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Abstract and Applied Analysis
Volume 2011, Article ID 947908, 15 pages
http://dx.doi.org/10.1155/2011/947908
Research Article

On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Department of Mathematics, Zanjan University, Zanjan 45195-313, Iran

Received 11 June 2011; Accepted 16 August 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 I. Akbarbaglu and S. Maghsoudi. 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.

Abstract

Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

1. Introduction

Throughout this paper, let denote a locally compact Hausdorff group with a fixed left Haar measure . By a weight function on , we mean an arbitrary strictly positive measurable function on , and, by a system of weights on , a set of weight functions such that given , in and , there is an such that () for locally almost all .

For a weight function and , let denote the space of all complex-valued measurable functions on such that , the usual Lebesgue space on with respect to ; see [1] for more details. Then, with the norm defined by for all is a Banach space. We also denote by the space of all measurable complex-valued functions on such that , the space defined in [1]. Then, with the norm defined by for all is a Banach space. Furthermore, for , the topological dual of coincides with , where is the exponential conjugate to defined by . In fact, the mapping from to defined by is an isometric isomorphism; see for example [2]. For measurable functions and on , the convolution multiplication is defined at each point for which this makes sense. The algebraic and topological properties of weighted -spaces have been studied extensively; see for example [25].

Let and be a system of weights on , we set

In this paper, we equip the space with the natural locally convex topology generated by the family of norms , where runs through . For a similar study in other contexts, see [68]. We investigate certain algebraic and topological properties of the locally convex space . Our results generalize and improve some interesting results of [5] and partially answer a question raised in [3].

2. Preliminaries and Some Basic Results

Let be a locally compact Hausdorff group with a fixed left Haar measure and be a system of weights on . We equip with the locally convex topology generated by the family of norms and denote this topology by . So has a basis of closed absolutely convex neighbourhoods at the origin of the form

Note that the topology is Hausdorff, because if and , we have . Put and fix . Then, and thus is Hausdorff.

If and are two systems of weights on and for every , there is a such that (pointwise locally almost everywhere on ), then we write . In the case which and , we write .

Proposition 2.1. Let and be two systems of weights on and be a measurable mapping such that . If the Radon-Nikodym function belongs to , then the mapping is a continuous linear map from into .

Proof. Given and , choose such that . Then we have Hence, . Since was arbitrary, . Continuity also follows from the above relations.

The space of all bounded Borel measurable functions on with compact support will be denoted by . Let us remark that if , then is norm dense in for any weight on ; see for example [9].

Corollary 2.2. Let and be two systems of weights on . Then,(i)If , then the induced topology on is weaker than .(ii)If and , then . In particular, if and only if .

Proof. (i) is trivial. For (ii), we observe that for any , there is a such that . So the identity map from into is continuous. Since is dense in , can be extended continuously to a continuous linear mapping on . The extension map is again the identity map. So . Hence, there exists a constant such that locally almost everywhere; see Lemma 2.1 in [10]. This proves that .

Let us recall the definition of the projective limit of a family of locally convex spaces. Let be a partially ordered set and be a family of locally convex spaces, and for , be a linear map from into . Suppose that for all and be the identity map on for all . Then, the projective limit of the family is defined as for more details see for example [11].

Proposition 2.3. Let be a system of weights on . Then is a complete space.

Proof. We note that for any two weights with , . Let the mapping be the canonical injection. Then, it is clear that is isomorphic to the projective limit system of the Banach spaces , , and, hence, is complete; see Lemma in [12].

Proposition 2.4. The locally convex space is normable if and only if the topology is generated by for some .

Proof. If is normable, then it has a neighbourhood of zero that is norm bounded with respect to for every . Hence, there is so that is norm bounded in the space for every . This implies that there is a positive constant so that , and our claim is proved. The converse is clear.

3. The Dual and Bidual of

In this section we deal with the dual space of and, among other things, characterize its equicontinuous subsets.

Theorem 3.1. If and , then the dual space of is with .

Proof. Let . We define the linear functional by , then .
Conversely, let . First, we know that for every . So there is a such that whenever . As is bounded in the intersection of the unit ball of with , is continuous on with the topology induced by the norm . Since is dense in , can be extended continuously to a continuous linear form on which we denote by . Then, we have , and hence there is a unique so that therefore, we obtain the following isomorphism: defined by , where for all .

Lemma 3.2. Let be a system of weights on . For every , define the mapping by . Then, for , where is the closed unit ball of and denotes its polar.

Proof. It is clear that is a well-defined continuous linear map. Also, is dense in . Therefore (the adjoint of ) is continuous and one to one linear map from into , where . Now, since is -compact by the Alaoglu theorem and so is -compact, while is obviously convex. So we find that Form which it follows that

We have the following characterization of the equicontinuous subsets of .

Theorem 3.3. Let and be a subset of . The following are equivalent.(a) is -equicontinuous.(b)There are and an equicontinuous subset of so that .(c)There are and such that whenever .

Proof. By (a), there is so that , where . According to Lemma 3.2, we have , where is the closed unit ball of . Hence .
There is so that by (b). So , and .
If , it is clear that and if , by Hölder’s inequality, for and we have Hence, , and this guarantees that is -equicontinuous in both cases.

Proposition 3.4. Let be a system of weights on . Then, the set of extreme points of is the set for , and for .

Proof. Fix and let be the map defined in Lemma 3.2. From Lemma 3.2, it follows that for any extreme point of , there is an extreme point of so that .
Conversely, let be arbitrary and let , where is an extreme point of . Clearly, , and if for some and , then there are such that and . Thus, and since is one to one, . However is an extreme point of , which implies that , and hence , that is, is an extreme point of . Now the rest of the proof is easy to complete; see for example Section 2.14 in [13].

Let us recall that a locally convex space is called semireflexive if .

Theorem 3.5. Let be a system of weights on . Then is semireflexive.

Proof. If , then the restriction of to , for every , belongs to , where was considered with the induced strong topology on . Now if and for some in the norm , then for every weakly bounded set in ,
This means that in the strong topology of . Hence, for every , there is a unique so that
Now note that if with , then and . Therefore for every , and hence almost everywhere. This implies that
Conversely, if , then it is obvious that the linear form is continuous with respect to the strong topology on . So the canonical imbedding is onto. Hence is semireflexive.

4. As a Topological Algebra

In this section, we study conditions on a system of weights for that with the convolution multiplication to be a topological algebra. We commence with some definitions.

If is a function on , the left translate of by is the function given by . A subset of functions on is called left translation invariant if for all and .

A weight function on a locally compact group is called left moderate if for all . It is easy to see that , ; see [4] or [9]. Let us remark that any submultiplicative and any locally integrable left moderate measurable function is bounded and bounded away from zero on any compact subset of ; see Theorem 2.7 in [10]. In particular, is bounded on compact sets. The condition that is left moderate is equivalent to that the space (for ) being translation invariant; see for more details [4]. Observe that for and ,

Lemma 4.1. Let be a system of weights on . Then is left translation invariant if and only if every element of is left moderate.

Proof. The “if” part is clear by the remarks above. For the converse, we need only to note that is dense in for .

Theorem 4.2. Let be a system of locally integrable left moderate weights on and . Then, the map from into is continuous.

Proof. Assume first that with . Let , , and be a net in convergent to . Choose a compact neighbourhood of , then whenever . Let Choose such that for all and . Then for all .
Finally, let be an arbitrary element of and . Let be an upper bound for the function on the compact neighbourhood of ; recall that is submultiplicative. For every , there exists such that . By the first part, we can choose such that for all . One can conclude that for all . This finishes the proof.

We now focus on some systems of weights for that to be an algebra under usual convolution whenever this integral makes sense. For , it is well known that is a convolution algebra if and only if is weakly submultiplicative; that is, for all , for some .

For any two weight functions and on , we set In the case where , we simply write .

The following lemma is similar to Lemma 2.2 in [9].

Lemma 4.3. Let and be a system of weights on . If is a convolution algebra, then is locally integrable for each .

The next result gives a sufficient condition for that to be a convolution algebra.

Theorem 4.4. Let and be a system of weights on . If for every , there is a such that , where is the conjugate exponent to , then the space is a complete locally convex algebra with continuous multiplication.

Proof. We must show that for all . By Lemma 4.3, is dense in , thus for any , it suffices to show that for all . For this, let . Writing and using Hölder's inequality, we obtain This shows that Whence This completes the proof.

The following corollary is a direct consequence of Theorem 4.4.

Corollary 4.5. Let and be a system of weights on such that for every , . Then is a complete locally multiplicative convex algebra.

The next result provides us with a class of weights on the additive group for which the usual weighted Lebesgue space becomes a Banach algebra.

Proposition 4.6. Let and be a natural number. Let be a function such that
(i)If , then .(ii).(iii)There exists a positive number such that for all .
Then is a Banach algebra, where is the conjugate exponent to .

Proof. For any , let and observe that
Hence, for ,
Thus, , and now the result follows from Corollary 4.5.

Example 4.7. Let , be the conjugate exponent to , and . Set where and . Then is a Banach algebra.

We are going to prove the converse of Theorem 4.4. For this, we fix some notation. If be two complex-valued functions on , then denotes the function on given by for all . Also for any two sets and of functions on we set . For a locally compact group , note that the cartesian product is a locally compact group by defining the product for all .

We need the following easy lemma in the sequel.

Lemma 4.8. Let and be a system of weights on such that . Then is dense in .

Proof. Since is norm dense in , then is projective tensor norm dense in , where is the projective tensor product. Hence is -dense in . On the other hand, it is known that is isometric with . In fact, the linear map for all and , can be extended to a surjective isometry; for more details, see for example [14]. Now we conclude that is -dense in .

The next theorem is our main result in this section.

Theorem 4.9. Let , be -compact, and be a system of weights on . If the space is an algebra with continuous multiplication, then for every there exists a such that .

Proof. Choose an arbitrary . Then, by assumption, there exists some such that for every , . Now for every , defines a continuous linear functional on with the norm . Also for every , , and we have
Set for . By Lemma 4.8, can be extended to a -continuous functional on . Since is -compact, by Exercise 15.14 in [1], it follows that the function belongs to . But Since is arbitrary, we conclude that ; see Section 14 in [15] or Theorem  20.15 in [1].

As an immediate consequence of Theorem 4.9, we obtain the following corollary that partially answers a question raised in [3].

Corollary 4.10. Let be a weight on -compact group and . Then is a convolution algebra if and only if .

5. Ideals and the Spectrum of the Algebra

We commence this section with the following proposition.

Proposition 5.1. Let and let be a translation invariant algebra. Then(i) has an approximate identity.(ii) has a bounded approximate identity or an identity if and only if is discrete.

Proof. (i) Let be a fixed relatively compact neighbourhood of the identity element , and let be the family of all neighbourhoods of contained in directed by reverse inclusion. Set , and note that since elements of are locally integrable, . Given and , then by Theorem 4.2, there exists a neighbourhood of the identity such that for . Now, for with , and , we have Hence, for all neighborhoods , from which it follows that in -topology.
(ii) Let be a bounded left approximate identity for . Fix an , then for some positive number . Let . Since is dense in with the norm , then given , there exists such that . Choose such that for all . Then it follows that for all . This means that has a bounded left approximate identity. But according to Theorem 4.2 in [9], this is equivalent to that is discrete.

The next theorem shows that closed ideals of the algebra are exactly translation invariant subspaces.

Theorem 5.2. Let and be a translation invariant algebra. Then a closed linear subspace of is an ideal in if and only if it is two-sided translation invariant.

Proof. Suppose that is a -closed two-sided translation invariant subspace of . We have to show that and for all and . Let , for some , such that for all . Then, for and any , Since , the Hahn-Banach theorem implies that for all and . Thus is a left ideal, and using the right translation invariance of , it is readily seen, in the same way, that is also a right ideal.
Conversely, let be a closed ideal of , and . Let be an approximate identity for . Then for each , we have Hence, in -topology. As is a closed left ideal, it follows that ; that is, is left translation invariant. Similarly, it is shown that is also right translation invariant.

We denote by the spectrum of consisting of all -continuous nonzero linear functionals on which are multiplicative; that is,

We conclude this work with the following result which is a characterization of the spectrum of .

Proposition 5.3. Let be a system of weights on 6-compact group . Then where

Proof. Let for some such that for almost all . Then, is -continuous and so -continuous. Moreover, for , That is, .
Conversely, let . Then is bounded on a -neighbourhood of zero. Thus is bounded on the set for some . Therefore can be extended to an element in . It follows that there exists a function such that for all . Since for , , we infer that By an argument similar to the proof of Theorem 4.9, we deduce that for almost all .

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