#### Abstract

Let be a locally compact group an arbitrary family of the weight functions on and . The locally convex space as a subspace of is defined. Also, some sufficient conditions for that space to be a Banach space are provided. Furthermore, for an arbitrary subset of and a positive submultiplicative weight function on , Banach subspace of is introduced. Then some algebraic properties of , as a Banach algebra under convolution product, are investigated.

#### 1. Introduction

Throughout the paper, let be a locally compact group with a fixed left Haar measure or . We call any Borel measurable function a weight function. For , the weighted -space with respect to is the set of all complex valued measurable functions on such that , the usual Lebesgue space as defined in [1]. This space will be denoted by , when is discrete. Two functions in are considered equal if they are equal -almost everywhere on . If , then is a locally convex space endowed with the topology generated by the seminorms defined by

For -measurable functions and on , the convolution multiplication is defined by at each point for which this makes sense. Then is said to exist if exists for almost all . Several authors have studied the convolution properties on the space and , where is positive and submultiplicative. It has been shown in [2–4] that the convolution of elements in and also does not exist in general. If this is the case, then it is desirable to study the closedness of these spaces under the convolution. For related results on the subject related to see also [5]. Also we refer to [3, 6–12] for the more general case of weighted -spaces. Besides these significant issues, some authors considered and investigated the intersection of the -spaces to each other and also together with other Banach spaces; for example, see [13–15].

It should be noted that weighted -spaces and its intersections have been studied more completely years ago, especially in the decade of 1970. We first refer to the Ph.D. thesis of Feichtinger titled by “subconvolutive functions” for a survey, which also contains many invaluable information related to the weight functions. Moreover, we found a lot of invaluable results related to weighted -spaces and also weight functions in many earlier publications. We refer to some of them such as [16–21]. Note that [17] (downloadable as from http://www.univie.ac.at/nuhag-php/bibtex/) is a technical report which contains many remarkable results related to the weight functions. Moreover, we found more complete results related to distributions and weighted spaces in [22]. In fact, some of our results in the present work, have been inspired by the results given in [22].

Also recently we considered an arbitrary intersection of the -spaces denoted by , where . Then we introduced the subspace of as and studied as a Banach algebra under convolution product, for the case where ; see [23].

The purpose of the present work is to generalize the results of [23] to the weighted case. We first give general information about the weight functions and collect most of the available results in a more concise way. Then for an arbitrary family of the weight functions on and , we introduce the subspace of the locally convex space . Moreover, we provide some sufficient conditions on and also to construct a norm on . Particularly, we show the deficiency of this space in taking a norm in the general case, with presenting some fundamental examples. The third section is assigned to the Lorentz spaces, which are suggested to us by the referee. We first give some preliminaries related to Lorentz spaces . Then for the case where is fixed and runs through , we introduce as a subspace of . As the main result, we prove that , where is positive.

Stimulated by these results, in the last two sections, we assume consists just of one positive and submultiplicative weight function . Then we introduce the Banach space and also the space , where , to imitate of the recent work of the authors [23]. Then we generalize the results of the third section in [23] to the space . The last section is essentially devoted to as a Banach algebra under convolution product. We first show that is always an abstract Segal algebra with respect to . At the end we obtain some results on the amenability of and its second dual.

#### 2. Weighted -Algebra

Let be a locally compact group and a weight function on and . It is plain to verify that the function defines a norm on if and only if is almost everywhere positive on . Due to the importance of this subject, most of the time the authors assume positivity in the general definition of a weight function. Thus in this and the last two sections, all weight functions are assumed to be positive. The present section is completely devoted to Banach space . In fact, some important results connected to the properties of convolution product on are gathered. First we recall two important kinds of positive weight functions which play an essential role in this survey. We refer to [17, 24] and also [21] which contain many valuable information related to the weight functions.

The weight function is called submultiplicative if for all The class of weights defining convolution algebras admits a complete description, and it turns out that every weight is equivalent to a continuous function. Moreover, it should be noted that is closed under convolution product if and only if is equivalent to a continuous submultiplicative weight function; see [19, 24, 25] for a full description. But the condition of submultiplicativity of is not a necessary condition, whenever ; see [11, Example 2.1]. However, for all , on a discrete group, a weight function of any that are Banach algebra is submultiplicative; indeed, for all where is the Dirac measure at .

The weight function is called of moderate growth if for all . It is remarkable to note that if is of moderate growth, then inclusion (6) implies that Also the condition of moderate growth for is equivalent to the space , for all , being left translation-invariant.

##### 2.1. Local Integrable Property of the Positive Weight Functions

Let be a locally compact group and a positive weight function on and . We say that is locally integrable if , for all compact subsets of . This property is very vital in this research. Thus most of the authors take it as an assumption. However, this is redundant if is a Banach algebra [11, Lemma 2.1]. It also should be emphasized that if is submultiplicative, then it is bounded and bounded away from zero on every compact subset of [24, Proposition 1.16]. It follows that is obviously locally integrable. It is required for the progress to give some special properties of the class of locally integrable positive weight functions. Two important results created by the local integrability of are obtained in the following. See [11, 12, 19, 24] for more information. (1) If is locally integrable, then where is the space of all bounded compactly supported functions on . It implies that is dense in [11, Lemma 2.2]. Note that when is continuous, one can easily replace rather than , where is the space consisting of all continuous functions with compact support. (2) An important result related to the weight functions has been proved in [19, Theorem 2.7]. Indeed, let be both of moderate growth and also locally integrable. Then is equivalent to a continuous weight function ; that is, for some constants , locally almost everywhere on . It follows that is bounded and bounded away from zero on every compact subset of .

##### 2.2. The Main Results

Let us recall the function on for , from [7], as the following: where is the exponential conjugate of , defined by . It is known that for , if , then is closed under convolution; see [26] as a more general case and also [7, Theorem 2.2]. This result has also been pointed out in [10]. Furthermore, we asked about the converse of this result whenever is a submultiplicative weight function in [7]. It is noticeable to know that this conjecture has been rejected by the examples given by Kuznetsova in [10], for an arbitrary positive weight function. Also the conjecture is rejected in a simultaneous work with [7], for a suitable submultiplicative weight function; see [12, Theorem 1.1]. It is unlike the result [27, Corollary 4.10]. In fact, as it is shown by this counterexample, the claim given in [27, Corollary 4.10] is not true. However, in the following proposition we show that the conjecture holds for the case where . Anyway, the conjecture given in [7] had been settled only for . It should be noted that one can extract this result from [28, Lemma 1] and also [17, page 12].

Proposition 1. *Let be a locally compact group and a positive weight function on . Then is closed under convolution if and only if .*

*Proof. *First let be closed under convolution. Since , it follows that , and so the function
belongs to , clearly. For the converse, suppose that and . Then for each , the function defined by
belongs to and so
almost everywhere on . Consequently
and the result is obtained.

Concisely, we indicate the results of almost all surveys done by the mathematicians in this field in the following remark.

*Remarks*. Let be a locally compact group and a positive weight function on and . (i) is closed under convolution if and only if is equivalent to a continuous submultiplicative weight function [11, Theorem 3.1]. (ii)If , for some , then the function belongs to , for almost everywhere . Since this function is positive, it follows that is -compact. (iii)If , is submultiplicative, and exists as a function for all , then is -compact [3, Theorem 2.5]. (iv)If , is amenable, and is closed under convolution, then is -compact [12, Corollary 3.3]. (v) is -compact if and only if for some , there exists a weight satisfying [11, Theorem 1.1]. (vi)If , then the -compactness of is not in general a necessary condition for the closedness of under convolution [11, Theorem 1.1 and Proposition 1.2]. (vii)If and , then [7, Theorem 2.2] and also [10] imply that is closed under convolution. (viii) is closed under convolution if and only if , as we proved in Proposition 1. Also [28, Lemma 1] and [17, page 12].

#### 3. General Properties of Arbitrary Weight Functions

Let be a locally compact group and . Take to be an arbitrary family of the weight functions on such that the function defined as is finite everywhere on . Then is in fact a weight function on . Set We equip the space with the natural locally convex topology generated by the family of seminorms , where and runs through . We will explain that the topology on is generated by the neighborhoods where , is any positive real number, and . In general, is not necessarily Hausdorff under . In fact, a locally convex space is Hausdorff if and only if it has a separated family of seminorms; see [29] for full information about the locally convex vector spaces. Now, consider the following subset of : It is obvious that in general and , for each . Also some elementary calculations show that if is a finite set, then and for each , The following example shows that the inclusions (20) can be proper.

*Example 2. *Take to be the additive group of the real numbers endowed with the discrete topology. Set , where , the characteristic function of the set . Obviously is the space of all complex valued functions on , which is in fact a locally convex space. Moreover, and . Also . It follows that

The main purpose of this section is to provide some conditions for that acts as a norm function on . Although all of them are sufficient conditions and occur naturally in applications, they can be useful in their own right. Let us first turn the attention to the following example.

*Example 3. *Consider the additive group of real numbers endowed with its standard topology, and let
Suppose that . Thus for all . Also
and so
whereas . It follows that is not a normed space.

According to Example 3, may not be treated as a norm function, even for a countable set of the weight functions. The following result shows that countability of can be a sufficient condition for normability of , whenever is positive almost everywhere on .

Proposition 4. *Let be a locally compact group and and let be a countable family of weight functions such that almost everywhere on . Then is a normed space.*

*Proof. *Assume that and . It follows that and so almost everywhere on , for all . Let , and for each , , and put . Since is countable and , then . Now let . Since , there exists at least one such that . It follows that . Consequently, almost everywhere on and the result is obtained.

In the following examples, we determine for two families of the weight functions.

*Example 5. *Take , the additive group of real numbers endowed with the usual topology. (1) Let , , and . Since is finite and also, for every , , then
We explain this example in detail. For each , we have
and so . It follows that . Now suppose that . Then
and so
(2) Let , where . Since, for each , , it follows that . Now let , the constant function of value and , the bracket function on . Then, but . Also and since , it follows that . Therefore
Moreover, is a normed space by Proposition 4.

It is clear that the existence of at least one positive weight in is enough for normability of with the topology induced by . Such a condition is not imposed in the present section. Instead, we introduce a more delicate framework for as the following.

*Definition 6. *Let be a locally compact group and a family of weight functions. Then is called locally positive if for each there exist and an open neighborhood of such that is positive on .

It is obvious that if consists of just one element , then local positivity of is equivalent to the fact that is positive. In this situation, is always a normed space under .

Note that if is locally positive, then for each . Thus the following result is obtained clearly from Proposition 4.

Corollary 7. *Let be a locally compact group and and let be countable and locally positive. Then is a normed space. *

We give another criterion for the normability of under in the next result. It shows that countability of can be removed in Corollary 7, in the case where is -compact.

Proposition 8. *Let be a -compact locally compact group and let be locally positive. Then is a normed space.*

*Proof. *Suppose that , where is a compact subset of , for each . Take such that . Thus , for each , and so almost everywhere on . Hence, almost everywhere on , for each . By the local positivity of , for each , there exists the neighborhood of and the weight function such that is positive on . So almost everywhere on . Since is compact, there exist the elements of such that . It follows that almost everywhere on . Therefore, almost everywhere on , and so the result is obtained.

Nevertheless, in the following example we show that -compactness of and also countability of are not necessary conditions for normability of .

*Example 9. *Consider again the additive group of real numbers endowed with the discrete topology, and let . Then is clearly locally positive and uncountable. Suppose that with . Thus , for all , and so everywhere on . It follows that acts as a norm on , whereas the group is not -compact.

As the final result in this section, we provide some sufficient conditions for to be a Banach space under the norm .

Theorem 10. *Let be a locally compact group and let be a countable family of positive weight functions. Then is a Banach space.*

*Proof. *By Corollary 7, is a norm on . To that end, let , and let be a Cauchy sequence in . Then for each , is a Cauchy sequence in . So there exists a net such that () and , for each . Therefore there exists a subnet of such that in the pointwise sense, outside a measurable subset of with . Continuallty, there exists a subnet such that in the pointwise sense, outside a measurable subset of with . Inductively for each , there exists a subnet of such that tends in the pointwise sense to , outside a measurable subset of with . Set . It follows that for all , outside the set , and so all the functions () are almost everywhere equivalent to a function on . Thus
We prove that as a sequence in converges to the function . Since is a Cauchy sequence in , for each , one can find a positive integer such that for all ,
where . It follows that
and so for each ,
Choosing , for each ,
Then,
Consequently . Also inequality (35) implies that , and the proof is completed.*Remarks*. It is worth noting that we point out here to the paper of Beurling [16] in this field. This valuable work was introduced to us by the referee. Because of the value of this work, let us mention it again briefly. Let be an abelian locally compact group and and let be a collection of the positive locally integrable weight functions on . Also suppose that is a function on such that for each , takes a finite value and also satisfies the conditions till in [16]. Consider the subset of consisting of all with (such a weight function is called normalized). For a fixed , let . Set
where is the exponential conjugate of defined by . Note that in [16], the definition of the norm functions and also has been given in a slightly different form from the usual way. In fact for each and ,
Now for each and , let
Then (resp., ) acts as a norm function on (resp., ). More importantly, by [16, Theorem 1], is a Banach algebra under convolution and is a Banach space which is the dual of . Also, we refer to the examples given in [16, Section 2] for making these spaces more clear. Indeed, in these examples, the spaces and are investigated and characterized for some suitable classes of the weight functions on the Euclidean space .

#### 4. Some Intersections of the Lorentz Spaces

In this section, we investigate the intersection of Lorentz spaces, which in fact was suggested to us by the referee. First we give some preliminaries and definitions that will be used throughout the section. See [30] for complete information in this field. Let be a complex valued measurable function on . For each , let The decreasing rearrangement of is the function defined by We adopt the convention , thus having whenever for all . By [30, Proposition ] for each we have where is the Lebesgue measure. For , define The set of all with is denoted by and is called the Lorentz space with indices and . As in -spaces, two functions in are considered equal if they are equal to -almost everywhere on . Note that (43) implies that .

We recall here [30, Proposition ] which is very useful in our main results.

Proposition 11. *Suppose and . Then there exists a constant such that
**. In other words, is a subspace of . *

Now let be fixed and an arbitrary subset of with . We introduce as a subset of by The main result of the present section is provided in the following.

Theorem 12. *Let be a locally compact group and let be an arbitrary subset of such that . Then , as two sets. Moreover, for each ,
*

*Proof. *Proposition 11 implies that , for each . Also for each
It follows that and
Thus . Now we prove the reverse of the inclusion. If , then , obviously. Moreover , for each . Now let . Thus there is a sequence in , converging to . For each , Fatou's lemma implies that
Consequently
It follows that and , as claimed.

By [30, Theorem ], the spaces are always quasi-Banach spaces (i.e., a complete quasi-normed space). Moreover, [30, Exercise ] implies that is a Banach space in the case where and . We end this section with the following result, which is immediately obtained from Theorem 12 and [30, Exercise ].

Corollary 13. *Let be a locally compact group and let be an arbitrary subset of . Then is a Banach space. *

#### 5. Introducing Some Intersection of Weighted -Spaces

Let be a locally compact group and and let consist of just one weight function that is submultiplicative and positive. Thus is a Banach space under , as we mentioned in the first section. Moreover, its dual space is the Banach space under the duality where and , and is the exponential conjugate of . The aim of this section is investigating an arbitrary intersection of the weighted -spaces, where runs through a subset of . It is performed in a similar way to the structure of and introduced in [23]. Since the dual of each -space may be participated in , then the structure of the dual of leads us to include it in our definition. Also our expectation of the behavior of this space as a Banach algebra under convolution necessitates us to insert . We turn the attention to this fact that is a Banach algebra under convolution whenever is submultiplicative. It justifies the assumption of submultiplicativity of . All these reasons justify that this space should be defined in a slightly different way from . We first introduce the space , where . Set where , for each . Then the function defined by is clearly a norm on . Furthermore, we have the next result that is in fact a partial case of the classical results on interpolation spaces; see [31].

Proposition 14. *Let be a locally compact group and a positive weight function on and . Then is a Banach space under . *

Let us recall from [32, Theorem 1] that (resp., ), for some if and only if is discrete (resp., compact). Similar arguments can be applied to get the same consequences in the weighted case.

Proposition 15. *Let be a discrete group and a positive submultiplicative weight function on and . Then , as Banach spaces. Moreover, , for each .*

*Proof. *Let . If , since is submultiplicative, then for each we have
Thus and , and consequently . It follows that and also . Now let . We first show that . Again the submultiplicativity of yields that
It follows that , and so . Also the explanation preceding the proposition implies that . Consequently and .

Proposition 16. *Let be a compact group and a positive submultiplicative weight function on and . Then the following assertions hold. *(i)If , then , as Banach spaces. (ii)If , then , as Banach spaces.

*Proof. * To get the result, it is sufficient to show that . First let and . Since is also a positive submultiplicative weight on , thus there is a positive constant such that , for each [24, Proposition 1.16]. By normalizing Haar measure on appropriately, we may assume that and thus
It follows that , and so . Also
Now let and . By some easy calculations we have
Thus, , and so . It follows that
Also
It is obtained by some similar arguments above that
Also , for each , and so the result is provided.

Corollary 17. *Let be a compact group and a positive submultiplicative weight function on and . Consider the following. *(i)*If , then *(ii)*If , then . *(iii)*. *

##### 5.1. The Banach Space

For a locally compact group and , set Let be a positive submultiplicative weight function on . Similarly to our recent work [23], we introduce by as a subspace of . Then is obviously a norm on . The main purpose of the present section is describing the properties of as a Banach space under the norm function . We will discuss first Proposition 2.2 in [23] for that is in fact a partial usage of the Riesz convexity Theorem [33, Theorem 13.19].

Proposition 18. *Let be a locally compact group and a positive submultiplicative weight function on and . Then
**
and for each and , one has
*

*Proof. *Let . Then
Thus, [23, Proposition 2.2] implies that
Hence,
and since and , then
and the proof is complete.

The next proposition shows an intimate relation between the spaces , whenever runs in an arbitrary subset of . The proof is immediate.

Proposition 19. *Let be a locally compact group and a subset of . Then the following assertions hold. *(1)If , then . (2)If and , then . (3)If and , then . (4)If and , then .

Using similar tools to the proof of [23, Lemma 3.1], it is obtained that and also . It follows that [23, Theorem 3.2] is also valid for the weighted case. It is given in the next result.

Theorem 20. *Let be a locally compact group and a subset of . Then
**
and all are equal to . Furthermore, is a Banach space under the following norm:
*

*Note.* In [23, Example 2.4] and also the explanation after [23, Proposition 2.3] of our recent paper, there are four misprints. All four inclusions have been printed in reverse. We correct them as follows. Suppose that is a locally compact group and . Then
Also in the example,

#### 6. as a Banach Algebra under Convolution Product

Let be a locally compact group and a positive submultiplicative weight function on and . It is appropriate to recall from the first section that is a Banach algebra under convolution product if and only if is equivalent to a submultiplicative weight function. Furthermore, we provided some satisfactory results for closedness of under convolution, in the case where . According to these results, it also is noticeable to know that is always closed under convolution. It is provided in the next proposition.

Proposition 21. *Let be a locally compact group and a positive submultiplicative weight function on and . Then is a Banach algebra under convolution product and norm .*

*Proof. *We first show that is a Banach algebra, for each . If and , then
Now let and . Since is submultiplicative, by [11, Theorem 3.1], and so
Also