Structure Theorem for Functionals in the Space S

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,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 Sω1,ω2 of all C^∞ functions ϕ such that sup_|α|≤me^kω1∂^αϕ_∞ and sup_|α|≤me^kω2∂^αϕ _∞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 Sω1,ω2without conditions on the derivatives. For functionals in the dual spaceSω₁,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 differ- 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 e^kw, k ≥ 0. Whereas the decay of the functions in Sw1,w2 and

their Fourier transform are measured by two different submultiplicative functionse^kw1 and e^kw2, 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 spaceS_w1,w2of the spaceSw1,w2.

The symbolsC^∞,C^∞₀ ,L^p, 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 L^pR, 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, x^α x^α₁¹, . . . , x^α_nⁿ. Withα ≤ β, we mean that α_j ≤ β_j 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

Rⁿe^2πixξgξdξ. 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 x^d 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.

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 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, . . .,p_k,0ϕ<∞, p_k,0◦ Fϕ<∞}, 2.7

wherep_k,0ϕ e^kwϕ_∞andp_k,0◦ Fϕ e^kwϕ _∞.

We observe thatSwbecomes the Schwartz spaceSwhen

wx ln1 |x|. 2.8

Forα, β >0, the Gelfand-Shilov spaceS^β_αof typeSis characterized in6by the space of allC^∞ functionsϕ:Rⁿ→Cfor which the seminorms

e^k|x|1/αϕ_∞, e^m|x|1/βϕ_∞ 2.9 are finite for somek, m∈N0.

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

p_k,mϕ sup

|β|≤m

e^kw1∂^βϕ_∞, π_k,mϕ sup

|β|≤m

e^kw2∂^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

S

pk,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 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 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 byx^αare 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

Lϕ

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, Q≥0 such that

fx≤P,

fx≤Q, 2.14 for allx∈R. Then

fx≤

2P Q 2.15

for allx∈R.

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

wherep_k,0ϕ e^kw1ϕ_∞, π_k,0ϕ e^kw2ϕ _∞.

Proof. Let us denote byBw1,w2 the space defined in3.1. The conditionsp_k,0ϕandπ_k,0ϕ imply the smoothness ofϕandϕ. The spaceBw1,w2becomes 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 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 thate^kw1x∂^βϕ_∞ande^kw2ξ∂^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 e^kw1∂^βϕ_∞. Assume thatβ β₁ 1, β₂, . . . , β_nwithβ₁ β₂ · · · β_nm,m0,1,2, . . . .We also indicateβ β₁, β₂, . . . , β_n,∂^βϕ∂x1∂^βϕ,fxx1 ∂^βϕx1, xforx x2, . . . , xnfixed,

∂^βϕx f_xx1. Moreover, ifh/0, we have

f_x x₁ h

f_x x₁

f_x x₁

h 1 2f_x

y

h², 3.3

whereyis a number betweenx₁andx₁ h. Thus, f_xx1≤ |fxx1 h| |fxx1|

|h|

|h|

2 f_xy. 3.4

We can write

e^{kw1x1 h,x}f_xx1 h≤e^{kw1x1 h,x}∂^βϕx1, x≤q_k,mϕ,

e^kw1xf_xx≤q_k,mϕ. 3.5 If we takehwith the same sign asx₁, we have

w1x≤w1

x1 h, x

. 3.6

That is,

fxx1 h fxx1≤Cmpk,mϕe^−kw1x. 3.7

To estimatef_xy ∂x1∂^βϕy, we write ∂_x1∂^βϕy

∂_x1∂^βϕy

≤

Rⁿ

2πiξ₁2πiξ^βϕξ dξ

≤Cβ,m

Rⁿ1 |ξ|^{m 2}e^−rw2ξe^rw2ξ|ϕξ|dξ,

3.8

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

∂x1∂^βϕy≤Cmπr,0ϕ. 3.9 Thus, we have

∂_x1∂^βϕy≤C_mπ_r,0ϕ, 3.10 that is,

∂^βϕx≤Cm

1

tpk,mϕe^−kw1x tπr,0ϕ

3.11

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

t

p_k,mϕe^−kw1x1/2

π_r,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,

e^kw1x∂^βϕx≤C_m

p_2k,mϕ_1/2

π_r,0ϕ_1/2

. 3.14

An argument, similar to the one leading to3.14, produces e^kw2ξ∂^βϕξ ≤C_m

π_2k,mϕ_1/2

p_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

e^kwx∂^βϕx≤C_m

p₂m 1k,0ϕ2^−m−1

π_r,0ϕ1−2^−m−1

e^kwξ∂^βϕξ ≤C_m ,

π₂m 1k,0◦ Fϕ2^−m−1

p_r,0ϕ1−2^−m−1

. 3.16

This completes the proof ofTheorem 3.1.

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 spacesS_w1,w2andS_w2,w1.

Note that the dual spacesS_w1,w2andS_w2,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 ⊆S^β_α. As a consequence,S^βα⊆S_|x|1/α,|x|^1/β.

4. A representation theorem for functionals in the spaceS_w1,w2

FromTheorem 3.1, we can write Sw1,w2

ϕ:Rⁿ−→C:ϕis continuous and for allk0,1,2, . . . ,Nk,ϕ<∞

, 4.1

whereNkϕ e^kw1ϕ_∞ e^kw2ϕ _∞.

Theorem 4.1. GivenL:Sw1,w2→C, the following statements are equivalent:

iL∈S_w1,w2;

iithere exist two regular complex Borel measures μ₁ and μ₂ of finite total variation and k ∈ {0,1,2, . . .}such that

Le^kw1μ₁ F e^kw2μ₂

, 4.2

in the sense ofS_w1,w2.

Proof. i⇒ii.GivenL∈S_w1,w2, according to4.1there existkandCso that Lϕ≤Ce^kw1ϕ_∞ e^kw2ϕ_∞

4.3 for allϕ∈Sw1,w2. Moreover, the map

Sw1,w2−→ C0× C0, ϕ−→

e^kw1ϕ, e^kw2ϕ 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

where f e^kw1ϕ,g e^kw2ϕ 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 Lϕ

Rⁿe^kw1ϕdμ₁

Rⁿe^kw2ϕdμ ₂ 4.7

for allϕ∈Sw1,w2. In the sense ofS_w1,w2,

Le^kw1μ₁ F e^kw2μ₂

. 4.8

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

Lϕ

Rⁿe^kw1ϕdμ₁

Rⁿe^kw2ϕdμ ₂. 4.9 This implies that

|Lϕ| ≤

Rⁿe^kw1ϕdμ₁

Rⁿe^kw2ϕdμ ₂

≤μ₁Rⁿe^kw1ϕ_∞ |μ₂|

Rⁿe^kw2ϕ_∞

≤Ce^kw1ϕ_∞ e^kw2ϕ_∞ .

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 inS^b_aas functionals in the spaceS_w1,w2. Then as a result we can characterizeS^βαusingTheorem 4.1.

Corollary 4.3. Letα, β >1. Then anyL∈S^βαcan be written as Le^k|x|1/αμ₁ F

e^k|ξ|1/βμ₂

4.12

which characterizes the dual spaceS^βα.

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