Convolution Algebraic Structures Defined by Hardy-Type Operators

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 ℝ⁺). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spaces 𝐿^𝑝_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.

1. Introduction

Let 𝐿^𝑝(ℝ⁺) be the set of Lebesgue 𝑝-integrable (class of) functions 𝑓, that is, a measurable function 𝑓∈𝐿^𝑝(ℝ⁺) when 𝑓_𝑝=^∞₀|𝑓(𝑡)|^𝑝𝑑𝑡^1/𝑝<∞, (1) for 1≤𝑝<∞. The classical Hardy inequality establishes that ^∞₀1𝑥^𝑥₀𝑓(𝑦)𝑑𝑦^𝑝𝑑𝑥^1/𝑝≤𝑝𝑓𝑝−1^𝑝,𝑓∈𝐿^𝑝ℝ⁺, (2) for 1<𝑝<∞; that is, the so-called Hardy operator ℋ, defined by (𝑥)1ℋ𝑓∶=𝑥^𝑥₀𝑓𝑦𝑑𝑦,𝑥≥0, (3) is a bounded operator on 𝐿^𝑝(ℝ⁺) with ‖ℋ‖≤𝑝/(𝑝−1) for 1<𝑝<∞.

Let 𝐿¹_loc(ℝ⁺) be the set of locally integrable functions. Given 𝑓,𝑔∈𝐿¹_loc(ℝ⁺) the (usual) convolution product ∗ is defined by 𝑓∗𝑔(𝑡)∶=^𝑡₀𝑓(𝑡−𝑠)𝑔(𝑠)𝑑𝑠,𝑡≥0. (4)

Given 𝑓∈𝐿¹(ℝ⁺) and 𝑔∈𝐿^𝑝(ℝ⁺), then 𝑓∗𝑔∈𝐿^𝑝(ℝ⁺) and 𝑓∗𝑔_𝑝≤𝑓₁𝑔_𝑝, (5) for 1≤𝑝≤∞. Note that the Hardy operator may be written in the following way: (𝑥)=1ℋ𝑓∫^𝑥₀𝜒_(0,∞)(𝑡)𝑑𝑡𝑓∗𝜒_(0,∞)(𝑥),𝑥≥0, (6) and the Hardy inequality may be written by 1𝜒_(0,∞)∗𝜒_(0,∞)𝑓∗𝜒_(0,∞)𝑝≤𝑝𝑓𝑝−1^𝑝,𝑓∈𝐿^𝑝ℝ⁺, (7) where 𝜒_(0,∞) is the characteristic function in the interval (0,∞). In fact, it is also known that if 𝑗_𝛼(𝑡)∶=𝑡^𝛼−1/(Γ(𝛼)) for 𝛼>0, then 1𝜒_(0,∞)∗𝑗_𝛼𝑓∗𝑗_𝛼𝑝≤Γ(𝛼+1)Γ1−1/𝑝Γ𝑓𝛼+1−1/𝑝^𝑝,𝑓∈𝐿^𝑝ℝ⁺, (8) for 1<𝑝<∞; 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 𝑘 such that the Hardy-type operator ℋ_𝑘, given by ℋ_𝑘𝑓=1𝜒_(0,∞)∗𝑘𝑓∗𝑘, (9) is a bounded operator in 𝐿^𝑝(ℝ⁺), that is, a weighted inequality of Hardy-type 1𝜒_(0,∞)∗𝑘𝑓∗𝑘_𝑝≤𝐶_𝑘,𝑝𝑓_𝑝,𝑓∈𝐿^𝑝ℝ⁺ (10) holds for 1<𝑝<∞. 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 𝑘), it is possible to conclude that the operator ℋ_𝑘 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_𝑘 (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 𝐿^𝑝_𝐾(ℝ⁺) for 𝑝≥1, (with 𝐾 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^𝑝_𝑘(ℝ⁺) (for 1<𝑝<∞) and to show that they are modules for the corresponding Banach algebras 𝒯¹_𝑘(ℝ⁺): given 𝑓∈𝒯^𝑝_𝑘(ℝ⁺) and 𝑔∈𝒯¹_𝑘(ℝ⁺), then 𝑓∗𝑔∈𝒯^𝑝_𝑘(ℝ⁺) (Theorem 13(i)). In the particular case of 𝑘=𝑗_𝛼 for some 𝛼>0, we obtain that the fractional Sobolev spaces 𝒯^𝑝_𝑗𝛼(ℝ⁺) (1<𝑝<∞) are modules for the corresponding Banach algebras 𝒯¹_𝑗𝛼(ℝ⁺) (Corollary 16). Similar results hold for other convolution products, as the dual convolution product ∘, as follows: 𝑓∘𝑔(𝑡)∶=^∞_𝑡𝑓(𝑠−𝑡)𝑔(𝑠)𝑑𝑠,𝑡≥0,𝑓,𝑔∈𝐿¹ℝ⁺, (11) (Corollary 12) and the cosine convolution product ∗_𝑐, as follows: 𝑓∗_𝑐1𝑔∶=2𝑓∗𝑔+𝑓∘𝑔+𝑔∘𝑓,𝑓,𝑔∈𝐿¹ℝ⁺, (12) (Theorem 13(ii)).

Note that subalgebras 𝒯¹_𝑘(ℝ⁺) (contained in 𝐿¹(ℝ⁺) and depending of a function 𝑘∈𝐿¹_loc(ℝ⁺)) were recently introduced in [8]. Some aspects of these Banach algebras (for 𝑘=𝑗_𝛼 with 𝛼>0) were studied in [9, 10]. These algebras (for the ∗ convolution) are canonical to define some algebra homomorphisms whose kernels are 𝑘-convoluted semigroups; see [8, Theorem 5.5]. Roughly speaking, given 𝑋 a Banach space and ℬ(𝑋) the set of linear and bounded operators, a 𝑘-convoluted semigroup, (𝑆_𝑘(𝑡))_𝑡≥0⊂ℬ(𝑋), may be thought as a “regularization by 𝑘” of a 𝐶₀-semigroup (𝑆(𝑡))_𝑡≥0 (possibly unbounded operators) acting on 𝑋, as follows: 𝑆_𝑘(𝑡)𝑥=^𝑡₀𝑘(𝑡−𝑠)𝑆(𝑠)𝑥𝑑𝑠,forsome𝑥∈𝑋. (13)

For 𝑘=𝑗_𝛼, the Banach algebra 𝒯¹_𝑗𝛼(ℝ⁺) was already introduced in [9] where the authors gave its connection with 𝛼-times integrated semigroups.

Given 𝑝≥1, it is said that 𝑝^′≥1 is its conjugate exponent if 1/𝑝+1/𝑝^′=1. For 𝑝=1, we follow the usual convention 𝑝^′=∞. In many occasions throughout this paper, we will use the variable constant convention, in which 𝐶 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 𝜔∶ℝ⁺→ℝ⁺ as a measurable function and 1≤𝑝<∞, let 𝐿^𝑝_𝜔(ℝ⁺) be the set of weighted Lebesgue 𝑝-integrable functions 𝑓, that is, 𝑓 is a measurable function and 𝑓_𝑝,𝜔∶=^∞₀𝜔(𝑡)𝑓(𝑡)^𝑝𝑑𝑡^1/𝑝<∞. (14) (in fact 𝐿^𝑝_𝜔(ℝ⁺) is formed by a class of functions which are equal except on Lebesgue null sets). For 𝑝=∞, 𝐿^∞_𝜔(ℝ⁺) is the set of Lebesgue measurable (class of) functions such that 𝑓_∞,𝜔∶=esssup_𝑡>0𝜔(𝑡)𝑓(𝑡)<∞. (15)

In the case 𝜔(𝑡)=1 for 𝑡>0, we simplify this notation and write 𝐿^𝑝(ℝ⁺) and ‖‖_𝑝 as in Section 1.

If 𝜔(𝑡+𝑠)≤𝐶𝜔(𝑡)𝜔(𝑠), a.e. for 𝑡,𝑠≥0 and 𝐶>0, then 𝑓∗𝑔∈𝐿^𝑝_𝜔(ℝ⁺) and 𝑓∗𝑔_𝑝,𝜔≤𝐶𝑓_1,𝜔𝑔_𝑝,𝜔, (16) for 𝑓∈𝐿¹_𝜔(ℝ⁺) and 𝑔∈𝐿^𝑝_𝜔(ℝ⁺); if 𝜔(𝑡−𝑠)≤𝐶𝜔(𝑡)𝜔(𝑠) a.e. for 𝑡≥𝑠≥0 and 𝐶>0, then 𝑓∘𝑔∈𝐿^𝑝_𝜔(ℝ⁺) and 𝑓∘𝑔_𝑝,𝜔≤𝐶𝑓_1,𝜔𝑔_𝑝,𝜔, (17) for 𝑓∈𝐿¹_𝜔(ℝ⁺) and 𝑔∈𝐿^𝑝_𝜔(ℝ⁺); see [11]. We prove similar inequalities in the next theorem.

We use the following notation. Let (𝑋,‖⋅‖_𝑋), (𝑌,‖⋅‖_𝑌), and (𝑍,‖⋅‖_𝑍) be three Banach algebras and ⊙ a binary operation ⊙∶𝑋×𝑌→𝑍. We write 𝑋⊙𝑌↪𝑍 to mean that there exists 𝑀>0 such that ‖𝑥⊙𝑦‖_𝑍≤𝑀‖𝑥‖_𝑋‖𝑦‖_𝑌 for all 𝑥∈𝑋 and 𝑦∈𝑌.

Theorem 1. Let 𝜔∶ℝ⁺→ℝ⁺ be a nonnegative and nondecreasing a.e. function and 1≤𝑝<∞. Then (i)𝐿^𝑝(ℝ⁺)∘𝐿¹_𝜔(ℝ⁺)↪𝐿^𝑝_𝜔(ℝ⁺); (ii)𝐿¹(ℝ⁺)∘𝐿^𝑝_𝜔(ℝ⁺)↪𝐿^𝑝_𝜔(ℝ⁺).

Proof. (i) Let 𝑓∈𝐿^𝑝(ℝ⁺) and 𝑔∈𝐿¹_𝜔(ℝ⁺); then 𝑓∘𝑔_𝑝,𝜔≤^∞₀^∞_𝑡𝑓(𝑠−𝑡)𝑔(𝑠)𝜔(𝑠)𝑑𝑠^𝑝𝑑𝑡^1/𝑝=𝑡↦^∞₀𝜒_(𝑡,∞)(𝑠)𝑓(𝑠−𝑡)𝜔(𝑠)𝑔(𝑠)𝑑𝑠_𝑝≤^∞₀𝑡↦𝜒_(𝑡,∞)(𝑠)𝑓(𝑠−𝑡)_𝑝𝜔(𝑠𝑔(𝑠=))𝑑𝑠_∞0𝑔(𝑠)𝜔(𝑠)^𝑠₀𝑓(𝑠−𝑡)^𝑝𝑑𝑡^1/𝑝𝑑𝑠≤𝑓_𝑝𝑔_1,𝜔, (18) where we have used Minkowski’s integral inequality.
 (ii) Let 𝑓∈𝐿¹(ℝ⁺) and 𝑔∈𝐿^𝑝_𝜔(ℝ⁺). We use similar ideas as in (i) to obtain ‖𝑓∘𝑔‖_𝑝,𝜔≤^∞₀^∞_𝑡𝜔(𝑠)𝑓(𝑠−𝑡)𝑔(𝑠)𝑑𝑠^𝑝𝑑𝑡^1/𝑝=^∞₀^∞₀𝑓(𝑢)𝑔(𝑢+𝑡)𝜔(𝑢+𝑡)𝑑𝑢^𝑝𝑑𝑡^1/𝑝≤^∞₀𝑓(𝑢)^∞_𝑢𝑔(𝑠)^𝑝𝜔^𝑝(𝑠)𝑑𝑠^1/𝑝𝑑𝑢≤𝑓₁𝑔_𝑝,𝜔, (19) 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 𝑘 [2, Definition 2.5] such that inequality (20) holds. See also [4, Section I.11.1]. Note that our approach does not fall into all these studies.

Definition 2. Let 𝑝≥1, let 𝑘∈𝐿¹_loc(ℝ⁺) be a nonnegative measurable function, and let 𝐾=𝜒_(0,∞)∗𝑘.(i)We say that 𝑘 satisfies the (HC)_𝑝 condition (Hardy-type condition) if there exists 𝐶_𝑘,𝑝>0 such that 𝑘∗𝑔_𝑝,1/𝐾≤𝐶_𝑘,𝑝𝑔_𝑝,∀𝑔∈𝐿^𝑝ℝ⁺. (20)(ii)We say that 𝑘 satisfies the (dHC)_𝑝 condition (dual Hardy-type condition) if there exists 𝐶^′_𝑘,𝑝>0 such that 𝑘∘𝑓_𝑝≤𝐶^′_𝑘,𝑝𝑓_𝑝,𝐾,∀𝑓∈𝐿^𝑝_𝐾ℝ⁺. (21)

For 𝑝=∞, inequality (20) holds for any measurable and positive function 𝑘∈𝐿¹_loc(ℝ⁺). Similarly, for 𝑝=1, inequality (21) holds for any measurable and positive function 𝑘∈𝐿¹_loc(ℝ⁺) (without additional conditions). However, for 𝑝=1 and 𝑘=𝜒_(0,∞), inequality (20) does not hold: take 𝑔(𝑦)=𝜒_(0,1)(𝑦)(1/√𝑦).

The products ∗ and ∘ are dual convolution products in the following sense: the equality 𝑘∗𝑓(^∞₀𝑡)𝑔(𝑡)𝑑𝑡=^∞₀𝑘∘𝑔(𝑠)𝑓(𝑡)𝑑𝑡 (22) holds for some “good” functions 𝑘,𝑓, and𝑔. In fact the following theorem may be present in terms of the boundedness of the Hardy-type operator ℋ_𝑘 and its adjoint.

Theorem 3. Let 𝑘 be a nonnegative measurable function, and let 1<𝑝<∞. Then 𝑘 satisfies the (HC)_𝑝 condition if and only if 𝑘 satisfies the (dHC)_𝑝′ condition for 𝑝^′ the conjugate exponent of 𝑝.

Proof. Suppose that 𝑘 satisfies the (HC)_𝑝 condition. Take 𝑓∈𝐿^𝑝′_𝐾(ℝ⁺) and let 𝑓(𝑥)∶=^∞_𝑥𝑘𝑦−𝑥𝑓𝑦𝑑𝑦=𝑘∘𝑓(𝑥),𝑥≥0. (23)
Let 𝑔∈𝐿^𝑝(ℝ⁺). Then 𝑓𝑔₁≤^∞₀^𝑦₀𝑘𝑔(𝑥𝑓𝑦≤𝑦−𝑥)𝑑𝑥𝑑𝑦_∞01∫^𝑦₀𝑘(𝜏)𝑑𝜏^𝑝𝑘∗𝑔𝑦^𝑝𝑑𝑦^1/𝑝×^∞₀^𝑦₀𝑘(𝜏)𝑑𝜏^𝑝′𝑓𝑦^𝑝′𝑑𝑦^1/𝑝′≤𝐶_𝑘,𝑝𝑔_𝑝^∞₀𝑘∗𝜒_(0,∞)𝑦𝑓𝑦^𝑝′𝑑𝑦^1/𝑝′=𝐶_𝑘,𝑝𝑔_𝑝𝑓_𝑝′,𝐾, (24) where Fubini’s theorem has been applied in the first equality, Hölder’s inequality in the second one, and the (HC)_𝑝 condition in the third one. This implies that 𝑘∘𝑓∈𝐿^𝑝′(ℝ⁺), ‖𝑘∘𝑓‖_𝑝′≤𝐶_𝑘,𝑝‖𝑓‖_𝑝′,𝐾, and 𝑘 satisfies the (dHC)_𝑝′ condition. Similarly, we prove the converse result.

Example 4. (i) It is well-known that the function 𝑗_𝛼(𝑡)∶=𝑡^𝛼−1/Γ(𝛼) for 𝛼>0 satisfies the (HC)_𝑝; that is, 1𝜒_(0,∞)∗𝑗_𝛼𝑓∗𝑗_𝛼𝑝≤Γ(𝛼+1)Γ1−1/𝑝Γ𝑓𝛼+1−1/𝑝^𝑝,𝑓∈𝐿^𝑝ℝ⁺, (25) for 1<𝑝<∞; in fact; the constant Γ(𝛼+1)Γ(1−1/𝑝)/Γ(𝛼+1−1/𝑝) is optimal in this inequality, see [1, Theorem 329].
By Theorem 3, the function 𝑗_𝛼 also verifies (dHC)_𝑝 for 1<𝑝<∞. In fact, this is a well-known result which may be found in [1, Theorem 329, page 245], ^∞₀^∞_𝑥(𝑡−𝑥)^𝛼−1Γ(𝛼)𝑓(𝑡)𝑑𝑡^𝑝≤Γ()Γ𝑑𝑥𝛼+11/𝑝Γ𝛼+1/𝑝^𝑝∞₀𝑥^𝛼𝑓Γ(𝛼+1)(𝑥)^𝑝𝑑𝑥. (26)
Also the constant Γ(𝛼+1)Γ(1/𝑝)/Γ(𝛼+1/𝑝) is optimal in the above inequality ([1, Theorem 329, page 245]).
(ii) The characteristic function 𝜒_(0,1) satisfies the (HC)_𝑝 for 1<𝑝<∞. Note that 𝜒_(0,1)∗𝜒_(0,∞)(𝑡)=min{1,𝑡} for 𝑡>0, and 1𝜒_(0,1)∗𝜒_(0,∞)𝜒_(0,1)∗𝑓^𝑝_𝑝≤¹₀1𝑡^𝑡₀𝑓(𝑠)𝑑𝑠^𝑝+𝑑𝑡_{∞1𝑡𝑡−1}𝑓(𝑠)𝑑𝑠^𝑝≤𝑑𝑡_∞01𝑡^𝑡₀𝑓(𝑠)𝑑𝑠^𝑝+𝑑𝑡_{∞1𝑡𝑡−1}𝑓(𝑠)^𝑝≤𝑝𝑑𝑠𝑑𝑡𝑝−1^𝑝𝑓^𝑝_𝑝+¹₀𝑓(𝑠)^𝑝𝑠𝑑𝑠+^∞₁𝑓(𝑠)^𝑝𝑑𝑠≤𝑀_𝑝𝑓^𝑝_𝑝 (27) for 𝑓∈𝐿^𝑝(ℝ⁺). It is clear that the characteristic function 𝜒_(1,∞) does not satisfy the (HC)_𝑝 for any 1<𝑝<∞.
(iii) Exponential functions {𝑒_𝜆∶0≠𝜆∈ℝ} (𝑒_𝜆(𝑡)∶=𝑒^𝜆𝑡 for 𝑡>0) do not satisfy the (HC)_𝑝 for any 1<𝑝<∞. In fact, we check that {𝑒_𝜆∶0≠𝜆∈ℝ} do not satisfy the (dHC)_𝑝 for any 1<𝑝<∞. Take 𝜆,𝜇>0, we get 𝑒_−𝜆∘𝑒_−𝜇=(1/(𝜆+𝜇))𝑒_−𝜇 and 𝑒_−𝜆∘𝑒_−𝜇𝑝=1𝑒𝜆+𝜇^−𝜇𝑝=11𝜆+𝜇𝜇𝑝^1/𝑝. (28)

Note that 𝑒_−𝜆∗𝜒_(0,∞)(𝑡)=(1−𝑒^−𝜆𝑡)/𝜆 for 𝑡>0 and 𝑒_−𝜇^𝑝_{𝑝,𝑒−𝜆∗𝜒(0,∞)}=1𝜆^𝑝∞₀𝑒^{−𝜇𝑝𝑡}1−𝑒^{−𝜆𝑡𝑝}≤𝑑𝑡_∞0𝑒^{−𝜇𝑝𝑡}𝑡^𝑝Γ𝑑𝑡=𝑝+1𝜇𝑝^𝑝+1 (29)

for 𝜇>0. Then there does not exist 𝐶>0 such that 11𝜆+𝜇𝜇𝑝^1/𝑝Γ≤𝐶𝑝+1^1/𝑝𝜇𝑝^1+1/𝑝 (30) for every 𝜇>0.

Now, take 𝜆>0 and 𝜇>𝜆. Then 𝑒_𝜆∘𝑒_−𝜇=(1/(𝜇−𝜆))𝑒_−𝜇 and 𝑒_𝜆∗𝜒_(0,∞)(𝑡)=(𝑒^𝜆𝑡−1)/𝜆 for 𝑡>0. Note that 𝑒_−𝜇^𝑝_{𝑝,𝑒𝜆∗𝜒(0,∞)}=1𝜆^𝑝∞₀𝑒^{−𝜇𝑝𝑡}𝑒^𝜆𝑡−1^𝑝≤1𝑑𝑡𝜆^𝑝∞₀𝑒^{−(𝜇−𝜆)𝑝𝑡}1𝑑𝑡=𝜆^𝑝1𝑝,𝜇−𝜆 (31) and there does not exist 𝐶^′>0 such that 11𝜇−𝜆𝜇𝑝^1/𝑝≤𝐶^′1𝜆1𝜇−𝜆𝑝^1/𝑝, (32) for any 𝜇>𝜆.

The next theorem is a particular case of [6, Theorem 4.4]: the condition 𝐴_𝑘,𝑝(𝑟)<∞ is the condition (4.7) given in [6, Theorem 4.4] for 𝑟∈(1,𝑝). We have decided to include this proof to avoid the lack of completeness of the paper.

Theorem 5. Let 𝑘 be a nonnegative measurable function with ∫^𝜀₀𝑘(𝑥)𝑑𝑥>0 for all 𝜀>0, and there exists 𝑟∈ℝ such that esssup_{𝑠∈(0,∞)}𝑠^{(𝑟−1)/𝑝∞}_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^𝑝−𝑟𝑑𝑢^1/𝑝∶=𝐴_𝑘,𝑝(𝑟)<∞, (33) for some 𝑝>1 and 1<𝑟<𝑝. Then 𝑔∗𝑘_𝑝,1/𝐾≤𝐴_𝑘,𝑝(𝑟)𝑔_𝑝,∀𝑔∈𝐿^𝑝ℝ⁺, (34) where 𝐾=𝜒_(0,∞)∗𝑘; that is, the function 𝑘 satisfies the (HC)𝑝 condition and ℋ_{𝑘ℬ(𝐿𝑝(ℝ+))}≤inf_1<𝑟<𝑝𝑝−1𝑝−𝑟^1/𝑝′𝐴_𝑘,𝑝(𝑟), (35) where ℋ_𝑘(𝑔)=(1/𝜒_(0,∞)∗𝑘)(𝑘∗𝑔), for 𝑔∈𝐿^𝑝(ℝ⁺).

Proof. Take 𝑔∈𝐿^𝑝(ℝ⁺), and then 𝑔∗𝑘_𝑝,1/𝐾=^∞₀1∫^𝑥₀𝑘(𝑟)𝑑𝑟^𝑝𝑥0𝑘(𝑥−𝑠)𝑔(𝑠)𝑑𝑠^𝑝𝑑𝑥^1/𝑝=¹₀𝑘(𝑥(1−𝑡))∫^𝑥₀𝑘(𝑟)𝑔()𝑑𝑟𝑥𝑡𝑥𝑑𝑡_𝑝≤¹₀𝑥𝑘(𝑥(1−𝑡))∫^𝑥₀𝑘(𝑟)𝑔()𝑑𝑟𝑥𝑡_𝑝=𝑑𝑡_10∞0𝑠𝑘((𝑠/𝑡)(1−𝑡))𝑡∫₀^𝑠/𝑡𝑘(𝑟)𝑑𝑟^𝑝𝑔(𝑠)^𝑝𝑑𝑠𝑡^1/𝑝𝑑𝑡, (36) where the variable has been changed (𝑠=𝑥𝑡) in the first, and Minkowski’s integral inequality has been used in the second step. Now, take 𝑝^′ the conjugate exponent of 𝑝 and apply the Hölder inequality and Fubini’s theorem to get 𝑔∗𝑘_𝑝,1/𝐾≤¹₀𝑡^{−(𝑟−1)/𝑝}𝑡^{(𝑟−1)/𝑝}×^∞₀𝑠𝑘((𝑠/𝑡)(1−𝑡))𝑡∫₀^𝑠/𝑡𝑘(𝑟)𝑑𝑟^𝑝𝑔(𝑠)^𝑝𝑑𝑠𝑡^1/𝑝≤𝑑𝑡₁₀𝑡^{−𝑝′(𝑟−1)/𝑝}𝑑𝑡^1/𝑝′×^∞₀𝑔(𝑠)^𝑝𝑠^𝑟−1∞_𝑠(𝑘(𝑢−𝑠))^𝑝∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^𝑝−𝑟𝑑𝑢𝑑𝑠^1/𝑝≤𝑝−1𝑝−𝑟^1/𝑝′𝐴_𝑘,𝑝(𝑟)^∞₀𝑔(𝑠)^𝑝𝑑𝑠^1/𝑝, (37) where we have applied the assuption that 𝑘 satisfies (33), and we conclude that ℋ_{𝑘ℬ(𝐿𝑝(ℝ+))}≤inf_1<𝑟<𝑝𝑝−1𝑝−𝑟^1/𝑝′𝐴_𝑘,𝑝(𝑟), (38) and the theorem is proved.

Note that inequality (33) may be written in terms of ∘ product due to 𝑘^𝑝∘ℎ_{1−𝑟/𝑝}𝜒_(0,∞)∗𝑘^𝑝(𝑠)=^∞_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^𝑝−𝑟𝑑𝑢,𝑠≥0, (39) and ℎ_{1−𝑟/𝑝}(𝑢)=𝑢^{1−𝑟/𝑝} for 𝑢>0. In the next lemma, we give some properties of the function 𝐴_𝑘,𝑝(𝑟).

Lemma 6. Take 𝑘 such that 𝐴_𝑘,𝑝(𝑟)<∞ for some 𝑝>1 and 1<𝑟<𝑝. Then (i)𝐴_𝑘,𝑝(𝑟^′)≤𝐴_𝑘,𝑝(𝑟) for 𝑝>𝑟^′>𝑟>1; (ii)𝐴_𝑘,𝑞(𝑟)≤(𝐴_𝑘,𝑝(𝑟))/(𝑟−1)^{(𝑝−𝑞)/𝑞𝑝} for 1<𝑞<𝑝 and 𝑟>1.

Proof. (i) Take 𝑟^′>𝑟, and we get 𝑠^{(𝑟′−1)/𝑝∞}_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^{𝑝−𝑟′}𝑑𝑢^1/𝑝≤𝑠^{(𝑟−1)/𝑝∞}_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑥^{𝑟′−𝑟}𝑢^{𝑝−𝑟′}𝑑𝑢^1/𝑝, (40) for 𝑠>0, and then 𝐴_𝑘,𝑝(𝑟^′)≤𝐴_𝑘,𝑝(𝑟). To show the part (ii), take the pair of conjugate exponents (𝑝/𝑞,𝑝/(𝑝−𝑞)) with 𝑝>𝑞 and apply the Hölder inequality, as follows: ^∞_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑞𝑢^𝑞−𝑟≤𝑑𝑢_∞𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^𝑝−𝑟𝑑𝑢^𝑞/𝑝∞_𝑠1𝑢^𝑟𝑑𝑢^{(𝑝−𝑞)/𝑝}, (41) and then 𝑠^{(𝑟−1)/𝑞∞}_𝑠𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑞𝑢^𝑞−𝑟𝑑𝑢^1/𝑞≤𝑠^{(𝑟−1)/𝑝}(𝑟−1)^{(𝑝−𝑞)/𝑞𝑝∞𝑠}𝑘(𝑢−𝑠)∫^𝑢₀𝑘(𝑥)𝑑𝑥^𝑝𝑢^𝑝−𝑟𝑑𝑢^1/𝑝, (42) for 𝑠>0. We conclude that 𝐴_𝑘,𝑞(𝑟)≤𝐴_𝑘,𝑝(𝑟)/𝑟−1^{(𝑝−𝑞)/(𝑞𝑝)}.

Example 7. (i) Let 𝑘 be function for which it is possible to find constants 0<𝑚≤𝑀 and 𝛼>0 such that 𝑚𝑡^𝛼−1Γ(𝛼)(𝑡)𝑡≤𝑘≤𝑀^𝛼−1Γ(𝛼). (43) We get 𝑠^{(𝑟−1)/𝑝∞}_𝑠𝛼^𝑝(𝑢−𝑠)^{(𝛼−1)𝑝}𝑢^𝛼𝑝𝑢^𝑝−𝑟𝑑𝑢^1/𝑝=𝛼^∞₀𝑥^{(𝛼−1)𝑝}(1+𝑥)^{𝛼𝑝−𝑝+𝑟}𝑑𝑥^1/𝑝Γ()Γ()=𝛼𝛼−1𝑝+1𝑟−1Γ𝛼𝑝−𝑝+𝑟^1/𝑝, (44) and 𝑘 satisfies condition (33) for 𝑟>1 and 1≤𝑝≤1/(1−𝛼) when 0<𝛼<1; 𝑘 satisfies the condition (33) for 𝑟>1 and 1≤𝑝 when 𝛼≥1. In all these cases, we obtain 𝑚𝑀𝛼Γ(𝛼−1)𝑝+1Γ(𝑟−1)Γ𝛼𝑝−𝑝+𝑟^1/𝑝≤𝐴_𝑘,𝑝(𝑟)≤𝑀𝑚𝛼Γ(𝛼−1)𝑝+1Γ(𝑟−1)Γ(𝛼𝑝−𝑝+𝑟)^1/𝑝, (45)

In fact condition (43) implies that the function 𝑘 may be written as 𝑘=ℎ𝑗_𝛼, where ℎ∈𝐿^∞(ℝ⁺) and inf_𝑡≥0ℎ(𝑡)≱0; then 𝑚=inf_𝑡≥0ℎ(𝑡) and 𝑀=‖ℎ‖_∞. Particular cases are (a)the trivial case 𝑘∶=𝑗_𝛼 for 𝛼>0 and 𝐴_{𝑗𝛼,𝑝}(𝑟)=𝛼((Γ((𝛼−1)𝑝+1)Γ(𝑟−1))/Γ(𝛼𝑝−𝑝+𝑟))^1/𝑝; (b)the family 𝑘(𝑡)∶=((𝐴𝑡+𝐵)/(𝐶𝑡+𝐷))^𝛾𝑗_𝛼(𝑡), for 𝐴,𝐵,𝐶,𝐷,𝛼,𝛾>0. In this case, (1)if 𝐴𝐷−𝐵𝐶<0, then 𝑚=(𝐴/𝐶)^𝛾 and 𝑀=(𝐵/𝐷)^𝛾; (2)if 𝐴𝐷−𝐵𝐶=0, then 𝑘=(𝐵/𝐷)^𝛾𝑗_𝛼; (3)if 𝐴𝐷−𝐵𝐶>0, then 𝑚=(𝐵/𝐷)^𝛾 and 𝑀=(𝐴/𝐶)^𝛾.

(ii) Let 𝜆>0, and consider functions 𝑒_𝜆,𝑒_−𝜆. Then 𝐴_{𝑒−𝜆,𝑝}(𝑝)=𝐴_{𝑒𝜆,𝑝}(𝑝)=∞ for 𝑝≥1: take 𝑠>0, and consider 𝑠^{1−1/𝑝∞}_𝑠𝑒^{𝜆(𝑢−𝑠)𝑝}∫^𝑢₀𝑒^𝜆𝑟𝑑𝑟^𝑝𝑑𝑢^1/𝑝=𝜆𝑠^1−1/𝑝𝑒^{−𝜆𝑠∞}_𝑠11−𝑒^{−𝜆𝑢𝑝}𝑑𝑢^1/𝑝=∞, (46) and we have proved the claim for the function 𝑒_𝜆. For the function 𝑒_−𝜆, note that lim_𝑠→∞𝑠^{1−1/𝑝∞}_𝑠𝑒^{−𝜆(𝑢−𝑠)𝑝}∫^𝑢₀𝑒^−𝜆𝑟𝑑𝑟^𝑝𝑑𝑢^1/𝑝=lim_𝑠→∞𝑠^1−1/𝑝𝜆∫^∞₀𝑒^{−𝜆𝑥𝑝}𝑑𝑥1−𝑒^{−𝜆(𝑥+𝑠)𝑝1/𝑝}=∞. (47)

(iii) The characteristic function 𝜒_(0,1) satisfies the assumption that 𝐴_{𝜒(0,1),𝑝}(𝑝)=∞ for 𝑝≥1, such that sup_𝑠≥1𝑠^{1−1/𝑝∞}_𝑠𝜒_(0,1)(𝑢−𝑠)∫^𝑢₀𝜒_(0,1)(𝑟)𝑑𝑟^𝑝𝑑𝑢^1/𝑝=sup_𝑠≥1𝑠^1−1/𝑝∫_𝑠^𝑠+1𝑑𝑢^1/𝑝=∞. (48) Note that the characteristic function 𝜒_(1,∞) verifies ∫^𝜀₀𝜒_(1,∞)(𝑠)𝑑𝑠=0 for 0<𝜀≤1.

The next theorem gives the boundedness of the operator of 𝑓↦𝑘∘𝑓 in 𝐿^𝑝-spaces. Similar results may be found in the literature, for example, [6, Theorem 4.3].

Theorem 8. Let 𝑘 be a nonnegative measurable function with ∫^𝜀₀𝑘(𝑥)𝑑𝑥>0 for all 𝜀>0, and there exists 𝑟∈ℝ such that esssup_{()𝑠∈0,∞}𝑠^{(𝑟−1)/𝑝}∫^𝑠₀𝑘(𝑥)∫𝑑𝑥^𝑠0𝑘^𝑝(𝑠−𝑢)𝑢^𝑝−𝑟𝑑𝑢^1/𝑝∶=𝐵_𝑘,𝑝(𝑟)<∞, (49) for some 𝑝>1 and 𝑝+1>𝑟>𝑝. Then 𝑘∘𝑓_𝑝≤𝐵_𝑘,𝑝(𝑟)𝑓_𝑝,𝐾∀𝑓∈𝐿^𝑝_𝐾ℝ⁺, (50) where 𝐾=𝜒_(0,∞)∗𝑘; that is, the function 𝑘 satisfies the (dHC)_𝑝 condition and 𝑇^′_{𝑘ℬ(𝐿𝑝𝐾(ℝ+),𝐿𝑝(ℝ+))}≤inf_{𝑝+1>𝑟>𝑝}𝑝−1𝑟−𝑝^1/𝑝′𝐵_𝑘,𝑝(𝑟), (51) where 𝑇^′_𝑘(𝑓)=𝑘∘𝑓 for 𝑓∈𝐿^𝑝_𝐾(ℝ⁺).

Proof. Take 𝑓∈𝐿^𝑝_𝐾(ℝ⁺) and then 𝑘∘𝑓_𝑝≤^∞₀^∞_𝑥𝑘(𝑠−𝑥)𝑓(𝑠)𝑑𝑠^𝑝𝑑𝑥^1/𝑝=^∞₁𝑘(𝑥(𝑡−1))𝑓(𝑥𝑡)𝑥𝑑𝑡_𝑝≤^∞₁𝑘(𝑥(𝑡−1))𝑓(𝑥𝑡)𝑥_𝑝=𝑑𝑡_∞1∞0𝑘(𝑥(𝑡−1))|𝑓(𝑥𝑡)|𝑥^𝑝𝑑𝑥^1/𝑝𝑑𝑡, (52) where we change the variable 𝑠=𝑥𝑡 and we apply Minkowski’s integral inequality. Take 𝑝^′ as the conjugate exponent of 𝑝 and apply the Hölder inequality and Fubini’s theorem to get 𝑘∘𝑓_𝑝≤^∞₁𝑡^{−(𝑟−1)/𝑝}𝑡^{(𝑟−1)/𝑝}×^∞₀𝑘(𝑥(𝑡−1))𝑓(𝑥𝑡)𝑥^𝑝𝑑𝑥^1/𝑝≤𝑑𝑡_∞1𝑡^{−(𝑟−1)𝑝′/𝑝}𝑑𝑡^1/𝑝′×^∞₀𝑓(𝑠)^𝑝𝑠^𝑟−1𝑠₀(𝑘(𝑠−𝑢))^𝑝𝑢^𝑝−𝑟𝑑𝑢𝑑𝑠^1/𝑝≤𝑝−1𝑟−𝑝^1/𝑝′𝐵_𝑘,𝑝(𝑟)^∞₀𝑓(𝑠)^𝑝𝑠₀𝑘(𝑥)𝑑𝑥^𝑝𝑑𝑠^1/𝑝, (53) where we have changed the variable and applied the assumption that 𝑘 satisfies (49). We conclude that 𝑇^′_{𝑘ℬ(𝐿𝑝(ℝ+))}≤inf_{𝑝+1>𝑟>𝑝}𝑝−1𝑟−𝑝^1/𝑝′𝐵_𝑘,𝑝(𝑟), (54) 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)_𝑝 and (dHC)_𝑝) and in the Appendix (conditions (DC), (DIC), and (AMC)_𝑝).

Table 1 

3. Convolution Banach Modules 𝒯^𝑝_𝑘(ℝ⁺)

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(𝑘) as the usual support of the function 𝑘 and the condition 0∈supp(𝑘) is equivalent to suppose that the function 𝑘 is not identically zero on [0,𝜀) for all 𝜀>0.

Let 𝑘∈𝐿¹_loc(ℝ⁺) be such that 0∈supp(𝑘). We define the operator 𝑇^′_𝑘∶𝒟₊→𝒟₊ given by 𝑓↦𝑇^′_𝑘(𝑓)∶=𝑘∘𝑓. (i) Then 𝑇^′_𝑘∶𝒟₊→𝒟₊ is an injective, linear, and continuous homomorphism such that 𝑇^′_𝑘𝑓∘𝑔=𝑓∘𝑇^′_𝑘𝑔,𝑓,𝑔∈𝒟₊. (55) (ii)  The map 𝑇^′_𝑘 extends to a linear and continuous map from 𝐿¹_𝐾(ℝ⁺) to 𝐿¹(ℝ⁺), which we denote again by 𝑇^′_𝑘∶𝐿¹_𝐾(ℝ⁺)→𝐿¹(ℝ⁺) such that ‖𝑇^′_𝑘‖≤1.  See [8, Theorem 2.5]. Then we define the space 𝒟_𝑘 by 𝒟_𝑘∶=𝑇^′_𝑘(𝒟₊) and the map 𝑊_𝑘∶𝒟_𝑘→𝒟₊ by 𝑓(𝑡)=𝑇^′_𝑘𝑊_𝑘𝑓𝑡)=_∞𝑡(𝑘(𝑠−𝑡)𝑊_𝑘𝑓(𝑠)𝑑𝑠,𝑓∈𝒟_𝑘,𝑡≥0; (56) see [8] for more details.

Example 9. (i) Take 𝛼>0 and 𝑘=𝑗_𝛼; the map 𝑊_𝑗𝛼 is the Weyl fractional derivative of order 𝛼, 𝑊_𝛼, and 𝒟_𝑗𝛼=𝒟₊; note that for 𝛼∈ℕ, 𝑊_𝛼=(−1)^𝛼𝑑^𝛼/𝑑𝑡^𝛼, the 𝛼-iterate of usual derivation; see more details, for example, in [9, 13].
(ii) Given 𝛼>0 and 𝑧∈ℂ, take 𝑘=𝑒_𝑧𝑗_𝛼; we have 𝒟_𝑘=𝒟₊ and 𝑊_{𝑗𝛼𝑒𝑧}𝑓=𝑒_𝑧𝑊_𝛼𝑒_−𝑧𝑓,𝑓∈𝒟₊. (57) See other examples in [8, Section 2].
(iii) For 𝑘=𝜒_(0,1), it is straightforward to check that 𝑇^′_𝜒(0,1)(𝑓)(𝑡)=∫_𝑡^𝑡+1𝑓(𝑠)𝑑𝑠 for 𝑓∈𝒟₊, 𝒟_𝜒(0,1)=𝒟₊ and 𝑊_𝜒(0,1)𝑓(𝑡)=−∑^∞_𝑛=0𝑓^′(𝑡+𝑛),𝑓∈𝒟₊,𝑡≥0. (58) Take 𝑓,𝑔∈𝒟_𝑘. Then 𝑓∗𝑔, 𝑓∘𝑔, 𝑓∗_𝑐𝑔∈𝒟_𝑘 and 𝑊_𝑘(𝑠)=𝑓∗𝑔_𝑠0𝑊_𝑘𝑔(𝑟)^𝑠_𝑠−𝑟𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)−𝑑𝑡𝑑𝑟_∞𝑠𝑊_𝑘𝑔(𝑟)^∞_𝑠𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡𝑑𝑟; (59)𝑊_𝑘(𝑓∘𝑔)=𝑓∘𝑊_𝑘𝑔;  𝑊_𝑘(𝑓∗_𝑐𝑔)=(1/2)(𝑊_𝑘(𝑓∗𝑔)+𝑓∘𝑊_𝑘𝑔+𝑔∘𝑊_𝑘𝑓); see [8, Theorem 2.10].

Under some conditions of 𝑘, 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 𝑘∈𝐿¹_loc(ℝ⁺) with 0∈supp(𝑘) satisfies the doubling condition (DC) and 𝐾=𝜒_(0,∞)∗𝑘. Then the integral 𝑓_1,𝐾∶=^∞₀𝑊_𝑘𝑓(𝑡)𝐾(𝑡)𝑑𝑡,𝑓∈𝒟_𝑘 (60) defines an algebra norm on 𝒟_𝑘 for the convolution product ∗ and also for ∗_𝑐. We denote by 𝒯_𝑘(ℝ⁺) the Banach space obtained as the completion of 𝒟_𝑘 in the norm ‖⋅‖_1,𝐾, and then we have 𝒯_𝑘(ℝ⁺)↪𝐿¹(ℝ⁺).

These Banach algebras 𝒯_𝑘(𝜏) are the algebras for which we want to establish the module versus algebra relation. If they are somehow the analogues of 𝐿¹(ℝ⁺), we are going to define the Banach spaces that will act as the analogues of 𝐿^𝑝(ℝ⁺), but we need some tools to do this construction.

From now on, we consider 𝑘∈𝐿¹_loc(ℝ⁺) as a nonnegative function such that 0∈supp(𝑘) and 𝐾=𝑘∗𝜒_(0,∞). Let 1<𝑝<∞ and suppose that 𝑘 verifies the (dHC)_𝑝 condition. Take 𝐹∈𝐿^𝑝_𝐾(ℝ⁺). The function 𝑇^′_𝑘𝐹, given by 𝑇^′_𝑘𝐹(𝑡)=(𝑡)=𝑘∘𝐹)(_∞𝑡𝑘(𝑠−𝑡)𝐹(𝑠)𝑑𝑠,a.e.𝑡≥0, (61) belongs to 𝐿^𝑝(ℝ⁺); moreover, 𝑇^′_𝑘 is a bounded operator, 𝑇^′_𝑘∶𝐿^𝑝_𝐾(ℝ⁺)→𝐿^𝑝(ℝ⁺), which extends the operator 𝑇^′_𝑘∶𝒟₊→𝒟₊.

Definition 11. Let 𝒯^𝑝_𝑘(ℝ⁺) denote the Banach space formed as the set 𝑇^′_𝑘(𝐿^𝑝_𝐾(ℝ⁺)) endowed with the norm |‖⋅‖|_𝑝,𝑘 and obtained as the image of the norm ‖⋅‖_𝑝,𝐾 of 𝐿^𝑝_𝐾(ℝ⁺) through the operator 𝑇^′_𝑘∶𝐿^𝑝_𝐾(ℝ⁺)→𝐿^𝑝(ℝ⁺). For 𝑝=1, we keep the notation 𝒯_𝑘(ℝ⁺).

In accordance with Definition 11, 𝑇^′_𝑘∶𝐿^𝑝_𝐾(ℝ⁺)→𝒯^𝑝_𝑘(ℝ⁺) is a surjective isometry and 𝒯^𝑝_𝑘(ℝ⁺) is a Banach space. Let 𝑊_𝑘∶𝒯^𝑝_𝑘(ℝ⁺)→𝐿^𝑝_𝐾(ℝ⁺) be the inverse isometry of 𝑇^′_𝑘 and 𝑊_𝑘∶𝒯^𝑝_𝑘(ℝ⁺)→𝐿^𝑝_𝐾(ℝ⁺) extends the operator 𝑊_𝑘∶𝒟_𝑘→𝒟₊ defined in the beginning of this section. Note that given a function 𝑓∈𝒯^𝑝_𝑘(ℝ⁺), then 𝑓∈𝐿^𝑝(ℝ⁺) and there exists a unique element in 𝐿^𝑝_𝐾(ℝ⁺) (we denote by 𝑊_𝑘𝑓 ) such that 𝑓(𝑥)=𝑇^′_𝑘𝑊_𝑘𝑓𝑥)=_∞𝑥(𝑘𝑦−𝑥𝑊_𝑘𝑓𝑦𝑑𝑦,a.e.𝑥≥0. (62)

Then for every 𝑓∈𝒯^𝑝_𝑘(ℝ⁺), the norm is given by 𝑓_𝑝,𝑘=^∞₀𝑊_𝑘𝑓(𝑡)^𝑝𝐾^𝑝(𝑡)𝑑𝑡^1/𝑝. (63)

With these ideas, it is easy to show that the continuous inclusion 𝒯^𝑝_𝑘(ℝ⁺)↪𝐿^𝑝(ℝ⁺) holds.

Examples. (i) For 𝑘=𝑗_𝛼, we write 𝒯^𝑝_(𝛼)(ℝ⁺) instead of 𝒯^𝑝_𝑗𝛼(ℝ⁺), for 1≤𝑝<∞ and 𝑓_{𝑝,𝑗𝛼}1∶=Γ(𝛼+1)^∞₀𝑊_𝛼𝑓(𝑡)^𝑝𝑡^𝛼𝑝𝑑𝑡^1/𝑝. (64)

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^𝑝_(𝛼)(ℝ⁺) for 𝑝≥1 seems to be new; see Corollary 16. The case where 𝛼∈ℕ (weighted Sobolev spaces) and 𝑝=1 was introduced and studied in [15]; in this case, 𝑓_1,𝑗𝑛1∶=𝑛!^∞₀𝑓^(𝑛)(𝑡)𝑡^𝑛𝑑𝑡. (65) See Corollary 15.

(ii) In the case 𝑘=𝑒_−𝜆𝑗_𝛼, with 𝛼,𝜆>0 and 𝑝≥1, we obtain the Banach space 𝒯^𝑝_{𝑒−𝜆𝑗𝛼}(ℝ⁺) embedded with the norm 𝑓_{𝑝,𝑒−𝜆𝑗𝛼}1∶=Γ(𝛼)^∞₀𝑊_𝛼𝑒_𝜆𝑓(𝑡)^𝑝𝑡₀𝑠^𝛼−1𝑒^{−𝜆(𝑠+𝑡)}𝑑𝑠^𝑝𝑑𝑡^1/𝑝. (66)

(iii) Take 𝑘=𝜒_(0,1) and 𝐾(𝑡)=∫^𝑡₀𝜒_(0,1)(𝑠)𝑑𝑠=𝑡𝜒_(0,1)(𝑡)+𝜒_[1,∞)(𝑡), for 𝑡≥0. We obtain the Banach space 𝒯^𝑝_𝜒(0,1)(ℝ⁺) for 1≤𝑝<∞ embedded with the norm 𝑓_{𝑝,𝜒(0,1)}∶=¹₀^∞_𝑛=0𝑓^′(𝑡+𝑛)^𝑝𝑡^𝑝𝑑𝑡+^∞₁^∞_𝑛=0𝑓^′(𝑡+𝑛)^𝑝𝑑𝑡^1/𝑝, (67) for 𝑓∈𝒟₊.

An easy consequence of Theorem 1 and from the embedding of 𝒯^𝑝_𝑘(ℝ⁺)↪𝐿^𝑝(ℝ⁺) for 𝑝≥1, we get the next corollary.

Corollary 12. Let 𝑘∈𝐿¹_loc(ℝ⁺) be a nonnegative function such that 0∈supp(𝑘) and satisfy the Hardy-type condition (dHC)_𝑝 for some 1≤𝑝<∞. Then (i) 𝐿^𝑝(ℝ⁺)∘𝒯_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺), and 𝒯^𝑝_𝑘(ℝ⁺)∘𝒯_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺);(ii) 𝐿¹(ℝ⁺)∘𝒯^𝑝_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺), and 𝒯_𝑘(ℝ⁺)∘𝒯^𝑝_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺).

Now we set the main result of this section.

Theorem 13. Let 1<𝑝<∞, 𝑘 satisfy (HC)_𝑝′, (DC), and (AMC)_𝑝′, for 𝑝^′ such that 1/𝑝+1/𝑝^′=1. Then (i)𝒯_𝑘(ℝ⁺)∗𝒯^𝑝_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺); (ii)𝒯_𝑘(ℝ⁺)∗_𝑐𝒯^𝑝_𝑘(ℝ⁺)↪𝒯^𝑝_𝑘(ℝ⁺).

Proof. (i) Let 𝑓,𝑔∈𝒟_𝑘. According to (59), 𝑊_𝑘𝑓∗𝑔(𝑠)^𝑝≤𝐶^𝑠₀𝑊_𝑘𝑔(𝑟)^𝑠_𝑠−𝑟𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡𝑑𝑟^𝑝+𝐶^∞_𝑠𝑊_𝑘𝑔(𝑟)^∞_𝑠𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡𝑑𝑟^𝑝. (68) Therefore, 𝑓∗𝑔_𝑝,𝑘=^∞₀𝑊_𝑘𝑓∗𝑔(𝑠)^𝑝𝐾^𝑝(𝑠)𝑑𝑠^1/𝑝≤𝐶(𝐼+𝐽), (69) where 𝐼∶=^∞₀𝐾^𝑝(𝑠)×^𝑠₀𝑊_𝑘𝑔(𝑟)×^𝑠_𝑠−𝑟𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡𝑑𝑟^𝑝𝑑𝑠^1/𝑝,𝐽∶=^∞₀𝐾^𝑝(𝑠)×^𝑠₀𝑊_𝑘𝑔(𝑟×_∞𝑠)𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡𝑑𝑟^𝑝𝑑𝑠^1/𝑝. (70)
By Minkowski’s integral inequality, we get 𝐼≤^∞₀𝑊_𝑘𝑔(𝑟)𝐼₁+𝐼₂𝑑𝑟, (71) where 𝐼₁∶=_𝑟^2𝑟𝐾^𝑝(𝑠)^𝑠_𝑠−𝑟𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡^𝑝𝑑𝑠^1/𝑝,𝐼₂∶=^∞_2𝑟𝐾^𝑝(𝑠)^𝑠_𝑠−𝑟𝑘(𝑡+𝑟−𝑠)𝑊_𝑘𝑓(𝑡)𝑑𝑡^𝑝𝑑𝑠^1/𝑝. (72)
We apply Theorems A.2 and A.9 to complete the proof and get 𝑓∗𝑔_𝑝,𝑘≤𝐶𝑓_𝑝,𝑘𝑔_𝑝,𝑘. (73)
(ii) We use the definition of ∗_𝑐, (i) and Corollary 12.

Remark 14. For 𝑝=1, 𝑘∈𝐿¹_loc(ℝ⁺) with 0∈supp(𝑘) and verifying the (DC) condition, the following embeddings 𝒯_𝑘ℝ⁺∗𝒯_𝑘ℝ⁺↪𝒯_𝑘ℝ⁺,𝒯_𝑘ℝ⁺∗_𝑐𝒯_𝑘ℝ⁺↪𝒯_𝑘ℝ⁺, (74) hold; see Theorem 10. Note that the condition (dHC)₁ and Theorem A.9 hold for 𝑝=1.

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 𝑘 which have appeared before. We also give some final remarks and comments.

4.1. Weighted Sobolev Spaces

Take 𝑛∈ℕ, 𝑝>1, 𝑗_𝑛(𝑡)=𝑡^𝑛−1/(𝑛−1)! for 𝑡>0, and the weighted Sobolev space 𝒯^𝑝_(𝑛)(ℝ⁺) is embedding with the norm 𝑓_{𝑝,𝑗𝑛}=1𝑛!^∞₀𝑓^(𝑛)(𝑡)^𝑝𝑡^𝑛𝑝𝑑𝑡^1/𝑝𝑓∈𝒟₊. (75)

Corollary 15. The Banach space 𝒯^𝑝_(𝑛)(ℝ⁺) is a module for the algebra 𝒯¹_(𝑛)(ℝ⁺) (with usual convolution ∗ or the cosine convolution ∗_𝑐) for 1<𝑝<∞.

4.2. Weighted Fractional Sobolev Spaces

Let 𝛼>0, 𝑝>1, 𝑗_𝛼(𝑡)=𝑡^𝛼−1/Γ(𝛼) for 𝑡>0, and the weighted fractional Sobolev space 𝒯^𝑝_(𝛼)(ℝ⁺) is embedding with the norm 𝑓_{𝑝,𝑗𝛼}=1Γ()∫𝛼+1^∞0|𝑊_𝛼𝑓(𝑡)|^𝑝𝑡^𝛼𝑝𝑑𝑡^1/𝑝,𝑓∈𝒟₊, (76) where 𝑊_𝛼 is the Weyl fractional derivation.

Corollary 16. The Banach space 𝒯^𝑝_(𝛼)(ℝ⁺) is a module for the algebra 𝒯¹_(𝛼)(ℝ⁺) (with usual convolution ∗ or the cosine convolution ∗_𝑐) for 1<𝑝<∞.

4.3. Scattering Sobolev Spaces

Taking the function 𝜒_(0,1), 𝑊_𝜒(0,1)𝑓(𝑡)=−^∞_𝑛=0𝑓^′(𝑡+𝑛),𝑓∈𝒟₊,𝑡≥0, (77) and we consider the Banach space 𝒯^𝑝_𝜒(0,1)(ℝ⁺) for 1≤𝑝<∞ embedded with the norm 𝑓_{𝑝,𝜒(0,1)}∶=¹₀^∞_𝑛=0𝑓^′(𝑡+𝑛)^𝑝𝑡^𝑝𝑑𝑡+^∞₁^∞_𝑛=0𝑓^′(𝑡+𝑛)^𝑝𝑑𝑡^1/𝑝, (78) for 𝑓∈𝒟₊.

Corollary 17. The Banach space 𝒯^𝑝_𝜒(0,1)(ℝ⁺) is a module for the algebra 𝒯¹_𝜒(0,1)(ℝ⁺) (with the usual convolution ∗ or the cosine convolution ∗_𝑐) for 1<𝑝<∞.

Final Comments. Under some conditions of a nonnegative function 𝑘∈𝐿¹_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. (1) For 𝑝=2, the Banach space 𝒯²_𝑘(ℝ⁺) could be, in fact, a Hilbert space with the inner product 𝑓,𝑔∶=^∞₀𝑊_𝑘𝑓(𝑡)𝑊_𝑘𝑔(𝑡)^𝑡₀𝑘(𝑠)𝑑𝑠²𝑑𝑡,𝑓,𝑔∈𝒯²_𝑘ℝ⁺. (79) (2)  For 1<𝑝<∞, the dual Banach space of 𝒯^𝑝_𝑘(ℝ⁺) may be written by 𝔗^𝑝′_𝑘(ℝ⁺) embedded with the norm 𝑓_𝑝′,𝑘∶=^∞₀𝑘∗𝑓(𝑡)∫^𝑡₀𝑘(𝑠)𝑑𝑠^𝑝′𝑑