A R K I V FOR MATEMATIK Band 4 nr 4
C o m m u n i c a t e d 3 J u n e 1959 b y OTTO FROSTMAN
A r e m a r k o n a t h e o r e m b y F r o s t m a n By LARS LITHNER
The origin of t h i s r e m a r k is a l e c t u r e h e l d b y F r o s t m a n in H e l s i n k i 1957 [2].
L e t us i n t r o d u c e some definitions a n d n o t a t i o n s .
L e t K be a n a r b i t r a r y c o m p a c t set in t h e e u c l i d e a n space R ~ a n d let ~ be a n u m b e r such t h a t 0 < a < n. P u t
I1~11~ - x "-=
J J I -ul
w h e r e / x is a d i s t r i b u t i o n of m a s s in R = a n d where x a n d y d e n o t e p o i n t s in R ", x = (x 1 . . . xn), y = (Yl, - . , Yn)-
A is t h e set of all 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 K , t h a t is,
#>~o, # ( K ) = I , # ( R " - K ) = 0 .
L e t Ca (K) be t h e c a p a c i t y of K of o r d e r1
C~,(K)
inf ]]g]f~/xeA
I t is well k n o w n t h a t if C= (K) > 0 t h e n t h e r e exists a 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 g~ in A t h a t satisfies
4xeA
/ ~ is called 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 of order ~ on K . F r o s t m a n [2] has set t h e p r o b l e m w h e t h e r these e q u i l i b r i u m d i s t r i b u t i o n s v a r y c o n t i n u o u s l y w i t h zr or not.
Or, if ~ ' ~ f l ( ' ~ m e a n s " t e n d s n o n - i n c r e a s i n g l y t o " ) , is i t t h e n t r u e t h a t / x ~ con- v e r g e s t o w a r d s a u n i q u e l y d e t e r m i n e d l i m i t ? (Convergence here in t h e weak sense, t h a t is, ~t~ --> ,u is e q u i v a l e n t t o
f[a/u~, -+ ftdlu
for all c o n t i n u o u s f u n c t i o n s ] w i t h c o m p a c t s u p p o r t s . )I f C~ ( K ) > 0, t h e a n s w e r is yes. The l i m i t in t h i s case is /xp which is e a s y t o p r o v e [2]. On t h e o t h e r h a n d , if C~ (K) = 0, C~ (K) > 0 for ~ > fl, t h e n t h e p r o b l e m is n o t solved b u t for special cases. F r o s t m a n t r e a t s such a special case in [2] n a m e l y t h e ease t h a t ~'-~ 1 a n d t h a t K is a' curve in t h e p l a n e ( n = 2) which is rectifiable.
H e p r o v e s t h a t in t h i s ease ~u~ -+ P0 where P0 is t h e d i s t r i b u t i o n in A for which t h e m a s s which 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 to t h e l e n g h t of t h a t arc.
31
L. LITHNER, A remark on a theorem by Frostman
The a i m of this r e m a r k is t o p r o v e a n o t h e r special case which does n o t c o n t a i n t h e case of F r o s t m a n b u t which in a certain sense e x t e n d s his result.
L e t us suppose t h a t t h e m e a s u r e of K is g r e a t e r t h a n zero. W e shall p r o v e Theorem. I [ ~ \~ 0 then #~ -+ t~o where fto is the distribution in A which has constant density.
Proo/. I t is e v i d e n t l y sufficient to p r o v e t h a t if/x~,, ~ n ~ 0 , is a n a r b i t r a r y con- v e r g e n t sequence, t h e n its l i m i t is ~0. P u t
where
K , ( x ) = C ~ l / r ~ : ' , r = 1 / ~ - - - + x 2 ~ ,
d e n o t e s the F o u r i e r t r a n s f o r m of [, t h a t is,
H e n c c
i t
R,t t - - 1
R , (x) = C ~ It/2-" r -~ --~ 1 when ~',~0.
This implies t h a t Kffi --~ ~5 in t h e d i s t r i b u t i o n sense, d} is t h e D i r a c function. P u t It # [1~ = C a Jl # t[~, t[ ~t tJ~ = f l/~ (x)I z d x.
W ~
T h e n we h a v e
I1~, II~. = f R. (~)I/~ (~)I'dx
which is easily seen by Parseval's formula.
N o w l e t / ~ = be a sequence which converges t o w a r d s a d i s t r i b u t i o n p ' (necessarily in A). T h e n we have
A - ' ~ A t
P~n (x) p (x) everywhere.
By Fatou's lemma we conclude
n ..4. 0 0 n ,-*, ~ r , 1 ~
Observing that #-n is a minimal distribution also with respect to the kernel K , . we get
32
ARKIV FOR MATEMATIK. B d 4 nr 4
[l~'l[~0< lim il~=~[l~= < Jim [l~0[l~ =
Or, I1#' 11~ < II/~o 11o ~,
B u t /~o is the uniquely determined equlibrium distribution of K with respect to tbe kernel ~. This implies
11/~o1[o= inf II/~11o (see [1]).
Hence #'=/~o and the theorem is proved.
R E F E R E N C E S
1. DENy, J.: Les potentiels d'~nergie finie. A e t a M a t h e m a t i c a 82 (1950).
2. FROSTMA1% O.: Suites convergentes de distributions d'~quilibre. Treizi~me Congr~s des Math.
Seandinaves (Helsinki) 1957.
Tryckt den 15 september 1959 Uppsala 1959. Almqvist & Wiksells Boktryckeri AB
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