# Structure Theorem for Functionals in the Space S

## Full text

(1)

International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 756834,9pages

doi:10.1155/2008/756834

## Structure Theorem for Functionals inthe SpaceS

ω12

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 Sω12 of all C functions ϕ such that sup|α|≤me1αϕ and sup|α|≤me2αϕ are finite for allk ∈ N0,α ∈ Nn0, whereω1andω2are two weights satisfy- ing the classical Beurling conditions. Moreover, we give a topological characterization of the space Sω12without conditions on the derivatives. For functionals in the dual spaceSω12, 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 spaceSw, as defined in2, consists ofCfunctions such that the functions and their Fourier transform jointly with all their derivatives decay ultrarapidly at infinity.

In this paper, we introduce the spaceSw1,w2ofCfunctions 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 dualSw1,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 Sw1,w2 and

(2)

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,C0 ,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 denotedbyα, 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 xα11, . . . , xαnn. Withαβ, we mean that αjβj for everyj. The Fourier transform of a functiong will be denoted by Fgor g and it will be defined as

Rne−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,

3

RΩt/1 t2dt <∞,

4 Ωt≥a bln1 tfor somea∈Rand someb >0.

Standard classes of functionswinMcare given by wx xd for 0< d <1, wx pln

1 |x|

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

CN

Rne−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.

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

(3)

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|

|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 ultradistributions.

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 ekwϕandpk,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 are finite for somek, m∈N0.

(4)

Definition 2.5. Givenw1, w2 ∈ Mc,the spaceSw1,w2is the space of allCfunctionsϕ:Rn→C for which the seminorms

pk,mϕ sup

|β|≤m

ekw1βϕ, πk,mϕ sup

|β|≤m

ekw2βϕ 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

S

pk,mk,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 S

pk,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 ofCfunctions 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 fromSw1,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

Rnϕdμ. 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, Q0 such that

fxP,

fx≤Q, 2.14 for allx∈R. Then

fx≤

2P Q 2.15

for allx∈R.

(5)

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

B

pk,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 thatekw1xβϕandekw2ξβϕ 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 ekw1βϕ. Assume thatβ β1 1, β2, . . . , βnwithβ1 β2 · · · βnm,m0,1,2, . . . .We also indicateβ β1, β2, . . . , βn,βϕ∂x1βϕ,fxx1 βϕx1, xforx x2, . . . , xnfixed,

βϕx fxx1. Moreover, ifh/0, we have

fx x1 h

fx x1

fx x1

h 1 2fx

y

h2, 3.3

whereyis a number betweenx1andx1 h. Thus, fxx1≤ |fxx1 h| |fxx1|

|h|

|h|

2 fxy. 3.4

We can write

ekw1x1 h,xfxx1 hekw1x1 h,xβϕx1, xqk,mϕ,

ekw1xfxx≤qk,mϕ. 3.5 If we takehwith the same sign asx1, we have

w1x≤w1

x1 h, x

. 3.6

That is,

fxx1 h fxx1Cmpk,mϕe−kw1x. 3.7

(6)

To estimatefxy x1βϕy, we write x1βϕy

x1βϕy

Rn

2πiξ12πiξβϕξ

Cβ,m

Rn1 |ξ|m 2e−rw2ξerw2ξ|ϕξ|dξ,

3.8

wherer >m n 2/bis an integer andbis the constant in condition 4 ofDefinition 2.1:

x1βϕyCmπr,0ϕ. 3.9 Thus, we have

x1βϕyCmπr,0ϕ, 3.10 that is,

βϕxCm

1

tpk,mϕe−kw1x r,0ϕ

3.11

for allt >0. As a function oft, the right side of3.11has a global minimum at

t

pk,mϕe−kw1x1/2

πr,0ϕ−1/2

. 3.12

Thus, we obtain the inequality βϕxCm

pk,mϕ1/2

πr,0ϕ1/2

e−k/2w1x, 3.13 that is,

ekw1xβϕxCm

p2k,mϕ1/2

πr,0ϕ1/2

. 3.14

An argument, similar to the one leading to3.14, produces ekw2ξβϕξCm

π2k,mϕ1/2

pr,0ϕ1/2

. 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

ekwxβϕxCm

p2m 1k,0ϕ2−m−1

πr,0ϕ1−2−m−1

ekwξβϕξCm ,

π2m 1k,0◦ Fϕ2−m−1

pr,0ϕ1−2−m−1

. 3.16

This completes the proof ofTheorem 3.1.

(7)

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/βSβα. 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

Lekw1μ1 F ekw2μ2

, 4.2

in the sense ofSw1,w2.

Proof. i⇒ii.GivenL∈Sw1,w2, according to4.1there existkandCso that 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

l1f, g Lϕ, 4.5

(8)

where f ekw1ϕ,g ekw2ϕ for a unique ϕ ∈ Sw1,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

Rnfdμ1

Rngdμ2 4.6

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

Rnekw1ϕdμ1

Rnekw2ϕdμ 2 4.7

for allϕ∈Sw1,w2. In the sense ofSw1,w2,

Lekw1μ1 F ekw2μ2

. 4.8

ii⇒i. Ifμ1andμ2are two regular complex Borel measures satisfyingiiandϕ ∈ Sw1,w2, then

Rnekw1ϕdμ1

Rnekw2ϕdμ 2. 4.9 This implies that

|Lϕ| ≤

Rnekw1ϕdμ1

Rnekw2ϕdμ 2

μ1Rnekw1ϕ2|

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 L

1 |x|k

μ1 F

1 |ξ|k

μ2

, 4.11

which gives a representation for the tempered distributions.

As consequence ofLemma 3.4, we can view the functionals inSbaas 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 F

ek|ξ|1/βμ2

4.12

which characterizes the dual spaceSβα.

(9)

References

1 A. Beurling, Quasi-Analyticity and General Distributions, Lectures 4 and 5, Multigraphed Lecture Notes, American Mathematical Society Summer Institute, Stanford, Calif, USA, 1961.

2 G. Bj ¨orck, “Linear partial diﬀerential operators and generalized distributions,” Arkiv f¨or Matematik, vol. 6, pp. 351–407, 1966.

3 L. H ¨ormander, Linear Partial Diﬀerential Operators, vol. 116 of Die Grundlehren der mathematischen Wis- senschaften, Springer, Berlin, Germany, 1963.

4 M. Andersson and B. Berndtsson, “Almost holomorphic extensions of ultradiﬀerentiable functions,”

Journal d’Analyse Math´ematique, vol. 89, pp. 337–365, 2003.

5 J. Alvarez and H. M. Obiedat, “Characterizations of the Schwartz spaceSand the Beurling-Bj ¨orck spaceSW,” Cubo, vol. 6, no. 4, pp. 167–183, 2004.

6 S.-Y. Chung, D. Kim, and S. Lee, “Characterization for Beurling-Bj ¨orck space and Schwartz space,”

Proceedings of the American Mathematical Society, vol. 125, no. 11, pp. 3229–3234, 1997.

7 W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, New York, NY, USA, 2nd edition, 1991.

8 J. J. Duistermaat, Fourier Integral Operators, Courant Institute of Mathematical Sciences, New York, NY, USA, 1973.

9 J. Hadamard, “Sur le module maximum d’une function et de ses d´eriv´es,” Comptes Rendus de l’Acad´emie des Sciences, vol. 41, pp. 68–72, 1914.

10 E. Landau, “Einige ungleichungen f ¨ur zweimal diﬀentiierbare Funktionen,” Proceedings of the London Mathematical Society, vol. 13, no. 1, pp. 43–49, 1914.

Updating...

## References

Related subjects :