A R K I V F O R M A T E M A T I K B a n d 7 n r 46 1.68 07 1 Communicated 24 April 1968 by LENNART CARLESO~
A problem of Newman on the eigenvalues of operators of convolution type
By
ANDERS WIKI n [1] N e w m a n has studied the problem of uniqueness of the class of equations
~F(x) - fE F(t) K ( x - t) dt = G(x) for x E E
under the restriction t h a t K has compact support. However, the result in [1] is true also without t h a t restriction:
Theorem 1. Let G be a locally compact abelian group with Haar measure dt and let K(x) E LI(G). Then/or any measurable set E
2 F ( x ) - f F ( t ) K ( x - t ) d t = O for x E E and FEL~C ~F=--O
i/ ,~ ~ H k = CH(K(~)]~ E G} = the closed convex hull o/the values assumed by the Fou.
rier trans/orm 1~ of K.
An equivalent theorem is obtained b y looking at the class of operators on L ~r KE F = { F-)e K for x E E,
0 for x r E,
where the kernel K E L 1. The theorem t h e n states t h a t for a n y measurable set E, K~ has all its eigenvalues inside HR. Thus HK is a bound, uniform in E, for the eigenvalues of K~. The question of the " b e s t " uniform bound has not been settled.
The eigenvalue problem when G = It or Z has been solved in the cases E = ( - ~ , ), ( - c~, 0) and (0, c~) (see e.g. Krein [2]). Together these eigenvalues form the set
AK = (/~($) I ~ e ~)U (~lind (4 - ~)= (2~) -1S-~ d,
arg (~t --/~($)) * 0}, i.e. the set of points on or "inside" the curve described b y the Fourier transform /~. Consequently, if MK is the best uniform bound for the eigenvalues t h e n 601A. WIK,
Eigenvalues of operators
AK~_Mr:~HK.
I n [1] there is an example where G = Z a n d KE for suitable E has eigenvalues outside AK. I n t h a t example MK is strictly between A~ and HK.F o r the proof of Theorem 1 the.following expression for the integral equation in question will be useful
]F(x)I~=F(x)(F-)eK)(x)
for allxEG.
(1)Proo/.
I t is sufficient to s t u d y the case ~t= 1. Following [1] we observe t h a t 1 r HK implies the existence of a complex n u m b e r ~ such t h a tRe (~(1 - / ~ ( ~ ) ) ) >7 1 for all ~ E G. (2) L e t x 0 denote an a r b i t r a r y point of G and let V be a compact symmetric subset of G such t h a t
fc(v) [ K(x)] dx< 89
For abbreviation we define/(x)
= Zn(x)" F(x)
where Z~ is the characteristic func- tion ofxo+V=={Xo+~_ix~lx, E V}.
With these notations Parseval's theorem gives us]0~]-1 iF(x)l dx= i l-i ]/(x)]~dx= ]~, l f~ ,](~),2d~"
F r o m the inequality (2) and from Parseval's theorem once again it follows t h a t
= l f oll(x)l'- /(x) (l , K) (x)dx = l f, o+, lF(x)12- P(X) fx.+, F(') K(x- )d dz
:Now b y the relation (1) this equals
f ~,+ vnF(x) f x,+,(vn)F(t) K(x- t) dt dx "
After a
y - - x e - x o + C ( V n)
change of variables (y = x - t ; x = x) we can use Fubini's theorem to get
-F(x) F(x - y) K(y) dy dx = [ f fxEx,+ Ey F(x) F(x - y) K(y) dx dy
y E V n + ~ ( V n )
where m y = V n f3 { y + C(Vn)}. Thus we have arrived at the inequality
fxo+v [F(x)l~dx<l~l f lK(Y)l fx~ [F(x)F(x--y)ldxdy"
(3)602
Define a n d
Thus (3) yields
~~176 ~[~1
f v IK(y)ldY fXo+E~ IF(x) F(x-y)Idxdy
§
I=[
Now E:_~ V" so
f,<v) I K(U) I v(n) dy < 89 v(n).
12
I"
Schwarz's inequality gives us
2 +
fxo+E IF(x)F(x--y)]dx<-<{ fxo+Ey[F(x)[2dx fx,_y+E IF(x)[ dx} .
(4) B u t fory e VEy = V n N {y + C(Vn)}- Vn~V n-l~- vn+I~V n-1
and
- y + Ey=C(V n) N
{ - - y +Vn} ~_
Vn+l~vnc. vn+I~v n-1 so the right member of (4) is less t h a n ~%o(n + 1 ) - ~xo(n-1). Thereforewan)< I~l" f lK(y)ldy{~=o(n+ 1)- ~=o(n- 1)}+ 89
< I[gll~{~xo(n + 1)- ~=0(n- 1)}+ 89
B u t as c v~o(n) and ~v(n) are increasing this yields
([] g [[1 + 1) c v~o(n - 1) < [[ g [h ~V~o(n + 1) + 89 + 1).
Varying x o we get
~ v ( n - 1 ) <
IIg[[l+ 89
(/~< 1).II Kill +
1S o ~v(2) < ~n-l~v(2n). (5)
We need the following simple lemma of N e w m a n [1].
Lemma.
Let V be a compact subset o/ G. Then there exist constants c and d such that m( V ~) < c " n a.
603 ARKIV FOR M A T E M A T I K , Bd 7 nr 46
~(n) = sup f~~
x o c G
A. WIK, Eigenvalues of operators
A p p l y i n g it in this s i t u a t i o n we get ~(2n) ~< C" M " (2n) d a n d t h u s f r o m (5)
~ ( 2 ) < C . M . / x n - l . ( 2 n ) d - + O ( n ~ c ~ ) .
F r o m t h e definition of ~v we get t h a t F ( x ) ~ O a n d the t h e o r e m is proved.
Remark. T h e proof is valid w i t h a m i n o r modification also in the case F E LP(G), 2 ~< p < ~ . The only place where we used IF(x) [ ~< M was t o p r o v e t h a t ~(2n) ~< C"
M . (2n) ~. W e can write I F I ~ (~ L v/~, F I ~ = F1 + F2 where F~ e 51 a n d F~ E L ~.
T h e n v(2n)~< C" (2n) d- IIF~ ~ + IIFl/11 a n d the result follows just as above.
T h e only r e m a i n i n g case of interest is F E L v, 1 ~< p < 2. I t is r e d u c e d to t h e a b o v e b y t h e following
L e m m a . I] I F ( x ) l ~ = F ( x ) . ( F ~ K) (x) where K E LI(G) and F E Lv(G), 1 ~ p < 2 then 2' e L2( G).
Proo/. Suppose p = 1. O t h e r cases are t r e a t e d similarly. I t follows t h a t I F I ~<
IF
~- K 9 N o w K c a n be w r i t t e n K = K 1 + K S where K~ E L 1, K2E L 1 f3L z a n d I K l 1 1 1 < ~ < 8 9 I g ~ ~ = M < ~ . T h e n12"1 < 12"1 ~-Igxl + IFI ~-Ig~l 9 (6)
Using this estimate of 12"1 in the first term of ~he right member we get a new estimate 12"1 < g~ + h~ where g~ = 12"1 ~ I K~I ~- I K~I
and h~ = 12"1 ~ IK~I ~-IKII + 12"1~ IK~I.
W e r e p e a t this procedure on t h e t e r m
12"1
~ I K ~ I in (6) to get successively new esti- m a t e s I F I ~< g~ + h, whereg i + i = g , ~ l K i l a n d h,+~=h,~lK~l+12,1~lK~l .
Therefore IIg,+~ I~< ~llg, II and [h,+~ll~< ~llh, ll~+MII2"11. It follows that IIg, lli<
e~+l. IIFII1 a n d Ih~ll~< 2 M . II F 1 a n d we get
112,-h;ll~<~'+~.ll2,111~o as i ~
whereIlh;ll~<llh, ll<2illFill .
W e can choose a subsequence {h,k } such t h a t hjk(x ) ~12"(~)1 a.e. T h e conclusion t h a t 2" E L 2 n o w follows f r o m F a t o u ' s l e m m a :
f l2"1~dx<liminfflh~kl~dx<(2MIIF~ll) 2.
T h u s we h a v e p r o v e d t h e m o r e general f o r m of T h e o r e m 1.
T h e o r e m 2. I / K ELI(G) then the operator KE de/ined in any one o/ the spaces L~(G), 1 <~ p <~ ~ , by
K E 2 , = K ~ 2 " /or x e E , K E F = O /or x ~ E has no eigenvalue outside the set H~.
Department o] Mathematics, University o] Uppsala, Uppsala, Sweden.
6O4
ARKIV FOR MATEMATIK. B d 7 n r 46
R E F E R E N C E S
1. NEWMAN, D. J., Uniqueness theorems for convolution t y p e equations. Proc. Amer. Math.
Soc. 16, 629-634 (1965).
2. Km~IN, M. G., Integral equations on a half-line with kernel depending upon t h e difference of t h e arguments. Uspehi Mat. ~Tauk 13, 3-120 (1958) ( = A m e r . Math. Soc. Transl. (2) 22, 163-288 (1962)).
Tryekt den 25 november 1968 Uppsala 1968. Almqvist & Wiksells Boktryckeri AB
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