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

Abstract

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

1. Introduction

The classical theory of Hardy spaces on ℝ^𝑛 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 operator Λ_𝛼, 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 operator Λ_𝛼, 𝛼>−1/2, associated with the reflection group ℤ₂ on ℝ: Λ_𝛼𝑓(𝑥)𝑑∶=𝑓(𝑥)+𝑑𝑥2𝛼+1𝑥𝑓(𝑥)−𝑓(−𝑥)2, (1.1) 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 Λ_𝛼, the Dunkl transform ℱ_𝛼, the Dunkl convolution ∗_𝛼, and a certain measure 𝜇_𝛼 on ℝ.

In this paper we define a Herz-type Hardy spaces ℋ^𝑝_𝛼,𝑞, 0<𝑝≤1<𝑞≤∞, in the Dunkl setting. Next, we consider the Dunkl convolution operators 𝑇_𝑘𝑓∶=𝑘∗_𝛼𝑓, where 𝑘 is a locally integrable function on ℝ. We use the atomic decomposition of the Herz-type Hardy spaces ℋ^𝑝_𝛼,𝑞 to study the ℋ^𝑝_𝛼,2−ℋ^𝑞_𝛼,2-bounded and the ℋ¹_𝛼,2−𝐿¹-bounded of the operators 𝑇_𝑘. Finally, we establish a version of multiplier theorem for the maximal Bochner-Riesz operators 𝜎^𝜂_𝛼, 𝑡>0, 𝜂>𝛼+1/2: 𝜎^𝜂_𝛼𝑓∶=sup_𝑡>0Φ^𝜂_𝛼,𝑡∗_𝛼𝑓, (1.2) where Φ^𝜂_𝛼,𝑡(𝑥)Γ∶=𝜂+12^𝛼+1𝑡^∞2𝛼+2_𝑛=0(−1)^𝑛2^2𝑛()𝑛!Γ𝑛+𝛼+𝜂+2𝑡𝑥^2𝑛, (1.3) on the Herz-type Hardy spaces ℋ^𝑝_𝛼,∞. In this version we prove the ℋ^𝑝_𝛼,∞−𝐿^𝑝-bounded of the operators 𝜎^𝜂_𝛼, for 𝛼+1/2<𝜂<𝛼+3/2.

The content of this work is the following. In Section 2, we recall some results about harmonic analysis and we define a Herz-type Hardy spaces ℋ^𝑝_𝛼,𝑞, 0<𝑝≤1<𝑞≤∞ for the Dunkl operator Λ_𝛼. In Section 3, we study the ℋ^𝑝_𝛼,2−ℋ^𝑞_𝛼,2-bounded and the ℋ¹_𝛼,2−𝐿¹-bounded of the convolution operators 𝑇_𝑘. In Section 4, we prove the ℋ^𝑝_𝛼,∞−𝐿^𝑝-bounded of the maximal Bochner-Riesz operators 𝜎^𝜂_𝛼.

Throughout the paper we use the classic notation. Thus 𝒮(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 on ℝ

For 𝛼≥−1/2 and 𝜆∈ℂ, the initial problem Λ_𝛼𝑓(𝑥)=𝜆𝑓(𝑥),𝑓(0)=1, (2.1) has a unique analytic solution 𝐸_𝛼(𝜆𝑥) called Dunkl kernel [4–6] given by 𝐸_𝛼(𝜆𝑥)=ℑ_𝛼(𝜆𝑥)+𝜆𝑥2()ℑ𝛼+1^𝛼+1(𝜆𝑥),𝑥∈ℝ, (2.2) where ℑ_𝛼(𝜆𝑥)∶=Γ(𝛼+1)^∞_𝑛=0(𝜆𝑥)^2𝑛2^2𝑛(),𝑛!Γ𝑛+𝛼+1 (2.3) is the modified spherical Bessel function of order α.

We notice that, the Dunkl kernel 𝐸_𝛼(𝜆𝑥) can be expanded in a power series [7] in the form 𝐸_𝛼(𝜆𝑥)=^∞_𝑛=0(𝜆𝑥)^𝑛𝑏_𝑛(𝛼), (2.4) where 𝑏_2𝑛(𝛼)=2^2𝑛𝑛!Γ()Γ()𝛼+1𝑛+𝛼+1,𝑏_2𝑛+1(𝛼)=2(𝛼+1)𝑏_2𝑛(𝛼+1). (2.5)

Note. Let 𝜇_𝛼 be the measure on ℝ given by 𝑑𝜇_𝛼(𝑥)1∶=2^𝛼+1Γ()|𝑥|𝛼+1^2𝛼+1𝑑𝑥. (2.6) We denote by 𝐿^𝑝(ℝ,𝜇_𝛼), 𝑝∈]0,∞], the space of measurable functions 𝑓 on ℝ, such that 𝑓_𝐿^𝑝_𝛼∶=_ℝ𝑓(𝑥)^𝑝𝑑𝜇_𝛼(𝑥)^1/𝑝][,𝑓<∞,𝑝∈0,∞_𝐿∞𝛼∶=esssup_𝑥∈ℝ𝑓(𝑥)<∞. (2.7)
The Dunkl kernel gives rise to an integral transform, called Dunkl transform on ℝ, 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 function 𝑓∈𝐿¹(ℝ,𝜇_𝛼), is given by ℱ_𝛼𝑓(𝜆)∶=_ℝ𝐸_𝛼(−𝑖𝜆𝑥)𝑓(𝑥)𝑑𝜇_𝛼(𝑥),𝜆∈ℝ. (2.8)
For 𝑇∈𝒮′(ℝ), we define the Dunkl transform ℱ_𝛼(𝑇) of 𝑇, by ℱ_𝛼(𝑇),𝜑∶=𝑇,ℱ_𝛼𝜑,𝜑∈𝒮(ℝ). (2.9)

Note. For all 𝑥,𝑦,𝑧∈ℝ, we put 𝑊_𝛼𝑥,𝑦,𝑧∶=1−𝜎_{𝑥,𝑦,𝑧}+𝜎_{𝑧,𝑥,𝑦}+𝜎_{𝑧,𝑦,𝑥}Δ_𝛼|𝑥|,𝑦,|𝑧|, (2.10) where 𝜎_{𝑥,𝑦,𝑧}𝑥∶=²+𝑦²−𝑧²{0}Δ2𝑥𝑦,if𝑥,𝑦∈ℝ⧵0,otherwise_𝛼|𝑥|,𝑦,|𝑧|𝑑∶=^𝛼|𝑥|+𝑦²−𝑧²𝑧²−|𝑥|−𝑦^2𝛼−1/2𝑥𝑦𝑧^2𝛼if|𝑧|∈𝐴_𝑥,𝑦,𝑑0,otherwise,_𝛼=2^1−𝛼(Γ(𝛼+1))²√𝜋Γ(𝛼+1/2),𝐴_𝑥,𝑦=|𝑥|−𝑦,|𝑥|+𝑦. (2.11) We denote by 𝜈_𝑥,𝑦 the following signed measure: 𝑑𝜈_𝑥,𝑦(𝑧)𝑊∶=^𝛼𝑥,𝑦,𝑧𝑑𝜇_𝛼(𝑧)if𝑥,𝑦∈ℝ⧵{0},𝑑𝛿_𝑥(𝑧)if𝑦=0,𝑑𝛿_𝑦(𝑧)if𝑥=0. (2.12) The Dunkl translation operators 𝜏_𝑥, 𝑥∈ℝ (see [6]) are defined for 𝑓∈𝒞(ℝ) (the space of continuous functions on ℝ), by 𝜏_𝑥𝑓𝑦∶=_𝑥|−𝑦^{|𝑥|+|𝑦|}_|𝑓(𝑧)𝑑𝜈_𝑥,𝑦(𝑧)+_{−𝑥|+𝑦}^{−||𝑥|−|𝑦||}_|𝑓(𝑧)𝑑𝜈_𝑥,𝑦(𝑧). (2.13)
Let 𝑓 and 𝑔 be two functions in 𝒮(ℝ). We define the Dunkl convolution product ∗_𝛼 of 𝑓 and 𝑔 by𝑓∗_𝛼𝑔(𝑥)∶=_ℝ𝜏_𝑥𝑓−𝑦𝑔𝑦𝑑𝜇_𝛼𝑦,𝑥∈ℝ. (2.14)
For 𝑇∈𝒮′(ℝ) and 𝑓∈𝒮(ℝ), we define the Dunkl convolution product 𝑇∗_𝛼𝑓 by𝑇∗_𝛼𝑓(𝑥)∶=𝑇𝑦,𝜏_𝑥𝑓−𝑦,𝑥∈ℝ. (2.15)
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.
Let 0<𝑝≤1<𝑞≤∞. A measurable function 𝑎 on ℝ is called a (𝑝,𝑞) atom, if 𝑎 satisfies the following conditions:
(i)there exists 𝑟>0 such that supp(𝑎)⊂[−𝑟,𝑟];(ii)‖𝑎‖_𝐿^𝑞_𝛼≤𝑟^{−2(𝛼+1)(1/𝑝−1/𝑞)}, where 𝑟 is given in (i);(iii)∫_ℝ𝑎(𝑥)𝑥^𝑗𝑑𝜇_𝛼(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)).
Let 0<𝑝≤1<𝑞≤∞. Our Herz-type Hardy space ℋ^𝑝_𝛼,𝑞 is constituted by all those 𝑓∈𝒮′(ℝ) that can be represented by𝑓=^∞_𝑗=0𝜆_𝑗𝑎_𝑗, (2.16) where 𝜆_𝑗∈ℂ and 𝑎_𝑗 is a (𝑝,𝑞) atom, for all 𝑗∈ℕ, such that ∑^∞_𝑗=0|𝜆_𝑗|^𝑝<∞ and the series in (2.16) converges in 𝒮′(ℝ).
We define on ℋ^𝑝_𝛼,𝑞 the norm ‖⋅‖_ℋ^𝑝_𝛼,𝑞 by𝑓_ℋ^𝑝_𝛼,𝑞∶=inf^∞_𝑗=0|𝜆_𝑗|^𝑝1/𝑝, (2.17) where the infimum is taken over all those sequences {𝜆_𝑗}_𝑗∈ℕ⊂ℂ such that 𝑓 is given by (2.16) for certain (𝑝,𝑞) atoms 𝑎_𝑗, 𝑗∈ℕ.
As the same in [7], we prove the following theorem.

Theorem 2.1. Let 0<𝑝≤1<𝑞≤∞ and 𝑓∈ℋ^𝑝_𝛼,𝑞. Then ℱ_𝛼𝑓𝑦≤𝐶𝑦^{2(𝛼+1)(1/𝑝−1)}𝑓_ℋ^𝑝_𝛼,𝑞,𝑦∈ℝ. (2.18)

3. Dunkl Convolution Operators on ℋ^𝑝_𝛼,2

In the following, we study on ℋ^𝑝_𝛼,2, 0<𝑝≤1 the Dunkl convolution operators defined by 𝑇_𝑘𝑓∶=𝑘∗_𝛼𝑓, where 𝑘 is a locally integrable function on ℝ.

Theorem 3.1. Let 0<𝑝≤𝑞≤1. Assume that for every 𝑛∈ℕ we are given 𝜉_𝑛>0 and a function 𝑔_𝑛 such that (i)𝑔_𝑛(𝑥)=0,|𝑥|≥2^−𝑛,(ii)‖𝑔_𝑛‖_𝐿¹_𝛼≤𝜉_𝑛2^{2(𝛼+1)(1/𝑞−1/𝑝)𝑛},(iii)‖𝑥^2𝑠+2ℱ_𝛼(𝑔_𝑛)‖_𝐿²_𝛼≤𝜉_𝑛2^{2(𝛼+1)(1/𝑞−1/2)𝑛},𝑠=[(𝛼+1)(1/𝑝−1)].
Suppose also that ∑^∞_𝑛=0(𝜉_𝑛)^𝑞<∞ and define 𝑘=∑^∞_𝑛=0𝑔_𝑛. Then 𝑇_𝑘 defines a bounded linear mapping from ℋ^𝑝_𝛼,2 into ℋ^𝑞_𝛼,2.
To prove this theorem the following lemma is needed.

Lemma 3.2. For 𝑦,𝑧∈ℝ and 𝑗∈ℕ, there are a constants 𝐴_𝑗,𝛼, such that 𝜏_𝑦𝑥^𝑗(𝑧)=𝐴_𝑗,𝛼^𝑗_𝑖=0𝑗!𝑏_𝑖(𝛼)𝑏_𝑗−𝑖(𝛼)𝑦^𝑖𝑧^𝑗−𝑖, (3.1) where 𝑏_𝑗(𝛼) are the constants given by (2.4).

Proof. Let 𝑦, 𝑧∈ℝ. By dominated convergence theorem, we can write 𝜏_𝑦𝑥^𝑗(𝑧)=lim_𝑡→0ℝ𝑥^𝑗𝐸_𝛼(𝑡𝑥)𝑑𝜈_𝑦,𝑧(𝑥), (3.2) and by derivation under the integral sign, we get 𝜏_𝑦𝑥^𝑗(𝑧)=lim_𝑡→0Λ^𝑗_𝛼,𝑡ℝ𝐸_𝛼(𝑡𝑥)𝑑𝜈_𝑦,𝑧(𝑥). (3.3) Then, from Theorem 2.4, [6] we obtain 𝜏_𝑦𝑥^𝑗(𝑧)=lim_𝑡→0Λ^𝑗_𝛼,𝑡𝐸_𝛼𝑡𝑦𝐸_𝛼(𝑡𝑧). (3.4) Let 𝐹(𝑡)=𝐸_𝛼(𝑡𝑦)𝐸_𝛼(𝑡𝑧) for |𝑡|<𝜉, where 𝜉>0. According to (2.11), [9], Λ^𝑗_𝛼,𝑡𝐹(𝑡)=𝐹^(𝑗)(𝑡)+^𝑗−1_𝑖=12^𝑖𝑃_𝑗−1𝑡⁰₁,…,𝑡⁰_𝑖𝐹^(𝑗)𝑡⋅𝑡⁰₁…𝑡⁰_𝑖+2^𝑗𝑄_𝑗−1𝑡¹₁,…,𝑡¹_𝑗−1𝐹^(𝑗)𝑡⋅𝑡¹₁…𝑡¹_𝑗, (3.5) where 𝑃_𝑗−1(𝑡⁰₁,…,𝑡⁰_𝑖), 𝑖=1,2,…,𝑗−1, and 𝑄_𝑗−1(𝑡¹₁,…,𝑡¹_𝑗−1) are polynomials of degree at most 𝑗−1 with respect to each variable. Moreover, 𝐹^(𝑗)(𝑡)=^𝑗_𝑖=0𝑗𝑖𝑦^𝑖𝑧^𝑗−𝑖𝐸_𝛼^(𝑖)𝑡𝑦𝐸_𝛼^{(𝑗−𝑖)}(𝑡𝑧). (3.6) Then, from (2.4) we deduce lim_𝑡→0𝐹^(𝑗)(𝑡)=^𝑗_𝑖=0^𝑗_𝑖𝑖!𝑗−𝑖!𝑏_𝑖(𝛼)𝑏_𝑗−𝑖(𝛼)𝑦^𝑖𝑧^𝑗−𝑖=^𝑗_𝑖=0𝑗!𝑏_𝑖(𝛼)𝑏_𝑗−𝑖(𝛼)𝑦^𝑖𝑧^𝑗−𝑖. (3.7) Therefore, 𝜏_𝑦𝑥^𝑗(𝑧)=lim_𝑡→0Λ^𝑗_𝛼,𝑡𝐹(𝑡)=𝐴^𝑗_{𝑗,𝛼𝑖=0}𝑗!𝑏_𝑖(𝛼)𝑏_𝑗−𝑖(𝛼)𝑦^𝑖𝑧^𝑗−𝑖, (3.8) where 𝐴_𝑗,𝛼=1+^𝑗−1_𝑖=12^𝑖𝑃_𝑗−1𝑡⁰₁,…,𝑡⁰_𝑖+2^𝑗𝑄_𝑗−1𝑡¹₁,…,𝑡¹_𝑗−1. (3.9) This finishes the proof of the lemma.

Proof of Theorem 3.1. Firstly, notice that ‖𝑔_𝑛‖_𝐿¹_𝛼≤𝜉_𝑛, 𝑛∈ℕ. Hence, the series defining 𝑘 converges in 𝐿¹(ℝ,𝜇_𝛼) and 𝑘∈𝐿¹(ℝ,𝜇_𝛼).
Let 𝑎 be a (𝑝,2) atom. Suppose that 𝑎(𝑥)=0,|𝑥|>𝑟 and that ‖𝑎‖_𝐿²_𝛼≤𝑟^{−2(𝛼+1)(1/𝑝−1/2)}, where 𝑟>0. We can write 𝑇_𝑘𝑎=^∞_𝑛=0𝑔_𝑛∗_𝛼𝑎. (3.10)
Step 1. Let 𝑛∈ℕ. From (2.13), for ||𝑦|−|𝑧||≥2^−𝑛, we have 𝜏_𝑦𝑔_𝑛(𝑧)=_𝑦−|𝑧^{|𝑦|+|𝑧|}_|𝑔_𝑛(𝑥)𝑑𝜈_𝑦,𝑧(𝑥)+_{−𝑦+|𝑧|}^{−||𝑦|−|𝑧||}𝑔_𝑛(𝑥)𝑑𝜈_𝑦,𝑧(𝑥)=0. (3.11) Hence, for |𝑦|>𝑟+2^−𝑛, we deduce 𝑔_𝑛∗_𝛼𝑎𝑦=^𝑟_−𝑟𝑎(−𝑧)𝜏_𝑦𝑔_𝑛(𝑧)𝑑𝜇_𝛼(𝑧)=0. (3.12)Step 2. Firstly, let us consider that 𝑟≥2^−𝑛. From (Proposition 3(i), [10]) and condition (ii) of the theorem, we have 𝑔_𝑛∗_𝛼𝑎_𝐿2𝛼≤4𝑔_{𝑛𝐿1𝛼}‖𝑎‖_𝐿²_𝛼≤4𝜉_𝑛𝑟+2^{−𝑛−2(𝛼+1)(1/𝑞−1/2)}. (3.13)
Assume now that 𝑟<2^−𝑛. Since ∫_ℝ𝑎(𝑥)𝑥^𝑗𝑑𝜇_𝛼(𝑥)=0, 𝑗=0,1,…,2𝑠+1, with 𝑠=[(𝛼+1)(1/𝑝−1)], we have𝑔_𝑛∗_𝛼𝑎(𝑥)=_ℝ𝑎−𝑦𝜏_𝑦𝑔_𝑛(𝑥)−^2𝑠+1_𝑗=0𝑦^𝑗𝑏_𝑗(𝛼)Λ^𝑗_𝛼𝑔_𝑛(𝑥)𝑑𝜇_𝛼𝑦,𝑥∈ℝ, (3.14) where 𝑏_𝑗(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 𝑔_𝑛∗_𝛼𝑎_𝐿2𝛼≤∫_ℝ𝑎−𝑦𝜏_𝑦𝑔_𝑛−∑^2𝑠+1_𝑗=0𝑦^𝑗𝑏_𝑗(𝛼)Λ^𝑗_𝛼𝑔_{𝑛𝐿2𝛼}𝑑𝜇_𝛼𝑦=∫_ℝ𝑎−𝑦ℱ_𝛼𝜏_𝑦𝑔_𝑛−∑^2𝑠+1_𝑗=0𝑦^𝑗𝑏_𝑗(𝛼)ℱ_𝛼Λ^𝑗_𝛼𝑔_{𝑛𝐿2𝛼}𝑑𝜇_𝛼𝑦=∫_ℝ𝑎−𝑦𝐸_𝛼𝑖𝑥𝑦−∑^2𝑠+1_𝑗=0𝑖𝑥𝑦^𝑗𝑏_𝑗(𝛼)ℱ_𝛼𝑔_{𝑛𝐿2𝛼}𝑑𝜇_𝛼𝑦. (3.15) According to page 302, [7] 𝐸_𝛼𝑖𝑥𝑦−^2𝑠+1_𝑗=0𝑖𝑥𝑦^𝑗𝑏_𝑗(𝛼)≤12^2𝛼Γ(𝛼+1)𝑥𝑦^2𝑠+2. (3.16) Using condition (iii) of the theorem and Hölder's inequality, we get 𝑔_𝑛∗_𝛼𝑎_𝐿2𝛼≤12^2𝛼Γ()𝑥𝛼+1^2𝑠+2ℱ_𝛼𝑔_{𝑛𝐿2𝛼}^𝑟_−𝑟𝑎𝑦𝑦^2𝑠+2𝑑𝜇_𝛼𝑦≤12^2𝛼Γ()𝑥𝛼+1^2𝑠+2ℱ_𝛼𝑔_{𝑛𝐿2𝛼}‖𝑎‖_𝐿²_𝛼2^𝑟₀𝑦^4𝑠+4𝑑𝜇_𝛼𝑦^1/2≤𝛾_𝑛2^{2(𝛼+1)(1/𝑞−1/2)𝑛}𝑟^{2𝑠−2(𝛼+1)(1/𝑝−1)+2}, (3.17) where 𝛾_𝑛𝜉∶=^𝑛2^2𝛼Γ(𝛼+1)(2𝑠+𝛼+3)2^𝛼+1Γ(𝛼+1)=𝑐𝜉_𝑛. (3.18) Using the fact that 𝑠−(𝛼+1)(1/𝑝−1)+1>0 and 𝑟<2^−𝑛, we obtain 𝑔_𝑛∗_𝛼𝑎_𝐿2𝛼≤𝛾_𝑛2^{2(𝛼+1)(1/𝑞−1/2)𝑛}≤𝛾_𝑛𝑟+2^{−𝑛−2(𝛼+1)(1/𝑞−1/2)}. (3.19)Step 3. We now prove for all 𝑗=0,…,2𝑠+1; 𝑠=[(𝛼+1)(1/𝑝−1)], that _ℝ𝑥^𝑗𝑔_𝑛∗_𝛼𝑎(𝑥)𝑑𝜇_𝛼(𝑥)=0. (3.20) Fubini's theorem and [6] (see also page 20, [10]) lead to _ℝ𝑥^𝑗𝑔_𝑛∗_𝛼𝑎(𝑥)𝑑𝜇_𝛼(𝑥)=_ℝ𝑥^𝑗_ℝ𝑎𝑦_ℝ𝑔_𝑛(𝑧)𝑑𝜈_𝑥,−𝑦(𝑧)𝑑𝜇_𝛼𝑦𝑑𝜇_𝛼(𝑥)=_ℝ𝑎𝑦_ℝ𝑔_𝑛(𝑧)_ℝ𝑥^𝑗𝑑𝜈_𝑦,𝑧(𝑥)𝑑𝜇_𝛼(𝑧)𝑑𝜇_𝛼𝑦=_ℝ𝑎𝑦_ℝ𝑔_𝑛(𝑧)𝜏_𝑦𝑥^𝑗(𝑧)𝑑𝜇_𝛼(𝑧)𝑑𝜇_𝛼𝑦. (3.21) Hence, by Lemma 3.2 and by taking into account that ∫_ℝ𝑎(𝑥)𝑥^𝑗𝑑𝜇_𝛼(𝑥)=0, 𝑗=0,1,…,2𝑠+1, with 𝑠=[(𝛼+1)(1/𝑝−1)], we get _ℝ𝑥^𝑗𝑔_𝑛∗_𝛼𝑎(𝑥)𝑑𝜇_𝛼(𝑥)=𝐴_𝑗,𝛼^𝑗_𝑖=0𝑗!𝑏_𝑖(𝛼)𝑏_𝑗−𝑖(𝛼)_ℝ𝑦^𝑖𝑎𝑦𝑑𝜇_𝛼𝑦_ℝ𝑔_𝑛(𝑧)𝑧^𝑗−𝑖𝑑𝜇_𝛼(𝑧)=0. (3.22)
According to the previous three steps, we conclude that (1/𝛾_𝑛)(𝑔_𝑛∗_𝛼𝑎) is a (𝑞,2) atom. Then, 𝑇_𝑘𝑎∈ℋ^𝑞_𝛼,2, and𝑇_𝑘𝑎_ℋ^𝑞_𝛼,2≤𝑐^∞_𝑛=0𝜉_𝑛^𝑞1/𝑞. (3.23)
Let now 𝑓 be in ℋ^𝑝_𝛼,2. Assume that 𝑓=∑^∞_𝑗=0𝜆_𝑗𝑎_𝑗, where 𝜆_𝑗∈ℂ and 𝑎_𝑗 is a (𝑝,2) atom, for every 𝑗∈ℕ, and such that ∑^∞_𝑗=0|𝜆_𝑗|^𝑝<∞. The series defining 𝑓 converges in 𝐿¹(ℝ,𝜇_𝛼). In fact, it is sufficient to note that ‖𝑎‖_𝐿¹_𝛼≤(1/2^𝛼Γ(𝛼+1))𝑟^{2(𝛼+1)(1−1/𝑝)}. Hence 𝑓∈𝐿¹(ℝ,𝜇_𝛼). Moreover, 𝑘∈𝐿¹(ℝ,𝜇_𝛼). Then by (Proposition 3(i), [10]), the operator 𝑇_𝑘 is bounded from 𝐿¹(ℝ,𝜇_𝛼) into itself, and from this, we deduce that 𝑇_𝑘𝑓=∑^∞_𝑗=0𝜆_𝑗𝑇_𝑘𝑎_𝑗. Using the fact that ∑^∞_𝑗=0|𝜆_𝑗|≤(∑^∞_𝑗=0|𝜆_𝑗|^𝑝)^1/𝑝, we obtain𝑇_𝑘𝑓_ℋ^𝑞_𝛼,2≤𝑐^∞_𝑛=0𝜉_𝑛^𝑞1/𝑞𝑓_ℋ^𝑝_𝛼,2. (3.24) This completes the proof of the Theorem 3.1.

We now study the Dunkl convolution operators 𝑇_𝑘 on the Herz-type Hardy spaces ℋ¹_𝛼,2.

Theorem 3.3. Let k be a locally integrable function on ℝ. Assume that the following three conditions are satisfied:(i)𝑇_𝑘 defines a bounded linear operator from 𝐿²(ℝ,𝜇_𝛼) into itself.(ii)𝑇_𝑘 defines a bounded linear operator from 𝐿¹(ℝ,𝜇_𝛼) into 𝒮′(ℝ).(iii)There exist 𝐴 and 𝑐>1 such that ∫_{|𝑧|>𝑐𝑅}𝜏_𝑥𝑘(𝑧)−𝑘(𝑧)𝑑𝜇_𝛼(𝑧)≤𝐴,|𝑥|∈(0,𝑅),𝑅>0. (3.25) Then 𝑇_𝑘 defines a bounded linear mapping from ℋ¹_𝛼,2 into 𝐿¹(ℝ,𝜇_𝛼).

Proof. Let 𝑎 be a (1,2) atom. We choose 𝑟>0 such that supp(𝑎)⊂[−𝑟,𝑟] and ‖𝑎‖_𝐿²_𝛼≤𝑟^−(𝛼+1). We can write _ℝ𝑇_𝑘𝑎(𝑥)𝑑𝜇_𝛼(𝑥)=_{|𝑥|<𝑐𝑟}+_{|𝑥|≥𝑐𝑟}𝑇_𝑘𝑎(𝑥)𝑑𝜇_𝛼(𝑥)∶=𝐼₁+𝐼₂. (3.26) Here 𝑐>1 is the one given in (iii).
From condition (i) of the theorem and Hölder's inequality, we deduce that 𝐼₁≤_ℝ|𝑇_𝑘𝑎(𝑥)|²𝑑𝜇_𝛼(𝑥)^1/22₀^𝑐𝑟𝑑𝜇_𝛼(𝑥)^1/2≤𝐶‖𝑎‖_𝐿²_𝛼𝑟^𝛼+1≤𝐶. (3.27) Also, by taking into account that ∫_ℝ𝑎(𝑦)𝑑𝜇_𝛼(𝑦)=0, the condition (iii) of the theorem allows us to write 𝐼₂=_{|𝑥|≥𝑐𝑟ℝ}𝜏_𝑥𝑘𝑦𝑎−𝑦𝑑𝜇_𝛼𝑦𝑑𝜇_𝛼(𝑥)=_{|𝑥|≥𝑐𝑟ℝ}𝜏_𝑥𝑘𝑦−𝑘(𝑥)𝑎−𝑦𝑑𝜇_𝛼𝑦𝑑𝜇_𝛼(𝑥)≤^𝑟_−𝑟𝑎−𝑦_{|𝑥|≥𝑐𝑟}𝜏_𝑦𝑘(𝑥)−𝑘(𝑥)𝑑𝜇_𝛼(𝑥)𝑑𝜇_𝛼𝑦≤𝐶^𝑟_−𝑟𝑎𝑦𝑑𝜇_𝛼𝑦≤𝐶‖𝑎‖_𝐿²_𝛼2^𝑟₀𝑑𝜇_𝛼𝑦^1/2≤C. (3.28) Hence, it concludes that 𝑇_𝑘𝑎_𝐿1𝛼≤𝐶. (3.29) Note that the positive constant 𝐶 is not depending on the (1,2) atom 𝑎.
Let now 𝑓 be in ℋ¹_𝛼,2. Then 𝑓∈𝒮′(ℝ) and 𝑓=∑^∞_𝑗=0𝜆_𝑗𝑎_𝑗, where 𝜆_𝑗∈ℂ and 𝑎_𝑗 is a (1,2) atom, for every 𝑗∈ℕ and ∑^∞_𝑗=0|𝜆_𝑗|<∞.
The series defining 𝑓 converges in 𝐿¹(ℝ,𝜇_𝛼). In fact, it is sufficient to note that ‖𝑎‖_𝐿¹_𝛼≤1/2^𝛼Γ(𝛼+1), for every (1,2) atom 𝑎. Hence 𝑓∈𝐿¹(ℝ,𝜇_𝛼). Then the condition (ii) of the theorem implies that 𝑇_𝑘𝑓=^∞_𝑗=0𝜆_𝑗𝑇_𝑘𝑎_𝑗. (3.30) By (3.29) the series in (3.30) converges in 𝐿¹(ℝ,𝜇_𝛼) and ‖𝑇_𝑘𝑓‖_𝐿¹_𝛼≤𝐶∑^∞_𝑗=0|𝜆_𝑗|. Hence ‖𝑇_𝑘𝑓‖_𝐿¹_𝛼≤𝐶‖𝑓‖_ℋ¹_𝛼,2.

4. Maximal Bochner-Riesz Operators on ℋ^𝑝_𝛼,∞

The Bochner-Riesz mean 𝜎^𝜂_𝛼,𝑡, for 𝑡>0 and 𝜂>𝛼+1/2 associated to the Dunkl transform ℱ_𝛼 is defined by 𝜎^𝜂_𝛼,𝑡𝑓(𝑥)∶=^𝑡_−𝑡𝑦1−²𝑡^2𝜂𝐸_𝛼𝑖𝑥𝑦ℱ_𝛼𝑓𝑦𝑑𝜇_𝛼𝑦,𝑓∈𝒮(ℝ). (4.1) The maximal operator 𝜎^𝜂_𝛼, 𝜂>𝛼+1/2 associated to the Bochner-Riesz means 𝜎^𝜂_𝛼,𝑡, 𝑡>0, is defined by 𝜎^𝜂_𝛼𝑓∶=sup_𝑡>0𝜎^𝜂_𝛼,𝑡𝑓. (4.2)

Lemma 4.1. For 𝑡>0 and 𝜂>𝛼+1/2, one has(i)𝜎^𝜂_𝛼,𝑡(𝑓)=Φ^𝜂_𝛼,𝑡∗_𝛼𝑓, where Φ^𝜂_𝛼,𝑡(𝑥)Γ∶=𝜂+12^𝛼+1Γ𝑡𝛼+𝜂+2^2𝛼+2ℑ_𝛼+𝜂+1(𝑖𝑡𝑥). (4.3) Here ℑ_𝛼 is the modified spherical Bessel function given by (2.3).(ii)The operator 𝜎^𝜂_𝛼 is bounded from 𝐿^𝑝(ℝ,𝜇_𝛼), 1≤𝑝≤∞ into itself.Proof. Let 𝑡>0 and 𝜂>𝛼+1/2.(i) By taking into account that the functions |𝑧|^𝛼+1/2ℑ_𝛼(𝑖𝑧) and ℑ_𝛼(𝑖𝑧) are bounded on ℝ it is not hard to see that Φ^𝜂_{𝛼,𝑡𝐿1𝛼}=Φ^𝜂_{𝛼,1𝐿1𝛼}=Γ𝜂+12^𝛼+1Γ𝛼+𝜂+2_ℝℑ_𝛼+𝜂+1(𝑖𝑥)𝑑𝜇_𝛼(𝑥)<∞. (4.4) On the other hand, from [11], we have ¹₀1−𝑦^2𝜂ℑ_𝛼𝑖𝑡𝑥𝑦𝑦^2𝛼+1Γ()Γ𝑑𝑦=𝛼+1𝜂+1ℑ2Γ𝛼+𝜂+2^𝛼+𝜂+1(𝑖𝑡𝑥). (4.5) Thus, Φ^𝜂_𝛼,𝑡(𝑥)=𝑡^2𝛼+22^𝛼Γ(𝛼+1)¹₀1−𝑦^2𝜂ℑ_𝛼𝑖𝑡𝑥𝑦𝑦^2𝛼+1=𝑑𝑦_𝑡−𝑡𝑦1−²𝑡^2𝜂𝐸_𝛼𝑖𝑥𝑦𝑑𝜇_𝛼𝑦. (4.6) Applying Inversion Theorem[5], we obtain ℱ_𝛼Φ^𝜂_𝛼,𝑡𝑦=𝑦1−²𝑡^2𝜂𝜒_{(−𝑡,𝑡)}𝑦, (4.7) where 𝜒_{(−𝑡,𝑡)} is the characteristic function of the set (−𝑡,𝑡). Thus, 𝜎^𝜂_𝛼,𝑡𝑓(𝑥)=_ℝ𝐸_𝛼𝑖𝑥𝑦ℱ_𝛼Φ^𝜂_𝛼,𝑡𝑦ℱ_𝛼𝑓𝑦𝑑𝜇_𝛼𝑦, (4.8) and from (Proposition 3 (ii), [10]), we deduce that 𝜎^𝜂_𝛼,𝑡𝑓(𝑥)=Φ^𝜂_𝛼,𝑡∗_𝛼𝑓(𝑥). (4.9)(ii)Using (i) and (Proposition 3 (i), [10]), we obtain 𝜎^𝜂_𝛼𝑓_𝐿^𝑝_𝛼≤4Φ^𝜂_{𝛼,1𝐿1𝛼}𝑓_𝐿^𝑝_𝛼. (4.10) This clearly yields the result.Theorem 4.2. Let 0<𝑝≤1<𝑞≤∞ and 𝑓∈ℋ^𝑝_𝛼,𝑞. For 𝑡>0 and 𝜂>𝛼+1/2, the operator 𝜎^𝜂_𝛼,𝑡 extended to a bounded operator from ℋ^𝑝_𝛼,𝑞 into 𝒮′(ℝ).

Proof. According to Theorem 2.1, if 𝑓∈ℋ^𝑝_𝛼,𝑞, then 𝜎^𝜂_𝛼,𝑡(𝑓) is in 𝒮′(ℝ) and it is defined by 𝜎^𝜂_𝛼,𝑡𝑓,𝜑=^𝑡_−𝑡𝑦1−²𝑡^2𝜂ℱ_𝛼𝑓𝑦ℱ_𝛼𝜑𝑦𝑑𝜇_𝛼𝑦,𝜑∈𝒮(ℝ). (4.11) Moreover, 𝜎^𝜂_𝛼,𝑡𝑓,𝜑≤𝐶𝑓_ℋ^𝑝_𝛼,𝑞ℝ𝑦^{2(𝛼+1)(1/𝑝−1)}ℱ_𝛼𝜑𝑦𝑑𝜇_𝛼𝑦,𝜑∈𝒮(ℝ). (4.12) Hence 𝜎^𝜂_𝛼,𝑡 is a bounded operator from ℋ^𝑝_𝛼,𝑞 into 𝒮′(ℝ).

We now study the behavior of the maximal Bochner-Riesz operator 𝜎^𝜂_𝛼 on ℋ^𝑝_𝛼,∞.

Theorem 4.3. Let 2(𝛼+1)/(𝛼+𝜂+3/2)<𝑝≤1. Then the maximal Bochner-Riesz operator 𝜎^𝜂_𝛼, 𝛼+1/2<𝜂<𝛼+3/2 is bounded from ℋ^𝑝_𝛼,∞ into 𝐿^𝑝(ℝ,𝜇_𝛼).
To prove this theorem we need the following lemma.

Lemma 4.4. (i) For 𝑥,𝑦∈ℝ and 𝜂>𝛼+1/2, 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦≤𝐶𝑡^{𝛼−𝜂+1/2}|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}. (4.13)
(ii) For 0<|𝑦|<|𝑥| and 𝜂>𝛼+1/2,𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)≤𝐶𝑦𝑡^{𝛼−𝜂+3/2}|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}. (4.14)

Proof. (i) Since the function |𝑧|^𝛼+1/2ℑ_𝛼(𝑖𝑧) is bounded on ℝ, it follows that Φ^𝜂_𝛼,𝑡(𝑧)≤𝐶𝑡^{𝛼−𝜂+1/2}|𝑧|^{−(𝛼+𝜂+3/2)};𝑡>0,𝑧∈ℝ. (4.15) According to [6] and the fact that Φ^𝜂_𝛼,𝑡 is even, we obtain 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦=^𝜋₀Φ^𝜂_𝛼,𝑡𝑥,𝑦_𝜃𝑑𝜌_𝑥,𝑦(𝜃), (4.16) where 𝑑𝜌_𝑥,𝑦(𝜃)Γ()∶=𝛼+1√𝜋Γ(𝛼+1/2)1−sgn𝑥𝑦cos𝜃sin^2𝛼𝜃𝑑𝜃,𝑥,𝑦_𝜃∶=𝑥²+𝑦²−2𝑥𝑦cos𝜃. (4.17) By (4.15) and using the fact that (𝑥,𝑦)_𝜃≥||𝑥|−|𝑦||, we deduce that 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦≤𝐶𝑡^{𝛼−𝜂+1/2𝜋}₀𝑥,𝑦_𝜃^{−(𝛼+𝜂+3/2)}𝑑𝜌_𝑥,𝑦(𝜃)≤𝐶𝑡^{𝛼−𝜂+1/2}|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}. (4.18)(ii)From [11], we have 𝑑Φ𝑑𝑧^{𝜂𝛼,𝑡}(𝑧)Γ=−𝜂+12^𝛼+2Γ𝑡𝛼+𝜂+3^2𝛼+4𝑧ℑ_𝛼+𝜂+2(𝑖𝑡𝑧). (4.19) Then, similarly to the proof in (i), we have 𝑑Φ𝑑𝑧^{𝜂𝛼,𝑡}(𝑧)≤𝐶𝑡^{𝛼−𝜂+3/2}|𝑧|^{−(𝛼+𝜂+3/2)}𝜏;𝑡>0,𝑧∈ℝ,_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)=^𝜋₀Φ^𝜂_𝛼,𝑡𝑥,𝑦_𝜃−Φ^𝜂_𝛼,𝑡(𝑥,0)_𝜃𝑑𝜌_𝑥,𝑦(𝜃), (4.20)(4.21) which can be written as 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)=^𝜋₀¹₀𝑑Φ𝑑𝑠^{𝜂𝛼,𝑡}𝑥,𝑠𝑦_𝜃𝑑𝑠𝑑𝜌_𝑥,𝑦(𝜃)≤𝑦¹₀^𝜋₀𝑑Φ𝑑𝑠^{𝜂𝛼,𝑡}𝑥,𝑠𝑦_𝜃𝑑𝜌_𝑥,𝑦(𝜃)𝑑𝑠. (4.22) Then from (4.20), it follows 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)≤𝐶𝑦𝑡^{𝛼−𝜂+3/21}₀^𝜋₀𝑥,𝑠𝑦_𝜃^{−(𝛼+𝜂+3/2)}𝑑𝜌_𝑥,𝑦(𝜃)𝑦𝑡𝑑𝑠≤𝐶^{𝛼−𝜂+3/21}₀|𝑥|−𝑠𝑦^{−(𝛼+𝜂+3/2)}𝑑𝑠. (4.23) Hence, if 0<|𝑦|<|𝑥|, we obtain 𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)≤𝐶𝑦𝑡^{𝛼−𝜂+3/2}|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}, (4.24) which completes the proof of the lemma.

Proof of Theorem 4.3. Let us first show that 𝐶>0, exists such that 𝜎^𝜂_𝛼(𝑎)_𝐿^𝑝_𝛼≤𝐶, (4.25) for every (𝑝,∞) atom 𝑎.
Let 𝑎 be an (𝑝,∞) atom. Suppose that 𝑎(𝑥)=0, |𝑥|>𝑟 and ‖𝑎‖_𝐿^∞_𝛼≤𝑟^{−2(𝛼+1)/𝑝}. We choose ℓ∈ℤ such that 2^ℓ−1<𝑟≤2^ℓ, we write _{|𝑥|≥4𝑟}𝜎^𝜂_𝛼(𝑎)(𝑥)^𝑝𝑑𝜇_𝛼(𝑥)≤𝐼₁+𝐼₂, (4.26) where 𝐼₁∶=^∞_{𝑖=1(𝑖+1)2ℓ≤|𝑥|≤(𝑖+2)2ℓ}sup_{𝑡≥𝛿𝑖}𝜎^𝜂_𝛼,𝑡(𝑎)(𝑥)^𝑝𝑑𝜇_𝛼(𝑥),𝐼₂∶=^∞_{𝑖=1(𝑖+1)2ℓ≤|𝑥|≤(𝑖+2)2ℓ}sup_{𝑡<𝛿𝑖}𝜎^𝜂_𝛼,𝑡(𝑎)(𝑥)^𝑝𝑑𝜇_𝛼(𝑥), (4.27)(4.28) being 𝛿_𝑖=2^−ℓ/𝑖^𝑏, where 𝑏 will be specified later.
According to Lemma 4.4 (i), for |𝑥|∈[(𝑖+1)2^ℓ,(𝑖+2)2^ℓ], 𝑖=1,2,…, we get 𝜎^𝜂_𝛼,𝑡(𝑎)(𝑥)≤^𝑟_−𝑟𝑎−𝑦𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦𝑑𝜇_𝛼𝑦≤𝐶𝑡^{𝛼−𝜂+1/2}𝑟^{−2(𝛼+1)/𝑝2ℓ}₀|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}𝑑𝜇_𝛼𝑦2≤𝐶^ℓ𝑡^{𝛼−𝜂+1/2}𝑖^{𝛼+𝜂+3/2}𝑟^{2(𝛼+1)/𝑝}. (4.29)
Then, using the fact that 2^ℓ−1<𝑟≤2^ℓ, we obtain 𝐼₁≤𝐶^∞_𝑖=1𝛿_𝑖^{𝛼−𝜂+1/2}2^{ℓ{𝛼−𝜂+1/2−2(𝛼+1)/𝑝}}𝑖^{𝛼+𝜂+3/2𝑝}𝑖^2𝛼+12^2ℓ(𝛼+1), (4.30) and hence, it concludes that 𝐼₁≤𝐶^∞_𝑖=1𝑖^{2𝛼+1−{𝛼+𝜂+3/2+(𝛼−𝜂+1/2)𝑏}𝑝}. (4.31) 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)𝑑𝜇_𝛼(𝑦)=0, from Lemma 4.4 (ii) we have 𝜎^𝜂_𝛼,𝑡(𝑎)(𝑥)≤^𝑟_−𝑟𝑎−𝑦𝜏_𝑥Φ^𝜂_𝛼,𝑡𝑦−Φ^𝜂_𝛼,𝑡(𝑥)𝑑𝜇_𝛼𝑦≤𝐶𝑡^{𝛼−𝜂+3/2}𝑟^{−2(𝛼+1)/𝑝2ℓ}₀|𝑥|−𝑦^{−(𝛼+𝜂+3/2)}𝑦𝑑𝜇_𝛼𝑦2≤𝐶^ℓ𝑡^{𝛼−𝜂+3/2}𝑖^{𝛼+𝜂+3/2}𝑟^{2(𝛼+1)/𝑝}. (4.32)
Then, for |𝑥|≥4𝑟, we obtain 𝐼₂≤𝐶^∞_𝑖=1𝛿_𝑖^{𝛼−𝜂+3/2}2^{ℓ{𝛼−𝜂+3/2−2(𝛼+1)/𝑝}}𝑖^{𝛼+𝜂+3/2𝑝}𝑖^2𝛼+12^2ℓ(𝛼+1), (4.33) and in fact, we deduce that 𝐼₂≤𝐶^∞_𝑖=1𝑖^{2𝛼+1−{𝛼+𝜂+3/2+(𝛼−𝜂+3/2)𝑏}𝑝}. (4.34) The last series converges provided that 𝑏>𝑝(𝛼+𝜂+3/2)−2(𝛼+1)𝑝(𝜂−𝛼−3/2).
Note that we can find 𝑏 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 _{|𝑥|≥4𝑟}𝜎^𝜂_𝛼(𝑎)(𝑥)^𝑝𝑑𝜇_𝛼(𝑥)≤𝐶, (4.35) for a certain 𝐶>0 that is not depending on 𝑎.
From Lemma 4.1 (ii) and (4.35) we deduce that𝜎^𝜂_𝛼(𝑎)^𝑝_𝐿𝑝𝛼≤_|𝑥|<4𝑟+_{|𝑥|≥4𝑟}𝜎^𝜂_𝛼(𝑎)(𝑥)^𝑝𝑑𝜇_𝛼(𝑥)≤𝐶‖𝑎‖^𝑝_{𝐿∞𝑘|𝑥|<4𝑟}𝑑𝜇_𝛼(𝑥)+1≤𝐶, (4.36) that is, (4.25) holds. Let now 𝑓 be in ℋ^𝑝_𝛼,∞. Assume that 𝑓=∑^∞_𝑗=0𝜆_𝑗𝑎_𝑗, where the series converges in 𝒮′(ℝ), and for every 𝑗∈ℕ, 𝑎_𝑗 is a (𝑝,∞) atom and 𝜆_𝑗∈ℂ, such that ∑^∞_𝑗=0|𝜆_𝑗|^𝑝<∞. From Theorem 4.2, for 0<𝑝≤1, we can write 𝜎^𝜂_𝛼,𝑡𝑓(𝑥)=^∞_𝑗=0𝜆_𝑗𝜎^𝜂_𝛼,𝑡𝑎_𝑗(𝑥);𝑡>0,𝑥∈ℝ. (4.37) Hence, from (4.25) it follows ‖𝜎^𝜂_𝛼(𝑓)‖^𝑝_𝐿𝑝𝛼≤𝐶∑^∞_𝑗=0|𝜆_𝑗|^𝑝. Thus we conclude that ‖𝜎^𝜂_𝛼(𝑓)‖_𝐿^𝑝_𝛼≤𝐶‖𝑓‖_ℋ^𝑝_𝛼,∞.

Acknowledgment

A. Gasmi and F. Soltani partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.