Convolution Algebraic Structures Defined by Hardy-Type Operators

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 R). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spaces Lp ω (R+) for p ≥ 1. 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.


Introduction
Let   (R + ) be the set of Lebesgue -integrable (class of) functions , that is, a measurable function  ∈   (R + ) when for 1 <  < ∞; that is, the so-called Hardy operator H, defined by is a bounded operator on   (R + ) with ‖H‖ ≤ /( − 1) for 1 <  < ∞.
Here, we concern to functions  such that the Hardy-type operator H  , given by is a bounded operator in   (R + ), that is, a weighted inequality of Hardy holds for 1 <  < ∞.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 H  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 H  (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    (R + ) for  ≥ 1, (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 T   (R + ) (for 1 <  < ∞) and to show that they are modules for the corresponding Banach algebras In the particular case of  =   for some  > 0, we obtain that the fractional Sobolev spaces are modules for the corresponding Banach algebras T 1   (R + ) (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)).
Given  ≥ 1, it is said that   ≥ 1 is its conjugate exponent if 1/ + 1/  = 1.For  = 1, 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.

Convolution and Hardy-Type Operators
Given  : R + → R + as a measurable function and 1 ≤  < ∞, let    (R + ) be the set of weighted Lebesgue -integrable functions , that is,  is a measurable function and In the case () = 1 for  > 0, we simplify this notation and write   (R + ) and ‖ ‖  as in Section 1.
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 H  and its adjoint.
Theorem 3. Let  be a nonnegative measurable function, and let 1 <  < ∞.Then  satisfies the (HC)  condition if and only if  satisfies the (dHC)   condition for   the conjugate exponent of .
Proof.Suppose that  satisfies the (HC)  condition.Take  ∈ Let  ∈   (R + ).Then where Fubini's theorem has been applied in the first equality, Hölder's inequality in the second one, and the (HC)  condition in the third one.This implies that  ∘  ∈    (R + ), ‖ ∘ ‖   ≤  , ‖‖   , , and  satisfies the (dHC)   condition.Similarly, we prove the converse result.
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  ∈ (1, ).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 ∫  0 () > 0 for all  > 0, and there exists  ∈ R such that ess sup for some  > 1 and 1 <  < .Then where  =  (0,∞) * ; that is, the function  satisfies the (HC)  condition and where  −  (−1)/ ) 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 ℎ 1−/ () =  1−/ for  > 0. In the next lemma, we give some properties of the function  , ().
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 (HC)  and (dHC)  ) and in the Appendix (conditions (DC), (DIC), and (AMC)  ).

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 D + the set of C (∞) functions with compact support on [0, ∞).We write by supp() as the usual support of the function  and the condition 0 ∈ supp() is equivalent to suppose that the function  is not identically zero on [0, ) for all  > 0.
(i) Then    : D + → D + is an injective, linear, and continuous homomorphism such that (ii) The map    extends to a linear and continuous map from  1  (R + ) to  1 (R + ), which we denote again by (56) see [8] for more details.
(ii) Given  > 0 and  ∈ C, take  =     ; we have D  = D + and (59) Under some conditions of , some Banach algebras under the convolution product may be considered as shown in 10. defines an algebra norm on D  for the convolution product * and also for *  .We denote by T  (R + ) the Banach space obtained as the completion of D  in the norm ‖ ⋅ ‖ 1, , and then we have These Banach algebras T  () are the algebras for which we want to establish the module versus algebra relation.If they are somehow the analogues of  1 (R + ), we are going to define the Banach spaces that will act as the analogues of   (R + ), but we need some tools to do this construction.
From now on, we consider  ∈  1 loc (R + ) as a nonnegative function such that 0 ∈ supp() and  =  *  (0,∞) .Let 1 <  < ∞ and suppose that  verifies the (dHC)  condition.Take  ∈    (R + ).The function    , given by belongs to   (R + ); moreover,    is a bounded operator,    : Then for every  ∈ T   (R + ), the norm is given by With these ideas, it is easy to show that the continuous inclusion T   (R + ) →   (R + ) holds.
Examples.(i) For  =   , we write T 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 T  () (R + ) for  ≥ 1 seems to be new; see Corollary 16.The case where  ∈ N (weighted Sobolev spaces) and  = 1 was introduced and studied in [15] for  ∈ D + .An easy consequence of Theorem 1 and from the embedding of T   (R + ) →   (R + ) for  ≥ 1, we get the next corollary.

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.
where   is the Weyl fractional derivation.
Final Comments.Under some conditions of a nonnegative function  ∈  1 loc (R + ), 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.
where   is the conjugate exponent of .
(3) It seems to be natural that reflexivity and interpolation properties hold in Banach spaces T   (R + ) for 1 <  < ∞.A.2.The Decreasing Integral Condition.Let  ∈  1 loc (R + ) be a nonnegative function.We say that  satisfies the decreasing