Abstract

We introduce the space of all functions such that and are finite for all , , where and are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space without conditions on the derivatives. For functionals in the dual space , 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. BjΓΆrck [2] studied and expanded the theory of Beurling on ultradistributions to extend the work of HΓΆrmander [3] on existence, nonexistence, and regularity of solutions of constant coefficient linear partial differential equations.

The Beurling-BjΓΆrck space , as defined in [2], consists of 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 of 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 and its dual

The main difference between the Beurling-BjΓΆrck space and the space is that the decay of the functions in and their Fourier transform are measured by the same submultiplicative function , Whereas the decay of the functions in and their Fourier transform are measured by two different submultiplicative functions and .

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

The symbols , , , and so forth indicate the usual spaces of functions defined on , with complex values. We denote by the Euclidean norm on , while indicates the norm in the space . When we do not work on the general Euclidean space , we will write , and so forth as appropriate. Partial derivatives will be denoted by , where is a multi-index . 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 , . With , we mean that for every . The Fourier transform of a function will be denoted by or and it will be defined as . The inverse Fourier transform is then . The letter 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 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 , we denote the space of functions of the form , where (1) is increasing, continuous, and concave,(2),(3),(4) for some and some .

Standard classes of functions in are given by

Remark 2.2. Let us observe for future use that if we take an integer then where 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 is subadditive for all .

Proof. Let . Since is increasing, we obtain Since is concave on and , we have If we take then we have This completes the proof of Lemma 2.3.

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

Theorem 2.4 (see [5]). Given , the space can be described both as a set and as a topology by where and .

We observe that becomes the Schwartz space when For , the Gelfand-Shilov space of type is characterized in [6] by the space of all functions for which the seminorms are finite for some .

Definition 2.5. Given the space is the space of all functions for which the seminorms are finite, for and .

We can assign to a structure to FrΓ©chet space by means of the countable family of seminorms

Since for all is integrable, so is well defined and the formulation of the condition makes sense for all .

The space , equipped with the family of seminorms is a FrΓ©chet space.

We observe that the space becomes the Beurling-BjΓΆrck space , when . When , the space of functions with compact support is dense subspace of for all . The conditions imposed on the function assure that the space satisfies the properties expected from a space of testing functions. For instance, the operators of differentiation and multiplication by are continuous from intothemselves, the space is a topological algebra under pointwise multiplication and convolution. Unfortunately, the Fourier transformation on is not a topological isomorphism from into itself for some , . For Example, if we take , , and , then but ; see [1, 2].

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

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 be a continuous function with continuous derivatives of order . Assume that there exist such that for all . Then for all .

3. Topological Characterization of the Space

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

Theorem 3.1. Given , the space can be described as a set and as a topology by where

Proof. Let us denote by the space defined in (3.1). The conditions and imply the smoothness of and The space becomes a FrΓ©chet space with respect to the family of norms From these definitions, it is clear that and that the inclusion is continuous. To prove the converse, we use the induction on and the general idea of Landau's inequality. Fix . We want to show that and are finite, for every and every multi-index , which is true for all , when . We assume that it is true for all , when , and we want to prove it for all and for . We start with . Assume that with , We also indicate , , for fixed, . Moreover, if , we have where is a number between and . Thus, We can write If we take with the same sign as , we have That is, To estimate , we write where is an integer and is the constant in condition 4 of Definition 2.1: Thus, we have that is, . As a function of , the right side of (3.11) has a global minimum at Thus, we obtain the inequality that is, An argument, similar to the one leading to (3.14), produces Combining (3.14), (3.15), the inductive hypothesis implies that . The open mapping theorem can provide once again the continuity of the inclusion. However, solving the recursive inequalities (3.14), (3.15) , we obtain This completes the proof of Theorem 3.1.

When , the characterization of given by Theorem 3.1 reduces to the characterization of Beurling-BjΓΆrck space given by Theorem 2.4. In particular, when , the characterization of reduces to the characterization of Schwartz space

Remark 3.2. The Fourier transform is a topological isomorphism between and . As a consequence, the Fourier transform is also a topological isomorphism between the dual spaces and .

Note that the dual spaces and are assigned to the weak topologies. For different pairs of admissible functions, the space has the following embedding properties.

Lemma 3.3. For every and , one has

Lemma 3.4. For , one has . As a consequence, .

4. A Representation Theorem for Functionals in the Space

From Theorem 3.1, we can write where .

Theorem 4.1. Given , the following statements are equivalent: (i); (ii) there exist two regular complex Borel measures and of finite total variation and such that in the sense of .

Proof. Given , according to (4.1) there exist and so that for all . Moreover, the map is well defined, linear, continuous, and injective. Let be the range of this map, on which we define the map where , for a unique . The map is linear and continuous. By the Hahn-Banach theorem, there exists a functional in the topological dual of such that and the restriction of to is .
Since the spaces and are isomorphic as Banach spaces, we can write . Using Theorem 2.6, there exist regular complex Borel measures and of finite total variation such that for all . If , then we conclude that for all . In the sense of , . If and are two regular complex Borel measures satisfying and , then This implies that It may be noted that and employed to obtain the above inequality, are of finite total variations. This completes the proof of Theorem 4.1 .

Remark 4.2. When , (4.2) becomes which gives a representation for the tempered distributions.

As consequence of Lemma 3.4 , we can view the functionals in as functionals in the space . Then as a result we can characterize using Theorem 4.1.

Corollary 4.3. Let . Then any can be written as which characterizes the dual space .