View of Quasiconformality and boundary correspondence

T,ÍATIIEMATICA

_

REVUE D,ANALYSE ^NUMÉ'RIQUE ET DE THÉORIE DE L,APPROXIMATION

I'ANALYSE

NUN,IÉRIQUA

ET LA TI{ÉORIE DE

L'APPROXIMATION Tome

5,

No

2,

1976,

pp.

¹¹⁷

-12ß

SUASICONFORMAI,ITY AND BOUNDARV

CORRESPONDENCE

by

P. CARAMA.N (Iaçt)

Introduction

Let D be a

domain

in the

Buclidean Ø-space

R, B the unit

^bal1,

: ^(qc), E the set of ^{poirr.ts oL} ôD inaccc- E* the corresponding boundary poinls

tt:at n :3, _.f is

differentiable and

that the

⁽ⁿ

- ^l) -

dimensional Hausdorff measure

Hn

¹

(E*) :

0.

T

esult was proved

by n stoRvrcx

l7l ^{lor n} :3,

_.f

. C', ^{ted and mn} <-. 4"

^{gives alèo}

(ìn

the

same paper)

an

^ng

^{to F'}

^Gehring

^only

^{under the}

hypotheses

that n :3

We shall

^begin

by giving a proof of this

theorern

lor f

^:

_?

= ^B

^q',

rvithout any othõr

additional-

restrictive

condition.

Then, we shall

^{pfov^e}

theE'kis

of-conformal capacity and even of ø-capacity zero lot ever-y ^¿

)0.

Gehring's conjecture ^is thât

E*

is evèn

of

logarithmic capacity ^{zerc : C} ¡(E*) .='

: 0

^and. M. r{EADE

[6]

^assertion

that E* is of

nervtonian capacily zero:

cl(E*) :0,

Now

we sha1l introduce

a

ferv concepts ^:

Let I

be an arc

family

^and.

F(Il)

be

a farnily of

admissible functions p(ø) satisfying

the

following properties:

lo

p(x)

ì ^{0 in}

^Rn,

2o

p(*) is Borel

ureasurable

in

^R",

30 _J' [tasZ ^{1 for}

^every

Tel,

Y

118

P"d",

R1'

rvhere dc

is the

rrolume element

(1)

A

qc accordipg

to väisälä's

geometric

defi'itiou is

characterized by

i4(r)

_I(

<¡1fl'*) <Kx,r |\

s'hich is

supposed

A function

¿¿

to

hold for

every

I

contained

in D,

rvhere

l* (f)

l'ines)

in

a domain

D if for

each

interval I : _{x; a-i<x¿<þ'(i _l,

3 QUASICONFORMALITY 119

then C"(E)

:

finf/"(¡r)]-1,

where

the infinum is taken

over

all

positive meastlfes ¡.r.

rvith total

^mass

1 and the support of _¡r contained

in.Efis

^cal1ed.

^by war,r,rN f9]

æ-cøþacity.

When

o,

:

d,-

E is

supposed

to

have

the

diameter less

than 1. For

any

athittarv

Borel set E,-c _o(E)

: 0 irr"3ï!uu,.î

f ,lo:.0,tî,jl"åitBitir"¿"ñ

to

capacity

of

,E* (i.e.

the

n-capactty)

is ^{city of}

^{E^'}

^is

^zero

^{for evely}

^c¿

)

^0'

|

^'

H"-rE* :0'

o s

i t i

o

n

1.

If ^lo'is

^the fømi@ of t.tnrcctifiøble ^ørcs

of

R", 'tlt'en

(J.

vfisÄLÄ

_[B]).

TI{EOREM 1.

f1"-18* :

0.

But,

since

f ^{is K} -

^qc,

it

^satisfies

M(l*) ' _Ño*, :0. _{let y¿- be a radial}

_segment

_{with an endpoint [*}

_e

_S'

Then

clearly,

xn'(€*) ^S

yv P, C.A.RAMAN

1

Then

the

modulus M

(l) of I is

given as

a[(l): ia¡

pe.F(t)

D

* Rl is

said

to be

ACL ( absolutely contiu,u,ous

f

on (i.e.

1C

(alurost

n)j, I C C

^D

D),uisA

C (absolu sense) on a.e. every) line

The

tuo

closed,

where Co is

(2)

cap,,E:cap E:inf _IV

segment d,isjoint

and

C

. ^is

^{compact, sets Co,}

^is

^defined

^CLC

^as

^{D relative to}

^D,

tely

continuous) (in the

parallel

to the

coordinaordinary

te

axes

M(to): Pro

^0p

cap¡(D, C6, Cr)

: i¡f

J ^l\/ultd'c,

I âtt

wÌ1ere

\/,,:

l*, . ,#)is ^the ,r"u,:;";T") _^oathe ^infimu'r is take'

over

all

ø, rvhich are continuous or1

Dl)-CoU

C.,

ACL

on

D

^__

(CoU

Cr) an_d assume

thè

boundary values

0 o'Zo unã r;" C;So"n

rurr"tions are

called

admissible

fol

capo

V),

^{C0, Ct).}

'lhe

caþacity o.f

ø

^bound,ed sct

E C

^IL,,

is

definecl

to

^be

lvhere

the infintun _{is taken over all functions}

_zr,

which are co¡tinuotr

and

ACL iu_R",

have

a co'rpact

support containãd

in a fixed ball

an

afe:1on.Ð.

The u-þotentiø\,0S øS n of a neasure

_¡,r.

is

denotecJ.

by

uu;, wher.e

(f

for every

p* e

F(l*)

and, integrating over S, on account

of

tr'ubini's theo-

rem, we

obtain

Hn-l(E*)< H"-,(Ei): I xo.{A*)rl"=

[1og

;)-' f

e*"d,",

where do

is

the superficial

"r"-"to, ^of

^S.

^Finally,

^taking

ttt" inti-rr- o,,"' all'

_Oi' e

F(l*),

yields

Hn-r(E*)= (.* ï)-' *1r.) :0,

i

if 0<a<n,

)"s

log

* d,s

1r " ^d,r, p ^p

1

n

llr

f

^{*ny*n- d,r*}

u!1x¡:

^drt(Y)

lx - ^!1"

and

^Rn

u[(x): Iton-'

eo \þJ

- )*r"ts l,

_ nt ^dt-tb,),

which is

called also

the

logørithmic þotentiø|.

rf

1"(¡-r,) d.enotes the encrgy

integral of. _¡t

1*(p)

: _I

uid,s(x),

Rn

2, Ex is

closed.

F'or any two points

%,y e

D, we shall

define

the relative

distance dr(x,

y) to

6e

the

gratest lo¡n'er bound.

of the

lengtþ

of all

arcs

joining

ø

tõ'y äa

^lying

in DItt

^{is clear}

that

^do@,

y)

^{is a}

metric

an;d Lhat doØ,

Y)¿

as desired.

120 P, C¡,RAMAN QUASICONFORMALITY

t2l

4 5

Z _lx - yl ^rvith

^{cqua.lity tf-} ry,

! ^{lie on}

some convex subdomain of

D.

For

points

x e

D,

E e'âD,

*e

defín-e

¿r(*, z) t" ¡1irråì"rì-îrî,

"t

rirn do(x, x,,,)

Arr _arc

\ c D rvith c'dpoints

_E

_e

_{0D is calred an end,cut}

_{of D}

_{from (.}

the

sets

Er, E, C D is thc

iufirnurrr ning .E',

to E, in

D.

sanre

endpoint

_E

e ôD

ate called Ua

ol

1, dok,,

O Ue,y, O

^Ue)

: Pr opositiorr ? L-:t/:_D=; B

bc

a

l{_qc uløþþing

(l< K < *), thut, d(Ei_, D;) =0 .irtþtic.i

^arla:,,

"D",i'"

0^,r;r;:,;i{tEî, Di C B

^anrt

En: Í-,(Ei)(h _ 1,2,). i '\-

^'L!

Ngrv

*'c shall i'troduce

.(accorcrirrg

to zo*r0

Lr l

]) ^{thc conccpt}

^or

boundary

clc'rents

_(a

gcnerali)utior,-ài'1,r;;"'?;,ä;i. t' ^',

,{

sequence

of domains {(J,,},

^(J,uC

D(nt,:7,2,...) is

saicl

to

bc regular iL

a)

ú,n,_1C ^(i,,,(tn.

: 1,2,

. . .),

(@ \

o)t,f, _ú,,,)c

_an,

, ^{c) 6,,:= 0(J,,fl D}

^(the r.elative bounclary

of

^(J,,,

in D) js a co¡lectcd

set,

d) d'oþ*,

6,,,+r)

)

^o,

. ^e)

there is at rnost an accessible boundary point of D,

',hich

^{is accessibre} boundarv

poi't for

each

of the

dornain"

"i ih;-;"qÃ"" {u,:}- --*"' , ïyo

seqrle_nc(,s

of d.omairs.!Ur,j,{(J,,,,}

_are

cailåa

aquiaale-nt

if

ever1.

term

of

each of thern

contai's alithó'termá

_of

the

othcr óne b"gioiug

b;ä

sufficienty great

_index.

A

bound'øry element

of a clomai'D is

the

pair (F, {u,,})

consisting of

a regula'

seqrlcncc

{U,,,}

^and

a cortinuum F : fi ^{ú,,,. Two}

^,boundary

elements (þ-; {(1,,,}),

(F', tul,})

^are consiclered as

¡aririilraiff

_{the two regular}

sequences

{u,,}

^and

{uíl defining them

are equivalent.

r' this

way-any

of the

equivalent _sequences determine uniquery

a

boundary _element.

Proposition ^3.

^Foreaerit

K -

Sctttnþþirr'gf

tP¿,8;itisþossibla

to ^estøbtisË an one-to-one corresponíence betiseen the boir'nd'ary-!2i.nt1- (F, _{U,,})

"if-rí-i"iin,

ponrts'o/S, so that,,to

eac.lt, _þ::yt

(F'"19,,\),there ^corrcs-

lrla-i'åi i ^{[t" po;å}

dercrm'inrtd bv

the {ui,,) : _Í(u*\)'

(For the

^{proof, see lemma}

3 in ^aper

[11])'

Let

us consider the class {4,

y}

of D-equivalent endcuts y

from

a boun-

dary point \e

^ôD.

On

the

other hand, given

a

^clas

a ^the

^corresp

c O B(t,

r,u)

I

sequence is

s t is

easy to

let

r { !r,nunð.yr C y, lrCy'

^be

^{st,chthat the}

diameters d("¡r)' d'(yl)S

r

arrd

f ^{is the}

^{common endpoint}

^of

^y1,

^{^¡í.}

^Since'¡',

^Y-ar¡

D-eqttivalent aud

re is

an

arc

a

L D Þtntng tnelll

allo

e Point

l*

e S, as desired'

L

^{e m rn}

a l. E* is

^cl'osed'.

I'et

r*:u{s-

^{tp xo, m)}

In

ord.er

to prove that E* is

^closed.,

it

^is _{_enough}

^to

^{show lJnaL}

^E* : F*.

We shall estäblish

first that E* C F*.

^{Indeed, suppose} 6* ^{e S,} but _E'r'eF*,

122 P, CARAMÄN

then _there is an

in

teger

_tn,n

such that (*

e

q*eS-

_fi0,

^To (4)

^complete

^tire

^proof

^of

^(3),

^a,t(t)

^{\'\¡e must}

^{s M(r).}

^shor'v

^that

Norv,

the

'Lact

that

Co and.

C,

are

disjoint

^irnplies

that M(l) <

^oo'

^Iìix o: t >landchoosepe F(l) so that pisl-"-integrabie' Bythepre-

1-e

cedirrg proposition, ^\4¡e call choose

I >

^{0 sg} t}Iat,ap e

r[Il(l)]. we

^may-assu-

*À, f'ot'"oä.,"ni"uó" of

notations,

that D is the half space r" > ^0.

^Set

plx) :

ap@

+

lø,,) (where ^e,,

is

the -versor

on the axis o*2, ^let _1t:.1 liìà'f"t y'È" i¡""átò y, tra'slated ^through the

'ector

^{lø,,. Then}

^y

^e

^f(l)

aud r,ve have

QUA SI CONFORMALITY t23

6

s- ^xo,

^r11

_],

hence

f)

^S;

^i.e. 4*

^betongs _to

nP

the b of

endpoin

t at _{*,

^tþ@

mo)

o, nt.o)l and

with

an

and u'ill

halre an endpoi

fiable

nt(eôI)

arcs from

l)

^accessible

b)'

recti-

dS

x

l

p

^{\ or(*}

^{)dt 1}

-[

Ilence

p. e

F(l),

rifll)

<

t

^{?id, ct¡7}

I

^{p"d' '}

Rtr

3.

Cap

E* :

0"

Propositio.n 4.^Lct CoCÐ,,Cr_CD

_be

_tuo

non-ernþly d.isioirtt closcd' sets,t the

family of

^arcs

*irùn-¡Llttbf,"la"òr";;

_{o," o}

rr(r),

^p e 2,,.

Then, giaen

e>

0, there

is øt(e)>0

such

int ,1_.f

^¡¡12¡

I for t <t(u),

'y'i;,'oÏI,oè,tl'

^ørcs

ioíning co(t): {*t

^a@,

^c0)s 4 ^to c,(t) -.

(For the p r

1>aper f2 ^|

r,emma _tnfuTr¡iì

sþace D, C, ø

of

Co

in

âD.

the

fantily

^of

(3) a,r(t):i*F,).

th"tAtguiog

^{as rì.}

^(ìTisnrxc

^{and. ¡r. r¡ÄrsÄr,Ä}

^{in le[rma}

^3.8

of

_[S], r,ve obtaiu

M(r) s

^J'

^nt¡r,¡,

Ncxt, lct Ir

denotc

thc

'arnily of

arcs

,"!i:l joi'

Co and C,

in Ð.

Then, by

:l;"ìiä:" ^ar'ument

^as

in

Gãrrrin;

;;d îàl;åh;; ;&;# [lot"a

^{abovc, rvc}

)-

*F,) s M(t)'

and taking the irrfimum over all such ^p yields A,I(l)

= ø"M(l).

F^inally,

if

^{u'e 1et a}

_-,. I,

^rve

obtaiir

^{(4) as desired.}

iót o11ary.

l.'Suþþose

E C'S ^is ø

^cl,osed. _þroþer ^subset

of S

ttnd'

A:\x;r <lxl a+Ì,

^where0

<r <L IÍI. isttoefamitv of

arcs ahich

ioin l|l ^{4 r}

^to

^{E in B}

^and,

l,

^tke

_famity

^{of ørcs uhiclt'}

ioin ^CA

^to

^{E¿n R,}

"tlrcn,'fònnulø (3) stitl, kotds

for

the new rneaning of

I

^and'

l''

I,et

xo e S

- ^E

^and.

^x' :

g(x) be an inversion

with

respecl

to l

^sphere

with

the cénter of inversion

xo.I,ei

us denote

by B(r): _{y; lll-1r}:-Co:

:9lB(r)1, Cr: q(E), l' :"ç(f) and l{ :.9(lJ1 ^Then, _-q(S): fI is

^a

ptai å

anii'C, - n

äà¿

*"

^are

in

the hypothesis of the preceding 1emma, so

that

M(r'') : _i *Fl).

But

iuequality (1) implies the invariance of the modulus rvith respect

to

the conformãl mappings,

allowing us to

conclud.e

that

M(t): I[(t'): I*["): i*('')

as desired.

P

r op ositi on ^5. IÍl -9t',n,ttte'n ilI(l-)=Ð,

(1,,).(u.ËuGI,IIDD

r-31.)

'

^"

'L.t _M(E*)

_{be the mod.ulus of the}

_family

_{of arcs rvith} an endpoint

in E*,

where

E* is

d.efined as above,

124 P. CARAMAN

is

tlte

føntit1

^{of arcs}

aíth

an end.þoitr.t belonging _to

tt,I(E*): n4(l*¡ :

s.

g OUASICoNFoRMALITY ^1'25

on the

other hand,

let

^{p e}

F(I])

and ^æ

an arbittaty

continuum ^of- x. ^TJren, there exists an arc

yei' ^such'that YC

^{ø, hence}

^peF(X),

^so

^that

Õ

I,emma 3. t¡i*

E*,

then

M(x)

< P"d",

I.et

_{{r,,,} be} an incrcasing sequeuce of nur¡.be ts ï,,,

}0

^such that 1im r,n

:

Rfr

-=

^{1, and}

{li;}

^{be a scqrlcnce}

^of

arc I

'llien, _{from the}

definitiorr

of E*,

^we

\r, =.!?, ^{..-.) are not} rcctifiaúle,

r

1, _M(1,,,)

:0 (m:7,2,...) _and'(l

I,et ñ;,

be the

family of

arcs rvhich

j

'lhen, the precedi'g corotary

allows

us to conclude that MFî,,) :o

(rn Next,

: 7,2, if li c s

. .

.),

I{ence,.taking

uld _{!-.í is} tïe into r"ttlv

account- of propositio

n

S,

M(¿ i;l':

O.

,thä ;i*;}-^;ilh the e'dpoi'rs

belongiug

to E*

ancl

rvith

_Vð

fl ^CS+'h,

^tn"u

f

rvhence,

taking the infimum over {]

^p

^e{(?), -'ve ^obtain M(x)= M(I')' which, togethõr with (6), gives

^(5), as desired'

'rHuoREM

2.

CaP

E* :0.

1f n.t j ^{ís an}

r^-neighbourhood

oI E*

(i.e.

the

set

of points rvithin

^a distance

z'from E*),

then, clearlY

(7)

^caP

ICE*(r), E*, R"l2

caP

E*

since the class of admissible functions

for

cap

lcE*(r),

^E'k,

^R"] is

^containecl

in

_^^

that of -ñ;"i;fãiï;:

cap

E*.

d.enote

the family of

^arcs,

wliich join_E*

and

CE*(r)

in

Rl,

an¿

f*

of

th"

preceding lemma, then evid,ent t-;:

C ^l-x

and the preceding lemma implies

MICE*(r), E*,

^R''f

: M(li

)

= U(f*) :

O

for all r > 0,

hence and

by (7), taking into

accorrnt also

the

^preceding corollary,

we

obtain

cap

E* l cap |CE*(r), E.*,

^R',1

: M|CE*(ï), E'.,

^11,,]

s jlf(Ì.''.) :0,

as desired..

I{.

wATrr,rN [9] ^gives the following definition of the coilf<¡rmal capacity ^:

-,-,nàï^f'6.

a Ëoînded set

in R",

cap

E

is.d'efined

bv

(2), where thc

infiáim ^{is taken}

^over

^ail

^{lunctíons 1,0}

e

CL, which have compact support

^¡ãr"rgi"g

to a

certain

fixed

sphere B(Ro) which

is

independent

of E

^ancl

ul, "" Þ -eigoiog

^7."

as

in

F. GErrRrr\G

ls

^p1pcr.

([4],.lemma,lì, it can

casily bc

sno*. ih"t"the ^infimurn

appearitrg

in trte-a"finitiou

⁽²⁾

of the

^conformal

ããp."rty

"f " ¡oo"ã"à

set ^{L-^is} nolincreased

if it is

taken

over all ^u

^{e Cl}

in

^R".

-corollary.

cap Et,:0,

^{ukere tke} conformø|, caþøcity

is

ta'hen in' Wall,in's ^sense,

i.i.

^ui'th,

ul" Þ I

^{(not ülB}

,- ^l)',

reõding theoren,

^{since the conformal}

n the introduction is not

lessthanthe

ty.

set

in R"

uíth' caP

F :0

(tloe con'for-

fl(ffty_

Oionctusict'tts

ure true:

>0.

(f9'j,

theorem B).

ò- càPøcitY Co(En)

:0. If

ⁿ

>

^2,

then

Cn(E*):0lor

^eaerY

a>0.

!.

M(l$ ₎ : i¡¡

p

p"d,'c

: inf

p

d't

and since

lö- : _{U ñ;,}

prulrositions 5 _vields

,L 0

p s

M(rö) _ M(l rj,) : _\, uçr¡;,¡ :

o.

clearly, F*

c ^fs u ^rô',

^so

that, from

above, and

by

propositiorr 5, rve con- clude

that

M(ix¡s _{M(tg )} + ttr(ri) :

o,

as desired,

ï ^t

o-p 9.s.i

t i

o

r17.^If 1is

^{the set of}

^ail

continuø itø

Rn

that interesccr

luo

closerl, d'isloittt srls^c.,"_ci, _{zaloere co"}

co"tol", iiii""rå'*iìrrrrnt _of a

bøil, thcn

M(y):

õap(Co, Cr,"'R"i'.

(For the proof,

sce

w.

zrpr{ÐR

[10],

^{theorern 3.8.)}

Corollary. _{M(Co, Ct,}

_R,,)

:

_{cap (Co,}

_{Cr, R).}

It is

enough

to

observe

that

(5) _{M(co, cr,}

_R,')

:

¡,1(x).

rndeed,

iÎ l'is

the

family of

arcs wrrich joirr-co _and.

c, andcri'R,, _the',

clearly, I _{C X,} hence,

proposition -S "yields

(6) _{M(cr, cr,}

_R")

: M(t) s

^M(x).

t26 Þ. canaMnñ

10 I\IA1'FIEMATICA

_

RPVUÍT DiANALYSE ^NUMÉRIQUÉ

ET DE THEORIE Dtr L'APPROXIMATION

r,'aNALySENUMÉRIQ*'ät-ir:i#ä:litt"ljrlu^pp*oxlMArloN

RE¡'ERENCES

[1 j C a 1' a m R. R. a u, S. S. Române ^p Rornâne,. e t r a, n-d,inrcnsional quøsiconfortnøl 1074.Bucure¡ti t9ôe; _,,ÀLacus È;"*",,, xËít (ecf) rtaþþings. Þdit. an¿ Edit. Acad.¡\cad.

'

ri;:;i;:i",y;;h;,#'ï"#nlÍ'il åli:lr,

^_

uas,iconformal møþþi,ngs in ^sþace. Trans. Ame¡, cient of quøsiconformatity. Acta Math. ll4, I maþþings in thyee sþøce. (pretintinary report).

( I e57).

rresþotr.dence

_of . a q-uasi,aottforntal ntaþþittg in Army. 1.he Univ. of Wjsconsin. uni,,IeSfrrii_

1e63).

n.taþþings

,in

sþace. Ãnn, Acad. gci. ["orr.

ses

of

d,ifferenti.able Junctions. Arkiv för Math.

nd p_caþaci.ty. Ilichigan t,Iath. J. ^{lG, 43_S1}

[ll]

^{3 o p u} AxaÃ

u, B. Havx

(1962)._{oôuo¿o rcìacca} omoôpameuutí s npocmpaHcnee, ficxa.

[12] "On

¿lrcttettmo, nocpeôcmaotrt ceqcttuti.l{otr,r. Axa^ Hayx

CCCP

_).

ANTIPROXIMINAL ^SËTS IN BANACH SPACES OF

CONTINUOUS ^FUNCTIONS

by

s. coBzas

(Cluj-Napoca)

Receiued l.VII.1974.

I.et x

be a normed linear ^space,

LI

attof'.-void. subset

of x and

x e

x.

We

use

the following

notations:

rt(x,

M) : _inf

_{l

_lx - ^{yll: )'}

^e

^M} -

the distance rro'rn x

to

IVI ;

P*(*) : þ

^e

^M:||x - yll:

^d'(x,

^I,l)} - ^the

^metric

plojectiou;

E(IVI)

: _{x

e

X: P*(x) A

^Ø}_

'r'lre set ¡Z is òatled þroxim.inal ^í1 E(

: M

(see

tsl) -

Follor,ving v.

of

all

normed linear ^{spaces r,vhich} set, and

by N,

^{the class}

^{of all}

^norme

oroximinal

bounded closed convex

Lr,

^.u,. ot,rJg

16], a

Banach spacc ^b

is rrou-reflexive.'I'he

chatacterizati

is

rnore complicated. The

first

exam

given by

M. EDDLSTETN

and ¡.

lho."u that

^{the space c} belongs al

tion; all

undefined terms

are

^as

ces

X, Y

are isomorþhic (notation phic bijection g :

X -' ^Y.

^{'l'he map}

^If

further, llqiø)ll : _lløll for

^all

ornorþkisnl, and we say

tlral

the ^spaces ù'c (notatíon'

X:Y)'

I,et

S be a compact Hausdorff spate and C(S) be the.!aga-cþ ^{space of}

all real -

valued, cåntinuous functidns defined òri S.

en

id,eøl

I

^{is a linear}

;;br;;"" ^or õijf't""tt tt'"t

^{xy e}

l r.or all x

e

I

^a''ð'

_v

^e

^{c(s)' rf S' is}

^a