#### Abstract

The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products on ). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spaces for . We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

#### 1. Introduction

Let be the set of Lebesgue -integrable (class of) functions , that is, a measurable function when for . The classical Hardy inequality establishes that for ; that is, the so-called Hardy operator , defined by is a bounded operator on with for .

Let be the set of locally integrable functions. Given the (usual) convolution product is defined by

Given and , then and for . Note that the Hardy operator may be written in the following way: and the Hardy inequality may be written by where is the characteristic function in the interval . In fact, it is also known that if for , then for ; see [1, Theorem 329]. From this point, there exists a wide literature about weighted inequalities of Hardy-type, in particular we mention monographs [2–4] and also [5–7].

Here, we concern to functions such that the Hardy-type operator , given by is a bounded operator in , that is, a weighted inequality of Hardy-type holds for . This kind of inequality may be considered as a weighted inequality for Hardy-Volterra integral operators; see [3, Section 9.B] and [6, Section 4]. Under some sufficient conditions (about the integrability of ), it is possible to conclude that the operator is bounded; see Theorem 5. The proof of this result is short and elegant and is inspired by the original Hardy inequality’s proof (see [3, page 24]).

We use the boundedness of operator (or its adjoint) to show our main result in the Section 3 Theorem 13. In Section 4, we apply our results to some concrete function spaces which, in fact, are modules for certain Banach algebras, in particular weighted Sobolev spaces, weighted fractional Sobolev spaces, and scattering Sobolev spaces. We also give some final remarks and comments about further studies.

In the Appendix, we present some new results in weighted Lebesgue spaces for , (with satisfying some integrability conditions, as the doubling condition or the Ariño-Muckenhoupt condition; see Theorems A.2 and A.9). These results are also essential in the proof of Theorem 13.

As we have commented, our principal aim in this paper is to introduce some Banach spaces (for ) and to show that they are modules for the corresponding Banach algebras : given and , then (Theorem 13(i)). In the particular case of for some , we obtain that the fractional Sobolev spaces () are modules for the corresponding Banach algebras (Corollary 16). Similar results hold for other convolution products, as the dual convolution product , as follows: (Corollary 12) and the cosine convolution product , as follows: (Theorem 13(ii)).

Note that subalgebras (contained in and depending of a function ) were recently introduced in [8]. Some aspects of these Banach algebras (for with ) were studied in [9, 10]. These algebras (for the convolution) are canonical to define some algebra homomorphisms whose kernels are -convoluted semigroups; see [8, Theorem 5.5]. Roughly speaking, given a Banach space and the set of linear and bounded operators, a -convoluted semigroup, , may be thought as a “regularization by ” of a -semigroup (possibly unbounded operators) acting on , as follows:

For , the Banach algebra was already introduced in [9] where the authors gave its connection with -times integrated semigroups.

Given , it is said that is its conjugate exponent if . For , we follow the usual convention . In many occasions throughout this paper, we will use the variable constant convention, in which denotes a constant which may not be the same in different lines. Subindexes in the constant will emphasize that it depends on parameters or functions.

#### 2. Convolution and Hardy-Type Operators

Given as a measurable function and , let be the set of weighted Lebesgue -integrable functions , that is, is a measurable function and (in fact is formed by a class of functions which are equal except on Lebesgue null sets). For , is the set of Lebesgue measurable (class of) functions such that

In the case for , we simplify this notation and write and as in Section 1.

If , a.e. for and , then and for and ; if a.e. for and , then and for and ; see [11]. We prove similar inequalities in the next theorem.

We use the following notation. Let , , and be three Banach algebras and a binary operation . We write to mean that there exists such that for all and .

Theorem 1. *Let be a nonnegative and nondecreasing a.e. function and . Then *(i)*;
*(ii)*. *

* Proof. *(i) Let and ; then
where we have used Minkowski’s integral inequality.

(ii) Let and . We use similar ideas as in (i) to obtain
and we conclude the result.

As we have mentioned in Section 1, the Hardy inequality has been deeply studied, from the original one [12] to recent works, see; for example, the monograph [2] and the references therein. In [2, Theorem 2.10], authors characterize some particular functions [2, Definition 2.5] such that inequality (20) holds. See also [4, Section ]. Note that our approach does not fall into all these studies.

*Definition 2. *Let , let be a nonnegative measurable function, and let .(i)We say that satisfies the condition (Hardy-type condition) if there exists such that
(ii)We say that satisfies the condition (dual Hardy-type condition) if there exists such that

For , inequality (20) holds for any measurable and positive function . Similarly, for , inequality (21) holds for any measurable and positive function (without additional conditions). However, for and , inequality (20) does not hold: take .

The products and are dual convolution products in the following sense: the equality holds for some “good” functions , and. In fact the following theorem may be present in terms of the boundedness of the Hardy-type operator and its adjoint.

Theorem 3. * Let be a nonnegative measurable function, and let . Then satisfies the condition if and only if satisfies the condition for the conjugate exponent of . *

*Proof. *Suppose that satisfies the condition. Take and let

Let . Then
where Fubini’s theorem has been applied in the first equality, Hölder’s inequality in the second one, and the condition in the third one. This implies that , , and satisfies the condition. Similarly, we prove the converse result.

*Example 4. *(i) It is well-known that the function for satisfies the ; that is,
for ; in fact; the constant is optimal in this inequality, see [1, Theorem 329].

By Theorem 3, the function also verifies for . In fact, this is a well-known result which may be found in [1, Theorem 329, page 245],

Also the constant is optimal in the above inequality ([1, Theorem 329, page 245]).

(ii) The characteristic function satisfies the for . Note that for , and
for . It is clear that the characteristic function does not satisfy the for any .

(iii) Exponential functions ( for ) do not satisfy the for any . In fact, we check that do not satisfy the for any . Take , we get and

Note that for and

for . Then there does not exist such that for every .

Now, take and . Then and for . Note that and there does not exist such that for any .

The next theorem is a particular case of [6, Theorem 4.4]: the condition is the condition (4.7) given in [6, Theorem 4.4] for . We have decided to include this proof to avoid the lack of completeness of the paper.

Theorem 5. *Let be a nonnegative measurable function with for all , and there exists such that
**
for some and . Then
**
where ; that is, the function satisfies the (HC) condition and
*

*where , for .*

*Proof. *Take , and then
where the variable has been changed in the first, and Minkowski’s integral inequality has been used in the second step. Now, take the conjugate exponent of and apply the Hölder inequality and Fubini’s theorem to get
where we have applied the assuption that satisfies (33), and we conclude that
and the theorem is proved.

Note that inequality (33) may be written in terms of product due to and for . In the next lemma, we give some properties of the function .

Lemma 6. *Take such that for some and . Then *(i)* for ; *(ii)* for and . *

*Proof. *(i) Take , and we get
for , and then . To show the part (ii), take the pair of conjugate exponents with and apply the Hölder inequality, as follows:
and then
for . We conclude that .

*Example 7. *(i) Let be function for which it is possible to find constants and such that
We get
and satisfies condition (33) for and when ; satisfies the condition (33) for and when . In all these cases, we obtain

In fact condition (43) implies that the function may be written as , where and ; then and . Particular cases are (a)the trivial case for and ; (b)the family , for . In this case, (1)if , then and ; (2)if , then ; (3)if , then and .

(ii) Let , and consider functions . Then for : take , and consider and we have proved the claim for the function . For the function , note that

(iii) The characteristic function satisfies the assumption that for , such that Note that the characteristic function verifies for .

The next theorem gives the boundedness of the operator of in -spaces. Similar results may be found in the literature, for example, [6, Theorem 4.3].

Theorem 8. * Let be a nonnegative measurable function with for all , and there exists such that
**
for some and . Then
**
where ; that is, the function satisfies the condition and
**
where for .*

*Proof. *Take and then
where we change the variable and we apply Minkowski’s integral inequality. Take as the conjugate exponent of and apply the Hölder inequality and Fubini’s theorem to get
where we have changed the variable and applied the assumption that satisfies (49). We conclude that
and the theorem is shown.

To finish this section, we present Table 1 where you may find functions and their behavior with respect to several conditions considered in this section (condition and ) and in the Appendix (conditions , , and ).

#### 3. Convolution Banach Modules

In the beginning of this section, we collect some definitions and properties that will be used throughout this section. We will denote by the set of functions with compact support on . We write by as the usual support of the function and the condition is equivalent to suppose that the function is not identically zero on for all .

Let be such that . We define the operator given by . (i) Then is an injective, linear, and continuous homomorphism such that (ii) The map extends to a linear and continuous map from to , which we denote again by such that . See [8, Theorem 2.5]. Then we define the space by and the map by see [8] for more details.

*Example 9. *(i) Take and ; the map is the Weyl fractional derivative of order , , and ; note that for , , the -iterate of usual derivation; see more details, for example, in [9, 13].

(ii) Given and , take ; we have and
See other examples in [8, Section 2].

(iii) For , it is straightforward to check that for , and
Take . Then , , and
; ; see [8, Theorem 2.10].

Under some conditions of , some Banach algebras under the convolution product may be considered as shown in 10.

Theorem 10 (see [8, Theorems 3.4 and 3.5]). * Let with satisfies the doubling condition and . Then the integral
**
defines an algebra norm on for the convolution product and also for . We denote by the Banach space obtained as the completion of in the norm , and then we have . *

These Banach algebras are the algebras for which we want to establish the module versus algebra relation. If they are somehow the analogues of , we are going to define the Banach spaces that will act as the analogues of , but we need some tools to do this construction.

From now on, we consider as a nonnegative function such that and . Let and suppose that verifies the condition. Take . The function , given by belongs to ; moreover, is a bounded operator, , which extends the operator .

*Definition 11. *Let denote the Banach space formed as the set endowed with the norm and obtained as the image of the norm of through the operator . For , we keep the notation .

In accordance with Definition 11, is a surjective isometry and is a Banach space. Let be the inverse isometry of and extends the operator defined in the beginning of this section. Note that given a function , then and there exists a unique element in (we denote by ) such that

Then for every , the norm is given by

With these ideas, it is easy to show that the continuous inclusion holds.

*Examples.* (i) For , we write instead of , for and

These families of spaces may be considered as Sobolev spaces of fractional order. There is huge literature about this topic; we only mention the monographs [13, 14] and reference therein. However, the result about the module algebra of for seems to be new; see Corollary 16. The case where (weighted Sobolev spaces) and was introduced and studied in [15]; in this case, See Corollary 15.

(ii) In the case , with and , we obtain the Banach space embedded with the norm

(iii) Take and , for . We obtain the Banach space for embedded with the norm for .

An easy consequence of Theorem 1 and from the embedding of for , we get the next corollary.

Corollary 12. *Let be a nonnegative function such that and satisfy the Hardy-type condition for some . Then *(i)* , and ;*(ii)* , and .*

Now we set the main result of this section.

Theorem 13. * Let , satisfy , , and , for such that . Then *(i)*;
*(ii)*. *

*Proof. *(i) Let . According to (59),
Therefore,
where

By Minkowski’s integral inequality, we get
where

We apply Theorems A.2 and A.9 to complete the proof and get

(ii) We use the definition of , (i) and Corollary 12.

*Remark 14. *For , with and verifying the condition, the following embeddings
hold; see Theorem 10. Note that the condition and Theorem A.9 hold for .

#### 4. Examples, Applications, and Final Remarks

In this section, we apply the main theorem of this paper, Theorem 13, to several particular examples of function which have appeared before. We also give some final remarks and comments.

##### 4.1. Weighted Sobolev Spaces

Take , , for , and the weighted Sobolev space is embedding with the norm

Corollary 15. *The Banach space is a module for the algebra (with usual convolution or the cosine convolution ) for .*

##### 4.2. Weighted Fractional Sobolev Spaces

Let , , for , and the weighted fractional Sobolev space is embedding with the norm where is the Weyl fractional derivation.

Corollary 16. *The Banach space is a module for the algebra (with usual convolution or the cosine convolution ) for .*

##### 4.3. Scattering Sobolev Spaces

Taking the function , and we consider the Banach space for embedded with the norm for .

Corollary 17. *The Banach space is a module for the algebra (with the usual convolution or the cosine convolution ) for .*

*Final Comments.* Under some conditions of a nonnegative function , we have introduced some function spaces which are module (for the usual and cosine convolution product) with respect to some function algebras. Now we comment on other points which might be considered in further studies, and we wish to mention here the following. (1) For , the Banach space could be, in fact, a Hilbert space with the inner product
(2) For , the dual Banach space of may be written by embedded with the norm