On convergent and divergent sequences of equilibrium distributions

A R K I V F O R MATEMATIK Band 4 nr 42

Communicated 10 May 1961 b y O. FROSTMAN a n d L. CARLESON

On convergent and divergent sequences of equilibrium distributions

By H A N S WALLIN

l. Introduction

We shall deal with sets belonging to an m-dimensional Euclidean space R m and we suppose all the time t h a t the sets are bounded Borel sets.

Let K=(r), n = 1, 2 . . . and K(r) denote non-negative, non-increasing functions, defined for r >~ 0, which are continuous for r > 0. We also suppose that Kn(r)->K(r) when n - + ~ . The problem which we shall discuss, is of the following type.

Suppose that Kn(r), n = 1, 2 . . . are such t h a t the equilibrium problem is pos- sible for every K~(r). 1 This is, for instance, the case if every Kn(r) is of the form r -~, m - 2 ~ z c < m . When is it true t h a t the equilibrium distributions be- longing to the kernels K=(r) and a certain closed set /v converge, when n - > ~ , on the assumption that the Kn-capacity 2 of the set F is positive? B y con- vergence we always mean convergence in the weak sence. We shall here above all deal with the case when the K-capacity of F is zero. This question is of interest because a positive answer would, to sets of capacity zero, assign a distribution of mass which would be an analogue to the equilibrium distribution for sets of positive capacity. The case when the K-capacity of F is positive is easily settled. Namely, if the K-capacity of F is positive and Kn(r)SK(r) - - t h a t is K=(r) tends non-decreasingly to K ( r ) - - i t immediately follows t h a t the equilibrium distributions belonging to K~(r) and the set E converge, when n--> c~, towards the equilibrium distribution belonging to K ( r ) a n d F. This is proved by Frostman in [2] for kernels of the type r ~ and his proof remains valid for general kernels. An assumption like K n ( r ) S K ( r ) is essential which is seen by the following counter-example) There exists a closed linear set F with I-Iausdorff dimension 4 larger than a chosen number ~0, ao < 1, such t h a t the equilibrium distributions belonging to the kernels r -~ and F fail to converge to the equilibrium distribution belonging to r -~~ and F, when :r + 0 - - t h a t is zr tends to ~0 from above. We give a proof of this in Theorem 7.

I t is possible to determine the equilibrium distribution exactly only in a few simple cases. This problem has been treated by Polya and Szeg5 [6], who have

F o r d e f i n i t i o n s see [1].

2 A s r e g a r d s t h e d e f i n i t i o n of c a p a c i t y see b e l o w .

a T h i s c o u n t e r - e x a m p l e h a s b e e n s h o w n t o m e b y P r o f e s s o r C a r l e s o n . 4 F o r t h e d e f i n i t i o n see [1], p. 90.

H. WALLIrr

On convergent and divergent sequences of equilibrium distributions

d e t e r m i n e d t h e e q u i l i b r i u m d i s t r i b u t i o n b e l o n g i n g t o t h e k e r n e l r -~, a < l , w h e n t h e set is, for i n s t a n c e , a l i n e a r i n t e r v a l . T h e i r r e s u l t shows t h a t t h e e q u i l i b r i u m d i s t r i b u t i o n s converge t o t h e d i s t r i b u t i o n of m a s s w h i c h h a s c o n s t a n t d e n s i t y in t h e g i v e n i n t e r v a l , w h e n a - - ~ l . A n a n a l o g o u s r e s u l t is v a l i d for a c i r c u l a r d i s k a n d t h e (3-dimensional) sphere. B y m a p p i n g a l i n e a r i n t e r v a l i s o m e t r i c a l l y on a r e c t i f i a b l e c u r v e in t h e p l a n e , F r o s t m a n [2] has shown, using t h e r e s u l t s of P o l y a a n d Szeg6, t h a t t h e e q u i l i b r i u m d i s t r i b u t i o n s b e l o n g i n g t o r ~ a n d a r e c t i f i a b l e p l a n e curve converge, w h e n a - > l , t o t h e d i s t r i b u t i o n of mass, w h e r e t h e m a s s w h i c h is s i t u a t e d on a n arc is p r o p o r t i o n a l t o t h e l e n g t h of t h a t arc.

A n o t h e r case has been t r e a t e d b y L i t h n e r [3] using m e t h o d s f r o m F o u r i e r a n a - lysis. H e h a s p r o v e d t h a t t h e e q u i l i b r i u m d i s t r i b u t i o n s b e l o n g i n g t o r ~ a n d F converge, w h e n a - - > m - 0 , t o w a r d s t h e d i s t r i b u t i o n which h a s c o n s t a n t d e n s i t y on F , if F is a c o m p a c t set in R m which h a s p o s i t i v e m - d i m e n s i o n a l L e b e s g u e m e a s u r e . W e shall p r o v e ( T h e o r e m 1 ) - - u s i n g q u i t e o t h e r m e t h o d s t h a n t h o s e of L i t h n e r - - t h a t t h i s r e s u l t r e m a i n s v a l i d for m u c h m o r e general k e r n e l s

Kn(r)

a n d

K(r).

T h u s we g e t a n a n a l o g o u s r e s u l t e v e n if we do n o t a s s u m e t h a t t h e k e r n e l s a r e such t h a t t h e e q u i l i b r i u m p r o b l e m is possible. I n t h i s case t h e r e e x i s t s no longer a n e q u i l i b r i u m d i s t r i b u t i o n , b u t we still consider a (not neces- s a r i l y u n i q u e l y d e t e r m i n e d ) d i s t r i b u t i o n of unit, m a s s on F , #n, w h i c h realizes

infer

f f Fr Kn(Ix-yl)dv(x)dv(Y)'

w h e r e F is t h e class of at] p o s i t i v e d i s t r i b u t i o n s of u n i t m a s s on F . T h e con- clusion is t h e n , j u s t as before, t h a t {/~n) converges t o t h e d i s t r i b u t i o n w h i c h has c o n s t a n t d e n s i t y on F . T h i s is t h e m a i n r e s u l t of t h e p a p e r .

T h e m e t h o d w h i c h we shall use in t h e p r o o f of T h e o r e m 1 can also be u s e d t o p r o v e t h e following r e s u l t ( T h e o r e m 2). I f 2' is a c o m p a c t set in R m a n d

C~,(F)

d e n o t e s t h e a - c a p a c i t y of F a n d

re(F)

t h e m - d i m e n s i o n a l Lebesgue m e a s u r e of F , we h a v e

l i m

C~(F) _ km(F),

~ , - - , m - o m - a

where k is a c o n s t a n t w h i c h

only

d e p e n d s on t h e d i m e n s i o n m of t h e space Rm. 1 T h e r e s t of t h e p a p e r chiefly consists of c o u n t e r - e x a m p l e s . T h e a i m of these is t o show t h a t t h e r e s u l t in T h e o r e m 1 is, in a c e r t a i n sense, t h e b e s t possible.

T h u s t h e r e is no n a t u r a l a n a l o g u e t o T h e o r e m 1 for sets h a v i n g L e b e s g u e m e a s u r e zero. D u e t o t h i s s o m e w h a t n e g a t i v e c h a r a c t e r of t h e r e s t of t h e p a p e r , we shall s o m e t i m e s give o n l y a s k e t c h of t h e p r o o f s of t h e c o u n t e r - e x a m p l e s . I n o r d e r t o k e e p t h e c a l c u l a t i o n s as simple as possible we shall also as a rule give t h e c o u n t e r - e x a m p l e s for l i n e a r sets.

I n T h e o r e m 3 we p r o v e t h a t t h e r e e x i s t s a closed l i n e a r set F w i t h a pre- s c r i b e d H a u s d o r f f d i m e n s i o n a0, 0 < a0-~< 1, so t h a t t h e e q u i l i b r i u m d i s t r i b u t i o n s b e l o n g i n g to t h e k e r n e l s r ~ a n d F do n o t converge w h e n a-->a 0 - 0 .

T h e q u e s t i o n t h a t is a n s w e r e d b y T h e o r e m 1 can in a n a t u r a l w a y be gen- eralized t o sets h a v i n g H a u s d o r f f d i m e n s i o n less t h a n 1. G i v e n a n a r b i t r a r y

1 Compare [2] where a similar formula is proved when F is a rectifiable plane curve.

ARKIV FOR MATEMATIK. B d 4 n r 4 2 positive n u m b e r :c0, 0 < :c0 < 1, a n d a closed linear set F w h i c h has positive a n d finite :c0-dimensional H a u s d o r f f measure, A~0(F), is it t h e n true t h a t the equi- librium distributions belonging t o the kernels r -~ a n d the set F converge, when :c-->ao-0, to a distribution where the mass situated on a subset of F is propor- tional to the :c0-dimensional H a u s d o r f f measure of t h a t subset? A negative an- swer to this question is given in T h e o r e m 4. I t should be noticed t h a t we m u s t here assume t h a t :co < 1, because if a0 = 1 A~0(F) coincides with t h e Lebesgue measure of F , as F is a linear set, a n d t h e n the answer to the question is in the affirmative according to T h e o r e m 1. B u t hence there appears, in a n a t u r a l w a y the question whether it is possible to c o n s t r u c t a closed set F in the plane h a v i n g positive a n d finite 1-dimensional H a u s d o r f f measure, 0 < A I ( F ) < oo, so t h a t the equilibrium distributions belonging to r ~ a n d F do n o t converge to a c o n s t a n t times the 1-dimensional H a u s d o r f f measure, when :c-->l - 0 . I n T h e o r e m 5 we prove t h a t this is possible. This result is t h u s an instance of the well-known fact t h a t there exists a f u n d a m e n t a l difference in the s t r u c t u r e of linear a n d plane sets with positive a n d finite 1-dimensional H a u s d o r f f measure.

The above counter-examples show t h a t there is no result which is analogous to Theorem 1 if the set F has Lebesgue measure zero. I n order to get a posi- tive solution of the convergence problem for sets with Lebesgue measure zero, we m u s t consequently limit ourselves to sets which satisfy some suitable con- dition of regularity. We shall prove (Theorem 6) t h a t if the set is a linear C a n t o r z set, t h e n we get a positive answer t o the convergence question. On the other hand, there exist simple sets for which t h e answer is in the negative, which is illustrated b y t h e fact t h a t the sequence of equilibrium distributions does n o t necessarily converge if t h e set F is the union of two C a n t o r sets. We sketch a proof of this in the r e m a r k to T h e o r e m 6.

A question which is related to the above is the following. L e t /zn be the distribution of the mass

1In

in each of t h e n points (not necessarily uniquely determined) which realize

1 , . . . , n

5 g(Ix,-zjl)

m i n ~<s _ K(D(K)),

where F is a closed set. If the K - c a p a c i t y of F is positive a n d the equilibrium problem is possible for

K(r),

then it is true t h a t {/Ln} converges to the equi- librium distribution belonging to

K(r)

and F , when n - - > ~ . 2 Does {/in} converge also if the K - c a p a c i t y of F is zero? The fact t h a t this is n o t necessarily t h e case is an immediate consequence of a result of Terasaka [8], who, in the case when

K ( r ) = l / r ,

c o n s t r u c t e d a closed enumerable set, such t h a t the se- quence

u~(x)= I K(Ix-yl)d/~n(y),

n = 1, 2, . . . , does n o t converge for every x be- longing to the c o m p l e m e n t of F . I t is, however, possible to m a k e T e r a s a k a ' s

z For definitions see [4], pp. 152 ff.

See [1], pp. 46 if, where this is proved for

K(r)= r -~ .

H. WALLIN,

On convergent and divergent sequences of equilibrium distributions

c o n s t r u c t i o n for a n a r b i t r a r y kernel

K(r)

a n d hence we can, to e v e r y kernel

K(r),

find a n enumerable set such t h a t {/~n} does n o t converge. F i n a l l y we can note t h a t the same negative r e s u l t also remains valid if instead of considering the n points which realize

K(D(n ~))

we t a k e the n points {y~}~ which realize

?t

K(R'~)=max

m i n _1 ~.

K([x_x~l)=mi n 1 ~ K(ix_y~l).

xc~eF x ~ F n i = l x e F T6 4=1

2. D e f i n i t i o n s a n d n o t a t i o n s

x = (x t, x 2, ..., x m) denotes a point in R ~. B y a closed m-dimensional interval we m e a n t h e set of points which satisfy t h e inequalities

a, 4x*~ b,,

where a~

a n d b, are a n y n u m b e r s such t h a t a , < bt, i = 1, 2 . . . . , m. S(x0, r) denotes t h e closed sphere I x - x o[

<~r. I(H, v; F)

denotes t h e energy integral

f f H(tx-yl)dv(x)dv(y).

T h e H - c a p a c i t y of a set F ,

CH(F),

we define as

CH(F)=

{inf

I(H,

v; F)} -1, where

y e F

F is t h e class of all positive distributions of u n i t mass on F , i.e. t h e class of all completely additive, non-negative set functions t a k i n g the value 1 on F a n d vanishing outside F . Particularly, we write the a - c a p a c i t y of F , i.e. t h e case when

H(r)=r -~,

as

C~(F).

A set E is said to be regular with respect to the distribution #, or shorter regular #, if # does n o t distribute a n y mass on the b o u n d a r y of E. T h e com- p l e m e n t of E we d e n o t e b y

E'

a n d for the m-dimensional Lebesgue measure of E we write

m(E).

H E 1 a n d E 2 are t w o sets, we denote by

EI~,E ~

t h e set of points belonging to E 1 b u t n o t to E 2. If E l s e 2 we write E ~ - E ~ instead of

E l S E 2.

.

I n this section we collect some inequalities which will be of c o n s t a n t use when we prove the lemmas required for the proof of T h e o r e m 1. L e t

H(r)

be defined for r~>0, continuous for r > 0 , non-increasing a n d non-negative. L e t lira

H(r)=H(O)~<

~ . L e t F be a closed set of positive H - c a p a c i t y . As F is

r-->0

closed there exists a n o t necessarily uniquely determined distribution, 3, of u n i t mass on F which realizes

inf

I(H, ~; F)= VH(F),

~er"

where F is the class of all positive distributions of unit mass on F . W e t h u s h a v e

I(H, 3; F ) = V.(F)= {C~(F)} -1.

W e call ~ a c a p a c i t a r y distribution belonging to

H(r)

a n d F . P a r t i c u l a r l y if t h e equilibrium problem is possible for H(r), a c a p a c i t a r y distribution is identical with

AnKIV FSn MATEMATIK. Bd 4 nr 42 t h e equilibrium distribution a n d accordingly unique. 1 We n o w form t h e potential belonging to the c a p a c i t a r y distribution ~ a n d t h e kernel

H(r)

u(x) = f F H(Ix

- y ] )

dr(y).

T h e n t h e following inequalities are true:

u(x) >~ VH(F)

for all x E F except p e r h a p s for a set with H - c a p a c i t y zero. (3.1)

u(x) <~ VH(F)

e v e r y w h e r e on t h e support of 3. (3.2)

u(x) <~ A . VH(F)

everywhere, where A is a c o n s t a n t which o n l y depends

on t h e dimension m of t h e space R ~. (3.3)

The inequalities (3.1) a n d (3.2) follow f r o m t h e fact t h a t

u(x)

is lower semi- c o n t i n u o u s b y using t h e " v a r i a t i o n m e t h o d of Gauss". F o r this we refer t o F r o s t m a n [1], pp 35 ff. (3.3) eusily follows f r o m t h e fact t h a t

H(r)

is m o n o t - o n o u s l y decreasing. This is a result of U g a h e r i [9].

4.

W e n o w prove a l e m m a which we shall use in the proofs of all our theorems.

First, however, we collect the conditions on our functions

K n ( r ) a n d K(r).

K~(r), n = 1, 2 . . . . , a n d

K(r)

are defined for r ~> 0, continuous for r > 0, ] are n o n - n e g a t i v e a n d non-increasing a n d satisfy l ~

Kn(r) = Kn(O) <<- 0% ~ (a)

lim _r--~0

K ( r ) = 0% ]

lim

K~(r) = K(r). (b)

n - - > o o

L e m m a 1.

Suppose that K~(r), n = 1, 2 . . . and K(r) satis/y conditions (a) and (b) and that F is a closed set satis/ying CK~(F)>0 and CK(F)=0. Let /~, be a capacitary distribution belonging to Kn(r) and F and suppose that {/zn} con.

verges to a certain distribution 1~. Denoting by S an m-dimensional sphere it is true that

CK~(S n F)

/~(S N F ) = lim

n-,~ CKn(F) i/ the sphere S is regular [~ and /a(S)>0.

3

Proo/.

We first prove t h a t

lim

I(Kn,/~n;

F) = ~ . (4.1)

W e introduce the functions [Kn(r)]N a n d [K(r)]N, where

[Kn(r)]N = Kn(r )

if

K~(r) <~ IV

a n d [Kn(r)]

= N

if

K ~ ( r ) > N ,

a n d where [K(r)]~ is defined in an analogous w a y .

i I t s h o u l d be o b s e r v e d t h a t , a c c o r d i n g to o u r c o n v e n t i o n , b o t h a e a p a c i t a r y d i s t r i b u t i o n a n d t h e e q u i l i b r i u m d i s t r i b u t i o n are d i s t r i b u t i o n s of u n i t m a s s .

2 T h i s l e m m a s h o u l d be c o m p a r e d t o t h e c a l c u l a t i o n s in F r o s t m a n [2].

H , WALLIN, On convergent and divergent sequences of equilibium distributions

The fact t h a t [Kn(r)]N and [K(r)]N are non-increasing and t h a t [K(r)JN is con- tinuous implies t h a t [Kn(r)]N converges uniformly to [K(r)]~ in every compact interval. Hence

lim I(Kn,

~an; F)>~

lim

I([Kn]N, /an; F)=I([K]N,/a; F),

for all N. B u t since C K ( F ) = 0 we have lim I([K]N, /a; F ) = oo, and so (4.1) is

proved. N-*~

We now suppose t h a t S has radius r where r is chosen so t h a t S is regular with respect to # and /a(S)>O. Choose a sphere S o with the same centre as S and with radius to, r o > r, so t h a t S o is regular #. Then the following esti- m a t e holds

>~ I(Kn, /an; S) >~ ! Kn(r o - r) (4.2)

~an(S) ~ I(Kn--, ~an; F) #n(S) -/an(So - S) I(Kn, ~an; F)

This is an easy consequence of the inequalities (3.1)and (3.2). I n fact, we h a v e

I (Kn, ~n; s) < f s f Kn(,x-Y,) d/an(x) d~n(Y) <<-l(Kn, /an; F) " f sd/an(x),

according to (3.2). This gives one of the inequalities (4.2). We get the other b y the following division of the energy integral

S F S S o - S S F~\So

I >1 I(Kn, ~an; F ) . fsd/an(x),

b y (3.1), because /an does not distribute a n y mass on a set of Kn-capacity zero.

>~

- I(Kn, /an; F) f d#n(x),

I I

J S o - S

according to (3.2).

I I I ~> - Kn(r o - r).

This gives the other inequality (4.2). Now letting n--> ~o in (4.2) we get b y (4.1) I(Kn, #n; S ) > lim I(Kn, ~an; S) >~ #(S) - # ( S o - S ) .

/a(S) >/lira

n-~:r ( n,/~n; F) ~ I(Kn, ~an; F)

If ro--r is small # ( S o - S ) is, however, arbitrarily small and hence we get

I(Kn, ~an; S)

/a(S) = n-*~lim I(Kn,/an; F) (4.3)

ARKIV F 6 R MATEMATIK. B d 4 n r 4 2

I n an analogous w a y we realize t h a t

I(K~, /~n; S')

(4.4)

# ( S ' ) = n-~oolim

l ( K n , / ~ ; F)

I t is e a s y to see t h a t t h e limit in (4.3) is e x a c t l y

lim

CK~(S N F)

n-.oo CK,(F)

N a m e l y , if vn is d i s t r i b u t e d on S O F with t h e s a m e relative d e n s i t y as a c a p a c i t a r y distribution belonging to

K~(r)

a n d S fl F a n d with the t o t a l m a s s rn(S fl F )

= / ~ ( S n F ) = / ~ ( S ) , we get

lim

I(Kn,

v~; S N F ) = 1. (4.5)

n-~oo I(Kn, /~n; S)

For, suppose t h a t lim

I(Kn,

~ ; S fl F ) = h < 1. (4.6)

~ _ ~ I ( K n , #~; S)

As /~n is a d i s t r i b u t i o n of u n i t m a s s which realizes inf

I(Kn, v; F),

we get 1 - I(K~, /a~; F)

~(g~, ~ ; F) I(Kn,

I(Kn, /~;J " F)'J ~

(4.7)

B u t if A is the c o n s t a n t which occurs in (3.3), we h a v e

<. A . I(Kn, /an; F) "

{~n(S)} -1"/~=(So

- S) + K=(r o - r)

which follows f r o m t h e fact t h a t

sup I

g=(Ix--

Yl)

dye(x) <. A "

{ v n ( S ) } - ' '

I(g=, v~; S N F)

y J S

~< A . {v=(S)}-'.

I(gn, /~,; S) < A

{~n(S)} -~.

I(K~, i~=; F),

where t h e first i n e q u a l i t y is o b t a i n e d f r o m (3.3). I f we use t h e e s t i m a t e which we h a v e just obtained, (4.7) gives b y first letting n-->c~ a n d t h e n

ro-->r,

533

H. WALLIN,

On convergent and divergent sequences of equilibrium distributions

1 ~< l i m I(K~, v~; S N F) + ^{l i m} I(K~,/~,; S')

n-.~r I(K~, ~ ; F) ~-~oo I(K~, /~, F)

A c c o r d i n g t o (4.3), (4.4) a n d (4.6) t h i s gives

1 <~ h/~(S) + #(S').

B u t as h < 1 a n d # ( S ) > 0 we h a v e 1 ~<

h#(S)

+/~(S') < 1, w h i c h gives a c o n t r a - d i c t i o n . T h e r e l a t i o n (4.5) is h e n c e true, a n d this, c o m b i n e d w i t h (4.3), i m m e - d i a t e l y gives

CK,,(S n F)

~(S) = # ( S N F ) = n-*~olim CK~(F) a n d so t h e l e m m a is p r o v e d .

Remark.

T h e conclusion of t h e l e m m a r e m a i n s v a l i d also if S d e n o t e s cer- t a i n m o r e g e n e r a l sets t h a n spheres. F o r i n s t a n c e , for S we c a n choose t h e i n t e r s e c t i o n b e t w e e n a s p h e r e a n d a n m - d i m e n s i o n a l i n t e r v a l ( c o m p a r e L e m m a 2) or t h e u n i o n of a f i n i t e n u m b e r of m - d i m e n s i o n a l i n t e r v a l s ( c o m p a r e T h e o r e m 2).

Clearly, t h e o n l y difference in t h e p r o o f in t h e s e cases is t h a t we h a v e t o choose S O in a s o m e w h a t d i f f e r e n t w a y t h a n a b o v e . F o r S O we choose a set, c o n t a i n i n g S, w h i c h is s i m i l a r to S, a n d t h e b o u n d a r y of which h a s a p o s i t i v e d i s t a n c e f r o m t h e b o u n d a r y of S.

.

B e f o r e we can p r o v e T h e o r e m 1 we n e e d some m o r e l e m m a s . W e f i r s t p r o v e a l e m m a w h i c h is i d e n t i c a l to T h e o r e m 1 in t h e special case w h e n F is t h e u n i o n of a f i n i t e n u m b e r of closed i n t e r v a l s .

L e m m a 2.

Suppose that Kn(r) and K(r) satis/y conditions (a) and

(b),

that

f" l [*1

t~K~(r)r'~-ldr< ~ /or n =

1 , 2 . . .

and that l ~ K ( r ) r ' n - l d r = c ~ . Suppose also

wd v

that F is the union o/ a finite number o/ closed m-dimensional intervals. Let t~n be a capacitary distribution belonging to Kn(r) and F. Then it is true that/~-->a, when n---~cr where a ( E ) = m(E)/m(F) /or every Borel set Eta--F.

Proo/.

T h e c o n d i t i o n s of t h e l e m m a g u a r a n t e e t h a t

CK~(F)>

0 for e v e r y n a n d t h a t

CK(F)=0. #~

e x i s t s as

Cx,(F)>

0 a n d we s u p p o s e t h a t {#n} c o n v e r g e s t o a c e r t a i n d i s t r i b u t i o n #. (If n e c e s s a r y , we choose a c o n v e r g e n t subsequence.) W e d e n o t e b y G t h e i n t e r i o r of F .

I t is e a s y to realize t h a t # ( S ( x , r ) ) > 0 for e v e r y s p h e r e

S(x, r)

w h e r e

x EG

a n d r > 0 . N a m e l y , if

#(S(x,

r ) ) = 0 for s o m e r > 0 , we could, for l a r g e v a l u e s of n, o b t a i n a s m a l l e r v a l u e t h a n

I(Kn, #~; F)

of t h e e n e r g y i n t e g r a l b e l o n g i n g t o

K,~(r)

a n d F b y a r e d i s t r i b u t i o n of / ~ on F so t h a t m o r e m a s s were distri- b u t e d in

S(x, r). 1

B u t t h i s w o u l d be a c o n t r a d i c t i o n to t h e f a c t t h a t / ~ is a c a p a c i t a r y d i s t r i b u t i o n b e l o n g i n g t o

K~(r)

a n d F a n d h e n c e we h a v e / ~ ( S ( x ,

r)) > 0

if

x E G

a n d r > 0 .

W e now t a k e t w o s p h e r e s

S(xl, r)

a n d

S(x2, r)

which a r e c o n t a i n e d in G. As

CK,(S(xl, r))=C~n(S(x2, r))

we h a v e , a c c o r d i n g to L e m m a 1

1 Compare the redistribution of the mass in the proof of Lemma 4.

ARKIV FOR MATEMATIK. B d 4 n r 4 2

~(S(xi, r ) ) = f ( S ( x 2 , r)),

(5.1)

except possibly for a n enumerable set of values r. Consequently (5.1) is valid for all values of r such t h a t S(x,, r ) = G , i = 1, 2. Two spheres belonging to G a n d h a v i n g t h e same radii are hence carrying t h e same q u a n t i t y of mass /x, which means t h a t on G # is distributed with c o n s t a n t density. H e n c e it o n l y remains to prove t h a t t h e b o u n d a r y F - G does n o t c a r r y a n y mass. L e t t h e centre x 0 of t h e sphere 8(x0, r) belong t o F - G a n d choose r so t h a t S ( x o, r) is regular with respect t o /x. (If t h e b o u n d a r y of S ( x o, r ) c a r r i e s a n y mass, this mass is situated on F - G . ) Choosing r small e n o u g h there is a subset of G which is c o n g r u e n t to F N S ( x o, r) a n d t h e conclusion of L e m m a 1 is, b y t h e r e m a r k following L e m m a 1, valid also for this set. This set a n d F N S(xo, r) hence c a r r y t h e same q u a n t i t y of mass. B u t this means t h a t t h e b o u n d a r y F - G does n o t c a r r y a n y mass, a n d t h u s the l e m m a is proved.

W e n o w consider t h e interval A, which is d e t e r m i n e d f r o m the inequalities 0-<, x * ~<a~, i = 1 . . . . , m, a n d a positive distribution of mass, f , on A such t h a t A is regular /~. We e x t e n d t h e d o m a i n of definition of f b y m a k i n g # periodic with periods a~, i.e. we p u t

f[a~ + (n 1 a 1 . . . n m am)] = pt(eo),

for a r b i t r a r y integers (ni}[ n a n d intervals eo c A. r -[- ( n 1 a 1 .. . . . nm am) d e n o t e s the interval in which to is carried b y t h e translation ( n l a 1 .. . . . nm am). W e c a n n o w define a class of distributions of mass, {fx}, where fix denotes t h e distri- b u t i o n which arises f r o m f , w h e n we translate the distribution f b y t h e v e c t o r x, i.e.

fx(co) = f(~o - x), (5.2)

for e v e r y interval co. T h e n t h e following l e m m a is valid.

L e m m a 3. L e t f be a positive distribution o/ m a s s with total m a s s M or~

A = ( x [ 0 < x t <<.a~, i = 1 . . . m } a n d let F be a closed subset o/ A h a v i n g positive m - d i m e n s i o n a l Lebesgue measure. T h e n there is a translation as above o / t h e distri- bution f , f * , so t h a t

f * ( F ) >~ m ( F ) 9 M . a 1 9 a 2 ... ^{a m}

L e m m a 3 follows from t h e relation

f~

' ... f ~ m f x ( F ) d x 1 ... dx m = M . m ( F ) ,

which is easily p r o v e d b y means of (5.2) for instance b y i n t r o d u c i n g t h e char- acteristic function of t h e set F in the integral a n d b y t h e n using F u b i n i ' s theorem.

Using L e m m a 2 a n d L e m m a 3 we now prove

H. WALLIN,

On convergent and divergent sequences of equilibrium distributions

Lerm~aa 4.

Let F 1 and F 2 be closed sets such that F2-~ 2' 1 and re(F1)> O. Let 2'~ be the union o/ a /inite number o/m-dimensional intervals. Suppose that K~(r) and K(r) satis/y the conditions (a) and (b) and that

S~ Kn(r)rm-l dr < oo and S~ K(r)r'~-l dr= c~.

Let /~tn be a capacitary distribution belonging to Kn(r)

and F~, i = 1, 2,

and suppose that /~1~--->(~1. Then it is true that

m(E)

al(E ) >~ m(F2), E c El, E Borel set.

Proo/.

Suppose t h a t t h e l e m m a is wrong. T h e n there is a n interval A such t h a t al(A) = h . m(A N F1)

m(F~)

, h < l a n d m ( A N F 1 ) > 0 . (5.3) T h e idea of the proof is as follows.

F 2 ~ F 1

implies t h a t

I(Kn,#ln;F1)>~

I(Kn,/~2n;

2'2) for all n. B u t this indicates t h a t if the limit distribution of (ju2n}

distributes more mass on A N F 1 t h a n the limit distribution of (#1~), t h e n it should be possible to reduce t h e e n e r g y integral

I(K~,

juln; F 1 ) b e redistributing t h e p a r t of the mass /~ln which falls in A so t h a t t h e relative d e n s i t y of the mass on A N F1 coincides with t h a t of ~u2~. As we cannot, however, g a u r a n t e e t h a t /~2n(A N F 1 ) > 0 - - w h i c h is required for a redistribution of the a b o v e men- tioned k i n d - - w e first h a v e to u n d e r t a k e a translation of t h e distribution ~u2~

according t o L e m m a 3 so t h a t t h e t r a n s l a t e d distribution distributes mass on A N F 1. This t r a n s l a t i o n introduces, as we shall see, faults in our estimates which we c a n neglect. N o w we t u r n t o t h e details.

W e can suppose t h a t A is regular with respect to t h e distributions 01 a n d /~en, n = 1, 2 . . . W e can also suppose t h a t A is covered b y one interval from F2, i.e. t h a t A = F 2.

L e t juan be the restriction of ~u2~ to A. F o r e v e r y n we can, according to L e m m a 3, find a translation, x~, of the distribution f~2n so t h a t t h e t r a n s l a t e d distribution, /~*~, satisfies

m ( A n

~'1).

* A

~2~( nF1)~>/~2n(A), m(5) This gives, b y L e m m a 2

l i r a / ~ ( A N El) >~ m(A n F1). (5.4)

n-~:r m(F~)

(Actually it is true t h a t lira #2~(A N F1) = m ( A fl *

El)Ira(F2).)

W e n o w define a sequence (h~)~ r b y

~in(A) =

hn~a~n( A

N ~1), (5.5)

a n d introduce new distributions (ju~) of u n i t mass on F r

ARKIV FOR MATEMATIK. B d 4 n r 42

h

i / ~ on A A F 1 1, on F I \ A

on $'~.

To get a contradiction it is enough to prove t h a t

I(Kn, /~;

/ ~ 1 ) <

I(Kn, /~n; F~)

when n is sufficiently large. To prove this we shall use the inequalities

I(Kn, #2,; F2) <~I(K,~, ~ul,~; F1)

(5.6)

a n d lim hn ~< h, (5.7)

n--> oo

the latter of which follows from (5.3), {5.4) a n d (5.5).

We shall estimate the integral

I(Kn,/~; F 1)

b y dividing up the domain of integration F~ a n d b y separating a certain r e s t set, Ra = R(~ ) U R~ ), d > 0, which is defined in the following way. We first notice t h a t the translations b y x~, n = 1, 2 . . . can be supposed to converge to a certain translation b y x*. (If necessary, we choose a convergent subsequence.) We can also assume t h a t one of the corners of the interval A is situated in the origin and t h a t the point x* belongs to A. Then R~ ) shall consist of those points which are situated a t a distance ~<d from the intersection of A a n d the set which consists of the union of the ( m - 1)-dimensional planes which on one hand pass through the point x* a n d where, on the other hand, each plane is parallel to one of the edge planes to A. ~a~(2) shall consist of those points which are situated at a distance ~<d from the b o u n d a r y of A.

f

P u t

un(x) = iF, Kn(Ix -- Yl) dttn(y).

We shall estimate

I(K~,

#~;

F1)

b y the following division.

I(Kn, tt,~; F1)= f F u,dx)d/u,dx)= (fa\ad+ fR + f(av~),) u,dx)dlt,dY)=

I + I I + I I I .

F o r x belonging to the intersection of A ~ R d and the support of /~n we have, b y 3.2, supposing n so large t h a t

Ix,~-x*l<d/2

d *

and hence, b y (5.6), for a fixed d

Ho WALLIN, On convergent and divergent sequences of equilibrium distributions

un(x) <~ hn I(K,, #in;

F1) + O(1).

For x belonging to the intersection Of (A U

R,~)'

and the support o f / t i n , we get in an analogous way

un(x) <~ I(K,,

jUln; FI) + 0(1), when d is fixed.

Finally, for x ERa, we shall prove t h a t

fA d * f Kn(lx-yl)dt~ln(y)<

Un(X) =hn g~(]x-y])

tt2~(y) +

fi F, A '

<~ h,~. A . 2 m. I(Kn, tt2n; F2) + A" I(Kn, ~ln; F1) = O(I(Kn, #1,; FI)),

where A is the constant in (3.3). The estimate

fa" g,~(]z - y])

d#l~(y)

<~ A . I(Kn,

/~1~;

F1)

follows immediately from (3.3). I n order to show t h a t

f ~ g,~(Ix-yl)dtt*n(y)<~A'2m'I(g,, tt2,~; f~),

FIFL

we proceed in the following way. Consider the domains into which A is sub- divided by the ( m - 1)-dimensional planes which, on the one hand, pass through the point xn and where, on the other hand, each plane is parallel to one of the edge planes to A. tt~n coincides, in each of these domains, with a transla- tion of the part of the capacitary distribution /~2~ which is situated in A, and as there are 2 m such domains the desired inequality follows by means of (3.3).

(5.5) and the three estimates which we have obtained for

u,(x)

give after simplification

I <~ hn I(Kn, rein; F~)./~I,(A)

+ 0(1).

* R

I I ~<

O(I(Kn,/tl,;

F1)) [ p , , ( R ~ A ) + hn/t~( d N A)].

I I I ~< O(1) +

I(Kn, /u1,~; F~).

#I~(A').

This gives

I(KT,, /~.;

FI)

I(g~, ~uln; FI) <~ h,~

pl,(A) +/tl~(A') + 0(1).

{I(K~, ttln; F~))-I §

+ o(1) [~I~(R~\A) + h~ ~ ( R ~ n A)].

B y (5.7)and the fact that

ARKIV FSR MkTEMATIK. B d 4 nr 42

we h a v e

* R m ( R ~ , N A ) 1 lim/~2.( a f3 A) m ( F z ) ,

--lira I(K,,,I(K"' ~tl.; la'; F1)F 1) <~ h a l ( A ) + a l ( A ' ) + const. 9 [ a l ( R d ~ A ) ~ m(Ram(F2)N A)lj,

n - - > ~

if d is chosen so t h a t Rg is r e g u l a r a l . d - > 0 gives, a s h < 1, l i m I(K,~, /-tn; F 1) < 1,

. - - , ~ I(Kn, ~tl.; F 1)

i n t h e case w h e n a l ( A ) > 0. A c c o r d i n g l y we h a v e o b t a i n e d a c o n t r a d i c t i o n t o (5.3) for t h i s case. I f we s u p p o s e t h a t a l ( A ) = 0 a n d m(A N F 1) > 0 , we e a s i l y o b t a i n a c o n t r a d i c t i o n b y m o v i n g some m a s s ~tl, t o A f3 F 1 a n d t h e r e d i s t r i - b u t i n g i t in t h e s a m e w a y as #~n is d i s b r i b u t e d a n d b y t h e n d o i n g a n d esti- m a t e of t h e e n e r g y i n t e g r a l w h i c h is a n a l o g o u s t o t h e one we h a v e d o n e a b o v e . B y t h a t L e m m a 4 is p r o v e d .

W e a r e n o w in a p o s i t i o n t o p r o v e T h e o r e m 1 b y using L e m m a 2 a n d L e m m a 4.

T h e o r e m 1. Suppose that K~(r), n = 1, 2 . . . and K(r) are de/ined /or r>~O, continuous /or r > O, satis/y lim K~(r) = K~(O) <<, oo, are non-negative and non-in-

r-->O

creasing. Also suppose that lira K n ( r ) = K ( r ) ,

K~(r) r m-1 dr < co and K(r) r 'n-1 dr = cr

0

Let ~t~ be a capacitary distribution belonging to Kn(r) and F, where F is a com- pact set o/ positive m-dimensional Lebesgue measure. Then {/~n} converges weakly to the distribution a which has constant density on F, a ( E ) = m ( E ) / m ( F ) , E c F , E Borel set.

Proo/. Choose a sequence {F,}, where e a c h F , is t h e u n i o n of a f i n i t e n u m b e r

co

of closed m - d i m e n s i o n a l i n t e r v a l s , so t h a t F 1 ~ F 2 ~ ... ~ F , ~ ... ~ F a n d F = N F , . 1 L e t /~n, be a c a p a c i t a r y d i s t r i b u t i o n b e l o n g i n g to Kn(r) a n d F , . A c c o r d i n g t o L e m m a 2 {/zn,} converges, w h e n n-->oo, t o t h e d i s t r i b u t i o n a~ which h a s con- s t a n t d e n s i t y on F , . B y L e m m a 4 we h a v e a,(E)~</z(E) for e v e r y E c F w h e r e /x d e n o t e s t h e l i m i t d i s t r i b u t i o n of a c o n v e r g e n t s u b s e q u e n c e t o {/z~}. B u t {a,}

c o n v e r g e s t o t h e d i s t r i b u t i o n a w i t h c o n s t a n t d e n s i t y on F . T h i s i m p l i e s a(E) <~#(E), E c F . B u t as a ( F ) = # ( F ) = 1 t h e r e m u s t be e q u a l i t y , i.e.

1 This equality easily follows if we recall that/~* is the translation of a distribution from a _2~t sequence which converges to a constant times the Lebesgue measure and that RdOA is regular with respect to the Lebesgue measure.

H. WALLIN, On convergent and divergent sequences of equilibrium distributions m ( E )

#(E) = a(E) - m (F)'

a n d t h i s is t r u e for e v e r y l i m i t d i s t r i b u t i o n /~. H e n c e {#n} c o n v e r g e s t o a a n d t h e t h e o r e m is p r o v e d .

.

L e m m a 1 can be p r o v e d in a g e n e r a l i z e d f o r m w h i c h can be u s e d t o p r o v e t h e following t h e o r e m

T h e o r e m 2. Let F be a compact set in R m. Then we have, i/ k =2-1:~-m/2F(m/2),

l i m C ~ ( F ) = k m ( F ) . (6.1)

~-~m- o m -

Proo/. W e first p r o w e t h e t h e o r e m in t h e case w h e n F is a sphere, F = S ( x o , R ) = S . F o r t h i s case M. Riesz ([7],,p. 16) h a s g i v e n a n e x p l i c i t for- m u l a for t h e e q u i l i b r i u m p o t e n t i a l b e l o n g i n g t o t h e k e r n e l r ~, m - 2 < ~ < m . N a m e l y , l e t dm(y) d e n o t e t h e e l e m e n t of v o l u m e a n d p u t

U(X)=-(89 sin~( m-~)2 fs(R2 ]y__ XO,2)-(m a)/2]~_~dm(y).

T h e n we h a v e u ( x ) = 1 for all x ES.

This f o r m u l a gives Ca(S) = - 2

7~

^sin ze(m - a) f n _{2 ,10} (R 2 ^-- r2)-(m-~)12 r m-I dr 2 sin ze(m - a) R~ ~1

2 3o

( 1 - - r ~ ) - ( m - ~ ) l ~ r m - 1 d r .

H e n c e

2-1~ rn/2F m(S),

, ~ m m - a m

i.e. (6.1) is t r u e in t h i s case.

W e n o w consider t h e g e n e r a l case. I t is c l e a r l y e n o u g h t o consider t h e case w h e n m ( F ) > O . W e choose a s p h e r e S = S ( x 0 , R ) so t h a t S D F . L e t / ~ b e t h e e q u i l i b r i u m d i s t r i b u t i o n b e l o n g i n g t o r -~ a n d S. T h e n {/~} converges, w h e n :r t o t h e d i s t r i b u t i o n w h i c h h a s c o n s t a n t d e n s i t y o n S, a n d t h e following f o r m u l a - - w h i c h is L e m m a 1 in t h e r e q u i r e d g e n e r a l i z e d f o r m - - i s v a l i d

,. C.(F) < ~ C.(F) < re(F) l i m / z ~ ( F ) ~< n m - -

~-~ ~_~,/c~(s) ~-~ ca(s) m(S) ^(6.2)

I n o r d e r to p r o v e (6.2), we choose, as in t h e p r o o f of T h e o r e m 1, a sequence {Fn} so t h a t S D F n ~ F ~ + I ~ F , N F ~ = F , w h e r e e v e r y F ~ consists of a f i n i t e

ARKIV FOR MATEMATIK. B d 4 n r 4 2 n u m b e r of i n t e r v a l s . A c c o r d i n g t o t h e r e m a r k following L e m m a 1 t h e conclu- sion of t h e l e m m a r e m a i n s v a l i d also in t h i s m o r e g e n e r a l s i t u a t i o n , i.e. we h a v e

lim C~(Fn) _ m(F~)

~-.m C~(S) m(S)

H e n c e _lim C=(F) <~ l i m C~,(Fn) m(Fn)

a n d , finally, l e t t i n g n - ~ c ~ , we g e t one of t h e i n e q u a l i t i e s in (6.2). I n o r d e r t o p r o v e t h e o t h e r i n e q u a l i t y we o b s e r v e t h a t

. . . . l ( r -~,

~; F) > C~(S). ~i(F),

a n d c o n s e q u e n t l y we h a v e ju~(F)<~C~(F)/C=(S), w h i c h gives t h e o t h e r i n e q u a l i t y in (6.2).

O u r e x p l i c i t f o r m u l a for /& m a k e s i t possible t o s h o w t h a t

ira(F)

¹

l i m / z ~ ( F ) = 9 (6.3)

N a m e l y ,

~ ( F , = 2 - 1 7 ~ - m / 2 R - a ~ ( 2 ) { ; ( 1 - r 2 ) - ( m - a ) / 2 r m - l d r } - l f F ( R 2 - J y - X o J 2 ) - ( m - ~ ) / 2 d m ( y ) ,

which g i v e s _lim ~ u ~ ( F ) = 2 - 1 ~ - m / 2 R - m p

(~)

. m . m t r ) = ~ )

,_, re(F)

9

~ - - - > m

(6.2) a n d (6.3) g i v e C~(F) m ( F ) (6.4)

lira C~(S) - re(S)

gc---> m

H e n c e l i m C:,(F) _{- - - =} ,. n m - - . C~(F) ,. n m - - - C~(S) m ( F ) km(S) = km(F),

~ m m - a ~-~m C~(S) ~-~m m - o~ m(S) a n d (6.1) is p r o v e d also in t h e g e n e r a l case.

Remark. T h e o r e m 1 is a n e a s y c o n s e q u e n c e of L e m m a 1 a n d (6.4) i n t h e case when t h e k e r n e l s K~(r), n = l , 2 . . . . , a r e all of t h e f o r m r -~ a n d K ( r ) = r - " . F o r g e n e r a l k e r n e l s K~(r) a n d K(r) we c a n n o t , however, p r o v e a r e l a t i o n of t h e form (6.4) since we do n o t h a v e a n e x p l i c i t f o r m u l a for a c a p a c i t a r y d i s t r i - b u t i o n b e l o n g i n g t o t h e k e r n e l K~(r) a n d a sphere, a n d so we h a d t o p r o v e T h e o r e m 1 b y t h e m o r e c o m p l i c a t e d m e t h o d w h i c h we h a v e used.

1 This formula is not a consequence of Theorem 1 as convergence in Theorem 1 means convergence in the weak sense.

~. WALLIN, On convergent and divergent sequences of equilibrium distributions

W e n o w t u r n t o t h e proofs of t h e T h e discussion in t h i s section is n u m b e r a, a < 1, t h e r e e x i s t s a closed

C~(F) > 0

a n d C~(F) = 0

o

c o u n t e r - e x a m p l e s .

b a s e d on t h e following fact.

l i n e a r set F s a t i s f y i n g for g ~ < a }.

for ~ > a

To a g i v e n

(7.1) F o r F i t is e v e n possible t o choose a C a n t o r set. To see this, we s u p p o s e t h a t F is a C a n t o r set w h e r e t h e n t h s t e p in t h e c o n s t r u c t i o n is a set w h i c h consists of 2 ~ i n t e r v a l s where e a c h i n t e r v a l has l e n g t h l~. As F h a s p o s i t i v e a - c a p a c i t y if a n d o n l y if ~ 2 -n l ~ < c~,1 (7.1) follows f r o m t h e f a c t t h a t we c a n c l e a r l y

1

choose {ln} so t h a t

ar

1

a n d ~ 2 -n l ; (~+~)= ~ for e v e r y c > 0 .

1

T h e o r e m 3. To every given number ~0, 0 < ~0 ~< 1, there exists a closed linear set F such that C~(F)> 0 /or ~ < ~o and C ~ ( F ) = 0 , and such that the e q u i l i b r i u m distributions ~ belonging to r -~ and F do not converge when ~--->r

Proo/. W e choose t w o sequences {~n} a n d (fin} where ~ n / : r t i n T a0 a n d a l < f l l < :r < . . . . O u r set F is to be t h e u n i o n of t w o sets A a n d B w h i c h c o m p e t e for t h e m a s s of t h e e q u i l i b r i u m d i s t r i b u t i o n s , a n d where A = U A ~ a n d

0

B = U Bn. W e w a n t t o c o n s t r u c t A a n d B so t h a t {/~} does n o t converge w h e n

0

~-->:r a n d r u n s t h r o u g h t h e sequences {~n} a n d {fin}. W e s t a r t w i t h t w o closed, d i s j o i n t i n t e r v a l s , I a n d J, a n d we shall c o n s t r u c t A a n d B in such a w a y t h a t A c I a n d B c J . W e first show t h a t i t is e n o u g h t o f i n d A a n d B so t h a t for i n s t a n c e

Can(A)

2 n - l > 2 n C~,(A)

C~,,(B) ~> ~> Cr (B)' n = l , 2 . . . (7.2) :Namely, if i t is t r u e t h a t t h e r e e x i s t s a d i s t r i b u t i o n /~ so t h a t t h e e q u i l i b r i u m d i s t r i b u t i o n s / ~ b e l o n g i n g t o r -~ a n d F = A 0 B converge t o # w h e n a-->~0, t h e n we h a v e , b y L e m m a 1, if / ~ ( A ) > 0 a n d # ( B ) > 0

. C~(A) C~(I N F ) C~,(I N F)

lm ~ - lim = lim - - - -

~--~m ~(B) ~-~m ~ ( J - N F ) , - ~ m C•(F)

. lim C C~(F) = #(A.)

~-~m ~( f3 F ) - / ~ ( I ) " { # ( j ) } - I ~t(B}

w h i c h is a c o n t r a d i c t i o n t o (7.2). I f one of t h e n u m b e r s # ( A ) a n d # ( B ) is zero, we g e t a c o n t r a d i c t i o n to (7.2) in a s i m i l a r w a y . C o n s e q u e n t l y , i t is e n o u g h t o f i n d A a n d B so t h a t (7.2) holds.

1 See for instance [5].

ARKIV FOR MATEMATIK. B d 4 n r 4 2 W e n o w c o n s t r u c t A a n d B , A ~ I, B ~ J . W e subdivide I into three equal smaller intervals. T h e first interval (counted from t h e left to t h e r i g h t ) w e

oO

denote b y I i a n d we shall do the construction so t h a t L I A , ~ c I r The second 2

interval is to belong to the c o m p l e m e n t of A a n d t h e t h i r d is to contain A r W e choose A i in such a w a y t h a t

C~,~(A1) > 0 a n d C a ( A 1 ) = 0 for :r > g r (7.3) Starting f r o m J we n o w c o n s t r u c t J1 in the same way, 5 B,, ~ J1, a n d B l as

s a subset of the t h i r d subinterval of J . We w a n t B 1 t o satisfy

C~, (Bi) > 0 a n d C a ( B 1 ) = 0 for a > fli- (7.4)

a n d C~,(B1) ~< C~,(A 0 9 2 -1. (7.5)

This can be realized, because if (7.5) is n o t satisfied for a certain choice of B1, we can replace B~ b y a subset of t h e t h i r d subinterval of J , a subset which is a translation of k . B ~ , k > 0 , where k . B 1 denotes the set consisting of the points k" x where x E B1, a n d choose k so small t h a t k "B 1 satisfies (7.5). Condi- tion (7.4) is n o t d i s t u r b e d b y this.

B y subdividing 11 into three equal smaller intervals, we get I s, [.JA, c I s,

a a n d we construct A s analogously. The condition (7.3) is replaced b y

C~,(A2)>0 a n d C~(A2)=0 for g > ~ 2

a n d C~,(A~) ~< 5~,(B1) 9 2 -2.

The construction is n o w c o n t i n u e d in the same way. T h e conditions on An are C~n(An)>0 a n d C ~ ( A n ) = 0 for a>cr

a n d C~k(A, ) < Ct~(Bk) " 2 -n, k = 1, 2 . . . . , n - 1.

The conditions on Bn are

C~,(B,~)>O a n d C~,(Bn)=O for r a n d C~k(Bn) <~ C,,~(Ak)" 2 -n, k = 1, 2, . . . , n.

F i n a l l y we choose t h e left endpoints of I a n d J as A 0 a n d B 0 respectively, which g u a r a n t e e s t h a t A a n d B will be closed.

We n o w have C=,(A) >1 C~,(An)

n. W),IJLII~, On convergent and divergent sequences of equilibrium distributions a n d C,.(B) <~ ~ C~.(B~) = ~ C~.(B~)<~ ~ C~.(An) .2 -~

v--O V = n ~)=?t

= C ~ . ( A . ) . 2 - n + ~ ~<

C~.(A).

2 - " + 1 .

n = l , 2, . . . ,

I n exactly t h e same w a y we get n = l , 2, . . . ,

which is t h e other inequality, a n d hence T h e o r e m 3 is proved.

This implies C , , ( A ) >1 2,_1, C~. (B)

which is one of the inequalities in (7.2).

C~-(A) < 2 - , ' Cz,(B)

.

B y doing a c o n s t r u c t i o n similar to the one we have used in t h e proof of T h e o r e m 3 we also get the counter-example which is f o r m u l a t e d in T h e o r e m 4.

However, we first have to prove the existence of sets h a v i n g s o m e w h a t different qualities t h a n those which have been f o r m u l a t e d in (7.1). W e do this in t h e following lemma.

L e m r ~ a 5, Suppose that two numbers l and ~o are given, where l > 0 and 0 < ~o < 1. Then there exists a constant r(1, o~o) which only depends on l and ~o so that, to any given positive number oq satis/ying a 1 < a o, it is possible to con- struct a closed linear set F satis/ying

F belongs to an interval with length I.

C~,(F) > r(I, ao).

The Hausdor// dimension o/ F is less than o~ o.

Proo/, The set F shall be a generalized Cantor set, 2' = N Fn.

1

(8.1) (8.2)

(8.3) We start with a closed interval ~o with length l. L e t k be an a r b i t r a r y positive n u m b e r a n d a a n a r b i t r a r y positive integer. We divide t h e interval ~o into 2 a - 1 subintervals with lengths

1 . ( a § 1, k l . ( a § 1 . ( a § . . . . k l . ( a §

1 . ( a + a k - k ) 1 c o u n t e d from t h e left to the right, a n d separate the a (closed) intervals with lengths l(a + a k - k ) 1~ The union of the separated intervals constitutes F r E v e r y interval belonging to F~ is n o w subdivided in an analogous w a y into 2 a - 1 subintervals a n d a intervals are separated. I n this w a y we get F 2 which con- sists of a 2 intervals with length l ( a + a k - k ) -2 each. I n the n t h step of this pro- cedure we obtain Fn which hence consists of a n intervals with length l(a + ak - k) -~

each. 9

ARKIV F/JR MATEMATIK. B d 4 nr 42

W e begin b y showing t h a t t h e H a u s d o r f f d i m e n s i o n of F is

log a 1

fl - l o g (a + a k - k)"

I t is i m m e d i a t e l y realized t h a t t h e H a u s d o r f f d i m e n s i o n of F c a n n o t be l a r g e r t h a n fl a n d in o r d e r t o realize t h a t i t is e q u a l t o fl, we can, for i n s t a n c e , con- s t r u c t a b o u n d e d y - p o t e n t i a l , ~ < fl, b e l o n g i n g t o a d i s t r i b u t i o n of u n i t m a s s o n F . T o do t h i s we i n t r o d u c e t h e set f u n c t i o n s {#n}~ r w h e r e /~n is t h e u n i t m a s s u n i f o r m l y d i s t r i b u t e d o n F~. {/~n} c o n v e r g e s t o a d i s t r i b u t i o n of t h e u n i t m a s s on F a n d we g e t t h e following e s t i m a t e , if ln = l ( a + a k - k ) -~,

f v ,~ o,~ d#(Y) <~ 1 l i m [" lv, lx_:y], d (y) 1

n - - > o o

< ~ 2 f l "/~ ~ 2 ~_ ~ ! a + a k - k ) " "

n=o l~ "a~ r-r d r = l " (1 - y ) n=o a ~

2" a

- l " ( 1 - ~ ) a - ( a + a k - k ) ~< 0%

if y < l o g a / l o g (a + a k - k)=ft. H e n c e t h e H a u s d o r f f d i m e n s i o n of F is ft.

The e s t i m a t e w h i c h we h a v e d o n e also gives us t h e following i n e q u a l i t y , if C,(F) > F(12 ~- y) . a - (a +aak - k)" _ M(y).

H e n c e t h e l e m m a follows if we can f i n d a c o n s t a n t r(l, a0) so t h a t

< ~0 (8.4)

a n d M(~I) >1 r(1, O~o) , (8.5)

where ~1 is t h e n u m b e r w h i c h is g i v e n in t h e l e m m a , ~1 < ~0. H o w e v e r , a s i m p l e c a l c u l a t i o n shows t h a t we can f i n d a c o n s t a n t r(l, O~o) o n l y d e p e n d i n g on 1 a n d

~0 so t h a t (8.4) a n d (8.5) a r e s a t i s f i e d if k a n d a a r e c h o s e n i n a s u i t a b l e w a y a n d large e n o u g h 3 L e m m a 5 is h e n c e p r o v e d a n d we c a n n o w show

T h e o r e m 4. Let ~o be an arbitrary number satis/ying 0 < ~o < 1. Then there exists a closed linear set F with positive and finite ~o-dimensional Hausdor// meas- ure, 0 < A ~ . ( F ) < oo, so that the equilibrium distributions ~ belonging to the kernels r -~ and F do not converge, when ~-->~o- 0 to a distribution where the mass situ- ated on a subset o/ F is proportional to the ~o-dimensional Hausdor// measure of that subset.

1 This is a consequence of a theorem by Ohtsuka [5] on the capacity of generalized Cantor sets. We give, however, a short direct proof which will also permit us to estimate C,(F), ~< ft.

2 The choice of k and a will naturally depend on ~1. For r(l, oeo) we can for instance choose any number which is smaller than [/~(1 - ar

H. WALLZN, On convergent and divergent sequences of equilibrium distributions

Proo/. A g a i n F s h a l l be t h e u n i o n of t w o sets A a n d B, F = A U B, w h e r e for B we choose a n a r b i t r a r y c o m p a c t set h a v i n g p o s i t i v e a n d f i n i t e ~0-dimen- sional H a u s d o r f f m e a s u r e , 0 < A ~ . ( B ) < co, a n d where A shall be c o n s t r u c t e d so t h a t A ~ o ( A ) = 0 . I n o r d e r t o c o n s t r u c t A we choose d i s j o i n t i n t e r v a l s cot w i t h

5

l e n g t h s l~, i = 1, 2, . . . , so t h a t Z li < ~ , a n d coi is a t a p o s i t i v e d i s t a n c e f r o m

1 1

B. W e also a s s u m e t h a t t h e choice is such t h a t t h e i n t e r v a l s eo~ c o n v e r g e t o one p o i n t . W e shall c o n s t r u c t A so t h a t A = 5 At, w h e r e As is closed a n d

0

A ~ c w ~ for i = l , 2 . . . F o r A 0 we choose t h e p o i n t t o w h i c h t h e i n t e r v a l s cot c o n v e r g e w h i c h m a k e s A closed.

A s A ~ ( B ) < ~ we h a v e C ~ , ( B ) = 0 w h i c h i m p l i e s lim C ~ ( B ) = O.

H e n c e we c a n choose a sequence {a~}~ r ~ i f l a 0 so t h a t for i n s t a n c e C~I(B) < 2 -1" r(l~, ~ ) , i = 1, 2, . . . ,

w h e r e r(l~, s0) is t h e c o n s t a n t w h i c h occurs in L e m m a 5. T h e l e m m a t h e n g u a r - a n t e e s t h a t we c a n c o n s t r u c t A~ so t h a t t h e c o n d i t i o n s w h i c h we h a v e f o r m u - l a t e d a b o v e are v a l i d a n d so t h a t

C~(Az) > r(l~, so), i = 1, 2 . . . T h e l a s t t w o i n e q u a l i t i e s i m p l y t h a t

C~,(A) > 2 . C~,(B), i = 1, 2, . . . .

B u t t h i s m e a n s , a c c o r d i n g t o L e m m a 1, t h a t t h e l i m i t d i s t r i b u t i o n of e v e r y c o n v e r g e n t s u b s e q u e n c e of {#~) d i s t r i b u t e s m a s s on A . As A ~ ( A ) = 0 we c a n h e n c e c o n c l u d e t h a t {/~} does n o t converge t o a d i s t r i b u t i o n w h e r e t h e m a s s s i t u a t e d on a s u b s e t of F is p r o p o r t i o n a l t o t h e s 0 - d i m e n s i o n a l H a u s d o r f f m e a s - u r e of t h a t subset.

.

W e shall now p r o v e t h a t t h e r e is a c o r r e s p o n d e n c e t o T h e o r e m 4 e v e n w h e n s 0 = 1. A s h a s b e e n p o i n t e d o u t in t h e i n t r o d u c t i o n , we h a v e to use p l a n e sets t o g e t such a c o r r e s p o n d e n c e .

T h e o r e m 5. There exists a closed set F in the plane having positive a n d / i n i t e 1-dimensional Hausdor// measure, A I ( F ) , so that the equilibrium distributions be- longing to the kernels r -~ and F do not converge, when ~ - - > 1 - 0 , to a distribution where the mass situated on a subset o/ F is proportional to the 1-dimensional Hausdor// measure o/ that subset.

T h e p r o o f of t h i s is a n a l o g o u s t o t h e proof of T h e o r e m 4. T h e difference is t h a t i n s t e a d of considering i n t e r v a l s o~i w i t h l e n g t h s l~, we consider s q u a r e s eo~

w i t h sides of l e n g t h s l~, i = 1 , 2, . . . . I n o r d e r t o be a b l e to c o n s t r u c t t h e sets A~ we also h a v e t o r e f o r m u l a t e L e m m a 5 for p l a n e sets a n d t h e case ce 0 = 1.

ARKIV FOR MATEMATIK. B d 4 nr 42 L e m m a 5'. To a given positive number l there exists a constant r(1) so that, to any given positive number ~1 satis/ying :r < 1, it is possible to construct a closed plane s e r f satis/ying

F belongs to a square with side o/ length I. (9.1)

C~I(F ) > r(1). (9.2)

The Hausdor// dimension o/ F is less than 1. (9.3)

P r o o / o / L e m m a 5'. Again we c o n s t r u c t F as a generalized C a n t o r set, 2, = I'l 2,n.

1 We start with a square co with side of length 1 a n d with n u m b e r s k a n d a as before. We d e r i d e ~o into subsquares b y dividing e v e r y side of eo into 2a - 1 sub- intervals h a v i n g t h e same lengths as before, i.e. the same lengths as in t h e proof of L e m m a 5. W e o b t a i n F 1 b y separating t h e a 2 squares with area l 2. (a + a k - k ) -2. Generally Fn consists of a 2n squares with area 1 2 . ( a + a k - k ) -2n each.

B y doing calculations similar to those in t h e proof of L e m m a 5, we can show t h a t F has H a u s d o r f f dimension

2 log a fl = log (a + ak - k)"

T h e estimate of Cr(F) becomes

/~(2 - y ) a 2 - ( a + a k - k ) ~

C,,,(F) > 2r+l a2 - M(~),

if y <ft. This finally gives t h a t we can o b t a i n fl < 1 a n d M(~I) >~ r(l),

b y choosing k, a a n d r(1) suitably. B y this L e m m a 5' a n d hence also T h e o r e m 5 is proved.

Remark. L e m m a 5' can of course be f o r m u l a t e d for a n a r b i t r a r y n u m b e r :r satisfying O < % < 2 a n d a closed plane set F. This implies t h a t T h e o r e m 5 can also be f o r m u l a t e d for sets F h a v i n g positive a n d finite ~0-dimensional H a u s - dorff measure, where a0 < 2. Similar extensions of t h e counter-examples to higher dimensions are of course also possible.

10.

T h e above counter-examples show t h a t t h e set F has to satisfy suitable conditions of regularity in order t o give convergence of the sequence of equi- librium distributions in t h e case when t h e Lebesgue measure of 2' is zero. As a n example we prove t h e simple t h e o r e m t h a t t h e C a n t o r sets are regular e n o u g h t o give convergence.