EARS INGE HEDBERG Removable singularities and condenser capacities

Removable singularities and condenser capacities

EARS INGE HEDBERG

1. Introduction

I n [3] Ahlfors a n d B e u r l i n g g a v e a c h a r a c t e r i z a t i o n in t e r m s o f e x t r e m a l distances o f t h e r e m o v a b l e singularities for t h e class o f a n a l y t i c f u n c t i o n s w i t h f i n i t e Dirichlet integral. T h e p u r p o s e of this p a p e r is to generalize a n d e x t e n d this r e s u l t in several directions.

L e t A(G) be a class o f a n a l y t i c or h a r m o n i c f u n c t i o n s d e f i n e d for a n y o p e n set G in t h e c o m p l e x p l a n e C or in d-dimensional E u c l i d e a n space Il a. W e s a y t h a t a c o m p a c t set E is r e m o v a b l e for A if for some o p e n G c o n t a i n i n g E e v e r y f u n c t i o n in A ( G ~ E) can be e x t e n d e d to a f u n c t i o n in A(G).

I f G c {3 we d e n o t e b y ADP(G), :p > O, t h e class of a n a l y t i c f u n c t i o n s f in G such t h a t f G Ef'(z)Epdm(z) < 0% m being p l a n e L e b e s g u e measure.

Ahlfors a n d B e u r l i n g [3] p r o v e d t h a t a set E is r e m o v a b l e for A D 2 i f a n d o n l y if t h e r e m o v a l o f E does n o t c h a n g e e x t r e m a l distances. (See T h e o r e m 7 below for a m o r e precise s t a t e m e n t . ) T h e i r p r o o f uses t h e c o n f o r m a l i n v a r i a n c e o f t h e class A D 2, so it does n o t i m m e d i a t e l y generalize to p # 2.

I n o r d e r t o t r e a t this p r o b l e m for all p, 1 < p < oo, we f i r s t r e f o r m u l a t e it b y m e a n s o f d u a l i t y as a n a p p r o x i m a t i o n p r o b l e m in t h e S o b o l e v space W~, q ~ :p/(p - - 1). ( T h e o r e m 1.) I t is t h e n q u i t e e a s y to give a n e c e s s a r y a n d sufficient c o n d i t i o n for a set to be r e m o v a b l e for ADs, 1 < p < oo. Our c o n d i t i o n ( T h e o r e m 4) is t h a t E be a >>null set>> for a c e r t a i n c o n d e n s e r c a p a c i t y , w h i c h for p ~- 2 is ~)eonjugate>> to t h e e x t r e m a l d i s t a n c e c o n s i d e r e d b y Ahlfors a n d B e u r l i n g ( L e m m a s 3 a n d 4).

Our m a i n result, h o w e v e r , is t h a t this n e c e s s a r y a n d sufficient c o n d i t i o n is e q u i v a l e n t t o a local, a p p a r e n t l y m u c h w e a k e r c o n d i t i o n ( T h e o r e m 6). T h u s , con- denser c a p a c i t y has an i n s t a b i l i t y p r o p e r t y similar to t h e i n s t a b i l i t y t h a t a wide class o f capacities is k n o w n to h a v e .

T h e a b o v e a p p r o x i m a t i o n p r o b l e m in W~ can j u s t as well be f o r m u l a t e d a n d solved in d dimensions, a n d this d-dimensional p r o b l e m is also e q u i v a l e n t t o a

1 8 2 L A R S I ~ G E H E D B E R G

problem of removability. I n fact, instead of considering functions in AD~ in plane regions it is equivalent to consider their real parts, i.e. real valued harmonic functions u with vanishing periods and such t h a t

f

^{[grad u ledm} < co. This class of functions can also be defined in R d.

Thus, following Rodin and Sario [25; p. 254] we denote for G c R ~ by FDe(G) the class of real valued harmonie functions u in G such t h a t

f.

^lgrad ul~dme <

(m,i is d-dimensional Lebesgue measure), and such t h a t u has no flux, i.e.

fc,d =fo /O dS---o

for all ( d - - 1 ) - c y c l e s c in G. B y t h e d e R h a m theorem the last condition is equivalent to saying t h a t the (d -- 1)-form *du is exact. Since our results extend to d dimensions at little extra cost, this is the generality in which we shall t r e a t the removability problem.

For comparison we also include characterizations of the removable singularities for the spaces ALe of analytic functions in plane L p and (which is equivalent for d = 2) HDe of harmonic functions u with

f

^{Igrad u]edmd} < ~ . (Theorems 1 and 2.)

There are also possibilities of generalizing to solutions of more general elliptic equations, but we leave these aside. Cf. H a r v e y and Polking [11].

I n the last section of the paper we discuss mainly the case of linear sets in the plane. We give conditions for removability (Theorem 13) which for p = 2 improve a theorem of Ahlfors and Beurling [3].

For more information about removable singularities we refer in addition to [3]

to the book by Sario and Nakai [27], and the bibliography given there. Some results on _FD 2 are due to Y a m a m o t o [31].

2. Preliminaries

For a n y open G in R d and 1 _<q < oo we denote by W~(G) the Sobolev space of locally integrable real valued functions f on G whose derivatives in the distribution sense are functions in Lq(G). We write Wxq(R d) = W~. As usual C~(R ~) = C~ and C~(G) denote the infinitely differentiable functions with compact support (in G). When G is bounded [[]gradfIlILq(G ) is a norm on C~(G) by the Poincar6 inequality, and the closure of C~(G) in this norm is denoted

W~(e).

o

I t is well known t h a t for q > d functions in Wq~ are continuous b u t for q _< d this is no longer true. The deviation from continuity is measured by a q-capacity which in a natural way is associated with the space W~. For compact sets K this q-capacity is defined by

Cq(K) = inf f lgrad 09 [qdm,

~o d R d

I ~ E M O V A B L E S I N G U L A R I T I E S A N D C O N D E N S E I ~ C A P A C I T I E S 183 where the i n f i m u m is t a k e n over all co E C [ such t h a t ~o > 1 on K. I n the case q < d the co are restricted to C[(B) for some f i x e d large ball B which contains K in its interior.

The definition is e x t e n d e d to a r b i t r a r y E C 1R e b y setting C~(E) = sup {C~(K); K c E, K compact}.

F o r q > d only the e m p t y set has q-capacity zero. Ca is a conformal invariant in R e, called conformal capacity. C2 is classical Newtonian (logarithmic for d = 2) capacity.

F o r an account of some of the properties of C~ and Wql we refer to T. B a g b y [4]. See also Meyers [22], Adams and Meyers [1], Maz'ja and H a v i n [21], H e d b e r g

[~5].

3. The dual problem

We show b y means of d u a l i t y t h a t the problem of characterizing the removable sets for F D p (ADP), 1 < p < 0% is equivalent to an a p p r o x i m a t i o n problem in W~, q = p/(p -- 1). F o r comparison we include the corresponding characterization for HDP (ALP). F o r p = 2 the result is found in R o y d e n [26].

C ~ R a ~ C~(G) the subset of Co ~ or C~(G) which We denote b y E( ) = CE or

consists of functions r such t h a t g r a d e belongs to C~(CE). I.e. C~ is the subalgebra of C~ which consists of functions t h a t are constant on each component of some neighborhood of E.

THEOREM 1.

a)

E is removable for HDP or ALP, 1 < p < co, i f and only if

o

C ~ ( G ~ E) is dense in W~(G), q = p / ( p - - 1), for some bounded open G ~ E.

b) E is removable for FDP or A D p, 1 < p < 0% if and only if C~(G) is dense

o

in W~(G), q = p/(p -- 1), for some bounded open G ~ E.

Proof. We prove the t h e o r e m for HDp and FDP. F o r A L p and ADP the proof is simpler and is omitted.

o

Suppose C~(G) is dense in W~(G). T h e n d e a r l y rodE= O, so if u C HDP(G ~ E) the partial derivatives ui are defined almost everywhere in G and belong to LP(G). We claim t h a t there is a distribution T in G whose partial derivatives D~T equal ui. B y a t h e o r e m of L. Schwartz a necessary and sufficient condition for this is t h a t t h e distribution partial derivatives Dju~ satisfy Diul -~

Dfuj for all i and j. I n other words, we claim t h a t

fG u,DjCdm =

u~D,r for all r C C~(G). B u t for all r E C~(G) it is easily seen t h a t

G G \ E G ~ E G \ E G

1 8 4 LAICS I N G E I t E D B E R G

I f

C~(G)

is dense in

C~(G)

the assertion follows from Hhlder's inequality. Thus there is a distribution T in G which coincides with u in G ~ E. B y a n o t h e r t h e o r e m o f L. Schwartz T is a function in Wf(G), i.e. u has an extension to

Wf(G), in particular u is locally in

L v.

Suppose now

C~(G ~ E)

is dense in VVlq(G). F o r a n y

u E HDP(G ~ E)

and

r E C~(G ~, E)

we have b y Green's formula

f u A C d m = - f grad u-grad Cdm= f cAudm=O.

G \ E G ~ E G \ E

I t follows t h a t

f . ~ r =

^-

f~

grad u . grad

Cdm= 0

for all r E

C[(G).

B y

Weyl's lemma u is harmonic in G, i.e.

u E HDP(G). o

T h e n let

u C FDP(G ~ E)

and suppose t h a t

C~(G)

is dense in

W~(G).

L e t

r E C~(G).

The support of grad r is a compact set in G ~ E. Only finitely m a n y components of its complement z9 intersect E, and we denote these b y s On 9 each of these r equals a c o n s t a n t , a , L e t y b e a ( d - - 1)-cyclein ( G ~ E ) N-(2 which is homologous t o zero in G ~ E, and set 7 f~ ~9i = y~. B y Green's formula

fuACd.~=-- f gradu.gradCdm= f Cdudm-- f ~(*~)=

G \ E G ~ E G ~ E 7

7i

I t follows as before t h a t

f. uACdm

= 0 for all r C

C~(G),

so again u is harmonic

in G, i.e.

u E FDe(G). o

Suppose conversely t h a t

C~(G ~ E)

is n o t dense in

Wql(G).

T h e n there is a

o

non-zero distribution T with support in E and continuous on

W~(G).

Clearly S = T * Ix] 2 d is harmonic in G ~ E . We claim t h a t S belongs to

HDP(G~E).

There are functions u, E

LP(G)

such t h a t for r e

C:(G) (T, r = f~,,u,D,r

Then (S, r = (T, r 9 Ixl 2-d) =

f ~, u,D,(r 9 [x?-')dm.

^Thus

r = - ( s . . . r = - f y

[xl2-d)dm.

J

B y the CMder6n-Zygmund t h e o r y

IlDjD~(r * ]xf-d)[ILq <

Cl[r so I t follows t h a t

DjS

E L ~, i.e.

S E W~(G).

I f in addition

C~(G)

is n o t dense in

i(DjS, r --< c Z il~&Pllr

(See also T h e o r e m 2 below.)

W~(G)

o there is a distribution T with the same properties as above and which also annihilates

C~(G). S = T * Ixl 2-~

again belongs to

W~(G)

and to

HDP(G~,E),

and we claim t h a t S now has vanishing periods, i.e.

S E FDP(G ~ E).

~ E 3 / I O V A B L ] ~ S I ~ T G U L A R I T I E S A : N D CO3'r C A P A C I T I E S 185 L e t r E C~(G) a n d let y be a (d - - l ) - c y c l e as a b o v e o f f t h e s u p p o r t o f g r a d r T h e n (S, A t ) = (T, [x[ 2-d 9 A t ) = C(T, r = 0, so b y G r e e n ' s f o r m u l a

G V G ~ E Y

Since t h e ai are a r b i t r a r y , 9 d S = 0 for all i. I t follows t h a t f~ * dS = 0 for all (d - - 1)-cycles in G ~ , E .

T h e following corollaries are obvious.

COROLLARY 1. Let A be H D ~, ALp, FDP or ADv for 1 < p < o0, and let E be compact. I f all functions f in A(G ~ E) can be extended to A(G) for some bounded open G containing E, then the same is true for all such G.

COROLLARY 2. The property of being removable for HDP, A L e, F D p or ADP, 1 < p < co, is local, i.e. E is removable if and only if every x in E has a compact neighborhood whose intersection with E is removable.

T h e o r e m 1 says t h a t p r o v i n g t h a t E is r e m o v a b l e for FDP (or A D Q is

o

e q u i v a l e n t to p r o v i n g a )>Stone-Weierstrass property)) for C~(G) in W~(G). F o r

o o

q > d W~(G) is a n algebra, b u t for q _ ~ d it is n o t . T h e s u b s p a c e Wq~(G) f l L ~~

h o w e v e r , is a B a n a c h a l g e b r a u n d e r t h e n o r m llfJ] = J]fJ]~ q - { f lgradflqdm} l/q, a n d t h e closure o f C~(G) in t h e same n o r m is again an algebra, w h i c h we d e n o t e

o

b y ~ ( G ) . I t consists o f t h e f u n c t i o n s in W~(G) t h a t are c o n t i n u o u s a n d t e n d t o zero a t t h e b o u n d a r y . Such ~)t~oyden algebras)> h a v e been s t u d i e d b y R o y d e n a n d r e c e n t l y b y L. Lewis [18] a n d J. L e l o n g - F e r r a n d [17]. W e shall l a t e r p r o v e a )>Stone- W e i e r s t r a s s theorem)) ( T h e o r e m 12) for t h e s e algebras also, i.e. we shall c h a r a c t e r i z e t h e c o m p a c t sets E such t h a t C~(G) is dense in c)J~q0(G), 1 < q < oo. A n e c e s s a r y c o n d i t i o n is o b v i o u s l y t h a t C~ is p o i n t s e p a r a t i n g , w h i c h is t h e ease i f a n d o n l y if E is c o m p l e t e l y disconnected.

4. Necessary and sufficient conditions for removability

T h e r e m o v a b l e sets for HDP a n d ALP, 1 < p < 0% allow a simple charac- t e r i z a t i o n w h i c h is u n d o u b t e d l y k n o w n . H o w e v e r , it does n o t seem to be g i v e n e x p l i c i t l y in t h e l i t e r a t u r e , e x c e p t for p = 2 w h i c h is classical (see e.g. Carleson

[6; Th. V I I : I ] ) , so for c o m p l e t e n e s s we include it here.

1 8 6 LARS I N G E H E D B E R G

T~EO~EM 2. A compact set E C R d is removable for H D s (ALe), 1 < p < 0%

i f and only if Cq(E) = 0 (i.e. i f and only if E = 0 in the case p < d/(d -- 1)).

Pro@ First assume Cq(E) > 0. Then there is a measure # > 0 supported b y E such t h a t the potential U~(x) = f Ix -- yll-dd/~(y) is in L~or (see e.g. Meyers [22; Th. 14]. The potential V~(x) = f Ix -- yl2-dd#(y) ( f log 1/[x -- yldtt(y) in case d = 2) is harmonic in C E, and its gradient in the distribution sense is Yl #(y), which is majorized by Uf(x). Thus U~

grad U~(x) = f grad~ lx -- :-dd

belongs to HDS(G ~ E) for all bounded G, b u t clearly not to HDP(G). Note however t h a t U~ C Wf(G).

I n the other direction the theorem follows from Theorem I and the following lemma.

LEMlV[A 1. Suppose E is compact and Cq(E) = O, 1 < q < oo. Then C[(G ~ E)

o

is dense in Wql(G) for all bounded open G.

Proof. I t is enough to show t h a t a n y q~ in C~(G) can be approximated. L e t e > 0 be arbitrary. There exists an r E C~ ~ such t h a t ~ = 1 in a neighborhood of s u p p ~ [ ' l E and f Igrado)Iqdm < e . Then r162 C C ~ ( G ~ E ) , and

/ igrad r lqdm =

f

grad ~o d- ~o grad r lqdm <_

__< 2 q-1 m a x Ir + 2 ~-1 max Igrad r q

f

lolqdm <_ eonst. 9 e

G

by the Poincar6 inequality.

The removable sets for F D s (and ADs) cannot be characterized in such simple terms. However the following simple theorem m a y be worth recording.

T H E O R ~ 3. Let E c C be compact and let G ~ E be open. Then every function f in Wf(G) f ' I A D S ( G ~ E ) , I < p < 0% belongs to ADS(G) i f and only i f m2E = O.

Pro@ Let f e Wf(G) fl ADS(G ~ E), G D E. Then Of/~5 C LS(G) and vanishes on G ~ E. I f m2E = 0 it follows by Weyl's lemma t h a t f is analytic in G.

Conversely, suppose m2E > 0. Then f(z) = f E (~ -- z)-ldm2(~) is analytic off E, a n d f is not identically zero, because limz~ ~ Izf(z) l =- m2E. B y the Calder6n- Z y g m u n d theorem f C Wf(G) for all p > 1.

I n order to characterize the removable sets for F D s and A D s we introduce a kind of condenser q-capacity, q = p / ( p - 1).

: R E M O V A B L E S I N G U L A R I T I E S A N D C O N D E N S E I ~ C A P A C I T I E S 187

Definition. Let R be a d-dimensionM, open rectangle with, say, sides parallel to the coordinate planes. Let E be compact. The d condenser q-capacities of R with respect to E are

I~O(R/E) = inf f lgrad ~o lqdm, i = 1, 2 , . . . , d,

r , ] n

where the infimum is taken over all ~o E C~ such that ~o(x) = 0 on one of the sides of R parallel to the coordinate plane xl = 0, and o)(x) = 1 on the opposite side. I f E = O we write F~i)(R), which is the ordinary condenser q-capacity of R.

B y the usual strict convexity and variation argument one sees t h a t there is a unique extremal function u in W~(R) which satisfies the equation

d i v ( [ g r a d u l q - 2 g r a d u ) = 0 in R ~ E ( A u = O for q = 2 ) .

When E = O the solution is linear. As is well known it follows t h a t if R has edges of length a~ perpendicular to the plane x i = 0, then / ' ~ 0 ( R ) = ai-qmdR.

THEOREM 4. Let E be a compact set in tl d. For E to be removable for F D p (ADe), 1 < p < oo, it is necessary and sufficient that

l~(~)(R/E) = F(~)(R), i = 1, 2 , . . . , d, q = p / ( p -- 1), for some open rectangle (or all open rectangles) R containing E.

Proof. Suppose first E is removable. Then b y Theorem 1 C~ is dense in C~

in the W~ sense. I t follows immediately t h a t the capacities I~O(R/E) and F~)(R) are the same for all R containing E.

Suppose conversely t h a t there is a rectangle R containing E such t h a t F~O(R/E) = F~O(R), i = 1, 2 , . . . , d. B y Theorem 1 it is clearly enough to show t h a t the restriction of C~ to R is dense in W~(R). For each i, i = 1 , 2 , . . . , d , there is a sequence {r in C~, such t h a t the r take the prescribed b o u n d a r y values, and such t h a t f R Ig rad ~ nlqdm ~ I~O(R/E) = I~')(R)" B y strict convexity {r tends strongly in W~ to the extremal function for /~)(R), which is of the form az~ ~ b. I t follows t h a t all polynomials of the first degree can be approximated in W~(R) b y functions in C~. Since W~ is closed under truncation (see e.g. Deny and Lions [7; Th. 3.2]) we can also assume t h a t the approximating functions are bounded, and the theorem then follows from the following lemma and the density of polynomials in Wql(R).

L~MMA 2. Suppose r and ~f belong to W~(R), and that IIr ~_ M , and }I~vI[o ~ ~ M . Let {r ~ and {%}~ be sequences such that

188 L A I R S I N G E I ~ E D B E R G

lim

f

igrad (r -- r = O,

R

lim

f/grad

^{(~ -- %)Iqdrn}

n ---~ oo R

= O, ]ir ~ M , and IlF.IJ~ < 3 1 . T h e n there is a subsequence {r lim,_.~

f.

^{fgrad (r - - r} qdm = O.

such that

Proof. grad (r -- r = C g r a d w + w grad r -- r grad w- -- % grad r = (r -- r grad W ~- r grad (~ -- ~,) ~- (~ -- %) grad r -~ ~ grad (r -- r

f [ r grad (W -- W-)l qdm ~ Mq f [grad (W -- %)i d m - ~ O, and similarly for the fourth term. We can choose a sequence {n~} such t h a t both r and %i converge pointwise almost everywhere to r and W. The convergence to zero of

f [ ( r -- r grad ~l~dm and f 1(~ -- %~) grad r follows from the Lebesgue convergence theorem.

The following corollaries to Theorem 4 are due to Ahlfors and Beurling [3] for p = d = = 2 .

COROLLARY 1. E is removable for F D s (ADs), 1 < p < ~ , i f the projection

of

E on each of the coordinate axes has one-dimensional measure zero.

C O R O L L A R ~ ~ 2. Let E b e a Cartesian product of a compact one-dimensional set with itself, e.g. a d-dimensional Cantor set. T h e n E is removable for F D s (ADs), 1 < p < 0% i f and only i f maE = O.

Corollary 1 follows easily from the theorem. The sufficiency in Corollary 2 then follows. The necessity was already observed in the course of the proof of Theorem 1.

T h a t the vanishing of rodE is by no means sufficient for removability in general is well known (see Ahlfors-Beurling [3; Th. 14]).

Since removability is a local property it is desirable to have a local condition for removability in terms of condenser capacity. For this reason we need to apply the definition of F~O(R/E) in a situation where E is allowed to intersect the b o u n d a r y of R.

Now, one complication arises from the fact t h a t when q < d - 1 there are continua with zero q-capacity. (For q > d -- 1 this cannot happen, because for a continuum E with diameter ~ one t h e n has Cg(E) > const. 8. (See e.g. Maz'ja [20; L e m m a 5].)) This :means t h a t a set E can be removable for FDP although there are rectangles R such t h a t no function in C~ can be 1 on one side of R and 0 on the opposite side. On the other hand, no continuum E with Cq(E) > 0

R E M O V A B L E S I 2 C G U L A I ~ I T I E S A N D C O : N D E k N S E I ~ C A P A C I T I E S 189 can be r e m o v a b l e for FDP. I n o r d e r to see this one o n l y has t o set # =/~1 --/z2 w h e r e /~ a n d #2 are p o s i t i v e m e a s u r e s w i t h s u p p o r t in d i f f e r e n t p a r t s o f E such t h a t #~(E) = #2(E) a n d such t h a t U~'~(x) ~ f [ x - - yll-~d#r i ~ 1, 2, are in Lp. This is possible b y M e y e r s [22; Th. 14]. T h e n Uf(x) ~ f Ix - - yl2-dd#(y) is in F D p for some n e i g h b o r h o o d of E a n d does n o t v a n i s h identically.

T h e r e f o r e , if a set E is r e m o v a b l e for FDP it has to be c o m p l e t e l y d i s c o n n e c t e d a f t e r a set o f q - c a p a c i t y zero has b e e n r e m o v e d . W e can t h u s w i t h o u t loss o f g e n e r a l i t y a s s u m e t h a t E is c o m p l e t e l y disconnected.

W e p r e f e r n o w t o w o r k w i t h capacities o f ring d o m a i n s i n s t e a d of rectangles.

B y a ring we m e a n a b o u n d e d d o m a i n G such t h a t t h e c o m p l e m e n t C G has e x a c t l y t w o c o m p o n e n t s .

Definition. L e t G be a ring in R d, a n d let E be closed. T h e c o n d e n s e r q - c a p a c i t y o f G w i t h r e s p e c t to E is

I~(G/E) = i n f f Igrad m [~dmd,

09 " / G

w h e r e t h e i n f i m u m is t a k e n o v e r all m E C~ such t h a t m = 1 c o m p o n e n t o f C G a n d m = 0 on t h e u n b o u n d e d c o m p o n e n t .

T h e c o n d i t i o n for r e m o v a b i l i t y t h e n t a k e s t h e following form.

on t h e b o u n d e d

THEOREM 5. Let E C R d be compact and completely disconnected. T h e n E is removable for F D e (ADe), 1 < p < Go, i f and only i f _Fq(G/E) = Fq(G), q = p / ( p - - 1), for all rings G.

T h e n e c e s s i t y is p r o v e d as in T h e o r e m 4. A p r o o f of t h e s u f f i c i e n c y of t h e con- d i t i o n can easily be given, b u t since this also follows f r o m T h e o r e m 6 below, we do n o t give it here.

T h e main r e s u l t of this p a p e r is t h a t t h e c o n d i t i o n in T h e o r e m 5 is e q u i v a l e n t t o a n a p p a r e n t l y m u c h w e a k e r condition. T h e s i t u a t i o n is c o m p a r a b l e t o t h e ))in- stability)) of a n a l y t i c c a p a c i t y a n d p o t e n t i a l t h e o r e t i c capacities (see e.g. Vitu~kin [30; Ch. VI: 1], GonSar [10], H e d b e r g [14; p. 162] a n d [15; Th. 9]).

W e d e n o t e t h e a n n u l u s { y C R d ; r < l Y - - X ] < R } b y A ( x , r , R ) .

THEOREM 6. Let E c R d be compact with m~E ~ O, and let 1 < q < Go.

Suppose that for all x E E , with the possible exception of a compact set Eo C E with Cq(Eo) = O, there exist a number K ( x ) < oo, and sequences {rn(x)}~ and {R,,(x)}~

decreasing to zero in such a way that

1 Rn(x)

1 + K(x) < to(x) < K(x), (1)

and so that

190 LARS ^{I N G E} HEDBERG

Fq(A(x, r,(x), R,(x))/E) < K(x) . F~(A(x, r,~(x), I~(x)) (2) for all n.

o

Then C~ is dense in W~, i.e. E is removable for FDP (ADP), p = q / ( q - 1), and I~(G/E) = I~(G) for all rings G.

The proof of this theorem is somewhat technical. We shall therefore first discuss how Theorems 4 and 5 compare with the characterization of removable sets for A D S given b y Ahlfors and Beurling, and then give the proof of Theorem 6 and some related results.

5. C o m p a r i s o n w i t h t h e A h l f o r s - B e u r l i n g t h e o r e m s

I n [3] Ahlfors and Beurling gave a characterization of the removable sets for A D S in terms of extremal lengths. L e t the plane compact set E be contained in a rectangle R with, say, sides parallel to the coordinate axes. We denote the extremal distance between the vertical sides of /~ with respect to R ~ E b y ~(~)(R ~ E), and between the horizontal sides by 2(2)(R ~ E). Then the theorem of Ahlfors a n d Beurling is the following.

T~IEOm~ 7. E is removable for A D 2 if and only if for some rectangle (or all rectangles) R containing E in its interior

X(')(R ~ E) = 2(O(R), i = 1, 2.

The sufficiency of this condition was obtained as a consequence of the following theorem, also proved in [3].

T ~ O g E M 8. E is removable for A D ~ if and only i f every region which is con- formally equivalent with 0 E has a complement of zero area.

We shall show directly t h a t Theorem 4, when specialized, is equivalent to Theorem 7, a n d we thus obtain new proofs of Theorems 7 and 8.

We first define, for a d-dimensional rectangle /~, another condenser p-capacity with respect to E.

Definition. F(0(R ~ E) = inf~ f R \ E ]grad ~o ]Pdx, i = 1, 2 , . . . , d, where the infimum is t a k e n over all co E C~(C E) such t h a t ~o = 1 on one of the sides of R parallel to the hyperplane xl = 0, and co = 0 on the other one.

Note t h a t always

F(O(R ~ E) < F(')(R) < F(~)(R/E).

The bridge between Theorems 4 a n d 7 is provided by Lemmas 3 and 4 below.

T h e y are of course not new (see the remarks below), but for the reader's convenience

I ~ E M O V A B L E S I l g G U L . k I ~ I T I E S A N D C O N D E N S E R , C A P A C I T I E S 191 we give some details. W e a g a i n a s s u m e t h a t E is a p l a n e c o m p a c t set c o n t a i n e d in t h e i n t e r i o r o f a r e c t a n g l e R.

L n ~ t 3. I'(~)(R/E)= 1/F(2J)(R'~ E), i = 1, j = 2 or i = 2, j = 1.

Proof. L e t R h a v e v e r t i c a l sides J ~ l a n d F~ of l e n g t h a a n d h o r i z o n t a l sides F2 a n d av~ of l e n g t h b. F i r s t a s s u m e t h a t E is b o u n d e d b y f i n i t e l y m a n y a n a l y t i c c u r v e s .

T h e sets o f f u n c t i o n s o) c o m p e t i n g in t h e e x t r e m a l p r o b l e m s d e f i n i n g /~1)(R/E) a n d /'~:)(R ~ E) are c o n v e x in W~(R) a n d W~(R ~ , E ) r e s p e c t i v e l y . T h e r e f o r e t h e r e are u n i q u e e x t r e m a l f u n c t i o n s u in W~(R) a n d v in W ~ ( R ~ E ) . B o t h are h a r m o n i c in _ R ~ E , u = 0 on F1, u = 1 on F'I, a n d v = 0 on F : , v = 1 on F ' 2. u is clearly c o n s t a n t o n e a c h c o m p o n e n t o f t h e i n t e r i o r o f E . B o t h u a n d v can be c o n t i n u e d h a r m o n i c a l l y b y r e f l e c t i o n o v e r t h e b o u n d a r i e s , so t h e y h a v e c o n t i n u o u s b o u n d a r y v a l u e s a n d b o u n d a r y derivatives.

T h e u s u a l v a r i a t i o n shows t h a t

f. gradu-gradCdm

= 0 for all r in C~

w i t h s u p p o r t o f f F1 O F'I. B y G r e e n ' s f o r m u l a

f ^4(,

^{a u ) = r = 0,}

OR U OE OR U ON

for all such r a n d it follows t h a t Ou/On-~ 0 on F 2 U F~, a n d t h a t

f (Ou/On)d

^{= 0} for e v e r y c o m p o n e n t c o f OE, i.e. u has no periods in _R ~ E . Similarly g r a d v 9 g r a d Cdm = 0 for all r in C ~ w i t h s u p p o r t o f f F 2 U F ~ . T h u s

foRooEr

for all such r a n d it follows t h a t a v i a n = 0 o n OE a n d on F 1 U F 1. I

Moreover, G r e e n ' s f o r m u l a

f Igrad ( u l - u2)[2dm =

f (u, .)

0 ( u l - - u2) On ds

G OG

shows t h a t b o t h u a n d v are u n i q u e l y d e t e r m i n e d b y t h e s e b o u n d a r y conditions.

Since u does n o t h a v e a n y periods it has a single-valued c o n j u g a t e h a r m o n i c f u n c t i o n u* in R ~ E . T h e f l o w lines for u are level lines for u*, a n d c o n v e r s e l y . T h e r e f o r e Ou*/On= 0 on a E a n d on F 1 U F ~ , a n d u * = e o n s t , on F 2 a n d on F ' 2. W e can a s s u m e t h a t u* ---- 0 on F2, a n d t h e n it follows f r o m t h e u n i q u e n e s s of v t h a t u * = cv for some c o n s t a n t c.

Again a p p l y i n g G r e e n ' s f o r m u l a we f i n d

f

IV~I)(R/E) = lgrad ul2dm = u ~n ds = ds ~- u*(F'~) =- c.

R ~ E OR U OE FI"

On t h e o t h e r h a n d

1 9 2 LA~S INGE ~ E D B E R G

f

^[grad

^{u 12din}

⁼

}"

^]grad

^{u* 12din}

^{= c 2}

f

]grad v ]2dm =

e2F~2)(R ~ E).

R \ E R \ E ~ \ E

T h u s r ~ : ) ( n / E ) = c = 1/r(~)(R \ E).

Moreover the analytic function

f = u 4- iu*

maps R \ E univalently onto a region bounded by a rectangle S and finitely m a n y vertical slits. The length of the horisontal sides of S is 1 and of the vertical sides c. Clearly /'~I)(R) alri:)(R) = a/b, so ~ >_ alb.

I n order to finish the proof we have to prove t h a t if E = N~ E,, E,+a C E,, then l i m , ~

l~O(R \ E,) = F~O(R ~ E),

and l i m ~ |

I~(~)(R/E,) = I'(O(R/E).

This is easily done by arguments similar to those used by Ziemer [32; L e m m a 3.9].

LnM~A 4. /~(j)(R ~ E) = 1/2(J)(R ~ E), j ---- 1, 2.

Proof.

Assume first, as before, t h a t E is bounded by finitely m a n y analytic curves. Let v be the extremal function defined above (for j =- 2), i.e. v is harmonic in R ~ , E, v = 0 on F2, v ---- 1 on f'2, and 0v/0n = 0 on 0E O F1 O

E'l,

and

f

^Igrad

vl2dm ~-- F(22)(R ~ E).

Then 2(2)(R ~ E) --

' f

[grad

vl2dm.

n \ E R \ E

This is Theorem 4--5 in Ahlfors [2; p. 65], and ~)almost certainly due to Beurling)).

(See [2; p. 81].)

The extension to general E is again easy. Details are found in Ziemer [32;

Th. 2.5.1] or [33; L e m m a 2.3].

I n a similar w a y one obtains a counterpart to Theorem 5. I f G is a ring domain in R d we denote b y

Mp(G ~ E)

the p-modulus of the family of hypersurfaees (curves for d = 2) in G \ E t h a t separate the components of C G. I.e. for

p = d = 2 M 2 ( G ~ E ) = 2 ( G ~ E ) -~,

where 2 ( G ~ E ) is the extremal length

of this family of curves.

THEOlCEM

9. I f E C (~ and G is a ring then F2(G/E ) = M2(G ~ E) -1.

E is removable for A D 2 if and only if

M2(G \ E) = l~2(G)

for all rings G.

Thus

Remarlcs.

Capacities of the types F(i)(R ~ E) and

Fp(G ~ E)

have been studied extensively. Thus L e m m a 4 is a special case of a theorem of J. Hesse [16]

who extended earlier work of Beurling, Fuglede [8], Gehring [9], Ziemer [33] and others. See also Ohtsuka [23]. Nullsets for these capacities have been investigated by Vi~isg~l~ [29] and Ziemer [32; Th. 3.14]. Capacities of the type

I'(~)(R/E)

and

: R E M O V A B L E S I N G U L A R I T I E S A N D C O N D E N S E R C A P A C I T I E S 1 9 3

Fp(G/E) on the other h a n d do not seem to have been studied before. A theorem similar to L e m m a 3 in a general situation on open R i e m a n n surfaces, b u t in terms of conjugate extremal lengths, was given by Marden and Rodin [19]. See also the monograph by Rodin and Sario [25; p. 124]. Generalizations of L e m m a 3 and Theorem 9 to p # 2 and to higher dimensions do not seem to be known. A )>con- jugate~> problem was solved, at least for p ~ 2, by Ziemer [32], and Bardet and Lelong-Ferrand [5], who extended earlier results of Fuglede [8], Gehring [9], and others.

In view of these results the following conjecture seems reasonable: For a n y compact E and a n y ring G C R e I'q(G/E) ~/q = Mp(G ~ E) -l/p, 1 < p < oo,

6. Proof of Theorem 6

In order to prove Theorem 6 it is by Theorem 1 enough to show t h a t every

o

function in C~~ E0) can be approximated in the W~ q norm by functions in C~.

The proof is by a direct construction of the @proximation functions. The con- struction is similar to one used previously by the author in [13].

oo o oo

We denote the closure of C E in W~ by C~. Various constants independent of x are denoted by C.

LI~sI~IA 5. Under the assumptions of Theorem 6

F~(A(x, r.(x), Rn(x))) < CK(x)~ 'R.(x) e-~, 1 < q < oo.

Proof. I t is well known t h a t

R

{f

Fq(A(x, r, R)) = C t(d-1)/(1-q)dt ,

r

corresponding to the extremal function u ( y ) = C l y - xl (e q)/(1 q). The assump- tion Rn(x)/rn(x) > 1 + 1/K(x) gives the result after a short computation.

We shall start by covering the set E ~ E 0 with balls in a special way by means of the following well-known lemma. See e.g. Stein [28; I. 1.7] for a proof.

LEMMA 6. Let E be a measurable subset of R d which is covered by the union of a f a m i l y of balls {B(x, r)} with bounded diameter. T h e n f r o m this f a m i l y a subsequence {B(xl, ri)}~=l can be selected so that E c [.J~ B(x~, ri) but the balls B(xi, ri/5) are all disjoint.

For every integer i ~ 1 we set E ~ = { x C E ; i <_K(x) < i + 1}, E = [.Ji~o Ei, where by assumption C~(Eo) = O.

SO

194 L A ~ S I N G E H E D B E ~ G

F o r c3> 0 we set G~ ---- {x; dist (x, E ) < 5 } . Since

m a E = O

b y a s s u m p t i o n , h a v e lim~+ 0

maG ~

= O. W e choose a n a r b i t r a r y e > 0 w h i c h will be k e p t

w e

f i x e d , a n d t h e n we choose a s e q u e n c e {~}~=1 such t h a t

~ id+qmeG~i < e.

i=1

}'or x C E i , i > 1, we can also assume t h a t /~n(x) < dl for all n.

I t follows f r o m L e m m a 6 t h a t e a c h E~, i = 1, 2 . . . can be c o v e r e d b y a (possibly infinite) sequence o f balls

B(xi, rj), xj E El, rj = r~(xj),

so t h a t

B(xi, Ri) C G~, Rj = R~(xj),

a n d so t h a t

<_ (i + 1) <

J J

T a k i n g t h e u n i o n o f t h e s e coverings a n d r e e n u m e r a t i n g we o b t a i n a c o v e r i n g of

E ~ E o

b y balls

B(xj, rj)

so t h a t /?j < ~1 a n d

j = l i=1

F o r each o f t h e s e balls t h e r e is b y a s s u m p t i o n a n d L e m m a 5 a f u n c t i o n Cj. C C~

so t h a t C j z 1 on

B(xi,~?.),

C y = 0 o f f

B(xj, Rj),

a n d

f Igrad

C j l ' d ~

< CK(xSRJ- .

T h u s

Rj j

f ^tgrad

CjLqdm < Cs.

(3)

L e t g be t h e f u n c t i o n to be a p p r o x i m a t e d . W e can a s s u m e t h a t 0 _< g < 1 a n d t h a t ]grad gI ~< 1. W e shall c o n s t r u c t an a p p r o x i m a t i o n t o g in an i n d u c t i v e w a y b y using t h e f u n c t i o n s r a n d t h e following l e m m a .

LEM~A 7. I f r and

r

belong to C~, then the functions

m a x (r162

and

rain (r r

belong to C~.

A similar l e m m a was p r o v e d in [13; L e m m a 2] a n d we o m i t t h e p r o o f here.

D e n o t e t h e s u p p o r t o f g b y S. T h e n E I 3 S is c o m p a c t , so we can select a f i n i t e s u b s e q n e n c e

{B(xj, rj)}~=l

t h a t covers E gl S. W e choose V so

0 < ~ ] _ _ < m i n { R j ; 1

< _ j < J } ,

a n d so t h a t 1/7 is a n integer.

T h e n we set

L k = { x ; g ( x )

> k~}, k = 0 , 1 , 2 . . . . , 1/~],

a n d d e f i n e

gk(x)

for each k b y

I%:EMOVABLE S I N G U L A R I T I E S A N D COI~DEIqSEI% C A P A C I T I E S 195

~0, x ~ L k

gk(x) ~-- ~g(x) -- kv, x 6 L k ~ Lk+ 1 [~, x

6

Lk+i.

F o r e~ch j --~ 1, 2, . . . , J t h e r e is a n integer #(j) such t h a t

B(x1, Rj) c L,(i),

b u t

B(x~.,Rj) ~ L,(j)+~,

a n d there is a n integer v(j) > / ~ ( j ) such t h a t B ( ~ , r~) [3 L~(i) # O, b u t B(x~, r~) f3 L(~)+~ = O.

Set

VJ ~- (v(j) --

#(J))vr if

v(j)

> #(j), a n d Y~i ---- Vr otherwise. Since Ig ta d gl --< 1, dist (~ L~, L~+i) >_ V- Thus

(v(j) -- #(j) -- 1)V _< dist (O L,(j)+I,

L~(i) ) < Bj + r~.

Since V <--B~ we h a v e ~ v j < 3 R j r so b y (3)

f

^Igrad

vjIqdm <_ Cs.

(4)

Set

~2 ---- [.J~B(x~,rj),

a n d set e ---- dist (0 f2, E VI S). Define a f u n c t i o n

Z(x)

b y

[0, dist (x, E [3 S) _< ~/2

Z ( x ) = ] 2 / e d i s t ( x , E V I S ) - - 1, e / 2 < d i s t ( x , E 9 1 S ) <

[1, dist (x, E f3 S) >_

T h e n gkz = 0 in a neighborhood of

E,

so

gkg 6 CE,

a n d

gkz =- gk

outside f2.

Now, set T0(x ) ---- m a x {~/(x);

B(xi, Ri) c

L0}, a n d set

ho(x )

---- m a x

{go(X)Z(x),

T0(x)}. T h e n

h o 6 C ~

b y L e m m a 7,

ho(x) >--V

on L1 a n d

ho(x ) = g ( x )

on C L1 \

(supp

T0).

Set

lo(x ) =

min{h0(x), V}- T h e n 10 6 C~, b y L e m m a 7 a n d

lo(x ) ----V

on L1.

N e x t , set Tx(X) ---- m a x {Vi(x);

B(xi, R]) c LI},

a n d set

hl(x ) ----

m a x

{g~(x)x(x),

Tl(x)}. T h e n hi 6 C~, a n d hl(X ) = 0 outside LI. Set kl(x) = m a x {h0(x),

lo(x) +

hi(x)}. T h e n also kl 6 C~, a n d we claim t h a t kl(x ) > 2 v on L 2. I n fact, if x 6 L2, a n d x belongs to some

B(xj, ri)

such t h a t

B(xj, R~)

intersects G LI, t h e n Vj(x) > 2 v, so

ho(x )

> 2 v. I f x 6 L 2 a n d x belongs to some B(xj.,ri) such t h a t

B(xi, Ri) C L 1,

t h e n Vi(x) ~ V so

lo(x ) + h ~ ( x )

> 2 v. I f x 6 L 2 \ 9 , t h e n

hi(x) > _ g x ( x ) z ( x ) = g x ( x ) = v ,

so again

ka(x)

> 2 v. On C L 1 we h a v e

hl(x ) = O,

so k~(x) = ko(x) = ho(X).

Moreover, if x 6 L~, i > 2, a n d x belongs to some

B(xj, rj)

such t h a t

B(xj, Rj)

intersects C L 2, t h e n kj(x) > i~. I n fact, either B(xj,_Rj) intersects CL1, a n d t h e n

Vi(x) >_iv,

so

ho(x ) > i v,

or else

B(xi, Rj) c L1,

a n d t h e n

Tl(X) >

~pj(x) ~ (i --

I)V, a n d

]Cl(X ) ~ 10(X ) -~ TI(X )

~ i v.

Set

ll(x) =

rain {kl(x), 2V} , a n d continue t h e construction in t h e same w a y . Assume t h a t k~_ 1 6 C ~ has been constructed, so t h a t

k~_1(x ) = k~_2(x)

on

G L~_I, k ~ _ l ( x ) > n v

on L~, a n d

k,_l(x ) > i v

if x 6 L , i > n, a n d i if x belongs to some

B(xj, ri)

such t h a t

B(xj, Rj)

intersects C L~.

196 L A R S I N G E H E D B E E G

T h e n set

a n d

ln_~(x) = rain {kn ~(x), nv},

T~(x) = m a x { ~ / x ) ; B ( % ~ ) C L~}, h~(x) = m a x {g~(x)z(x), V~.(x)},

~o(x) = m a x {kn_l(x), Zo_l(.) + h.(x)}.

Clearly

]cn(x ) C C~,

a n d

k~(x)= kn_~(x )

on G L~.

I f x C L ~ + I , we claim t h a t

k , ~ ( x ) ~ ( n J r

1)~. I n fact, if x E L f + l , a n d x belongs t o some

B(xj, rj)

such t h a t

B(xj, Rj)

intersects CL~, t h e n b y t h e i n d u c t i o n hypothesis k~ ~(x) ~_ (n q- 1)~, so ]c,~(x) ~ (n -k 1)~. I f

x E B(x/, rj),

a n d

B(x], Rj) C L.,

t h e n T.(x) > V, so

k~(x) ~_

I~_I(X ) ~- V = (n ~ 1)~. F i n a l l y , i f

x C L~+I ~ .(2,

t h e n

gn(x)z(x ) = g.(x)

= ~, so again

kn(X) ~ (n ~-

1)~.

I f x E L . i > n ~ 1, a n d if x belongs to some

B(xj, rj)

such t h a t

B(xj, Ri)

intersects 0 L~+I, we claim t h a t

k.(x) ~_ i~.

I n fact, either

B(xj, Rj)

intersects C L . , a n d t h e n k,~_l(X) ~ i~, b y t h e i n d u c t i o n h y p o t h e s i s , so

k,,(x) ~ i~,

or else

B(Xj, R/) C L,,

in which c a s e ~l)j(x) ~ (i - -

n)v,

so k~(x) ~

1,_l(x) Jr (i -- n)v = i v.

I f u = lfl~ we f i n d t h a t ]c~+~ = ]c~. W e f i n a l l y set ]c = / q / ~ , a n d claim t h a t /c a p p r o x i m a t e s t h e given f u n c t i o n g.

I t is clear t h a t

]c(x) = g(x)

outside G~ = G O . ~ o r e o v e r , for almost all

x E L n ~ L , + >

n = 0, 1 . . . . , we h a v e either grad]c(x) = g r a d y ~ i ( x ) for some

j,

or grad/c(x) = g r a d

(g,(x)z(x))

(see e.g. D e n y - L i o n s [7; Th. 3.2], a n d

!grad

(g~(x)z(x)) ] =

]g.(x) grad Z(x) ~- Z(x) grad

g(x)] ~_ 3.

Thus

f i g r a d ^(g

]c) ]~dm

] ^f (Ig r a d g] ~- Igrad

kl)~dm CmdGo ~-

Go

-~- C ~j f Igrad y3j[qdm ~ CmdGo -~- Ce

b y (4). Since s a n d

m~Go

are a r b i t r a r i l y small, t h e t h e o r e m follows.

W e also observe t h a t

Ig(x)

-- k(x) l < 3 m a x / R j < 3~1, so ]c also a p p r o x i m a t e s g u n i f o r m l y .

Variants of t h e p r o o f of T h e o r e m 6 also give the following results.

THEOREM 10. a ) S u p p o s e

E is contained in a Cl-submanifold M C R ~ of dimension a, and that m~(E) = O. Then the conclusion cf Theorem 6 is still true if

(2)

is replaced by

Fq(A(x, r.(x), Rn(x))/E) < K(x)Rn(z) ~-q

(5)

for all n.

: R E M O V A B L E S I N G U L A R I T I E S A N D C O N D E N S E I ~ C A I ~ A C I T I E S 197 b) Suppose that the a-dimensional Hausdorff measure A s ( E ) ~ 0 for some

< d. Then the conclusion of Theorem 6 is true i f there exists a compact Eo with Cq(Eo) ~ 0 so that for all x C E ~ Eo there is a K = K(x), 0 < K ( x ) < ], so that

- - Fq(A(x, K R , R ) / E )

lim < oo:

R-~O R S q

Proof. To p r o v e t h e t h e o r e m one o n l y has to m o d i f y t h e covering a r g u m e n t in t h e f i r s t p a r t o f t h e p r o o f o f T h e o r e m 6. W e keep t h e n o t a t i o n f r o m t h a t proof.

I n ease a) we k n o w t h a t l i m ~ 0 ms(G ~ fl M ) = O. W e n o w choose {5;} so t h a t

•

iS+qm~(G~i N M ) < e,

i = 1

a n d t h e n (3) follows as before.

I n ease b) we w r i t e E ' ~ E o = [3~ El, w h e r e E~ is t h e set w h e r e i -1 < K ( x ) < 1 - - i -1, a n d w h e r e _Pq(A(x, K ( x ) R , R ) / E ) < i R s-q for all R < i -1.

W e choose a s e q u e n c e {d/} so t h a t

•

i~+ldl < e

1

a n d c o v e r e a c h Ei w i t h balls B(x;,r~) so t h a t rj. < i -2 can assume, b y d o u b l i n g t h e rj if necessary, t h a t xj C E~.

Rj = r/K(x~), t h a t

('

JR]Fq(A(xj, rj, Ri)/E ) < i ~+1 ~ r? < i~+ld,.

J J

(3) follows, a n d t h e n t h e t h e o r e m follows as before.

a n d ~ i r ~ < bi. W e I t follows, if we set

THEORE~I 11. a) I f in Theorems 6 and 10 a) the function K ( x ) is uniformly bounded, the conclusion is still true i f the hypotheses me(E ) ~-- 0 or m s ( E ) = 0 are removed.

b) Suppose that A s ( E ) < oo for some ~ < d. Then the conclusion of Theorem 6 is true i f for every e > 0 there exists a compact E o with Cq(Eo) < e and a number M < oo s o t h a t f o r a l l x C E ~ E o there i s a K = - K ( x ) , 1 / M < K ( x ) < 1 - - 1/M, so that

lira" l~q(A(x' K R , R ) / E ) < M .

R-->O j ~ s - - q

Proof. W e can no longer claim t h a t

f

^]grad (g -- k)lqdm in t h e p r o o f of T h e o r e m 6 is small, b u t in all cases f Igrad k lqdm is b o u n d e d i n d e p e n d e n t l y o f e, so t h e r e is a w e a k l y c o n v e r g e n t s e q u e n c e of f u n c t i o n s k. B y t h e B a n a c h - S a k s t h e o r e m t h e r e is a s u b s e q u e n e e k (1) such t h a t Kn = 1/n ~ k (i) converge s t r o n g l y . T h e

1 9 8 L A ~ S I N G E ~ E D B E R G

k (0, and therefore the

Kn,

also converge uniformly to g, so g is also the strong limit of the K=.

THEOREM 12. Under the additional assumption that E is completely disconnected the equivalent conditions in Theorems 5 and 6 are necessary and sufficient, and the conditions in Theorems 10 and 11 are sufficient for C~(G) to be dense in the Royden q-algebra c)'lZ~(G) for any bounded domain G ~ E.

I n fact, a modification of the proof of L e m m a 1 shows t h a t C~(G) is dense in c)Jlq(G) if E is completely disconnected and Cq(E) -~ O. Theorem 12 then follows easily, since, as we noted above, the approximating functions in the proof of Theorem 6 actually converge uniformly.

We finally remark t h a t a Baire category argument (see Sario and Nakai [27;

Th. VI. 1. L, p. 371]) shows t h a t if E is compact, and E = [J~ E~, where the Ei are compact and removable for FDe, t h e n E is also removable.

7. Sets on hyperplanes

I n this section we give some further results in the case when the set E is con- tained in a hyparplane, especially when d = 2 and E is a linear set, in which ease we improve a theorem of Ahlfors and Beurling. The discussion also serves to illuminate the difference between the capacities I'~(R/E) and _Pq(R ~ E).

We denote the hyperplane {x ERa; x d = O} by R s-a, and the halfspaces {x E Ild; x~ > 0 (x d < 0)} by R~ (Rd_). Suppose the compact set E belongs to R d-1. I t is well known t h a t the restriction of W q to R d 1 can be identified with

A q , q / ' l ~ d - - l'~

the Besov or Lipschitz space ~11p~l, j, i.e. the space of b o u n d a r y values of harmonic functions u E wq(R~). See e.g. Stein [28; VI. 4.4] and references given there. I t is easy to see t h a t C~(R a) is dense in W~(R d) if and only if C~(R d 1) is dense in Aq, V~d-l~ I.e. the problem can be reduced to the same problem in a ₉ ~Xl/p\~,~, ].

Lipschit'z space, and for d > 2 a solution requires a s t u d y of condenser capacities in these spaces. F o r d ~ i a linear functional on A1/p(tl ) t h a t annihilates C~(It) q,q

P , P

can be identified with a function f C A1/q (R) such t h a t

fRfr

^{= 0 for all}

C E C ~ ( R ) . I t follows t h a t f = 0 on C E , i.e. C~ is dense in A[i ~ if and only

P , P

if every function f C A1/q (tl) with support on E has to be identically zero. We

P , P

say t h a t E is a set of uniqueness for A1/q (R). A good description of these sets of uniqueness can be given in terms of >>Bessel eapacities~> C~,e (see e.g. Meyers [22] for definitions and properties), which are equivalent to the classical t{iesz capacities for 2o == 2. We denote the space of Bessel potentials of order ~ (or equivalently lgiesz potentials with respect to the kernel lxl 1-~) of functions in LP(tl) b y ~ ( t l ) , and note the following inclusion relations (see Stein [28; V. 5.3], P

where there are misprints in the s t a t e m e n t of the theorem, however).

R E M O V A B L E S I N G U L A R I T I E S A N D C O N D E N S E R C A P A C I T I E S 199 A~'P G ~ c A~ "2, 1 < p < 2 .

AP~'2cc-~3cAP~ 'p, 2 < p < oo.

/ + ' p, A~',/ ~ ,

Moreover A [ ' 2 c A ~ , < p , and C A [ ' p < p , ut least locally.

Necessary and sufficient conditions for a set to be a uniqueness set for ~ have been given b y Polking [24] and the author [15; Th. 9]. For p~ > d functions in QPP are continuous, and then the condition is t h a t E has no interior.

We summarize the result. We denote the interval [x -- 8, x ~ 8] b y I,(8).

THEORn~[ 13. Let E C C be compact, and suppose E C R, the real axis.

For E to be removable for A D p, 1 < p < 2, it is sufficient that E is a set of uniqueness for ~ / q ( R ) . I t is necessary that E is a set of uniqueness for ?~iq(R) for all p' < p.

For E to be removable for A D 2 it is necessary and sufficient that E is a set of uniqueness for ~ / 2 ( R ) .

For E to be removable for ADP, p > 2, it is necessary and sufficient that E is totally disconnected.

E is a set of uniqueness for ~ / q , 1 < p < 2, i f and only if one of the following equivalent conditions is satisfied:

(a) Cl,q,p(I ~ E) ~ C1/q,r(I ) for some interval I containing E.

(b)

C1/q,p(I

\ , E) = C1/q,p(I ) for every interval I.

(c)

lira

&+O 8

- - G / + / = ( 8 ) \ E)

> 0 for almost all x E E .

The capacity C~/2, 2(E) is equivalent to the plane logarithmic capacity C2(E), and the result is t h a t E is removable for A D z if and only if either

(a) C2(I ~ E) = C2(I ) for some interval I containing E, (b) C2(I ~ E) = C2(I ) for every interval I,

o r

- - Cdlx(8 ) ~ E)

(c) lim > 0 for ahnost all x C E .

Note t h a t Q(I~(8)) = 1/(log 4/8). Similar results can be given for sets on circles.

The characterization (a) is due to Ahlfors and Beurling [3]. Their result was extended to sets on C 2 curves b y Carleson [6; VI. 3]. See also [12; Th. 1] where a related theorem of Carleson was published.

Sharp comparison theorems between the C~,p and H a u s d o r f f measures, and between the C~,p for different ~ and p have been given b y Maz'ja and IIavin [21], and b y Adams and Meyers [1].

Suppose again t h a t E ~ R e-1 C R e, and consider F(+(R ~ E) for a rectangle /~ t h a t is symmetrically situated with respect to R e-1. Let u be the corresponding

200 LA~S INGE HEDBERG

e x t r e m a l f u n c t i o n . T h e n 1 - u ( . , - x~) is also an e x t r e m a l f u n c t i o n a n d since t h e e x t r e m a l is u n i q u e it follows t h a t u ~ 89 on R ~-1 ~ E.

p d

T h e r e s t r i c t i o n o f u t o R d + belongs t o

WI(R+),

a n d it can be e x t e n d e d to W[(R d) b y s e t t i n g u ( . , - x d ) ~ u(',xd). T h e r e s t r i c t i o n o f W[(R d) to R d-1 is again

A1/q,

P,P a n d it follows t h a t

F(pd)(R ~ E) z F(pd)(R)

if E is a set o f u n i q u e - ness for A~]~(Rd-~). B y t h e inclusion r e l a t i o n s a b o v e a n d [15; Th. 9] this is t h e case if

lim~_>oC1/q,p(B(x,(~)~E)~l-d> 0

for (md ~) a.e. x in

E,

in p a r t i c u l a r if m d I(E) ~ 0. F o r p ~- 2 C1/2, 2(E) is again e q u i v a l e n t to

C2(E),

i.e.

to classical c a p a c i t y w i t h r e s p e c t to t h e N e w t o n kernel ]x] 2-d in R d. Cf. V/~iss [29], a n d Z i e m e r [32; Th. 3.14].

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R E M O V A B L E S I N G U L A R I T I E S AND CONDI~NSER CAPACITIES 201

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.Received November 29, 1973 Lars Inge I-Iedberg

D e p a r t m e n t of Mathematics University of Stockholm [Box 6701

S-113 85 Stockholm, Sweden