A connection between a,capacity and LP-classes o f differentiable functions

1.64051 C o m m u n i c a t e d 11 M a r c h 1964 b y O. FROSTMAN a n d L . CARLESON

A connection between a,capacity and LP-classes o f differentiable functions

B y HANS WALLIN

.

Let x = (x 1 .. . . . x m) be a point in the m-dimensional Euclidean space R m. The following measure was recently used b y Serrin [6] for investigating removable singularities of a class of quasi-linear partial differential equations:

Definition. Let E be a bounded set in R m. M , (E), where 1 <~ s < oo, is defined by

M, (E) = inf

f Igrad 18

^{d~, (1.1)}

where the in/imum is taken over all continuously di//erentiable /unctions y~ which have compact supports and are >1 1 on E. I / s >1 m we also require the support o/

v 2 to belong to a certain fixed sphere ix[ < R 0 < oo which is independent o/ E.

We intend to investigate the connection between Ms(E) and the potential theoretic ~-capacity of E. As M s ( E ) = Ms(E), where E is the closure of E, the only case of interest is to consider compact sets. The investigation has a close connection with [7], to which we shall refer concerning some details of the proofs.

Let us first introduce some notations. The support of a measure # and of a function / is denoted b y S~ and S r respectively. S(r), r > 0, is the closed sphere J x I ~< r. The ~-potential, 0 ~< ~ < m, of a measure ~u is denoted b y u~, where

u"~(x)- (d~(y)

- 3 1 ~ . , if O<~<m,

and u~ (x) = f log [ x l--_ y j d/x(Y) 9

Here and elsewhere, the integration is to be extended over the whole space, if no limits of integration are indicated. If # is absolutely continuous and has a density /, dlu = / d x , we also write u~ instead of u~.

If I~(~u) denotes the energy integral of lu,

L(~) = f u"~ d~(x),

331

H. WALLIN, :c-capacity and LV-classes of differentiable functions we define t h e ~ - c a p a c i t y of a b o u n d e d Borel set E, C~(E), b y

C~(E) = {inf I~(/x)} -1,

where t h e i n f i m u m is t a k e n over all positive measures /x w i t h t o t a l mass 1 a n d S~ c E. W h e n ~ = 0 we m a k e this definition o n l y if the d i a m e t e r of E is less t h a n 1. F o r a n a r b i t r a r y Borel set E we p u t Co(E ) = 0 if a n d o n l y if Co(E f3 S ) = 0 for e v e r y sphere S w i t h d i a m e t e r less t h a n 1.

W e shall use t h e well-known fact t h a t if F is a c o m p a c t set w i t h C~,(F)>0 - - w e suppose t h e d i a m e t e r of F less t h a n 1 if ~ = 0 - - t h e n there exists a u n i q u e positive measure v w i t h t o t a l mass k, k > 0 , a n d S ~ c F such t h a t inf, I~(u) is a t t a i n e d for v = v where v ranges o v e r t h e class of all positive measures w i t h t o t a l mass k a n d S~ ~ F . ~ is called t h e capacitary distribution w i t h t o t a l mass k of order ~ of F . u~ has t h e following properties:

u~(x)>~k{C~(F)} -1 for e v e r y x E F except w h e n x belongs

to a set Of ~ - c a p a c i t y zero. (1.2) u~(x) <~ k{C~(F)} -1 for e v e r y x E S~. (1.3)

u~(x) <~M.k{C~,(F)} -1 everywhere, (1.4)

where M is a c o n s t a n t which m a y be chosen o n l y d e p e n d i n g on m. I f a > 0 w e m a y in fact choose M = 2 ~ < 2 m. W e shah also use t h e fact t h a t if F is t h e u n i o n of a finite n u m b e r of closed spheres, t h e n

u~(x) ~ k{C~(F)} -1 for e v e r y x E F . (1.5) T h e fl-dimensional measure, 0 < fl < m, of a b o u n d e d set E, L~ (E), is defined as

inf ~ r~,

V

where t h e i n f i m u m is t a k e n over all the coverings of E b y families of open spheres w i t h radii {r,}.

L e t C r be t h e class of all infinitely differentiable functions in R z a n d C~

those functions in C ~ t h a t h a v e c o m p a c t supports. Lror p>~ 1, is t h e class of all Lebesgue m e a s u r a b l e f u n c t i o n s / in R ~ such t h a t j'y I/(x)[ p d r < ~ for e v e r y c o m p a c t set F a n d L ~, p >/1, is t h e class of all m e a s u r a b l e f u n c t i o n s f such t h a t ~ I/(x)l ~ dx < oo. W e use t h e n o t a t i o n

a n d we write I l f l l - instead of II/[ILp(R~).

.

Theorem. (A). Let F be a compact set in R m with C~(F)=0, where 0 <~o~ < m.

I / o~=0 we suppose that F is a subset o/ the sphere I x l < R 0 , where R o is the constant occurring in the de/inition o/ Ms(F). The [ollowing conclusions are true:

332

I / 0 ~ < ~ < m - 2 , then M ~ _ ~ ( F ) = 0 . (2.1) 1/ O < ~ m - 2 < o ~ < m - 1 , then M~_~_~(F)=0 /or every s > 0

such that m - cr ~ 7> 1. (2.2)

(B). Let F be a compact set in R m with M p ( F ) = O , where l <.p<.m. The /ol- lowing conclusions are true:

I [ l <~p~2, then Cm_~(F)=O. (2.3)

I / 2 < p < ~ m , then C~_~+~(F)=0 ~or every e > 0 . (2.4) For the proof we need the following ]emma.

Lemma 1. Let 0 < or < fl < m. Let 1 a be a positive measure with i~(R m) < r Then

II u~ II- < MI"

{/~(Rm)} l/p"

( sup u~(z)} (p-1)/p,

x E R m

= ; ( 2 . 5 ) provided 2<.p f l - a

and /or every sphere S with radius r we have

m - 6 r

II u~ [I-<s, < Ms" {#(Rm)} ~/'" ( s u p u~(x)} ('-a)/p, provided 1 ~<p < ~ < 2. (2.6) M 1 is a constant depending on m, p and cr and M 2 is a constant depending on m, p, cr fl and r.

References to papers where this lemma is proved can be found in [7], p. 70.

Proo/ o/ (A) o~ the theorem. We first t r e a t the case cr L e t F~, for n = 1, 2, 3 . . . . , be t h e union of finitely m a n y closed spheres such t h a t C~(F=)< n -1 and F~ D F, where F is the given compact set with C~(F)= 0. L e t /~= be the capacitary distribution of order :r of Fn with total mass

~.

( F . ) = 2 n . C ~ ( F . ) .

This means t h a t 0 < / t n ( F ~ ) < 2 . According to (1.5) and (1.4) we have

and

u~"(x)>~2n for every x E F ~ , (2.7)

u~"(x) < 2 M n everywhere. /2.8)

(2.7) and (1.3) give t h a t u~- is constant on S t , , i.e. the restriction of u~" to St, is continuous and consequently u~" is continuous everywhere according to the continuity principle.

We now form v2~ = r 0e/~., i.e. ~n (x) = S q- (x - y) d/~= (y), where ~= E C~ ~ is a non-negative function with S q~dx = 1. This means t h a t ~ y ~ d x < 2 . B y choosing S% belonging to a sufficiently small neighborhood of the origin we can make 3 3 3

H. WALLIN, a-capacity and

LP-classes of differentiable functions

Sv, a subset of a given n e i g h b o r h o o d of F , a n d accordingly also of F . As u~" is con- tinuous we can, in this way, also m a k e t h e difference

l u~"(x)- u~"(x)l

less t h a n a n y given positive n u m b e r e v e r y w h e r e (of. [7], p. 59). D u e to (2.7) a n d (2.8) we can c o n s e q u e n t l y choose ~0, such t h a t

u~.(x)>~n

for e v e r y

x e F .

(2.9)

a n d u~"(x) < M - n everywhere, (2.10)

where M is a n e w c o n s t a n t which is i n d e p e n d e n t of n. W e also observe t h a t u ~ - ~ C ~ as ~ . e C ~ .

~OW we choose r 0 such t h a t F U S ~ ~ S ( r 0 / 2 ) f o r e v e r y n. [According to t h e a b o v e we can m a k e t h e c o n s t r u c t i o n of ~0, so t h a t this choice of r 0 is possible.]

L e t ~0 E Cff be a function, i n d e p e n d e n t of n, which is identically equal to 1 in

S(ro)

and p u t

/n(X)= u~"(x)

a n d

gn (X)

= Tb - 1 " fn (X) " Cf(X).

W e observe t h a t

g ~ E C ~ a n d gn(x)>~l for e v e r y x E F . (2.11) F o r every p >/1 we obtain, w i t h c o n s t a n t s M w h i c h are i n d e p e n d e n t of n:

f ' grad g. ,~' dx << M " n- P f , /. grad w ,~ dx + M " n- P f , cf grad L [~ dx = ln + I I. .

As ~0 is identically equal to 1 in

S(ro)

a n d

[L(x)[<2.(r~ -~

if [ x l ~ r 0,

\ 2 /

we get f

In ~< M - n - P . m a x [/n (x) l | J g r a d ~

[" dx

< const 9 n-~.

Ixl~>ro J

L e t r 1 be i n d e p e n d e n t of n a n d chosen so t h a t

S(rl)D S~.

As

II~

< e o n s t , n - ~

f s

[ g r a d / n I ~

dx,

(rl)

we w a n t to estimate I grad/nl. D u e t o t h e properties of yJ~ we o b t a i n

]grad/n(x)[ < c o n s t , u~_ 1 (X) for e v e r y x. (2.12) W e n o w choose p :

p = m - ~

if 0 < ~ < m - 2

a n d

p = m - a - e

if

m - 2 < ~ t < m - 1 .

where e > 0 is chosen satisfying m - ~ - e/> 1.

334

D u e t o (2.12) a n d this choice of p we can use (2.5) or (2.6) in L e m m a 1, w i t h fl e q u a l t o ~ + 1 , t o e s t i m a t e / / ~ . This gives, as ~y)~dx<2,

I I n < e o n s t - n -p { sup u ~ (x)) ~-1,

X E R m

and, according t o (2.10),

II~ < const 9 n -1, w i t h c o n s t a n t s t h a t a r e i n d e p e n d e n t of n.

T h e e s t i m a t e s of I~ a n d I I n show t h a t

lim

flgradg.l d =O

w i t h our choice of p . C o m b i n e d w i t h (2.11) this gives M ~ ( F ) = O , which m e a n s t h a t (A) of t h e t h e o r e m is p r o v e d in t h e case w h e n a > 0.

W e n o w p r o v e (A) w h e n a = 0 . W e can c o v e r F b y finitely m a n y closed spheres (S~}~ w i t h d i a m e t e r s less t h a n 1 such t h a t (J ~ S~ is a s u b s e t of I xl < Ro- I t is clearly enough to p r o v e t h a t M m ( F ~ S 0 = 0 for i = 1, 2, ..., N , a n d it is c o n s e q u e n t l y e n o u g h to consider t h e case w h e n F itself has d i a m e t e r less t h a n 1.

W e can t h e n r e p e a t t h e c o n s t r u c t i o n which we used when ~ > 0 b u t w i t h ob- vious modifications. F o r instance, t h e choices of ~ a n d ~ are m a d e so t h a t S~, a n d Sv are subsets of Ix I < R 0 a n d we use t h e following l e m m a [el. F u g l e d e 3, p. 301] i n s t e a d of L e m m a 1:

L e m m a 2. Let 0 < f l < m and 2<<.p=m/fl. Let It be a positive measure with compact support, S , ~ S ( r l i . I / S i8 a sphere o/ radius r 2 and o),n denotes the sur- /ace o/ the unit sphere in R m, then

fs{

u~(x)}P dx <~ {it(Rm)} ~-2 "eomIo(/a ) + M " {it(Rm)) p, where M is a constant depending on m, r I and r 2.

Remark. T h e following result a n d its p r o o f h a s b e e n c o m m u n i c a t e d t o m e b y P r o f e s s o r L e n n a r t Carleson:

I / F is a compact set with L~(E)=O, 0 < a ~ m - 1 , then M m _ ~ ( F ) = 0 .

As C ~ ( F ) = 0 implies L ~ + ~ ( F ) = 0 for e v e r y e > 0 , this gives a b e t t e r result t h a n (A) of t h e t h e o r e m w h e n 0 ~< m - 2 < ~ < m - 1.

T h e proof t h a t L ~ ( F ) = O implies M m - ~ ( F ) = 0 proceeds in t h e following w a y . L e t {x~)[ be g i v e n p o i n t s a n d S~ (r), r > 0, t h e o p e n sphere Ix - x, I < r, v = 1, 2 , . . . , n, a n d s u p p o s e t h a t {r~)~ are chosen so t h a t [J ~ S~(r,)~ F.

W e define linear f u n c t i o n s l~ b y 2rv -- r

l v ( r ) - , r ~ < r ~ < 2 r ~ r~

3 3 5

H. WALLIN,

or-capacity and LV-classes of differentiable functions

and p u t

1 when x ES,(r,),

~ , ( x ) =

I I~(Ix-x,[)

when

xeS,(2r~)-S~(r,),

[ 0 when x belongs to the complement of S~(2r~).

Then we have

]grad~0~]=r; 1 in the interior of

S,(2r,)-S~(r~).

If we p u t

~(x)=

m a x ~ ( x ) ,

l ~ i ~ n

then ~p(x)~> 1 on F and

Igrad Vl m." e x

~< ~

J ~,,r.) r: ~m-~ e x < const. ~-~ r:.

where the constant only depends on m. If L~ (F) = 0 we can make ~ r~ arbitrarily small, which means t h a t

f lgra d ~[m-~

dx

will be arbitrarily small. B y using standard methods to approximate v 2 it is possible to prove the existence of a continuously differentiable function / with compact support and

/(x) 7>

1 on F so t h a t

f lgrad /['~-~dx

is less t h a n a n y given positive number. This means t h a t L , ( F ) = 0 implies

i m - ~ ( F ) = 0.

.

Proo/ o/ (B) o/ the theorem.

L e t F be a compact set with

Mv(F )=0

where

l<~p<~m.

We m a y assume m > 1 because if m = p = l , then My(F) > 0 for e v e r y F.

We define ~ b y

g = m - p

if

l <.p<~2,

and g = m - p + e if

2<p<~m,

where e > 0 , (3.1)

and we shall prove t h a t

C~(F)= O.

As M p ( F ) = 0 there exists a sequence {/,}F of continuously differentiable func- tions with compact supports and

3 3 6

/ = ( x ) > n f o r e v e r y x E F ,

f I g r a d / n ]P dx < const., n = 1, 2 . . . (3.2) a n d

d

I n t h e case w h e n m = p we suppose f u r t h e r m o r e , as we m a y , t h a t SI, is a subset of I x l < R o for e v e r y n, where R 0 is t h e c o n s t a n t occurring in t h e definition of M , ( F ) .

Considered as a distribution, /~ belongs to t h e class BL~(L~oo)of distributions in R ~ s u c h t h a t all t h e partial derivatives (in t h e d i s t r i b u t i o n s e n s e ) o f t h e first order are functions in LaPoc. This f a c t a n d t h e f a c t t h a t Sr~ is c o m p a c t m e a n [see for instance 7, p. 71] t h a t there exist c o n s t a n t s b~ a n d d~, n o t d e p e n d i n g on n, such t h a t

51b,

m

-yl

^{/n(y) dy a.e.}

a n d

In(x) =,~d~ log[ -yl.U~y,/.(y)dy ^a.e.

for n = l , 2 , . . . , if m > 2 , (3.3)

for n = 1 , 2 . . . if m = 2 . (3.4)

H o w e v e r , since all t h e partial derivatives of t h e first order o f / ~ are continuous, we conclude t h a t also t h e integrals in (3.3) a n d (3.4) are c o n t i n u o u s and, con- sequently, t h a t t h e relations (3.3) a n d (3.4) are t r u e e v e r y w h e r e in R m.

To finish t h e proof of (B) of t h e t h e o r e m we n e e d a n e s t i m a t e of t h e a - c a p a c i t y of t h e set H(~ n) where t h e r i g h t m e m b e r s of ( 3 . 3 ) a n d ( 3 . 4 ) a r e larger t h a n a, a > 0. B y m a j o r i z i n g t h e integrals in (3.3) a n d (3.4) we o b t a i n t h a t there exists a sequence {g~} of n o n - n e g a t i v e c o n t i n u o u s functions such t h a t H(a n) is a subset of t h e set G(a n) where

u g. x f gn(Y)

m-l( ): JIx~m-idy

is larger t h a n a. W e m a y also a s s u m e t h a t Sg~ c Ss. and, due t o (3.2), t h a t

fg

^{V dx} < const, n = 1, 2 . . . (3.5) where t h e c o n s t a n t is i n d e p e n d e n t of n.

C~ (G(~ n)) is e s t i m a t e d b y s t a n d a r d m e t h o d s . T h e two eases p ~< 2 a n d p > 2 give different calculations. F o r t h e sake of completeness we t r e a t one of t h e m , t h e case p > 2, in detail.

T h e estimate of C~,(G~ '~ is s o m e w h a t facilitated for certain values of m a n d p if I.JnSr~ is a b o u n d e d set. I.JnSr~ is b o u n d e d if m = p as Sis is a subset of I x l < R o, n = 1 , 2 , 3 . . . in this case. B u t even w h e n m > p > ~ l we m a y choose {/n} so t h a t LIn Sf,, is a b o u n d e d set. Because if I.Jn Sis is n o t b o u n d e d for t h e sequence {]~} which was chosen originally, we replace {/n} b y a sequence {/*}

defined b y /* (x)=/n (x)'~v(x), where ~p(x) is a f u n c t i o n in C~ which is identically

a. WALLI1%

O~-capacity and L~-classes of differentiable functions

equal to 1 in a neighborhood of F. Then /* is continuously differentiable,

/*(x) > n

on F and the set (J~ $I: is bounded. F r o m the estimate

f ,grad/*,'dx<const f ,/~gradw,~dx+const f ,v/grad/.,~dx,

(3.6) we m a y , finally, prove t h a t (3.2) is true with fn replaced b y /*: according t o (3.2) the second t e r m of the right m e m b e r of (3.6) is less t h a n a constant which is independent on n. I n order to realize t h a t the whole expression (3.6) is less t h a n a constant it is hence enough to prove t h a t

suplll=ll ( )<oo m>p l,

n

for every bounded set E. B u t this is an immediate consequence of (3.2), (3.3) a n d (3.4) if p = l and of (3.2) a n d the following inequality b y Sobolev if m > p > l .

II/~ II~, ~

const

II

grad/~ I1--,

r = mp , m - p

where the constant is independent of n.

I n the calculations below, we assume, as we accordingly m a y , t h a t [J~ Sin is bounded. This means t h a t also (J~ So. is bounded.

Now let m >~p > 2. Since U n Sgn is bounded, we can choose a finite n u m b e r r 0 such t h a t

(J, Sg~ c S(ro) = So.

L e t /x be a positive measure with S~ ~ G(a ~) a n d /x(R z) = 1. We obtain b y means of H61der's inequality, if

p ' = p / ( p - 1 ) ,

a < fu~_,(x)d/~(x) = fsU~m_I(X) gn(x)dx~H~nl]I2"HU~m_IlIL~'(S.).

^(3.7)

To estimate the last norm we use formula (2.6) of L e m m a 1 with f l = m - 1.

An easy calculation shows t h a t the conditions of the l e m m a are satisfied if we choose e small enough in (3.1), a choice which we m a y obviously m a k e w i t h o u t limitation. (2.6) gives t h e n

II 4 const {sup u:(x)} (v'-l)'v'.

~ E R m

R e m e m b e r i n g (3.5), we obtain, after simplification, from (3.7) a n d the e s t i m a t e above,

a v

< const {sup u~(x)}.

l~ow let ~t be the capacitary distribution with total mass 1 of order a of an a r b i t r a r i l y chosen closed subset

F(~ n)

of G(~ n). This and (1.4) give

a ~ < const (C~ (F(~n))} -1.

Hence the same inequality is true with F(~ ~) replaced b y G(~ ~) and we h a v e p r o v e d the following inequality when m/> p > 2:

338

I f m > p > ~ l or r e = p > 2 , t h e n there exists a positive constant M, not de- pending on a a n d n, such t h a t

C~(G~n)) < M a -~, a > 0 , n = l , 2 . . . (3.8) T h e proof of (3.8) when 1 ~ p < 2 , m > p ~ w h i c h m a y be completed even w i t h o u t t h e assumption t h a t U . Sr~ is b o u n d e d - - i s first carried t h r o u g h when p = 2 or 1, a f t e r which the case 1 < p < 2 is reduced to the case p = 2 b y an application of HSlder's inequality. Compare for instance [7] formulas (8.11) a n d (8.15) where, however, the presence of a function q complicates the proof.

When r e = p = 2 we h a v e the following inequality instead of (3.8): I f S is an a r b i t r a r y sphere with d i a m e t e r less t h a n 1, t h e n there exist constants M a n d %, such t h a t

Co(G(a~)nS)<M'a -~ if a > a 0 , r e = p = 2 , n = l , 2 . . . (3.9) R e m e m b e r i n g t h a t [ . ( x ) > n on F, t h a t H(~)~ G(a ~), a n d t h a t (3.3) a n d (3.4) are true everywhere, we obtain

c ~ ( F ) < c~(o(2'), n = 1, 2 , . . . . (3.10) (When a = 0 , i.e. when r e = p = 2 , F is to be replaced b y F N S a n d G(~ ~) b y G~ n~ N S where S is a sphere having d i a m e t e r less t h a n 1.) (3.10) combined with (3.8) or (3.9) give t h a t C~(F)= O, a n d (B) of the t h e o r e m is proved.

Remark 1. The same methods of proofs also give an analogous theorem if we introduce derivatives of higher orders in (1.1).

Remark 2. Restricting ourselves to the case m > p we observe t h a t the result (2.4) of (B) of the t h e o r e m is best possible in the following sense:

I [ m > p > 2 there exists a compact set F satis/yin 9

M A F ) = O, Cm_~(F) > o. (3.11)

To prove this we shall use the following result b y du Plessis [5, T h e o r e m 4 a n d p. 131ff.]:

L e t ~ a n d q be given numbers, 0 < ~ < m , 2 < q < ~ . There exists a compact set E with C m _ ~ ( E ) > 0 a n d a function / E L q with c o m p a c t support such that, if ~ = m - o c / q , then u~(x)= c~ everywhere on E.

I t should be noted t h a t the proof of this fact which is illustrated for the case m = 2 in [5] is incomplete. The set E which is constructed in [5], p. 132, (where it is denoted b y M) can not be used if l < a < 2 = m . However, for E it is possible to use the m-dimensional Cantor set which is the Cartesian p r o d u c t of m equal 1-dimensional Cantor sets, G, where G is the usual Cantor set which is obtained starting f r o m a n interval of length 1 a n d a sequence (~n} such t h a t 0 < ~n < 1/2; i.e. G = N G~, where Gn consists of 2 n closed intervals each of length

~l~z ... ~n. I t is well known t h a t if 0 < fl < m, t h e n C~(E)> 0 if and only if 339

:u. W-a.LHN, o~-capacity and LV-classes o f differentiable functions

oo

2 ^{. . . .} (}1~...#.)-~< ~ .

n = l

B y using this it is possible to construct the function / and to carry through the proof b y obvious modifications of the proof given b y du Plessis for the case m = l [4, p. 896ff.].

We now turn to the proof of the existence of a compact set F satisfying (3.11), where p is given, m > p > 2 . According to the above there exists a com- pact set F with C ~ _ ~ ( F ) > 0 and a non-negative function g E L ~ with compact support such t h a t u ~ _ l ( x ) = c~ on F. We shall prove t h a t . M p ( F ) = 0 . qLet, for n = 1, 2 . . . ~n E C~ be a non-negative function with .~ cfn d x = 1 such t h a t U ~ S~, is a bounded set. As u ~ _ l ( x ) > n off an open set containing F we have u~_l-)eq),~(x)>n on F if we choose Sr in a sufficiently small neighborhood o f

g~ F since

the origin. P u t t i n g g n = g ~ C f n this means t h a t U,n_l(x ) > n on

1 1

Um_ = r m-1 -~ gn = ~ -~ g -)S q) n = Ugm-1-)e qPn 9

Furthermore, we have u g- E C ~. We now choose a function ~ E C~ which is _{m - - 1} identically equal to 1 on a set, the interior of which contains F and U.Sgn, and p u t

In

(x) = n -1 u~% (x). ~(x).

Hence In E C ~ , In(x)/> 1 on F. (3.12)

There exists a constant M such t h a t

f lgradLpdz<Mn -~ f N~

~grad~l~dx+M-

-~ f l~gradu~%l'dz.

^(3.13)

I n the same way as in the proof of (A) of the theorem we realize t h a t the first term of the right member tends to zero when n - - > ~ . The second term of the right member may, for instance, be estimated b y means of the theory of singular integrals. We have [1, p. 129]

~u~% l (x)__ _ ( x ~ _ y~

lira (1 - m ) Jl I 'm+lg'~(y)dy

~ X i ~-~o x - yl>~e

a . e . ~

and from this we infer [1, p. 116],

II

grad u ~ - ~ l l .

<

const

II g. Ik,,

where the constant is independent of n. B u t an application of I-I61der's in- equality shows t h a t (see for instance [2, p. 192]),

and consequently we have proved t h a t also the second term of the right member of (3.13) tends to zero when n-->oo. Hence

lira ~ [ g r a d [,~ I ~' d x = O.

~ - - - ~ o 0 3

This combined with (3.12) finally gives t h a t 3 / p ( F ) = 0 and so we have proved the existence of a compact set $' satisfying (3.11) if r e > p > 2 .

R E F E R E N C E S

1. CALDERO~, A. P., a n d ZYGMU_~D, A., On t h e existence of certain singular integrals. Acta Math. 88, 85-139 (1952).

2. FUGLEDE, B., E x t r e m a l length and functional completion. A e t a Math. 98, 171-219 (1957).

3. FUGLEDE, B., On generalized potentials of functions in t h e Lebesgue classes. Math. Seand. 8, 287-304 (1960).

4. I)u PLESSIS, N., A t h e o r e m about fractional integrals. Proc. Amer. Math. Soc. 3, 892-898 (1952).

5. Dv PLESSIS, N., Some theorems a b o u t t h e Riesz fractional integral. Trans. Amer. Math.

Soe. 80, 124-134 (1955).

6. SERRXN, J., Local behaviour of solutions of quasi-linear equations. Acta Math. 111, 247-302 (1964).

7. WALLIN, H., Continuous functions a n d potential theory. Arkiv Matematik, vol. 5, No. 5, 55-84 (1963).

Tryekt den 25 juni 1964

Uppsala 1964. Ahnqvist & Wiksells Boktryckeri AB

3 ~ 1