Hörmander’s Topological Paley–Wiener Theorem
(Informal class notes, S. Helgason)
The space D = D(Rn ) = Cc∞ (Rn ) is given the inductive limit topology
of the spaces DBj (0) of functions ’ ∈ D with support in the ball B j (0) =
{x ∈ Rn = |x| ≤ 1}. This topology can be characterized by the following
result of Schwartz (Distributions, p. 67).
Theorem 1. Given two monotonic sequences
{ }:
0;
1;:::
i
→0
{N } : N0 ; N1 ; : : :
Ni → ∞
let V ({ }{n}) denote the set of functions ’ ∈ D satisfying for each j ≥ 0
the conditions:
(1)
|D α ’(x)| ≤
j
for | | ≤ Nj ;
|x| ≥ j :
Then the sets V ({ } ; {N }) form a fundamental system of neighborhoods of
0 in D.
Let A ≥ 0 and DA the space DBA (0) topologized by the seminorms
(2)
kf km =
X
sup |(D α f )(x)| :
|α|≤m |x|<A
Also let HA = HA (Cn ) denote the space of holomorphic functions of exponential type A, that is the space of holomorphic functions ’ such that for
each N ∈ Z+
(3)
|k’k|N = sup (1 + | |)N e−A| Im ζ| |’( )| < ∞ :
ζ∈Cn
Im denoting the imaginary part of . We topologize H A with the seminorms |k k|N .
Theorem 2. The Fourier transform f → fe where
Z
e
f (x)e−ihx,ζi dx ;
∈ Cn
f( ) =
Rn
is a homeomorphism of DA onto HA .
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Proof:
The Paley–Wiener theorem states that
eA = H A :
D
The continuity statements follow easily from the formulas
Z
|β| β e
(4)
i
f ( ) = (D β f )(x)e−ihx,ζi dx
Rn
and the inversion
(5)
(D α f )(x) = (2 )−n
Z
Rn
(i )α fe( )eihx,ζi d :
The space DA is complete. If fei is a Cauchy sequence in HA , replacing f
in (5) by fi −fj we see
Proof:
Let W ({ } ; {M }) denote the set of u ∈ D satisfying (6). Given k ∈ Z +
the set
1
Wk = {u ∈ DB (0) : |e
ek| Im | }
u( )| ≤ k
k
(1 + | |)Mk
is by Theorem 2 a neighborhood of 0 in DB k (0) and is clearly contained in
W ({0}{M }). If V is a convex set containing W ({ } ; {M }) then V ∩ DB k (0)
contains the neighborhood Wk of 0 in DB (0) so by the definition of inductive
k
limit V is a neighborhood of 0 in D.
Proving the converse amounts to proving that given V ({ }; {N }) there
exist sequences { } {M } such that
W ({ }{M }) ⊂ V ({ }; {N }) :
For this we shift the path of integration in
Z
−n
u
e( )eihx,ξi d
(7)
u(x) = (2 )
Rn
to another one, in which the two weight factors in (3) are comparable. We
write
x = (x1 ; : : : ; xn ) ; x0 = (x1 ; : : : ; xn−1 )
0 = ( ;:::;
= ( 1; : : : ; n) ;
1
n−1 )
0
= ( 1; : : : ; n) ;
= ( 1 ; : : : ; n−1 )
; ∈ Rn :
= +i
Then
(8)
Z
Rn
u
e( )e
ihx,ξi
Z
d =
e
ihx0 ,ξ 0 i
Rn−1
d
0
Z
R
eixn ξn u
e( 0 ;
n) d n
:
In the last integral we shift from R to the contour in C given by
(9)
m
:
n
=
n
+ im log(2 + [| 0 |2 +
2 1/2
);
n]
m being arbitrary. We claim that, by Cauchy’s theorem
Z
Z
e( 0 ; n ) d
(10)
eixn ξn u
e( 0 ; n ) d n = eixn ζn u
R
γm
3
n
:
ηn
γm
m log(1 + |ξ′|)
ξn
For this we must estimate the right integrand in the “strip” between the
n -axis and the curve m .
The function n → u
e( 0 ; n ) satisfies
|e
u( 0 ;
(11)
n )|
≤ CN
eA| Im ζn |
(1 + | n |)N
for some A, all N , the constant CN depending only on N . On the vertical
line joining ( n ; 0) to ( n ; n ), u
e( 0 ; n ) (with 0 fixed) decays faster than any
−1
power of | n | . Secondly,
|eixn ζn | ≤ e|xn ||ηn | ;
which is bounded by a polynomial in | n |. Also on m
d n 1
@(| |) (12)
d n = 1 + im 2 + | | @ n ≤ 1 + m (m > 0)
thus (10) follows from Cauchy’s theorem in one variable. Putting
Γm = { ∈ C n |
and d = d
(13)
1
:::d
0
∈ Rn−1 ;
n
∈
m}
n−1 d n
we thus have for each m > 0
Z
−n
u(x) = (2 )
u
e( )eihx,ζi d :
Γm
Now suppose the sequences { }, {N } and V ({ }; {N }) are given as in
Theorem 1. We have to construct sequences { } {M } such that (6) implies
4
(1). By rotational invariance we may assume x = (0; : : : ; 0; x n ) with xn > 0.
For each n-tuple we have
Z
(14)
(D α u)(x) = (2 )−n
u
e( )(i )α eihx,ζi d :
Γm
Starting with positive sequences { }, {M } we shall modify them successively such that (6) ⇒ (1). Note that for ∈ Γ m
ek| Im ζ| ≤ (2 + | |)km
(15)
(16)
|
α
| ≤ | ||α| ≤ ([| |2 + m2 (log(2 + | |))2 ]1/2 )|α| :
For (1) with j = 0 we take xn = |x| ≥ 0, | | ≤ N0 so
|eihx,ζi | = e−hx,Im ζ| ≤ 1
(17)
for
∈ Γm :
Thus if u satisfies (6) we have by (12), (15), (16)
(18)
∞
X
≤
0
|(D α u)(x)|
Z
(1 + [| |2 + m2 (log(2 + | |))2 ]1/2 )N0 −Mk (2 + | |)km (1 + m) d :
k
Rn
We can choose sequences { }, {M } (all k , Mk > 0) such that this expression
is ≤ 0 . This then verifies (1) for j = 0. We now fix 0 and M0 . Next we
want to prove (1) for j = 1 by shrinking the terms in 1 ; 2 ; : : : and increasing
the terms in M1 ; M2 ; : : : ( 0 , M0 having been fixed).
Now we have xn = |x| ≥ 1 so (17) is replaced by
(19)
|eihx,ζi | = e−hx,Im ζi ≤ (2 + | |)−m for
∈ Γm
so in the integrals in (18) the factor (2 + | |) km is replaced by (2 + | |)(k−1)m .
In the sum we separate out the term with k = 0. Here M 0 has been fixed
but now we have the factor (2 + | |)−m which assures that this k = 0 term
is < 21 for a sufficiently large m which we now fix. In the remaining terms
in (18) (for k > 0) we can now increase 1= k and Mk such that the sum is
< 1 =2. Thus (1) holds for j = 1 and it will remain valid for j = 0. We now
fix this choice of 1 and M1 .
Now the inductive process is clear. We assume 0 ; 1 ; : : : ; j−1 and M0 ; M1 ;
: : : ; Mj−1 having been fixed by this shrinking of the i and enlarging of the
Mi .
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We wish to prove (1) for this j by increasing 1= k , Mk for k ≥ j. Now
we have xn = |x| ≥ j and (19) is replaced by
|eihx,ζi | = e−hx,Im ζi ≤ (2 + | |)−jm
(20)
and since | | ≤ Nj , (18) is replaced by
|(D α f )(x)|
(21)
≤
j−1
X
k
k=0
+
X
k≥j
k
Z
ZR
(1 + [| |2 + m2 (log(2 + | |))2 ]1/2 )Nj −Mk (2 + | |)(k−j)m (1 + m) d
n
(1 + [| |2 + m2 (log(2 + | |))2 ]1/2 )Nj −Mk (2 + | |)(k−j)m (1 + m) d :
Rn
In the first sum the Mk have been fixed but the factor (2 + | |) (k−j)m decays
exponentially. Thus we can fix m such that the first sum is < 2j .
In the latter sum the 1= k and the Mk can be increased so that the total
sum is < 2j . This implies the validity of (1) for this particular j and it
remains valid for 0; 1; : : : j − 1. Now we fix j and Mj .
This completes the induction. With this construction of { }, {M } we
have proved that W ({ }; {M }) ⊂ V ({ }; {N }). This proves Theorem 3.
Differential Operators with Constant Coefficients
e given in TheThe description of the topology of D in terms of the range D
orem 3 has important consequences for solvability of differential equations
on Rn with constant coefficients.
Theorem 4. Let D 6= 0 be a differential operator on R n with constant
coefficients. Then the mapping f → Df is a homeomorphism of D onto
DD.
Proof: This proof was shown to me by Hörmander in 1972. A related proof
appears in Ehrenpries, loc. cit.
It is clear from Theorem 2 that the mapping f → Df is injective on D.
The continuity is also obvious.
For the continuity of the inverse we need the following simple lemma.
Lemma 5. Let P 6= 0 be a polynomial of degree m, F an entire function on
Cn and G = P F . Then
|F ( )| ≤ C sup |G(z + )|;
|z|≤1
where C is a constant.
6
∈ Cn ;
Pm
k
Proof: Suppose
first n = 1 and that P (z) =
0 ak z (am 6= 0). Let
P
m
−k
m
Q(z) = z
. Then, by the maximum principle,
0 ak z
(22)
|am F (0)| = |Q(0)F (0)| ≤ max |Q(z)F (z)| = max |P (z)F (z)| :
|z|=1
|z|=1
For general n let A be an n × n complex matrix, mapping the ball | | < 1
in Cn into itself and such that
m−1
X
m
Pk ( 2 ; : : : ; n ) 1k ; a 6= 0 :
P (A ) = a 1 +
0
Let
F1 ( ) = F (A );
G1 ( ) = G(A );
P1 ( ) = P (A ) :
Then
G1 (
1
+ z;
2; : : : ; n)
= F1 (
1
+ z;
2 ; : : : ; n )P1 ( 1
+ z;
2; : : : ; n)
and the polynomial
z → P1 (
1
+ z; : : : ;
n)
has leading coefficient a. Thus by (22)
|aF1 ( )| ≤ max |G1 (
|z|=1
1
+ z;
2 ; : : : ; n )|
≤ maxn |G1 ( + z)| :
z∈C
|z|≤1
Hence by the choice of A
|aF ( )| ≤ sup |G( + z)|
z∈Cn
|z|≤1
proving the lemma.
For Theorem 4 it remains to prove that if V is a convex neighborhood
of 0 in D then there exists a convex neighborhood W of 0 in D such that
(23)
f ∈ D; Df ∈ W ⇒ f ∈ V :
We take V as the neighborhood W ({ }; {M }). We shall show that if W =
W ({ }; {M }) (same {M }) then (26) holds provided the j in { } are small
enough. We write u = Df so u
e( ) = P ( ) fe( ) where P is a polynomial. By
Lemma 5
(24)
|fe( )| ≤ C sup |e
u( + z)| :
|z|≤1
But |z| ≤ 1 implies
(1 + |z + |)−Mj ≤ 2Mj (1 + | |)−Mj ;
so if
C2Mj ej
j
≤
j
| Im (z + )| ≤ | Im | + 1 ;
then (23) holds.
Q.e.d.
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Corollary 6. Let D 6= 0 be a differential operator on R n with constant
(complex) coefficients. Then
(25)
D D0 = D0 :
In particular, there exists a distribution T on R n such that
(26)
DT = :
Definition A distribution T satisfying (26) is called a fundamental solution
for D.
To verify (25) let L ∈ D 0 and consider the functional D ∗ u → L(u) on
(∗ denoting adjoint). Because of Theorem 2 this functional is well
defined and by Theorem 4 it is continuous. By the Hahn-Banach theorem
it extends to a distribution S ∈ D 0 . Thus S(D ∗ u) = Lu so DS = L, as
claimed.
D∗ D
Corollary 7. Given f ∈ D there exists a smooth function u on R n such
that
Du = f :
In fact, if T is a fundamental solution one can put u = f ∗ T .
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