# Convolution Algebraic Structures Defined by Hardy-Type Operators

Authors: Pedro J. Miana; Juan J. Royo; Luis Sánchez-Lajusticia
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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 onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spaces Lωpℝ+ 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.


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## Body

## 1. Introduction

LetLp(ℝ+) be the set of Lebesgue p-integrable (class of) functions f, that is, a measurable function f∈Lp(ℝ+) when
(1)∥f∥p=(∫0∞|f(t)|pdt)1/p<∞,
for 1≤p<∞. The classical Hardy inequality establishes that
(2)(∫0∞‍|1x∫0xf(y)dy|pdx)1/p≤pp-1∥f∥p,f∈Lp(ℝ+),
for 1<p<∞; that is, the so-called Hardy operator ℋ, defined by
(3)ℋf(x):=1x∫0xf(y)dy,x≥0,
is a bounded operator on Lp(ℝ+) with ∥ℋ∥≤p/(p-1) for 1<p<∞.LetLloc1(ℝ+) be the set of locally integrable functions. Given f,g∈Lloc1(ℝ+) the (usual) convolution product * is defined by
(4)(f*g)(t):=∫0t‍f(t-s)g(s)ds,t≥0.Givenf∈L1(ℝ+) and g∈Lp(ℝ+), then f*g∈Lp(ℝ+) and
(5)∥f*g∥p≤∥f∥1∥g∥p,
for 1≤p≤∞. Note that the Hardy operator may be written in the following way:
(6)ℋf(x)=1∫0x‍χ(0,∞)(t)dt(f*χ(0,∞))(x),x≥0,
and the Hardy inequality may be written by
(7)∥1χ(0,∞)*χ(0,∞)f*χ(0,∞)∥p≤pp-1∥f∥p,f∈Lp(ℝ+),
where χ(0,∞) is the characteristic function in the interval (0,∞). In fact, it is also known that if jα(t):=tα-1/(Γ(α)) for α>0, then
(8)∥1χ(0,∞)*jαf*jα∥p≤Γ(α+1)Γ(1-1/p)Γ(α+1-1/p)∥f∥p,f∈Lp(ℝ+),
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 functionsk such that the Hardy-type operator ℋk, given by
(9)ℋk(f)=1χ(0,∞)*kf*k,
is a bounded operator in Lp(ℝ+), that is, a weighted inequality of Hardy-type
(10)∥1χ(0,∞)*kf*k∥p≤Ck,p∥f∥p,f∈Lp(ℝ+)
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 ℋk 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 operatorℋk (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 spacesLKp(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 spaces𝒯kp(ℝ+) (for 1<p<∞) and to show that they are modules for the corresponding Banach algebras 𝒯k1(ℝ+): given f∈𝒯kp(ℝ+) and g∈𝒯k1(ℝ+), then f*g∈𝒯kp(ℝ+) (Theorem 13(i)). In the particular case of k=jα for some α>0, we obtain that the fractional Sobolev spaces 𝒯jαp(ℝ+) (1<p<∞) are modules for the corresponding Banach algebras 𝒯jα1(ℝ+) (Corollary 16). Similar results hold for other convolution products, as the dual convolution product ∘, as follows:
(11)(f∘g)(t):=∫t∞‍f(s-t)g(s)ds,t≥0,f,g∈L1(ℝ+),
(Corollary 12) and the cosine convolution product *c, as follows:
(12)f*cg:=12(f*g+f∘g+g∘f),f,g∈L1(ℝ+),
(Theorem 13(ii)).Note that subalgebras𝒯k1(ℝ+) (contained in L1(ℝ+) and depending of a function k∈Lloc1(ℝ+)) were recently introduced in [8]. Some aspects of these Banach algebras (for k=jα 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, (Sk(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:
(13)Sk(t)x=∫0t‍k(t-s)S(s)xds,forsomex∈X.Fork=jα, the Banach algebra 𝒯jα1(ℝ+) was already introduced in [9] where the authors gave its connection with α-times integrated semigroups.Givenp≥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

Givenω:ℝ+→ℝ+ as a measurable function and 1≤p<∞, let Lωp(ℝ+) be the set of weighted Lebesgue p-integrable functions f, that is, f is a measurable function and
(14)∥f∥p,ω:=(∫0∞‍|ω(t)f(t)|pdt)1/p<∞.
(in fact Lωp(ℝ+) is formed by a class of functions which are equal except on Lebesgue null sets). For p=∞, Lω∞(ℝ+) is the set of Lebesgue measurable (class of) functions such that
(15)∥f∥∞,ω:=ess supt>0|ω(t)f(t)|<∞.In the caseω(t)=1 for t>0, we simplify this notation and write Lp(ℝ+) and ∥∥p as in Section 1.Ifω(t+s)≤Cω(t)ω(s), a.e. for t,s≥0 and C>0, then f*g∈Lωp(ℝ+) and
(16)∥f*g∥p,ω≤C∥f∥1,ω∥g∥p,ω,
for f∈Lω1(ℝ+) and g∈Lωp(ℝ+); if ω(t-s)≤Cω(t)ω(s) a.e. for t≥s≥0 and C>0, then f∘g∈Lωp(ℝ+) and
(17)∥f∘g∥p,ω≤C∥f∥1,ω∥g∥p,ω,
for f∈Lω1(ℝ+) and g∈Lωp(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.
Letω:ℝ+→ℝ+ be a nonnegative and nondecreasing a.e. function and 1≤p<∞. Then (i)
Lp(ℝ+)∘Lω1(ℝ+)↪Lωp(ℝ+);(ii)
L1(ℝ+)∘Lωp(ℝ+)↪Lωp(ℝ+).Proof.
(i) Letf∈Lp(ℝ+) and g∈Lω1(ℝ+); then
(18)∥f∘g∥p,ω≤(∫0∞‍(∫t∞‍|f(s-t)||g(s)|ω(s)ds)pdt)1/p=∥t↦∫0∞‍χ(t,∞)(s)|f(s-t)|ω(s)|g(s)|ds∥p≤∫0∞‍∥t↦χ(t,∞)(s)|f(s-t)|∥pω(s)|g(s)|ds=∫0∞‍|g(s)|ω(s)(∫0s‍|f(s-t)|pdt)1/pds≤∥f∥p∥g∥1,ω,
where we have used Minkowski’s integral inequality.
(ii) Letf∈L1(ℝ+) and g∈Lωp(ℝ+). We use similar ideas as in (i) to obtain
(19)∥f∘g∥p,ω≤(∫0∞‍(∫t∞‍ω(s)|f(s-t)||g(s)|ds)pdt)1/p=(∫0∞‍(∫0∞‍|f(u)||g(u+t)|ω(u+t)du)pdt)1/p≤∫0∞‍|f(u)|(∫u∞‍|g(s)|pωp(s)ds)1/pdu≤∥f∥1∥g∥p,ω,
and we conclude the result.As we have mentioned in Section1, 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.
Letp≥1, let k∈Lloc1(ℝ+) be a nonnegative measurable function, and let K=χ(0,∞)*k.(i)
We say thatk satisfies the (HC)p condition (Hardy-type condition) if there exists Ck,p>0 such that
(20)∥k*g∥p,1/K≤Ck,p∥g∥p,∀g∈Lp(ℝ+).(ii)
We say thatk satisfies the (dHC)p condition (dual Hardy-type condition) if there exists Ck,p′>0 such that
(21)∥k∘f∥p≤Ck,p′∥f∥p,K,∀f∈LKp(ℝ+).Forp=∞, inequality (20) holds for any measurable and positive function k∈Lloc1(ℝ+). Similarly, for p=1, inequality (21) holds for any measurable and positive function k∈Lloc1(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
(22)∫0∞(k*f)(t)g(t)dt=∫0∞(k∘g)(s)f(t)dt
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 ℋk and its adjoint.Theorem 3.
Letk 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 thatk satisfies the (HC)p condition. Take f∈LKp′(ℝ+) and let
(23)f~(x):=∫x∞k(y-x)f(y)dy=(k∘f)(x),x≥0.
Letg∈Lp(ℝ+). Then
(24)∥f~g∥1≤∫0∞(∫0y‍k(y-x)|g(x)|dx)|f(y)|dy≤(∫0∞1(∫0y‍k(τ)dτ)p(k*|g|(y))pdy)1/p×(∫0∞‍(∫0y‍k(τ)dτ)p′|f(y)|p′dy)1/p′≤Ck,p∥g∥p(∫0∞‍(k*χ(0,∞)(y)|f(y)|)p′dy)1/p′=Ck,p∥g∥p∥f∥p′,K,
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∈Lp′(ℝ+), ∥k∘f∥p′≤Ck,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 functionjα(t):=tα-1/Γ(α) for α>0 satisfies the (HC)p; that is,
(25)∥1χ(0,∞)*jαf*jα∥p≤Γ(α+1)Γ(1-1/p)Γ(α+1-1/p)∥f∥p,f∈Lp(ℝ+),
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 Theorem3, the function jα 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],
(26)∫0∞‍(∫x∞‍(t-x)α-1Γ(α)|f|(t)dt)pdx≤(Γ(α+1)Γ(1/p)Γ(α+1/p))p∫0∞‍(xαΓ(α+1)|f|(x))pdx.
Also the constantΓ(α+1)Γ(1/p)/Γ(α+1/p) is optimal in the above inequality ([1, Theorem 329, page 245]).
(ii) The characteristic functionχ(0,1) satisfies the (HC)p for 1<p<∞. Note that χ(0,1)*χ(0,∞)(t)=min{1,t} for t>0, and
(27)∥1χ(0,1)*χ(0,∞)χ(0,1)*f∥pp≤∫01‍(1t∫0t‍|f(s)|ds)pdt+∫1∞‍(∫t-1t‍|f(s)|ds)pdt≤∫0∞‍(1t∫0t‍|f(s)|ds)pdt+∫1∞‍∫t-1t‍|f(s)|pdsdt≤(pp-1)p∥f∥pp+∫01‍|f(s)|psds+∫1∞‍|f(s)|pds≤Mp∥f∥pp
for f∈Lp(ℝ+). It is clear that the characteristic function χ(1,∞) does not satisfy the (HC)p for any 1<p<∞.
(iii) Exponential functions{eλ:0≠λ∈ℝ} (eλ(t):=eλt for t>0) do not satisfy the (HC)p for any 1<p<∞. In fact, we check that {eλ: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
(28)∥e-λ∘e-μ∥p=1λ+μ∥e-μ∥p=1λ+μ(1μp)1/p.Note thate-λ*χ(0,∞)(t)=(1-e-λt)/λ for t>0 and
(29)∥e-μ∥p,e-λ*χ(0,∞)p=1λp∫0∞‍e-μpt(1-e-λt)pdt≤∫0∞‍e-μpttpdt=Γ(p+1)(μp)p+1forμ>0. Then there does not exist C>0 such that
(30)1λ+μ(1μp)1/p≤CΓ(p+1)1/p(μp)1+1/p
for every μ>0.Now, takeλ>0 and μ>λ. Then eλ∘e-μ=(1/(μ-λ))e-μ and eλ*χ(0,∞)(t)=(eλt-1)/λ for t>0. Note that
(31)∥e-μ∥p,eλ*χ(0,∞)p=1λp∫0∞‍e-μpt(eλt-1)pdt≤1λp∫0∞‍e-(μ-λ)ptdt=1λp1(μ-λ)p,
and there does not exist C′>0 such that
(32)1μ-λ(1μp)1/p≤C′1λ1((μ-λ)p)1/p,
for any μ>λ.The next theorem is a particular case of [6, Theorem 4.4]: the condition Ak,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.
Letk be a nonnegative measurable function with ∫0ɛ‍k(x)dx>0 for all ɛ>0, and there exists r∈ℝ such that
(33)ess sups∈(0,∞)s(r-1)/p(∫s∞‍(k(u-s)∫0uk(x)dx)pup-rdu)1/p:=Ak,p(r)<∞,
for some p>1 and 1<r<p. Then
(34)∥g*k∥p,1/K≤Ak,p(r)∥g∥p,∀g∈Lp(ℝ+),
where K=χ(0,∞)*k; that is, the function k satisfies the (HC)p condition and
(35)∥ℋk∥ℬ(Lp(ℝ+))≤inf1<r<p((p-1p-r)1/p′Ak,p(r)),
where ℋk(g)=(1/χ(0,∞)*k)(k*g), for g∈Lp(ℝ+).Proof.
Takeg∈Lp(ℝ+), and then
(36)∥g*k∥p,1/K=(∫0∞‍1(∫0x‍k(r)dr)p(∫0x‍k(x-s)|g(s)|ds)pdx)1/p=||∫01‍k(x(1-t))∫0x‍k(r)drg(xt)xdt||p≤∫01‍||xk(x(1-t))∫0x‍k(r)drg(xt)||pdt=∫01‍(∫0∞‍(sk((s/t)(1-t))t∫0s/t‍k(r)dr)p|g(s)|pdst)1/pdt,
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
(37)∥g*k∥p,1/K≤∫01‍t-(r-1)/pt(r-1)/p××(∫0∞‍(sk((s/t)(1-t))t∫0s/t‍k(r)dr)p|g(s)|pdst)1/pdt≤(∫01‍t-p′(r-1)/pdt)1/p′×(∫0∞‍|g(s)|psr-1∫s∞‍(k(u-s))p(∫0u‍k(x)dx)pup-rduds)1/p≤(p-1p-r)1/p′Ak,p(r)(∫0∞‍|g(s)|pds)1/p,
where we have applied the assuption that k satisfies (33), and we conclude that
(38)∥ℋk∥ℬ(Lp(ℝ+))≤inf1<r<p((p-1p-r)1/p′Ak,p(r)),
and the theorem is proved.Note that inequality (33) may be written in terms of ∘ product due to
(39)kp∘(h1-r/p(χ(0,∞)*k))p(s)=∫s∞‍(k(u-s)∫0u‍k(x)dx)pup-rdu,s≥0,
and h1-r/p(u)=u1-r/p for u>0. In the next lemma, we give some properties of the function Ak,p(r).Lemma 6.
Takek such that Ak,p(r)<∞ for some p>1 and 1<r<p. Then (i)
Ak,p(r′)≤Ak,p(r) for p>r′>r>1;(ii)
Ak,q(r)≤(Ak,p(r))/(r-1)(p-q)/qp for 1<q<p and r>1.Proof.
(i) Taker′>r, and we get
(40)s(r′-1)/p(∫s∞‍(k(u-s)∫0u‍k(x)dx)pup-r′du)1/p≤s(r-1)/p(∫s∞‍(k(u-s)∫0u‍k(x)dx)pxr′-rup-r′du)1/p,
for s>0, and then Ak,p(r′)≤Ak,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:
(41)∫s∞(k(u-s)∫0u‍k(x)dx)quq-rdu≤(∫s∞‍(k(u-s)∫0u‍k(x)dx)pup-rdu)q/p(∫s∞‍1urdu)(p-q)/p,
and then
(42)s(r-1)/q(∫s∞‍(k(u-s)∫0u‍k(x)dx)quq-rdu)1/q≤s(r-1)/p(r-1)(p-q)/qp(∫s∞‍(k(u-s)∫0u‍k(x)dx)pup-rdu)1/p,
for s>0. We conclude that Ak,q(r)≤Ak,p(r)/r-1(p-q)/(qp).Example 7.
(i) Letk be function for which it is possible to find constants 0<m≤M and α>0 such that
(43)mtα-1Γ(α)≤k(t)≤Mtα-1Γ(α).
We get
(44)s(r-1)/p(∫s∞‍αp(u-s)(α-1)puαpup-rdu)1/p=α(∫0∞‍x(α-1)p(1+x)αp-p+rdx)1/p=α(Γ((α-1)p+1)Γ(r-1)Γ(αp-p+r))1/p,
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
(45)mMα(Γ((α-1)p+1)Γ(r-1)Γ(αp-p+r))1/p≤Ak,p(r)≤Mmα(Γ((α-1)p+1)Γ(r-1)Γ(αp-p+r))1/p,In fact condition (43) implies that the function k may be written as k=hjα, where h∈L∞(ℝ+) and inft≥0h(t)≱0; then m=inft≥0h(t) and M=∥h∥∞. Particular cases are (a)
the trivial casek:=jα for α>0 and Ajα,p(r)=α((Γ((α-1)p+1)Γ(r-1))/Γ(αp-p+r))1/p;(b)
the familyk(t):=((At+B)/(Ct+D))γjα(t), for A,B,C,D,α,γ>0. In this case,
(1)
ifAD-BC<0, then m=(A/C)γ and M=(B/D)γ;(2)
ifAD-BC=0, then k=(B/D)γjα;(3)
ifAD-BC>0, then m=(B/D)γ and M=(A/C)γ.(ii) Letλ>0, and consider functions eλ,e-λ. Then Ae-λ,p(p)=Aeλ,p(p)=∞ for p≥1: take s>0, and consider
(46)s1-1/p(∫s∞‍eλ(u-s)p(∫0u‍eλrdr)pdu)1/p=λs1-1/pe-λs(∫s∞‍1(1-e-λu)pdu)1/p=∞,
and we have proved the claim for the function eλ. For the function e-λ, note that
(47)lims→∞s1-1/p(∫s∞‍e-λ(u-s)p(∫0u‍e-λrdr)pdu)1/p=lims→∞s1-1/pλ(∫0∞e-λxpdx(1-e-λ(x+s))p‍)1/p=∞.(iii) The characteristic functionχ(0,1) satisfies the assumption that Aχ(0,1),p(p)=∞ for p≥1, such that
(48)sups≥1(s1-1/p(∫s∞‍χ(0,1)(u-s)(∫0u‍χ(0,1)(r)dr)pdu)1/p)=sups≥1(s1-1/p(∫ss+1‍du)1/p)=∞.
Note that the characteristic function χ(1,∞) verifies ∫0ɛ‍χ(1,∞)(s)ds=0 for 0<ɛ≤1.The next theorem gives the boundedness of the operator off↦k∘f in Lp-spaces. Similar results may be found in the literature, for example, [6, Theorem 4.3].Theorem 8.
Letk be a nonnegative measurable function with ∫0ɛ‍k(x)dx>0 for all ɛ>0, and there exists r∈ℝ such that
(49)ess sups∈(0,∞)s(r-1)/p∫0s‍k(x)dx(∫0s‍kp(s-u)up-rdu)1/p:=Bk,p(r)<∞,
for some p>1 and p+1>r>p. Then
(50)∥k∘f∥p≤Bk,p(r)∥f∥p,K∀f∈LKp(ℝ+),
where K=χ(0,∞)*k; that is, the function k satisfies the (dHC)p condition and
(51)∥Tk′∥ℬ(LKp(ℝ+),Lp(ℝ+))≤infp+1>r>p((p-1r-p)1/p′Bk,p(r)),
where Tk′(f)=k∘f for f∈LKp(ℝ+).Proof.
Takef∈LKp(ℝ+) and then
(52)∥k∘f∥p≤(∫0∞‍(∫x∞‍k(s-x)|f(s)|ds)pdx)1/p=||∫1∞‍k(x(t-1))|f(xt)|xdt||p≤∫1∞‍||k(x(t-1))|f(xt)|x||pdt=∫1∞‍(∫0∞‍(k(x(t-1))|f(xt)|x)pdx)1/pdt,
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
(53)∥k∘f∥p≤∫1∞‍t-(r-1)/pt(r-1)/p××(∫0∞‍(k(x(t-1))|f(xt)|x)pdx)1/pdt≤(∫1∞‍t-(r-1)p′/pdt)1/p′×(∫0∞‍|f(s)|psr-1∫0s‍(k(s-u))pup-rduds)1/p≤(p-1r-p)1/p′Bk,p(r)(∫0∞‍|f(s)|p(∫0s‍k(x)dx)pds)1/p,
where we have changed the variable and applied the assumption that k satisfies (49). We conclude that
(54)∥Tk′∥ℬ(Lp(ℝ+))≤infp+1>r>p((p-1r-p)1/p′Bk,p(r)),
and the theorem is shown.To finish this section, we present Table1 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
Func.\cond.
(
HC
)
p
(
d
HC
)
p
(DC)
(DIC)
(
AMC
)
p
j
α
p
>
1
p
≥
1
✓
✓
p
>
1
χ
(
0,1
)
p
>
1
p
≥
1
✓
✓
p
>
1
e
-
λ
×
×
✓
✓
p
>
1
e
λ
×
×
×
✓
×
χ
(
1
,
∞
)
×
×
×
×
×
forα,λ>0.


## 3. Convolution Banach Modules𝒯kp(ℝ+)

In the beginning of this section, we collect some definitions and properties that will be used throughout this section. We will denote by𝒟+ 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,ɛ) for all ɛ>0.Letk∈Lloc1(ℝ+) be such that 0∈supp(k). We define the operator Tk′:𝒟+→𝒟+ given by f↦Tk′(f):=k∘f.(i)
ThenTk′:𝒟+→𝒟+ is an injective, linear, and continuous homomorphism such that
(55)Tk′(f∘g)=f∘Tk′(g),f,g∈𝒟+.(ii)
The mapTk′ extends to a linear and continuous map from LK1(ℝ+) to L1(ℝ+), which we denote again by Tk′:LK1(ℝ+)→L1(ℝ+) such that ∥Tk′∥≤1.See [8, Theorem 2.5]. Then we define the space 𝒟k by 𝒟k:=Tk′(𝒟+) and the map Wk:𝒟k→𝒟+ by
(56)f(t)=Tk′(Wk(f))(t)=∫t∞‍k(s-t)Wkf(s)ds,f∈𝒟k,t≥0;see [8] for more details.Example 9.
(i) Takeα>0 and k=jα; the map Wjα is the Weyl fractional derivative of order α, Wα, and 𝒟jα=𝒟+; note that for α∈ℕ, Wα=(-1)αdα/dtα, the α-iterate of usual derivation; see more details, for example, in [9, 13].
(ii) Givenα>0 and z∈ℂ, take k=ezjα; we have 𝒟k=𝒟+ and
(57)Wjαezf=ezWα(e-zf),f∈𝒟+.
See other examples in [8, Section 2].
(iii) Fork=χ(0,1), it is straightforward to check that Tχ(0,1)′(f)(t)=∫tt+1‍f(s)ds for f∈𝒟+, 𝒟χ(0,1)=𝒟+ and
(58)Wχ(0,1)f(t)=-∑n=0∞‍f′(t+n),f∈𝒟+,t≥0.
Take f,g∈𝒟k. Then f*g, f∘g, f*cg∈𝒟k and
(59)Wk(f*g)(s)=∫0s‍Wkg(r)∫s-rs‍k(t+r-s)Wkf(t)dtdr-∫s∞‍Wkg(r)∫s∞‍k(t+r-s)Wkf(t)dtdr;Wk(f∘g)=f∘Wkg;  Wk(f*cg)=(1/2)(Wk(f*g)+f∘Wkg+g∘Wkf); see [8, Theorem 2.10].Under some conditions ofk, some Banach algebras under the convolution product may be considered as shown in 10.Theorem 10 (see [8, Theorems 3.4 and 3.5]).
Letk∈Lloc1(ℝ+) with 0∈supp(k) satisfies the doubling condition (DC) and K=χ(0,∞)*k. Then the integral
(60)∥f∥1,K:=∫0∞‍|Wkf(t)|K(t)dt,f∈𝒟k
defines an algebra norm on 𝒟k for the convolution product * and also for *c. We denote by 𝒯k(ℝ+) the Banach space obtained as the completion of 𝒟k in the norm ∥·∥1,K, and then we have 𝒯k(ℝ+)↪L1(ℝ+).These Banach algebras𝒯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 Lp(ℝ+), but we need some tools to do this construction.From now on, we considerk∈Lloc1(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∈LKp(ℝ+). The function Tk′F, given by
(61)Tk′F(t)=(k∘F)(t)=∫t∞‍k(s-t)F(s)ds,a.e.t≥0,
belongs to Lp(ℝ+); moreover, Tk′ is a bounded operator, Tk′:LKp(ℝ+)→Lp(ℝ+), which extends the operator Tk′:𝒟+→𝒟+.Definition 11.
Let𝒯kp(ℝ+) denote the Banach space formed as the set Tk′(LKp(ℝ+)) endowed with the norm |∥·∥|p,k and obtained as the image of the norm ∥·∥p,K of LKp(ℝ+) through the operator Tk′:LKp(ℝ+)→Lp(ℝ+). For p=1, we keep the notation 𝒯k(ℝ+).In accordance with Definition11, Tk′:LKp(ℝ+)→𝒯kp(ℝ+) is a surjective isometry and 𝒯kp(ℝ+) is a Banach space. Let Wk:𝒯kp(ℝ+)→LKp(ℝ+) be the inverse isometry of Tk′ and Wk:𝒯kp(ℝ+)→LKp(ℝ+) extends the operator Wk:𝒟k→𝒟+ defined in the beginning of this section. Note that given a function f∈𝒯kp(ℝ+), then f∈Lp(ℝ+) and there exists a unique element in LKp(ℝ+) (we denote by Wkf ) such that
(62)f(x)=Tk′(Wkf)(x)=∫x∞‍k(y-x)Wkf(y)dy,a.e.x≥0.Then for everyf∈𝒯kp(ℝ+), the norm is given by
(63)|∥f∥|p,k=(∫0∞‍|Wkf(t)|pKp(t)dt)1/p.With these ideas, it is easy to show that the continuous inclusion𝒯kp(ℝ+)↪Lp(ℝ+) holds.Examples. (i) For k=jα, we write 𝒯(α)p(ℝ+) instead of 𝒯jαp(ℝ+), for 1≤p<∞ and
(64)|∥f∥|p,jα:=1Γ(α+1)(∫0∞‍|Wαf(t)|ptαpdt)1/p.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(α)p(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,
(65)|∥f∥|1,jn:=1n!∫0∞‍|f(n)(t)|tndt.
See Corollary 15.(ii) In the casek=e-λjα, with α,λ>0 and p≥1, we obtain the Banach space 𝒯e-λjαp(ℝ+) embedded with the norm
(66)|∥f∥|p,e-λjα:=1Γ(α)(∫0∞‍|Wα(eλf)(t)|p(∫0t‍sα-1e-λ(s+t)ds)pdt)1/p.(iii) Takek=χ(0,1) and K(t)=∫0t‍χ(0,1)(s)ds=tχ(0,1)(t)+χ[1,∞)(t), for t≥0. We obtain the Banach space 𝒯χ(0,1)p(ℝ+) for 1≤p<∞ embedded with the norm
(67)|∥f∥|p,χ(0,1):=(∫01‍|∑n=0∞‍f′(t+n)|ptpdt+∫1∞‍|∑n=0∞‍f′(t+n)|pdt)1/p,
for f∈𝒟+.An easy consequence of Theorem1 and from the embedding of 𝒯kp(ℝ+)↪Lp(ℝ+) for p≥1, we get the next corollary.Corollary 12.
Letk∈Lloc1(ℝ+) be a nonnegative function such that 0∈supp(k) and satisfy the Hardy-type condition (dHC)p for some 1≤p<∞. Then (i)
Lp(ℝ+)∘𝒯k(ℝ+)↪𝒯kp(ℝ+), and 𝒯kp(ℝ+)∘𝒯k(ℝ+)↪𝒯kp(ℝ+);(ii)
L1(ℝ+)∘𝒯kp(ℝ+)↪𝒯kp(ℝ+), and 𝒯k(ℝ+)∘𝒯kp(ℝ+)↪𝒯kp(ℝ+).Now we set the main result of this section.Theorem 13.
Let1<p<∞, k satisfy (HC)p′, (DC), and (AMC)p′, for p′ such that 1/p+1/p′=1. Then (i)
𝒯k(ℝ+)*𝒯kp(ℝ+)↪𝒯kp(ℝ+);(ii)
𝒯k(ℝ+)*c𝒯kp(ℝ+)↪𝒯kp(ℝ+).Proof.
(i) Letf,g∈𝒟k. According to (59),
(68)|Wk(f*g)(s)|p≤C(∫0s‍|Wkg(r)|∫s-rs‍k(t+r-s)|Wkf(t)|dtdr)p+C(∫s∞‍|Wkg(r)|∫s∞‍k(t+r-s)|Wkf(t)|dtdr)p.
Therefore,
(69)|∥f*g∥|p,k=(∫0∞‍|Wk(f*g)(s)|pKp(s)ds)1/p≤C(I+J),
where
(70)I≔(∫0∞‍Kp(s)×××(∫0s|Wkg(r)|‍××××××∫s-rsk(t+r-s)|Wkf(t)|dtdr)pds∫0s|Wkg(r)|‍)1/p,J≔(∫0∞Kp(s)×××(∫0s|Wkg(r)|‍××××××∫s∞‍k(t+r-s)|Wkf(t)|dtdr)pds∫0s|Wkg(r)|‍)1/p.
By Minkowski’s integral inequality, we get(71)I≤∫0∞‍|Wkg(r)|(I1+I2)dr,
where
(72)I1:=(∫r2r‍Kp(s)(∫s-rs‍k(t+r-s)|Wkf(t)|dt)pds)1/p,I2:=(∫2r∞‍Kp(s)(∫s-rs‍k(t+r-s)|Wkf(t)|dt)pds)1/p.
We apply TheoremsA.2 and A.9 to complete the proof and get
(73)|∥f*g∥|p,k≤C|∥f∥|p,k|∥g∥|p,k.
(ii) We use the definition of*c, (i) and Corollary 12.Remark 14.
Forp=1, k∈Lloc1(ℝ+) with 0∈supp(k) and verifying the (DC) condition, the following embeddings
(74)𝒯k(ℝ+)*𝒯k(ℝ+)↪𝒯k(ℝ+),𝒯k(ℝ+)*c𝒯k(ℝ+)↪𝒯k(ℝ+),
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, Theorem13, to several particular examples of function k which have appeared before. We also give some final remarks and comments.


### 4.1. Weighted Sobolev Spaces

Taken∈ℕ, p>1, jn(t)=tn-1/(n-1)! for t>0, and the weighted Sobolev space 𝒯(n)p(ℝ+) is embedding with the norm
(75)|∥f∥|p,jn=1n!(∫0∞‍|f(n)(t)|ptnpdt)1/pf∈𝒟+.Corollary 15.
The Banach space𝒯(n)p(ℝ+) is a module for the algebra 𝒯(n)1(ℝ+) (with usual convolution * or the cosine convolution *c) for 1<p<∞.


### 4.2.  Weighted Fractional Sobolev Spaces

Letα>0, p>1, jα(t)=tα-1/Γ(α) for t>0, and the weighted fractional Sobolev space 𝒯(α)p(ℝ+) is embedding with the norm
(76)|∥f∥|p,jα=1Γ(α+1)(∫0∞‍|Wαf(t)|ptαpdt)1/p,f∈𝒟+,
where Wα is the Weyl fractional derivation.Corollary 16.
The Banach space𝒯(α)p(ℝ+) is a module for the algebra 𝒯(α)1(ℝ+) (with usual convolution * or the cosine convolution *c) for 1<p<∞.


### 4.3. Scattering Sobolev Spaces

Taking the functionχ(0,1),
(77)Wχ(0,1)f(t)=-∑n=0∞‍f′(t+n),f∈𝒟+,t≥0,
and we consider the Banach space 𝒯χ(0,1)p(ℝ+) for 1≤p<∞ embedded with the norm
(78)∥|f|∥p,χ(0,1):=(∫01‍|∑n=0∞‍f′(t+n)|ptpdt+∫1∞‍|∑n=0∞‍f′(t+n)|pdt)1/p, for f∈𝒟+.Corollary 17.
The Banach space𝒯χ(0,1)p(ℝ+) is a module for the algebra 𝒯χ(0,1)1(ℝ+) (with the usual convolution * or the cosine convolution *c) for 1<p<∞.Final Comments. Under some conditions of a nonnegative function k∈Lloc1(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)
Forp=2, the Banach space 𝒯k2(ℝ+) could be, in fact, a Hilbert space with the inner product
(79)〈f,g〉≔∫0∞‍Wkf(t)Wkg(t)¯(∫0t‍k(s)ds)2dt,f,g∈𝒯k2(ℝ+).(2)
For1<p<∞, the dual Banach space of 𝒯kp(ℝ+) may be written by 𝔗kp′(ℝ+) embedded with the norm
(80)∥∥f∥∥p′,k:=(∫0∞‍|k*f(t)∫0t‍k(s)ds|p′dt)1/p′,f∈𝔗kp′(ℝ+),wherep′ is the conjugate exponent of p.(3)
It seems to be natural that reflexivity and interpolation properties hold in Banach spaces𝒯kp(ℝ+) for 1<p<∞.


## 4.1. Weighted Sobolev Spaces

Taken∈ℕ, p>1, jn(t)=tn-1/(n-1)! for t>0, and the weighted Sobolev space 𝒯(n)p(ℝ+) is embedding with the norm
(75)|∥f∥|p,jn=1n!(∫0∞‍|f(n)(t)|ptnpdt)1/pf∈𝒟+.Corollary 15.
The Banach space𝒯(n)p(ℝ+) is a module for the algebra 𝒯(n)1(ℝ+) (with usual convolution * or the cosine convolution *c) for 1<p<∞.


## 4.2.  Weighted Fractional Sobolev Spaces

Letα>0, p>1, jα(t)=tα-1/Γ(α) for t>0, and the weighted fractional Sobolev space 𝒯(α)p(ℝ+) is embedding with the norm
(76)|∥f∥|p,jα=1Γ(α+1)(∫0∞‍|Wαf(t)|ptαpdt)1/p,f∈𝒟+,
where Wα is the Weyl fractional derivation.Corollary 16.
The Banach space𝒯(α)p(ℝ+) is a module for the algebra 𝒯(α)1(ℝ+) (with usual convolution * or the cosine convolution *c) for 1<p<∞.


## 4.3. Scattering Sobolev Spaces

Taking the functionχ(0,1),
(77)Wχ(0,1)f(t)=-∑n=0∞‍f′(t+n),f∈𝒟+,t≥0,
and we consider the Banach space 𝒯χ(0,1)p(ℝ+) for 1≤p<∞ embedded with the norm
(78)∥|f|∥p,χ(0,1):=(∫01‍|∑n=0∞‍f′(t+n)|ptpdt+∫1∞‍|∑n=0∞‍f′(t+n)|pdt)1/p, for f∈𝒟+.Corollary 17.
The Banach space𝒯χ(0,1)p(ℝ+) is a module for the algebra 𝒯χ(0,1)1(ℝ+) (with the usual convolution * or the cosine convolution *c) for 1<p<∞.Final Comments. Under some conditions of a nonnegative function k∈Lloc1(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)
Forp=2, the Banach space 𝒯k2(ℝ+) could be, in fact, a Hilbert space with the inner product
(79)〈f,g〉≔∫0∞‍Wkf(t)Wkg(t)¯(∫0t‍k(s)ds)2dt,f,g∈𝒯k2(ℝ+).(2)
For1<p<∞, the dual Banach space of 𝒯kp(ℝ+) may be written by 𝔗kp′(ℝ+) embedded with the norm
(80)∥∥f∥∥p′,k:=(∫0∞‍|k*f(t)∫0t‍k(s)ds|p′dt)1/p′,f∈𝔗kp′(ℝ+),wherep′ is the conjugate exponent of p.(3)
It seems to be natural that reflexivity and interpolation properties hold in Banach spaces𝒯kp(ℝ+) for 1<p<∞.



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*Source: 212465-2013-11-20.xml*