# Convolution Operators and Bochner-Riesz Means on Herz-Type Hardy Spaces in the Dunkl Setting

Authors: A. Gasmi; F. Soltani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We study the Dunkl convolution operators on Herz-type Hardy spacesℋα,2p and we establish a version of multiplier theorem for the maximal Bochner-Riesz operators on
the Herz-type Hardy spaces ℋα,∞p.


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## Body

## 1. Introduction

The classical theory of Hardy spaces onℝn has received an important impetus from the work of Fefferman and Stein, Lu and Yang [1, 2]. Their work resulted in many applications involving sharp estimates for convolution and multiplier operators.By using the technique of Herz-type Hardy spaces for the Dunkl operatorΛα, we are attempting in this paper to study the Dunkl convolution operators, and we establish a version of multiplier theorem for the maximal Bochner-Riesz operators on these spaces.The Dunkl operatorΛα, α>-1/2, associated with the reflection group ℤ2 on ℝ: (1.1)Λαf(x):=ddxf(x)+2α+1x[f(x)-f(-x)2],
is the operator devised by Dunkl [3] in connection with a generalization of the classical theory of spherical harmonics. The Dunkl analysis with respect to α≥-1/2 concerns the Dunkl operator Λα, the Dunkl transform ℱα, the Dunkl convolution *α, and a certain measure μα on ℝ.In this paper we define a Herz-type Hardy spacesℋα,qp, 0<p≤1<q≤∞, in the Dunkl setting. Next, we consider the Dunkl convolution operators Tkf:=k*αf, where k is a locally integrable function on ℝ. We use the atomic decomposition of the Herz-type Hardy spaces ℋα,qp to study the ℋα,2p-ℋα,2q-bounded and the ℋα,21-L1-bounded of the operators Tk. Finally, we establish a version of multiplier theorem for the maximal Bochner-Riesz operators σαη, t>0, η>α+1/2: (1.2)σαη(f):=sup⁡t>0|Φα,tη*αf|,
where (1.3)Φα,tη(x):=Γ(η+1)2α+1t2α+2∑n=0∞(-1)n22nn!Γ(n+α+η+2)(tx)2n,
on the Herz-type Hardy spaces ℋα,∞p. In this version we prove the ℋα,∞p-Lp-bounded of the operators σαη, for α+1/2<η<α+3/2.The content of this work is the following. In Section2, we recall some results about harmonic analysis and we define a Herz-type Hardy spaces ℋα,qp, 0<p≤1<q≤∞ for the Dunkl operator Λα. In Section 3, we study the ℋα,2p-ℋα,2q-bounded and the ℋα,21-L1-bounded of the convolution operators Tk. In Section 4, we prove the ℋα,∞p-Lp-bounded of the maximal Bochner-Riesz operators σαη.Throughout the paper we use the classic notation. Thus𝒮(ℝ) and 𝒮′(ℝ) are the Schwartz space on ℝ and the space of tempered distributions on ℝ, respectively. Finally, C will denote a positive constant not necessary the same in each occurrence.


## 2. The Dunkl Harmonic Analysis onℝ

Forα≥-1/2 and λ∈ℂ, the initial problem (2.1)Λαf(x)=λf(x),f(0)=1,
has a unique analytic solution Eα(λx) called Dunkl kernel [4–6] given by (2.2)Eα(λx)=Iα(λx)+λx2(α+1)Iα+1(λx),x∈R,
where (2.3)Iα(λx):=Γ(α+1)∑n=0∞(λx)2n22nn!Γ(n+α+1),
is the modified spherical Bessel function of order α.We notice that, the Dunkl kernelEα(λx) can be expanded in a power series [7] in the form (2.4)Eα(λx)=∑n=0∞(λx)nbn(α),
where (2.5)b2n(α)=22nn!Γ(α+1)Γ(n+α+1),b2n+1(α)=2(α+1)b2n(α+1).Note.
Letμα be the measure on ℝ given by
(2.6)dμα(x):=12α+1Γ(α+1)|x|2α+1dx.
We denote by Lp(ℝ,μα), p∈]0,∞], the space of measurable functions f on ℝ, such that
(2.7)‖f‖Lαp:=[∫R|f(x)|pdμα(x)]1/p<∞,p∈]0,∞[,‖f‖Lα∞:=esssup⁡x∈R|f(x)|<∞.
The Dunkl kernel gives rise to an integral transform, called Dunkl transform onℝ, which was introduced by Dunkl in [8], where already many basic properties were established. Dunkl's results were completed and extended later on by de Jeu in [5].
The Dunkl transform of a functionf∈L1(ℝ,μα), is given by
(2.8)Fα(f)(λ):=∫REα(-iλx)f(x)dμα(x),λ∈R.
ForT∈𝒮′(ℝ), we define the Dunkl transform ℱα(T) of T, by
(2.9)〈Fα(T),φ〉:=〈T,Fα(φ)〉,φ∈S(R).Note.
For allx,y,z∈ℝ, we put
(2.10)Wα(x,y,z):={1-σx,y,z+σz,x,y+σz,y,x}Δα(|x|,|y|,|z|),
where
(2.11)σx,y,z:={x2+y2-z22xy,ifx,y∈R∖{0}0,otherwiseΔα(|x|,|y|,|z|):={dα[((|x|+|y|)2-z2)(z2-(|x|-|y|)2)]α-1/2|xyz|2αif|z|∈Ax,y,0,otherwise,dα=21-α(Γ(α+1))2πΓ(α+1/2),Ax,y=[||x|-|y||,|x|+|y|].
We denote by νx,y the following signed measure:
(2.12)dνx,y(z):={Wα(x,y,z)dμα(z)ifx,y∈R∖{0},dδx(z)ify=0,dδy(z)ifx=0.
The Dunkl translation operators τx, x∈ℝ (see [6]) are defined for f∈𝒞(ℝ) (the space of continuous functions on ℝ), by
(2.13)τxf(y):=∫||x|-|y|||x|+|y|f(z)dνx,y(z)+∫-(|x|+|y|)-||x|-|y||f(z)dνx,y(z).
Letf and g be two functions in 𝒮(ℝ). We define the Dunkl convolution product *α of f and g by(2.14)f*αg(x):=∫Rτxf(-y)g(y)dμα(y),x∈R.
ForT∈𝒮′(ℝ) and f∈𝒮(ℝ), we define the Dunkl convolution product T*αf by(2.15)T*αf(x):=〈T(y),τxf(-y)〉,x∈R.
We begin by recalling the definition of the Herz-type Hardy space in the Dunkl setting. Firstly we introduce a class of fundamental functions that we will call atoms.
Let0<p≤1<q≤∞. A measurable function a on ℝ is called a (p,q) atom, if a satisfies the following conditions:
(i)
there existsr>0 such that supp(a)⊂[-r,r];(ii)
∥a∥Lαq≤r-2(α+1)(1/p-1/q), where r is given in (i);(iii)
∫ℝa(x)xjdμα(x)=0, for all  j=0,1,…,2s+1,where s=[(α+1)(1/p-1)], (the integer part of (α+1)(1/p-1)).
Let0<p≤1<q≤∞. Our Herz-type Hardy space ℋα,qp is constituted by all those f∈𝒮′(ℝ) that can be represented by(2.16)f=∑j=0∞λjaj,
where λj∈ℂ and aj is a (p,q) atom, for all j∈ℕ, such that ∑j=0∞|λj|p<∞ and the series in (2.16) converges in 𝒮′(ℝ).
We define onℋα,qp the norm ∥·∥ℋα,qp by(2.17)‖f‖Hα,qp:=inf⁡(∑j=0∞|λj|p)1/p,
where the infimum is taken over all those sequences {λj}j∈ℕ⊂ℂ such that f is given by (2.16) for certain (p,q) atoms aj, j∈ℕ.
As the same in [7], we prove the following theorem.Theorem 2.1.
Let0<p≤1<q≤∞ and  f∈ℋα,qp. Then
(2.18)|Fα(f)(y)|≤C|y|2(α+1)(1/p-1)‖f‖Hα,qp,y∈R.


## 3. Dunkl Convolution Operators onℋα,2p

In the following, we study onℋα,2p, 0<p≤1 the Dunkl convolution operators defined by Tkf:=k*αf, where k is a locally integrable function on ℝ.Theorem 3.1.
Let0<p≤q≤1. Assume that for every  n∈ℕ we are given  ξn>0 and a function  gn such that  (i)
gn(x)=0,|x|≥2-n,(ii)
∥gn∥Lα1≤ξn22(α+1)(1/q-1/p)n,(iii)
∥x2s+2ℱα(gn)∥Lα2≤ξn22(α+1)(1/q-1/2)n,s=[(α+1)(1/p-1)].
Suppose also that∑n=0∞(ξn)q<∞ and define  k=∑n=0∞gn. Then  Tk defines a bounded linear mapping from  ℋα,2p into  ℋα,2q.
To prove this theorem the following lemma is needed.Lemma 3.2.
Fory,z∈ℝ and  j∈ℕ, there are a constants  Aj,α, such that
(3.1)τy(xj)(z)=Aj,α∑i=0jj!bi(α)bj-i(α)yizj-i,
where bj(α) are the constants given by (2.4).Proof.
Lety, z∈ℝ. By dominated convergence theorem, we can write
(3.2)τy(xj)(z)=lim⁡t→0∫RxjEα(tx)dνy,z(x),
and by derivation under the integral sign, we get
(3.3)τy(xj)(z)=lim⁡t→0Λα,tj∫REα(tx)dνy,z(x).
Then, from  Theorem 2.4, [6] we obtain
(3.4)τy(xj)(z)=lim⁡t→0Λα,tj(Eα(ty)Eα(tz)).
Let F(t)=Eα(ty)Eα(tz) for |t|<ξ, where ξ>0. According to (2.11), [9],
(3.5)Λα,tjF(t)=F(j)(t)+∑i=1j-1{2iPj-1(t10,…,ti0)F(j)(t⋅t10…ti0)+2jQj-1(t11,…,tj-11)F(j)(t⋅t11…tj1)},
where Pj-1(t10,…,ti0), i=1,2,…,j-1, and Qj-1(t11,…,tj-11) are polynomials of degree at most j-1 with respect to each variable. Moreover,
(3.6)F(j)(t)=∑i=0j(ji)yizj-iEα(i)(ty)Eα(j-i)(tz).
Then, from (2.4) we deduce
(3.7)lim⁡t→0F(j)(t)=∑i=0j(ij)i!(j-i)!bi(α)bj-i(α)yizj-i=∑i=0jj!bi(α)bj-i(α)yizj-i.
Therefore,
(3.8)τy(xj)(z)=lim⁡t→0Λα,tjF(t)=Aj,α∑i=0jj!bi(α)bj-i(α)yizj-i,
where
(3.9)Aj,α=1+∑i=1j-1{2iPj-1(t10,…,ti0)+2jQj-1(t11,…,tj-11)}.
This finishes the proof of the lemma.Proof of Theorem3.1.
Firstly, notice that∥gn∥Lα1≤ξn, n∈ℕ. Hence, the series defining k converges in L1(ℝ,μα) and k∈L1(ℝ,μα).
Leta be a (p,2) atom. Suppose that a(x)=0,|x|>r and that ∥a∥Lα2≤r-2(α+1)(1/p-1/2), where r>0. We can write
(3.10)Tka=∑n=0∞gn*αa.
Step 1.
Letn∈ℕ. From (2.13), for ||y|-|z||≥2-n, we have
(3.11)τygn(z)=∫||y|-|z|||y|+|z|gn(x)dνy,z(x)+∫-(|y|+|z|)-||y|-|z||gn(x)dνy,z(x)=0.
Hence, for |y|>r+2-n, we deduce
(3.12)gn*αa(y)=∫-rra(-z)τygn(z)dμα(z)=0.Step 2.
Firstly, let us consider thatr≥2-n. From (Proposition 3(i), [10]) and condition (ii) of the theorem, we have
(3.13)‖gn*αa‖Lα2≤4‖gn‖Lα1‖a‖Lα2≤4ξn(r+2-n)-2(α+1)(1/q-1/2).
Assume now thatr<2-n. Since ∫ℝa(x)xjdμα(x)=0, j=0,1,…,2s+1, with s=[(α+1)(1/p-1)], we have(3.14)gn*αa(x)=∫Ra(-y)[τygn(x)-∑j=02s+1yjbj(α)Λαjgn(x)]dμα(y),x∈R,
where bj(α) are the constants given by (2.4).
Using the properties of the Dunkl transform established by de Jeu [5] (see also [7, 10]), we deduce
(3.15)‖gn*αa‖Lα2≤∫R|a(-y)|‖τygn-∑j=02s+1yjbj(α)Λαjgn‖Lα2dμα(y)=∫R|a(-y)|‖Fα(τygn)-∑j=02s+1yjbj(α)Fα(Λαjgn)‖Lα2dμα(y)=∫R|a(-y)|‖[Eα(ixy)-∑j=02s+1(ixy)jbj(α)]Fα(gn)‖Lα2dμα(y).
According to  page 302, [7]
(3.16)|Eα(ixy)-∑j=02s+1(ixy)jbj(α)|≤122αΓ(α+1)|xy|2s+2.
Using condition (iii) of the theorem and Hölder's inequality, we get
(3.17)‖gn*αa‖Lα2≤122αΓ(α+1)‖x2s+2Fα(gn)‖Lα2∫-rr|a(y)|y2s+2dμα(y)≤122αΓ(α+1)‖x2s+2Fα(gn)‖Lα2‖a‖Lα2[2∫0ry4s+4dμα(y)]1/2≤γn22(α+1)(1/q-1/2)nr2s-2(α+1)(1/p-1)+2,
where
(3.18)γn:=ξn22αΓ(α+1)(2s+α+3)2α+1Γ(α+1)=cξn.
Using the fact that s-(α+1)(1/p-1)+1>0 and r<2-n, we obtain
(3.19)‖gn*αa‖Lα2≤γn22(α+1)(1/q-1/2)n≤γn(r+2-n)-2(α+1)(1/q-1/2).Step 3.
We now prove for allj=0,…,2s+1; s=[(α+1)(1/p-1)], that
(3.20)∫Rxj(gn*αa)(x)dμα(x)=0.
Fubini's theorem and [6] (see also page 20, [10]) lead to
(3.21)∫Rxj(gn*αa)(x)dμα(x)=∫Rxj∫Ra(y)[∫Rgn(z)dνx,-y(z)]dμα(y)dμα(x)=∫Ra(y)∫Rgn(z)[∫Rxjdνy,z(x)]dμα(z)dμα(y)=∫Ra(y)∫Rgn(z)τy(xj)(z)dμα(z)dμα(y).
Hence, by Lemma 3.2 and by taking into account that ∫ℝa(x)xjdμα(x)=0, j=0,1,…,2s+1, with s=[(α+1)(1/p-1)], we get
(3.22)∫Rxj(gn*αa)(x)dμα(x)=Aj,α∑i=0jj!bi(α)bj-i(α)[∫Ryia(y)dμα(y)][∫Rgn(z)zj-idμα(z)]=0.
According to the previous three steps, we conclude that(1/γn)(gn*αa) is a (q,2) atom. Then, Tka∈ℋα,2q, and(3.23)‖Tka‖Hα,2q≤c(∑n=0∞(ξn)q)1/q.
Let nowf be in ℋα,2p. Assume that f=∑j=0∞λjaj, where λj∈ℂ and aj is a (p,2) atom, for every j∈ℕ, and such that ∑j=0∞|λj|p<∞. The series defining f converges in L1(ℝ,μα). In fact, it is sufficient to note that ∥a∥Lα1≤(1/2αΓ(α+1))r2(α+1)(1-1/p). Hence f∈L1(ℝ,μα). Moreover, k∈L1(ℝ,μα). Then by (Proposition 3(i), [10]), the operator Tk is bounded from L1(ℝ,μα) into itself, and from this, we deduce that Tkf=∑j=0∞λjTkaj. Using the fact that ∑j=0∞|λj|≤(∑j=0∞|λj|p)1/p, we obtain(3.24)‖Tkf‖Hα,2q≤c(∑n=0∞(ξn)q)1/q‖f‖Hα,2p.
This completes the proof of the Theorem 3.1.We now study the Dunkl convolution operatorsTk on the Herz-type Hardy spaces ℋα,21.Theorem 3.3.
Letk be a locally integrable function on ℝ. Assume that the following three conditions are satisfied:(i)
Tk defines a bounded linear operator from L2(ℝ,μα) into itself.(ii)
Tk defines a bounded linear operator from L1(ℝ,μα) into 𝒮′(ℝ).(iii)
There existA and c>1 such that(3.25)∫|z|>cR|τxk(z)-k(z)|dμα(z)≤A,|x|∈(0,R),R>0.
Then Tk defines a bounded linear mapping from ℋα,21 into L1(ℝ,μα).Proof.
Leta be a (1,2) atom. We choose r>0 such that supp(a)⊂[-r,r] and ∥a∥Lα2≤r-(α+1). We can write
(3.26)∫R|Tka(x)|dμα(x)=[∫|x|<cr+∫|x|≥cr]|Tka(x)|dμα(x):=I1+I2.
Here c>1 is the one given in (iii).
From condition (i) of the theorem and Hölder's inequality, we deduce that(3.27)I1≤[∫R|Tka(x)|2dμα(x)]1/2[2∫0crdμα(x)]1/2≤C‖a‖Lα2rα+1≤C.
Also, by taking into account that ∫ℝa(y)dμα(y)=0, the condition (iii) of the theorem allows us to write
(3.28)I2=∫|x|≥cr|∫Rτxk(y)a(-y)dμα(y)|dμα(x)=∫|x|≥cr|∫R{τxk(y)-k(x)}a(-y)dμα(y)|dμα(x)≤∫-rr|a(-y)|∫|x|≥cr|τyk(x)-k(x)|dμα(x)dμα(y)≤C∫-rr|a(y)|dμα(y)≤C‖a‖Lα2[2∫0rdμα(y)]1/2≤C.
Hence, it concludes that
(3.29)‖Tka‖Lα1≤C.
Note that the positive constant C is not depending on the (1,2) atom a.
Let nowf be in ℋα,21. Then f∈𝒮′(ℝ) and f=∑j=0∞λjaj, where λj∈ℂ and aj is a (1,2) atom, for every j∈ℕ and ∑j=0∞|λj|<∞.
The series definingf converges in L1(ℝ,μα). In fact, it is sufficient to note that ∥a∥Lα1≤1/2αΓ(α+1), for every (1,2) atom a. Hence f∈L1(ℝ,μα). Then the condition (ii) of the theorem implies that
(3.30)Tkf=∑j=0∞λjTkaj.
By (3.29) the series in (3.30) converges in L1(ℝ,μα) and ∥Tkf∥Lα1≤C∑j=0∞|λj|. Hence ∥Tkf∥Lα1≤C∥f∥ℋα,21.


## 4. Maximal Bochner-Riesz Operators onℋα,∞p

The Bochner-Riesz meanσα,tη, for t>0 and η>α+1/2 associated to the Dunkl transform ℱα is defined by (4.1)σα,tη(f)(x):=∫-tt(1-y2t2)ηEα(ixy)Fα(f)(y)dμα(y),f∈S(R).
The maximal operator σαη, η>α+1/2 associated to the Bochner-Riesz means σα,tη, t>0, is defined by (4.2)σαη(f):=sup⁡t>0⁡|σα,tη(f)|.Lemma 4.1.
Fort>0 and η>α+1/2,  one has(i)
σα,tη(f)=Φα,tη*αf, where
(4.3)Φα,tη(x):=Γ(η+1)2α+1Γ(α+η+2)t2α+2Iα+η+1(itx).
Here ℑα is the modified spherical Bessel function given by (2.3).(ii)
The operatorσαη is bounded from Lp(ℝ,μα),  1≤p≤∞ into itself.Proof.
Lett>0 and η>α+1/2.(i)
By taking into account that the functions|z|α+1/2ℑα(iz) and ℑα(iz) are bounded on ℝ it is not hard to see that(4.4)‖Φα,tη‖Lα1=‖Φα,1η‖Lα1=Γ(η+1)2α+1Γ(α+η+2)∫R|Iα+η+1(ix)|dμα(x)<∞.
On the other hand, from [11], we have
(4.5)∫01(1-y2)ηIα(itxy)y2α+1dy=Γ(α+1)Γ(η+1)2Γ(α+η+2)Iα+η+1(itx).
Thus,
(4.6)Φα,tη(x)=t2α+22αΓ(α+1)∫01(1-y2)ηIα(itxy)y2α+1dy=∫-tt(1-y2t2)ηEα(ixy)dμα(y).
Applying  Inversion Theorem[5], we obtain
(4.7)Fα(Φα,tη)(y)=(1-y2t2)ηχ(-t,t)(y),
where χ(-t,t) is the characteristic function of the set (-t,t). Thus,
(4.8)σα,tη(f)(x)=∫REα(ixy)Fα(Φα,tη)(y)Fα(f)(y)dμα(y),
and from (Proposition 3 (ii), [10]), we deduce that
(4.9)σα,tη(f)(x)=Φα,tη*αf(x).(ii)
Using (i) and (Proposition 3 (i), [10]), we obtain(4.10)‖σαη(f)‖Lαp≤4‖Φα,1η‖Lα1‖f‖Lαp.
This clearly yields the result.Theorem 4.2.
Let0<p≤1<q≤∞ and  f∈ℋα,qp. For  t>0 and  η>α+1/2, the operator  σα,tη extended to a bounded operator from  ℋα,qp into  𝒮′(ℝ).Proof.
According to Theorem2.1, if f∈ℋα,qp, then σα,tη(f) is in 𝒮′(ℝ) and it is defined by
(4.11)〈σα,tη(f),φ〉=∫-tt(1-y2t2)ηFα(f)(y)Fα(φ)(y)dμα(y),φ∈S(R).
Moreover,
(4.12)|〈σα,tη(f),φ〉|≤C‖f‖Hα,qp∫R|y|2(α+1)(1/p-1)|Fα(φ)(y)|dμα(y),φ∈S(R).
Hence σα,tη is a bounded operator from ℋα,qp into 𝒮′(ℝ).We now study the behavior of the maximal Bochner-Riesz operatorσαη on ℋα,∞p.Theorem 4.3.
Let2(α+1)/(α+η+3/2)<p≤1. Then the maximal Bochner-Riesz operator σαη,  α+1/2<η<α+3/2 is bounded from ℋα,∞p into Lp(ℝ,μα).
To prove this theorem we need the following lemma.Lemma 4.4.
(i) Forx,y∈ℝ and η>α+1/2,
(4.13)|τxΦα,tη(y)|≤Ctα-η+1/2||x|-|y||-(α+η+3/2).
(ii)  For0<|y|<|x| and η>α+1/2,(4.14)|τxΦα,tη(y)-Φα,tη(x)|≤C|y|tα-η+3/2||x|-|y||-(α+η+3/2).Proof.
(i) Since the function|z|α+1/2ℑα(iz) is bounded on ℝ, it follows that
(4.15)|Φα,tη(z)|≤Ctα-η+1/2|z|-(α+η+3/2);t>0,z∈R.
According to [6] and the fact that Φα,tη is even, we obtain
(4.16)τxΦα,tη(y)=∫0πΦα,tη((x,y)θ)dρx,y(θ),
where
(4.17)dρx,y(θ):=Γ(α+1)πΓ(α+1/2){1-sgn⁡(xy)cos⁡θ}sin⁡2αθdθ,(x,y)θ:=x2+y2-2|xy|cos⁡θ.
By (4.15) and using the fact that (x,y)θ≥||x|-|y||, we deduce that
(4.18)|τxΦα,tη(y)|≤Ctα-η+1/2∫0π(x,y)θ-(α+η+3/2)dρx,y(θ)≤Ctα-η+1/2||x|-|y||-(α+η+3/2).(ii)
From [11], we have(4.19)ddzΦα,tη(z)=-Γ(η+1)2α+2Γ(α+η+3)t2α+4zIα+η+2(itz).
Then, similarly to the proof in (i), we have
(4.20)|ddzΦα,tη(z)|≤Ctα-η+3/2|z|-(α+η+3/2);t>0,z∈R,(4.21)τxΦα,tη(y)-Φα,tη(x)=∫0π[Φα,tη((x,y)θ)-Φα,tη((x,0)θ)]dρx,y(θ),
which can be written as
(4.22)τxΦα,tη(y)-Φα,tη(x)=∫0π∫01dds[Φα,tη((x,sy)θ)]dsdρx,y(θ)≤|y|∫01∫0πddsΦα,tη((x,sy)θ)dρx,y(θ)ds.
Then from (4.20), it follows
(4.23)|τxΦα,tη(y)-Φα,tη(x)|≤C|y|tα-η+3/2∫01∫0π(x,sy)θ-(α+η+3/2)dρx,y(θ)ds≤C|y|tα-η+3/2∫01||x|-s|y||-(α+η+3/2)ds.
Hence, if 0<|y|<|x|, we obtain
(4.24)|τxΦα,tη(y)-Φα,tη(x)|≤C|y|tα-η+3/2||x|-|y||-(α+η+3/2),
which completes the proof of the lemma.Proof of Theorem4.3.
Let us first show thatC>0, exists such that
(4.25)‖σαη(a)‖Lαp≤C,
for every (p,∞) atom a.
Leta be an (p,∞) atom. Suppose that a(x)=0, |x|>r and ∥a∥Lα∞≤r-2(α+1)/p. We choose ℓ∈ℤ such that 2ℓ-1<r≤2ℓ, we write
(4.26)∫|x|≥4r|σαη(a)(x)|pdμα(x)≤I1+I2,
where
(4.27)I1:=∑i=1∞∫(i+1)2l≤|x|≤(i+2)2lsup⁡t≥δi|σα,tη(a)(x)|pdμα(x),(4.28)I2:=∑i=1∞∫(i+1)2l≤|x|≤(i+2)2lsup⁡t<δi|σα,tη(a)(x)|pdμα(x),
being δi=2-ℓ/ib, where b will be specified later.
According to Lemma4.4 (i), for |x|∈[(i+1)2ℓ,(i+2)2ℓ], i=1,2,…, we get
(4.29)|σα,tη(a)(x)|≤∫-rr|a(-y)||τxΦα,tη(y)|dμα(y)≤Ctα-η+1/2r-2(α+1)/p∫02l||x|-|y||-(α+η+3/2)dμα(y)≤C(2lt)α-η+1/2iα+η+3/2r2(α+1)/p.
Then, using the fact that2ℓ-1<r≤2ℓ, we obtain
(4.30)I1≤C∑i=1∞(δiα-η+1/22l{α-η+1/2-2(α+1)/p}iα+η+3/2)pi2α+122l(α+1),
and hence, it concludes that
(4.31)I1≤C∑i=1∞i2α+1-{α+η+3/2+(α-η+1/2)b}p.
Note that the last series is convergent provide that b<(p(α+η+3/2)-2(α+1)/p(η-α-1/2)).
On the other hand, since∫ℝa(y)dμα(y)=0, from Lemma 4.4 (ii) we have
(4.32)|σα,tη(a)(x)|≤∫-rr|a(-y)||τxΦα,tη(y)-Φα,tη(x)|dμα(y)≤Ctα-η+3/2r-2(α+1)/p∫02l||x|-|y||-(α+η+3/2)|y|dμα(y)≤C(2lt)α-η+3/2iα+η+3/2r2(α+1)/p.
Then, for|x|≥4r, we obtain
(4.33)I2≤C∑i=1∞(δiα-η+3/22l{α-η+3/2-2(α+1)/p}iα+η+3/2)pi2α+122l(α+1),
and in fact, we deduce that
(4.34)I2≤C∑i=1∞i2α+1-{α+η+3/2+(α-η+3/2)b}p.
The last series converges provided that b>p(α+η+3/2)-2(α+1)p(η-α-3/2).
Note that we can findb such that the series in (4.31) and (4.34) converge if and only if p>(2(α+1)/α+η+3/2). By combining (4.31) and (4.34) we show that
(4.35)∫|x|≥4r|σαη(a)(x)|pdμα(x)≤C,
for a certain C>0 that is not depending on a.
From Lemma4.1 (ii) and (4.35) we deduce that(4.36)‖σαη(a)‖Lαpp≤[∫|x|<4r+∫|x|≥4r]|σαη(a)(x)|pdμα(x)≤C[‖a‖Lk∞p∫|x|<4rdμα(x)+1]≤C,
that is, (4.25) holds. Let now f be in ℋα,∞p. Assume that f=∑j=0∞λjaj, where the series converges in 𝒮′(ℝ), and for every j∈ℕ, aj is a (p,∞) atom and λj∈ℂ, such that ∑j=0∞|λj|p<∞. From Theorem 4.2, for 0<p≤1, we can write
(4.37)σα,tη(f)(x)=∑j=0∞λjσα,tη(aj)(x);t>0,x∈R.
Hence, from (4.25) it follows ∥σαη(f)∥Lαpp≤C∑j=0∞|λj|p. Thus we conclude that ∥σαη(f)∥Lαp≤C∥f∥ℋα,∞p.



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