International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 756834,9pages
doi:10.1155/2008/756834
Research Article
Structure Theorem for Functionals in
the Space
S
ω1,ω2
Hamed M. Obiedat, Wasfi A. Shatanawi, and Mohd M. Yasein
Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
Correspondence should be addressed to Hamed M. Obiedat,[email protected]
Received 19 August 2007; Revised 30 September 2007; Accepted 22 November 2007
Recommended by Manfred H. Moller
We introduce the space Sω1,ω2 of all C∞ functions ϕ such that sup|α|≤mekω1∂ αϕ
∞ and
sup|α|≤mekω2∂αϕ
∞are finite for allk ∈ N0,α ∈ Nn0, whereω1andω2are two weights
satisfy-ing the classical Beurlsatisfy-ing conditions. Moreover, we give a topological characterization of the space Sω1,ω2without conditions on the derivatives. For functionals in the dual spaceS
ω1,ω2, we prove a
structure theorem by using the classical Riesz representation thoerem.
Copyrightq2008 Hamed M. Obiedat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The theory of ultradistributions introduced by Beurling1was to find an appropriate con-text for his work on almost holomorphic extensions. Beurling proved that ultradistributions are limits of holomorphic functions in the upper and lower half-planes. Bj ¨orck2studied and expanded the theory of Beurling on ultradistributions to extend the work of H ¨ormander3on existence, nonexistence, and regularity of solutions of constant coefficient linear partial diff er-ential equations.
The Beurling-Bj ¨orck spaceSw, as defined in2, consists ofC∞functions such that the
functions and their Fourier transform jointly with all their derivatives decay ultrarapidly at infinity.
In this paper, we introduce the spaceSw1,w2ofC∞functions such that the functions and
their Fourier transform jointly with all their derivatives decay ultrarapidly at infinity. More-over, we give a characterization of the spaceSw1,w2and its dualS
w1,w2.
The main difference between the Beurling-Bj ¨orck spaceSwand the spaceSw1,w2 is that
the decay of the functions in Sw and their Fourier transform are measured by the same
submultiplicative function ekw, k ≥ 0. Whereas the decay of the functions in S
their Fourier transform are measured by two different submultiplicative functionsekw1 and
ekw2, k≥0.
This paper is organized in three sections. InSection 2, we give preliminary definitions and results and introduce the spaceSw1,w2.InSection 3, we give a topological characterization
of the spaceSw1,w2without conditions on the derivatives. InSection 4, we use the topological
characterization of the spaceSw1,w2that is given inSection 3to prove a representation theorem
for functionals in the dual spaceSw1,w2of the spaceSw1,w2.
The symbolsC∞,C∞0 ,Lp, and so forth indicate the usual spaces of functions defined on
Rn, with complex values. We denote by|·|the Euclidean norm onRn, while·
∞indicates the
norm in the spaceL∞. When we do not work on the general Euclidean spaceRn, we will write
LpR, and so forth as appropriate. Partial derivatives will be denotedby∂α, whereαis a
multi-indexα1, . . . , αn. If it is necessary to indicate on which variables we are taking the derivative,
we will do so by attaching subindexes. We will use the standard abbreviations|α|α1 · · · αn,
xα xα1
1 , . . . , x
αn
n . Withα ≤ β, we mean that αj ≤ βj for everyj. The Fourier transform of
a functiong will be denoted by Fgor g and it will be defined asRne−2πixξgxdx. The inverse Fourier transform is thenF−1g
Rne2πixξgξdξ. The letterCwill indicate a positive constant, that may be different at different occurrences. If it is important to indicate that a constant depends on certain parameters, we will do so by attaching subindexes to the constant. We will not indicate the dependence of constants on the dimensionnor other fixed parameters.
2. Preliminary definitions and results
In this section, we give definitions and results which we will use later.
Definition 2.1see2. With Mc, we denote the space of functionsw : Rn → Rof the form
wx Ω|x|, where
1 Ω:0,∞→0,∞is increasing, continuous, and concave, 2 Ω0 0,
3RΩt/1 t2dt <∞,
4 Ωt≥a bln1 tfor somea∈Rand someb >0.
Standard classes of functionswinMcare given by
wx xd for 0< d <1, wx pln1 |x| forp >0. 2.1
Remark 2.2. Let us observe for future use that if we take an integerN >n/b,then
CN
Rn
e−Nwxdx <∞, ∀w∈ Mc, 2.2
wherebis the constant in condition 4 ofDefinition 2.1.
The following lemma was observed in2without proof. Our proof is an adaptation of 4, Proposition 4.6.
Proof. Let 0< k <1. SinceΩis increasing, we obtain
wx y≤Ω
k k|x|
1−k 1−k|y|
≤ max
Ω
|x| k ,Ω
|y| 1−k .
2.3
SinceΩis concave on0,∞andΩ0 0, we have
Ωk k|x|
≥kΩ|x| k
, Ω
|y|
1−k
≥ 1 1−kΩ
|y|. 2.4
If we take
k Ω|x|
Ω|x| Ω|y|, 2.5
then we have
wx y≤ max
Ω|x| k
,Ω
|y|
1−k
≤wx wy.
2.6
This completes the proof ofLemma 2.3.
We now recall a topological characterization of the Beurling-Bj ¨orck spaceSwof test
func-tions for tempered ultradistribufunc-tions.
Theorem 2.4see5. Givenw∈ Mc, the spaceSwcan be described both as a set and as a topology
by
Sw{ϕ:RnC:ϕis continuous and for allk0,1,2, . . .,pk,0ϕ<∞, pk,0◦ Fϕ<∞}, 2.7
wherepk,0ϕ ekwϕ∞andpk,0◦ Fϕ ekwϕ∞.
We observe thatSwbecomes the Schwartz spaceSwhen
wx ln1 |x|. 2.8
Forα, β >0, the Gelfand-Shilov spaceSβαof typeSis characterized in6by the space of allC∞
functionsϕ:Rn→Cfor which the seminorms
ek|x|1/αϕ∞, em|x|1/βϕ∞ 2.9
Definition 2.5. Givenw1, w2 ∈ Mc,the spaceSw1,w2is the space of allC∞functionsϕ:Rn→C
for which the seminorms
pk,mϕ sup
|β|≤m
ekw1∂βϕ
∞, πk,mϕ sup
|β|≤m
ekw2∂βϕ
∞ 2.10
are finite, fork, m∈N0andβ∈Nn0.
We can assign toSw1,w2a structure to Fr´echet space by means of the countable family of
seminorms
Spk,m,πk,m
∞
k,m0. 2.11
Since pk,mϕ < ∞for all k 0,1,2, . . . , ϕ is integrable, soϕ is well defined and the
formulation of the conditionπk,mϕmakes sense for allk0,1,2, . . ..
The spaceSw1,w2, equipped with the family of seminorms
Spk,m, πk,m:k, m∈N0
, 2.12
is a Fr´echet space.
We observe that the spaceSw1,w2becomes the Beurling-Bj ¨orck spaceSw1, whenw1w2.
Whenw2x ln1 |x|, the space ofC∞functions with compact supportDis dense subspace ofSw1,w2for allw1∈ Mc. The conditions imposed on the functionwassure that the spaceSw1,w2
satisfies the properties expected from a space of testing functions. For instance, the operators of differentiation and multiplication byxαare continuous fromS
w1,w2into themselves, the space
Sw1,w2is a topological algebra under pointwise multiplication and convolution. Unfortunately,
the Fourier transformation onSw1,w2is not a topological isomorphism fromSw1,w2into itself for
somew1,w2∈ Mc.For Example, if we takew1x |x|1/2,w2x ln1 |x|, andf∈D\Dw1,
thenf∈Sw1,w2butf/∈Sw1,w2; see1,2.
Theorem 2.6Riesz representation theorem7. Given a functionalLin the topological dual of the spaceC0, there exists a unique regular complex Borel measureμsuch that
Lϕ
Rn
ϕdμ. 2.13
Moreover, the norm of the functionalLis equal to the total variation|μ|of the measureμ. Conversely, any such measureμdefines a continuous linear functional onC0.
We conclude this section with Lemma 2.7 8, the version of which is due to Hadamard9, see also10.
Lemma 2.7see8,10. Letf:R→Rbe a continuous function with continuous derivatives of order
≤2. Assume that there existP, Q≥0 such that
fx≤P,
fx≤Q, 2.14
for allx∈R. Then
fx≤2P Q 2.15
3. Topological characterization of the spaceSw1,w2
In this section, we present the following characterization of the spaceSw1,w2, which imposes
no conditions on the derivative.
Theorem 3.1. Givenw1, w2 ∈ Mc, the spaceSw1,w2can be described as a set and as a topology by
Sw1,w2
ϕ:Rn−→C:ϕis continuous and for allk0,1,2, . . . , pk,0ϕ<∞, πk,0ϕ<∞, 3.1
wherepk,0ϕ ekw1ϕ∞, πk,0ϕ ekw2ϕ∞.
Proof. Let us denote byBw1,w2 the space defined in3.1. The conditionspk,0ϕandπk,0ϕ
imply the smoothness ofϕandϕ. The spaceBw1,w2becomes a Fr´echet space with respect to the
family of norms
Bpk,0, πk,0
∞
k0. 3.2
From these definitions, it is clear thatSw1,w2⊆Bw1,w2and that the inclusion is continuous. To
prove the converse, we use the induction on|β|and the general idea of Landau’s inequality. Fixϕ∈Bw1,w2\ {0}. We want to show thatekw1x∂
βϕ
∞andekw2ξ∂βϕ∞are finite, for every
k0,1,2, . . . and every multi-indexβ, which is true for allk, whenβ0. We assume that it is true for allk, when|β| ≤m, and we want to prove it for allkand for|β|m 1. We start with ekw1∂βϕ
∞. Assume thatβ β1 1, β2, . . . , βnwithβ1 β2 · · · βnm,m0,1,2, . . . .We
also indicateβ β1, β2, . . . , βn,∂βϕ∂x1∂
βϕ,f
xx1 ∂β
ϕx1, xforx x2, . . . , xnfixed,
∂βϕx fxx1. Moreover, ifh/0, we have
fx
x1 h
fx
x1
fxx1
h 1 2f
x
yh2, 3.3
whereyis a number betweenx1andx1 h. Thus,
fxx1≤ |
fxx1 h| |fxx1| |h|
|h| 2 f
xy. 3.4
We can write
ekw1x1 h,xf
xx1 h≤ekw1x1 h,x
∂βϕx1, x≤qk,mϕ, ekw1xf
xx≤qk,mϕ.
3.5
If we takehwith the same sign asx1, we have
w1x≤w1x1 h, x. 3.6
That is,
To estimatefxy ∂x1∂
βϕy, we write
∂x1∂
βϕy∂
x1
∂βϕy
≤
Rn
2πiξ12πiξβϕξ dξ
≤Cβ,m
Rn
1 |ξ|m 2e−rw2ξerw2ξ|ϕξ |dξ,
3.8
wherer >m n 2/bis an integer andbis the constant in condition 4 ofDefinition 2.1:
∂x1∂
βϕy≤C
mπr,0ϕ. 3.9
Thus, we have
∂x1∂
βϕy≤C
mπr,0ϕ, 3.10
that is,
∂βϕx≤Cm
1
tpk,mϕe
−kw1x tπ
r,0ϕ
3.11
for allt >0. As a function oft, the right side of3.11has a global minimum at
tpk,mϕe−kw1x 1/2
πr,0ϕ
−1/2. 3.12
Thus, we obtain the inequality
∂βϕx≤Cm
pk,mϕ 1/2
πr,0ϕ
1/2
e−k/2w1x, 3.13
that is,
ekw1x∂βϕx≤C
m
p2k,mϕ 1/2π
r,0ϕ
1/2. 3.14
An argument, similar to the one leading to3.14, produces
ekw2ξ∂βϕξ ≤C
m
π2k,mϕ 1/2p
r,0ϕ
1/2. 3.15
Combining3.14,3.15, the inductive hypothesis implies thatϕ ∈ Sw. The open mapping
theorem can provide once again the continuity of the inclusion. However, solving the recursive inequalities3.14,3.15, we obtain
ekwx∂βϕx≤Cm
p2m1k,0ϕ2
−m−1
πr,0ϕ
1−2−m−1
,
ekwξ∂βϕξ ≤Cm
π2m1k,0◦ Fϕ2
−m−1
pr,0ϕ
1−2−m−1
.
3.16
Whenw1x w2x, the characterization ofSw1,w2 given byTheorem 3.1reduces to
the characterization of Beurling-Bj ¨orck spaceSw1 given byTheorem 2.4. In particular, when
w1x w2x ln1 |x|, the characterization ofSw1,w2 reduces to the characterization of
Schwartz spaceS.
Remark 3.2. The Fourier transform is a topological isomorphism betweenSw1,w2 andSw2,w1.
As a consequence, the Fourier transform is also a topological isomorphism between the dual spacesSw1,w2andSw2,w1.
Note that the dual spacesSw1,w2andSw2,w1are assigned to the weak topologies. For dif-ferent pairs of admissible functions, the spaceSw1,w2has the following embedding properties.
Lemma 3.3. For everyw1< w1 andw2< w2, one has
Sw1,w2 →Sw1,w2. 3.17
Lemma 3.4. Forα, β >1, one hasS|x|1/α,|x|1/β ⊆Sβα. As a consequence,Sβα⊆S|
x|1/α,|x|1/β.
4. A representation theorem for functionals in the spaceSw1,w2
FromTheorem 3.1, we can write
Sw1,w2
ϕ:Rn−→C:ϕis continuous and for allk0,1,2, . . . ,Nk,ϕ<∞
, 4.1
whereNkϕ ekw1ϕ∞ ekw2ϕ∞.
Theorem 4.1. GivenL:Sw1,w2→C, the following statements are equivalent:
iL∈Sw1,w2;
iithere exist two regular complex Borel measures μ1 and μ2 of finite total variation and k ∈
{0,1,2, . . .}such that
Lekw1μ
1 F
ekw2μ
2
, 4.2
in the sense ofSw1,w2.
Proof. i⇒ii.GivenL∈Sw 1,w2, according to4.1there existkandCso that
Lϕ≤Cekw1ϕ
∞ ekw2ϕ∞
4.3
for allϕ∈Sw1,w2. Moreover, the map
Sw1,w2−→ C0× C0,
ϕ−→ekw1ϕ, ekw2ϕ
4.4
is well defined, linear, continuous, and injective. LetRbe the range of this map, on which we define the map
where f ekw1ϕ,g ekw2ϕ for a unique ϕ ∈ Sw
1,w2. The mapl1 : R → C is linear and
continuous. By the Hahn-Banach theorem, there exists a functionalL1in the topological dual C0× C0ofC0× C0such thatL1l1and the restriction ofL1toRisl1.
Since the spacesC0× C0andC0× C0are isomorphic as Banach spaces, we can write L1f, g L1f,0 L10, g. UsingTheorem 2.6, there exist regular complex Borel measuresμ1 andμ2of finite total variation such that
L1f, g
Rn fdμ1
Rn
gdμ2 4.6
for allf, g∈ C0× C0. Iff, g∈ R, then we conclude that
Lϕ
Rn
ekw1ϕdμ
1
Rn
ekw2ϕdμ
2 4.7
for allϕ∈Sw1,w2. In the sense ofS
w1,w2,
Lekw1μ
1 F
ekw2μ
2
. 4.8
ii⇒i. Ifμ1andμ2are two regular complex Borel measures satisfyingiiandϕ ∈ Sw1,w2,
then
Lϕ
Rn
ekw1ϕdμ
1
Rn
ekw2ϕdμ
2. 4.9
This implies that
|Lϕ| ≤
Rn
ekw1ϕdμ
1 Rn
ekw2ϕdμ
2
≤μ1Rnekw1ϕ ∞ |μ2|
Rnekw2ϕ ∞
≤Cekw1ϕ
∞ ekw2ϕ∞
.
4.10
It may be noted that μ1 andμ2,employed to obtain the above inequality, are of finite total variations. This completes the proof ofTheorem 4.1.
Remark 4.2. Whenw1x w2x 1 |x|k,4.2becomes
L1 |x|kμ1 F1 |ξ|kμ2, 4.11
which gives a representation for the tempered distributions.
As consequence ofLemma 3.4, we can view the functionals inSb
aas functionals in the
spaceSw1,w2. Then as a result we can characterizeSβαusingTheorem 4.1.
Corollary 4.3. Letα, β >1. Then anyL∈Sβαcan be written as
Lek|x|1/αμ1 Fek|ξ|1/βμ2 4.12
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