On the Projective Description of Weighted (LF)-Spaces of Continuous Functions

Abstract

We solve the problem of the topological or algebraic description of countable inductive limits of weighted Fréchet spaces of continuous functions on a cone. This problem is investigated for two families of weights defined by positively homogeneous functions. Weights of this form play the important role in Fourier analysis.

1. Introduction

The problem of the projective description of the topology of inductive limits of weighted spaces of holomorphic and continuous functions has attracted the attention of mathematicians after Ehrenpreis [1] proved the fundamental principle. This problem has applications in partial differential equations and convolution equations, (ultra-) distribution theory, in the theory of quasianalytic functionals, and representations of functions by exponential series. The systematic investigation of it was started by Bierstedt and Meise and Bierstedt et al. [2, 3]. The aim of the projective description is to find the conditions under which an inductive limit coincides algebraically with its projective hull or it is its topological subspace. In the case of an inductive limit of weighted Banach spaces of continuous functions the problem of their projective descriptions is very well studied. By now effective conditions are obtained that the topology of a countable inductive weighted limit of continuous functions coincides with the one of its projective hull . The algebraic identity always holds in this case. The case of (LF)-spaces is more complicated. The development in projective descriptions of countable inductive weighted limits of continuous functions had been surveyed in [4, 5]. Abstract conditions for the algebraic and topological description of weighted (LF)-spaces of continuous functions were obtained by Bierstedt and Bonet [6]. Their realizations in concrete situations are also interesting. In this connection we mention the article of Bonet and Meise [7]. In [7] weighted (LF)-spaces of continuous functions defined by weights which arise in the theory of ultradistributions and qusianalytic functionals of Roumieu type were investigated (see Remark after Theorem 9). Bonet et al. [8] have studied the problem of the projective description of (LF)-spaces of continuous functions for weights which are connected with a convex locally closed set in .

In our article this problem is solved for two families of weights defined by positively homogeneous functions on cones in normed linear spaces. Weights of this form play the important role in Fourier analysis.

2. The Problem of the Projective Description and Notations

We recall the necessary notations and definitions and state the problem of the projective description [3, 6].

Let be a locally compact Hausdorff space, and let be a double sequence of strictly positive continuous functions on such that for each . For an upper semicontinuous function we introduce the Banach space of continuous functions: The weighted inductive limit of Fréchet spaces of continuous functions on is defined by The space is a Hausdorff (LF)-space.

The system of weights assosiated with consists of all upper semicontinuous functions such that for each there are and with on . The projective hull of the inductive limit is defined by The topology of is defined by the system of seminorms , . The space is contained in its projective hull with continuous inclusion.

The problem of the projective description for the spaces is to determine conditions under which(A)the spaces and coincide algebraically, or(T)the space is a topological subspace of its projective hull .

3. Case of Weights Defined by Positively Homogeneous Functions

Let be a normed (complex or real) linear space with the norm . We assume that is a cone in ; that is, for all and . The set is endowed with the induced topology. It is assumed that is locally compact. Then each set , , is compact in . Let , , be continuous and positively homogeneous of degree functions such that for each . Further is a continuous function with Put , . We define weight functions by

Theorem 1. The space is a topological subspace of .

Proof. Since , then for each the Fréchet space coincides algebraically and topologically with the Fréchet space By [3, Theorem  1.3.] is a topological subspace of .
We investigate now the algebraic identity .

Theorem 2. The following conditions are equivalent.(i)The space coincides algebraically with .(ii)The condition (RD) holds:

Proof. By [6, Proposition 4] algebraic identity is equivalent to the following condition (wQ) which was introduced by Vogt [9]: that is, Since , the latter is equivalent to Thus the algebraic identity holds if and only if or
. Suppose that (RD) holds. Fix . By (RD) : We choose and fix . If then Hence the inequality (13) holds with such that . Thus the condition (wQ) is valid and consequently the algebraic identity is fulfilled.
. Suppose that , , , the inequality (12) or (13) holds. Assume that the condition (RD) does not hold. Taking into account the positive homogeneity of functions we obtain that We select and for as in (12) and (13). Set and choose for as in (17). We take and as in (12) and (13). Put , (from the continuity of it follows that ). For every by (17), Hence, We set . It is clear that . In consequence of (17) and (19) there is such that Since the function is continuous on and , there is such that
We put . Then and and also for both inequalities (12) and (13) do not hold. It is a contradiction.

Example 3. (a) Let be convex compact subsets of such that is contained in the interior of for each . We denote by , , , the supporting function of . The functions are positively homogeneous of order 1 and convex on . We put .
The sequence satisfies the condition (RD) of Theorem 2. Indeed, we can take in the condition (ii) of Theorem 2 for . The necessary inequality follows from , from the positive homogeneity and continuity of functions .
(b) Let be a continuous and positively homogeneity of order function and , , where is a strictly increasing sequence of positive numbers. The sequence satisfies the condition (RD).
We will show that the condition (RD) is equivalent to the condition of the regular decrease of the sequence which plays the important role in projective descriptions in the (LB) case [3].

Definition 4 (see [3, Definition  2.1]). The sequence of functions such that on is called regularly decreasing if

Theorem 5. Let be continuous functions which are positively homogeneous of degree , on , . The following assertions are equivalent:(i)The condition (RD) holds.(ii)The sequence is regularly decreasing.

Proof. . Suppose that the sequence decreases regularly. Then , Assume that the condition (RD) does not hold. Then For we choose as in the condition (23) and for this select as in the condition (24).
We will prove that from the equality it follows that . Indeed, if for some , then also for each the equality holds. Therefore From this it follows that . Hence for each . Since for all by (24) It is a contradiction with (23).
The implication is obvious.

4. Case of Weights Defined by a Composition with Positively Homogeneous Functions

Let be a cone in a normed (real or complex) linear space , that is, for all and . Set for where is the norm in . We endow with the induced topology.

Let , , be continuous and positively homogeneous of degree functions such that on for each . Suppose that is a continuous function such that We put , .

Further we will use functions with a covering property.

Definition 6. A function satisfies the condition (SJ) if there is such that for all .

Example 7. The following functions satisfy the condition (SJ) with :(a) if , , ,(b), , if , , , .
We fix and define weight functions by where is a continuous function which satisfies the condition (SJ).

Theorem 8. The space is a topological subspace of .

The assertion follows from [3, Theorem  1.3] (see the proof of Theorem 1).

We will investigate further when the algebraic identity holds.

Theorem 9. The following assertions are equivalent:(i) holds algebraically.(ii)There is such that for each .

Proof. By [3, Proposition  4] the algebraic identity is equivalent to the condition (wQ): that is, Since , this condition is equivalent to Thus the algebraic equality holds if and only if the following condition (wQ1) is fulfilled: or
. Assume that the condition (i) holds but (ii) is not satisfied. We select and for as in (wQ1). Since (ii) does not hold, there are and such that and . Choose so large that We fix some . Define and for as in (wQ1). Since and , there is such that where is a constant as in condition (SJ).
Let . Then . By the condition (SJ) there is such that and . From it follows that There is a contradiction with (wQ1).
The implication is obvious.
We give a corollary of Theorem 9. Let , ; , ; , ; , , be continuous and positively homogeneous of degree functions such that for each ; let function be such as above. We put , . The function satisfies the condition (SJ).
Put where is fixed.

Corollary 10. The following conditions are equivalent:(i) holds algebraically.(ii)There is such that for each .

Remark 11. Let be a convex open subset of ; let be a fundamental sequence of convex compact subsets of such that for each . For each we denote by , , the supporting functions of . The functions are convex and positively homogeneous of order 1. Let be a quasianalytic weight function as in Braun et al. [10] (see [7, p. 125] too). In [7, Proposition 7] it was proved that the algebraic identity does not hold for the functions , . This assertion follows from Corollary 10, since for each and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the referee for valuable remarks.