 3 9
T h e m i n i m u m of F 1 is a t t a i n e d for r = Val/bl, a n d we c o n c l u d e t h a t t h i s m i n i m u m is g r e a t e r t h a n t h e c o r r e s p o n d i n g v a l u e of F 2 if
(a, - %) b 1 § (b I b2) a I > 0.
A f t e r a l e n g t h y c o m p u t a t i o n one finds t h a t t h i s e x p r e s s i o n e q u a l s ( ~ - l )
~(~§247
8 (~ + 1)a a n d hence t h a t i t is i n d e e d p o s i t i v e .
A c c o r d i n g l y , we h a v e p r o v e d t h a t
§ l = 2 V a l b l ,
K
a n d h e n c e t h a t K _< }/a 1 b t ~ V'a~ b I - 1.
W e p r e f e r to r e p l a c e t h i s c o m p l i c a t e d r e s u l t b y t h e s i m p l e r i n e q u a l i t y K ~ o 2 a n n o u n c e d in t h e t h e o r e m . To o b t a i n it., it is sufficient to show t h a t
( ' ; 4 a l b t ~ ~2+~.2 9
E x p l i c i t l y , t h i s i n e q u a l i t y r e a d s
(~ - 1)(3~o 7 + ~6 -i- 895 + 1294 4 e - 4 ) > 0 , a n d i t is o b v i o u s l y s a t i s f i e d for all o > 1.
7. Absolute Continuity
7.1. I t has b e e n a n o p e n q u e s t i o n w h e t h e r t h e b o u n d a r y c o r r e s p o n d e n c e i n d u c e d b y a q u a s i c o n f o r m a l m a p p i n g is a l w a y s g i v e n b y a n a b s o l u t e l y c o n t i n u o u s f u n c t i o n fl- Our T h e o r e m 1 r e d u c e s t h i s q u e s t i o n t o one t h a t d e a l s o n l y w i t h m o n o t o n e f u n c t i o n s of a r e a l v a r i a b l e . The a n s w e r is in t h e n e g a t i v e , a n d e v e n t h e following s t r o n g e r s t a t e m e n t is true:
THEOREM 3. There exists a quasicon/ormal mapping o / t h e hall-plane on itsel/ whose boundary correspondence is given by a completely singular [unction ,u with ~ (tt) arbitrary close to 1.1
1 We wish to acknowledge that Mr. Errett Bishop has also constructed a function which satisfies a ~-condition without being absolutely continuous. The construction that we shall use is essentially different from his.
1 4 0 A. BEUI~LII~G A N D L . A H L F O R S
F r o m a f u n c t i o n - t h e o r e t i c v a n t a g e p o i n t t h e m o s t s t r i k i n g c o n s e q u e n c e of t h i s t h e o - r e m is t h a t t h e d i s t i n c t i o n b e t w e e n sets of z e r o a n d p o s i t i v e h a r m o n i c m e a s u r e is n o t p r e s e r v e d u n d e r q u a s i c o n f o r m a l m a p p i n g s .
7.2. L e t ~ > 1 b e g i v e n . W e c h o o s e a n i n c r e a s i n g s e q u e n c e of n u m b e r s o,,, 1 < o~ < ~, a n d a f i x e d n u m b e r ~, 0 < ~ < ( ~ i - 1)j'(~l § 1).
W e a r e g o i n g t o c o n s t r u c t a n i n f i n i t e s e q u e n c e of i n t e g e r s 0 < n I < n 2 < ... w i t h t h e p r o p e r t y t h a t t h e f u n c t i o n s ,u~(x), d e f i n e d b y
v
# , , ( x ) = 0 f ~l~I 1~ (1 + 2
cosn~x) dx,
s a t i s f y ~ (/&) _< o, a n d c o n v e r g e t o a s i n g u l a r f u n c t i o n # (X) w i t h ~ (#) _< ~.
To s i m p l i f y t h e n o t a t i o n we s h a l l use t h e s a m e s y m b o l s / ~ , ~ n d l~ for t h e c o r r e s p o n d - ing i n t e r v a l a n d s e t f u n c t i o n s . S i n c e
p,.~ ~ (x) = i (1 + 2 cos n,.: ~ x)/u,((x)
dx
o< / ~ + 1 (oJ) ~ 1 +
we h a v e 1 - ) . = , u T ( 0 ) ) - :
for a n y i n t e r v a l ~o. F o r a n y p a i r of i n t e r v a l s co, co' we o b t a i n 1 - ) ~ < /~,, ~ 1 (co) #,, (~o) 1 + ) ~
1-~-),--tl~l(o./)~
:---/t, (~o') "-< 1 - ) ~ " (21)7 . 3 . W e c a n c h o o s e a n a r b i t r a r y n 1 > 0, for i t is o b v i o u s t h a t Q (/q) < (1 + 2 ) / ( 1 - 2)
< O~. S u p p o s e t h a t n a , . . . , n , , h a v e b e e n d e t e r m i n e d . I f
i - 1 2~
t h e n f #~ (x) cos
n,.~lxdx=O.
0
F o r t h i s r e a s o n , a n d b y R i e m a n n - L e b e s g u e ' s L e m m a ,
3~
f ~ : (X) COS nv+i x d x - - > O o
w h e n n,,-1-->o0, u n i f o r m l y for all x. H e n c e #~+1 t e n d s u n i f o r m l y t o / ~ .
L e t ~o a n d o~' d e n o t e t w o n e i g h b o r i n g i n t e r v a l s (x - t, x) a n d (x, x § t) of l e n g t h t.
S i n c e #~ (x) is u n i f o r m l y c o n t i n u o u s we c a n f i n d ~1,, > 0 so t h a t
T H E B O U N D A R Y C O R R E S I ~ O N D E N C E U ] S T D E R Q U A S I C O N F O R M A L M A P P I N G S 141 l + ) t 1 _ < f f . ( ~ o ) < l - - 2
1 - ) . q,,+l ff~ (aY i = 1 + ) . ~.+1 w h e n e v e r t <~/~. T o g e t h e r w i t h (21) i t follows t h a t
1 ,u,+l(w)_<e.+l
q,+i
ff.+1(r
(22)u n d e r t h e s a m e c o n d i t i o n on t.
B e c a u s e if,,+1 t e n d s u n i f o r m l y t o #~, a n d b e c a u s e ~(#~) =< ~ < ~+a, we can choose n,+l so t h a t (22) is also t r u e for t > ~/,. I n o t h e r words, we can choose n,,+l so t h a t ~ (ff~+l)
< ~+1. I n a d d i t i o n , we can a n d will m a k e sure t h a t
I if,,+1 (x) - if, (x) l < y2 1 .N,, (23)
for all x.
7.4. F o r v~>oo t h e sequence {#,~} converges u n i f o r m l y to a n o n - d e c r e a s i n g l i m i t f u n c t i o n p. I t is clear t h a t ff (0) = 0, ff (x + 2~) - / t
(x) +
2 z , a n d ~ (if) ~ ~. W e claim t h a t ,u is p u r e l y s i n g u l a r .To see t h i s we w r i t e
oo
g ( x ) = log (1 + 2 cos x ) = ~ 7ke~'~,
oo
w h e r e
a n d oo.
-oo
if
Y 0 = ~
l o g ( l + 2 c o s x ) d x = - a < O
0
W e d e t e r m i n e q so t h a t
ikl>q
a eikx
T h e n g ( x ) < - 2 + ~ yk
1_-<.1 k I:.<q
a n d log ff~ (x) = ~ g (nj x) < - j=l 2- + ~ ~k e 'knJ ~ (241
l < l k l < q =
- va+qb,(x).
2
T h e F o u r i e r e x p a n s i o n of r c o n t a i n s a t m o s t 2 q,, d i f f e r e n t powers, a n d in each t e r m t h e coefficient is
142 A . B E U I ~ L I : N G A N D L . A t I L F O R S
'f
H e n c e ._~ . r
dx
"/5 '2 q~ S 2.0
As a r e s u l t t h e s u b s e t of (0, 2 7~) on which r > a v / 4 is of m e a s u r e <
c/r,
where c is a c o n s t a n t t h a t d e p e n d s o n l y on 2. B e c a u s e of (24) t h i s set c o n t a i n s t h e set E , d e f i n e d b y t h e i n e q u a l i t ya i1
~: (x) > e - 7 .
C o n s e q u e n t l y t h e m e a s u r e m(E,.) of t h e l a t t e r set t e n d s to 0, while on t h e o t h e r h a n d
# (E,.)-->2 ~.
W e m a k e use of t h e f a c t t h a t
#'~ (x)
is a t r i g o n o m e t r i c p o l y n o m i a l of d e g r e e _< N~ =i,
: ~ . hi.
F o r t h i s r e a s o n E,., c o n s i d e r e d on t h e u n i t circle, consists of a t m o s t N,.1
arcs. F r o m (23) we o b t a i n
]f~(x)-/,,.(x)l<l/(v-1)N,,,
a n d we conclude t h a ti#(E~)--/i,,(E,.)l
< 1 / ( v - 1). H e n c e /.t(E~)-+2u, a n d sincem(E,,)--~O
it follows t h a t/t
is p u r e l y s i n g u l a r on (0,2~), a n d c o n s e q u e n t l y on t h e whole real axis.
