Structure Theorem for Functionals in the Space 𝔖^′𝜔_1,𝜔2

Abstract

We introduce the space 𝔖_𝜔1,𝜔2 of all 𝐶^∞ functions 𝜙 such that 𝑠𝑢𝑝_{|𝛼|≤𝑚}‖𝑒^𝑘𝜔1𝜕^𝛼𝜙‖∞ and 𝑠𝑢𝑝_{|𝛼|≤𝑚}‖𝑒^𝑘𝜔2𝜕^𝛼𝜙‖∞ are finite for all 𝑘∈ℕ₀, 𝛼∈ℕ^𝑛₀, where 𝜔₁ and 𝜔₂ are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space 𝔖_𝜔1,𝜔2 without conditions on the derivatives. For functionals in the dual space 𝔖^′_𝜔1,𝜔2, we prove a structure theorem by using the classical Riesz representation thoerem.

1. Introduction

The theory of ultradistributions introduced by Beurling [1] was to find an appropriate context for his work on almost holomorphic extensions. Beurling proved that ultradistributions are limits of holomorphic functions in the upper and lower half-planes. Björck [2] studied and expanded the theory of Beurling on ultradistributions to extend the work of Hörmander [3] on existence, nonexistence, and regularity of solutions of constant coefficient linear partial differential equations.

The Beurling-Björck space 𝔖_𝑤, as defined in [2], 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 space 𝔖_𝑤1,𝑤2 of 𝐶^∞ functions such that the functions and their Fourier transform jointly with all their derivatives decay ultrarapidly at infinity. Moreover, we give a characterization of the space 𝔖_𝑤1,𝑤2 and its dual 𝔖^′_𝑤1,𝑤2.

The main difference between the Beurling-Björck space 𝔖_𝑤 and the space 𝔖_𝑤1,𝑤2 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_𝑤1,𝑤2 and their Fourier transform are measured by two different submultiplicative functions 𝑒^𝑘𝑤1 and 𝑒^𝑘𝑤2,𝑘≥0.

This paper is organized in three sections. In Section 2, we give preliminary definitions and results and introduce the space 𝔖_𝑤1,𝑤2. In Section 3, we give a topological characterization of the space 𝔖_𝑤1,𝑤2 without conditions on the derivatives. In Section 4, we use the topological characterization of the space 𝔖_𝑤1,𝑤2 that is given in Section 3 to prove a representation theorem for functionals in the dual space 𝔖^′_𝑤1,𝑤2 of the space 𝔖_𝑤1,𝑤2.

The symbols 𝐶^∞, 𝐶^∞₀, 𝐿^𝑝, and so forth indicate the usual spaces of functions defined on ℝ^𝑛, with complex values. We denote by |⋅| the Euclidean norm on ℝ^𝑛, while ‖⋅‖_∞ indicates the norm in the space 𝐿^∞ . When we do not work on the general Euclidean space ℝ^𝑛, we will write 𝐿^𝑝(ℝ), and so forth as appropriate. Partial derivatives will be denoted by 𝜕^𝛼, where 𝛼 is a multi-index (𝛼₁,…,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|=𝛼₁+⋯+𝛼_𝑛, 𝑥^𝛼=𝑥^𝛼1₁,…,𝑥^𝛼𝑛_𝑛. With 𝛼≤𝛽, we mean that 𝛼_𝑗≤𝛽_𝑗 for every 𝑗. The Fourier transform of a function 𝑔 will be denoted by ℱ(𝑔) or 𝑔 and it will be defined as ∫_ℝ𝑛𝑒^{−2𝜋𝑖𝑥𝜉}𝑔(𝑥)𝑑𝑥. The inverse Fourier transform is then ℱ⁻¹(𝑔)=∫_ℝ𝑛𝑒^{2𝜋𝑖𝑥𝜉}𝑔(𝜉)𝑑𝜉. The letter 𝐶 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 (see [2]). With ℳ_𝑐, we denote the space of functions 𝑤∶ℝ^𝑛→ℝ of the form 𝑤(𝑥)=Ω(|𝑥|), where (1)Ω∶[0,∞)→[0,∞) is increasing, continuous, and concave,(2)Ω(0)=0,(3)∫_ℝΩ((𝑡)/(1+𝑡²))𝑑𝑡<∞,(4)Ω(𝑡)≥𝑎+𝑏ln(1+𝑡) for some 𝑎∈ℝ and some 𝑏>0.

Standard classes of functions 𝑤 in ℳ_𝑐 are given by 𝑤(𝑥)=|𝑥|^𝑑for0<𝑑<1,𝑤(𝑥)=𝑝ln(1+|𝑥|)for𝑝>0. (2.1)

Remark 2.2. Let us observe for future use that if we take an integer 𝑁>(𝑛/𝑏), then 𝐶_𝑁=_ℝ𝑛𝑒^{−𝑁𝑤(𝑥)}𝑑𝑥<∞,∀𝑤∈ℳ_𝑐, (2.2) where 𝑏 is the constant in condition 4 of Definition 2.1.

The following lemma was observed in [2] without proof. Our proof is an adaptation of [4,Proposition 4.6].

Lemma 2.3. Conditions 1 and 2 in Definition 2.1 imply that 𝑤 is subadditive for all 𝑤∈ℳ_𝑐.

Proof. Let 0<𝑘<1. Since Ω is increasing, we obtain 𝑘𝑤(𝑥+𝑦)≤Ω(𝑘|𝑥|+1−𝑘1−𝑘|𝑦|)≤max{Ω(|𝑥|𝑘),Ω(|𝑦|1−𝑘)}. (2.3) Since Ω is concave on [0,∞) and Ω(0)=0, we have 𝑘Ω(𝑘|𝑥|)≥𝑘Ω(|𝑥|𝑘),Ω(|𝑦|11−𝑘)≥1−𝑘Ω(|𝑦|). (2.4) If we take 𝑘=Ω(|𝑥|)Ω(|𝑥|)+Ω(|𝑦|), (2.5) then we have 𝑤(𝑥+𝑦)≤max{Ω(|𝑥|𝑘),Ω(|𝑦|1−𝑘)}≤𝑤(𝑥)+𝑤(𝑦). (2.6) This completes the proof of Lemma 2.3.

We now recall a topological characterization of the Beurling-Björck space 𝔖_𝑤 of test functions for tempered ultradistributions.

Theorem 2.4 (see [5]). Given 𝑤∈ℳ_𝑐, the space 𝔖_𝑤 can be described both as a set and as a topology by 𝔖_𝑤={𝜑∶ℝ^𝑛ℂ∶𝜑𝑖𝑠𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑛𝑑𝑓𝑜𝑟𝑎𝑙𝑙𝑘=0,1,2,…,𝑝_𝑘,0(𝜑)<∞,𝑝_𝑘,0∘ℱ(𝜑)<∞}, (2.7) where 𝑝_𝑘,0(𝜑)=‖𝑒^𝑘𝑤𝜑‖_∞ and 𝑝_𝑘,0∘ℱ(𝜑)=‖𝑒^𝑘𝑤𝜑‖_∞.

We observe that 𝔖_𝑤 becomes the Schwartz space 𝔖 when 𝑤(𝑥)=ln(1+|𝑥|). (2.8) For 𝛼,𝛽>0, the Gelfand-Shilov space 𝑆^𝛽_𝛼 of type 𝑆 is characterized in [6] by the space of all 𝐶^∞ functions 𝜑∶ℝ^𝑛→ℂ for which the seminorms ‖𝑒^{𝑘|𝑥|1/𝛼}𝜑‖_∞,‖𝑒^{𝑚|𝑥|1/𝛽}𝜑‖_∞ (2.9) are finite for some 𝑘,𝑚∈ℕ₀.

Definition 2.5. Given 𝑤₁,𝑤₂∈ℳ_𝑐, the space 𝔖_𝑤1,𝑤2 is the space of all 𝐶^∞ functions 𝜑∶ℝ^𝑛→ℂ for which the seminorms 𝑝_𝑘,𝑚(𝜑)=sup_{|𝛽|≤𝑚}‖𝑒^𝑘𝑤1𝜕^𝛽𝜑‖_∞,𝜋_𝑘,𝑚(𝜑)=sup_{|𝛽|≤𝑚}‖𝑒^𝑘𝑤2𝜕^𝛽𝜑‖_∞ (2.10) are finite, for 𝑘,𝑚∈ℕ₀ and 𝛽∈ℕ^𝑛₀.

We can assign to 𝔖_𝑤1,𝑤2 a structure to Fréchet space by means of the countable family of seminorms

𝑆={𝑝_𝑘,𝑚,𝜋_𝑘,𝑚}^∞_𝑘,𝑚=0. (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(𝜑) makes sense for all 𝑘=0,1,2,….

The space 𝔖_𝑤1,𝑤2, equipped with the family of seminorms 𝑆={𝑝_𝑘,𝑚,𝜋_𝑘,𝑚∶𝑘,𝑚∈ℕ₀}, (2.12) is a Fréchet space.

We observe that the space 𝔖_𝑤1,𝑤2 becomes the Beurling-Björck space 𝔖_𝑤1, when 𝑤₁=𝑤₂. When 𝑤₂(𝑥)=ln(1+|𝑥|), the space of 𝐶^∞ functions with compact support 𝔇 is dense subspace of 𝔖_𝑤1,𝑤2 for all 𝑤₁∈ℳ_𝑐. The conditions imposed on the function 𝑤 assure that the space 𝔖_𝑤1,𝑤2 satisfies the properties expected from a space of testing functions. For instance, the operators of differentiation and multiplication by 𝑥^𝛼 are continuous from 𝔖_𝑤1,𝑤2 intothemselves, the space 𝔖_𝑤1,𝑤2 is a topological algebra under pointwise multiplication and convolution. Unfortunately, the Fourier transformation on 𝔖_𝑤1,𝑤2 is not a topological isomorphism from 𝔖_𝑤1,𝑤2 into itself for some 𝑤₁, 𝑤₂∈ℳ_𝑐. For Example, if we take 𝑤₁(𝑥)=|𝑥|^1/2, 𝑤₂(𝑥)=ln(1+|𝑥|), and 𝑓∈𝔇⧵𝔇_𝑤1, then 𝑓∈𝔖_𝑤1,𝑤2 but 𝑓∉𝔖_𝑤1,𝑤2; see [1, 2].

Theorem 2.6 (Riesz Representation Theorem [7]). Given a functional 𝐿 in the topological dual of the space 𝒞₀, there exists a unique regular complex Borel measure 𝜇 such that 𝐿(𝜑)=_ℝ𝑛𝜑𝑑𝜇. (2.13) Moreover, the norm of the functional 𝐿 is equal to the total variation |𝜇| of the measure 𝜇. Conversely, any such measure 𝜇 defines a continuous linear functional on 𝒞₀.

We conclude this section with Lemma 2.7 [8], the version of which is due to Hadamard [9], see also [10].

Lemma 2.7 (See [8, 10]). Let 𝑓∶ℝ→ℝ be a continuous function with continuous derivatives of order ≤2. Assume that there exist 𝑃,𝑄≥0 such that |𝑓(𝑥)|≤𝑃,|𝑓^′′(𝑥)|≤𝑄, (2.14) for all 𝑥∈ℝ. Then |𝑓^′(𝑥)|≤2𝑃𝑄 (2.15) for all 𝑥∈ℝ.

3. Topological Characterization of the Space 𝔖_𝑤1,𝑤2

In this section, we present the following characterization of the space 𝔖_𝑤1,𝑤2, which imposes no conditions on the derivative.

Theorem 3.1. Given 𝑤₁,𝑤₂∈ℳ_𝑐, the space 𝔖_𝑤1,𝑤2 can be described as a set and as a topology by 𝔖_𝑤1,𝑤2={𝜑∶ℝ^𝑛→ℂ∶𝜑iscontinuousandforall𝑘=0,1,2,…,𝑝_𝑘,0(𝜑)<∞,𝜋_𝑘,0(𝜑)<∞}, (3.1) where 𝑝_𝑘,0(𝜑)=‖𝑒^𝑘𝑤1𝜑‖_∞,𝜋_𝑘,0(𝜑)=‖𝑒^𝑘𝑤2𝜑‖_∞.

Proof. Let us denote by 𝔅_𝑤1,𝑤2 the space defined in (3.1). The conditions 𝑝_𝑘,0(𝜑) and 𝜋_𝑘,0(𝜑) imply the smoothness of 𝜑 and 𝜑. The space 𝔅_𝑤1,𝑤2 becomes a Fréchet space with respect to the family of norms 𝐵={𝑝_𝑘,0,𝜋_𝑘,0}^∞_𝑘=0. (3.2) From these definitions, it is clear that 𝔖_𝑤1,𝑤2⊆𝔅_𝑤1,𝑤2 and 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_𝑤1,𝑤2⧵{0}. We want to show that ‖𝑒^{𝑘𝑤1(𝑥)}𝜕^𝛽𝜑‖_∞ and ‖𝑒^{𝑘𝑤2(𝜉)}𝜕^𝛽𝜑‖_∞ are finite, for every 𝑘=0,1,2,… and every multi-index 𝛽, which is true for all 𝑘, when 𝛽=0. We assume that it is true for all 𝑘, when |𝛽|≤𝑚, and we want to prove it for all 𝑘 and for |𝛽|=𝑚+1. We start with ‖𝑒^𝑘𝑤1𝜕^𝛽𝜑‖_∞. Assume that 𝛽=(𝛽₁+1,𝛽₂,…,𝛽_𝑛) with 𝛽₁+𝛽₂+⋯+𝛽_𝑛=𝑚, 𝑚=0,1,2,…. We also indicate 𝛽^′=(𝛽₁,𝛽₂,…,𝛽_𝑛), 𝜕^𝛽𝜑=𝜕_𝑥1𝜕^𝛽′𝜑, 𝑓_𝑥′(𝑥₁)=𝜕^𝛽′𝜑(𝑥₁,𝑥^′) for 𝑥^′=(𝑥₂,...,𝑥_𝑛) fixed, 𝜕^𝛽𝜑(𝑥)=𝑓^′_𝑥′(𝑥₁). Moreover, if ℎ≠0, we have 𝑓_𝑥′(𝑥₁+ℎ)=𝑓_𝑥′(𝑥₁)+𝑓^′_𝑥′(𝑥₁1)ℎ+2𝑓_𝑥^′′_′(𝑦)ℎ², (3.3) where 𝑦 is a number between 𝑥₁ and 𝑥₁+ℎ. Thus, |𝑓^′_𝑥′(𝑥₁)|≤|𝑓_𝑥′(𝑥₁+ℎ)|+|𝑓_𝑥′(𝑥₁)|+|ℎ||ℎ|2|𝑓_𝑥^′′_′(𝑦)|. (3.4) We can write |𝑒^{𝑘𝑤1(𝑥1+ℎ,𝑥′)}𝑓_𝑥′(𝑥₁+ℎ)|≤|𝑒^{𝑘𝑤1(𝑥1+ℎ,𝑥′)}𝜕^𝛽′𝜑(𝑥₁,𝑥^′)|≤𝑞_𝑘,𝑚(𝜑),|𝑒^{𝑘𝑤1(𝑥)}𝑓_𝑥′(𝑥)|≤𝑞_𝑘,𝑚(𝜑). (3.5) If we take ℎ with the same sign as 𝑥₁, we have 𝑤₁(𝑥)≤𝑤₁(𝑥₁+ℎ,𝑥^′). (3.6) That is, |𝑓_𝑥′(𝑥₁+ℎ)|+|𝑓_𝑥′(𝑥₁)|≤𝐶_𝑚𝑝_𝑘,𝑚(𝜑)𝑒^{−𝑘𝑤1(𝑥)}. (3.7) To estimate 𝑓_𝑥^′′_′(𝑦)=𝜕_𝑥1𝜕^𝛽𝜑(𝑦), we write |𝜕_𝑥1𝜕^𝛽𝜑(𝑦)|=|𝜕_𝑥1𝜕^𝛽≤𝜑(𝑦)|_ℝ𝑛|2𝜋𝑖𝜉₁(2𝜋𝑖𝜉)^𝛽𝜑(𝜉)|𝑑𝜉≤𝐶_{𝛽,𝑚ℝ𝑛}(1+|𝜉|)^𝑚+2𝑒^{−𝑟𝑤2(𝜉)}𝑒^{𝑟𝑤2(𝜉)}|𝜑(𝜉)|𝑑𝜉, (3.8) where 𝑟>(𝑚+𝑛+2)/𝑏 is an integer and 𝑏 is the constant in condition 4 of Definition 2.1: |𝜕_𝑥1𝜕^𝛽𝜑(𝑦)|≤𝐶_𝑚𝜋_𝑟,0(𝜑). (3.9) Thus, we have |𝜕_𝑥1𝜕^𝛽𝜑(𝑦)|≤𝐶_𝑚𝜋_𝑟,0(𝜑), (3.10) that is, |𝜕^𝛽𝜑(𝑥)|≤𝐶_𝑚[1𝑡𝑝_𝑘,𝑚(𝜑)𝑒^{−𝑘𝑤1(𝑥)}+𝑡𝜋_𝑟,0(𝜑)] (3.11)forall𝑡>0. As a function of 𝑡, the right side of (3.11) has a global minimum at 𝑡=(𝑝_𝑘,𝑚(𝜑)𝑒^{−𝑘𝑤1(𝑥)})^1/2(𝜋_𝑟,0(𝜑))^−1/2. (3.12) Thus, we obtain the inequality |𝜕^𝛽𝜑(𝑥)|≤𝐶_𝑚(𝑝_𝑘,𝑚(𝜑))^1/2(𝜋_𝑟,0(𝜑))^1/2𝑒^{(−𝑘/2)𝑤1(𝑥)}, (3.13) that is, |𝑒^{𝑘𝑤1(𝑥)}𝜕^𝛽𝜑(𝑥)|≤𝐶_𝑚(𝑝_2𝑘,𝑚(𝜑))^1/2(𝜋_𝑟,0(𝜑))^1/2. (3.14) An argument, similar to the one leading to (3.14), produces |𝑒^{𝑘𝑤2(𝜉)}𝜕^𝛽𝜑(𝜉)|≤𝐶_𝑚(𝜋_2𝑘,𝑚(𝜑))^1/2(𝑝_𝑟,0(𝜑))^1/2. (3.15) Combining (3.14), (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 inequalities (3.14), (3.15) , we obtain |𝑒^{𝑘𝑤(𝑥)}𝜕^𝛽𝜑(𝑥)|≤𝐶_𝑚(𝑝_{2𝑚+1𝑘,0}(𝜑))^2−𝑚−1(𝜋_𝑟,0(𝜑))^{1−2−𝑚−1},|𝑒^{𝑘𝑤(𝜉)}𝜕^𝛽𝜑(𝜉)|≤𝐶_𝑚(𝜋₂^𝑚+1_𝑘,0∘ℱ(𝜑))^2−𝑚−1(𝑝_𝑟,0(𝜑))^{1−2−𝑚−1}. (3.16) This completes the proof of Theorem 3.1.

When 𝑤₁(𝑥)=𝑤₂(𝑥), the characterization of 𝔖_𝑤1,𝑤2 given by Theorem 3.1 reduces to the characterization of Beurling-Björck space 𝔖_𝑤1 given by Theorem 2.4. In particular, when 𝑤₁(𝑥)=𝑤₂(𝑥)=ln(1+|𝑥|), the characterization of 𝔖_𝑤1,𝑤2 reduces to the characterization of Schwartz space 𝔖.

Remark 3.2. The Fourier transform is a topological isomorphism between 𝔖_𝑤1,𝑤2 and 𝔖_𝑤2,𝑤1. As a consequence, the Fourier transform is also a topological isomorphism between the dual spaces 𝔖^′_𝑤1,𝑤2 and 𝔖^′_𝑤2,𝑤1.

Note that the dual spaces 𝔖^′_𝑤1,𝑤2 and 𝔖^′_𝑤2,𝑤1 are assigned to the weak topologies. For different pairs of admissible functions, the space 𝔖_𝑤1,𝑤2 has the following embedding properties.

Lemma 3.3. For every 𝑤₁<𝑤^′₁ and 𝑤₂<𝑤^′₂, one has 𝔖_{𝑤′1,𝑤′2}↪𝔖_𝑤1,𝑤2. (3.17)

Lemma 3.4. For 𝛼,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^′_𝑤1,𝑤2

From Theorem 3.1, we can write 𝔖_𝑤1,𝑤2={𝜑∶ℝ^𝑛→ℂ∶𝜑iscontinuousandforall𝑘=0,1,2,…,𝒩_𝑘,(𝜑)<∞}, (4.1) where 𝒩_𝑘(𝜑)=‖𝑒^𝑘𝑤1𝜑‖_∞+‖𝑒^𝑘𝑤2𝜑‖_∞.

Theorem 4.1. Given 𝐿∶𝔖_𝑤1,𝑤2→ℂ, the following statements are equivalent: (i)𝐿∈𝔖^′_𝑤1,𝑤2; (ii) there exist two regular complex Borel measures 𝜇₁ and 𝜇₂ of finite total variation and 𝑘∈{0,1,2,…} such that 𝐿=𝑒^𝑘𝑤1𝜇₁+ℱ[𝑒^𝑘𝑤2𝜇₂], (4.2) in the sense of 𝔖^′_𝑤1,𝑤2.

Proof. (𝑖)⇒(𝑖𝑖). Given 𝐿∈𝔖^′_𝑤1,𝑤2, according to (4.1) there exist 𝑘 and 𝐶 so that 𝐿(𝜑)≤𝐶(‖𝑒^𝑘𝑤1𝜑‖_∞+‖𝑒^𝑘𝑤2𝜑‖_∞) (4.3) for all 𝜑∈𝔖_𝑤1,𝑤2. Moreover, the map 𝔖_𝑤1,𝑤2→𝒞₀×𝒞₀,𝜑→(𝑒^𝑘𝑤1𝜑,𝑒^𝑘𝑤2𝜑) (4.4) is well defined, linear, continuous, and injective. Let ℛ be the range of this map, on which we define the map 𝑙₁(𝑓,𝑔)=𝐿(𝜑), (4.5) where 𝑓=𝑒^𝑘𝑤1𝜑, 𝑔=𝑒^𝑘𝑤2𝜑 for a unique 𝜑∈𝔖_𝑤1,𝑤2. The map 𝑙₁∶ℛ→ℂ is linear and continuous. By the Hahn-Banach theorem, there exists a functional 𝐿₁ in the topological dual (𝒞₀×𝒞₀)^′ of 𝒞₀×𝒞₀ such that ‖𝐿₁‖=‖𝑙₁‖ and the restriction of 𝐿₁ to ℛ is 𝑙₁.
Since the spaces (𝒞₀×𝒞₀)^′ and 𝒞^′₀×𝒞^′₀ are isomorphic as Banach spaces, we can write 𝐿₁(𝑓,𝑔)=𝐿₁(𝑓,0)+𝐿₁(0,𝑔). Using Theorem 2.6, there exist regular complex Borel measures 𝜇₁ and 𝜇₂ of finite total variation such that 𝐿₁(𝑓,𝑔)=_ℝ𝑛𝑓𝑑𝜇₁+_ℝ𝑛𝑔𝑑𝜇₂ (4.6) for all (𝑓,𝑔)∈𝒞₀×𝒞₀. If (𝑓,𝑔)∈ℛ, then we conclude that 𝐿(𝜑)=_ℝ𝑛𝑒^𝑘𝑤1𝜑𝑑𝜇₁+_ℝ𝑛𝑒^𝑘𝑤2𝜑𝑑𝜇₂ (4.7) for all 𝜑∈𝔖_𝑤1,𝑤2. In the sense of 𝔖^′_𝑤1,𝑤2, 𝐿=𝑒^𝑘𝑤1𝜇₁+ℱ[𝑒^𝑘𝑤2𝜇₂]. (4.8)(ii)⇒(i). If 𝜇₁ and 𝜇₂ are two regular complex Borel measures satisfying (ii) and 𝜑∈𝔖_𝑤1,𝑤2, then 𝐿(𝜑)=_ℝ𝑛𝑒^𝑘𝑤1𝜑𝑑𝜇₁+_ℝ𝑛𝑒^𝑘𝑤2𝜑𝑑𝜇₂. (4.9) This implies that |𝐿(𝜑)|≤|_ℝ𝑛𝑒^𝑘𝑤1𝜑𝑑𝜇₁|+|_ℝ𝑛𝑒^𝑘𝑤2𝜑𝑑𝜇₂|≤|𝜇₁|(ℝ^𝑛)‖𝑒^𝑘𝑤1𝜑‖_∞+|𝜇₂|(ℝ^𝑛)‖𝑒^𝑘𝑤2𝜑‖_∞≤𝐶(‖𝑒^𝑘𝑤1𝜑‖_∞+‖𝑒^𝑘𝑤2𝜑‖_∞). (4.10) It may be noted that 𝜇₁ and 𝜇₂, employed to obtain the above inequality, are of finite total variations. This completes the proof of Theorem 4.1 .

Remark 4.2. When 𝑤₁(𝑥)=𝑤₂(𝑥)=(1+|𝑥|)^𝑘, (4.2) becomes 𝐿=(1+|𝑥|)^𝑘𝜇₁+ℱ[(1+|𝜉|)^𝑘𝜇₂], (4.11) which gives a representation for the tempered distributions.

As consequence of Lemma 3.4 , we can view the functionals in (𝑆^𝑏_𝑎)^′ as functionals in the space 𝔖^′_𝑤1,𝑤2. Then as a result we can characterize (𝑆^𝛽_𝛼)^′ using Theorem 4.1.

Corollary 4.3. Let 𝛼,𝛽>1. Then any 𝐿∈(𝑆^𝛽_𝛼)^′ can be written as 𝐿=𝑒^{𝑘|𝑥|1/𝛼}𝜇₁+ℱ[𝑒^{𝑘|𝜉|1/𝛽}𝜇₂] (4.12) which characterizes the dual space (𝑆^𝛽_𝛼)^′.