(C.I.M.E.)
SOLVABILITY OF INVARIANT DIFFERENTIAL
OPERATORS ON HOMONOGEOUS
MANIFOLDS
S. Helgason
Varenna. August 25 /September 2, 1975
-282S. Helgason
SOLVABILITY OF INVARIANT DIFFERENTIAL
OPERATORS ON HOMOGENEOUS
MANIFOLDS
Sigurdur Helgason
MIT Cambridge Mass .
§1 .
Introduction and summary .
Over a 100 years ago S . Lie [18] raised the
following question:
Given a differential equation, how can knowledge
about its invariance group be utilized towards its solution?
Special case .
~ =
(*)
dx
in the plane,
group
Consider the differential equation
cj>t
Y(x,y)
X(x,y)
It is called stable under a one- parameter
(t E ffi)
the integral curves
of diffeomorphisms if each
<Pt
permutes
(all concepts are here local) .
Example .
The left hand side represents the tangent of the
angle between the radius vector and the integral curve .
Thus
the equation shows that the integral curves intersect each
circle x 2 + y 2 = r 2 under a fixed angle . Hence the group
-283S. Helgason
of rotations around the origin permutes the integral curves,
i.e., leaves the equation, stable.
For a one-parameter group
in the plane let
Up
=
U denote the induced vector field
[·d ~ t • P)J
a
= ~(p)~x
C
dt
Theorem (Lie).
t=O
o
+
a
n(p)~Y
o
Equation(*) is stable under
Z = x-l
ax + y_l
ay
only
- if the vector field
[U,Z] = t..Z
In this case
equation
of transformations
(~t)
(Xn - Y~)-l
(>..
p ~
JR 2 •
(~t)
if and
satisfies
a function)
is an integrating factor for the
Xdy - Ydx = 0.
Thus knowing a stability group for a differential
equation provides a way to solve it.
For the example above
the equation is stable under the group
$t(x,y) = (x cos t - y sin t, x sin t + y cos t)
for which
u
= -
a
a
Yax + xay
and the theorem gives the solution
In spite of extensions of this theorem to higher
order partial differential equations these methods have had
-284S. Helgason
a limited influence on the general theory of differential
equations.
Nevertheless, Lie ' s question has been of enormous
importance in mathematics, in fact leading to the theory of
Lie groups.
Aiming for less generality we can pose a few
problems in Lie's spirit.
Let
being a closed subgroup of
~
G/H
be a coset space,
Lie group
differential operator on the manifold
invariant under
G
D(f
for all
g
T(g):xH
~
origin
eH
A.
c
T(g)) = (Df)
o
of
of
G/H
G/H .
T(g)
o
T(g)
T
o
Is
Let
on
G/H
which is
f
00
i C (G/H)
o
denote the
D
D, i.e., a
such that
o,
locally solvable, i.e., is
for some neighborhood
Is
be a
We can then ask the following questions.
denoting the delta-distribution at
D
D
denoting the translation
onto itself.
DT =
C.
G/H
Does there exist a fundamental solution for
dist~ibution
B.
and let
in the sense that
G, the mapping
gxH
G
H
V
of
o?
globally solvable, i.e., is
DC""(G/H) = c""(G/H) ?
o
on
G/H?
-285S . Helgason
Let
D(G/H)
denote the algebra of all G- invariant
differential operators
o~
G/H.
group of translations
of
Rn
For example, if
and
H = {0}
G
then
is the
D(G/H)
consists of all differential operators with constantv
coefficients; by results of Ehrenpreis and Malgrange,
questions A)
case .
E)
and
C)
have positive answers in this
We shall see however that in general the answers
depend on the space
G/H .
We shall now survey - in chronological order - some
of the principal results which have been obtained for the
questions above.
First if
X = G/K
is a symmetric space of the
noncompact type (i.e . ,
G
noncompact, connected semisimple
with finite center and
K
a maximal compact subgroup) then
as proved in [11] each
D
E
D(G/K)
has a fundamental solu-
tion, and consequently, by convolution,
solvable .
Since in solving the equation
by the K-invariance of
B
and
o,
assume
D
is locally
DT =
T
o
we may,
to be
K-invariant, the problem can be handled within Fourier
analysis of K- invariant functions on
X.
solvability discussed later does not
see~
The global
to be accessible
by this method .
It was proved in Peetre [19] (cf . also Hormander
[15] , p. 170) that a differential operator of constant
strength is necessarily locally solvable .
.
Without entering
into the precise definition, a differential operator of
constant strength is in a certain sense a bounded perturba-
-286-
S. Helgason
tion of a constant coefficient operator.
One might wonder
whether an invariant differential operator is not necessarily
of constant strength.
This can not be so because then it
would be locally solvable and the operator
which is essentially the non-locally solvable Levi-operator
is (cf. [4]) a left invariant differential operator on the
Heisenberg group
x,y,zE,lR.
Thus we see that
answer.
B has in general a negative
In view of this, one can modify the problem and ask:
For what Lie groups
L
are all left invariant differential
operators (i.e., all members of
solvable.
D(L) = D(L/{e})
locally
Using Hormander's necessary condition for the
local solvability of a differential operator [15], p. 157,
he, and independently
order that each
C~rezo-Rouvi~re
D ( D(L)
[4], proved that in
should be locally solvable it is
necessary and sufficient that the Lie algebra of
L
be
either abelian or isomorphic to the algebra of
n x n
matrices where each row except the first is
(and thus has
0
an abelian ideal of codimension 1).
The contrast between this negative result for
and the positive result for the symmetric space
G/K
L
-287S. Helgason
disappears when we view
spaces.
A coset space
L
in the context of symmetric
~lc
~·Lie
(B
group,
C a closed
subgroup) is called a symmetric coset space if there exists
an involutive automorphism
C.
The spaces
cr
of
B with fixed point set
G/K
considered above have this property.
L
as a homogeneous space under left and
Now we consider
right translations simultaneously, i.e., we let the product
group
L
L
x
act on
L
by
g
The subgroup leaving
e
E L.
fixed is the diagonal
L* - L
so
we have the coset space representation
L = (L
Then the algebra
the algebra
L
Z(L)
X
D(L x L/L*)
L)/L*.
is canonically isomorphic to
of hi-invariant differential operators on
and the natural problem becomes:
locally (globally) solvable on
Given
DE Z(L), is it
L?
This problem was considered by Rals for simply
connected nilpotent Lie groups
D E Z(L)
L.
He proved that each
has a fundamental solution and hence is locally
solvable .
The method is based on harmonic analysis on
L.
Shortly afterwards [13] I proved local solvability for each
D E Z(L)
for the case when
L
is semisimple.
an easy consequence of a structure theorem of
for the operators in
analysis on
L.
Z(L)
The proof was
Harish~Chandra
but did not require any harmonic
-288 S . Helgason
At about the same time I proved global solvability,
00
00
DC (G/K) = C (G/K)
D
for each invariant differential operator
on the symmetric space
G/K .
The proof [13] is based on
a characterization of the image of
transform on
C~(G/K)
under the Radon
G/K .
What about global solvability on
L?
A remarkable
example of Cerezo-Rouviere [5] shows that for any complex
semisimple Lie group
L
there exists an element
which is not globally solvable .
This
w
w € Z(L)
can be taken as the
imaginary part of the complex Casimir operator on
L
L.
viewed as a real Lie group has a Casimir operator
which they showed, using harmonic analysis on
solvable .
But
C
L, is globally
This global solvability was more recently proved
by Rauch- Wigner [21], for all non compact real semisimple
Lie groups
methods .
L, with finite center, using entirely different
The main step in their proof is verifying that no
null bicharacteristic of
C
lies over a compact subset of
G.
The case of a solvable Lie group
left out .
found by
has so far been
Rais' method for the nilpotent case was extended
by Duflo - Rals [7] to solvable
D € Z(L)
L
i s locally solvable .
Rouvi~re
L
and they proved that each
An alternative proof was
[22], modifying in an interesting manner
methods which Hormander had introduced for constant
coefficient operators .
His method, being local, does not
give the supplementary result of [7] that if
exponential solvable Lie group then
D
L
is an
has a fundamental
-289S . Helgason
solution .
The results I have quoted on the Lie groups
on the symmetric space
G/K
L
and
suggest the following more
general question.
Let
B/C
be a symmetric coset space and
B- invariant differential operator on it .
Is
D
D
a
necessarily
locally solvable?
§2 .
Solvability results.
Indications of proofs.
We shall now indicate the proofs of the principal
results already stated.
We emphasize that the proofs of
Theorems 1 - 5 are quite independent and socan be read in any
order.
We have already mentioned the fact that a leftinvariant differential operator on the Heisenberg group is
not necessarily locally solvable .
In order to describe the
proof of Rais ' positive result for nilpotent groups I recall
the main results from harmonic analysis on nilpotent Lie
groups [17] .
Let
~
be a simply connected nilpotent Lie group ,
its Lie algebra ;
sentati6~
~·
G
G
acts on
and by duality on
To each
fE
representation
~*
rrf
OJ-*,
~
via the adjoint repre -
the dual vector space to
is associated a unitary irreducible
as follows :
Let
H
be a closed
connected subgroup with the property that its Lie algebra
-290-
S. Helgason
·"9
f( [~ ,~])
satisfies
= 0.
with these properties.
of
xf(exp X)
nf
= e 2nlf(X) ,
nf
·f 1 , f 2 E
·Of*,
nf
""
G,
and
then
nf
Xr ·
H;
are unitarily
2
fl
G actin.g on '}* .
identified with
of
G induced by
is independent of · the choice of
equivalent if and only. if
orbit· of
T by
denote the representation of
It is known that
moreover if
H of maximum 'dime!"lsion
Then we can define a homomorphism
into the circle group
H
and let
Ass.ume
f
Thus the orbit space 0}*/G
the· dual spac·e of
is a distribution on
are in the same
2
G
The character
G.
is
Tf
(Dixmier [6]) which is given
by the formula
(1)
where
denotes the Euciidean Fourier transform and
a G-invariant measure on the orbit
Plancherel formula:
G• f.
there exists a measure
on tJ* IG
( 2)
0
~ o
denoting the orbit
G•f ,
Using (2) and (1) for
exp = ·ll' . we get for a Haar measure
is
Finally there is a
such that
f
~
df
on
OJ*
-291S. Helgason
1~ .(f)df =I~*Ia ri.a •f•<f')dp(f'))dA(fo) .
(3)
a
For
s
g;eneral Lie algebra
S(~)
of the symmetric algebra
enveloping algebra
s( tn) = tn . (m
a
action of
onto
th~
U(,).
U(Uj)
E
U(,);
s
z+, X~ Of).
OJ
there is a bijection
onto the universal
is determined by the property
Also
s
commutes with the
so maps the ring of a-invariants
ring of a-invariants in
I(CJ} C S(Oj)
U(Oj), the center
Z(Oj>
of
Recall now that in terms of differential operators,
= D(a)
and
Z("J)
= Z(a).
eigend1str1but1on of each
The character'
Tf
is an
Z E Z(a)
(4)
and the eigenvalue
evaluated at
Z(f)
is just the polynomial
s- 1 (Z)
f.
Let us also recall the following general fact
([2], [3], [20]):
En.
<P
+
Let
Then if Re s > 0,
1Plfl.n
8
(x)Hx)dx
P > 0
P8
be a positive polynomial on
defines a distribution
on Rn, the mapping
meromorphic function on
s
+
ps
extends to a
c with values in the space s I (E~)
of tempered distribution on lRn' and the poles occur among
the numbers
on
(5)
P).
-1/N, -2/N, ...
Moreover
(N
being an integer depending
-292-
S. Helgason
This can be used to construct a fundamental solution for a
constant coefficient differential operator
Taking Fourier transform
f
+
f
we have
D
on
mn:
(Df)- = Pf
where
P
is a polynomial, which we may assume positive, replacing
D
by
DD 1
if necessary (*=adjoint).
The tempered
distribution
then satisfies
s = -1.
we get
T1 * Ts = Ts+l
Expanding
Ts
which is holomorphic near
in a Laurent series near this point
Ts
Iak(s+l)k where each ak is a distribution.
-r
The relation above then gives T 1 a = o and up to a
0
1
is a fundamental solution for D.
factor,
We can now prove the following result (cf. [20]).
Theorem l.
Each bi-invariant differential operator
nilpotent Lie group
G
We may assume
notation above.
Z
on a
is locally solvable.
G
Replacing
simply connected and use the
Z
by
ZZ
we may assume the
A
polynomial
P = Z(f)
the distribution
~ +
in (4) to be positive.
(Z•~)(e)
(• =adjoint).
Let
~
denote
Then by (2) -
( 4)
( 6)
In analogy with
Ts
above we define
~s
for
Re s > 0
by
-293-
S. Helgason
By the result about
P
quoted,
s
extends to a
s
meromorphic d i stribution-valued function on C and by (5)
~
~
and ( 6),
~l = ~ .
Moreover , by (l) and (3),
( 7)
whence
b
0
z~s
= ~s+l'
i.e .,
in the Laurent series
~
~s
*
~s = ~s+l "
00
Again the term
L bk(s+l)
k
gives a
-r
fundamental solution for
..
Z.
Duflo and Rais [7] extepded this argument to
solvable groups giving
.
Theorem 2 .
A hi-invariant differential operator
D
on a
solvable Lie group is locally solyable.
A completely different proof was found by
We sketch his argument below .
[22] .
x1 ,
Let
and let
l < j
by
Rouvi~re
< n
. . . , Xn
be a bas i s of any Lie algebra
denote the derived algebra
let
aj
[~, '}] .
denote the endomorphism of
'1'
For any
U(~)
j1
given
-294-
S. Helgason
( 8)
Lemma 1 .
Assume
and
Xj ~ DOf·
is a derivation of
Then
aJ z<<J ) c z cop .
In fact the restriction of
on IR
and extends to a derivation
T(~) .
Since
d
d([1,qJ)
= 0
induces a derivation of
immediately by induction, so
X€
OJ ,
Z E. Z(Op
and
U(,) .
to
d
or +
vanishes
IR
of the tensor algebra
d(X ® Y- Y ®X)= 0
Then (8) follows for
d = aj .
d
Finally, if
then
so
Remark.
The proof in [22] is quite different and proceeds
by showing that in a suitable neighborhood of
e
in
G,
(9)
where
e.
is a suitably chosen function which vanishes at
This will be used below .
Next we have the general identity for the
12
~P ' uU 2 = - (P ' u,fPu) + (P*u,f(P ' ) *u) - (P *u,(P") *u)
·+ ([P * ,P']u,fu)
norm,
-295S. Helgason
on an open set
differential operator on
P'
=
[P,f],
=
P"
f E c""(u),
u E cJ(u)
U C G,
U with adjoint
[P' ,f] .
P*, where
PE
Using this on
any
P
z<op,
= fj
f
in (9) we get easily by Lemma 1 and (9)
where
C
m of
P
= supjfj I .
This implies by induction on the order
( 10)
Lemma 2.
P
$
Let
Z(Dr~).
r
be an integer
and
+0
Q
and a neighborhood
II Qu II
e
5.,
II Pu II
u E. C~(U) .
for all
This is proved by induction on
comes from (10) and Lemma 1 .
= dim
~
- dim
Then
D~.
p
nonzero derivative of
Then
U of
such that
G
( 11 )
j
Z(~),
P €
Then there exists an operator
Q E Z(Dr~) 1\ Z(Dr-l~),
in
> 1
Q E.
z<op
and
the passage from
the analytic group
to
H
The case
and we take for
Q
a
of maximal order in
r+l
so
Q E Z(OJ-} A
-b
r = 1
=
DrOJ·
yields (11) after some analysis on the factor group
G is solvable let
For
Z(D, .
one uses the case
C G with Lie algebra
Now if the group
r = 1
In fact, let
j > 1
Q E. U(DOj)
r
r.
r
on
This
G/H.
be such
-296S. Helgason
that
DrGf =
{O}
so
Z(Dr~) =
c.
Then Lemma 2 gives an
inequality
Jl u II ~ C IIPu II
which as well known [14] implies local solvability, i . e.,
Theorem 2.
Next we consider the semisimple case .
Here we have
by [10]'
Theorem 3.
Each hi-invariant differential operator
semisimple Lie group
G is locally solvable .
u,
For this let
of
0
Cf
in
such that
be a connected open neighborhood
exp
onto an open neighborhood
is a diffeomorphism of
UG
of
e
assume, as we can (cf . [9]) that
u,.
that is, for each compact subset
cc
Cl(Ad(G) • C) C U,. Let
D on a
n
in
G.
In addition we
is completely
invarian~
UOJ-' the closure
denote the Jacobian of the
exponential mapping, i . e . ,
-ad X
1
n ( x) = det ( - :d x· )
Then according to Harish-Chandra there exists a specific
Ad( G) - invariant differential operator
Pn
on
't
with
constant coefficients such that
(Df)
(12)
whenever
f
is a
o
exp
00
C
= n- l/ 2pD (n 112 f
function on
under inner automorphisms .
o
exp)
UG' locally invariant
(This formula is proved in [9]
-
297 _:__
S. Helgason
by computing the radial parts of the operators
D and
p0 ;
for an analogous global formula for the symmetric space
G/K
(with
G complex) see [11]).
Denoting by
A the mapping
we have by (12),
( 13)
If
tions on
G is compact, the
Ad(G)-invariant distr!bu -
form in a natural way the dual to the space of
Ad(G) - invariant functions in
C~(U~) .
The adjoint of
A
is
112 s)exp, the image of the distribution
given by A*S
(n
n1 1 2s under exp, so by (13)
if
S
is an
Ad(G) - invariant distribution on
U~.
Harish-
Chandra proved ([8], [9), p .· 477), that (14) is true even i f
G is noncompact; since the process of averaging over
G is
not available, entirely different methods are required.
(p 0 *)*
take
has a fundamental solution
on
~
Ad(G) - invariant (cf . the construction of
Theorem 1) .
on
S
UG'
DT
which we may
a0
before
Using (14) we get a fundamental solution for
= o,
and Theorem 3 follows by convolution.
Global solvability does not in general hold in
Theorem 3 as shown by the following example of Cerezo Rouviere [5) .
Let
Now
G be a complex semisimple Lie group,
D
-298S. Helgason
K a maximal compact subgroup and
Lie algebras.
Then
structure of~·
OJ = ~ +
Let
(Ti)
OJ, {l
J-k. where
their respective
J
is the complex
be a basis of~. orthonormal
forthe negative of the Killing form, and put
w =
L
(JTi)·Ti·
i
Then if
[T,Ti] =}: cijTJ' where
TE-i,
(cij)
is skew
j
so
[T,w] =
L [T,JTi].Ti
i
+ (JTi).[T,T 1 J
=
=
=
Similarly
w
L cij(JTJ)·Ti- L cij(JTJ)•Ti
i,j
i,j
=0
[JT,w]
= L Ti(JTi).
Thus
since
w
w
= 0.
can also be written
is a hi-invariant differential
i
operator on
G.
Since it annihilates all
G which are right invariant under
C®
functions on
K it is not globally
solvable.
Nevertheless we have by [5] for complex
[21] for real
Theorem 4 .
Let
G and by
G,
G be a connected noncompact semisimple Lie
group with finite center .
Then the Casimir operator
C
is
globally solvable .
This follows from general results of Hormander [16]
once the following three facts have been established [21].
-299S. Helgason
c
(I)
(II)
For any compact set
r C G there exists
another compact set
r'
r
Int f'
C
on
and if
such that
C G
u e E'(G)
(distributions
supp(Cu)C r
with compact support) with
G
supp u c f'.
then
(III)
c""(a).
c
is one-to-one on
c
No null characteristic of
compact subset of
lies over a
G.
The Holmgren uniqueness theorem implies for the
C that if a hyper-surface
analytic operator
non-characteristic at
x0
then a solution
which vanishes on one side of
hood of
in
S
u
S
in
to
G is
Cu
=0
must vanish in a neighbor-
This implies (I) and .(II) without
G.
difficulty because it is easy to determine plenty of noncharacteristic surfaces for
C.
In order to describe the proof of (III) let us
recall a few facts concerning null bicharacteristics.
Let
If
(x 1 , ... ,xn)
M be a manifold,
its cotangent bundle.
is a local coordinate system on
is a local coordinate system
T*(M).
T*M
(x 1 , •..
M then
,xn'~l'''''~n)
on
The two-form
n
S)
=
r d~ i "
1
dxi
is called the canonical 2-form on
T*M.
It has the property
-
300 -S . Helgason
that if
~
is a diffeomorphism of
diffeomorphism of
T*M
Given a 1 - form
on
on
w
let
X
(cf . e . g . [1], p . 97) .
denote the vector field
w
T*M given by
rl(• X)= w( • )
(15)
' w
w = Eaidxi + Ebjd~j
If
(16)
If
n
preserves
T*M
M then the lifted
w = df
2Xw
=
where
f
•
then
a
a
Ebiaxi - Eaja~j
00
EC
(T*M)
then
Xdf
Hamiltonian vector field; the function
f
be constant on the integral curves of
Xdf '
(x 1 (t), . ..
,~n(t))
is an integral curve of
is called a
is easily seen to
In fact, if
Xdf
then by
(16),
1 af
2
n-'
k
so
In particular, let
symbol of
c:T*G
~m
be the principal
if
C, i . e .,
metric tensor determined by the Killing form .
is the
The
b i characteristics of
c
curves in
to the Hamiltonian vector field
T*G - ( 0)
I f the constant value of
is
are by definition the integral
c
on an integral curve of
xdc '
xdc
0 , this curve is called a null bicharacteristic .
By the invariance property of
n
stated above it
-301-
S. Helgason
n,
is clear that
Since
is hi-invariant on
C
function on
T*G.
as a 2-form on . T*G, is hi-invariant.
T*G
so
Xdc
G,
c
is a hi-invariant
is a hi-invariant vector field on
Using in addition the fact that
Xdc
is Hamiltonian
the authors prove that its integral curves have the form
y(t) = (a exp tX, wa exp tX)
where
w
and
Now suppose
y
a left invariant 1-form on
G.
is a null bicharacteristic for
lying over a compact subset of
G.
C
Then the same is true of
the curve
y (t) = (exp tX, w
)
exp tX
0
so
group
c
=0
c(y 0 (t)
K
of
G.
and
exp tX
But then
lies in a maximal compact sub-
B(X,X) < 0
implies easily that
is strictly negative on
y , which is a contradiction.
0
Finally, we have by [11] and [13],
Theorem
5.
Let
X
compact type, D
=
G/K
be a symmetric space of the non-
a G-invariqnt differential operator on
X.
Then
(i)
D
has a fundamental solution.
(ii)
D
is
g~obally
solvable.
This theorem can be proved either by means of the
Fourier transform on
on
X.
X
or by means of the Radon transform
After (i) is proved, (ii) follows on the basis of
general functional analysis methods provided one can prove
for each ball
B C X:
-302-
S. Helgason
f ( C~(X),
(17)
supp(Df)~B
-1> supp(f)C B.
Here risupp" stands for "support".
We now sketch how (i) and (17) can be proved by
X.
means of the Radon transform on
Iwasawa decomposition of
G where
A horocycle in
a point in
X under a subgroup of
A is abelian and
N
G conjugate to
N.
The
G permutes the horocycles transitively; more precise~O
ly, if
denotes the horocycle
then each horocycle
~
= origin in X)
= ka·~o where
m in the centralizer
M of
A in
of all horocycles is naturally
Thus the space
manifold of
(o
k £ K is unique up to a right multi-
plication by an element
identified with
N•o
can be written
~
a£ A is unique and
K.
be an
X is by definition an orbit of
nilpotent.
group
G = KAN
Let
(K/M) x A.
Each horocycle
X so inherits a volume element
the Radon transform of a function
f(~) =
f
on
~
is a subdcr
from
X;
X is defined by
Jr(x)dcr(x),
~
whenever this integral exists.
differential operator
D
on the manifold
(Dr)"
(18)
moreover
fl
There exists a G-invariant
= B?
fEC~(X);
has the form
(19)
where
such that
k E K,a € A),
P
is a translation-invariant differential operator on
the Euclidean space
A
(cf. [10], §4-5).
Thus the Radon
-303S. Helgason
transform converts the operator
ent differential operator .
al solution for
D into a constant coeffici -
This principle gives a fundament-
D by a method similar to the one used for
Theorem 3 (cf . [11]) .
In order to use this principle to deduce (17) the
following lemma is decisive .
Lemma 3.
that
B.
Let
f( 0 = 0
Then
E
f
C~(X)
and
B C X a closed ball.
whenever the horoc;Ycle
No~
R denote its radius.
d(o~ka·~
(18) and (19) and
denoting distance .
B.
B
supp(Df)
centered at
~
0 ) ! d(o,a•o)
=0
Pa(f(ka · ~ 0 ))
d
is disjoint from
vanishes identically outside
f
To verify (17) we may assume
Let
~
if
Assume
o.
B implies by
that
d(o,a•o) > R,
But the function
a
-+
f(ka · ~ 0 )
has
compact support so by the Lions - Titchmarsh convexity theorem
0 =0
f(ka · ~ )
we conclude that
Lemma 3 implies
supp(f)
if
<: BR(Q)
d(o,a•o) > R.
But then
so (17) is verified,
proving Theorem 5.
Plausible as Lemma 3 looks, its proof (cf. [13])
is too technical to describe here.
Instead, I.sketch a prodf
of the analogue of Lemma 3 for mn .
Here the Radon transform
F
of a function
F(~)
is defined by
JF(x)dcr(x),
£:
being an arbitrary hyperplane and dcr
(20)
;
F(x)
element on it.
=
being the surface
-304S. Helgason
Lemma 4.
00
Let
FE C (Rn)
(i)
For each integer
satisfy the conditions:
k > 0,
F(x)lxlk
is
bounded.
(ii)
There exists a constant
F(~) = 0
A
d(O,~) > A,
for
such that
d
denoting
distance.
Then
Ixi
F
vanishes identically outside the ball
< A.
Proof (cf. [12]).
function, i.e.,
Suppose first
F(x) = ~(jxj)
Then there exists an even function
$(d(0,0) = F(O.
~
is a radial
~ E C (R)
where
A
F
00
£
is even.
00
C (R)
such that
Now (20) takes the form
is the area of the unit sphere in 'R n-1 • By a
n-1
simple change of variables (21) is transformed into Abel's
where
A
integral equation and is inverted by
where
c
is a constant.
-1\
0 < u < A
so·
Now by (ii),
1
~(s- )
lemma for the case when
0
F
for
for
0 < s < A- 1 , proving the
is radial.
Consider now the general case.
consider the function
~(u- 1 ) = 0
Fix
x ~ Rn
and
-- 305-
S. Helgason
= J(
n
F(x+k • y)dk,
y € 1R ,
O(n)
where
dk
group
O{n); gx(y)
center
x
is the normalized Haar-measure on the orthogonal
is the average of
IYI.
and radius
gx(~)
on the sphere with
We have
F(x+k·~)dk,
J
=
F
O(n)
x+k·~
where
k·~
is the hyperplane
translated by
x.
Clearly
d(O,x+k·~) ~ d(O,~) -
so if
d(O,~) > lxl + A
lxl
gx(~) = 0.
we have
is a
But
radial function so by the first part of the proof,
IYI
for
This means that if
S
~ lxl + A.
is a sphere with surface element
dw
then
JF ( s ) dw ( s )
(22)
= 0
s
if
S
encilioses the solid
ball
-,
radius
A), Lemma
F E C~(Rn)
Let
F(x) lxlk
is bounded.
BA ( 0).
BA(O) .
(with center
0
and
4 will therefore follow from another lemma.
Lemma 5.
solid ball
BA(O)
Assume (22) whenever
Then
k ~ 0,
such that for each
F
S
encloses the
vanishes identically outside
-306-
S. Helgason
The idea for proving this is to perturb
(22) a bit and
~1fferentiate
S
in
with respect to the parameter
of perturbation, thereby obtaining additional relations.
S
=
SR(x)
and
BR(x)
I
the corresponding ball, then by (22),
F(y)dy =
I
F(y)dy
Rn
BR(x)
so
a!
J
F(y)dy =
o,
i BR(x)
that is
(23)
Using the divergence theorem on the vector field
V(y) = F(y)ei
we obtain from (23)
I
F(x+s).:i dw(s) = 0.
SR(O)
Combining this with (22) we deduce
I
SR(x)
By iteration
If
F(s)sidw(s) = 0.
-307-
S. Helgason
J
F(s)p(s)dw(s) = 0
SR(x)
if
p(s)
Remark.
is an arbitrary polynomial, so the lemma follows .
The funct i on
F( z) =
k1
(z E C)
smoothed out
z
near the origin satisfies (by Cauchy's theorem) assumption
(ii) in Lemma 4 .
Similar remark
Thus condition (i) cannot be weakened.
applies to Lemma 5.
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310 --
S. Helgason
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