On the spectral synthesis problem for (n-1)-dimensional subsets of R n > 2
YNGVE ])OMAR University of Uppsala, Sweden
1. Introduction
L e t E be a closed subset of R n a n d K ( E ) t h e space of all functions in ~(Rn), vanishing in some neighborhood of E . ~ L I ( R ~) is t h e B a n a c h space of F o u r i e r t r a n s f o r m s of functions in LI(Rn). ~ ( R ") C ~LI(R~), a n d we denote b y K - - ~ t h e closure of K ( E ) in ~LI(Rn). The well-known concept of sets of spectral synthesis can be defined as follows: E is a set of spectral synthesis if K ( E ) contains e v e r y element in ~ L I ( R n) t h a t vanishes on E.
C. Herz [3] has p r o v e d t h a t S 1 C R 2 is a set of spectral synthesis. ]~is proof c a n u n f o r t u n a t e l y n o t be e x t e n d e d to o b t a i n t h e corresponding result for more general curves. I t is however possible to use a different approach to get t h e desired extension of t h e result of Herz (cf. [2]). W e shall here a p p l y basically t h e same m e t h o d to investigate a still more general problem.
As was discovered b y L. Schwartz [9], t h e sphere S ~-1 C R ~ is n o t of spectral synthesis, if n ~_ 3. N. Th. Varopoulos [10] has i n v e s t i g a t e d t h i s question in more detail, using m e t h o d s related to t h e Herz m e t h o d for n -~ 2. L e t us denote, for a n y closed set E a n d a n y positive integer m, b y J,~(E) t h e space of functions in
~ ( R n ) , n ~ 2, vanishing on E t o g e t h e r w i t h all their partial derivatives of order _~ m - 1. T a k i n g closures in ~/fl(Rn), we h a v e t h e n
J I ( S ~-1) ~ J2(S ~-1) ~ . . . ~ J[(~+1)/2](S ~-1) ---- K ( S ~-1) , (1.1) where all inclusions are strict. I t is v e r y easy to u n d e r s t a n d from this w h y there is a f u n d a m e n t a l difference b e t w e e n t h e case n = 2 a n d t h e case n > 3 in this context.
The cited p a p e r of Varopoulos does however c o n t a i n a considerably more precise description o f t h e s i t u a t i o n t h a n t h e one given above. L e t us b y B~(S~-I), m ~ 1, d e n o t e t h e linear space s p a n n e d b y all measures on S "-~ w i t h i n f i n i t e l y differentiable
24 AI~KIV :F5tr MATEM.ATIK. Vol. 9 No. 1
density function and by all partial derivatives of order < m -- 1 of those measures when considered as distributions on R". I t is possible to show t h a t
B,,,(S n-l)
is a subspace of the dual of ~LI(R'), if m ~ (n ~- 1)/2, and in t h a t case we defineI,,,(S ~-1)
as the annihilator ofB,,,(S n-l)
in ~LI(R"). I t is illuminating to regardIm(S ~-1)
as a subspace of ~LI(R ") characterized by the vanishing on S ~-1, in a generalized sense, of the elements together with their partial derivatives of order m -- 1. This notion of vanishing is then the same as ordinary vanishing, if the element belongs to C m in a neighborhood of S ~-1. I t follows from Theorem 3 in [10], t h a tIm(S = g (sn-1), (1.2)
for 1 < m < ( n - } - 1)/2. I n the case n = 2, (1.1) and (1.2) together imply the result of Herz.
Our aim is to generalize (1.1) and (1.2). The generalization is two-fold. I n the first place we replace La(R ") by the more general space L~(Rn), n _> 2, ~ real, of Lebesgue measurable functions f with the norm
(1 +
R ~
Secondly, and this is more i m p o r t a n t since it creates the need for a method different from the one developed b y Herz and Varopoulos, we consider sets E on an arbitrary (n -- 1)-dimensional manifold M in R n, infinitely differentiable, without multiple points and with non-vanishing Gaussian curvature. The sets E are assumed com- pact and, in the main theorem, satisfying the restricted cone property (Definition 3.3).
K(E)
andJm(E)
are then well defined, and for 1 _~m < ~ ~- (n ~- 1)/2, spacesI,,(E)
can be defined as in the case E = S n-1 (Definition 2.5 and Definition 2.8). Our results are formulated in Theorem 2.9 and Theorem 3.4, of which the second theorem is the most important.For sets E satisfying the restricted cone property it is possible to express some consequences of our results by generalizing some concepts from the t h e o r y of spectral synthesis. Thus it is natural to say t h a t E is of spectral synthesis with respect to L~(R"), if /I(E) =
K(E).
B y our theorems this is true, if [c~ ~- (n ~ 1)/2] : 1, t h a t is to say if 1 < 2~ ~- n < 3. Adopting a notion introduced b y Herz [4] one can say t h a t E is a smooth set with respect to L~(Rn), ifII(E ) = JI(E)).
This is always true if 2~ ~- n > 1.There are various possibilities for further generalizations. The spaces L~(R~ "),
- - p n
n > 2, can thus be replaced by spaces L~(R ), 1 _< p < oo, defined b y the norm
~ [ f ( ~ ) , " (1 ~-,~,)~'d~l ~/',
R ~
O1~ TI-IE S P E C T I ~ A L S Y N T H E S I S : P R O B L E M 2 5
and similar results hold for these spaces. The infinite differentiability of the mani- fold M can be exchanged to differentiability up to a certain order, as was the assumption made in the s t u d y [2]. In the cases when the Gaussian curvature of M vanishes in some subset of E, or when the manifold M is of lower dimension t h a n n -- 1, the corresponding problems can be stated but are in general still open.
2. Preliminaries
The following lemma of v a n der Corput type is of fundamental importance for our investigation. The lemma is essentially due to W. L i t t m a n [5].
LEMMA 2.1. Let q~ E ~(R~), m ~_ 1, be a function with its support contained in an open set B. Let yJ be real-valued and infinitely differentiable in B and such that
I a2~ F
the inverse of the Hessian determinant ~ ! exists and is bounded in B.
Then there exists, for every real number A, a positive constant D such that
e '(<x''*+~('):) ~(y)dy E <_ D(1 ]~[)-~/2 1 ~-
R ~
for every • E R m , ~ E R. For fixed m , cf , B and A, the same constant D can be chosen for all functions ~v for which we have uniform bounds on the absolute values of the functions, on the absolute values of each of their partial derivatives and on the inverse of the Hessian of %
Proof of Lemma 2.1. I t is possible to write ~ as a finite sum of functions in
!~(Rm), each of t h e m with its support included in some closed sphere included in B. Hence it is allowed to assume from the beginning t h a t B is a sphere.
For a n y set of (V, ~) such t h a t lyE/( 1 4- I~l) is uniformly bounded, the inequality follows from the above-mentioned paper of L i t t m a n . The only thing t h a t needs to be checked, since it is not explicitely stated b y L i t t m a n , is the claimed uniformity property of the constant D. An examination shows, however, t h a t this is a direct consequence of his proof.
Hence we can restrict our attention to the case when IV[ > 2(1 4- [~I)(1 4- sup Igrad ~vl).
0
We can then integrate partially p times in the direction t for which -~ <y, ~> = 1~], and this procedure gives t h a t the left h a n d member of (2.1) is dominated b y
D'(lVl-4- I ~ ] F ( I v l - l~l(sup lgrad ~1)) -2p _< D"lvl -~ ,
where D' and D" are constants with the same uniformity properties as those claimed for D. Choosing p ~ m / 2 4- A , we obtain (2.1).
2 6 A R K I V F O R M A T E M A T I K . V o l . 9 N o . 1
We are going to s t u d y distributions on a fixed space R n. Later on we assume t h a t n > 2, b u t the case n = 1 can as well be accepted in this introductory dis- cussion. Functions and distributions on the dual R ~ are denoted b y f , / t , etc., in order to distinguish t h e m from functions and distributions f , #, etc. on the original
R n ,
Definition 2.2. For every real ~ , L I ( R ~) denotes the Banach space of all Lebesgue measurable functions ] on (the dual) R ~ with a finite norm
a n d L ~ ( R ~) on (the dual)
[[J~12 -= ess. sup. If(~)I(1 + I~1) -~ .
~ 1 ~ n
We observe t h a t a n y function ] in L~(R") or L ~ ( R =)
R n
denotes the Banach space of all Lebesgue measurable functions f R ~ with a finite norm
can be considered as a distribution in the space 3'(Rn). Hence such a function f has, in ordinary distribution sense, a Fourier transform f, and we prefer to normalize the Fourier transformation in a w a y t h a t corresponds to the formal relation
f(x) = e-'~'~
R"
x E tl ~. We adopt the convention t h a t whenever pairs of functions or distributions f , f or /~, # etc. are mentioned in the same context t h e y denote pairs of Fourier
transforms.
Definition 2.3. ~L~(R ") and ~L~~ ~) denote the Banaeh spaces of Fourier transforms f of elements f in L~(tP) and L ~ ( R ") with the norms [ffiJ~ and ][fJI~, respectively, defined b y
A
lifiJ~ = Ilfll~, IlfIl~ ~ = [[fll2 .
L ~ ( R n) can be considered as the Banach space of bounded linear functionals on /)(Rn), the corresponding is thus true for ~ L ~ ( R =) and ~L~(Rn), and we define
" f I(~) "
( f , g) = ( f , ~) -~ g(--~)d~,
R n
whenever f E ~L~(R=), g e ~L~~ I t should be observed t h a t S ( R n) is a subspaee o f L~(R ") and L~ ~ (R") as well as of the transform spaces, and that, by our definition
O:N" T H E S P E C T R A L S Y N T H E S I S P R O B L E ~ r 27
( f , g) = (2~r)-"<f, g > , (2.2) w h e n e v e r f or g belongs t o 3(1t~), w h e r e < , > has t h e usual d i s t r i b u t i o n m e a n i n g . W e n e e d some simple p r o p e r t i e s o f t h e space ~ L I ( R ") a n d t h e y are collected in t h e following l e m m a . All t h a t is s t a t e d in it is k n o w n , b u t we shall give t h e proofs in o r d e r t o a v o i d t o o m a n y t r i v i a l references.
L:EMMA 2.4.
1 ~ Let g E ~L~~ ") and suppose that ( f , g) = 0 for some family of f E S ( l t ~ such that the family of f is translation invariant. Then the support of g is contained in the set of common zeros of the functions f.
2 ~ ~ ( l t n) is a dense subspace of ~LI(R=).
3 ~ Multiplication with a fixed ~ E 5(1t") is a bounded linear transformation from LI~(I! n) to itself and from L ~ ( R n) to itself.
4 ~ . ( f ~ v , g ) = (f,q~g), i f fE~LX~(Rn), g E L ~ C R " ) , ~ E
5 ~ Every
f e~L~(lt ~)
can be approximated arbitrarily closely in ~Ll~(lt n) by elements of the form ~vf, where ~ E ~ ( R ~ ) .Proof of Lemma 2.4. 1 ~ follows f r o m (2.2) a n d e l e m e n t a r y d i s t r i b u t i o n t h e o r y . A p p l y i n g 1 ~ t o t h e f a m i l y ~ ( t l n) we f i n d t h a t g has e m p t y s u p p o r t , h e n c e g ---- 0.
T h u s 2 ~ holds b y t h e H a h n - B a n a c h t h e o r e m .
T o p r o v e 3 ~ we o b s e r v e t h a t if f E ~L~(R"), ~ E S(R~), t h e n
E tT n.
a n d
(~) = r - $0)~(~0)dr
R rt
H e n c e b y a n e a s y a p p l i c a t i o n o f F u b i n i ' s t h e o r e m
a n d 3 ~ follows.
r co ~x A co
IIf~ll~ ~ Ilfll~ II~llt,~t,
(2.3)
if 0 < e < 1. 5 ~ is t h u s p r o v e d .
4 ~ is a d i r e c t c o n s e q u e n c e o f 3 ~ a n d F u b i n i ' s t h e o r e m .
F o r t h e p r o o f o f 5 ~ we choose a f u n c t i o n ~ E ~ ( 1 t " ) such t h a t ~(x) ---- 1, w h e n x belongs t o some o p e n set w h i c h c o n t a i n s x = 0. T h e n we d e f i n e ~ E ~ ( l t n ) , e > 0, b y t h e r e l a t i o n r (x) = ~(ex), x E 1t n, a n d o b s e r v e t h a t ]]f~0 - - fl[ia --> 0, as e - ~ 0, for e v e r y f E ~ ( l t n ) . B y 2 ~ it is t h u s e n o u g h t o show t h a t t h e o p e r a t o r n o r m o f ?~, w h e n this f u n c t i o n is c o n s i d e r e d as m u l t i p l i e r on ~L~(tln), is uni- f o r m l y b o u n d e d , as e--> 0. B y (2.3) this n o r m is
_< I,~(~/,~)l(l + I~1) ~ ~-~ < ll,~ll~,~,
p.,,,
28 AttKIV F(JR MATEMATIK. Vol. 9 :No. 1
I n t h e following we assume t h a t n > 2, a n d t h a t M is a n (n - - 1)-dimensionaI i n f i n i t e l y d i f f e r e n t i a b l e m a n i f o l d in R", w i t h o u t m u l t i p l e p o i n t s a n d w i t h n o n - v a n i s h i n g Gaussian c u r v a t u r e . E is a c o m p a c t subset o f M a n d E ~ d e n o t e s i t s i n t e r i o r w i t h r e s p e c t to M .
W e shall i n t r o d u c e some spaces o f d i s t r i b u t i o n s s u p p o r t e d b y E . T h e y a r e n e e d e d in o r d e r to c h a r a c t e r i z e d i f f e r e n t degrees o f v a n i s h i n g on E ~ for e l e m e n t s in ~L~(R~).
Definition 2.5. B I ( E ) d e n o t e s t h e space o f all measures on R ~ which can b e o b t a i n e d f r o m t h e u n i f o r m mass d i s t r i b u t i o n on E ~ b y m u l t i p l y i n g it w i t h f u n c t i o n s in ~ ( R ~ ) , v a n i s h i n g on E ~ E ~ t o g e t h e r w i t h all t h e i r d e r i v a t i v e s . F o r m ~ 2, B,,(E) is t h e linear space s p a n n e d b y t h e
p a r t i a l d e r i v a t i v e s o f o r d e r _~ m - - 1.
T h e e l e m e n t s in B~(E) are o b v i o u s l y T h e a s s u m p t i o n s on M h a v e m o r e o v e r x o E M t h e r e exists a n e i g b o r h o o d U~~ o f one of t h e c o o r d i n a t e s in R" b y z a n d o r d i n a t e v e c t o r b y y, can be w r i t t e n in
measures in B1(E ) t o g e t h e r w i t h t h e i r b o u n d e d a n d r e g u l a r B o r e l measures.
t h e following consequence. F o r e v e r y x 0 w i t h r e s p e c t t o M which, d e n o t i n g t h e r e m a i n i n g ( n - 1)-dimensional co-
t h e f o r m {(y,z)]z = y~(y) , y E V} ,
where V is a closed ball in R n-l, a n d w h e r e ~ is real a n d i n f i n i t e l y differentiable w i t h n o n - v a n i s h i n g H e s s i a n in V. H e n c e , if # C B I ( E ) has its s u p p o r t in U:~
its F o u r i e r t r a n s f o r m ~ can be r e p r e s e n t e d in t h e f o r m 7*(9, ~) = (2~)-~/e~(<x''~+~J~(x)~) ~ ( y ) d y ,
. ] R n - - 1
E R n-l, ~ E R, w h e r e cf E ~ ( R "-x) has its s u p p o r t in V ~ I t follows f r o m L e m m a 2.1, choosing 2A = m = n - - 1 in (2.1) t h a t ]~(~)I(1 § ]$])(=-~)/2 is b o u n d e d f o r E R n. T a k i n g a n a r b i t r a r y # E Bx(E), it can b y a s t a n d a r d c o m p a c t n e s s a r g u m e n t be p a r t i t i o n e d into a f i n i t e s u m o f measures o f t h e a b o v e t y p e , a n d we o b t a i n t h e r e - fore t h e following l e m m a .
L]~MMA 2.6. ~(~)(1 § l~l) (~-1)/2 is bounded for $ E R ~, i f # E B I ( E ).
L e t us n o w assume t h a t # E B I ( E ) a n d t h a t D # is a p a r t i a l d e r i v a t i v e o f /z
/ N
o f o r d e r p _~ 0. T h e n D # = P/~, w h e r e P is a m o n o m i a l o f degree p. T h u s ID~(~)I(1 + I~]) (n-D/2-p
is b o u n d e d for ~ E R ", b y L e m m a 2.6. F o r # C B,,(E) we can t h u s conclude t h a t Ih(~)[(1 + I~1) +-')/~==+~
is b o u n d e d , for ~ E R ~, a n d this p r o v e s t h e following l e m m a . n §
LEMMA 2.7. B,~(E) C ~L~~ i f m ~ o~ § 2
O ~ T [ t E S P E C T I ~ A L S Y N T t t E S I S P R O B L E ! ~ I 29 W e are n o w prepared to introduce t h e subspaces of L~(R n) which shall be the objects o f our investigation.
Definition 2.8.
1 ~ K ( E ) is t h e space of all functions in ~(Rn), which v a n i s h in a n e i g h b o r h o o d o f E w i t h respect to R n.
2 ~ F o r e v e r y integer m _~ 1, Jm(E) is t h e space of all elements in ~ ( I l n ) , w h i c h vanish on E t o g e t h e r w i t h all their partial derivatives u p to t h e order 3 ~ F o r a ~ -- (n -- 1)/2 a n d f o r e v e r y integer m , 1 _~ m < ~ + (n + 1)/2, I,,(E) is the a n n i h i l a t o r in ~ L ~ ( R n) of the subspace B,~(E) of ~ L ~ ( R ~) (cf. L e m m a 2.7).
4 ~ . F o r e v e r y integer m ~_ 1 , Ca(E) is t h e annihilator in ~ L ; ( R ) of the space Jm(E) in ~YL~(R~).
The following t h e o r e m contains some p r e l i m i n a r y results on t h e relations between t h e spaces i n t r o d u c e d in Definition 2.8. The t h e o r e m is n o t e n t i r e l y n e w a n d t h e m e t h o d s used in t h e p r o o f are known. Cf. Reiter [7, pp. 37--39] a n d Varopoulos
~10] for results of a similar kind.
THEO~E~ 2.9. n - 1
1 ~ 7LI~(R n) ~ Ix(E ) ~ . . . ~ I[~+(~+1)/2](E), i f ~
2 2 ~ J L ~ ( R ~) ~ JI(E) ~ J2(E) ~ . . . .
3 ~ Let ~(~) = o(1 ~- I~]) -(n-I)!2, as l~]--> ~ , and let g h a v e i t s s u p p o r t i n c l u d e d
~ L ~ ( R ).
in E. T h e n g = O. A s a consequence ~ < - (n - 1)/2 implies that K ( E) ~ 1 4 ~ . Let ~(~) = o(1 ~ ]~])m--(~--1)12, as I~l ~ ~ , when m is an integer ~ 1, and let g have its support included in E . T h e n < f , g> = 0 f o r every f E J,~(E). A s a consequence a ~ - - (n - - 1)/2 implies that K ( E ) = J[:+(n+l)/21(E).
5 ~ Let ~ ~ - - ( n - - 1)/2, l _ ~ m < ~ + ( n - ~ 1)/2, and suppose that f E ~ L ~ ( R n) VI C~-I(R~). T h e n f E Ira(E) i f and only i f f vanishes on E ~ together
with its p a r t i a l derivatives of order ~_ m - - 1.
P r o o f of theorem 2.9. 1 ~ a n d 2 ~ are i m m e d i a t e consequences of Definition 2.8.
To prove 3 ~ a n d 4 ~ we shall use a m e t h o d due to A. Beurling [1] a n d H. P o l l a r d [6] (cf. also Herz [3]). L e t t h e distribution g h a v e its s u p p o r t c o n t a i n e d in E, a n d Iet f E ~(Rn). Choose ~ E ~ ( R ~) so t h a t it vanishes outside {x I Ixl < 1} a n d s u c h t h a t
f q~(x)dx = 1.
(2~)-"
, J
F o r every s > 0 we define ~ b y t h e relation ~(x)~ = e-~q~(x/e), x E R ~, a n d p u t f = f ~ § where f~ = f on E2~ = {xldist(x, E ) _ ~ 2e} a n d f~ = 0 on CE2~.
We define f . % b y t h e relation
f * ~ (x) =
(2=)-off(x - (xo)dxo,
a /
30 ARKIV :FSR IVIATEI~kTIK, V o l . 9 :No. 1
x E R ", a n d define f ~ . q ~ a n d f ' * q ~ in t h e corresponding way. W e f i n d f * ~ E ~(R~), f~ * ~ E ~ ( R '~) a n d f~ 9 ~ E K(E), a n d f o r m
< f , g > = < f - - f , w ~ , g > + ( f ~ . ~ v ~ , g > - } - ( f ' * w ~ , g > . The last t e r m vanishes a n d hence
(2~)-"l(f, g>l <
. ] . ]
ir n ir ~
E 3 ( R ~) a n d hence t h e first t e r m of t h e right h a n d side t e n d s to 0, as s --> 0, b y d o m i n a t e d convergence. The second t e r m is, b y Schwarz' inequality, for a n y real fi,
_ I~d~) l/@(~)['[g(- \1/~
< $/e)12d~) <_
R,~ Rn
( / ;2
< ~-~-~ (2~)-~ IL(x)l~dx I~(~)1~(1 + l~l)vlO( - ~/~)l~(t + I~/~l)-Vd~ .
~a R"
The second factor of t h e last expression t e n d s to 0, b y b o u n d e d convergence, if 10(~)l = o(1 + IS1) ~, as
t h a t < f , g > = O , if
is finite.
t~L ~-> ~ - H e n c e t h e geometric properties of E 2 ~ show lim s - n/2--Z+ 1/2 sUp If(x) J
e->O x~E2e
Choosing fi ~ -- (n -- 1)/2, we h a v e ( f , g> ~- 0, for e v e r y f E ~ ( R ' ) , hence g = 0. The a n n i h i l a t o r of K(E) in L ~ ( R ' ) is b y L c m m a 2.4, 1 ~ a n d e l e m e n t a r y distribution t h e o r y t h e subspace of all g E L~~ ~) w i t h s u p p o r t included in E . I f a < - - ( n - - 1)/2, t h e o n l y such g is t h e element g : 0. H e n c e 3 ~ is proved.
4 ~ is p r o v e d in t h e same w a y .
I n order to prove 5 ~ we use a m e t h o d of L. Schwartz [8]. I f # E BI(E) a n d i f D is a partial derivation operator of order p, 0 ~ p < m -- 1, we o b t a i n b y p partial integrations
( f , D~) = ( 2 ~ ) - ~ ( f , D y > = (2~)-~( - 1)P(Df, #7 .
I t follows f r o m this t h a t ( f , D/~) vanishes for e v e r y # E Bx(E) if a n d o n l y if D f vanishes on E ~ Varying p a n d D we o b t a i n from this t h e p r o p e r t y 5 ~
O N T H E S P E C T R A L S Y N T H E S I S P R O B L E M 31
3. The m a i n theorem W e assume in t h e following t h a t ~ ~ - - (n - - 1)/2 .
I t is a c o n s e q u e n c e o f T h e o r e m 2.9, 5 ~ t h a t if E ~ is n o n - e m p t y , t h e n all I,~(E) are different w h e n e v e r t h e y are defined. A n o t h e r c o n s e q u e n c e is t h a t if E ~ is non- e m p t y , t h e n Jm(E) is i n c l u d e d in Ira(E), b u t J,~_I(E) is n o t i n c l u d e d in Ira(E), w h e n e v e r t h e spaces are defined. Since Ira(E) is closed, we can conclude t h a t all J,,(E) are different, if 1 ~ m _~ ~ + (n + 1)/2 a n d E ~ is n o n - e m p t y . As for t h e inclusion J,~(E) c I,~(E) it is e a s y t o see t h a t if t h e b o u n d a r y o f E in M is too irregular, t h e n t h e inclusion m a y be strict. This is for i n s t a n c e t h e case if ~ = 0 a n d E contains a n isolated p o i n t . W e shall h o w e v e r in this section i n t r o d u c e a regu- l a r i t y c o n d i t i o n o n E w h i c h g u a r a n t e e s t h a t Jm(E) = I,~(E), I ~ m ~ ~ + (n + 1)/2.
W e shall in f a c t p r o v e a slightly s t r o n g e r result, n a m e l y t h a t e v e r y g E C,~(E) is t h e limit in a(TL~~ n) , ~L~(Rn)) of a sequence {g~}~o , g, E B,~(E).
W e shall use t h e following simple localization l e m m a , w h e r e t h e n o t i o n o f w e a k c o n v e r g e n c e refers t o a ( ~ L~~ ~) , ~ L I ( R " ) ) .
LEMM& 3.1. Every g E C~(E), 1 ~ m ~ ~ + (n + 1)/2 , is the weak limit of a sequence {g.}~o, g, E B,,(E), if every point x E E has an open neighborhood U~ c I~ ~ such that every g E C,~(E f] ~ ) is the weak limit of a sequence {g~}~, g~ E BIn(E).
Proof of Lemma 3.1. T h e n e i g h b o r h o o d s U~ c o v e r t h e c o m p a c t set E a n d we can t h e r e f o r e select a f i n i t e s u b c o v e r i n g (Uv)l q. T h e r e exist f u n c t i o n s ~p E ~ ( R ~ ) , p = 1 . . . q , such t h a t t h e s u p p o r t o f ~p is i n c l u d e d in Up a n d such t h a t Z~e = 1 in a n o p e n set, c o n t a i n i n g E. B y L e m m a 2.4, 3 ~ ~eg E ~ + JL~ ( R ) and, b y L e m m a 2.4, 4 ~ q~pg E C,~(E [~ U~). N o w g : X~%g, a n d since e v e r y ~eg is a w e a k limit o f t h e desired kind, t h e same holds for g.
T h e r e g u l a r i t y c o n d i t i o n is i n t r o d u c e d b y t h e following t w o definitions.
Definition 3.2. A closed set F c R n-1 is said t o h a v e t h e restricted cone property at a point Yo E R "-1, if t h e r e exists a n e i g h b o r h o o d V 0 of Y0 a n d a cone K defined b y
K = ( y E R n-1 I (1 - - 5)lYl ~ ( Y , Yl~ ~
(~},
w h e r e 0 < 5 < 1 , y l E R n - l , [Y11= 1, such t h a t
y - - K c F , (3.1)
for e v e r y y E F f] V0.
Definition 3.3. T h e set E C M is said t o h a v e t h e restricted cone property, if for e v e r y x E E a n d e v e r y s u f f i c i e n t l y small n e i g h b o r h o o d V of x w i t h r e s p e c t t o R n, t h e o r t h o g o n a l p r o j e c t i o n o f E N V o n t o t h e t a n g e n t h y p e r p l a n e a t x has t h e r e s t r i c t e d cone p r o p e r t y a t t h e p o i n t x.
W e are n o w in a position t o f o r m u l a t e o u r m a i n t h e o r e m .
THnORW~ 3.4. I f E has the restricted cone property, then every g E C~(E),
32 AItKIV FOR lgATE~IATIK. Vol. 9 NO. 1 n + l
1 ~ m ~ - ~ - 2 , is the limit in a ( ~ L ~ ( R ) ~L~(Rn)) of a sequence (g~)~
of elements in B ~ ( E ) . I n particular, this implies that J,~(E) = Ira(E).
The projection onto the hyperplane in Definition 3.3 can obviously be substituted by a projection onto a suitably chosen coordinate hyperplane. Using such a modi- fication of Definition 3.3 and applying the localization lemma 3.1, we see t h a t Theorem 3.4 is proved, if we can prove the following proposition.
PROI'OSITIO~ 3.5. Let F , V o and K be subsets of R n-~, satisfying the properties requested in Definition 3.2, and let V 1 and V 2 be open sets in R "-1 such that
and such that
for
every Yo E V o , y l C V 1.Voc L c
Y 0 - - K c V I , y l - - K c V~ (3.2)
v,,.
Let ~f be real and infinitely differentiable with non-vanishing H e s s i a n in We define in R n the sets
M 0 ~ - { ( y , z ) l Y e V 2 , z = - ~ ( y ) } E 1 = {(y, z) J y e F F! V1, z = ~(y)}
E 0 ----{(y,z) l y e F A Vo, z = ~ ( y ) ) .
M o is then an (n - - 1)-dimensional manifold with the same properties as those requested f o r M . E o and E 1 are considered as subsets of M o.
n ~ - I
Let 1 ~ m ~ ~ ~ 2 ' g E Cm(Eo). T h e n there exists a sequence (g~)~, g~ E B,~(E1), which converges in a(JL~ ~ ~ ( R ) 7L'~(R")) n , to g.
4. Proof of Proposition 3.5
This entire section is devoted to the proof of Proposition 3.5, which as we have remarked earlier implies our main result, Theorem 3.4.
We shall first introduce two auxiliary functions fi and y. ~ E ~ ( R "-1) is a function with its support contained in K, and satisfying
(2~) -("-1) / fi(y)dy = 1 o (4.1)
Rn--1 J
Y = ~ ( R " - I ) is assumed to take the value 1 in a neighborhood of Vo and to vanish outside V r We use the notation ? as well for the function on R ~ with values y(y) for every ( y , z ) E R n.
O ~ T H E S P E G T I : t A L S Y ~ T ] ~ : E S ] : S I~I~OBLE1V~ 33 W e p u t
W o - ~ ( ( y , z ) l y e W ~ - - ( ( y , z ) l Y e W~.----((y,z) [ y C a n d i n t r o d u c e a b i j e c t i o n S o f W2 o n t o
V0, zeR}
itself b y t h e d e f i n i t i o n S ( y , z) ~- (y , z - - ~f(y)) ,
( y , z) C W 2. O b v i o u s l y S a n d its inverse S -1 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 on W 2.
F o r e v e r y differentiable f u n c t i o n ~, d e f i n e d in a subset Q o f W 2, ~ o S a n d
~ o S -1 are i n f i n i t e l y differentiable f u n c t i o n s d e f i n e d in S-I(Q) a n d S(Q) r e s p e c t i v e l y . F o r a d i s t r i b u t i o n v, s u p p o r t e d b y a subset Q o f W 2 we can in a n analogous w a y define d i s t r i b u t i o n s v o S a n d v o S -I, s u p p o r t e d b y S-I(Q) a n d S(Q), r e s p e c t i v e l y .
F o r a n y h, 0 ~ h ~ 1, we d e n o t e b y fih t h e m e a s u r e s u p p o r t e d b y { ( y , z) I Y e R n-l, z ~- 0} and w i t h d e n s i t y h-(n-1)fl(y/h), y e R n-1. fi~ d e n o t e s t h e m e a s u r e s u p p o r t e d b y t h e same h y p e r p l a n e , b u t w i t h d e n s i t y h-(n-1)fi( - y/h), y C R n-1.
D e f i n i n g c o n v o l u t i o n b e t w e e n f u n c t i o n a n d m e a s u r e in t h e u s u a l w a y , we see t h a t if ~ is i n f i n i t e l y differentiable on W2, t h e n b y (3.2) t h e r e exists a differentiable f u n c t i o n on W1, o b t a i n a b l e b y c o n v o l u t i n g ~ a n d /~h. W e call this f u n c t i o n
* fib. I f v is a d i s t r i b u t i o n , w i t h s u p p o r t i n c l u d e d in Wo, t h e n b y (3.2) v 9 fib*
has its s u p p o r t i n c l u d e d in W 1. Using this we can give t h e following definition, w h e r e g is a d i s t r i b u t i o n , s a t i s f y i n g t h e conditions o f P r o p o s i t i o n 3.5.
Definition 4.1. F o r e v e r y h, 0 ~ h ~ 1, we p u t T * g ~-- ((g o S - 1 ) . / ~ h ~) o S and, for e v e r y f e ~ ( R n ) , t h e f u n c t i o n T h f is d e f i n e d b y
T h f ( y , z) ~ y(y)[((f o S -1) 9 fib) o S ] ( y , z ) , if ( y , z) e W~, T h f ( y , z ) : O , if ( y , z ) ~ W 1 .
I t is e a s y t o see t h a t t h e s e definitions m a k e sense. W e h a v e use for t h e following l e m m a , w h e r e we assume 0 ~ h ~ 1:
LEMMA 4.2.
]% T*g C B~(E1).
2 ~ For every f e ~ ( R ' ) we have T ~ f e ~ ( R ' ) and iiTh f _ 7 f l l ~ - ~ 0 , as h - + 0 . 1
3 ~ There exists a constant C, independent of h, such that
~1 1
/ f f e %(R~).
34 AI~KIV FOR IIIiTEMATIK. Vol. 9 lqo. 1
W e p o s t p o n e t h e p r o o f o f t h e l e m m a a n d show first h o w t h e l e m m a can be u s e d t o p r o v e P r o p o s i t i o n 3.5.
B y L e m m a 4.2, 3 ~ Th can be e x t e n d e d t o a b o u n d e d linear o p e r a t o r f r o m YL~(R n) into itself, w i t h o p e r a t o r n o r m _< C, for e v e r y h. W e cull its a d j o i n t AI,.
F o r e v e r y f E %(R"), e l e m e n t a r y d i s t r i b u t i o n t h e o r y gives ( 2 ~ ) " ( f , Ahg) ---- ( 2 u ) " ( T h f , g) = ( y [ ( ( f o S 1 ) , fib) o S ] , g ) =
= ( ( ( f o S -1) , fin) o S , g} = ( f , ((g o S -1) * /~1~) ~ S ) : -- ( f , T Z g } = (2=)~ T ' e ) ,
w h e r e t h e s y m b o l ( , } refers to d i s t r i b u t i o n s on W 1. H e n c e A,g ~-- T*g.
N o w it is e a s y t o conclude t h a t T * g t e n d s t o g w e a k l y , as h--> 0. F o r t h e n o r m o f t h e o p e r a t o r Ah, c o n s i d e r e d as a b o u n d e d linear t r a n s f o r m a t i o n f r o m
~L~~ ") t o itself is _< C, h e n c e t h e e l e m e n t s Ahg h a v e u n i f o r m l y b o u n d e d n o r m in L~ ( R ) . H e n c e it suffices to show, t h a t ( f , Ahg) --> ( f , g), as h -+ O, for e v e r y f e %(R"). B u t for such e l e m n t s f
( f , A h g ) - - ( f , g) = ( T . f - - 7 f , g ) ,
a n d t h e r i g h t h a n d m e m b e r t e n d s to 0, b y L e m m a 4.2, 2 ~
A n d since, b y L e m m a 4.2, 1 ~ we h a v e T*g E BIn(E1), for e v e r y h, 0 < h < 1, P r o p o s i t i o n 3.5 is p r o v e d .
_Proof of Zemma 4.2.
1 ~ W e k n o w t h a t t h e s u p p o r t o f g is i n c l u d e d in E0, hence g o S -1 has its s u p p o r t i n c l u d e d in t h e o r t h o g o n M p r o j e c t i o n o f E 0 into t h e c o o r d i n a t e h y p e r - p l a n e { ( y , z) [ y C R "-1, z ---- 0}, in t h e following called t h e y - h y p e r p l a n e . B y (3.1) a n d (3.2) a n d b y t h e d e f i n i t i o n s o f E o a n d E 1 we f i n d t h a t ( g o S - 1 ) , f i * , for e v e r y h w i t h 0 < h < 1, has its s u p p o r t i n c l u d e d in t h e orthogonM p r o j e c t i o n o f E 1 i n t o t h e y - h y p e r p l a n e . H e n c e t h e s u p p o r t s o f t h e d i s t r i b u t i o n s T*g are i n c l u d e d in E r
I t is w e l l k n o w n (Schwartz [8], p. 101) t h a t we h a v e for some q a r e p r e s e n t a t i o n q--1
g o S -1 = ~ /tp @ (~(P) p=0
w h e r e /tp are d i s t r i b u t i o n s on R "-1, w i t h c o m p a c t s u p p o r t a n d w h e r e d (p) is t h e d e r i v a t i v e s o f o r d e r 29 o f t h e ] ) i r a c m e a s u r e o n R. This has t h e n to b e i n t e r - p r e t e d in t h e sense t h a t t h e r e p r e s e n t a t i o n o f R ~ as R "-1 • R c o r r e s p o n d s to t h e c o o r d i n a t e r e p r e s e n t a t i o n ( y , z), y E R "-1, z C R. F r o m this it is e a s y to u n d e r - s t a n d t h a t we h a v e a r e p r e s e n t a t i o n
q . 1 p=0
w h e r e F~ n o w b e l o n g t o ~ ( R " - ~ ) . I t is seen f r o m this t h a t t h e s u p p o r t o f e v e r y
O N T H E S P E C T R A L S Y N T H E S I S I~ROBLEI~I 35
~p @ d(P) is included in the support of (g o S -1) 9 fi*, and from this we see at once, by applying the mapping S to the two members, t h a t T * g C Bq(E1).
I t remains to show t h a t ~-1 v a 0 implies t h a t q < m. I f 90q_1 # 0, then it is possible to find a function q C ~ ( R ' ) with support included in W 1, vanishing on the y - h y p e r p l a n e together with all derivatives of order < q -- 2, b u t such t h a t
< ~ , ( g o S - 1 ) , f i ~ * > # 0 . H e n c e
<(~
* 8,,) o s , g} # o ,w h i c h s h o w s that there is a function in Jq_1(Eo), w h i c h is n o t annihilated b y g.
B y the a s s u m p t i o n g E Cm(Eo) , a n d thus q ~ m .
2 ~ The infinite differentiability of Thf, for f E ~(Rn), is a consequence of the infinite differentiability of (f o S -1) 9 flh in W 1 and of the assumption on the support of 7. The compactness of the support is evident. Hence T h f E ~(R~).
The supports of T h f for f fixed, h variable, are in fact included in a fixed com- pact subset of W e. Hence it suffices to show, due to well-known estimates, t h a t all partial derivatives of T h f converge uniformly to the partial derivative of yr.
I t is evident t h a t it suffices to show t h a t the partial derivatives of k *fih con- verge uniformly to /~ on every compact subset of W1, if k E %(R~), which is immediate.
3 ~ Let us agree in the following to interpret the product of y and a n y complex- valued function defined in l ~ , as a function on tl ~ with values determined b y the product in W1, and with the value 0 outside W 1. Then for f C %(R~), with the changes in the order of the integration m o t i v a t e d b y absolute convergence, we have, for ( y , z) E R ~,
Y ( Y ) / f ( Y - - Yo, z -~ y ( y - - Yo) - - ~(Y))h-(~-~)fl(Yo/h)dYo = T , f ( y ,
Z)
Rn-- 1
r ( v ) / f ( y - t ~ , z § w ( v - h~) - - V,(y))~(~)g~ =
/ i
, ]
Rn--I
R R rt-1 R n-1
Hence, b y Fourier's inversion formula, we have for every ( y , ~) E R ~ (2z)-1 / T h f ( y , z ) e ~ d z ~-
Ir
: ~'(y) f f e--~(<Y--h'~'~l~
Rn--1 Rn--1
and forming the Fourier transform of T h f we thus obtain
36 hItIr F61~ MATEMATIK. V o l . 9 N o . 1
(2:r), '-1 T h f ( v , $) = (2z) -1 T f f ( y , z)e ~zc+i<y''J* d z d y = (4.2)
R n - 1 R
f l
/ d v o f ( ~ ] o , $ ) E ( ~ , ~]o , $ , h ) ,
, ]
R n - 1
where ( ~ / , $ ) E R n and where, for e v e r y (V,~/o,$) C R 2"-1
J , ] Rn--1 Rn--1
The function ( y , a) --+ (~f(y - - ha) - - ~p(y))/h is infinitely differentiable on the s u p p o r t of ( x , y ) - + y(y)fi(a) because of (3.2), a n d each partial derivative has a bound, uniform in h. F u r t h e r m o r e , the Hessian of the function is - - H ( y - - h a ) H ( y ) , where H is the Hessian of ~, hence its inverse has bounds, which are uniform in h. A p p l y i n g L e m m a 2.1 w i t h m = 2 n - - 2 and A = n ~ - [~I, we obtain for e v e r y (~ , ~o, $) E R 2~-I,
( _o!t
I E ( v , V0, r h): ~ D(1 q- ISh:) -(~-:) 1 -~ 1 + t~h[! ' where D is a constant, i n d e p e n d e n t of h, a n d b y (4.2) we obtain
II ~f[l~
= d~IT~I(~,
$)1(1 -t- 1(~, $)l)~d~ --< (4.3)I~ ,,J i~n--1
< (2:r) ~-~ d$ d~0]f(~o, $)1 E(W,W0, ~, h)](1 § [(~, $)l)~d~.
R R n - 1 Rn--1
B u t e l e m e n t a r y inequalities show t h a t
1 + < (1 + l(~o, ~)1) ~ 1 + : ~ l / ' for e v e r y (~ , ~]o, $) ~ R~n-~ a n d hence
/
I E ( ~ , ~ o , h)l(1 + I(~, r <Rn--1
D(1 -~ I(Vo, $)I) ~ (1 + IShl)-(n-a) 1 q- f - ~ ~ i ] d~ =
Rn--1
= D1(1 q- ](r? o, $)[)~, for e v e r y (~0, $) ~ R~,
and 3 ~ is proved.
where D 1 is i n d e p e n d e n t of h. H e n c e b y (4.3) IIThfJ]~ ~ (2:r) 1-n Dl[[fll ~ ,
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