f Distributions with bounded potentials and absolutely convergent Fourier series

Distributions with bounded potentials and absolutely convergent Fourier series

TOI~B;16]aN ~EDBERG

Institu~ Mittag-Leffler, Djursholm, Sweden

1. Introduction

In the first part of this paper w e s h o w that the capacity of a c o m p a c t subset of the n-dimensional Euclidean space can he characterized b y means of distributions (in the sense of L. Schwartz) which are carried b y the set and which have hounded potentiMs.

I n the second part we consider compact subsets of the real line with the property t h a t no non-trivial function can locally be in the class of Fourier transforms of Ll-functions a n d y e t be constant on the intervals of the complement of the set.

Using the result from the first p a r t we shall show t h a t a sufficient condition for a set to have this property is t h a t it has logarithmic capacity zero. This improves a result b y K a h a n e and Katznelson [4, p. 21] concerning Cantor sets.

I t will Mso he shown t h a t a necessary condition is t h a t the set has capacity zero with respect to all kernels (log+ 1/]x[) 2+~, d > 0.

The author is v e r y grateful to prof. L. Carleson for his m a n y valuable suggestions.

2. Notations and definitions

We denote the n-dimensional Euclidean space b y R =, its points b y x

2~1/2

@ l , . - - , x - ) and we write IxI = ( x ~ + . . . + x n ) .

B y A 1~ we mean the space of all functions

f

~o~rier transform is an integrable function, so t h a t f = g in V.

The Fourier transform of a function, measure or tempered distribution S is denoted by S.

5 0 T O R B J O R N H E D B E R G

B y a kernel we m e a n , in t h e case w h e n n > 2, a n i n t e g r a b l e f u n c t i o n on R n, with c o m p a c t s u p p o r t , of t h e f o r m K ( x ) = H(cf(x)), w h e r e H is a n o n - n e g a t i v e , c o n t i n u o u s , increasing a n d c o n v e x f u n c t i o n o n R a n d w h e r e ~0 is a f u n d a m e n t a l solution of L a p l a e e ' s e q u a t i o n , i.e.

1

log - - , n = 2

~(x) = Ixl

[xl 2 . n , n __ > 3

W h e n n = 1 we m e a n b y a k e r n e l a n even, i n t e g r a b l e a n d positive f u n c t i o n

~vith c o m p a c t s u p p o r t on I t w h i c h is c o n v e x on (0, oo).

W e shall t h r o u g h o u t this p a p e r let K~ be a k e r n e l w h i c h in a n e i g h b o u r h o o d o f t h e origin equals Ix] -~ w h e n ~ > 0 or log l/ix [ w h e n ~ = 0 a n d w h i c h m o r e - o v e r is i n f i n i t e l y differentiable for x :~ 0.

W e d e f i n e as u s u a l t h e c a p a c i t y Cx~(E) o f a c o m p a c t set E with r e s p e c t t o a k e r n e l K b y

(CK(E)) -~ = i n f ( s u p U~'(x))

t~ I t n

w h e r e t h e i n f i m u m is t a k e n o v e r all positive m e a s u r e s on E o f mass 1. F o r t h e o t h e r basic c o n c e p t s o f classical p o t e n t i a l t h e o r y we r e f e r t h e r e a d e r t o [2].

3 . D i s t r i b u t i o n s w i t h b o u n d e d p o t e n t i a l s

L e t S be a d i s t r i b u t i o n (in t h e sense o f L. S c h w a r t z ) on R". W e d e f i n e t h e p o t e n t i a l o f S w i t h r e s p e c t t o t h e k e r n e l K as t h e c o n v o l u t i o n of S a n d K a n d d e n o t e it b y U s = - - S * K .

D e n y [3] has in g r e a t detail s t u d i e d p o t e n t i a l s o f d i s t r i b u t i o n s w i t h f i n i t e e n e r g y a n d has e.g. s h o w n t h a t a set of c a p a c i t y zero w i t h r e s p e c t t o some k e r n e l c a n n o t c a r r y a n o n - z e r o d i s t r i b u t i o n w i t h f i n i t e e n e r g y w i t h r e s p e c t t o t h e same kernel.

I n t h e special case w h e n n = 1 a n d for kernels K~ this was also p r o v e d for ~ ---- 0 in a n l J p p s a l a l e c t u r e b y B e u r l i n g in 1940 a n d l a t e r b y ]3roman [1] for 0 < ~ < 1.

Our aim is t o s t u d y distributions whose p o t e n t i a l s are b o u n d e d f u n c t i o n s a n d we shall o b t a i n some analogous results.

T H n O I ~ 1. Let E be a compact subset of 1t" and let S be a distribution with support on E. I f the potential of S with respect to a lcernel K is a function U s and i f furthermore S ( 1 ) = 1 then

ess sup ]US(x)] > (C~(S))-I

x E l l n

D I S T R I B U T I O N S W I T H B O U N D E D P O T E N T I A L S A N D A B S O L U T E L Y CONVEI:~GE~-NTT F O U l : t I E R S E R I E S 5 1

Proof. L e t s > 0 be given a n d let % be t h e equilibrium measure of t h e set~

E~ --- {x E R"; diet (x, E) < s}. L e t f u r t h e r m o r e k E C~ be a positive f u n c t i o n w i t h s u p p o r t in t h e n-dimensional u n i t bull a n d assume f kdx = 1. W r i t e k~(x) =

8 - n k ( X l / F ~ , . . . , X n / , 9 ) a n d let S = S 9 k~.

W e n o w easily o b t a i n t h e following inequalities b y using wellknown p r o p e r t i e s o f t h e equilibrium measure.

IUSl

IU s~] > .f IUS~Id~ >

e s s s u p

B u t CK(E~)--+CK(E) as s - - > 0 a n d f r o m S ( 1 ) ~ - 1 it follows t h a t

f S f l x = 1

which proves the theorem.

W e are latex" on going to use this t h e o r e m in t h e classical cases of l o g a r i t h m i c c a p a c i t y a n d a-capacity, i.e. c a p a c i t y w i t h respect to t h e kernels K a. F o r t h e s e kernels we can prove t h e following corollary.

COROLLARY. A compact set E C R ~ has positive ~x-capacity ( i f ~ ~ 0: logarithmic capacity) for m a x (0, n -- 2) < cr < n i f and only i f it carries a non-zero distribution whose potential with respect to Ka is a bounded function.

Proof. Assume t h a t S # 0 is carried b y t h e set E a n d t h a t S * K~ is a b o u n d e d function.

It is b y the t h e o r e m sufficient to find a distribution S O w h i c h satisfies t h e s a m e a s s u m p t i o n s as S a n d w h i c h in addition has the p r o p e r t y that S0(1 ) ~: O_

If S is n o n - v a n i s h i n g there m u s t exist Y0 C R ~ s u c h that S(e I('' Y~ r O. W e shall p r o v e that the distribution S o ~ e~("Y~ has the required properties. T h i s follows h n m e d i a t e l y if we can show t h a t S(y)(K~(y -- Yo) -- Kc~(Y)) is the F o u r i e r t r a n s f o r m of a b o u n d e d function.

W e claim t h a t we can write

K~(Y ^-- Yo) ^-- K~(Y) = K~(y)(P(y) + Q(y) + R(y)) (1) where P , Q e LI(Rn), R e L2(Rn).

Since / ~ ( y ) = C[yl ~-n +

O(Iyl -N)

K~(y -- Yo) -- K~(y) ~- K~(y) IY -- Y01 ~-~ -- 1 + 0(lyl -N)

--/(~(Y)(

-

))

1

where p~ are certain polynomials of degree k.

~ 2 T O R B J O R N H E D B E R G

W e can w i t h o u t loss o f g e n e r a l i t y assume t h a t Y0 ~ (1, 0 . . . 0) a n d w r i t e Ks(Y -- Yo) -- I~c~(Y) = Ka(Y)( ~ ~ ~xtkY~IY1-2k ~- O(IYl -[-~] -1))

k = l 120

W e claim t h a t we can f i n d a n L l - f u n c t i o n such t h a t t h e b e h a v i o u r of its F o u r i e r t r a n s f o r m a t i n f i n i t y is close t o t h e b e h a v i o u r o f t h e s u m in t h e last m e m b e r a b o v e . Consider t h e r e f o r e one t e r m , y~ly] -:k a n d let P l k e C ~ ( R ~ { 0 } ) be some f u n c t i o n w i t h c o m p a c t s u p p o r t w h i c h in a n e i g h b o u r h o o d o f t h e origin equals (-- i O/~xl)~Ix[ 2~-'~ w h e n n # 2k or ( - - i a/axl)tlog 1/]x] w h e n n ~ 2k. T h e n ))lk(y) -~ Clyt-Sky~ -}- O([yl -N) for all N as l y ] - + oo a n d Pll, e L l ( R " ) .

B y a d d i n g c o n s t a n t multiples o f such f u n c t i o n s P,~ we f i n a l l y o b t a i n a f u n c t i o n P eLI(R ~) which satisfies ~5(y) _

~p~(y)tyi-2~ _= O(lyl-~)

Since S~/~ a n d S , K s are b o u n d e d f u n c t i o n s a n d since S * K~ e L2(R ") it n o w follows t h a t N * K s 9 ( P ~- Q ~ / ~ ) is a b o u n d e d f u n c t i o n w h i c h p r o v e s t h e s u f f i c i e n c y p a r t of t h e corollary. T h e n e c e s s i t y follows d i r e c t l y f r o m t h e d e f i n i t i o n of c a p a c i t y .

I t seems p r o b a b l e t h a t t h e c o r o l l a r y also holds for general kernels K a l t h o u g h we h a v e n o t b e e n able t o p r o v e this.

L e t us n o w e n d this section b y showing h o w t h e corollary could be used to p r o v e t h e classical r e s u l t (see e.g. [1, ch. VII]) t h a t a set is >>removable>> for b o u n d e d h a r m o n i c functions i f it has c a p a c i t y zero w i t h r e s p e c t to / ~ - 2 .

L e t t h e r e f o r e D c 1t =, n > 2, be a b o u n d e d region whose b o u n d a r y /7 is a s m o o t h surface a n d let E C D be a closed set s t r i c t l y c o n t a i n e d inside /'. Assume t h a t E has c a p a c i t y zero w i t h r e s p e c t to K~_2 a n d let u be a b o u n d e d a n d h a r m o n i c f u n c t i o n on D ~ E . W e claim t h a t u can be e x t e n d e d to t h e whole of D.

Choose a f u n c t i o n v w i t h c o m p a c t s u p p o r t w h i c h coincides w i t h u on some n e i g h b o u r h o o d o f E a n d w h i c h is i n f i n i t e l y differentiable o u t s i d e E .

T h e n A v - ~ S Jr ~, w h e r e S is a d i s t r i b u t i o n c a r r i e d b y E a n d w h e r e ~ E C~.

B u t t h e p o t e n t i a l o f S w i t h r e s p e c t t o K ~ _ 2 is

S * K n _ 2 - - m - ZIv *K,~_ 2 ^{- - ~} *K,,_2-=- v * A K , , _ 2 - ^~ * K , _ 2 ~- v -~ v ,~v ^{- - ~} *K,,_2 where ~ C C~.

W e t h u s h a v e t h a t U s ---- S * K . _ 2 is a b o u n d e d f u n c t i o n w h i c h b y t h e c o r o l l a r y implies t h a t S = 0 a n d h e n c e t h a t u can be e x t e n d e d as claimed.

4. A class of thin sets

L e t E be a c o m p a c t subset o f t h e real line a n d assume t h a t f E A 1~ is eonstan~

on e a c h i n t e r v a l o f t h e c o m p l e m e n t o f E .

D I S T R I B U T I O N S W I T H B O U 2 ~ D E D 1 ) O T E 2 q T I A L S A ~ I ) A I 3 S O L U T E L Y C O N V E R G E 2 q T F O U I C l E R S E R I E S 5 3

Our aim is to t r y to characterize the sets E for which no nontrivial function f can have the properties just stated.

K a h a n e and :Katznelson [4, p. 21] have given an example of a Cantor set with the above property. The following theorem contains their result.

T ~ o n E ~ 2. Let E c R be a compact set of logarithmic capacity zero. I f f E A ~~176 is constant on each interval of the complement of E then

f

Proof. Let f be an arbitrary function t h a t fulfills the hypothesis. Then there exists a function g which satisfies:

(i) ~ ~ L~(R).

(ii) g---f on some interval I ~ E .

(iii) g ~ 0 outside some neighbourhood of i . (iv) g is infinitely differentiable outside I.

L e t S be t h e derivative of f in the sense of distributions and let U s be the logarithmic potential of S.

Then S = dg/dx ~- ? where ~ C C ~ and U s = U g" + U ~.

B u t ( U g ' ) ~ ( x ) ~ iz~(x)l~o(x ) e L~(R) since ~0(z)---- 1~Ix ] ~-O([xl -N) for a n y N as x -+ ~ and hence U g' is a bounded function. This is also true for U ~ a n d S is thus a distribution on E with bounded logarithmic potential.

B y the corollary this leads to a contradiction unless S--~ 0 and hence the theorem follows.

Our next theorem shows t h a t this result is the best possible in the sense t h a t logarithmic capacity cannot be replaced b y capacity with respect to a n y larger kernel.

Before we state the theorem we give the following lemma.

LEMMA

and {m~}~ are some given sequences. Write l, = ~n+lm~rr c~

(m n ~ 1 ) l ~ < l l n _ l for n = 1 , 2 . . .

Then E has capacity zero with respect to a kernel K i f

]-T (m, +

n : l 1

f:

1 K(t) dt < CK(x), x :/: 0 then the condition is also necessary.

I f x

co r

2 . . L e t E : { x e R ; x : ~ l ~ir" s ~ : O, 1 . . . m,} where {r,} 1 and assume that

Proof. The proof of the lemma is, apart from minor modifications, identical to the proof given in e.g. [2] of the corresponding theorem concerning the ordinary Cantor set (i.e. the case when m i --- 1, i = 1, 2 . . . . ) and is therefore omitted here.

The last condition in the lemma is clearly satisfied for all kernels Ks, 0 < ~ < 1.

~ 4 T O R B J O R N H E D B E R G

TH~OlCE~ 3. Let K be a kernel such that lim~+0K(x)(log 1 / I x ] ) - 1 = ~ . T h e n lhere exists a compact subset of R with capacity zero with respect to K and a non- zero measure carried by the set whose primitive function is in A ~~162

Proof. L e t E be a set of t h e form {x = ~ eirl, es = O, 1 . . . . , mi} a n d a s s u m e t h a t E has positive logarithmic c a p a c i t y a n d choose a measure /x on E w i t h f i n i t e e n e r g y w i t h respect to t h e logarithmic kernel. L e t v = # 9 # a n d let J(x) =--./idv. The s u p p o r t of v is obviously a subset of

I V = E - ~ E = { ~ u , r ,, U i = 0 , 1 . . . 2 m i } .

1

T h a t t h e energy of /x is finite implies t h a t

f

Iyl->

B u t f(y) = ~(y)/iy = (~(y))2/iy a n d hence f E A ~~162

W e n o w claim t h a t we can choose {r~} a n d {m~) in such a w a y t h a t

n

(]'-[ (m i ~- 1)) -1 log I/t. < oO (2)

1 1

(]"[ (2m i + 1)) -1 K(2/,) = ~ (3)

1 1

where

In

B o t h these relations are satisfied if we e.g. choose {r~) a n d {ml) so t h a t (i) l o g l / 1 , = n - 2 ] - [ ~ ( m , ~ - 1), n ~ - 1, Z , . . .

(ii) K(2/n) >__a n l o g l / I n , n - ~ 1 , 2 . . . . (iii) mi -+ ~ , i -+

(iv) (2mnA- 1)ln < l n _ l / 2 , n = 1 , 2 . . . .

Since ln+ ~ = In - - mn+lrn+~ it is clear t h a t this choice can be done.

Using L e m m a s 2 a n d 3 it n o w follows from (2) a n d (3) t h a t E (and t h u s F ) h a s positive logarithmic c a p a c i t y a n d t h a t F has c a p a c i t y zero w i t h respect to K which proves our theorem.

The n e x t t h e o r e m gives a necessary condition in t e r m s of c a p a c i t y for a set t o be in our class of t h i n sets.

THEOREM 4. Let E C R be a compact set of positive capacity with respect to the kernel (log + 1/]xD 2+~ for some ~ > o. Then E carries a (positive) measure # whose p r i m i t i v e function is in A 1~162

DISTRIBUTIONS W I T H B O U N D E D POTENTIALS AND ABSOLUTELY CONVERGENT F O U R I E R SERIES 55

Proof. P u t K(x) = (log + 1/Ix[) 2+~ a n d let # be a measure on E w i t h finite energy w i t h respect to K .

I t is easy to show that (log ]y])l+~Iy[-l(I~(y))-i t e n d s to a c o n s t a n t as [y[ - ~ oo a n d it therefore follows t h a t

f [~(Y)[~

Iyl-> 1 lYI

- - (log [y])I+~dy ^{< oo}

[ *

B y Schwarz's i n e q u a l i t y this implies t h a t ] ]~(y) l[yI-ldy ^{< oo} a n d hence g~y I>_i

f:

t h a t f ( x ) = d/~ E A 1~ which proves t h e theorem.

I t is possible t h a t a n y set E of positive logarithmic c a p a c i t y carries a non-zero distribution w i t h a primitive f u n c t i o n in A l~ The following result shows t h a t such a distribution could n o t always be chosen as a positive measure a n d t h e necessary construction would therefore p r o b a b l y h a v e to be complicated.

THEOREM 5. There exists a (~ ~ 0 and a compact set E~ c R of positive capacity with respect to the kernel K(x) -= (log + 1/Ixl) 1+~ such that if f E A l~176 is non-decreasing and constant on each interval of the complement of E~ then

f

Proof. L e t E~ be t h e Cantor set {x E R; x - = ~'1 e~r~, e~ ~ 0 or 1} where oo

~ n + l r~ ~ exp (-- 2"/(I+~)), n = 1, 2 , . . . a n d where ~ ~ 0 is a n u m b e r to be f i x e d later. E~ has positive c a p a c i t y with respect to a kernel (log + 1/tx[) I+' if a n d o n l y if 0 < t < 6 .

Assume t h a t f is a f u n c t i o n t h a t fulfills t h e hypothesis. I t s derivative in t h e sense of distributions is a positive measure # w i t h s u p p o r t in E~. Consider t h e convolution v = # 9 # which is a measure on F~ ~ E~ ~- E~ a n d suppose we k n o w t h a t ~ has b o u n d e d energy w i t h respect t o t h e kernel (log + l/lxl) 1+3~" for some n u m b e r ~' > ~.

This a s s u m p t i o n is e q u i v a l e n t t o

f l;(y)1

lyl >_ l ]Yl

- - ( l o g ]y])3~" d y < o~

f 0 ~

and, since d# E __A ~~176 we also k n o w t h a t

f lg(y) l

lyl >_ l

56 TORBaO~N ~EDBE~G

l~rom these two inequalities it follows b y means of I-I61der's i n e q u a l i t y t h a t lY] (log [yl) a' dy < oo

]y] e l

i.e. t h a t # has finite energy with respect to t h e kernel (log + 1/[XI) 1+~' where ~' > ~.

B n t this is impossible unless # = 0 a n d hence t h e t h e o r e m follows as soon as we h a v e p r o v e d t h e following lemma.

L ] ~ M A 3. Let E~ be a Cantor set as above and let _F~= E,~ ~ E~. Let # be a positive measure on Ez with bounded logarithmic potential. Then, i f ~ > 0 is suffi- ciently small, there exists d' > (~ such that the measure r = /t * tt has finite energy with respect to the kernel (log + l/Ix[) l+a~'.

Proof. B y m e a n s of a simple estimate we see t h a t /~(I) ~ C(log 1/1I[)-1 for all intervMs I of length ]I[ less t h a n 1.

W e also observe t h a t ~a = n ~ ~ Fn where each set ~n is the union o f 3" intervals

c o

I (n) of length In = ~ , + ~ 2rr a n d w i t h left endpoints x(~ "), k = 1 , . . . , 3 ~.

oo t !

E a c h p o i n t x @ can be wTitten x(k ") = ~ (e~+e,)r~, w i t h ~, ~ = 0 or 1, in N ~ ) different ways. L e t q be t h e n u m b e r of indices i for which W = s~ -~ s'~ = 1. I t t h e n follows t h a t x~) can be o b t a i n e d in N ~ ) = 2 q different w a y s as a s u m of two points in Ea. ~ o r a f i x e d q there are (~)2 "-q such points x(~ ") in Fa. W e f i n d t h a t ~ k (N(k"))2 = ~q=0 (~)2"-q" 22q = 6~" n

L e t ~(") be t h e measure whose restriction to a n y interval I(k ") is u n i f o r m l y d i s t r i b u t e d a n d whose mass on a n y interval I (") equals v(I@). I t is easy to see t h a t v(n) converges w e a k l y to v as n--> oo a n d it is therefore sufficient to prove t h a t t h e energy of v(") is b o u n d e d u n i f o r m l y in n.

The energy of v(~) w i t h respect to t h e kernel (log+ 1/Jxl) l+aa" is b y definition

ff(,o + 1

k., Ix -- y I/ dv(n)(x)dv(")(Y) " (4)

,i o)• n)

L e t us define

f f (

Dn ~ ~ log+ - dv(n)(x)&@(y)

Ix yl/

We claim t h a t

E(~ <n)) < C ~ D=.

(5)

D I S T R I B U T I O N S W I T H B O U N D E D 1 ) O T E N T I A L S A N D A : B S O : L U T E L Y C O N V E I ~ G E N T F O U R I E R S E R I E S 5 7

To prove this, le$ Q=, 1 _~ m < n, denote the sum of all terms in the right hand member of (4) that correspond to pairs of intervals with a mutual distance less than I~_~ and greater than 89 l~_~.

In other words, Q= is the sum over the set T= consisting of all indices (k, l) such that @ ) - ~ ~!') for 0 < i < ~n -- I and ~2) :/: ~ (where x(p ~) = ~ :

~!V)r,).

Then

1 ~I+3a"

- - / ~" r176 <

Q~ < C log 1~_1/ ~

< C(Iog 1 )1+3a'~

I t is obvious t h a t we get all terms in (4) b y summing over m and adding D . and hence (5) follows.

]~ut since ~(")(Ip)) < CNp)(log 1/1.) -2 and since

1 n lrt

1 ~1 +3,V

t _ U

z~ > - y i/ dx~y < c

0 0

we have t h a t

D,, < C log T.] Z (NP)) z = C log T./ 6" < C7"

k

where y = 6 9 2 -(3-3a')/(l+a) (since l= = 2 exp (-- 2"/(1+a))).

Choose now ( ~ > 0 so t h a t 3 ( 1 - - 8)/(1 -4- 8) > 2 1 o g 6 (i.e. 5 < 0 . 0 7 4 . . . ) and choose 3 ' > ~ so t h a t y < 1. Since E(v ( n ) ) _ ~ C ~ 0 y ~ the lemma follows. n

Relerenees

1. BROMA~, A., On two classes of trigonometric series. Thesis, Uppsala, 1947.

2. CA~Lasso~, L., Selected problems on exceptional sets. Van Nostrand, Princeton, 1967 (Mimeo- graphed, Uppsala, 1961).

3. ]DENY, J., Sur les distributions d'energie finie. Acta Math. 82 (1950).

4. •AKANE, J-.-:P., Sdries d~ _Fourier absolument convergentes. Springer-Verlag, ]3er]in, 1970.

Received J u n e 7, 1971 TorbjSrn IYedberg

H6gskolenheten Fack

S-951 O0 Luleg Sweden