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}

^{}

_{ω}_{1}

_{,ω}_{2}

**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,hobiedat@hu.edu.jo Received 19 August 2007; Revised 30 September 2007; Accepted 22 November 2007 Recommended by Manfred H. Moller

We introduce the space S*ω*1*,ω*2 of all *C*^{∞} functions *ϕ* such that sup_{|α|≤m}e^{kω}^{1}*∂*^{α}*ϕ*_{∞} and
sup_{|α|≤m}e^{kω}^{2}*∂*^{α}*ϕ* _{∞}are finite for all*k* ∈ N0,*α* ∈ N^{n}_{0}, where*ω*1and*ω*2are two weights satisfy-
ing the classical Beurling conditions. Moreover, we give a topological characterization of the space
S*ω*1*,ω*2without conditions on the derivatives. For functionals in the dual spaceS^{}*ω*_{1}*,ω*_{2}, 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 diﬀer- ential equations.

The Beurling-Bj ¨orck spaceS*w*, as defined in2, consists of*C*^{∞}functions such that the
functions and their Fourier transform jointly with all their derivatives decay ultrarapidly at
infinity.

In this paper, we introduce the spaceS*w*1*,w*2of*C*^{∞}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 spaceS*w*1*,w*2and its dualS^{}_{w}_{1}_{,w}_{2}*.*

The main diﬀerence between the Beurling-Bj ¨orck spaceS*w*and the spaceS*w*1*,w*2 is that
the decay of the functions in S*w* and their Fourier transform are measured by the same
submultiplicative function *e** ^{kw}*,

*k*≥ 0. Whereas the decay of the functions in S

*w*1

*,w*2 and

their Fourier transform are measured by two diﬀerent submultiplicative functions*e*^{kw}^{1} and
*e*^{kw}^{2}*, k*≥0.

This paper is organized in three sections. InSection 2, we give preliminary definitions
and results and introduce the spaceS*w*1*,w*2*.*InSection 3, we give a topological characterization
of the spaceS*w*1*,w*2without conditions on the derivatives. InSection 4, we use the topological
characterization of the spaceS*w*1*,w*2that is given inSection 3to prove a representation theorem
for functionals in the dual spaceS^{}_{w}_{1}_{,w}_{2}of the spaceS*w*1*,w*2*.*

The symbols*C*^{∞},*C*^{∞}_{0} ,*L** ^{p}*, and so forth indicate the usual spaces of functions defined on
R

*, with complex values. We denote by|·|the Euclidean norm onR*

^{n}*, while·*

^{n}_{∞}indicates the norm in the space

*L*

^{∞}. When we do not work on the general Euclidean spaceR

*, we will write*

^{n}*L*

*R, and so forth as appropriate. Partial derivatives will be denotedby*

^{p}*∂*

*, where*

^{α}*α*is a multi- indexα1

*, . . . , α*

*n*. 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|

*α*1 · · · α

*n*,

*x*

^{α}*x*

^{α}_{1}

^{1}

*, . . . , x*

^{α}

_{n}*. With*

^{n}*α*≤

*β, we mean that*

*α*

*≤*

_{j}*β*

*for every*

_{j}*j*. The Fourier transform of a function

*g*will be denoted by Fgor

*g*and it will be defined as

R^{n}*e*^{−2πixξ}*gxdx. The*
inverse Fourier transform is thenF^{−1}g

R^{n}*e*^{2πixξ}*gξdξ. The letterC*will 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 dimension*n*or other fixed parameters.

**2. Preliminary definitions and results**

In this section, we give definitions and results which we will use later.

*Definition 2.1*see2. With M*c*, we denote the space of functions*w* : R* ^{n}* → Rof the form

*wx Ω|x|, where*

1 Ω:0,∞→0,∞is increasing, continuous, and concave, 2 Ω0 0,

3

RΩt/1 *t*^{2}dt <∞,

4 Ωt≥*a* *b*ln1 *t*for some*a*∈Rand some*b >*0.

Standard classes of functions*w*inM*c*are given by
*wx x** ^{d}* for 0

*< d <*1,

*wx p*ln

1 |x|

for*p >*0. 2.1
*Remark 2.2. Let us observe for future use that if we take an integerN >*n/b,then

*C**N*

R^{n}*e*^{−Nwx}*dx <*∞, ∀w∈ M*c*, 2.2
where*b*is 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.

* Lemma 2.3. Conditions 1 and 2 inDefinition 2.1imply thatwis subadditive for allw*∈ M

*c*

*.*

*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

Ω*k*
*k*|x|

≥*kΩ*|x|

*k*

*,* Ω |y|

1−*k*

≥ 1
1−*k*Ω

|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 spaceS*w*of test func-
tions for tempered ultradistributions.

**Theorem 2.4**see5. Given*w*∈ M*c**, the space*S*w**can be described both as a set and as a topology*
*by*

S*w*{ϕ:R* ^{n}*C:

*ϕis continuous and for allk*0,1,2, . . .

*,p*

*ϕ*

_{k,0}*<*∞, p

*◦ Fϕ*

_{k,0}*<*∞}, 2.7

*wherep** _{k,0}*ϕ e

^{kw}*ϕ*

_{∞}

*andp*

*◦ Fϕ e*

_{k,0}

^{kw}*ϕ*

_{∞}

*.*

We observe thatS*w*becomes the Schwartz spaceSwhen

*wx *ln1 |x|. 2.8

For*α, β >*0, the Gelfand-Shilov space*S*^{β}* _{α}*of type

*S*is characterized in6by the space of all

*C*

^{∞}functions

*ϕ*:R

*→Cfor which the seminorms*

^{n}*e*^{k|x|}^{1/α}*ϕ*_{∞}*,* *e*^{m|x|}^{1/β}*ϕ*_{∞} 2.9
are finite for some*k, m*∈N0.

*Definition 2.5. Givenw*1*, w*2 ∈ M*c**,*the spaceS*w*1,*w*2is the space of all*C*^{∞}functions*ϕ*:R* ^{n}*→C
for which the seminorms

*p** _{k,m}*ϕ sup

|β|≤m

*e*^{kw}^{1}*∂*^{β}*ϕ*_{∞}*,* *π** _{k,m}*ϕ sup

|β|≤m

*e*^{kw}^{2}*∂*^{β}*ϕ*_{∞} 2.10

are finite, for*k, m*∈N0and*β*∈N^{n}_{0}.

We can assign toS*w*1,*w*2a structure to Fr´echet space by means of the countable family of
seminorms

*S*

*p**k,m*,π*k,m*

_{∞}

*k,m0*. 2.11

Since *p**k,m*ϕ *<* ∞for all *k* 0,1,2, . . . , ϕ is integrable, so*ϕ* is well defined and the
formulation of the condition*π**k,m*ϕmakes sense for all*k*0,1,2, . . ..

The spaceS*w*1,*w*2, equipped with the family of seminorms
*S*

*p**k,m**, π**k,m*:*k, m*∈N0

, 2.12

is a Fr´echet space.

We observe that the spaceS*w*1,*w*2becomes the Beurling-Bj ¨orck spaceS*w*1, when*w*1*w*2.
When*w*2x ln1 |x|, the space of*C*^{∞}functions with compact supportDis dense subspace
ofS*w*1,*w*2for all*w*_{1}∈ M*c*. The conditions imposed on the function*w*assure that the spaceS*w*1,*w*2

satisfies the properties expected from a space of testing functions. For instance, the operators of
diﬀerentiation and multiplication by*x** ^{α}*are continuous fromS

*w*1,

*w*2into themselves, the space S

*w*1,

*w*2is a topological algebra under pointwise multiplication and convolution. Unfortunately, the Fourier transformation onS

*w*1,

*w*2is not a topological isomorphism fromS

*w*1,

*w*2into itself for some

*w*1,

*w*2∈ M

*c*.For Example, if we take

*w*1x |x|

^{1/2},

*w*2x ln1 |x|, and

*f*∈D\D

*w*1, then

*f*∈S

*w*1,

*w*2but

*f/*∈S

*w1*

*,*

*w2*; see1,2.

**Theorem 2.6**Riesz representation theorem7. Given a functional*Lin the topological dual of the*
*space*C0*, there exists a unique regular complex Borel measureμsuch that*

*Lϕ *

R^{n}*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 on*C0*.*

We conclude this section with Lemma 2.7 8, the version of which is due to Hadamard9, see also10.

**Lemma 2.7**see8,10. Let*f*:R→R*be a continuous function with continuous derivatives of order*

≤*2. Assume that there existP, Q*≥*0 such that*

*fx*≤*P,*

*f*^{}x≤*Q,* 2.14
*for allx*∈R. Then

*f*^{}x≤

2P Q 2.15

*for allx*∈R.

**3. Topological characterization of the space**S*w*1*,w*2

In this section, we present the following characterization of the spaceS*w*1*,w*2, which imposes
no conditions on the derivative.

* Theorem 3.1. Givenw*1

*, w*2 ∈ M

*c*

*, the space*S

*w*1

*,w*2

*can be described as a set and as a topology by*S

*w*1

*,w*2

*ϕ*:R* ^{n}*−→C:

*ϕis continuous and for allk*0,1,2, . . . , p

*k,0*ϕ

*<*∞, π

*k,0*ϕ

*<∞*

*,*
3.1

*wherep** _{k,0}*ϕ e

^{kw}^{1}

*ϕ*

_{∞}

*, π*

*ϕ e*

_{k,0}

^{kw}^{2}

*ϕ*

_{∞}

*.*

*Proof. Let us denote by*B*w*1*,w*2 the space defined in3.1. The conditions*p** _{k,0}*ϕand

*π*

*ϕ imply the smoothness of*

_{k,0}*ϕ*and

*ϕ.*The spaceB

*w*1

*,w*2becomes a Fr´echet space with respect to the family of norms

*B*

*p*_{k,0}*, π** _{k,0}*∞

*k0*. 3.2

From these definitions, it is clear thatS*w*1*,w*2⊆B*w*1*,w*2and that the inclusion is continuous. To
prove the converse, we use the induction on|β|and the general idea of Landau’s inequality.

Fix*ϕ*∈B*w*1*,w*2\ {0}. We want to show thate^{kw}^{1}^{x}*∂*^{β}*ϕ*_{∞}ande^{kw}^{2}^{ξ}*∂*^{β}*ϕ* _{∞}are finite, for every
*k*0,1,2, . . . and every multi-index*β, which is true for allk, whenβ*0. We assume that it is
true for all*k, when*|β| ≤*m, and we want to prove it for allk*and for|β|*m* 1. We start with
e^{kw}^{1}*∂*^{β}*ϕ*_{∞}. Assume that*β* β_{1} 1, β_{2}*, . . . , β** _{n}*with

*β*

_{1}

*β*

_{2}· · ·

*β*

_{n}*m,m*0,1,2, . . . .We also indicate

*β*

^{}β

_{1}

*, β*

_{2}

*, . . . , β*

*,*

_{n}*∂*

^{β}*ϕ∂*

*x*1

*∂*

^{β}^{}

*ϕ,f*

*x*

^{}x1

*∂*

^{β}^{}

*ϕx*1

*, x*

^{}for

*x*

^{}x2

*, . . . , x*

*n*fixed,

*∂*^{β}*ϕx f*_{x}^{}x1. Moreover, if*h/*0, we have

*f*_{x}^{}
*x*_{1} *h*

*f*_{x}^{}
*x*_{1}

*f*_{x}^{}
*x*_{1}

*h* 1
2*f*_{x}^{}

*y*

*h*^{2}, 3.3

where*y*is a number between*x*_{1}and*x*_{1} *h. Thus,*
*f*_{x}^{}x1≤ |f*x*^{}x1 *h|* |f*x*^{}x1|

|h|

|h|

2 *f*_{x}^{}y. 3.4

We can write

*e*^{kw}^{1}^{x}^{1}^{ h,x}^{}^{}*f*_{x}^{}x1 *h*≤*e*^{kw}^{1}^{x}^{1}^{ h,x}^{}^{}*∂*^{β}^{}*ϕx*1*, x*^{}≤*q** _{k,m}*ϕ,

*e*^{kw}^{1}^{x}*f*_{x}^{}x≤*q** _{k,m}*ϕ. 3.5
If we take

*h*with the same sign as

*x*

_{1}, we have

*w*1x≤*w*1

*x*1 *h, x*^{}

. 3.6

That is,

*f**x*^{}x1 *h* *f**x*^{}x1≤*C**m**p**k,m*ϕe^{−kw}^{1}^{x}. 3.7

To estimate*f*_{x}^{}y *∂**x*1*∂*^{β}*ϕy, we write*
*∂*_{x}_{1}*∂*^{β}*ϕy*

*∂*_{x}_{1}*∂*^{β}*ϕy*

≤

R^{n}

2πiξ_{1}2πiξ^{β}*ϕξ* *dξ*

≤*C**β,m*

R* ^{n}*1 |ξ|

^{m 2}*e*

^{−rw}

^{2}

^{ξ}

*e*

^{rw}^{2}

^{ξ}|

*ϕξ|dξ,*

3.8

where*r >*m *n* 2/bis an integer and*b*is the constant in condition 4 ofDefinition 2.1:

∂*x*1*∂*^{β}*ϕy*≤*C**m**π**r,0*ϕ. 3.9
Thus, we have

*∂*_{x}_{1}*∂*^{β}*ϕy*≤*C*_{m}*π** _{r,0}*ϕ, 3.10
that is,

∂^{β}*ϕx*≤*C**m*

1

*tp**k,m*ϕe^{−kw}^{1}^{x} *tπ**r,0*ϕ

3.11

for all*t >*0. As a function of*t, the right side of*3.11has a global minimum at

*t*

*p** _{k,m}*ϕe

^{−kw}

^{1}

^{x}1/2

*π** _{r,0}*ϕ−1/2

. 3.12

Thus, we obtain the inequality
*∂*^{β}*ϕx*≤*C**m*

*p**k,m*ϕ_{1/2}

*π**r,0*ϕ_{1/2}

*e*^{−k/2w}^{1}^{x}, 3.13
that is,

*e*^{kw}^{1}^{x}*∂*^{β}*ϕx*≤*C*_{m}

*p*_{2k,m}ϕ_{1/2}

*π** _{r,0}*ϕ

_{1/2}

. 3.14

An argument, similar to the one leading to3.14, produces
*e*^{kw}^{2}^{ξ}*∂*^{β}*ϕξ* ≤*C*_{m}

*π*_{2k,m}ϕ_{1/2}

*p** _{r,0}*ϕ

_{1/2}

. 3.15

Combining3.14,3.15, the inductive hypothesis implies that*ϕ* ∈ S*w*. The open mapping
theorem can provide once again the continuity of the inclusion. However, solving the recursive
inequalities3.14,3.15, we obtain

*e*^{kwx}*∂*^{β}*ϕx*≤*C*_{m}

*p*_{2}*m 1**k,0*ϕ2^{−m−1}

*π** _{r,0}*ϕ1−2

^{−m−1}

*e*^{kwξ}*∂*^{β}*ϕξ* ≤*C** _{m}* ,

*π*_{2}*m 1**k,0*◦ Fϕ2^{−m−1}

*p** _{r,0}*ϕ1−2

^{−m−1}

. 3.16

This completes the proof ofTheorem 3.1.

When*w*1x *w*2x, the characterization ofS*w*1*,w*2 given byTheorem 3.1reduces to
the characterization of Beurling-Bj ¨orck spaceS*w*1 given byTheorem 2.4. In particular, when
*w*_{1}x *w*_{2}x ln1 |x|, the characterization ofS*w*1*,w*2 reduces to the characterization of
Schwartz spaceS.

*Remark 3.2. The Fourier transform is a topological isomorphism between*S*w*1*,w*2 andS*w*2*,w*1.
As a consequence, the Fourier transform is also a topological isomorphism between the dual
spacesS^{}_{w}_{1}_{,w}_{2}andS^{}_{w}_{2}_{,w}_{1}.

Note that the dual spacesS^{}_{w}_{1}_{,w}_{2}andS^{}_{w}_{2}_{,w}_{1}are assigned to the weak topologies. For dif-
ferent pairs of admissible functions, the spaceS*w*1*,w*2has the following embedding properties.

* Lemma 3.3. For everyw*1

*< w*

_{1}

^{}

*andw*2

*< w*

^{}

_{2}

*, one has*

S*w*^{}_{1}*,w*^{}_{2} →S*w*1*,w*2*.* 3.17
* Lemma 3.4. Forα, β >1, one has*S|x|

^{1/α}

*,|x|*

^{1/β}⊆

*S*

^{β}

_{α}*. As a consequence,*S

^{β}*α*

^{}⊆S

^{}

_{|x|}1/α

*,|x|*

^{1/β}

*.*

**4. A representation theorem for functionals in the space**S^{}_{w}_{1}_{,w}_{2}

FromTheorem 3.1, we can write
S*w*1*,w*2

*ϕ*:R* ^{n}*−→C:

*ϕ*is continuous and for all

*k*0,1,2, . . . ,N

*k*

*,*ϕ

*<*∞

*,* 4.1

whereN*k*ϕ e^{kw}^{1}*ϕ*_{∞} e^{kw}^{2}*ϕ* _{∞}.

* Theorem 4.1. GivenL*:S

*w*1

*,w*2→C, the following statements are equivalent:

i*L*∈S^{}_{w}_{1}_{,w}_{2}*;*

ii*there exist two regular complex Borel measures* *μ*_{1} *and* *μ*_{2} *of finite total variation and* *k* ∈
{0,1,2, . . .}*such that*

*Le*^{kw}^{1}*μ*_{1} F
*e*^{kw}^{2}*μ*_{2}

*,* 4.2

*in the sense of*S^{}_{w}_{1}_{,w}_{2}*.*

*Proof.* i⇒ii.Given*L*∈S^{}_{w}_{1}_{,w}_{2}, according to4.1there exist*k*and*C*so that
*Lϕ*≤*Ce*^{kw}^{1}*ϕ*_{∞} *e*^{kw}^{2}*ϕ*_{∞}

4.3
for all*ϕ*∈S*w*1*,w*2. Moreover, the map

S*w*1*,w*2−→ C0× C0*,*
*ϕ*−→

*e*^{kw}^{1}*ϕ, e*^{kw}^{2}*ϕ* 4.4

is well defined, linear, continuous, and injective. LetRbe the range of this map, on which we define the map

*l*1f, g *Lϕ,* 4.5

where *f* *e*^{kw}^{1}*ϕ,g* *e*^{kw}^{2}*ϕ* for a unique *ϕ* ∈ S*w*1*,w*2. The map*l*1 : R → C is linear and
continuous. By the Hahn-Banach theorem, there exists a functional*L*1in the topological dual
C0× C0^{}ofC0× C0such thatL1l1and the restriction of*L*_{1}toRis*l*_{1}.

Since the spacesC0× C0^{}andC^{}_{0}× C^{}_{0}are isomorphic as Banach spaces, we can write
*L*1f, g *L*1f,0 *L*10, g. UsingTheorem 2.6, there exist regular complex Borel measures*μ*_{1}
and*μ*_{2}of finite total variation such that

*L*1f, g

R^{n}*fdμ*_{1}

R^{n}*gdμ*_{2} 4.6

for allf, g∈ C0× C0. Iff, g∈ R, then we conclude that
*Lϕ *

R^{n}*e*^{kw}^{1}*ϕdμ*_{1}

R^{n}*e*^{kw}^{2}*ϕdμ* _{2} 4.7

for all*ϕ*∈S*w*1*,w*2. In the sense ofS^{}_{w}_{1}_{,w}_{2},

*Le*^{kw}^{1}*μ*_{1} F
*e*^{kw}^{2}*μ*_{2}

. 4.8

ii⇒i. If*μ*_{1}and*μ*_{2}are two regular complex Borel measures satisfyingiiand*ϕ* ∈ S*w*1*,w*2,
then

*Lϕ *

R^{n}*e*^{kw}^{1}*ϕdμ*_{1}

R^{n}*e*^{kw}^{2}*ϕdμ* _{2}. 4.9
This implies that

|Lϕ| ≤

R^{n}*e*^{kw}^{1}*ϕdμ*_{1}

R^{n}*e*^{kw}^{2}*ϕdμ* _{2}

≤*μ*_{1}R^{n}*e*^{kw}^{1}*ϕ*_{∞} |μ_{2}|

R^{n}*e*^{kw}^{2}*ϕ*_{∞}

≤*Ce*^{kw}^{1}*ϕ*_{∞} *e*^{kw}^{2}*ϕ*_{∞}
*.*

4.10

It may be noted that *μ*_{1} and*μ*_{2}*,*employed to obtain the above inequality, are of finite total
variations. This completes the proof ofTheorem 4.1.

*Remark 4.2. Whenw*_{1}x *w*_{2}x 1 |x|* ^{k}*,4.2becomes

*L*

1 |x|*k*

*μ*_{1} F

1 |ξ|*k*

*μ*_{2}

, 4.11

which gives a representation for the tempered distributions.

As consequence ofLemma 3.4, we can view the functionals inS^{b}_{a}^{}as functionals in the
spaceS^{}_{w}_{1}_{,w}_{2}. Then as a result we can characterizeS^{β}*α*^{}usingTheorem 4.1.

* Corollary 4.3. Letα, β >1. Then anyL*∈S

^{β}*α*

^{}

*can be written as*

*Le*

^{k|x|}^{1/α}

*μ*

_{1}F

*e*^{k|ξ|}^{1/β}*μ*_{2}

4.12

*which characterizes the dual space*S^{β}*α*^{}*.*

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