A R K I V F O R M A T E M A T I K B a n d 5 n r 5
Communicated 13 March 1963 b y O. F R O S T ~ a n d L. CARLESO~
Continuous functions and potential theory
By HANS WALLIN
Introduction
L e t F 0 be a c o m p a c t set in
R m.
I f F o h a s a - c a p a c i t y zero, 0 ~< ~ < m , 1 t h e r e exists, a c c o r d i n g t o a w e l l - k n o w n r e s u l t b y E v a n s [8], a p o s i t i v e m e a s u r e p c o n c e n t r a t e d o n F 0 such t h a t t h e p o t e n t i a l of o r d e r a of p is i n f i n i t e e v e r y w h e r e o n 2' 0. I n C h a p t e r I we c o n s i d e r a p r o b l e m r e l a t e d t o t h i s r e s u l t . W e s h a l l p r o v e t h a t a c o m p a c t s e t F o h a s a - c a p a c i t y zero if a n d o n l y if e v e r y c o n t i n u o u s f u n c t i o n coincides e v e r y w h e r e on F 0 w i t h a c o n t i n u o u s a - p o t e n t i a l of a m e a s u r e w i t h c o m p a c t s u p p o r t . This is a c o n s e q u e n c e of t h e T h e o r e m s 1 a n d 2. H o w e v e r , t h e s e t h e o r e m s c o n t a i n m u c h m o r e t h a n t h e a b o v e c h a r a c t e r i z a t i o n of c o m p a c t sets of a - c a p a c i t y zero. I n p a r t i c u l a r we consider t h e case w h e n we h a v e m o r e g e n e r a l k e r n e l s t h a n r -~.L e t h b e a p o s i t i v e i n t e g e r , p > ~ l , 0 ~ < ~ < m , a n d l e t F 0 b e a c o m p a c t set w i t h ~- c a p a c i t y zero. I n C h a p t e r I I we use T h e o r e m 1 t o d e d u c e c o n d i t i o n s on h, p a n d w h i c h g u a r a n t e e t h a t e v e r y f u n c t i o n ]o w h i c h is t h e r e s t i c t i o n t o F 0 of a c o n t i n u o u s function, c a n b e e x t e n d e d t o a f u n c t i o n [ h a v i n g t h e following p r o p e r t i e s : / is d e f i n e d a n d c o n t i n u o u s e v e r y w h e r e in R m a n d i n f i n i t e l y d i f f e r e n t i a b l e on t h e c o m p l e m e n t of F0; all t h e p a r t i a l d e r i v a t i v e s of / of o r d e r s less t h a n or e q u a l t o h a n d t h e f u n c t i o n / itself b e l o n g t o
LP(Rm).
T h e r e s u l t is s t a t e d in T h e o r e m 3. T h e c o n d i t i o n s of t h e t h e o r e m i m p l y in p a r t i c u l a r ~ < m - 1; i.e. all t h e c o m p a c t sets c o n s i d e r e d in T h e o r e m 3 h a v e H a u s d o r f f d i m e n s i o n less t h a n m - 1.W e f o r m u l a t e a c o n v e r s e of T h e o r e m 3 in w 7 ( T h e o r e m 4) w h e r e we also consider a c e r t a i n class of B e p p o L e v i f u n c t i o n s , which, i n t h e case a < m - 1, is m o r e g e n e r a l t h a n t h e class of f u n c t i o n s c o n s i d e r e d i n T h e o r e m 3. T h e case p = 1 is s t u d i e d f u r t h e r in w
As a b y - p r o d u c t of o u r i n v e s t i g a t i o n we o b t a i n a t h e o r e m on t h e e x i s t e n c e of u n i f o r m l y c o n t i n u o u s h a r m o n i c f u n c t i o n s w i t h f i n i t e D i r i c h l e t i n t e g r a l s in t h e u n i t sphere, w h i c h t a k e g i v e n c o n t i n u o u s v a l u e s on a c e r t a i n s u b s e t of t h e b o u n d a r y of t h e u n i t s p h e r e ( T h e o r e m 5).
I wish t o t h a n k P r o f e s s o r L. Carleson w h o s u g g e s t e d t h e s u b j e c t of t h i s p a p e r a n d c o n t r i b u t e d w i t h m a n y i d e a s t o t h e p r o o f s of t h e t h e o r e m s .
CHAPTER I. On the representation of continuous functions by potentials
1.
Notations anddefinitions
R m is t h e m - d i m e n s i o n a l E u c l i d e a n space, m >~ 1, w i t h p o i n t s x = (x 1 ... xm), Ix[ t h e d i s t a n c e f r o m x t o t h e origin. B y a closed c u b e in R m we m e a n t h e set of p o i n t s satis- f y i n g t h e i n e q u a l i t i e s
a ~ x ~ a t §
w h e r ea~,i=l
... m, a r e a n y n u m b e r s a n d / > 0 .1 ~ = 0 corresponds to the logarithmic capacity.
H. WALLX~, Continuous functions and potential theory
S(Xo, r ) d e n o t e s t h e closed s p h e r e [X-Xo[ ~<r. W e w r i t e m i n {a,b} for t h e s m a l l e r of t h e n u m b e r s a a n d b.
T h e c o m p l e m e n t of a set E we d e n o t e b y ~ E . I f E 1 a n d E 2 a r e t w o sets, t h e n E 1 \ E 2 is t h e set of p o i n t s b e l o n g i n g t o E 1 b u t n o t t o E 2. I f E I D E2, we w r i t e E I - E~
i n s t e a d of E 1 \ E~.
B y a kernel we m e a n a f u n c t i o n K s a t i s f y i n g t h e following c o n d i t i o n s :
(a) K is defined in the interval r > O, is finite and continuous, non-negative and non- increasing and satisfies lim,_~o K ( r ) = co.
(b) (r) r m-ldr < c~.
(a) a n d (b) a r e for i n s t a n c e s a t i s f i e d if K(r) = r -~, 0 < ~ < m.
L e t a be a r e a l m e a s u r e on R m, i.e. a c o m p l e t e l y a d d i t i v e r e a l set f u n c t i o n . T h e support of a is d e n o t e d b y S~. a + is t h e p o s i t i v e a n d a - t h e n e g a t i v e p a r t of a, a = a + - a -, a n d l a l = a + + a - .
T h e p o t e n t i a l of a m e a s u r e a b e l o n g i n g t o a k e r n e l K , t h e K - p o t e n t i a l of a, is d e n o t e d b y u~,
u~ (x) = ~ K ( I x - y I) da(y), 1 3
a n d t h e e n e r g y i n t e g r a l b y I~(a),
x - y l ) d a ( x ) d a ( y ) .
u ~ is well-defined a t t h e p o i n t x p r o v i d e d u~+(x) a n d u ~ - ( x ) a r e n o t b o t h infinite.
I f t h e r e will be no m i s u n d e r s t a n d i n g we w r i t e u" i n s t e a d of u~. I f a is a b s o l u t e l y c o n t i n u o u s a n d h a s a d e n s i t y g, d a = g d x , we s o m e t i m e s w r i t e u ~ i n s t e a d of u~.
T h e K - c a p a c i t y of a b o u n d e d B o r e l set E, C~:(E), is d e f i n e d as Cx(E) = { inf IK(v)} -1,
~eF~
where r E is t h e class of p o s i t i v e m e a s u r e s ~ w i t h t o t a l m a s s 1 a n d ~q,c E . T h e K - c a p a c i t y is a n i n n e r m e a s u r e , i.e.
C~(E) = s u p CK(F), (1.1)
FC:E where F is a closed s u b s e t of E .
T h e a - p o t e n t i a l a n d t h e l o g a r i t h m i c p o t e n t i a l , i.e. t h e p o t e n t i a l s in t h e cases K ( r ) = r -~, 0 < a < m , a n d K ( r ) = - l o g r , 2 we also d e n o t e b y u~ a n d u~ r e s p e c t i v e l y a n d a n a l o g o u s l y for t h e a - c a p a c i t y a n d t h e l o g a r i t h m i c c a p a c i t y .
1 T h e i n t e g r a t i o n is t o be e x t e n d e d o v e r t h e w h o l e space if n o l i m i t s of i n t e g r a t i o n are indi- cated.
W e shall also consider t h e case w h e n K(r)= - l o g r in s p i t e of t h e fact t h a t - l o g r t a k e s n e g a t i v e v a l u e s too. W e s o m e t i m e s o m i t t h e special simple t r e a t m e n t w h i c h is n e e d e d in t h e case K(r) = - log r.
56
ARKIY FOR MATEMATIK. Bd 5 n r 5 Our kernel K satisfies the continuity principle (see for instance Ugaheri [18]):
If # is a positive measure with compact support a n d the restriction of u~ to S~ is continuous, then u~ is continuous in the whole space.
L e t 2' be a compact set with positive K-capacity. Then there exists a positive measure ~--which is not necessarily uniquely d e t e r m i n e d - - w i t h ~(R a) = 1 , S , ~ 2", such t h a t
I~(~) = inf IK(v)={C~(F)) -1,
v eFF
where Fpis the class of positive measures v with v(R m) = I, S v c F. We call v a capacitary distribution and u~r a capacitary potential belonging to K and F. u~r satisfies the fol- lowing inequalities:
u~(x) >J(Gx(2")} -1 /or every xE2" except when x belongs to a set o/K.capacity
zero.
(1.2)
u~(x) < (~(2")}-~ /or every xzS,. (1.3)
U~(Z) ~.< A* {GK(~'~)} -1 everywhere, where A is a constant which only depends
on the dimension m o] the space R 'n.
(1.4)
As to (1.2) and (1.3) we refer to F r o s t m a n [9, pp. 35 ff.] and Fuglede [11, p. 159].
(1.4) is a result b y Ugaheri [18].
I f 2"0 is a compact set, we denote b y S(2"0) the class of functions which are restric- tions to F 0 of real functions defined a n d continuous everywhere on R m.
2. Representation o f continuous functions by potentials I n this section we also m a k e the following assumption on our kernel K:
(e) I / E is the union o / a finite number o/closed spheres and u~x a capacitary potential belonging to K and E, we have
u~(x) >~ (C~c(E)) -1, /or every x E E . (2.1) The condition (c) is for instance satisfied if K(r) = r -~, 0 < ~ < m, and, more generally if K satisfies
K(r) <MK(2r), for every r >0, where M is a constant. 1
We shall now prove the following theorem on the representation of continuous functions b y potentials.
Theorem 1. Suppose that K is a kernel satis/ying (c) and that F o is a compact set with CK(F0)=0. Then there exists, /or every /unction /oES(Fo), an absolutely continuous measure a with compact support having the /ollowing properties: The potential o / a , u~, is continuous in the whole space and equal to/o on 2"0, i.e.
u~(x) =/0(x), /or every xE2"0;
1 F o r a p r o o f of t h i s w e r e f e r t o K u n u g u i [13] or C a r l e s o n [4, p. 16]. T h e s e a u t h o r s h a v e m a d e f u r t h e r a s s u m p t i o n s o n t h e k e r n e l K . H o w e v e r , i t is e a s y t o see t h a t t h e r e s u l t is t r u e also for o u r k e r n e l s .
57
H. WALLIN, Continuous functions and potential theory
i//0(x) > 0 / o r every x E F0, then (~ is a positive measure and i / K is in/initely di//erentiable in the interval r > 0 then u~ is in/initely di//erentiable on CFo.
F o r t h e p r o o f we n e e d t h e following s i m p l e l e m m a .
L e m m a 1. Let K be a kernel satis/ying (c) and F o a compact set with C K ( F 0 ) = 0 . Suppose that a is given, a > O. Then there is a set E which is the union o / a finite number of closed spheres, E D Fo, so that C K( E) < a. I / u ~ is a capacitary potential belonging to K and E, then u~ is continuous everywhere and i / A is the constant in (1.4) we have
{CK(E)} -1 ~< u~(x) <. A(CK(E)} -1 /or every x E E (2.2) and thus in particular/or every x E F.
P r o o / o / L e m m a 1. W e first o b s e r v e t h a t t h e e x i s t e n c e of a set E follows i m m e d i a t e l y f r o m t h e f a c t s t h a t F 0 is c o m p a c t a n d C K ( P 0 ) = 0 . (2.2) is a c o n s e q u e n c e of (2.1) a n d (1.4). U s i n g (1.3) we also f i n d t h a t t h e r e s t r i c t i o n of u ~ t o S~ is c o n s t a n t a n d t h u s u ~ is c o n t i n u o u s e v e r y w h e r e a c c o r d i n g t o t h e c o n t i n u i t y principle.
Proo/ o/ Theorem 1. W e first s u p p o s e t h a t ]o(x) > 0 for e v e r y x E F 0. L e t ] b e a con- t i n u o u s e x t e n s i o n o f / 0 t o Rm a n d F a c o m p a c t set such t h a t e v e r y p o i n t of F 0 is a n i n t e r i o r p o i n t of F a n d / ( x ) > 0 for e v e r y x E F .
W e s t a r t b y p r o v i n g t h a t for a n y e > 0 t h e r e e x i s t s a p o s i t i v e m e a s u r e v w i t h v(R m) less t h a n a g i v e n p o s i t i v e n u m b e r so t h a t S~ is a s u b s e t of a g i v e n n e i g h b o r h o o d of F 0 a n d u" = u ~ is c o n t i n u o u s a n d satisfies t h e following i n e q u a l i t i e s , if M = 3mA + 1, m is t h e d i m e n s i o n of t h e s p a c e a n d A t h e c o n s t a n t in (1.4),
u'(x) </(x) /or every x e P , (2.3)
u'(x) >~ / o ( x ) - M e /or every x E F o. (2.4) W e i n t r o d u c e t h e sequence of n e t s ~ = ( ~ } , w h e r e ~0 consists of all closed cubes w i t h corners h a v i n g i n t e g e r c o o r d i n a t e s , a n d ~ , i > 0 , consists of all closed cubes w h i c h we o b t a i n b y d i v i d i n g t h e c u b e s in Tl~-i i n t o 2 m e q u a l cubes b y ( m - 1)-dimen- sional h y p e r p l a n e s p a r a l l e l t o t h e c o o r d i n a t e planes. L e t wl ... eor b e a n u m b e r of c o n g r u e n t cubes f r o m ~ , U w t D F , so t h a t t h e o s c i l l a t i o n of ] is less t h a n e in eo~ for i = 1 ... r. W e s e p a r a t e t h o s e c u b e s eo~,i = 1 ... r, w h i c h a r e such t h a t t h e m a x i m u m of/0 on F 0 N wt is l a r g e r t h a n Me. I. I n t h i s w a y we g e t t h e cubes ~o~ ... w~. U s i n g L e m m a 1 i t is e a s y t o realize t h a t , for i = 1 ... s, we c a n choose a p o s i t i v e m e a s u r e w i t h t o t a l m a s s less t h a n a g i v e n p o s i t i v e n u m b e r h a v i n g t h e following p r o p e r t i e s : T h e s u p p o r t of t h e m e a s u r e is a s u b s e t of a g i v e n n e i g h b o r h o o d of F 0 N w~; t h e K - p o t e n t i a l of t h e m e a s u r e is c o n t i n u o u s e v e r y w h e r e a n d t a k e s values" b e t w e e n ~ a n d A e o n F 0 N eo~;
i t is less t h a n or e q u a l t o A e o n eo~ a n d on t h o s e cubes 0) 1 ... wr w h i c h h a v e n o n - v o i d i n t e r s e c t i o n s w i t h w~ a n d i t is f i n a l l y less t h a n a g i v e n p o s i t i v e n u m b e r elsewhere.
I n t h i s w a y we g e t a p o s i t i v e m e a s u r e a s s o c i a t e d w i t h e v e r y c u b e eo~, i = 1 ... s. L e t v~ be t h e s u m of t h e s e m e a s u r e s . D u e t o t h e choice of t h e c o n s t a n t M , M = 3~A + 1, a n d t h e f a c t t h a t t h e o s c i l l a t i o n of / is less t h a n e in e v e r y c u b e o~, i = l ... r, i t is e a s y t o realize t h a t we c a n m a k e t h e a b o v e p r o c e d u r e so t h a t
u " ( x ) < / ( x ) /or every x E F , (2.5) u'~(x) >1 m i n ( / 0 ( x ) - - M ~ , e } /or every x E F o. (2.6) i If/o(X)<~Me for every x E F 0 there are no such cubes. In this ease, however, (2.3) and (2.4) are satisfied with v = 0.
58
ARKIV F f R MATEMATIK. B d 5 n r 5 Our procedure also g u a r a n t e e s t h a t we can get
vl(R m)
smaller t h a n a given positive n u m b e r a n d S,, as a subset of a given n e i g h b o r h o o d of F 0.I f (2.3) a n d (2.4) are n o t true for
V=Vl
we c o n s i d e r / - u " . A s / ( x )-u~'(x)
> 0 for e v e r y x E F a n d u ~' is continuous we can repeat t h e procedure leading to (2.5) a n d (2.6) b u t with / replaced b y / - u " a n d / 0 b y / o - u " .In
this w a y we get a positive measure v~ h a v i n g properties which are analogous to those of v I , a n d a continuous p o t e n t i a l u "2 satisfyingu~(x) </(x) -
u~'(x)/or every xE$',
(2.7)u"(x) >~
m i n {/o(X) -u"(x) - Me, e} /or every x E F o.
(2.8) (2.6) a n d (2.8) give, forXePo,
u"(x) +
u"(x) >~
m i n {/o(X) - M e , 2e}, a n d so (2.7) a n d (2.8) yieldu"(x) + u " ( x ) < / ( x ) for e v e r y
xEP,
u"(x) +
u~'(x) ~
m i n{[o(X) - Me,
2e} for e v e r yx E F o.
I f (2.3) a n d (2.4) are n o t t r u e for ~ = v l +v2 we r e p e a t o u r procedure anew. After n steps we h a v e o b t a i n e d n positive measures ~1 ... u, h a v i n g continuous potentials
u",
.... u ' - so t h a tu " ( x ) < / ( x )
/or every xEP,
(2.9)~. u'~(x)~
m i n { / 0 ( x ) -Ms, he} /or every xePo,
(2.10)t = 1
a n d so t h a t ~ v,(R ~) is less t h a n a given positive n u m b e r a n d U ~S,, is a subset of a given n e i g h b o r h o o d of F o. A s / 0 is b o u n d e d on F0, there exists, according to (2.10), a smallest n u m b e r n o so t h a t (2.4) holds with ~ =v~ +... +~,,. F r o m (2.9) we see t h a t also (2.3) is t r u e for this choice of u.
As t h e second step of t h e proof we show t h a t there even exists a n absolutely continuous positive measure,
yJdx,
where ~ is infinitely differentiable a n d t h e total mass of~2dx
is less t h a n a given positive n u m b e r a n d S~ is a subset of a given neigh- b o r h o o d of F0, so t h a t (2.3) a n d (2.4) are t r u e with v replaced b yy~dx.
I n fact, let be a n infinitely differentiable function with c o m p a c t support, ~ ~>0,Sq~(x)dx = 1,
a n d let v 2 be t h e convolution of ~ a n d ~, yJ = ~ - u . ~ is t h e n infinitely differentiable a n d t h e measurevjdx
has t h e same total mass as u. B y choosing r small e n o u g h a n d such t h a tS ~ c S(O,r),
whereS(O,r)
is t h e sphere with centre 0 a n d radius r, we can m a k e S~ a subset of a given n e i g h b o r h o o d of S,. Asu" = K ~ v
we h a v eu ~ = K~e~ = K~e~-x-~ = ~ e u ~.
B y choosing r small e n o u g h we can t h u s also m a k e t h e difference u ~ - u " less t h a n a given n u m b e r , u n i f o r m l y on F . As we can prove a n inequality (2.4) for e v e r y e > 0 we conclude t h a t we also can prove t h e same inequality a n d t h e inequality (2.3) with ~ replaced b y a measure ~0dx h a v i n g t h e properties s t a t e d a b o v e .
R. WAT.LIN,
Continuous functions and potential theory
T h u s we h a v e p r o v e d t h e e x i s t e n c e of a n a b s o l u t e l y c o n t i n u o u s m e a s u r e ~/-~1 h a v i n g a n i n f i n i t e l y d i f f e r e n t i a b l e d e n s i t y so t h a t
u'(x) </(x)
for e v e r y x E F ,u~"(x) >I/o(X)
- 2 -1 for e v e r y x E F o.B y c o n s i d e r i n g / - u " i n s t e a d of / we g e t a n a l o g o u s l y a m e a s u r e ~u~ so t h a t u"'(x) < / ( x ) - u ' ( x ) for e v e r y x E F ,
u~"(x) >1/o(x) -
ur - 2-2 for e v e r y x E F 0, w h i c h givesuZ'(x) + uI"(x) </(x)
for e v e r y x E F ,u ' ( x ) +
ut"(x) >~/o(x)
- 2 -2 for e v e r y x E F 0.A f t e r n s t e p s we h a v e o b t a i n e d n p o s i t i v e m e a s u r e s , ~ i . . .
[~n,
s a t i s f y i n gn
~, urn(x)
< / ( x )/or every x E F ,
(2.11)i = l
n
t~=lum(x) >~ /o(x)- 2 -n /or every xE F o.
(2.12) W e can f u r t h e r m o r e s u p p o s e t h a t / ~ is a b s o l u t e l y c o n t i n u o u s a n d h a s a n i n f i n i t e l y d i f f e r e n t i a b l e d e n s i t y for i = 1 ... n, t h a t ~ [btt(R m)
is less t h a n a g i v e n p o s i t i v e n u m - b e r w h i c h is i n d e p e n d e n t of n, a n d t h a t t h e m a x i m a l d i s t a n c e b e t w e e n F 0 a n d a p o i n t on t h e b o u n d a r y of S . n t e n d s t o zero as n t e n d s t o i n f i n i t y . U n d e r t h e s e as- s u m p t i o n s i t follows t h a t { ~ ' p ~ } ~ = l converges w e a k l y t o a n a b s o l u t e l y c o n t i n u o u s p o s i t i v e m e a s u r e / ~ such t h a t t h e d e n s i t y of/4, is i n f i n i t e l y d i f f e r e n t i a b l e o m 0 F 0 . U s i n g (2.12) we o b t a i n , as t h e k e r n e l K is n o n - n e g a t i v e ,n
u~'(x)>~um(x)>~/o(X)-2 -n,
n = 1, 2 . . . for e v e r yx E P o.
1
This gives
u~(x)>1/o(X)
for e v e r y u E F o. As { ~ / x ~ } c o n v e r g e s w e a k l y t o /x we h a v e b y (2.11):n
/(x) >~ lira ~.
um(x) >~ ui'(x)
for e v e r y x E F .n - - i . oo 1
I t follows t h a t
u~'(x) =/o(X) /or every xE Fo,
(2.13)u~(x) <~/(x) /or every x E F .
(2.14) F r o m t h e c o n s t r u c t i o n we c o n c l u d e a t once t h a t u" is c o n t i n u o u s on C F 0. T o p r o v e t h a t u ~ is c o n t i n u o u s a t a p o i n t x 0 b e l o n g i n g t o F 0 we use (2.14). I t is well k n o w n t h a tu"(Xo) ~
l i mu~(x).
x . . . . ~ x e
6 0
ARKIY FSR MATEMATIK. B d 5 nr 5
To prove t h a t u~'(Xo) >~ lim u~'(x)
x . - - ~ x o
we observe t h a t x 0 is an interior p o i n t of F a n d so (2.14) yields lim u~(x) < l i m / ( x ) =/(Xo) =/0(x0) = u"(Xo).
x . - ~ x o x...~x o
This proves t h a t u ~ is continuous a t x 0 a n d t h u s everywhere.
F r o m t h e c o n s t r u c t i o n of u" we easily realize t h a t u ~ is infinitely differentiable on C F 0 in t h e case when K is infinitely differentiable in the interval r > 0 . Because if x I is a point in ~ F 0 we can write u":
n 1 n !
u" = ~. u " + (u u - 2 u ' ) (2.15)
1 1
a n d choose n 1 so large t h a t t h e s u p p o r t of I t - ~.~'itt does n o t contain x 1. T h e second t e r m of t h e r i g h t m e m b e r of (2.15) is t h e n infinitely differentiable a t x I due t o our a s s u m p t i o n on K, a n d t h e first t e r m due to the fact t h a t ~ i t ~ i s absolutely continuous a n d has a n infinitely differentiable density.
B y t h a t our t h e o r e m is completely p r o v e d in t h e case when/o(X) > 0 for every x G F 0. T h e general case follows i m m e d i a t e l y b y writing )to as t h e difference between t w o functions which are strictly positive on F 0.
R e m a r k 1. F r o m t h e proof it is clear t h a t t h e measure ~ in T h e o r e m 1 can be chosen with a+(R m) + a - ( R z) less t h a n a given n u m b e r so t h a t S~ is a subset of a given n e i g h b o r h o o d of F 0. W e can also m a k e u" t h e difference between two continuous potentials generated b y absolutely continuous positive measures h a v i n g densities which are infinitely differentiable on C F o.
R e n m r k 2. T h e condition K n o n - n e g a t i v e is n o t necessary for the validity of T h e o r e m 1. F o r instance, it is easy to realize t h a t t h e t h e o r e m is t r u e also for the kernel - l o g r.
R e m a r k 3. F r o m T h e o r e m 1 we can deduce t h e following result: Suppose that g is a strictly positive function which is lower semi-continuous i n R m and that C~(Fo)=0, F o compact. T h e n there exists a potential u~, continuous on C F 0, which is generated by a positive absolutely continuous measure tt such that Sg is a subset o / a given neighborhood o / F o and
u"(x) = g(x) /or every x E F o.
I n fact, there exists a sequence of continuous functions {gt}~ r 0<g~_l(x)<gt(x) for e v e r y x e F 0 , 4 = 2 , 3 ... converging pointwise t o g o n F 0. H e n c e there is a positive measure Its h a v i n g suitable properties a n d a continuous potential u ~ so that, if g 0 = 0 ,
ut'~(x) = gt(x) - gt-l(X) for e v e r y x e F0, i = 1, 2 . . .
I t is possible to arrange so t h a t { ~ i t ~ } F converges w e a k l y to a positive measure It h a v i n g t h e required properties a n d in fact also those s t a t e d for a in R e m a r k 1. This gives t h e required properties t o u ~ including t h e equality u"(x)=g(x) for e v e r y xE F 0. {Compare the calculations leading f r o m (2.11) a n d {2.12) to (2.13).)
H. WALLIN,
Continuous functions and potential theory
This r e s u l t c l e a r l y c o n t a i n s t h e following r e s u l t b y R u d i n [16] as a special case (the case
g(x)=
oo for e v e r y x E F 0 ) : S u p p o s e t h a t G is a n o p e n set c o n t a i n i n g t h e c o m p a c t set F 0 a n d t h a t C ~ _ 2 ( F 0 ) = 0 , m >~ 2. T h e n t h e r e e x i s t s a p o s i t i v e a b s o l u t e l y c o n t i n u o u s m e a s u r e p h a v i n g c o m p a c t s u p p o r t such t h a t u~_2 is i n f i n i t e o n F0, c o n t i n u o u s on C F 0 a n d h a r m o n i c in t h e c o m p l e m e n t of t h e closure of G. 1Remark 4. Let U
b e t h e o p e n u n i t s p h e r e inR m.
I f F 0 is a closed s u b s e t of t h e b o u n d a r y of U, /0E$(F0) a n dCm_~(Fo)=0,
m>~2, i t is n o t h a r d t o p r o v e , b y m o d i - f i c a t i o n s of t h e p r o o f of T h e o r e m 1, t h a t t h e r e e x i s t s a m e a s u r e ~ s u c h t h a t S , isUm-2(x)-/o(X)
for c o m p a c t a n d does n o t c o n t a i n a n y p o i n t s f r o m U a n d such t h a t o _e v e r y
x E F o. As
before we c a n g e t u ~ - 2 a s t h e difference b e t w e e n t w o c o n t i n u o u s p o t e n t i a l s g e n e r a t e d b y p o s i t i v e m e a s u r e s w h i c h m e a n s t h a t t h e e n e r g y i n t e g r a lIra_2( ]a I )
is finite. T h e r e f o r e we c a n conclude: F o r a n y f u n c t i o n / 0 E $(F0) t h e r e e x i s t s a f u n c t i o n u,u(x)=/0(x)
for e v e r yxEFo,
w h i c h is h a r m o n i c in U, c o n t i n u o u s in t h e closure of U a n d h a s a f i n i t e D i r i c h l e t i n t e g r a l ,f m / o ~ \ z
A converse of t h i s p r o p o s i t i o n will be p r o v e d in w 8.
3. Modulus of continuity of potentials
W e consider t w o k e r n e l s K a n d K 0 a n d s u p p o s e t h a t
lim
Ko(r )
{K(r)} -1 = oo. (3.1)r-~0
W e shall show t h e e x i s t e n c e of a m o d u l u s of c o n t i n u i t y on t h e set where ~K."lal is b o u n d e d b y a g i v e n c o n s t a n t , a m o d u l u s of c o n t i n u i t y w h i c h is c o m m o n t o all p o t e n t i a l s u ~ w i t h
a+(R m) +a-(R m)
less t h a n a g i v e n c o n s t a n t . I n w 4 we shall t h e n use t h i s r e s u l t t o p r o v e a c o n v e r s e of T h e o r e m 1.L e m m a 2.
Suppose that the kernels K and K o satis/y
(3.1).Then there exists a non- negative /unction t defined in the "interval r
> 0 , limr_.0t(r)=0, only depending on K and K o so that, if a is a measure with compact support, a+(R m) § m) <<.M 1 and
u ~ ( x i ) < M 2, i = 1, 2,then we have
I u~(xl) - u~(x,) I < (M1 + M~) t(I Xl - x~ I). (3.2)
Proo/.
W e consider o n l y t h e case w h e n a is a p o s i t i v e m e a s u r e , f r o m w h i c h t h e g e n e r a l case is a n i m m e d i a t e consequence. L e t x I a n d x~ b e t w o p o i n t s w i t hU~o(X~)
M2, i -- 1,2, a n d p u t ] x I - x~ I = r0- L e t S b e t h e o p e n s p h e r e w i t h c e n t r e (x I § x2)- 2 -1 a n d r a d i u s
2-1r0-}-rl,
w h e r e r 1 is a n u m b e r w h i c h we shall choose l a t e r d e p e n d i n g on r 0 so t h a t r I t e n d s t o zero w h e n r 0 t e n d s t o zero.u~(xl)--u~(x~)= f (K([xl--y])-- K(]x~--yl))da(y)~ f s + fcs= I + II.
1 R u d i n also t r e a t s t h e s i m p l e e x t e n s i o n t o a r b i t r a r y closed sets.
62
ARKIV FOR MATEMATIK. B d 5 nr 5
I n order t o estimate I we i n t r o d u c e t h e f u n c t i o n t~:
tl(r )
= (Ko(r)}-x 9g(r).
t I satisfies limr_,0 t l ( r ) = 0 according t o (3.1). I f G(x;r) is t h e value of a for the open sphere with centre x a n d radius r, we h a v e
, II < f:~ r) + fi'+~'K(r)da(xz; r)
fr,+r~ fle+r,
= tx(r) Ko(r ) da(xx; r) + tl(r ) Ko(r ) da(x~; r)
0
sup tl(r ) {U~o(Xl) ~- U~fo(Z2) } < 2 M 2 sup tl(r),
0~<r~ro+r~ 0~<r~ro+ri
i.e. I I]~< 2 M s sup
tl(r ).
(3.3)O~r<~re+r~
To be able t o estimate
11
we l~ave t o examine t h e difference K( [Xl - y I ) - K( I xz - y I) w h e n y E C S . W e n e e d a result of t h e following kind: There exists a non-negative function $~ defined in the interval r > O , limr-.ot2(r)=O,
o n l y depending on K, so that, for e v e r y ~] > O,K ( r ) - K ( r + o ) <~t2(~]) /or r>~tz(~), 0 < 0 < 7.
(3.4) W e suppose for a m o m e n t t h a t this has been p r o v e d a n d use it t o estimateII.
AsI l x x - y l - Ix2-yll<~ Ixl-x~l =r o
a n dI x t - y l ~ r 1,
/ = 1 , 2 , w h e n yECS, we h a v e b y using (3.4) with ~ = r 0 a n d choosing r I=t2(ro),
]1II < f~.s IK(lxl-yl)-K([x~-yl)l da(y) < M 1.$2(r0).
W e define t b y t(r0) = 2 sup
tl(r ) +
t~(r0), r o > 0,(3.5)
where sup is t a k e n for those r which satisfy 0 ~< r ~< r 0 +
$2(r0).
t satisfies t h e d e m a n d s of t h e lemma; (3.3) a n d (3.5) give (3.2).I t remains t o prove t h e existence of a function t2, limr_,0
t~(r)
= 0 , only depending on K, such t h a t (3.4) is valid. W e observe t h a t K is u n i f o r m l y continuous in t h e interval a ~- r < oo for e v e r y a > 0. F r o m this we conclude t h a t for e v e r y e > 0 there exists a largest n u m b e r q(e), which is possibly + co, so t h a tK ( r ) - K ( r + ~ ) < e for r>~e,,
0~<0~<q(e), (3.6) a n d it is n o t h a r d to realize t h a t this implies t h e existence o f a function t 2 with t h e desired properties so t h a t (3.4) is true. W i t h t h a t t h e l e m m a is proved.4. A converse of Theorem 1
W e denote b y ~/(K, Fo) t h e class of functions which are restrictions to F 0 of K - p o t e n t i a l s u~ of measures ~ with c o m p a c t supports. A f u n c t i o n / 0 E S ( F 0) belongs to
"ll(K, Fo)
if there is a function gE~/(K, Fo) well-defined a t every p o i n t of F0,Ho WALLIN, Continuous functions and potential theory
such t h a t ]o(x)=g(x) for every x E F o. W e shall prove the following converse of Theorem 1:
Theorem 2. Let K be a kernel and F o a compact set, CK(Fo) > 0 . T h e n there exists a /unction 1o belonging to $(Fo) but not to ~ ( K , Fo).
R e m a r k 1. F r o m t h e proof of T h e o r e m 2 it will a p p e a r t h a t also t h e following more general proposition is true: We can find a / u n c t i o n / o E $(F0) such that there is no / u n c t i o n / r o m ~ ( K , Fo) coinciding w i t h / o on F o except on a subset o/ 2' 0 o / K - c a p a c i t y
z e r o .
R e m a r k 2. I t will a p p e a r f r o m t h e proof t h a t t h e a s s u m p t i o n K n o n - n e g a t i v e is n o t necessary for the v a l i d i t y of t h e theorem. F o r instance the t h e o r e m is t r u e also for t h e kernel - l o g r.
R e m a r k 3. T h e o r e m 1 a n d T h e o r e m 2 give in particular the following characteriza- tion of c o m p a c t sets of K - c a p a c i t y zero: Let K satis/y the condition (e) o[ w 2. A compact set F o has K-capacity zero i/ and only i / e v e r y /unction ]rom $(F0) is the restriction to F o o] a continuous K.potential o / a measure with compact support.
Proo/ o/ Theorem 2. As CK(Fo) > 0 there exists a kernel K 0 such t h a t
lim K o ( r ) 9 ( K ( r ) } - 1 = c ~ ( 4 . 1 )
r - ~ 0
a n d CK,(Fo) > 0.1 (4.2)
According to (4.2) there is a positive measure % with t o t a l mass 1, Sv, c F 0 a n d IK.(~0) < ~o.
W e first use L e m m a 2 to deduce a p r o p e r t y on F 0 of an a r b i t r a r y K - p o t e n t i a l u~ of a measure a with c o m p a c t s u p p o r t a n d after t h a t we shall c o n s t r u c t a continuous function / with r e s t r i c t i o n / 0 to F 0 such t h a t / 0 does n o t h a v e this p r o p e r t y . L e t a be a measure with c o m p a c t s u p p o r t a n d M l = a + ( R m ) + a - ( R m ) . A s u ~ is finite except on a set of K0-capacity zero a n d %, due t o t h e fact t h a t IK.(v0) < 0% does n o t concentrate a n y mass on such a set, we conclude t h a t t h e set where uK, is finite has lal
%-measure 1, i.e. measure 1 with respect t o ~0. Consequently, for e v e r y e > 0 there is a c o n s t a n t M 2 =M2(e ) so t h a t u ~ is less t h a n M 2 on F 0 except on a subset of F 0 h a v i n g %-measure less t h a n e. According t o (4.1) a n d L e m m a 2 this means t h a t there exists a function t defined in t h e interval r > 0 , limr-)0 t(r) = 0 , o n l y d e p e n d i n g on K a n d K o such t h a t
lug(x1) - u~(x,)] <~ ( M 1 § M2) t(I x 1 - x 2 I), (4.3) /or x 1 and x 2 belonging to F o except when x 1 and x~ belong to a subset o/ Po having
%-measure less than e.
W e c o n s t r u c t t h e function / in t h e proof of the following lemma:
L e m m a 3. Let K o be a kernel, F o a compact set with CK.(Fo) > 0 , % a positive measure with S~.c Fo, vo(R m) = 1, IK,(%) < oo and t* a non.decreasing /unction de/incd in the interval r > 0 such that t * ( r ) > 0 i/ r > 0 , limr_~ot*(r) = 0 . T h e n there exists a /unction
1 Compare Carleson [3, p. 405 I.
64
X~K~V r 6 n MATEMATIK. B d 5 n r 5
1, de]ined and continuous everywhere, having the/ollowing property /or all su[[iciently small positive values el ~: F o r every Borel set E, E c Fo, with uo(E) <~, there are points x 1 and x 2 belonging to F o - E with ] x 1 -- x2 ] arbitrarily small so that
[/(Xl) - / ( x 2 ) l >~ M a t * ( I x , - x ~ I), M3 positive eonstant.~ (4.4) Using L e m m a 3 we can easily finish t h e proof of T h e o r e m 2. L e t K0, F 0 a n d r 0 in t h e l e m m a be identical with t h e kernel K0, t h e set 2'0 a n d t h e measure uo occurring in t h e proof of the theorem. W e choose t h e function t* in t h e l e m m a so t h a t
lim t*(r) (t(r)} -1 = oo. (4.5)
r--~0
I f / is t h e function o c c u r r i n g in t h e l e m m a it is clear b y (4.3), (4.4) a n d (4.5) t h a t there is no measure a with c o m p a c t s u p p o r t such t h a t u~ a n d / coincide everywhere on F 0.
This proves our theorem.
Proo[ o / L e m m a 3. W e shall c o n s t r u c t / as a s u m , / ( x ) = ~.F/s(x). To c o n s t r u c t {/s}~
we use three sequences of positive n u m b e r s (as}, {b,} a n d {d,}.
We first suppose t h a t t h e intersection between F 0 a n d a n a r b i t r a r y (m - 1)-dimen- sional plane parallel t o some coordinate plane has %-measure zero.
To construct/~, f o r a fixed i, we star~ b y covering F o b y m e a n s of c o n g r u e n t cubes from t h e sequence of sets T l = ( ~ t } which we used in the proof of T h e o r e m 1, a n d we use cubes w i t h diameters less t h a n or equal to 2-1.as. ~ After t h a t we separate those cubes, s a y eo;1,oJ;2 ... which intersect F 0 in a set of positive %-measure. F o r e v e r y cube eo~j, where j is a fixed n u m b e r , j = 1,2, ..., we consider t h e infinite strip which is d e t e r m i n e d b y t h e p o i n t s (x I, , . , x m) where x 1 varies arbitrarily a n d x ~ .... ,x m v a r y in t h e same intervals as t h e corresponding coordinates for an a r b i t r a r y p o i n t in conj.
I f there is n o cube w~8, s ~ , which is c o n t a i n e d in this strip a n d has a n ( m - 1 ) - dimensional edge plane in c o m m o n with o);i, we divide w;i into t w o rectangles b y an ( m - 1)-dimensional h y p e r p l a n e perpendicular to t h e xl-axis so t h a t b o t h rectangles g e t t h e p r o p e r t y t h a t t h e y intersect F o in a set of positive r0-measure. (Compare (4.9) below.) T h e cubes o)~I,r ... are in this w a y replaced b y rectangles m~,o)~ ...
We n o w choose t h e n u m b e r ds satisfying
0 < ds < m i n % ( F 0 N eo~). (4.6)
i
0 t l . H
I f r0(F 0 co,s)>2ds we divide, for 1 = 1 , 2 ... w,j into t w o or m o r e parts b y ( m - 1 ) - dimensional hyperplanes perpendicular to the xl-axis in such a w a y t h a t ~o~;,eo~, ...
are replaced b y rectangles w~1 ... o)~ m with t h e following properties:
uo(Fo\ U eo~j) = 0. (4.7)
j r 1
d~<%(FoN~osj ) <2d~, ] = 1 ... ne (4.8)
Every ~o tj has an (m - 1)-dimensional edge plane in common with a rectangle a~ s8 , s ~=], in the in/inite strip which is determined by the points (x 1, ...,x ~) where x 1 is arbitrary and x ~ ... x m vary in the same intervals az the corresponding coordinates/or an
arbitrary point o[ ws~. (4.9)
1 In general we cannot make (4.4) true for all x 1 and x~ belonging to F 0. This is a consequence of a result by Besicovitch, [1, p. 183].
We suppose=for the moment that the number a s is given.
It. WALLIN,
Continuous functions and potential theory
As the intersection between F o and an ( m - 1)-dimensional hyperplane parallel to some coordinate plane has %-measure zero, we can form a subset A~s of cots containing the b o u n d a r y of cots such t h a t the distance from the b o u n d a r y of co~s to co~s-A~s is larger t h a n zero and, due to (4.8),
vo(Fofl
( w ~ j - A t j ) ) > d , , j = 1 ... ni.We can also m a k e the choice of A~j such that, if ~ is a given number, ~}>0, and we define A b y
CO n t
A = U O (Atj N F0), (4.10)
t = l J = l
then we have ~o(A) <~.
We now choose [~. F o r every rectangle cotj there is an infinite strip of the kind described in (4.9). I n each such strip we choose [4 identically equal to
t*(at)
in every second set c o t s - A , and identically equal to zero in every second, counted from sets cots-Ats with points having smaller xX-coordinates to sets with points having larger xl-coordinates. I n the remaining points of R m we define [t such t h a t / , becomes con-n t
tinuous a n d less t h a n or equal to
t*(at)
everywhere and zero on C Us~lco, 9We now fix two points x t and x~ from LJ ~l{cots fi F0) \ A belonging to two different rectangles co~s bordering on each other in an infinite strip of the kind described in
~(0 f^r the distance between cots-Ats a n d cot,-Ats (4.9). I f we introduce the notation oss u
a n d define bt b y
b~ = m i n ~}~, ( 4 . 1 1 )
j * s
then, clearly, we have
I/t(xt)-/~(x[) I =t*(at)
andb~ <. Ixt-x~l <.at.
As the function t* is non-decreasing, we obtain
I/,(~,)
- / , ( ~ ; )I >/t*(
I~, -~; I)-
(4.12) For j > i we have I/s(x,)-/s(x~) I ~< 2 m a x I/~(y) [ ~< 2t*(as),which gives
I/s(x,)-/s(x~) I <t*(Ix,-x~l).2t*(as).{t*(b,)}-~,
j > i . (4.13) I f we furthermore assume t h a tat<bt_x,
i = 2 , 3 ... (4.14)we get
/s(x~)-/s(x~)
= 0, i < i . (4.15)P u t t i n g / ( x ) =
~'/t(x)
we have b y (4.12), (4.13) and (4.15)I/(x,) -/(~;) I ~ t*(I
x , - x ;I)[1 - 2 (t*(b,)}-~ 9 ~ t*(~)].
I f the expression in square brackets is larger t h a n or equal to 2 -1 , i.e. if
t*(aj) < ~ t*(b,),
J > i
(4.16)
6 6
ARKIV FOR MATEMATIK. S d 5 D.r 5
then [/(x,) -/(x~) [/> 89 t*([ x~ - x; I), i = 1, 2 . . . (4.17) I t is easy to realize t h a t it is possible to choose the sequence {a~} such t h a t (4.14) and (4.16) are satisfied and this shows t h a t it is possible to perform our construction.
(4.16) guarantees t h a t ~. / converges uniformly and therefore / is continuous.
Suppose now t h a t E is a Borel set, E c F0, v0(E) <e. To finish the proof of the lemma it is, according to (4.17), enough to prove that, for all sufficiently small values of e and for all i, there exist points xi and x~ from U1~21(r162 n F0) \ (E U A) belonging to two different rectangles cots bordering on each other in an infinite strip of the kind described in (4.9). Suppose, on the contrary, t h a t this is not the case. Thus, for some i, at least one third of the rectangles ~o~1 ... eo~,~ do not contain any points from F 0 \ (E t) A). Using (4.8) and the fact t h a t we have chosen A so t h a t v0(A ) <~7 we get
> v 0 ( E U A ) > d i . 3 i, (4.18) 8-b~]
where ni is the number of rectangles wij for a fixed i. B u t (4.7) and (4.8) give 1 = vo(Fo) <2d~n~,
which, combined with (4.18), gives e + 7 >6-1. As ~/ can be chosen arbitrarily small we get a contradiction if e is small enough.
The lemma is thus proved in the case when the intersection between F 0 and an arbitrary (m-1)-dimensional hyperplane parallel to some coordinate plane has v0-measure zero. If this condition is not satisfied we choose an (m-1)-dimensional hyperplane P parallel to some coordinate plane such t h a t v0(P f3 F0) > 0 and carry through the above construction o f / - - w i t h F o replaced b y P N F0--in the ( m - 1)- dimensional hyperplane P. The fact t h a t we m a y have vo(P N F0) < 1 does not change the idea of the construction. After having constructed ] in P we extend ] to a con- tinuous function in R z. The extended function clearly satisfies the conditions of the lemma which thus is proved.
CHAPTER I I . A n extension problem for continuous functions 5. S t a t e m e n t o f the problem
We first introduce some more notations. We denote the sequence (s 1 ... sh) of indices between 1 and m b y s and its length h b y [ s ] and we p u t
a h Ox"... ~x "h"
For any number p >/1 we denote b y L v the class of all Lebesgue measurable func- tions / in R m such t h a t S ]/(x)]Vdx < co; we denote b y L~oo the class of all measurable f u n c t i o n s / i n R m such t h a t Sr]/(x)]Vdx<oo for every compact set F. We u s e t h e notation
and we write II/ll~ instead of i[/[[~',~,.
67
H. "WALL1N,
Continuous functions and potential theory
W e shall use a class of d i s t r i b u t i o n s i n R m (in t h e sense of Schwartz [17]) consisting of f u n c t i o n s from L~or and, i n order to a v o i d confusion w i t h t h e u s u a l p o i n t w i s e d e r i v a t i o n , we a l w a y s s t a t e e x p l i c i t l y w h e n it concerns d e r i v a t i o n i n t h e d i s t r i b u t i o n sense. ~ denotes t h e Dirac m e a s u r e a n d we write D s i n s t e a d of DSS.
W e shall deal w i t h t h e following e x t e n s i o n problem: L e t F 0 be a c o m p a c t set h a v i n g m - d i m e n s i o n a l Lebesgue m e a s u r e zero. W e are i n t e r e s t e d i n f i n d i n g c o n d i t i o n s o n F 0 which g u a r a n t e e t h a t e v e r y f u n c t i o n / 0 E $ ( F 0 ) has a c o n t i n u o u s e x t e n s i o n / to R m such t h a t all t h e d e r i v a t i v e s o f / - - i n a c e r t a i n s e n s e - - o f orders less t h a n or e q u a l h b e l o n g to L ~, where h a n d p are g i v e n n u m b e r s , p >~ I. I t t u r n s o u t t h a t a r e l e v a n t c o n d i t i o n is t h a t
C~(Fo)=O
for a c e r t a i n :r 0 ~ c r irrespective of t h e r e g u l a r i t y of t h e set F 0 i n other respects. T h e results are f o r m u l a t e d i n t h e T h e o r e m s 3, 4 a n d 6 as c o n d i t i o n s o n t h e c o n n e c t i o n s b e t w e e n h, p, cr a n d t h e d i m e n s i o n m of t h e space.I n t h e case w h e n we are searching for a positive s o l u t i o n of o u r e x t e n s i o n p r o b l e m it is n a t u r a l to r e q u i r e t h a t t h e e x t e n d e d f u n c t i o n / is to be i n f i n i t e l y differentiable o n C F 0 (Theorem 3). W h e n we t r y to f i n d a converse (Theorem 4) of T h e o r e m 3 we shall use a c e r t a i n class of B e p p o Levi f u n c t i o n s of order h, BLh(LlPoc), 1 p ~> 1. This is t h e class of d i s t r i b u t i o n s T i n R ~ such t h a t all t h e d e r i v a t i v e s (in t h e d i s t r i b u t i o n sense) of order h are f u n c t i o n s b e l o n g i n g to L~or i.e.
DSTEL~oc
for all s w i t h Is[ = h.W e s t a r t our i n v e s t i g a t i o n b y discussing some properties of t h e class
BLn(Lroo),
p >~ 1, a n d t h e class of f u n c t i o n s which are i n f i n i t e l y d i f f e r e n t i a b l e o n CF0, where
C:r
= 0 for some ~ < m - l :1 ~ Following D e n y - L i o n s [7, p. 314] we s a y t h a t a f u n c t i o n g, defined i n
R m,
has t h e p r o p e r t y(AC)
i n R ~ if g is a b s o l u t e l y c o n t i n u o u s o n a l m o s t e v e r y line 2 w i t h a g i v e n direction if this d i r e c t i o n coincides w i t h t h e d i r e c t i o n of some c o o r d i n a t e axis.U s i n g t h e p r o p e r t y
(AC)
we get t h e following c h a r a c t e r i z a t i o n of t h e d i s t r i b u t i o n s i nBLh(L~or
BLh(L~oo )
consists of those d i s t r i b u t i o n s T i n R ~ which h a v e t h e following pro- perties: E v e r y d e r i v a t i v eD~T
(in t h e d i s t r i b u t i o n sense) w i t h 0 ~< l sl < h a is a f u n c t i o n w h i c h - - p r o p e r l y defined o n a set of Lebesgue m e a s u r e z e r o - - g i v e s a f u n c t i o n g,~h a v i n g t h e p r o p e r t y (AC); all t h e d e r i v a t i v e s of t h e first order of g~ i n t h e u s u a l pointwise sense are i n L~oo a n d t h e y also c o n s t i t u t e t h e d e r i v a t i v e s of g~ i n t h e d i s t r i b u t i o n sense.
This c h a r a c t e r i z a t i o n is g i v e n b y D e n y a n d Lions [7, p. 315] for h = 1. T h e general case follows easily f r o m t h e case h = 1 if we use a t h e o r e m b y K r y l o f f [17, p a r t I I , p. 37] to conclude t h a t if all t h e d e r i v a t i v e s of t h e first order of a d i s t r i b u t i o n are f u n c t i o n s i n L~or t h e n t h e d i s t r i b u t i o n itself is a f u n c t i o n i n L~oo. 4
2 ~ F 0 is a c o m p a c t set w i t h
C~(Fo)=O
for some r162 s a t i s f y i n g ~ < m - 1 . Suppose t h a t t h e f u n c t i o n g is defined a n d i n f i n i t e l y differentiable o n C F o a n d t h a t all t h e p a r t i a l d e r i v a t i v e s of g of order h b e l o n g t o L~oc, where p >~ 1. C~(2'0) = 0 a n d cr < m - 1 m e a n t h a t a l m o s t e v e r y line w i t h a g i v e n d i r e c t i o n does n o t i n t e r s e c t F 0. Otherwise 1 The discussion of this chapter concerning Beppo Levi functions in the case h = 1, should be compared to the discussions in Deny [5] and Deny-Lions [7]. The definition ofBLh(L~oc)
is found in the work of Deny-Lions. Compare also Nikodym [14].2 A function is absolutely continuous on a line if the restriction of the function to an arbitrary compact interval of the line is absolutely continuous.
a DST
with Isl =0 denotes T.4 Observe that by means of Kryloff's theorem it is possible to enunciate more than we have done in our characterization of the distributions in
BLh(L~oc).
68
ARKIV FOR M&TEMATIK. Bd 5 n r 5
there would exist a direction such t h a t the orthogonal projection _F0 of F o on an ( m - 1 ) - d i m e n s i o n a l normal plane to this direction would satisfy Cp(F0)>0 for every f l < m - 1 , contrary to the fact t h a t C~(F0)=0.1 This shows t h a t g and all the partial derivatives of all orders of g have the p r o p e r t y (AC). As all the partial derivatives of order h of g are in L~oc we can use the result b y D e n y - L i o n s [7, p. 315]
and Kryloff's theorem as in 1 ~ to conclude t h a t g defines a distribution belonging to BLh(L~oc ). I n particular we obtain t h a t 9 a n d all the partial derivatives of g of orders less t h a n h are in L~oc too.
3 ~ . As we in the extension problem only are interested in continuous functions we
~*~L ~ F0), p ~> 1, of functions in the following introduce two classes Ah(Lror ~ h t lor
ways: Ah(L~or is the class o//unctions / defined and continuous everywhere which-- considered as distributions---belong to BLh(L~oc). ~4~(I_nvo~, Fo) is the class o//unctions / defined and continuous everywhere in R m, infinitely di//erentiable on ~Fo, and such that all the partial derivatives o / / o / o r d e r s less than or equal to h belong to L~or
F r o m the discussion in 2 ~ we conclude t h a t A~(LlVor F0), with C~(F0)=0 for some
< m - 1 , is a subclass of Aa(L~or
4 ~ We shall need the following fact: L e t /0E S(Fo). I f there exists a function /e,~*(/~Voc, Fo) (or Ah(LlVoc)) coinciding pointwise w i t h / 0 on Fo, t h e n there exists a function/*eJ4~(L~oo, Fo) (or Ah(L~oc)) which is zero outside a compact set and coin- cides pointwise w i t h / o on F 0.
I n fact, as the f u n c t i o n / * we only h a v e to choose/~, where ~ is identically 1 on F 0, infinitely differentiable and has a compact support.
6. A n extension theorem We need the following lemma:
I,e.mma 4. Let 0 < ~ < fl < m. Let ix be a positive measure with ix( R m) < oo and suppose that u~ is bounded. Then we have
u ~ e L v i / 2 ~ < p = f l _ ~ (6.1)
and u~EZ~o~i[ l~<p ~ r < ~ m - < 2. (6.2)
More exactly, /or p > 1 we have, i/ a = #(Rm),
m - ~ (6.3)
and /or every sphere S with radius r, we have
m - - 6~
Ilu~llLp(s,<~M~(a,r).{supu~(x)} (~-1)1v, i / l < p < ~ _ < 2 . (6.t) Ml(a ) is a constant depending/urther on m, p and cr and M2(a,r ) a constant depending /urther on m, p, ~ and ft.
1 Compare for instance F r o s t m a n [9, p. 91] a n d Brelo~ [2, p. 330].
69
m WALH~, Continuous functions and potential theory
F o r a proof of t h e lemma, based essentially on HSlders i n e q u a h t y , we refer to du Plessis [15], D e n y [6] a n d Fuglede [10]. A proof based on f u n c t i o n t h e o r y is given in Carleson [4, p. 61 a n d p. 80] for some special values of r162 a n d ft. T h e m e t h o d is, however, applicable in t h e general case, too.
Theorem 3. Let F o be a compact set, p >~ 1 and h a positive integer. Suppose that either C~(Fo)=O /or c r where m-ph>~O and p>~2 (6.5)
o r
C~(Fo)=O /or some ~ satis/ying ~ < m - p h where m - p h > O and l ~ p < 2 . (6.6) Then every/unction belonging to S(Fo) can be extended to a/unction which is defined and continuous everywhere in R "~ and infinitely di//erentiable on CFo and such that all the partial derivatives o/ orders less than or equal to h o/ the extended /unction and the extended/unction itsel/ are in 1.2.
Proo/. W e first t r e a t t h e case a > 0 a n d assume t h a t either (6.5) or (6.6) is valid.
Suppose t h a t / 0 is a f u n c t i o n f r o m S(Fo). According to w 5, 4 ~ it is e n o u g h t o prove t h a t there is a function from/4~(LFoo, Fo) coinciding pointwise w i t h / 0 on F 0.
As C~(Fo)=0 there exists a n absolutely continuous measure a, c o n s t r u c t e d as in T h e o r e m 1, with c o m p a c t s u p p o r t such t h a t u~ is continuous everywhere, infinitely differentiable on CFo, u I"1 is b o u n d e d a n d u~(x)=/o(X ) for e v e r y
XeFo.
W e shall prove t h a t u~ E A~(L~oo, F0). A n easy consequence of the properties of a is t h a tf ,
DSug(x)= /or every xeCFo, (6.'/)
/or all sequences s with I s I <~ h i / ~ + h < re. The index x in D~ denotes derivation with respect to x.
(6.'/) yields, if = h , ~ + h < ~ ,
IDSu~,(x)l<.<MuLS~h(x)
/or every x e C F o , (6.8) where M is a c o n s t a n t only depending on h, ~ a n d m. As ul~ ~ is b o u n d e d we can, b y means of (6.8) a n d L e m m a 4 used with j~ = ~ + h , conclude t h a t all t h e derivatives of order h of u~ belong to L~or we use (6.1) if (6.5) is valid a n d (6.2) if (6.6) is valid.u~E Ah(Lloo, Fo) , a n d so t h e F r o m t h e discussion in w 5, 2 ~ we finally conclude t h a t " * v
t h e o r e m is p r o v e d if a > 0.
The case a = 0 is t r e a t e d analogously to t h e case a > 0, b y using t h e following l e m m a instead of L e m m a 4:
L e m m a 5. Let 0 <fl < m. Let/x be a positive measure with compact support. I/either I0(#) is finite and 2 <~p=m/fl or 1 <<.p<m/fl, then u~EL~or
The proof of L e m m a 5 is analogous to t h a t of L e m m a 4 (see Fuglede [10]).
7. A converse of Theorem 3
T h e o r e m 3 gives a positive solution of o u r extension problem with t h e e x t e n d e d function in the class M~(L~or F0) if F 0 satisfies (6.5) or (6.6). T h e following t h e o r e m gives a converse of T h e o r e m 3 as 14h(Lloo, Fo) is a subclass of ~4h(L~o~ p C~(Fo)=0 for some :r < m - 1:
70
ARKIV FOR MATEMATIK. B d 5 n r 5 T h e o r e m 4. Let F o be a compact set. A su/ficient condition/or the existence o / a / u n c - tion in S(Fo) , which cannot be extended to a/unction in Ah(L~oc), is that either
C~(F0)>0 /or some o~ satis/ying c r where m-ph>~O and p > 2 (7.1) or C~(Fo) > 0 / o r or = m - p h where m - p h > 0 and 1 ~<p ~ 2. (7.2) I n order not to encumber the proof with details we shall not t r e a t the case when m - p h = 0 , 1 ~<p ~< 2. I t is, however, no difficulty to modify the proof which we shall give to obtain: The theorem is also true in the case when
Co(Fo) >O, m - p h = O , l~<p~<2. (7.3) An inspection of the Theorems 3 a n d 4 shows t h a t t h e y give a complete solution of our extension problem when p = 2 b u t not when p 4= 2.
Theorem 4 is proved in w 8. F o r the proof we need integral representations of those functions in BLh(L~or which are zero outside a compact set. To deduce these we use the following formulas (Schwart z [17, p a r t I, p. 47]): I f Aa is the Laplace operator iterated h times, h >~ 1, t h e n we have, in the distribution sense, if (~ is the Dirac measure and M 1 and M 2 are certain constants only depending on h and m,
M 1 . A h l x [ ~h-m =(~ i / m - 2 h > 0 or m - 2 h < 0 , m odd, (7.4) M 2 9 Aa(] x ]2h- m log Ix ]) = ~ i / m - 2h ~< 0, m even. (7.5)
We can write Ah = ~ a , DSD ~, (7.6)
where as are constants a n d the sum is extended over a n u m b e r of multiindices s with Is I = h. I f / e BLh(L~oc ) a n d / has compact support we obtain, b y means of (7.6), in the ease when (7.4) is valid,
/ = 6 ~ e / = M l & h l x p -'n ~e /~- M i ~ a,D'-)e DS]xl2~-m ~e / = M~ ~.a,DS[x[ 2h-m ~r D~/.
The distributions D~[xl 2a-'n and DS/ are functions and DS/ has compact support.
The convolution DS[x] 2a-m-)eD~/can consequently be written as an integral and as D~/EL', we get:
I / / E BLa(LI~oo), / has compact support and the/unction Ics. a is defined by b~.a(x ) = DS[xl 2h-m, then there are/unctions g~ with compact supports, gsEL ~, and constants b~
so that
](x) = ~ bs~lcs.h(x-y)g,(y)dy a.e., i / m - 2 h > 0 or m - 2 h < 0 , m odd. (7.7) ,]
Analogously we get by (7.5)--with other values on the constants b~--i/ the /unction
* x = D S ( I x l 2 ~ m l o g l x b* h is defined by k~. h( ) ]),
/.
l(x)
8.h(x--y)gs(y)dy a.e., i/ m - - 2 h < O, m even.J (7.8)
The sums in (7.7) and (7.8) are extended over a number o/multiindices s with Is[ = h.
I n the proof of Theorem 4 we eilso need the following lemma:
H, WALLIN, Continuous functions and potential
theory
Lemraa O. 1 Let K be a kernel. Suppose that g~, i = 1, 2, are/unctions having the property that/or every e > 0 there exists a Borel set E with C~(E) < e such that the restrictions o/
gl and gr to CE both are continuous. I] we,/urthermore, assume that gl(x) =g2(x) a.e. and that the set o/points x where g l ( x ) 4 g~(x) ks a Borel set, then gl(x) =g2(x) except on a set o/ K-capacity zero.
Proo/. L e t E 1 be t h e set of points x where gl(x)~g2(x) a n d suppose t h a t C~(E~)=
a > 0 . Choose e, 0 < e < a , a n d let E be a Borel set with C~(E) < e such t h a t t h e restric- tions of gl a n d g2 t o CE b o t h are continuous. W e f u r t h e r choose a c o m p a c t subset F of E 1 \ E with C~(F) > a - e . L e t F~, for n = 1,2 ... F . ~ F , be the u n i o n of finitely m a n y closed spheres h a v i n g radii ~<r,, where r,-->O w h e n n-->oo, a n d centres be- longing to F a n d let F * consist of t h e set of points situated a t a distance ~<r~ f r o m F , .
I f ~u, is a c a p a c i t a r y distribution belonging to t h e kernel K a n d F , , t h e n
u~-(x) < A. {0~(F.)}-~,
(7.9)where A is the c o n s t a n t in (1.4). Let rpn , ~ef.dx = 1, be a n o n - n e g a t i v e f u n c t i o n which is infinitely differentiable a n d let S% be a subset of S(0, rn). W e define yJ~ b y v/. =cf.-)e/an. yJndx is a positive measure with t o t a l mass 1 a n d S ~ c F*. Since E 1 has Lebesque measure zero there exists for e v e r y ~ > 0 a closed set H.(~), H.(~) c F * \ El, such t h a t t h e restriction of t h e measure yJndx to H.(~]) has t o t a l mass >/1 - ~ . H e n c e
IK(~)n)/~ (1 __~)2 {CK(Ha(~))}-I/~ (1 _~)2 {CK(.~n* \ j~1}-1 7_.+0 gives
IK(~n) ~ (C1c(F*n \ El)} -1.
(7.10)On the other h a n d we h a v e b y (7.9)
u~"(x) = K * ~ . -.x- ,u,,Cx) = ~ . * u p ( x ) < A . { O x ( F . ) } -~, which gives I K ( ~ , ) < . A . { O K ( F , ) } -1. This inequality a n d (7.10) give
C~(F*\E1) >i A - I C ~ ( F . ) >1 A-1Cx(F) > A -1" (a - e).
B y choosing e so small t h a t A - l ( a - e ) > e we get CK(F* \ El)>CK(E) as CK(E)<e, i.e. the set (F* \ El) \ E is n o n - e m p t y .
L e t x~E (F* \ E l ) \ E. According to our construction we h a v e
g l ( X . ) = g2(Xn), n = 1,2 ... a n d ( 7 . 1 1 )
There exists a point y, E F such that ] x , - y , [ ~<2r,, n = 1 , 2 ... (7.12) W e n o w prove t h e existence of a p o i n t x o f r o m F where gl a n d g2 coincide which gives a contradiction to t h e fact t h a t F c E 1. W e suppose t h a t t h e sequence of points (x,} converges to a p o i n t x 0. (If this is n o t t h e case, we choose a c o n v e r g e n t sub- sequence.) F r o m (7.12) we conclude t h a t (y,} converges t o x 0 too. H e n c e xoEF.
W e n o w use the estimate
I gl(Xo) - gu(Xo) l < ]gz(Xo)
-gl(Xn) I
-~ [gl(Xn) --g2(xn)] +
I - g .(xo) I 9 z A special case of this lemma is given in Deny and Lions [7, p. 353].72
ARKIV FOR MATEMATIK. B d 5 n r 5
The second t e r m of the right m e m b e r is zero according to (7.11). As xn and x o belong to CE and the restrictions of 91 and g2 to CE are continuous, we conclude t h a t the two remaining terms of the right m e m b e r are arbitrarily small when n is large.
This proves t h a t gx(Xo)=g2(x0) which gives a contradiction to the fact t h a t x o E F . 8. P r o o f o f T h e o r e m 4 a n d a theorem on h a r m o n i c f u n c t i o n s
The idea of the proof of Theorem 4 is as follows: We start b y observing that, due to w 5,4 ~ we only h a v e to prove the existence of a function ]o E $(Fo) which cannot be extended to a function in ~4h(L~or with compact support if (7.1) or (7.2) is satisfied.
B u t every function in ~4h(/~Voc) is--considered as a d i s t r i b u t i o n - - a n element in BLh(Lroc ). This means t h a t every function in Mh(LlVoc) with compact support can be written on the form (7.7) or (7.8). Following the method of the proof of Theorem 2 we shall deduce moduli of continuity of the right m e m b e r s of (7.7) and (7.8) having properties analogous to those of the modulus of continuity of the potentials in the proof of Theorem 2. This will also show t h a t the right m e m b e r s of (7.7) and (7.8) have such properties t h a t we can use L e m m a 6 to conclude t h a t every function in Mh(Lroo) with compact suppor~ has a representation (7.7) or (7.8) valid not only a.e.
b u t everywhere except on a set of K0-eapacity zero, where the kernel K 0 is defined below. (Compare (8.1) and (8.17).) The choice of K 0 is such t h a t C~o(Fo) > 0 a n d this will enable us to infer, from L e m m a 3, the existence of a function [0ES(F0) which cannot be extended to a function in Mh(/~Vor if (7.1) or (7.2) is valid.
The case 1--<19--<2. We s t a r t b y treating the case 1 ~<19~2 a n d we suppose conse- quently t h a t (7.2) is true for the given set F 0 a n d given values on h and 19, 1 <<-19 <~2.
Since C~(Fo)>0 for ~ = m - 1 9 h there exists a kernel K 0 satisfying
lira Ko(r ) r z - v ~ = co, CKo(Fo) > O, lira Ko(r ) > 0. (8.1) As CK.(Fo)>0 we can choose a measure v 0 with
v o >1 O, vo(R m) = 1, S,0c Fo, IK~ < cr (8.2) We formulate the information which we need a b o u t the right members of (7.7) and (7.8) in the following lemma:
L e m m a 7. Let the compact set Fo, the kernel Ko, and the measure v o satis/y (8.1) and (8.2) and sulx2aose that 19 and h are given numbers, h a positive integer, with m -19 h > 0,
1 <19<2. Using the notations o/(7.7) and (7.8) we de/ine t h e / u n c t i o n vh by
vh(x)=~bsfks.~(x-y)g~(y)dy i/
m - 2 h > 0 or m - 2 h < O , m odd, (8.3)by vh(x) ~ (8.4)
and = ~. b~ k * . h ( x - y ) g s ( y ) d y i / m - 2 h < ~ O , m even, .I
at those 1points where the right members are well de/ined. The s u m s are extended over the same multiindices s as in (7.7) and (7.8), i.e. over a number o / s with I sl = h; b s are constants and gs /unctions in L v with compact supports. Then there exists a non-negative /unction t h de/ined in the interval r > 0, limr+0 th(r ) = O, only depending on m, h and K o such that the/ollowing assertion is true:
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