mn header will be provided by the publisher
A support theorem for quasianalytic functionals
Tobias Heinrich∗ and Reinhold Meise∗∗
Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Key words ultradifferentiable functions, quasianalytic functionals, support
MSC (2000) 46F05,46F015
Dedicated to Klaus Bierstedt at the occasion of his 60th birthday and to José Bonet at the occasion of his 50th
birthday
For a weight function ω and an open set G in RN denote by E(ω) (G) (resp. E{ω} (G)) the ω-ultradifferentiable
functions of Beurling (resp. Roumieu) type on G. Using ideas of Hörmander it is shown that the functionals u
0
0
in E(ω)
(G) and E{ω}
(G) can be embedded into the realanalytic functionals on RN and that there is a smallest
supporting set for u in the corresponding class which coincides with the realanalytic (hyperfunction) support of
u. Moreover, if ω is quasianalytic and if a compact subset K of G is the union of the compact sets K1 and K2
0
then each u ∈ E{ω}
(G) which is supported by K can be decomposed as u = u1 + u2 , where uj is supported
by Kj for j = 1, 2.
Copyright line will be provided by the publisher
1
Introduction
Classes of ultradifferentiable functions have been used and investigated intensively since the twenties of the
last century. According to the theorem of Denjoy-Carleman they come in two groups: the quasianalytic and
the nonquasianalytic classes. There are various ways to define these classes (see e.g. Beurling [1], Komatsu
[9], Hörmander [8], Braun, Meise, and Taylor [5], section 8). Independently of the definition there are several
methods known to construct functions with prescribed properties in nonquasianalytic classes (see e.g. Bonet,
Braun, Meise, and Taylor [2], Franken [6], Hörmander [8], Petzsche [13]), while no such method seems to be
known for the quasianalytic case, except for realanalytic functions, for which complex methods can be applied.
More is known for quasianalytic functionals. Hörmander [7] showed that they behave similar to realanalytic
functionals in various respects. In particular he proved that for the elements in the class C L (RN )0 there is a
smallest supporting compact set which does not depend on the class L and coincides with the support of u,
considered as a realanalytic functional. In an early version of Bonet, Meise, and Melikhov [4], this result was used
to show that for certain weight functions ω (see Definition 2.1) and each convex open set G in RN the weighted
0
0
(LF)-space FE{ω}
(G) of Fourier-Laplace transforms of the functionals in E{ω}
(G) coincides algebraically with
its projective hull. It was clear that this result would extend to all weight functions if one could prove the analogue
0
of Hörmander’s theorem for the elements of E{ω}
(G).
In the present paper we show that suitable modifications of arguments in Hörmander [7] can be used to prove
the following: For each weight function ω satisfying ω(t) = o(t) as t tends to infinity and each open set G0 in
0
RN , the space E(ω)
(G0 ) embeds into A0 (RN ), the space of realanalytic functionals on RN , since the polynomials
0
are dense in E(ω) (G0 ). Moreover, for each u ∈ E(ω)
(RN ) the realanalytic support suppA0 (u) of u is a supporting
set for u in the class E(ω) and hence the smallest one with this property. More precisely, for each open set
0
G ⊃ suppA0 (u) the functional u admits a unique extension to an element of E(ω)
(G). Using a reduction argument
from Braun, Meise, and Taylor [5], the same assertion for the Roumieu case and all weight functions ω can be
deduced from the Beurling case, since the polynomials are also dense in E{ω} (G0 ). This result is used to obtain
one of the main theorems in Bonet, Meise, and Melikhov [4], Theorem 4.6. In order to be able to apply the
reduction argument, we need to know that for convex open sets G in RN the Fourier-Laplace transform provides
∗
∗∗
email: [email protected]
Corresponding author: email: [email protected]
Copyright line will be provided by the publisher
2
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
0
0
a linear topological isomorphism between E(ω)
(G) (resp. E{ω}
(G)) and an appropriate weighted space of entire
functions (see Theorem 3.6 and Theorem 3.7). In the Roumieu case this was proved by Rösner [14]. The proof
presented here is more elementary and easier.
As in Hörmander [7] a modification of the arguments that are used in the proof of the support theorem 4.11
can be modified to show the following decomposition result (see Theorem 5.10). Whenever ω is a quasianalytic
weight function and K is a compact subset of RN with K = K1 ∪ K2 , where K1 , K2 are compact, then each
0
u ∈ E{ω}
(RN ) for which K is {ω}-supporting can be decomposed as u = u1 + u2 , where Kj is {ω}-supporting
for uj , j = 1, 2. This result implies that for each quasianalytic weight function ω, each open set G in RN ,
0
and each u ∈ E{ω}
(G) there exist a weight function σ satisfying σ(t) = o(ω(t)) as t tends to infinity and
0
U ∈ E(σ) (G) such that u = U E (G) . In the nonquasianalytic case the corresponding result was proved in
{ω}
Braun, Meise, and Taylor [5], Proposition 7.6.
2
Preliminaries
In this preliminary section we fix the notation and provide a number of basic results that will be used in the
subsequent sections. Throughout this article | · | denotes the euclidian norm on RN or CN and B(a, r) denotes
the open ball of radius r and center a.
Definition 2.1 (a) Let ω : [0, ∞[ → [0, ∞[ be a continuous increasing function which satisfies ω|[0,1] ≡ 0.
It is called a weight function if it has the following properties:
(α) There exists K ≥ 1 such that ω(2t) ≤ K(1 + ω(t)) for t ≥ 0.
(β) log(1 + t) = o(ω(t)) as t tends to infinity.
(γ) ω(t) = O(t) as t tends to infinity.
(δ) ϕ : [0, ∞[ → [0, ∞[, ϕ(x) := ω(ex ) is convex.
(b) A weight function ω is called quasianalytic if
Z ∞
0
ω(t)
dt = ∞.
1 + t2
If this integral is finite, then ω is called nonquasianalytic.
Remark. It is no restriction to assume that ω|[0,1] ≡ 0, because for each function σ which satisfies (α) − (δ),
there exist a weight function ωσ and a > 1 such that ωσ and σ coincide on [a, ∞[.
Definition 2.2 Let ϕ : [0, ∞[ → [0, ∞[ be convex and increasing. Assume further that ϕ(0) = 0 and that
limx→∞ x/ϕ(x) = 0. Then the Young conjugate ϕ∗ of ϕ is defined as
ϕ∗ : [0, ∞[ → [0, ∞[, ϕ∗ (y) := sup(xy − ϕ(x)).
x≥0
Remark 2.3 For each function ϕ satisfying the conditions in 2.2, the following assertions are easy to verify:
(a) ϕ∗ is convex, increasing and satisfies ϕ∗ (0) = 0, limy→∞ y/ϕ∗ (y) = 0, and (ϕ∗ )∗ = ϕ.
(b) The function y → ϕ∗ (y)/y is increasing on ]0, ∞[.
(c) ϕ∗ (λx) ≤ λϕ∗ (x) for 0 ≤ λ ≤ 1 and x ≥ 0.
Definition 2.4 Let G be an open subset of RN and ω a weight function. Then we define
E(ω) (G) := {f ∈ C ∞ (G) : for each K ⊂ G compact and each m ∈ N
kf kω,m,K := sup sup |f (α) (x)| exp(−mϕ∗ (
x∈K
α∈NN
0
|α|
)) < ∞}
m
E{ω} (G) := {f ∈ C ∞ (G) : for each K ⊂ G compact there is m ∈ N
1
|f |ω,m,K := sup sup |f (α) (x)| exp(− ϕ∗ (m|α|)) < ∞}.
m
x∈K α∈NN
0
Copyright line will be provided by the publisher
mn header will be provided by the publisher
3
The elements of E(ω) (G) (resp. E{ω} (G)) are called ω-ultradifferentiable functions of Beurling type (resp.
Roumieu type). E(ω) (G) will be endowed with the locally convex topology which is induced by the seminorms
{k·kω,m,K : m ∈ N, K ⊂ G compact}. Then E(ω) (G) is a nuclear Fréchet space. The topology of E{ω} (G) is defined as proj←Q indn→ E{ω} (Q, n), where for each relatively compact open subset Q of G, the space E{ω} (Q, n)
is defined as
1
E{ω} (Q, n) := {f ∈ C ∞ (Q) : sup sup |f (α) (x)| exp(− ϕ∗ (n|α|)) < ∞}.
n
x∈Q α∈N0
Obviously, it is a Banach space.
In the sequel we will use the notation E∗ (G) if a statement holds whenever “∗” is replaced everywhere by (ω)
or by {ω}. Also, we will write k · km,K (resp. | · |m,K ) if it is clear to which weight function ω we refer.
By E∗0 (G) we denote the strong dual of E∗ (G). Its elements are called quasianalytic functionals of ∗-type
on G if the corresponding weight function is quasianalytic. Otherwise they are called ∗-ultradistributions with
compact support in G.
Lemma 2.5 Let ω be a weight function, let ϕ be defined by 2.1 (δ), and let K denote the constant in 2.1 (α).
Then the following assertions hold:
(a) ϕ(x + j) ≤ 2jK 2j (1 + ϕ(x)), x ≥ 0, j ∈ N.
(b) For each k, l, m ∈ N there exists µ ∈ N and C > 0 such that
ky +
1
1 ∗
ϕ (my + l) ≤ C + ϕ∗ (µy), y ≥ 0.
m
µ
(c) For each k, l, m ∈ N there exists µ ∈ N and C > 0 such that
ky + µϕ∗ (
y+l
y
) ≤ C + mϕ∗ ( ), y ≥ 0.
µ
m
P r o o f. (a) From 2.1 (α) and the definition of ϕ we get
ϕ(x + 1) ≤ K 2 (2 + ϕ(x)) ≤ 2K 2 (1 + ϕ(x))
and hence by induction
ϕ(x + j + 1) ≤ K 2 (2 + ϕ(x + j)) ≤ K 2 (2 + 2jK 2j (1 + ϕ(x))) ≤ 2(j + 1)K 2(j+1) (1 + ϕ(x)).
(b) If k, l, and m are given, we use (a) to find K 0 such that ϕ(x + k) ≤ K 0 (1 + ϕ(x)) for x ≥ 0. Then we
choose µ ∈ N so that µ ≥ 2K 0 m and get for x ≥ k:
1
1
K0
1
1
ϕ(x) = ϕ(x − k + k) ≤
(1 + ϕ(x − k)) ≤
+
ϕ(x − k).
µ
µ
µ
2m 2m
For y ≥ 0 we choose x0 = x0 (y) ≥ k satisfying x0 y −
the preceding estimate imply
ky +
1 ∗
ϕ (2my)
2m
1 ∗
ϕ (my + l)
m
− k) = supx≥k (xy −
1
2m ϕ(x
− k)). This and
1
1
ϕ(x)) = sup(xy −
ϕ(x − k))
2m
2m
x≥0
x≥k
1
1
1
= x0 y −
ϕ(x0 − k) ≤ x0 y +
− ϕ(x0 )
2m
2m µ
1
1
1
1
≤
+ sup(xy − ϕ(x)) =
+ ϕ∗ (µy).
2m x≥0
µ
2m µ
=
sup((x + k)y −
By the convexity of ϕ∗ we obtain for C :=
ky +
1
2m ϕ(x0
1
2m (1
+ ϕ∗ (2l))
1 ∗ 1
1
ϕ ( (2my) + (2l))
m
2
2
1 ∗
1 ∗
1
≤ ky +
ϕ (2my) +
ϕ (2l) ≤ C + ϕ∗ (µy).
2m
2m
µ
= ky +
(c) One starts with a sufficiently large µ ≥ m and applies the same arguments as in the proof of (b).
Copyright line will be provided by the publisher
4
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Example 2.6 (a) The function ω1 : [0, ∞[ → [0, ∞[, ω1 (t) := max(t − 1, 0), is a weight function. Since
ϕ1 (x) := ω1 (ex ) = ex − 1, x ≥ 0, we have
y
ϕ∗1 (y) = 0, 0 ≤ y ≤ 1, and ϕ∗1 (y) = y log + 1, y > 1.
e
From this it follows that for each open set G in RN the space E{ω1 } (G) coincides with the space A(G) of all
realanalytic functions on G, while E(ω1 ) (G) can be identified in a natural way with the space H(CN ) of all entire
functions on CN .
(b) For 0 < α < 1 the function ωα : [0, ∞[→ [0, ∞[, ωα (t) := max(tα − 1, 0), is a nonquasianalytic weight
function. Note that E(ωα ) (G) (resp. E{ωα } (G)) consists of all C ∞ -functions on the open set G in RN which are
in the Gevrey class of exponent 1/α of Beurling (resp. Roumieu) type.
1
t
(c) For 0 < β ≤ 1 the function σβ : [0, ∞[→ [0, ∞[, σβ (t) := max (log(e+t))
β − (log(1+e))β , 0 , is a
quasianalytic weight function.
For further examples of quasianalytic weight functions we refer to [4], Example 4.7.
Remark 2.7 (a) If ω is any weight function then the condition 2.1 (γ) implies the existence of L > 0 such
that
ω(t) ≤ L(1 + ω1 (t)), t ≥ 0,
where ω1 is defined in 2.6. Hence the definition of the Young conjugate gives that
1
(2.1)
ϕ∗1 (y) ≤ 1 + ϕ∗ (Ly), y ≥ 0.
L
From this estimate and Example 2.6 it follows easily that for each open set G in RN the space A(G) is continuously embedded into E{ω} (G) and that the restriction map %G : H(CN ) → E(ω) (G) is well-defined and
continuous.
(b) Let σ and ω be two weight functions. If σ(t) = O(ω(t)) then it follows as in part (a) that for each open set
G in RN
E(ω) (G) ⊂ E(σ) (G) and E{ω} (G) ⊂ E{σ} (G)
and that the corresponding inclusion maps are continuous. If σ(t) = o(ω(t)), then
E{ω} (G) ⊂ E(σ) (G)
with continuous inclusion map.
(c) If ω = o(t), then for each l ∈ N there is a constant Cl > 0 such that
y
y log y ≤ y + lϕ∗ ( ) + Cl , y ≥ 0.
l
N
∞
Remark 2.8 By S(R ) we denote the space of all C -functions on RN for which all derivatives are rapidly
decreasing (see e.g. Hörmander [8], 7.1.2). The Fourier transform
Z
F : S(RN ) → S(RN ), F(f ) : ξ 7→
f (x)e−ihx,ξi dx,
RN
is a linear topological isomorphism (see [8], 7.1.5).
For ψ ∈ C0∞ (RN ) we define
ψ(D) : S(RN ) → S(RN ),
ψ(D)f := F −1 (ψ(Ff )).
Since ψ(Ff ) has compact support, ψ(D)f extends to an entire function on CN . If we define Ψ ∈ H(CN ) by
N Z
1
ψ(ξ)eihξ,zi dξ
Ψ(z) :=
2π
RN
then we have for each z ∈ CN :
Z
Z Z
1
1
ihz,ξi
ψ(D)f (z) =
ψ(ξ)Ff (ξ)e
dξ =
ψ(ξ)f (x)e−ihx,ξi eihz,ξi dx dξ
(2π)N
(2π)N
Z
Z
Z
1
ihz−x,ξi
=
f
(x)
ψ(ξ)e
dξ
dx
=
f (x)Ψ(z − x) dx = f ∗ Ψ(z).
(2π)N
Copyright line will be provided by the publisher
mn header will be provided by the publisher
3
5
A density result and the Fourier-Laplace transform
In this section we show that H(CN ) is dense in E∗ (G) for each open set G in RN . This result is then used to
show that for each convex open set G in RN the Fourier-Laplace transform establishes an isomorphism between
E∗0 (G) and the appropriate weighted (LF)-space A∗ (CN , G) of entire functions on CN .
Notation 3.1 For j ∈ N denote by Ej the function
Ej : CN → C, Ej (z) := (j/π)N/2 exp(−jz 2 ).
Proposition 3.2 Let ω be a weight function and let G be an open set in RN . Then the following assertions
hold:
(a) H(CN ) is dense in E{ω} (G).
(b) H(CN ) is dense in E(ω) (G), provided that ω(t) = o(t) as t tends to infinity.
P r o o f. For f, χ ∈ C ∞ (G) assume that χ has compact support in G and that 0 ≤ χ ≤ 1. Then define
fj := Ej ∗ χf for j ∈ N and note that fj ∈ H(CN ). A standard induction argument shows that for ∂k :=
∂
∂xk , 1 ≤ k ≤ N , the following identity holds for n ∈ N0 and j ∈ N:
(3.1)
∂in . . . ∂i1 (Ej ∗ χf ) = Ej ∗ χ(∂in . . . ∂i1 f ) +
n
X
(∂in . . . ∂iν+1 Ej ) ∗ (∂iν χ)(∂iν−1 . . . ∂i1 f ),
ν=1
and hence
(3.2)
|∂in . . . ∂i1 (f − fj )| ≤
≤
|∂in . . . ∂i1 f − Ej ∗ χ(∂in . . . ∂i1 f )| +
n
X
|(∂in . . . ∂iν+1 Ej ) ∗ (∂iν χ)(∂iν−1 . . . ∂i1 f )|.
ν=1
To estimate the right hand side of (3.2) we assume that for a given compact set K ⊂ G and 0 < c < 1 the
set Q := K + B(0, c) is contained in G and that χ(ξ) = 1 for each ξ ∈ Q. Denote by Q1 ⊂ G the support
of χ. Then the generalized mean value theorem implies that for x ∈ K, y ∈ B(0, c), and α ∈ NN
0 we have the
following estimate
√
(3.3)
|f (α) (x − y) − f (α) (x)| ≤ N |y| sup sup |f (α+ej ) (x − ty)|.
t∈[0,1] 1≤j≤N
Now note that for each j ∈ N we have
Z
Z
(3.4)
Ej (y) dy = 1,
RN
RN
C0
|y|Ej (y) dy ≤ √ , and
j
Z
C0
Ej (y)dy ≤ √
j
|y|>c
for a constant C0 not depending on j. From this and (3.3) we get
Z
|Ej ∗ χf (α) (x) − f (α) (x)| = Ej (y)(χ(x − y)f (α) (x − y) − f (α) (x)) dy RN
Z
≤
Ej (y)|f (α) (x − y) − f (α) (x)| dy
|y|≤c
Z
(3.5)
+
Ej (y)(χ(x − y)|f (α) (x − y)| + |f (α) (x)|) dy
|y|>c
Z
Z
√
≤ N sup sup |f (α+ej ) (ξ)|
|y|Ej (y) dy + 2 sup |f (α) (ξ)|
Ej (y) dy.
ξ∈Q 1≤j≤N
|y|≤c
ξ∈Q1
|y|>c
Copyright line will be provided by the publisher
6
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Hence we estimated the first summand on the right of formula (3.2) on K. To obtain an estimate for the second
summand, note that for x ∈ K each term in the sum can be estimated as follows
|(∂in . . . ∂iν+1 Ej ) ∗ (∂iν χ)(∂iν−1 . . . ∂i1 f )(x)|
Z
≤ |∂in . . . ∂iν+1 Ej (y)| |∂iν χ(x − y)| |∂iν−1 . . . ∂i1 f (x − y)| dy
(3.6)
≤ sup |∂iν−1 . . . ∂i1 f (ξ)| sup sup |∂j χ(y)| λN (supp χ) sup |∂in . . . ∂iν+1 Ej (y)|
y∈G 1≤j≤N
ξ∈Q1
|y|>c
≤ M sup |∂iν−1 . . . ∂i1 f (ξ)| sup |∂in . . . ∂iν+1 Ej (y)|,
|y|>c
ξ∈Q1
where M := supy∈G sup1≤j≤N |∂j χ(y)|λN (supp χ) and λN denotes the Lebesgue measure on RN . Now note
that for each γ ∈ NN
0 and r > 0 the Cauchy inequalities imply
γ!
sup{|Ej (z)| : z ∈ D(y, r)},
r|γ|
√
QN
where D(y, r) = j=1 B(yj , r). For r := c/(4 N ), y ∈ RN , |y| ≥ c, and z = u + iv ∈ D(y, r) we get:
(γ)
|Ej (y)| ≤
(u − y)2 + v 2 = |z − y|2 =
N
X
|zj − yj |2 ≤ N r2 =
j=1
c2
.
16
This implies |u| = |y − (y − u)| ≥ |y| − |u − y| ≥ c − c/4 = 3c/4 and v 2 − u2 ≤ −c2 /2 and hence
(γ)
|Ej (y)|
(3.7)
≤
|γ|
r
N2
|γ| N2
2
j
j
−jc2 /2
|γ| log r1 +|γ| log |γ|
e
= e
e−jc /2 .
π
π
Now we distinguish the Roumieu and the Beurling case.
Roumieu case: Let f be an arbitrary element of E{ω} (G) and let U be any zero neighborhood in E{ω} (G). By
the definition of the topology of E{ω} (G) there exists a relatively compact open subset Q0 of G and a continuous
seminorm p on indn→∞ E{ω} (Q0 , n) such that {g ∈ E{ω} (G) : p(g) < 1} ⊂ U . We let K := Q0 and choose
m1 ∈ N such that |f |m1 ,Q1 is finite. From the convexity of ϕ∗ it follows for α ∈ NN
0
1 ∗
1 ∗
ϕ (m1 (|α| + 1)) =
ϕ
m1
m1
1
1 ∗
1
1 ∗
(2m1 |α|) + (2m1 ) ≤
ϕ (2m1 |α|) +
ϕ (2m1 ).
2
2
2m1
2m1
Hence we obtain from (3.5), (3.4), and Q ⊂ Q1 that for x ∈ K
!
Z
∗
1
C
0
ϕ
(m
|α|)
1
√ + 2e m1
|Ej ∗ χf (α) (x) − f (α) (x)| ≤ N |f |m1 ,Q1 e
Ej (y) dy
j
|y|>c
(3.8)
1 ∗
p
√
∗
1
≤ N e 2m1 ϕ (2m1 |α|) C0 |f |m1 ,Q1 e 2m1 ϕ (2m1 ) + 2 / j.
√
1
m1
ϕ∗ (m1 (|α|+1))
According to Lemma 2.5 (b) and Remark 2.7 (a) we can find m4 ≥ m3 ≥ m2 ≥ m1 in N and constants
C2 < C3 < C4 such that for each γ ∈ NN
0 we have
|γ| log
1
1
1 ∗
1 ∗
+ |γ| log |γ| ≤ C2 + 1 + log
|γ| +
ϕ (m2 |γ|) ≤ C3 +
ϕ (m3 |γ|)
r
r
m2
m3
as well as
C3 + |γ| +
1 ∗
1 ∗
ϕ (m3 |γ|) ≤ C4 +
ϕ (m4 |γ|).
m3
m4
Copyright line will be provided by the publisher
mn header will be provided by the publisher
7
By the convexity of ϕ∗ it follows from (3.6) and (3.7) with |γ| = n − ν that for x ∈ K we have
n
X
|(∂in . . . ∂iν+1 Ej ) ∗ (∂iν χ)(∂iν−1 . . . ∂i1 f )(x)| ≤
ν=1
≤ |f |m1 ,Q1 M
(3.9)
n
X
C3 + m1 ϕ∗ (m3 (ν−1))+ m1 ϕ∗ (m3 (n−ν))
e
3
3
ν=1
! N
j 2 −jc2 /2
e
π
N2
2
j
e−jc /2
π
N2
∗
1
2
j
≤ |f |m1 ,Q1 M eC4 + m4 ϕ (m4 n)
e−jc /2 .
π
1
≤ |f |m1 ,Q1 M neC3 + m3 ϕ
∗
(m3 n)
Next note that for l := max(2m1 , m4 ) there exists a constant M1 , not depending on j, such that from (3.2), (3.8),
and (3.9) we get
!
N/2
1
j
−jc2 /2
(3.10)
|f − fj |l,K ≤ M1 √ +
e
|f |m1 ,Q1 .
π
j
Since p|E{ω} (Q0 ,l) is continuous and since K = Q0 , this implies the existence of j0 ∈ N such that f − fj0 ∈ U .
By our choices, we proved (a).
Beurling case: Let f ∈ E(ω) (G) and k · km,K be a given continuous seminorm on E(ω) (G). The convexity of ϕ∗
implies
|α|
1
|α| + 1
∗
∗
∗
≤ mϕ
+ mϕ
, α ∈ NN
2mϕ
0 .
2m
m
m
Hence we obtain from (3.5), (3.4), and Q ⊂ Q1 that for x ∈ K
(3.11)
|Ej ∗χf (α) (x) − f (α) (x)| ≤
Z
√
|α|+1
|α|
mϕ∗ ( m )
2mϕ∗ ( 2m ) C0
√
≤ N kf k2m,Q e
+ 2kf km,Q1 e
Ej (y) dy
j
|y|>c
√
p
∗ |α|
∗ 1
≤ N emϕ ( m ) C0 kf k2m,Q1 emϕ ( m ) + 2 / j.
Since ω(t) = o(t), according to Lemma 2.5 (c) and Remark 2.7 (c), we can find m6 ≥ m5 ≥ m in N and
constants C5 < C6 < C7 such that for each γ ∈ NN
0 we have
1
1
|γ|
|γ|
∗
∗
|γ| log + |γ| log |γ| ≤ C5 + 1 + log
|γ| + m6 ϕ
≤ C6 + m5 ϕ
r
r
m6
m5
as well as
C6 + |γ| + m5 ϕ
∗
|γ|
m5
≤ C7 + mϕ
∗
|γ|
m
.
By the convexity of ϕ∗ it follows from (3.6) and (3.7) with |γ| = n − ν that for x ∈ K we have the estimate
n
X
|(∂in . . . ∂iν+1 Ej ) ∗ (∂iν χ)(∂iν−1 . . . ∂i1 f )(x)|
ν=1
≤ kf km5 ,Q1 M
(3.12)
n
X
C6 +m5 ϕ∗
“
e
ν−1
m5
”
“
”
+m5 ϕ∗ n−ν
m
5
ν=1
! N
j 2 −jc2 /2
e
π
N2
2
j
≤ kf km5 ,Q1 M ne
e−jc /2
π
N2
2
n
j
C7 +mϕ∗ ( m
)
≤ kf km5 ,Q1 M e
e−jc /2 .
π
C6 +m5 ϕ∗
“
n
m5
”
Copyright line will be provided by the publisher
8
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Because
√ of (3.11), (3.12) and (3.2) we can choose a constant M2 , not depending on j, such that kf − fj km,K ≤
M2 / j. Obviously, this implies (b).
Corollary 3.3 For each weight function ω and each open set G in RN the following sets are dense in E{ω} (G):
C[z1 , · · · , zN ], Exp := {exp(−ih·, zi) : z ∈ CN }.
If ω(t) = o(t) for t tending to infinity, these sets are also dense in E(ω) (G).
P r o o f. The density of C[z1 , · · · , zN ] in E{ω} (G) follows from Proposition 3.2, the density of C[z1 , · · · , zN ]
in H(CN ), and the continuity of the inclusion map H(CN ) ,→ E{ω} (G). Since also Exp is dense in H(CN ), the
same arguments show that it is dense in E{ω} (G). For E(ω) (G) the same arguments as for E{ω} (G) apply.
In order to introduce the Fourier-Laplace transform, we need some preparation.
Definition 3.4 Let ω be a weight function and G an open convex set in RN . For each compact set K in G,
the support functional of K is defined as
hK : RN → R, hK (x) := sup{hx, yi : y ∈ K}.
For K as above and λ > 0 let
A(K, λ) := {f ∈ H(CN ) : kf kK,λ := sup |f (z)| exp(−hK (Im z) − λω(|z|)) < ∞}
z∈C
and define
A(ω) (CN , G) := indK,n→ A(K, n)
A{ω} (CN , G) := indK→ proj←n A(K,
1
).
n
It is easy to check that A(K, λ) is a Banach space, that A(ω) (CN , G) is an (LB)-space, and that A{ω} (CN , G) is
an (LF)-space.
Notation 3.5 (Fourier-Laplace transform) Let ω be a weight function and let G be an open convex set in
RN . For each u ∈ E∗0 (G) it is easy to check that
u
b : CN → C, u
b(z) := ux (e−ihx,zi )
is an entire function which belongs to A∗ (CN , G) and that
F : E∗0 (G) → A∗ (CN , G), F(u) := u
b,
is linear and continuous.
Theorem 3.6 For each weight function ω satisfying ω(t) = o(t) as t tends to infinity and each convex open
set G ⊂ RN the Fourier-Laplace transform
0
F : E(ω)
(G) → A(ω) (CN , G)
is a linear topological isomorphism.
P r o o f. To derive this from a result in Meise and Taylor [10], fix a compact convex subset K in G with K̊ 6= ∅
and let
N
E(ω) (K) := {f = (fα )α∈NN
∈ C(K)N0 : f(0) ∈ C ∞ (K̊), fα |K̊ = (f(0) |K̊ )(α) , for each α ∈ NN
0 and
0
kf km = sup sup |fα (x)| exp(−mϕ∗ (
x∈K α∈NN
0
|α|
)) < ∞ for each m ∈ N}.
m
Copyright line will be provided by the publisher
mn header will be provided by the publisher
9
Then choose an increasing fundamental sequence (Km )m∈N of convex compact sets in G and note that
E(ω) (G) = proj←m E(ω) (Km ).
(3.13)
Next note that by Meise and Taylor [10], Proposition 3.6, we have
0
F(E(ω)
(Km )) = indn→ A(Km , n),
(3.14)
since the Rproof of the surjectivity of the map G defined in this proposition does not use that ω satisfies the
∞
condition 0 ω(t)/(1 + t2 )dt < ∞. Now fix an arbitrary element g of A(ω) (CN , G). Then there exists m ∈ N
0
such that g ∈ indn→∞ A(Km , n). Hence (3.14) implies the existence of v ∈ E(ω)
(Km ) satisfying F(v) = g.
0
From (3.13) it follows, that v induces u ∈ E(ω) (G) satisfying u(f ) = v(f |Km ) for f ∈ E(ω) (G). This implies
0
u
b = F(v) = g and hence the surjectivity of F : E(ω)
(G) → A(ω) (CN , G). By Corollary 3.3, F is also
injective and hence linear, continuous and bijective. Since a standard argument shows that E(ω) (G) is a Fréchet0
Schwartz space, E(ω)
(G) is a (DFS)-space (see Meise and Vogt [11], Proposition 25.20). Thus F is a topological
isomorphism by the open mapping theorem.
Theorem 3.7 For each weight function ω and each convex open set G in RN the Fourier-Laplace transform
0
(G) → A{ω} (CN , G)
F : E{ω}
is a linear topological isomorphism.
P r o o f. By Corollary 3.3, the Fourier-Laplace transform F is injective. To show that it is also surjective, fix
g ∈ A{ω} (CN , G). Then there exists a compact, convex set K ⊂ G so that g ∈ proj←n A(K, n1 ). Hence for
each n ∈ N there is Cn > 0 such that
|g(z)| ≤ Cn exp(hK (Im z) +
1
ω(|z|)).
n
This implies that h : [0, ∞[ → [0, ∞[, defined as
h(t) := sup{log+ (|g(z)| exp(−hK (Im z))) : |z| = t},
for each n ∈ N satisfies the estimate
h(t) ≤
1
ω(t) + log Cn , t ≥ 0,
n
or in other words, h(t) = o(ω(t)) as t tends to infinity. Therefore, we can apply Braun, Meise, and Taylor [5],
Lemma 1.7 and Remark 1.8 (1) to get a weight function σ such that σ = o(ω) and h = o(σ). It is easy to check
that this implies the existence of C > 0 such that
|g(z)| ≤ C exp(hK (Im z) + σ(z)), z ∈ CN .
0
Hence g ∈ A(σ) (CN , G) and by Theorem 3.6 there exists u ∈ E(σ)
(G) satisfying F(u) = g. Since σ satisfies
0
σ = o(ω), the inclusion map j : E{ω} (G) ,→ E(σ) (G) is continuous by Remark 2.7 (b). Thus u ◦ j ∈ E{ω}
(G)
N
and F(u ◦ j) = g. Since g was an arbitrary element of A{ω} (C , G), we proved that F is surjective and hence
bijective.
0
To show that F −1 is continuous, or equivalently that F is open, note first that E{ω}
(G) is ultrabornologic
by Meise and Vogt [11], Proposition 24.23, since it is the strong dual of a complete Schwartz space. Then fix
an increasing fundamental sequence (Qm )m∈N of convex, open, relatively compact subsets of G and note that
indn→ E{ω} (Qm , n) is an (LB)-space for each m ∈ N. Hence (indn→ E{ω} (Qm , n))0b is a Fréchet space. Next let
0
%m : E{ω} (G) → indn→ E{ω} (Qm , n) denote the restriction map and let %0m be its adjoint. Then Em
:= im%0m is
0
0
a linear subspace of E{ω} (G) which carries a natural Fréchet space topology. Moreover, E{ω} (G) = indm→ Em
as vector spaces, where the space on the right hand side carries a natural (LF)-space topology, which is stronger
0
than the topology of E{ω}
(G). Hence the open mapping theorem (see Meise and Vogt [11], 24.30) implies that
0
both topologies coincide. In particular, E{ω}
(G) is an (LF)-space. Since A{ω} (CN , G) is an (LF)-space by its
definition, the open mapping theorem now implies that F is open, which completes the proof.
Copyright line will be provided by the publisher
10
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Remark. Theorem 3.7 for N = 1 was proved by Meyer [12] and for N ∈ N by Rösner [14]. The present proof
is simpler since it uses the density result of Corollary 3.3.
0
(G)
Corollary 3.8 Let ω be a weight function and let G be a convex open set in RN . Then for each u ∈ E{ω}
0
there exist a weight function σ satisfying σ(t) = o(ω(t)) as t tends to infinity and U ∈ E(σ)
(G) such that
u = U |E{ω} (G) .
P r o o f. By Theorem 3.7 there exists g ∈ A{ω} (CN , G) with F(u) = g. In the proof of Theorem 3.7 we
showed the existence of a weight function σ satisfying σ = o(ω) such that g ∈ A(σ) (CN , G). By Theorem 3.6,
there is U ∈ A(σ) (CN , G) satisfying F(U ) = g. Now it follows from Corollary 3.3 and Remark 2.7 (b) that
U |E{ω} (G) = u.
Remark 3.9 For a weight function ω let
W0 (ω) = {σ : σ is a weight function, σ = o(ω)}.
Then for each convex open set G in RN , Corollary 3.8 can be stated as
[
0
0
(G) =
E(σ)
(G).
E{ω}
σ∈W0 (ω)
4
The support theorem
In this section we use a modification of the proof of Hörmander [7], Theorem 3.4, to show that for each u ∈
E∗0 (RN ) one can define the notion of support. To do so we first introduce the concept of supporting sets.
Definition 4.1 Let ω be a weight function. A compact set K in RN is said to be ∗-supporting for u ∈ E∗0 (RN )
if for each open set G ⊃ K there exists U ∈ E∗0 (G) satisfying U |E∗ (RN ) = u.
Lemma 4.2 Let ω be a weight function. Then the following assertions hold:
(a) For each u ∈ E∗0 (RN ) there exists a ∗-supporting compact set.
(b) If a compact set K is ∗-supporting for u ∈ E∗0 (RN ), then each compact set Q ⊃ K has this property, too.
(c) Let σ be a weight function which satisfies σ = O(ω). If K ist (σ)-supporting (resp. {σ}-supporting) for
0
0
u ∈ E(σ)
(RN ) (resp. u ∈ E{σ}
(RN )) then K is also (ω)-supporting (resp. {ω}-supporting) for u.
0
0
(d) Let σ and ω be weight functions satisfying σ = o(ω). Assume that for u ∈ E{ω}
(RN ) there is U ∈ E(σ)
(RN )
with U |E{ω} (RN ) = u. If the compact set K in RN is (σ)-supporting for U then K is {ω}-supporting for u.
0
P r o o f. (a) If u ∈ E(ω)
(RN ) is given, then there exist a compact set K in RN , m ∈ N and C > 0 such that
(4.1)
|u(f )| ≤ Ckf kω,m,K , f ∈ E(ω) (RN ).
Next fix an open set G in RN with G ⊃ K and let
E := {f ∈ E(ω) (G) : there is g ∈ E(ω) (RN ) with g|G = f }.
Then (4.1) implies that u induces a linear form ũ on E which is continuous for the topology induced on E by
0
E{ω} (G). By the theorem of Hahn-Banach, ũ can be extended to U ∈ E(ω)
(G). Obviously U |E(ω) (RN ) = u.
0
N
If u ∈ E{ω} (R ) then we note that by the definition of the topology of E{ω} (RN ) in 2.4, there exists a bounded
open set Q in RN and a continuous semi-norm p on indn→ E{ω} (Q, n) such that
(4.2)
|u(f )| ≤ p(f ), f ∈ E{ω} (RN ).
Now let K := Q and note that for each open set G ⊃ K we get from (4.2) and the theorem of Hahn-Banach the
0
existence of U ∈ E{ω}
(G) satisfying U |E{ω} (RN ) = u as is the previous case.
Copyright line will be provided by the publisher
mn header will be provided by the publisher
11
(b) This is obvious from the definition.
0
(c) We only treat the Beurling case. Let G be an open set in RN with G ⊃ K and assume that U ∈ E(σ)
(G)
satisfies U |E(σ) (RN ) = u. By Remark 2.7 (b), the hypothesis implies that the inclusion map J : E(ω) (G) ,→
0
E(σ) (G) is continuous. Hence V := J 0 (U ) is in E(ω)
(G) and satisfies V |E(ω) (RN ) = u|E(ω) (RN ) since E(ω) (RN ) ⊂
E(σ) (RN ).
0
(d) Let G be an open set in RN with G ⊃ K. By hypothesis there exists V ∈ E(σ)
(G) satisfying V |E(σ) (RN ) =
U and by Remark 2.7 (b) the inclusion map J : E{ω} (G) → E(σ) (G) is continuous. Hence W := J 0 (V ) is in
0
E{ω}
(G) and satisfies W |E{ω} (RN ) = V |E{ω} (RN ) = U |E{ω} (RN ) = u. This proves that K is {ω}-supporting for
u.
Of course one would like to know whether there exists a smallest compact set which is ∗-supporting for a
0
given functional u ∈ E∗0 (RN ). In the case E{ω
(RN ) = A0 (RN ) it is known since long, that this is the case.
1}
From Schapira [15], Theorem 111, or Hörmander [8], Theorem 9.1.6, we recall the following result.
0
Theorem 4.3 For each real analytic functional u ∈ A0 (RN ) = E{ω
(RN ) there is a smallest compact set K
1}
in RN which is {ω1 }-supporting for u.
Definition 4.4 For each u ∈ A0 (RN ) the smallest compact set K in RN which is {ω1 }-supporting for u is
called the support of u and is denoted by suppA0 (u).
Remark 4.5 Let ω be a weight function and let G be an open set in RN . Then the inclusion map i{ω} :
A(RN ) → E{ω} (RN ) and the restriction map %{ω},G : E{ω} (RN ) → E{ω} (G) are continuous by Remark 2.7 (b)
and have dense range by Proposition 3.2. Hence the adjoint map of their composition
0
j{ω} := (%{ω},G ◦ i{ω} )0 : E{ω}
(G) → A0 (RN )
0
is injective. Therefore, we can identify E{ω}
(G) with a linear subspace of A0 (RN ). Using this identification,
0
suppA0 (u) is defined for each u ∈ E{ω} (G).
0
If ω satisfies ω(t) = o(t) then the same arguments imply that we can also consider E(ω)
(G) as a linear subspace
0
N
of A (R ).
We are going to show that under the hypotheses of Remark 4.5 for each u ∈ E∗0 (RN ) the set suppA0 (u) is in
fact ∗-supporting for u. By Lemma 4.2 (c), this implies that suppA0 (u) is the smallest ∗-supporting compact set
for u. To prove this we need some preparation. We begin by recalling the following lemma from Hörmander [8],
section 1.4.
Lemma 4.6 For a compact set K in RN and t > 0 let K(t) := K + B(0, t). Then there exists C > 0 such
that for each t, ν > 0 there exists χ ∈ C0∞ (K(νt)) which has the following properties:
(a) 0 ≤ χ ≤ 1 and χ ≡ 1 on some neighborhood of K.
|α|
(b) |χ(α) (x)| ≤ Ct
for x ∈ RN and α ∈ NN
0 with |α| ≤ ν.
Lemma 4.7 Let ω be a weight function which satisfies ω(t) = o(t) as t tends to infinity and define ϕ∗ by
2.1 (δ) and 2.2. Let G, X, and Y be open subsets of RN , let K ⊂ RN be compact, X bounded, and assume
K ⊂ Y ⊂ Y ⊂ X ⊂ X ⊂ G.
Let (χν )ν∈N0 be a sequence in C0∞ (X) satisfying 0 ≤ χν ≤ 1 and χν ≡ 1 on some neighborhood of Y . Assume
further that there exists C1 > 0 such that for ν ∈ N0
(4.3)
|α|
sup |χ(α)
for |α| ≤ ν.
ν (x)| ≤ (C1 ν)
x∈X
Then for each n ∈ N there exists m0 > n such that for each m ∈ N, m ≥ m0 , there exists C > 0 such that for
each ν ∈ N0 and each f ∈ E(ω) (G) the following estimate holds:
(4.4)
|ξ|ν |F(χν f )(ξ)| ≤ C exp(nϕ∗ (ν/n))kf km,X , ξ ∈ RN .
Copyright line will be provided by the publisher
12
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
√
P r o o f. Given n ∈ N choose k ∈ N so that 2 + log N ≤ k. By Lemma 2.5 (c) there exist m0 ∈ N and
C0 > 0 such that
y
y
) ≤ C0 + nϕ∗ ( ), y ≥ 0.
ky + m0 ϕ∗ (
m0
n
Then fix m ∈ N, m ≥ m0 , and note that by 2.3 (b) y 7→ ϕ∗ (y)/y is increasing on ]0, ∞[. Hence we have
kν + mϕ∗ (ν/m) ≤ kν + m0 ϕ∗ (ν/m0 ) ≤ C0 + nϕ∗ (ν/n), ν ∈ N.
Since ω(t) = o(t) by hypothesis, we can choose C2 > 0 such that
mω(C1 ν) ≤ ν + C2 , ν ∈ N,
where C1 denotes the constant in (4.3). Then we have
√
ν(1 + log N ) + mω(C1 ν) + mϕ∗ (ν/m) ≤ kν + C2 + mϕ∗ (ν/m)
(4.5)
≤ C0 + C2 + nϕ∗ (ν/n), ν ∈ N.
Next fix ν ∈ N and α ∈ NN
0 satisfying |α| = ν. Then the Leibniz formula and (4.3) imply for f ∈ E(ω) (G)
and x ∈ K:
X α (α−β)
|(χν f )(α) (x)| ≤
|χ(β)
(x)|
ν (x)||f
β
β≤α
(4.6)
X α ≤
(C1 ν)|β| exp(mϕ∗ (|α − β|/m))kf km,X .
β
β≤α
From Remark 2.3 (c) we get for 0 ≤ β ≤ α
|β| |α|
|α − β| |α|
·
+ ϕ∗
·
ϕ∗ (|α − β|/m) + ϕ∗ (|β|/m) = ϕ∗
|α|
m
|α| m
|α − β| |β|
α
|α|
≤
+
ϕ∗
.
= ϕ∗
|α|
|α|
m
m
Using this estimate and Remark 2.3 (a), we get from (4.6)


X α ∗
|β|
|(χν f )(α) (x)| ≤ 
exp |β| log(C1 ν) − mϕ∗ ( )  emϕ (|α|/m) kf km,X
β
m
β≤α
!
∗
|β|
|β|
(4.7)
≤ 2ν exp m sup
log(C1 ν) − ϕ∗ ( )
emϕ (ν/m) kf km,X
m
m
β≤ν
≤ 2ν exp(mϕ∗∗ (log C1 ν) + mϕ∗ (ν/m))kf km,X
≤ exp(ν + mω(C1 ν) + mϕ∗ (ν/m))kf km,X .
Note that for ν = 0 the estimate (4.7) holds, too.
Next note that for ν = 2k and ξ ∈ RN we get from (4.7)
ν
|ξ| |F(χν f )(ξ)| =
X
N
ξj2
k
j=1
(4.8)
∂j2
k
(χν f ) (ξ)|
j=1
N
X
k
|
∂j2 (χν f )(x)||e−ihx,ξi | dx
Z
≤
X
≤N
|F(χν f )(ξ)| = |F
X
N
ν/2
j=1
exp(ν + mω(C1 ν) + mϕ∗ (ν/m))kf km,X ,
Copyright line will be provided by the publisher
mn header will be provided by the publisher
13
since we have N k summands under the integral. For ν = 2k + 1 a similar argument leads to the same estimate.
From this estimate, our choices, and (4.5) we now get
√
|ξ|ν |F(χν f )(ξ)| ≤ exp(ν(1 + log N ) + mω(C1 ν) + mϕ∗ (ν/m))kf km,X
≤ C exp(nϕ∗ (ν/n))kf km,X .
Hence (4.4) holds with C := exp(C0 + C2 ).
Lemma 4.8 For a weight function ω define ϕ∗ as in 2.2 and for n ∈ N define
ln : [0, ∞[ → [0, ∞[, ln (t) := sup tν exp(−nϕ∗ (ν/n)).
ν∈N0
Then for each t ∈ [0, ∞[ there exists a smallest number ν(n, t) ∈ N satisfying
ln (t) = tν(n,t) exp(−nϕ∗ (ν(n, t)/n))
and ln is an increasing function.
P r o o f. By Remark 2.3, the function y 7→ ϕ∗ (y)/y is increasing and unbounded on [0, ∞[. Hence there
exists yt > 0 such that ϕ∗ (yt )/yt > log t. For ν ≥ nyt this implies
n
nϕ∗ (ν/n) = ν ϕ∗ (ν/n) ≥ ν log t.
ν
Hence the supremum in the definition of ln is a maximum, which implies the first assertion. The second one
follows from it, since for t1 < t2 we have
ν(n,t1 )
ln (t1 ) = t1
ν(n,t1 )
exp(−nϕ∗ (ν(n, t1 )/n)) ≤ t2
exp(−nϕ∗ (ν(n, t1 )/n))
≤ sup tν2 exp(−nϕ∗ (ν/n)) = ln (t2 ).
ν∈N0
Lemma 4.9 Under the hypotheses of Lemma 4.7 let (χν )ν∈N0 be the sequence from that lemma. Let (ψn )n∈N0 be
a sequence in C0∞ (RN ) satisfying supp ψ0 ⊂ B(0, 2), ψn ≥ 0 for n ∈ N0 ,
supp ψn ⊂ {ξ ∈ RN : 2n−1 ≤ |ξ| ≤ 2n+1 }, n ∈ N,
P∞
and n=0 ψn (ξ) = 1 for each ξ ∈ RN . Then for each µ ∈ N there are k, m ∈ N, m > k > µ, such that
the following holds for ν(n) := ν(k, 2n−1 ), defined in Lemma 4.8. There exists C > 0 such that for each
f ∈ E(ω) (G) the function
R(f ) : RN → C, R(f )(x) :=
(4.9)
∞
X
ψn (D)(χν(n) f )(x)
n=0
is in C ∞ (RN ) and satisfies
(4.10)
sup sup |R(f )(α) (x)| exp(−µϕ∗ (|α|/µ)) ≤ Ckf km,X .
x∈RN α∈NN
0
P r o o f. If µ ∈ N is given, we apply Lemma 2.5 (c) to find k > µ and d > 0 such that
(4.11)
2y + kϕ∗ ((y + N + 1)/k) ≤ d + µϕ∗ (y/µ), y ≥ 0.
Applying Lemma 4.7 with n = k we choose m ∈ N so large that m > k and such that the conclusion of 4.7
holds. Then 4.7 implies the following assertion:
For t > 0 and each ξ ∈ RN satisfying |ξ| ≥ t, each ν ∈ N and each f ∈ E(ω) (G) we have
|F(χν f )(ξ)| ≤ C|ξ|−ν ekϕ
∗
(ν/k)
kf km,X ≤ Ct−ν ekϕ
∗
(ν/k)
kf km,X .
Copyright line will be provided by the publisher
14
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
By Lemma 4.8 and the definition of the function lk in Lemma 4.8, this gives for ν = ν(n):
|F(χν(n) f )(ξ)| ≤ Clk (t)−1 kf km,X , |ξ| ≥ t.
Since t ≤ |ξ| ≤ 4t implies lk (|ξ|/4) ≤ lk (t), we get from this estimate and the definition of the function lk that
for t ≤ |ξ| ≤ 4t and each p ∈ N we have
|F(χν(n) f )(ξ)| ≤ Clk (|ξ|/4)−1 kf km,X
−1
≤ C [(|ξ|/4)p · exp(−kϕ∗ (p/k))]
kf km,X .
Next fix α ∈ NN
0 and apply the previous estimate with p = |α| + N + 1 to get from the properties of (ψn )n∈N
and the choice of k that for ξ ∈ RN we have:
∞
X
|ξ α ||ψn (ξ)||F(χν(n) f )(ξ)|
n=1
≤C
(4.12)
4
|ξ|
|α|+N +1
exp(kϕ∗ ((|α| + N + 1)/k))|ξ|α (1 − ψ0 (ξ))kf km,X
≤ C4N +1 exp(2|α| + kϕ∗ ((|α| + N + 1)/k))
≤ C1 exp(µϕ∗ (|α|/µ))
1 − ψ0 (ξ)
kf km,X
|ξ|N +1
1 − ψ0 (ξ)
kf km,X .
|ξ|N +1
Since ψ0 F(χν(0) f ) has compact support, and since |ξ α ψ0 (ξ)F(χν(0) f )(ξ)| satisfies similar estimates, we proved
that
∞
X
|ξ α ψn (ξ)F(χν(n) f )(ξ)|
n=0
converges in L1 (RN ) for each α ∈ NN
0 . Hence
∞
X
ψn (D)(χν(n) f ) = F
−1
∞
X
!
ψn F(χν(n) f )
n=0
n=0
is in C ∞ (RN ) and satisfies
"
F
−1
∞
X
!#(α)
ψn F(χν(n) f )
n=0
=F
−1
∞
X
!
α
(iξ) ψn (ξ)F(χν(n) f )(ξ) .
n=0
Hence the estimates (4.12) and the similar estimates for ξ α ψ0 (ξ)F(χν(0) f )(ξ) imply the existence of C2 > 0
such that
|R(f )(α) (x)| ≤ C2 exp(µϕ∗ (|α|/µ))kf km,X , x ∈ RN , α ∈ NN
0 .
Thus, we proved formula (4.10) for C = C2 .
From the proof of Hörmander [7], Theorem 3.4, we recall the following lemma.
Lemma 4.10 Let G, X, Y , and K be subsets of RN satisfying
K⊂Y ⊂Y ⊂X⊂X⊂G
and assume that K and X are compact. Denote by C the constant from Lemma 4.6 and let δ :=
Y ). Then there exists a sequence (hn )n∈N0 in C ∞ (RN ) having the following properties:
1√
2eC N
dist(K, RN \
|α|
0 ≤ hn ≤ 1, hn |B(0,2n ) ≡ 1, hn |RN \B(0,2n+1 ) ≡ 0, and sup |h(α)
for |α| ≤ 2n δ.
n (x)| ≤ (C0 δ)
x∈RN
Copyright line will be provided by the publisher
mn header will be provided by the publisher
15
For ψ0 := h0 and ψn := hn − hn−1 , n ∈ N, let
N Z
1
Ψn (z) :=
ψn (ξ)eihz,ξi dλN (ξ).
2π
N
R
√
n
N
Then for C1 := 78 eN ! and each z ∈ C satisfying |z| ≥ eCδ N and |Im z| ≤ δ/13 the following estimate
holds for each n ∈ N0 :
|Ψn (z)| ≤ C1 δ −N e−δ2
(4.13)
n
/3
.
Theorem 4.11 Let ω be a weight function which satisfies ω(t) = o(t) as t tends to infinity. Then for each
0
u ∈ E(ω)
(RN ) the set suppA0 (u) is (ω)-supporting for u.
0
P r o o f. Let u be an arbitrary element of E(ω)
(RN ). Then there are a compact set Q in RN , µ ∈ N and Cu > 0
such that
0
(RN ).
|u(f )| ≤ Cu kf kµ,Q , f ∈ E(ω)
µ
the local Banach space for the seminorm k kµ,Q (see e.g. Meise and Vogt [11], p. 279) and let
Denote by E(ω)
µ 0
µ
N
) satisfying u = ũ ◦ ι.
denote the canonical map. Then there is a unique ũ ∈ (E(ω)
ι : E(ω) (R ) → E(ω)
0
Next let K := suppA0 (u) and fix any open set G ⊃ K. In order to show that there exists U ∈ E(ω)
(G)
N
satisfying U |E(ω) (R ) = u, we choose open sets X and Y satisfying
K⊂Y ⊂Y ⊂X⊂X⊂G
so that X is compact. Then we apply Lemma 4.10 to get a sequence (ψn )n∈N0 which has the properties given
in 4.10. Next we use Lemma 4.6 to choose a sequence (χν )ν∈N0 which satisfies the conditions stated in Lemma
4.7. By Lemma 4.9 there are m, k ∈ N with m > k > µ such that formula (4.9) defines a linear map
R : E(ω) (G) → C ∞ (RN ).
From the estimate (4.10) it follows, that the range of R is in fact contained in the following Banach space
µ
E(ω)
(RN ), defined as
∗
µ
E(ω)
(RN ) := {f ∈ C ∞ (RN ) : kf kµ := sup sup |f (α) (x)|e−µϕ
(|α|/µ)
< ∞}.
x∈RN α∈NN
0
Moreover, (4.10) shows that the map
µ
R : E(ω) (G) → E(ω)
(RN )
is continuous. From the definition of R and the proof of Lemma 4.9 it follows that R(f ) = limp→∞ Rp (f ) for
each f ∈ E(ω) (G), where
Rp (f ) =
p
X
ψn (D)(χν(n) f ) =
n=0
p
X
(χν(n) f ) ∗ Ψn
n=0
µ
is an entire function by Remark 2.8. In particular, Rp (f ) ∈ E(ω)
(RN ). It is easy to check that (ι ◦ Rp (f ))p∈N is
µ
a Cauchy sequence in E(ω) and that there exists a continuous linear map
µ
R∞ : E(ω) (G) → E(ω)
satisfying R∞ (f ) = lim ι ◦ Rp (f ), f ∈ E(ω) (G).
p→∞
0
Hence u1 := ũ ◦ R∞ is in E(ω)
(G).
∞
Next choose χ ∈ C0 (X) satisfying 0 ≤ χ ≤ 1 and χ ≡ 1 in some neighborhood of Y and denote by
(ν(n))n∈N0 the sequence defined in Lemma 4.9. We claim that for each f ∈ E(ω) (G) the series
(4.14)
S(f ) :=
∞
X
((χ − χν(n) ))f ∗ Ψn
n=0
Copyright line will be provided by the publisher
16
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
converges absolutely and uniformly on the set
U := {z ∈ CN : dist(z, K) < dist(K, CY )/2}
(4.15)
and defines a bounded holomorphic function on U . To prove this claim, fix z ∈ U and y ∈ RN \ Y . By the
definition of U there is x ∈ K satisfying |z − x| < dist(K, CY )/2. This implies
|x − y| ≥ |y − x| − |z − x| ≥ dist(K, CY )/2
and |Im(z − y)| = |Im z| < δ/13. From this and (4.13) we get for
Z
|((χ − χν(n) )f ) ∗ Ψn (z)| ≤
|(χ(y) − χν(n) (y))f (y)||Ψn (z − y)|dy
X\Y
Z
(4.16)
n
≤2
|f (y)||Ψn (z − y)|dy ≤ 2C1 λN (X \ Y )δ −N e−δ2 /3 kf km,X .
X\Y
Hence the series in (4.14) converges in H ∞ (U ). From (4.16) we get the existence of a constant C2 > 0 such that
sup |(Sf )(z)| ≤ C2 kf km,X , f ∈ E(ω) (G).
(4.17)
z∈U
Now choose an open subset H of RN which satisfies H ⊃ K and H ⊂ U ∩ RN . Since K = suppA0 (u) there
0
exists V ∈ E{ω
(H) satisfying V |E{ω1 } (RN ) = u|E{ω} (RN ) . It is easy to check that the Cauchy inequalities imply
1}
the existence of q ∈ N and C4 > 0 such that
|g|ω1 ,q,H ≤ C4 sup |g(z)|.
z∈U
From this estimate and (4.17) it follows that S : E(ω) (G) → E{ω1 } (H) is continuous. Consequently, u2 := V ◦ S
0
is in E(ω)
(G). If we define
p
p
X
X
Sp (f ) :=
ψn (D)(χf ) =
(χf ) ∗ Ψn ,
n=0
n=0
then for each f ∈ E{ω1 } (G) we have
u2 (f ) = V ( lim Sp (f )) = lim V (Sp (f )) = lim u(Sp (f )).
p→∞
p→∞
p→∞
Next we claim that
(u1 + u2 )|E(ω) (RN ) = u.
(4.18)
0
Since G is an arbitrary open neighborhood of K and since u1 + u2 ∈ E(ω)
(G), the proof is complete if we show
(4.18). Because of Corollary 3.3, (4.18) holds if we prove that
(u1 + u2 )(f ) = u(f ) for each f ∈ C[z1 , . . . , zN ].
To do this, fix f ∈ C[z1 , . . . , zN ], deg f = q, and note that by the definition of u1 and u2 we have
(u1 + u2 )(f ) = ũ ◦ R1 (f ) + V ◦ S(f ) = lim ũ ◦ ι ◦ Rp (f ) + lim u ◦ Sp (f )
p→∞
(4.19)
= lim {u(
p→∞
= lim u(
p→∞
p
X
ψn (D)(χν(n) f )) + u(
n=0
p
X
n=0
p→∞
p
X
ψn (D)((χ − χν(n) )f ))}
n=0
ψn (D)(χf )) = lim u(fp ),
p→∞
Copyright line will be provided by the publisher
mn header will be provided by the publisher
where fp :=
(4.20)
Pp
n=0
17
ψn (D)(χf ) for p ∈ N. Now we claim that the following assertion holds
There exists an open set U0 in Cn with K ⊂ U0 such that (fp )p∈N converges
to f in H ∞ (U0 ).
Assume for a moment that (4.20) holds. Then choose an open set H0 in RN satisfying H0 ⊃ K and H0 ⊂
0
U0 ∩ RN . Since K = suppA0 (u), there exists W ∈ E{ω
(H0 ) satisfying W |E{ω1 } (RN ) = u|E{ω1 } (RN ) . The
1}
same arguments as above show that (4.20) implies the convergence of (fp )p∈N to f in E{ω1 } (H0 ). Since f and
fp are in H(CN ), this and (4.19) imply
u(f ) = W (f ) = lim W (fp ) = lim u(fp ) = (u1 + u2 )(f ).
p→∞
p→∞
Hence (4.20) implies (4.18).
To prove (4.20) we note first that (fp )p∈N converges to χf in C ∞ (RN ). Since χf = F −1 (F(χf )) this follows
from a standard estimate using the integral formula for the inverse Fourier transform. Since χ|Y ≡ 1, this implies
that fp → f in C ∞ (Y ). Next let U be defined as in (4.15). To show that for α ∈ NN
0 with |α| > q = deg f the
(α)
sequence (fp )p∈N converges to zero in H ∞ (U ), note that
fp(α) =
p
X
(χf )(α) ∗ Ψn =
p
X


n=0
n=0
X α β≤α
β

χ(β) f (α−β)  ∗ Ψn .
(α)
Since |α| > q, each term in this sum for which f (α−β) 6≡ 0 satisfies |β| > 0. Hence fp |Y ≡ 0. By the estimate
for Ψn from Lemma 4.10 this implies
|fp(α) (z)
−
(α)
fp+r (z)|
(β)
≤ sup sup |χ
(y)f
(α−β)
N
(y)|λ (X \ Y )2̇
|α|
C1 δ
−N
y∈X β≤α
e−δ2
n
/3
.
n=p+1
(α)
(α)
p+r
X
Consequently, the sequence (fp )p∈N converges in H ∞ (U ). If gα = limp→∞ fp
in H ∞ (U ), then
gα |U ∩RN = f (α) |U ∩RN = 0
and hence gα ≡ 0. Now fix a point x0 ∈ K and consider the Taylor expansion of fp at x0 of order q with
remainder. Then it follows that the remainder tends to zero on some polydisk with center x0 as p tends to infinity.
Hence (4.20) holds locally at each point in K. From this we get (4.20) by a compactness argument.
0
Theorem 4.12 For each weight function ω and each u ∈ E{ω}
(RN ) the set suppA0 (u) is {ω}-supporting for
u.
0
P r o o f. Let u be an arbitrary element of E{ω}
(RN ). By Corollary 3.8 there exist a weight function σ satisfy0
N
ing σ = o(ω) and U ∈ E(ω) (R ) such that u = U |E{ω} (RN ) . Since ω satisfies condition 2.1 (γ), σ satisfies σ(t) =
o(t) as t tends to infinity. Hence Theorem 4.11 implies that suppA0 (U ) is (σ)-supporting for U . By Lemma
4.2 (d), suppA0 (U ) is {ω}-supporting for u. Since u|A(RN ) = U |A(RN ) , we have suppA0 (U ) = suppA0 (u) and
the proof is complete.
5
Decomposition of quasianalytic functionals of Roumieu type
In this section we show that for each ω-quasianalytic functional u of Roumieu type which is supported by the
union K1 ∪ K2 of two compact sets in RN , we can find ω-quasianalytic functionals u1 , u2 supported by K1 , K2
respectively such that u1 + u2 = u. Since this result holds only for quasianalytic weight functions ω, we assume
that throughout this section ω has this property.
Copyright line will be provided by the publisher
18
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Definition 5.1 For any compact interval I in R we define
E{ω} (I) := {f ∈ C ∞ (I) : there exists µ ∈ N so that |f |ω,µ,I < ∞ }.
Definition 5.2 Let ω be a weight function and n ∈ N. We set Mp := exp( n1 ϕ∗ (np)) for p ∈ N0 and
mp := Mp /Mp−1 for p ∈ N, and we define νn (t) := #{p ∈ N : mp ≤ t} for t ≥ 0.
Remark 5.3 For n ∈ N let Mp , mp and νn be defined as in the previous definition. We set Ln (t) :=
supp∈N0 tp /Mp , t ≥ 0. Then for each t ≥ 0 the supremum in the definition of Ln is attained for p = νn (t), i.e.,
Ln (t) = tνn (t) /Mνn (t) .
P r o o f. The convexity of ϕ∗ implies Mp2 ≤ Mp−1 Mp+1 for p ∈ N. Hence the sequence (mp )p∈N is increasing. If t > 0 and p ∈ N are given, then the estimate p ≤ νn (t) implies mp ≤ mp+1 ≤ . . . ≤ mνn (t) ≤ t and
consequently Mj /Mj−1 ≤ t for p ≤ j ≤ νn (t). From this we get
Mνn (t) ≤ Mνn (t)−1 t ≤ . . . ≤ Mp−1 tνn (t)−p+1
and hence tp /Mp ≤ tνn (t) /Mνn (t) for 0 ≤ p ≤ νn (t). Similar arguments show that for p > νn (t) we have
tp /Mp < tνn (t) /Mνn (t) .
Lemma 5.4 Let ω be a quasianalytic weight function. For each m ∈ N and each δ > 0 there exist
b0 , . . . , bn ∈ R, such that for each f ∈ E{ω} ([0, 1]) the following estimate holds:
|f (1) −
n
X
bj f (j) (0)| ≤ δ|f |ω,m,[0,1] .
j=0
P r o o f. To modify
the proof of Hörmander [8], Lemma 1.3.6, appropriately, we fix m ∈ N and let Mp :=
1 ∗
ϕ (mp) for p ∈ N. As already mentioned above the sequence (mp )p∈N is increasing and satisfies
exp m
limp→∞ mp = ∞. The associated function (see Komatsu [9], p. 48ff) M (t) := supp∈N0 log(tp /Mp ), t > 0,
satisfies
M (t) ≤ m−1 ω(t) ≤ M (t) + log(t),
t ∈ [1, ∞[.
R∞
P∞
Hence it follows from Komatsu [9], Lemma 4.1, that 1 ω(t)/t2 dt = ∞ implies p=1 m1p = ∞. Let aj :=
m−1
j0 +j for some large j0 , such that aj < 1 for all j ∈ N. Using the notation of Hörmander [8], section 1.3, the
identity
Ha ∗ af (j+1) = f (j) − τa f (j) − f (j) (0)χ[0,a[ + f (j) (T )χ]T,T +a]
for 0 < a < T permits to complete the proof as in [8], Lemma 1.3.6.
Applying Lemma 5.4 and the estimate (3.10) from the proof of Proposition 3.2, we can use the arguments
from the proof of Hörmander [7], Lemma 5.3, to derive the following approximation lemma. In it we use the
notation introduced in Lemma 4.6, i.e., for K ⊂ RN compact and t > 0 we let K(t) := K + B(0, t).
Lemma 5.5 Let ω be a quasianalytic weight function. Let K be a compact subset of RN and t > 0. For each
m ∈ N there exists µ ∈ N, µ ≥ m, such that for each δ > 0 the following assertion holds: For any linear form u
on E{ω} (RN ) satisfying
|u(f )| ≤ Cu |f |ω,µ,K(t/2) ,
f ∈ E{ω} (RN ),
for some Cu > 0, there exists v ∈ E 0 (RN ) with supp(v) ⊂ K such that
|(u − v)(f )| ≤ δ|f |ω,m,K(t) ,
f ∈ E{ω} (RN ).
The following statements correspond to those in Lemma 4.9, where the same operator R is defined in the
Beurling case.
Copyright line will be provided by the publisher
mn header will be provided by the publisher
19
Lemma 5.6 Let G be an open, bounded subset of RN , and (χν )ν∈N0 a sequence of non-negative functions in
such that for some constant C > 0
C0∞ (G),
|α|
sup |χ(α)
ν (x)| ≤ (Cν) ,
(5.1)
|α| ≤ ν.
x∈G
For a weight function ω and k ∈ N denote by νk the function defined in 5.2. Then for each m ∈ N one can find
k0 ∈ N, such that for each k ≥ k0 , ν(n) := νk (2n−1 ) for n ∈ N0 , and each sequence (ψn )n∈N0 in C0∞ (RN )
satisfying the conditions in Lemma 4.9, the map
Rm : E{ω} (RN ) → E{ω} (RN ),
Rm f (x) :=
∞
X
ψn (D) χν(n) f (x),
n=0
is well-defined. Moreover, for each m ∈ N and k ≥ k0 there exist m0 ∈ N and C 0 > 0 with
|Rm f |ω,m0 ,RN ≤ C 0 |f |ω,m,G ,
f ∈ E{ω} (RN ).
0
In particular, for each m ∈ N, each k ≥ k0 , and each u ∈ E{ω}
(RN ) there exists Cm > 0 such that
|u ◦ Rm (f )| ≤ Cm |f |ω,m,G ,
f ∈ E{ω} (RN ).
P r o o f. Since the proof is based on the same ideas as the proofs of Lemma 4.7 and 4.9, similar estimates lead
to the same result in the Roumieu case. Given m ∈ N one can use Lemma 2.5 (b) to find some larger n ∈ N, such
that the equations (4.7) and (4.8) hold with exp(nϕ∗ (ν/n)) replaced by exp( n1 ϕ∗ (nν)) and kf km,X by |f |m,X .
For t ≥ 0 we define Ln (t) := supp∈N0 tp /Mp as in Remark 5.3. Then Ln is an increasing function and the
supremum is attained for p = νn (t) (see Remark 5.3). Since Ln has the same properties as ln , we can proceed as
in Lemma 4.9 to define Rm . In the same way we conclude, that Rm satisfies the required estimates.
The following results can be derived from the proof of Theorem 4.11.
Lemma 5.7 Let ω be a weight function. For K, G ⊂ RN assume that K is compact and G is open and
bounded. Let X be an open neighborhood of K in RN . Then there exists a sequence (ψn )n∈N0 in C0∞ (RN ),
which satisfies the conditions in Lemma 4.9, such that for any sequence (gn )n∈N0 of non-negative functions in
C0∞ (G) satisfying gn ≡ 0 on X for n ∈ N0 and supn∈N0 gn ≤ C for some C > 0, the map
S : E{ω} (RN ) → E{ω} (RN ),
Sf (x) :=
∞
X
ψn (D) (gn f ) (x),
n=0
is well-defined. Furthermore, there exist a complex neighborhood U of K and some C0 > 0, such that the series
defining Sf converges normally on U to a holomorphic function and
sup |Sf (z)| ≤ C0 kf kL1 (G) ,
f ∈ E{ω} (RN ).
z∈U
0
Hence for any u ∈ E{ω}
(RN ) with supp(u) ⊂ K there exists C1 > 0 with
|(u ◦ S)(f )| ≤ C1 kf kL1 (G) ,
f ∈ E{ω} (RN ).
Lemma 5.8 Let ω be a weight function. Let K be a compact subset of RN and χ a non-negative function
in C0∞ (RN ) with χ ≡ 1 in a neighborhood of K. Let (ψn )n∈N0 be a sequence in C0∞ (RN ), which satisfies the
conditions
P∞in Lemma 4.9. Then there exists a complex neighborhood U of K, such that for each polynomial f the
series n=0 ψn (D)(χf ) converges normally to f in U .
Now we can perform the same steps in the Roumieu case as in the proof of Theorem 4.11.
Copyright line will be provided by the publisher
20
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
Lemma 5.9 Let ω be a quasianalytic weight function. Let K1 and K2 be compact subsets of RN with K1 ∩
0
K2 6= ∅, and assume for u ∈ E{ω}
(RN ) that supp(u) ⊂ K1 ∪ K2 . For j = 1, 2 let Xj be open, bounded subsets
m
N
of RN containing Kj . Then for each m ∈ N there exist linear forms um
1 , u2 on E{ω} (R ) and Cm > 0, such
m
m
that u = u1 + u2 , and
|um
f ∈ E{ω} (RN ).
j (f )| ≤ Cm |f |ω,m,Xj ,
Furthermore, for m, m0 ∈ N with m < m0 there exists Cm,m0 > 0, such that
0
m
|(um
j − uj )(f )| ≤ Cm,m0 |f |ω,m,X1 ∩X2 ,
f ∈ E{ω} (RN ),
j = 1, 2.
P r o o f. Choose open sets Yj and Zj with Kj ⊂ Yj ⊂ Ȳj ⊂ Zj ⊂ Z̄j ⊂ Xj for j = 1, 2. We apply
Lemma 4.6 to obtain for each j a sequence (χjν )ν∈N0 of non-negative functions in C0∞ (Zj ) satisfying χjν ≡ 1
in Yj , and a constant M1 depending only on dist(Yj , RN \ Zj ) such that
sup |(χjν )(α) (x)| ≤ (M1 ν)|α| ,
(5.2)
|α| ≤ ν.
x∈Xj
Let χ
e1ν := χ1ν and χ
e2ν := (1 − χ1ν )χ2ν . Then 1 − χ
e1ν − χ
e2ν = (1 − χ1ν )(1 − χ2ν ) = 0 on Y1 ∪ Y2 . Choose
χ ∈ C0∞ (Z1 ∪ Z2 ) with χ ≥ 0 and χ ≡ 1 on Y1 ∪ Y2 . It follows from Leibniz’ formula, that χ
ejν satisfies the same
estimate with M1 replaced by 2M1 . Hence Lemma 5.6 and 5.7 show that there exist ψn ∈ C0∞ (RN ), n ∈ N0 ,
such that for each m ∈ N we can define
Rjm : E{ω} (RN ) → E{ω} (RN ),
Rjm f (x) :=
S m : E{ω} (RN ) → E{ω} (RN ),
S m f (x) :=
∞
X
n=0
∞
X
ψn (D) χ
ejν(n) f (x)
ψn (D)
h
(j = 1, 2)
i
χ−χ
e1ν(n) − χ
e2ν(n) f (x),
n=0
where ν(n) is defined in Lemma 5.6. By Lemma 5.6 for each m ∈ N there exists M2 > 0, such that wjm :=
u ◦ Rjm satisfies
|wjm (f )| ≤ M2 |f |ω,m,Xj ,
f ∈ E{ω} (RN ), j = 1, 2.
m
:= u ◦ S m is continuous for the norm k · kL1 (X1 ∪X2 ) , there exists v0m ∈ L∞ (X1 ∪ X2 ) with v m (f ) =
RSince v m
v f dλN for each f ∈ E{ω} (RN ). We define
X1 ∪X2 0
v1m (f )
Z
v0m f
:=
dλ
N
and
v2m (f )
Z
:=
v0m f dλN ,
X2 \X1
X1
0
m
m
m
then vjm ∈ E{ω}
(RN ) with supp(vjm ) ⊂ X̄j . If we set um
j := wj + vj , then uj satisfies the required estimate
and it follows from Lemma 5.8 that for each polynomial f :
!
∞
X
m
m
u1 (f ) + u2 (f ) = u
ψn (D)(χf ) = u(f ).
n=0
m
N
Hence Proposition 3.2 implies um
1 (f ) + u2 (f ) = u(f ) for all f ∈ E{ω} (R ).
m0
m0
m
To show the remaining estimate, fix m, m0 ∈ N with m < m0 . Since um
1 − u1 = u2 − u2 , we may take
∞
j = 1. Choose a sequence (gν )ν∈N0 of non-negative functions in C0 (X2 ) satisfying gν ≡ 1 in Z2 , such that for
some M3 > 0
(5.3)
sup |gν(α) (x)| ≤ (M3 ν)|α| ,
|α| ≤ ν.
x∈X2
Then choose k and k 0 for m and m0 according Lemma 5.6 and note that we can assume k < k 0 . Next define
ν(n) := νk (2n−1 ) resp. ν 0 (n) := νk0 (2n−1 ) for n ∈ N0 . For each p ∈ N satisfying p ≥ k 0 one has kp ≤
Copyright line will be provided by the publisher
mn header will be provided by the publisher
21
k 0 (p − 1). Therefore, it follows from the convexity of ϕ∗ that
ϕ∗ (kp) − ϕ∗ (k(p − 1))
k
=
≤
ϕ∗ (kp) − ϕ∗ (k(p − 1))
kp − k(p − 1)
∗ 0
ϕ (k p) − ϕ∗ (k 0 (p − 1))
ϕ∗ (k 0 p) − ϕ∗ (k 0 (p − 1))
.
=
k 0 p − k 0 (p − 1)
k0
Thus mp ≤ m0p for all p ≥ k 0 , where mp , m0p depend on k, k 0 respectively. This implies for each t ≥ 0
νk (t) = #{p ∈ N : mp ≤ t} ≥ #{p ≥ k 0 : mp ≤ t} ≥ #{p ≥ k 0 : m0p ≤ t} ≥ νk0 (t) − k 0 ,
and hence ν 0 (n) ≤ ν(n) + k 0 for all n ∈ N. From this and the estimates (5.3) and (5.2) we get the existence of
M ≥ max(M3 , 2M1 ) such that for each n ∈ N0
h
i(α)
|α|
e1ν(n) − χ
e1ν 0 (n)
sup gν(n) χ
(x) ≤ (M ν(n)) , |α| ≤ ν(n).
x∈X1 ∪X2
Since in the proof of Lemma 5.6 the hypothesis (5.1) is used only for ν = ν(n), we can apply Lemma 5.6 to
define
!
∞
i
h
X
0
e1ν 0 (n) f
wm,m (f ) := u
ψn (D) gν(n) χ
e1ν(n) − χ
,
f ∈ E{ω} (RN ).
n=0
0
Then there exists M4 > 0 so that |wm,m (f )| ≤ M4 |f |ω,m,X1 ∩X2 for all f ∈ E{ω} (RN ). Let
!
∞
i
h
X
1
1
m,m0
eν(n) − χ
eν 0 (n) f
,
f ∈ E{ω} (RN ).
W
(f ) := u
ψn (D) 1 − gν(n) χ
n=0
Since 1 − gν(n)
χ
e1ν(n) − χ
e1ν 0 (n) has support in (RN \ Z2 ) ∩ (Z1 \ Y1 ) ⊂ Z1 \ (Y1 ∪ Y2 ), it follows from
0
0
0
0
Lemma 5.7, that W m,m is continuous for the norm k · kL1 (Z1 ) , and we have w1m − w1m = wm,m + W m,m .
0
e2ν 0 drop out,
We split v m − v m in the same way. Since 1 − gν vanishes in Z2 , the terms involving χ
e2ν and χ
0
0
0
where we insert a factor 1 − gν . We obtain v m − v m = v m,m − W m,m , where
!
∞
h
i
X
0
v m,m (f ) := u
ψn (D) gν(n) χ
e1ν 0 (n) + χ
e2ν 0 (n) − χ
e1ν(n) − χ
e2ν(n) f
,
f ∈ E{ω} (RN ),
n=0
0
0
0
0
is continuous for the norm k · kL1 (X2 ) . It follows that W m,m + v1m − v1m = v m,m − (v2m − v2m ) is in fact
0
0
m0
m,m0
continuous for the norm k · kL1 (X1 ∩X2 ) . Then um
+ W m,m + v1m − v1m satisfies the required
1 − u1 = w
estimate.
Theorem 5.10 Let ω be a quasianalytic weight function and K1 , K2 be compact subsets of RN . Then for any
0
0
u ∈ E{ω}
(RN ) with supp(u) ⊂ K1 ∪ K2 there exist u1 , u2 ∈ E{ω}
(RN ) with supp(uj ) ⊂ Kj for j = 1, 2, such
that u = u1 + u2 .
P r o o f. We assume K1 ∩ K2 6= ∅, for the statement of the theorem is trivial if K1 ∩ K2 = ∅. We prove first
0
that for fixed t > 0 one can find uj ∈ E{ω}
(RN ) with supp(uj ) ⊂ Kj (t) and u1 + u2 = u. Let K := K1 ∩ K2 .
It’s easy to check that we can find 0 < t0 < t with K1 (t0 ) ∩ K2 (t0 ) ⊂ K(t/2). Choose an increasing sequence
(µν )ν∈N in N, such that Lemma 5.5 holds with m := ν and µ = µν . With Xj := Kj (t0 ) we define uqj for q ∈ N
by Lemma 5.9 and set Ujν := uµj ν . Then for each ν ∈ N there exists Cν > 0, such that for all f ∈ E{ω} (RN )
|Ujν (f )| ≤ Cν |f |ω,µν ,Xj
and
|(Ujν+1 − Ujν )(f )| ≤ Cν |f |ω,µν ,K(t/2) .
0
Hence it follows from Lemma 5.5 that we can find v ν ∈ E{ω}
(RN ) with supp(vν ) ⊂ K so that
|(U1ν+1 − U1ν − v ν )(f )| ≤ 2−ν |f |ω,ν,K(t) ,
f ∈ E{ω} (RN ).
Copyright line will be provided by the publisher
22
T. Heinrich and R. Meise: A support theorem for quasianalytic functionals
This implies, that for f ∈ E{ω} (RN )
u1 (f ) := U11 (f ) +
∞
X
(U1ν+1 − U1ν − v ν )(f ) = U1k (f ) +
ν=1
∞
X
(U1ν+1 − U1ν − v ν )(f ) −
k−1
X
v ν (f )
ν=1
ν=k
0
exists and can be estimated by a constant times |f |ω,k,K1 (t) for every k ∈ N, in particular, u1 ∈ E{ω}
(RN ) and
supp(u1 ) ⊂ K1 (t). Set u2 = u − u1 = U1k + U2k − u1 . Then the properties of the linear forms Ujν imply that
u2 (f ) = U2k (f ) −
∞
X
(U1ν+1 − U1ν − v ν )(f ) +
k−1
X
v ν (f ),
f ∈ E{ω} (RN ).
ν=1
ν=k
0
Hence it follows as above that u2 ∈ E{ω}
(RN ) and supp(u2 ) ⊂ K2 (t).
0
So far we proved that for each t > 0 and j = 1, 2 there exist utj ∈ E{ω}
(RN ) with supp(utj ) ⊂ Kj (t) and
ut1 +ut2 = u. For 0 < t < T this implies ut1 −uT1 = uT2 −ut2 ∈ E{ω} (RN ) with supp(ut1 −uT1 ) ⊂ K1 (T )∩K2 (T ).
Set tn := 2−n . Then for each n, k ∈ N there exists Cn,k > 0 with
t
|(u1n+1 − ut1n )(f )| ≤ Cn,k |f |ω,k,K(tn−1 ) ,
f ∈ E{ω} (RN ).
0
We apply Lemma 5.5 to find wn ∈ E{ω}
(RN ) with supp(wn ) ⊂ K for all n ∈ N0 and
t
|(u1n+1 − ut1n − wn )(f )| ≤ 2−n |f |ω,m,K(2tn−1 ) ,
f ∈ E{ω} (RN ).
Hence we can define for f ∈ E{ω} (RN )
u1 (f ) := ut11 (f ) +
∞
X
t
(u1n+1 − ut1n − wn )(f ) = ut1k (f ) +
n=1
∞
X
t
(u1n+1 − ut1n − wn )(f ) −
n=k
k−1
X
wn (f ).
n=1
0
Then u1 ∈ E{ω}
(RN ) and supp(u1 ) ⊂ K1 (tk ) for all k ∈ N, thus supp(u1 ) ⊂ K1 . Let u2 := u − u1 =
ut1k + ut2k − u1 , then
u2 (f ) = ut2k (f ) −
∞
X
n=k
t
(u1n+1 − ut1n − wn )(f ) +
k−1
X
wn (f ),
f ∈ E{ω} (RN ).
n=1
0
The same argument shows u2 ∈ E{ω}
(RN ) and supp(u2 ) ⊂ K2 . Since u1 + u2 = u, the proof is complete.
Corollary 5.11 Let ω be a quasianalytic weight function and G be an open set in RN . Then for each u ∈
0
there exist a weight function σ satisfying σ(t) = o(ω(t)) as t tends to infinity and U ∈ E(σ)
(G) such
that u = U |E{ω} (G) .
0
E{ω}
(G)
open sets Gj in G with Ḡj ⊂ G such thatP
K ⊂
S P r o o f. Let K := supp(u). Choose finitely many convex
0
Gj . For each j we apply Theorem 5.10 to find uj ∈ E{ω}
(G) with supp(uj ) ⊂ Ḡj ∩ K such that u =
uj .
0
According to Corollary 3.8 there exist weight functions σj satisfying σj (t) = o(ω(t)) and Uj ∈ E(σ
(G)
such
j)
0
that uj = Uj |E{ω} (G) . Since for σ := max σj we still have σ(t) = o(ω(t)) and Uj ∈ E(σ)
(G) for each j, we may
P
take U := Uj .
References
[1] A. Beurling: Quasi-analyticity and general distribution, Lectures 4. and 5. AMS Summer Institute, Standford (1961).
[2] J. Bonet, R.W. Braun, R. Meise, B.A. Taylor: Whitney’s extension theorem for nonquasianalytic classes of ultradifferential functions, Studia Math 99 (1991), 155–184.
[3] J. Bonet, R. Meise: Quasianalytic functionals and projective descriptions, Math. Scand. 94 (2004), 249–266
Copyright line will be provided by the publisher
mn header will be provided by the publisher
23
[4] J. Bonet, R. Meise, S.N. Melikhov: Projective representations of spaces of quasianalytic functionals, Studia Math. 164
(2004), 91–102.
[5] R.W. Braun, R. Meise, B.A. Taylor: Ultradifferentiable functions and Fourier analysis, Result. Math. 17 (1990), 206–
237.
[6] U. Franken: Continuous linear extension of ultradifferentiable functions of Beurling type, Math. Nachr. 164 (1993),
119–139.
[7] L. Hörmander: Between distributions and hyperfunctions, Asterisque 131 (1985), 89–106.
[8] L. Hörmander: The Analysis of Linear Partial Differential Operators I, Springer, Berlin 1983.
[9] H. Komatsu: Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo Sec. IA 20 (1973) ,
25–105.
[10] R. Meise, B. A. Taylor: Whitney’s extension theorem for ultradifferentiable functions of Beurling type, Ark. Mat. 26
(1988), 265–287.
[11] R. Meise: D. Vogt: Introduction to Functional Analysis, Oxford Univ. Press 1997.
[12] T. Meyer: Die Fourier-Laplace Transformation quasianalytischer Funktionale und ihre Anwendung auf Faltungsoperatoren, Diplomarbeit, Düsseldorf 1989.
[13] H. J. Petzsche: On E. Borel’s theorem, Math. Ann. 282 (1988), 299–313.
[14] T. Rösner: Surjektivität partieller Differentialoperatoren auf quasianalytischen Roumieu-Klassen, Dissertation,
Düsseldorf 1997.
[15] P. Schapira: Théorie des Hyperfonctions, Springer LNM 126 (1970).
Copyright line will be provided by the publisher