A Multiplier Theorem for Herz-Type Hardy Spaces Associated with the Dunkl Transform

Abstract

The main purpose of this paper is to establish a Hörmander multiplier theorem for Herz-type Hardy spaces associated with the Dunkl transform.

1. Introduction

Let 𝑇_𝑚(𝑓) be a multiplier operator defined in terms of Fourier transforms by 𝑇_𝑚(𝑓)=ℱ⁻¹(𝑚ℱ(𝑓)) for suitable functions 𝑓. The multiplier theorem of Hörmander [1] gives a sufficient condition on 𝑚 for the operator 𝑇_𝑚 to be bounded on 𝐿^𝑝(ℝ^𝑛) whenever 1<𝑝<∞, namely, that 𝑚 is a bounded 𝐶^ℓ-function on ℝ^𝑛⧵{0} satisfying the Hörmander condition 𝑀(2,ℓ) as follows: _𝑅^2𝑅𝑚^(𝑠)(𝜉)²𝑑𝜉^1/2≤𝐶𝑅^{(𝑛+1)/2−𝑠},∀𝑅>0, (1) where ℓ is the least integer greater than 𝑛/2 and 𝑠=0,1,…,ℓ. In [2], the authors proved that if 𝑚 satisfies the Hörmander condition with ℓ>𝑛(1/𝑝−1/2), then 𝑇_𝑚 is bounded on the Hardy spaces 𝐻^𝑝(ℝ^𝑛) with 0<𝑝≤1.

In [3], the authors considered the following multiplier operator which is associated with the Dunkl transform: 𝑇^𝛼_𝑚𝑓=ℱ_𝛼⁻¹𝑚ℱ_𝛼𝑓, (2) where ℱ_𝛼 designs the Dunkl transform and using Hörmander’s technique proved the following theorem.

Theorem 1. Let ℓ be the least integer greater than 𝛼+1 and let 𝑚 be a bounded 𝐶^ℓ-function on ℝ⧵{0} which satisfies the Hörmander condition 𝑀_𝛼(2,ℓ) as follows: _𝑅^2𝑅𝑚^(𝑠)(𝜉)²𝑑𝜇_𝛼(𝜉)^1/2≤𝐶𝑅^{𝛼+1−𝑠},∀𝑅>0, (3) where 𝐶 is a constant independent of 𝑅 and 𝑠=0,1,…,ℓ. Then, the multiplier operator associated with the Dunkl transform can be extended to a bounded operator from 𝐿^𝑝(𝜇_𝛼) into itself for 1<𝑝<∞, where 𝐿^𝑝(𝜇_𝛼) is the Lebesgue space on ℝ with respect to the following measure: 𝜇_𝛼(𝑥)=2^𝛼+1Γ(𝛼+1)⁻¹|𝑥|^2𝛼+1,1𝛼>−2. (4)

The Hardy spaces associated with Herz spaces can be regarded as the local version at the origin of the classical Hardy spaces 𝐻^𝑝 and they are good substitutes for 𝐻^𝑝 when we study the boundedness of nontranslation invariant operators. To establish the boundedness of operators in hardy-type spaces on ℝ^𝑛, one usually appeals to the atomic decomposition characterization of these spaces. In [4, 5], the authors studied the Herz-type Hardy spaces 𝐻𝐾̇^𝛽,𝑝_𝛼,2 for the Dunkl operator in one-dimension and gave an atomic decomposition characterization of these spaces. The aim of this work is to prove the following Hörmander multiplier theorem on the spaces 𝐻𝐾̇^𝛽,𝑝_𝛼,2.

Theorem 2. Let 0<𝑝≤1, 𝛽=(1/𝑝)−(1/2), and ℓ be an integer greater than 2(𝛼+1)𝛽. If 𝑚 satisfies the Hörmander condition 𝑀_𝛼(2,ℓ), then the operator 𝑇^𝛼_𝑚 is bounded on 𝐻𝐾̇^𝛽,𝑝_𝛼,2.

The paper is organized as follows. In Section 2, we recall some results about harmonic analysis and Herz-type Hardy spaces associated with the Dunkl operator on ℝ. In Section 3, we give the proof of the main result of this work. Then, as an application, we obtain the boundedness of the generalized Hilbert transform on 𝐻𝐾̇^𝛽,𝑝_𝛼,2.

Throughout this paper, let 𝑆(ℝ) be the usual Schwartz space and let ℰ(ℝ) be the space of 𝐶^∞-functions on ℝ. We always use 𝐶 to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. We use the shorter notation ‖𝑓‖_𝑝,𝛼 instead of ‖𝑓‖_{𝐿𝑝(𝜇𝛼)}.

2. Preliminaries

In this section, we recapitulate some results about harmonic analysis on Dunkl hypergroups and the Herz-type Hardy space and its atomic decomposition which will be used later. For details, the reader is referred to [6–8].

Let 𝛼>−1/2. We consider the differential-difference operator introduced in [9] as follows: Λ_𝛼𝑓(𝑥)=𝑑𝑓(𝑥)+𝑑𝑥2𝛼+1𝑥⋅𝑓(𝑥)−𝑓(−𝑥)2,𝑓∈ℰ(ℝ), (5) and call it the Dunkl operator.

For 𝜆∈ℂ, the following initial value problem: Λ_𝛼𝑓(𝑥)=𝜆𝑓(𝑥),𝑓(0)=1,𝑥∈ℝ, (6) has a unique solution 𝐸_𝛼(𝜆⋅) (called the Dunkl kernel) given by 𝐸_𝛼(𝑧)=𝑗_𝛼()+𝑧𝑖𝑧2()𝑗𝛼+1^𝛼+1(𝑖𝑧),𝑧∈ℂ, (7) where 𝑗_𝛼 is the normalized Bessel function of the first kind (with order 𝛼) defined on ℂ by 𝑗_𝛼(𝑧)=Γ(𝛼+1)^∞_𝑛=0(−1)^𝑛(𝑧/2)^2𝑛().𝑛!Γ𝑛+𝛼+1 (8) The integral representation of 𝐸_𝛼 is given by 𝐸_𝛼()=Γ()𝑖𝜆𝑥𝛼+1√𝜋Γ(𝛼+(1/2))¹₋₁(1−𝑡)1−𝑡^{2𝛼−(1/2)}𝑒^{−𝑖𝜆𝑥𝑡}𝑑𝑡. (9) From which, we get 𝜕^𝑛_𝑥𝐸_𝛼(𝑖𝜆𝑥)≤|𝜆|^𝑛,𝜆,𝑥∈ℝ,𝑛∈ℕ. (10)

The Dunkl transform ℱ_𝛼, which was introduced by [10] and studied in [11], is defined for 𝑓∈𝐿¹(𝜇_𝛼) by ℱ_𝛼𝑓(𝑥)=_ℝ𝐸_𝛼−𝑖𝑥𝑦𝑓𝑦𝑑𝜇_𝛼𝑦,𝑥∈ℝ. (11) This transform satisfies the following properties.(i)For all 𝑓∈𝐿¹(𝜇_𝛼), we have ℱ_𝛼𝑓_∞,𝛼≤𝑓_1,𝛼. (12)(ii)For all 𝑓∈𝐿¹(𝜇_𝛼) such that ℱ_𝛼(𝑓)∈𝐿¹(𝜇_𝛼), we have the following inversion formula: ℱ_𝛼⁻¹𝑓(𝑥)=ℱ_𝛼𝑓(−𝑥),a.e.𝑥∈ℝ. (13)(iii)For all 𝑓∈𝒮(ℝ), ℱ_𝛼Λ_𝛼𝑓(𝑥)=𝑖𝑥ℱ_𝛼𝑓(𝑥). (14)(iv)ℱ_𝛼 is a topological isomorphism from 𝒮(ℝ) into itself. (v)ℱ_𝛼 is an isometric isomorphism of 𝐿²(𝜇_𝛼), and we have the following Parseval formula: _ℝ𝑓(𝑥)𝑔(𝑥)𝑑𝜇_𝛼(𝑥)=_ℝℱ_𝛼𝑓(𝑥)ℱ_𝛼(𝑔)(𝑥)𝑑𝜇_𝛼(𝑥),ℱ_𝛼𝑓_2,𝛼=𝑓_2,𝛼. (15)

The following lemma can be proved, similar to Lemma 7.25, page 343, in [12].

Lemma 3. Let ℓ be the least integer greater than 𝛼+1. If 𝑚 satisfies the Hörmander condition 𝑀_𝛼(2,ℓ), then there is a constant 𝐶 independent of 𝑚, such that if 𝑞=1 or 𝑠−ℓ+𝛼+1<(𝛼+1)/𝑞≤𝛼+1, the following inequality holds: _𝑅^2𝑅𝑚^(𝑠)(𝜉)^2𝑞𝑑𝜇_𝛼(𝜉)≤𝐶𝑅^{2(𝛼+1)−2𝑞𝑠},∀𝑅>0. (16) Furthermore, in case 𝑠−ℓ+𝛼+1<0, then |𝑥|^𝑠|𝑚^(𝑠)(𝑥)|≤𝐶 and 𝑚^(𝑠) is continuous on ℝ⧵{0}.

Notation. 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|, (17) where 𝜎_{𝑥,𝑦,𝑧}=𝑥²+𝑦²−𝑧²{0},Δ2𝑥𝑦,if𝑥,𝑦∈ℝ⧵0,otherwise,_𝛼|𝑥|,𝑦,|𝑧|=𝑑_𝛼|𝑥|+𝑦²−𝑧²𝑧²−|𝑥|−𝑦^2𝛼−1/2𝑥𝑦𝑧^2𝛼,if|𝑧|∈𝐴_𝑥,𝑦,𝑑0,otherwise,_𝛼=2^1−𝛼(Γ(𝛼+1))²√(),𝐴𝜋Γ𝛼+1/2_𝑥,𝑦=|𝑥|−𝑦,|𝑥|+𝑦. (18) The Dunkl translation operator 𝜏_𝑥, 𝑥∈ℝ is defined for a continuous function 𝑓 on ℝ by 𝜏_𝑥𝑓𝑦=_ℝ𝑓(𝑧)𝑑𝛾_𝑥,𝑦(𝑧),𝑦∈ℝ, (19) where 𝛾_𝑥,𝑦 is the signed measures given by 𝑑𝛾_𝑥,𝑦(𝑧)=𝑊_𝛼𝑥,𝑦,𝑧𝑑𝜇_𝛼(𝑧),if𝑥,𝑦∈ℝ⧵{0},𝑑𝛿_𝑥(𝑧),if𝑦=0,𝑑𝛿_𝑦(𝑧),if𝑥=0. (20) The operator 𝜏_𝑥 has the following properties. (i)For 𝑥,𝑦∈ℝ and a continuous function 𝑓 on ℝ, we have 𝜏_𝑥𝑓𝑦=𝜏_𝑦𝑓(𝑥). (21)(ii)For all 𝑥∈ℝ, the operator 𝜏_𝑥 can be extended to 𝐿^𝑝(𝜇_𝛼)(𝑝≥1), and for 𝑓∈𝐿^𝑝(𝜇_𝛼), we have 𝜏_𝑥𝑓_𝑝,𝛼≤3𝑓_𝑝,𝛼. (22)(iii)For all 𝑥,𝜆∈ℝ and 𝑓∈𝐿¹(𝜇_𝛼), we have ℱ_𝛼𝜏_𝑥𝑓(𝜆)=𝐸_𝛼(𝑖𝜆𝑥)ℱ_𝛼𝑓(𝜆). (23)
Let 𝑝,𝑞,𝑟∈[1,∞] such that 1/𝑝+1/𝑞=1/𝑟+1. The convolution product of 𝑓∈𝐿^𝑝(𝜇_𝛼) and 𝑔∈𝐿^𝑞(𝜇_𝛼) is defined by 𝑓∗_𝛼𝑔(𝑥)=_ℝ𝜏_𝑥𝑓−𝑦𝑔𝑦𝑑𝜇_𝛼𝑦,a.e.𝑥, (24) and we have 𝑓∗_𝛼𝑔_𝑟,𝛼≤3𝑓_𝑝,𝛼𝑔_𝑞,𝛼. (25) If 𝑓,𝑔∈𝐿¹(𝜇_𝛼), then ℱ_𝛼𝑓∗_𝛼𝑔=ℱ_𝛼𝑓ℱ_𝛼𝑔. (26)

Now, let us recall the definition of the Herz-type Hardy space and its atomic decomposition. For 𝑁∈ℕ being sufficiently large, we denote by 𝐹_𝑁 the subset of 𝑆(ℝ) constituted by all those 𝜙∈𝑆(ℝ) such that supp(𝜙)⊂[−1,1] and for all 𝑚,𝑛∈ℕ such that 𝑚,𝑛≤𝑁, we have 𝜌_𝑚,𝑛𝜙=sup_𝑥∈ℝ(1+|𝑥|)^𝑚Λ^𝑛_𝛼𝜙(𝑥)≤1. (27) Moreover, the system of seminorms {𝜌_𝑚,𝑛}_{𝑚,𝑛∈ℕ} generates the topology of 𝑆(ℝ).

Let 𝑓∈𝑆^′(ℝ). We define the 𝛼-grand maximal function 𝐺_𝛼(𝑓) of 𝑓 by 𝐺_𝛼𝑓(𝑥)=sup_{𝑡>0,𝜙∈𝐹𝑁}𝜙_𝑡∗_𝛼𝑓(𝑥),𝑥∈ℝ, (28) where 𝜙_𝑡 is the dilation of 𝜙 given by 𝜙_𝑡(𝑥)=𝑡^−2(𝛼+1)𝜙𝑥𝑡,𝑥∈ℝ. (29)

Definition 4. Let 𝛽∈ℝ, 𝑝∈]0,∞[, and 𝑞∈[1,∞].(i) The homogeneous weighted Herz space 𝐾̇^𝛽,𝑝_𝛼,𝑞 is the space constituted by all functions 𝑓∈𝐿^𝑞_loc(𝜇_𝛼), such that 𝑓_̇𝐾^𝛽,𝑝_𝛼,𝑞=^∞_𝑘=−∞2^{2(𝛼+1)𝛽𝑘𝑝}𝑓𝜒_𝑘^𝑝_𝑞,𝛼^1/𝑝<∞, (30) where 𝜒_𝑘 is the characteristic function of 𝐴_𝑘={𝑥∈ℝ/2^𝑘−1≤|𝑥|≤2^𝑘}.(ii) The nonhomogeneous weighted Herz space 𝐾^𝛽,𝑝_𝛼,𝑞 is defined, as usual, by 𝐾^𝛽,𝑝_𝛼,𝑞=𝐿^𝑞(𝜇_𝛼)∩𝐾̇^𝛽,𝑝_𝛼,𝑞. Moreover, ‖𝑓‖_𝐾^𝛽,𝑝_𝑞,𝛼=‖𝑓‖_𝑞,𝛼+‖𝑓‖_̇𝐾^𝛽,𝑝_𝛼,𝑞.
Note that 𝐾̇^0,𝑞_𝛼,𝑞=𝐾^0,𝑞_𝛼,𝑞=𝐿^𝑞(𝜇_𝛼).

Definition 5. Let 𝛽∈ℝ, 𝑝∈]0,∞], and 𝑞∈]1,∞]. The Herz-type Hardy space 𝐻𝐾̇^𝛽,𝑝_𝛼,𝑞 is the space of distributions 𝑓∈𝑆^′(ℝ) such that 𝐺_𝛼(𝑓)∈𝐾̇^𝛽,𝑝_𝛼,𝑞. Moreover, we define 𝑓_𝐻̇𝐾^𝛽,𝑝_𝛼,𝑞=𝐺_𝛼𝑓_̇𝐾^𝛽,𝑝_𝛼,𝑞. (31) In the same way, we define the space 𝐻𝐾^𝛽,𝑝_𝛼,𝑞 for the non-homogeneous case.

Definition 6. Let 𝑞∈]1,∞] and 𝛽≥1−1/𝑞. A measurable function 𝑎 on ℝ is called a (central) (𝛽,𝑞,𝑠)-atom if it satisfies the following: (i)supp(𝑎)⊂[−𝑟,𝑟], for some 𝑟>0,(ii)‖𝑎‖_𝑞,𝛼≤𝑟^{−2(𝛼+1)𝛽}, (iii)∫_ℝ𝑎(𝑥)𝑥^𝑘𝑑𝜇_𝛼(𝑥)=0, 𝑘=0,1,…,𝑠, where 𝑠=[2(𝛼+1)(𝛽−1+1/𝑞)] and [⋅] denotes the integer part function.

The following theorem is shown in [4].

Theorem 7. Let 0<𝑝≤1<𝑞≤∞ and 𝛽≥1−1/𝑞. Then, 𝑓∈𝐻𝐾̇^𝛽,𝑝_𝛼,𝑞 if and only if, for all 𝑗∈ℕ⧵{0}, there exist a (𝛽,𝑞,𝑠)-atom 𝑎_𝑗 and 𝜆_𝑗∈ℂ, such that ∑^∞_𝑗=1|𝜆_𝑗|^𝑝<∞ and 𝑓=∑^∞_𝑗=1𝜆_𝑗𝑎_𝑗. Moreover, 𝑓_𝐻̇𝐾^𝛽,𝑝_𝛼,𝑞∼inf^∞_𝑗=1𝜆_𝑗^𝑝1/𝑝, (32) where the infimum is taking over all atomic decompositions of 𝑓.

In the sequel, fix 𝑞=2 and 𝛽=1/𝑝−1/2.

Definition 8. For 0<𝑝≤1. Set 𝑠≥[2(𝛼+1)(1/𝑝−1)], 𝜀>𝑠/2(𝛼+1), 1𝑎=1−𝑝+𝜀, and 𝑏=1/2+𝜀. A central (𝑝,𝑠,𝜀)-molecule is a function 𝑀∈𝐿²(𝜇_𝛼) satisfying the following: (i)𝑀(𝑥)|𝑥|^{2(𝛼+1)𝑏}∈𝐿²(𝜇_𝛼), (ii)‖𝑀‖^𝑎/𝑏_2,𝛼‖𝑀(𝑥)|𝑥|^{2(𝛼+1)𝑏}‖^{1−𝑎/𝑏}_2,𝛼≡𝑁(𝑀)<∞, (iii)∫_ℝ𝑀(𝑥)𝑥^𝑘𝑑𝜇_𝛼(𝑥)=0, 𝑘=0,1,…,𝑠.

Proposition 9. Let (𝑝,𝑠,𝜀) be the triple cited in the previous definition. Every central (𝑝,𝑠,𝜀)-molecule 𝑀 belongs to 𝐻𝐾̇^𝛽,𝑝_𝛼,2 and ‖𝑀‖_𝐻̇𝐾^𝛽,𝑝_𝛼,2≤𝐶𝑁(𝑀), where the constant 𝐶 is independent of 𝑀.

Proof. Let 𝑀 be a central (𝑝,𝑠,𝜀)-molecule and suppose that ‖𝑀‖_𝐻̇𝐾^𝛽,𝑝_𝛼,2=1. In the general case, letting 𝑀=𝑀/‖𝑀‖_𝐻̇𝐾^𝛽,𝑝_𝛼,2, we have ‖𝑀‖_𝐻̇𝐾^𝛽,𝑝_𝛼,2=1.
Let 𝐸₀={|𝑥|≤1}, 𝐸_𝑘={2^𝑘−1<|𝑥|≤2^𝑘}, and 𝑀_𝑘=𝑀𝜒_𝑘, 𝑘=1,2,3,…, where 𝜒_𝑘 is the characteristic function of 𝐸_𝑘. For each 𝑘, there exists a unique polynomial 𝑄_𝑘, of degree at most 𝑠, such that if 𝑃_𝑘=𝑄_𝑘𝜒_𝑘; then _ℝ𝑀_𝑘−𝑃_𝑘𝑥^𝑗𝑑𝜇_𝛼(𝑥)=0,𝑗=0,1,…,𝑠. (33)
Using some ideas in [2], we can show that each (𝑀_𝑘−𝑃_𝑘) is a multiple of a central (𝛽,2,𝑠)-atom with a sequence of coefficients in 𝑙^𝑝. We also show that the sum ∑^+∞_𝑘=0𝑃_𝑘 can be written as an infinite linear combination of central (𝛽,∞,𝑠)-atom with a sequence of coefficients in 𝑙^𝑝. Since a (𝛽,∞,𝑠)-atom is also (𝛽,2,𝑠)-atom, hence, 𝑀=^+∞_𝑘=0𝑀_𝑘=^+∞_𝑘=0𝑀_𝑘−𝑃_𝑘+^+∞_𝑘=0𝑃_𝑘=^+∞_𝑖=0𝜆_𝑖𝑎_𝑖, (34) where 𝑎_𝑖 is a central (𝛽,2,𝑠)-atom and ∑^+∞_𝑘=0|𝜆_𝑖|^𝑝<∞. It follows from Theorem 7 that 𝑀∈𝐻𝐾̇^𝛽,𝑝_𝛼,2 and ‖𝑀‖_𝐻̇𝐾^𝛽,𝑝_𝛼,2≤𝐶𝑁(𝑀).

The following Lemma plays an important role in the proof of the main result of this work.

Lemma 10. Let 𝑎 be a (𝛽,2,𝑠)-atom. For all integer 0≤𝑘≤𝑠 and every 1≤𝑢≤∞, there exists a constant 𝐶 independent of 𝑎, such that (i)ℱ_𝛼(𝑎)^(𝑘)𝑦≤𝐶𝑦^{𝑠+1−𝑘}‖𝑎‖^𝐴_2,𝛼,1𝐴=1−𝛽12+𝑠+12()(ℱ𝛼+1𝑦∈ℝ,ii)_𝛼(𝑎)^(𝑘)𝑦²_𝑢′,𝛼≤𝐶‖𝑎‖^{2−(1/𝛽)((𝑘/𝛼+1)+(1/𝑢))}_2,𝛼,1𝑢+1𝑢^′=1𝑦∈ℝ. (35)

Proof. (i) Let 𝑎 be a (𝛽,2,𝑠)-atom. Consider that 𝑟>0 such that supp(𝑎)⊂[−𝑟,𝑟] and that ‖𝑎‖_2,𝛼≤𝑟^{−2(𝛼+1)𝛽}. From (9), (iii) of Definition (19), and the estimate for the remainder in Taylors’ formula, it follows that ℱ_𝛼(𝑎)^(𝑘)𝑦=𝐶_𝛼¹₋₁(1−𝑡)1−𝑡^2𝛼−1/2𝑡^𝑘×^𝑟_−𝑟(𝑖𝑥)^𝑘exp𝑖𝑥𝑦𝑡−^𝑠−𝑘_𝑛=0𝑖𝑥𝑦𝑡^𝑛(𝑥)𝑛!×𝑎𝑑𝜇_𝛼(𝑥)𝑦𝑑𝑡≤𝐶^{𝑠+1−𝑘𝑟}_−𝑟|𝑥|^𝑠+1|𝑎(𝑥)|𝑑𝜇_𝛼(𝑥)≤𝐶𝑦^{𝑠+1−𝑘}‖𝑎‖_2,𝛼^𝑟_−𝑟|𝑥|^2(𝑠+1)𝑑𝜇_𝛼(𝑥)^1/2≤𝐶𝑦^{𝑠+1−𝑘}‖𝑎‖_2,𝛼𝑟^𝑠+𝛼+2. (36) From (ii) of Definition (19), we obtain ℱ_𝛼(𝑎)^(𝑘)𝑦≤𝐶𝑦^{𝑠+1−𝑘}‖𝑎‖^𝐴_2,𝛼,1𝐴=1−𝛽12+𝑠+12().𝛼+1 (37) (ii) For 𝑢=1, ℱ_𝛼(𝑎)^(𝑘)𝑦=^𝑟_−𝑟𝜕^𝑘_𝑦𝐸_𝛼−𝑖𝑦𝑥𝑎(𝑥)𝑑𝜇_𝛼(𝑥). (38) Using (10), we get the following for all 𝑦∈ℝ: ℱ_𝛼(𝑎)^(𝑘)𝑦≤𝐶^𝑟_−𝑟|𝑥|^𝑘|𝑎(𝑥)|𝑑𝜇_𝛼(𝑥)≤𝐶^𝑟_−𝑟|𝑎(𝑥)|²𝑑𝜇_𝛼(𝑥)^1/2×^𝑟_−𝑟|𝑥|^2𝑘𝑑𝜇_𝛼(𝑥)^1/2≤𝐶‖𝑎‖_2,𝛼𝑟^𝑘+𝛼+1. (39) From (ii) of Definition (19), we obtain the following for all 𝑦∈ℝ: ℱ_𝛼(𝑎)^(𝑘)𝑦²≤𝐶‖𝑎‖^{2−(1/𝛽)(𝑘/(𝛼+1)+1)}_2,𝛼. (40) For 𝑢=∞, _ℝℱ_𝛼(𝑎)^(𝑘)(𝑥)²𝑑𝜇_𝛼(𝑥)≤𝐶^𝑟_−𝑟|𝑥|^2𝑘|𝑎(𝑥)|²𝑑𝜇_𝛼(𝑥)≤𝐶𝑟^2𝑘‖𝑎‖²_2,𝛼≤𝐶‖𝑎‖^{2−(1/𝛽)(𝑘/(𝛼+1))}_2,𝛼. (41) For 1<𝑢<∞, _ℝℱ_𝛼(𝑎)^(𝑘)𝑦^2𝑢′𝑑𝜇_𝛼(𝑥)=_ℝℱ_𝛼(𝑎)^(𝑘)𝑦²ℱ_𝛼(𝑎)^(𝑘)𝑦^2𝑢′−2𝑑𝜇_𝛼(𝑥)≤𝐶‖𝑎‖^{(𝑢′−1)(2−(1/𝛽)(𝑘/(𝛼+1))}_2,𝛼ℝℱ_𝛼(𝑎)^(𝑘)(𝑥)²𝑑𝜇_𝛼(𝑥)≤𝐶‖𝑎‖^{𝑢′(2−(1/𝛽)(𝑘/(𝛼+1)+(1/𝑢′))}_2,𝛼. (42) Finally, we get the following for all 𝑦∈ℝ: ((ℱ_𝛼(𝑎))^(𝑘)(𝑦))²_𝑢′,𝛼≤𝐶‖𝑎‖^{2−(1/𝛽)(𝑘/(𝛼+1)+(1/𝑢))}_2,𝛼. (43)

3. Proof of Theorem 2

Let 0<𝑝≤1 and ℓ be an integer greater than 2(𝛼+1)𝛽. Set 𝑠=[2(𝛼+1)(1/𝑝−1)], 𝜖=ℓ/2(𝛼+1)−(1/2), 𝑎=1−(1/𝑝)+𝜖, and 𝑏=𝜖+(1/2).

We have ℓ−1≥𝑠; then, according to Proposition 9 to prove Theorem 2 it suffices to prove that, for any (𝛽,2,ℓ)-atom 𝑓, 𝑇^𝛼_𝑚𝑓 is a central (𝑝,𝑠,𝜖)-molecule with 𝑁(𝑇^𝛼_𝑚𝑓)<𝐶 for some constant 𝐶 independent of 𝑓. In other words, we need to check that (i)𝑇^𝛼_𝑚𝑓(𝜉),𝑇^𝛼_𝑚𝑓(𝜉)𝜉^ℓ∈𝐿²𝜇_𝛼,(ii)𝑇^𝛼_𝑚𝑓^𝑎/𝑏_2,𝛼𝑇^𝛼_𝑚𝑓(𝜉)𝜉^{ℓ1−𝑎/𝑏}_2,𝛼≡𝑁𝑇^𝛼_𝑚𝑓()<𝐶,iii_ℝ𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗𝑑𝜇_𝛼(𝜉)=0∀𝑗=0,1,…,𝑠. (44) Firstly, we prove (i) and (ii).

𝑚 satisfies the Hörmander condition 𝑀_𝛼(2,ℓ); then, by Theorem 1, there exists a constant 𝐶 independent of 𝑓, such that 𝑇^𝛼_𝑚𝑓_2,𝛼≤𝐶𝑓_2,𝛼. (45) From (14) and (13), we have Λ^ℓ_𝛼ℱ_𝛼𝑇^𝛼_𝑚𝑓(𝜉)=ℱ_𝛼(−𝑖𝑥)^ℓ𝑇^𝛼_𝑚𝑓(𝑥)(𝜉). (46) Then, by Plancherel theorem to estimate ‖𝑇^𝛼_𝑚𝑓(𝜉)|𝜉|^ℓ‖_2,𝛼, it suffices to estimate ‖Λ^ℓ_𝛼(𝑚ℱ_𝛼(𝑓))‖_2,𝛼, which turns out to prove that Λ^ℓ_𝛼𝑚ℱ_𝛼𝑓_2,𝛼≤𝐶𝑓^{1−(ℓ/2(𝛼+1)𝛽)}_2,𝛼. (47)

By induction, we have Λ^ℓ_𝛼𝑚ℱ_𝛼𝑓(𝜉)=^𝑙_𝑟=0𝑎_𝑟𝜉^𝑟−ℓ𝑚ℱ_𝛼𝑓^(𝑟)(𝜉)+^𝑙_𝑟=0𝑏_𝑟𝜉^𝑟−ℓ𝑚ℱ_𝛼𝑓^(𝑟)(−𝜉), (48) where 𝑎_𝑟 and 𝑏_𝑟 are constants.

But, using Leibniz formula, we have the following for 𝑟∈{0,1,…,ℓ}: 𝑚ℱ_𝛼𝑓^(𝑟)(𝜉)=^𝑟_𝑘=0𝐶^𝑘_𝑟ℱ_𝛼𝑓^(𝑘)(𝜉)(𝑚)^{(𝑟−𝑘)}(𝜉). (49) So, to establish (47), it suffices to claim that 𝜉^𝑟−ℓℱ_𝛼𝑓^(𝑘)(𝜉)(𝑚)^{(𝑟−𝑘)}(𝜉)_2,𝛼≤𝐶𝑓^{1−(ℓ/2(𝛼+1)𝛽)}_2,𝛼forallintegers0≤𝑘≤𝑟≤ℓ. (50) For the case 𝑘=ℓ, we use Lemma 10 (ii) with 𝑢=∞ and Lemma 3 to get the following: 𝜉^𝑟−ℓℱ_𝛼𝑓^(𝑘)(𝜉)(𝑚)^{(𝑟−𝑘)}(𝜉)_2,𝛼≤𝐶ℱ_𝛼𝑓^(𝑘)_2,𝛼≤𝐶𝑓^{1−(ℓ/2(𝛼+1)𝛽)}_2,𝛼. (51) For 0≤𝑘<ℓ, we have 𝜉^𝑟−ℓℱ_𝛼𝑓^(𝑘)(𝜉)(𝑚)^{(𝑟−𝑘)}(𝜉)²_2,𝛼=_{𝑗∈ℤ2𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟−ℓ)}ℱ_𝛼𝑓^(𝑘)(𝜉)²×(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉)=𝑆₁+𝑆₂, (52) where 𝑆₁=^𝑗0_{𝑗=−∞2𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟−ℓ)}ℱ_𝛼𝑓^(𝑘)(𝜉)²×(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉),𝑆₂=^+∞_{𝑗=𝑗0+12𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟−ℓ)}ℱ_𝛼𝑓^(𝑘)(𝜉)²×(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉), (53) and 𝑗₀ is the integer, such that 2^{2(𝛼+1)𝑗0}<𝑓^1/𝛽_2,𝛼≤2^{2(𝛼+1)(𝑗0+1)}. (54) Firstly, we estimate 𝑆₁.

Using (i) of Lemma 10 and the fact that 𝑚 satisfies the Hörmander condition 𝑀_𝛼(2,ℓ), we get _{2𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟−ℓ)}ℱ_𝛼𝑓^(𝑘)(𝜉)²(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉)≤𝐶𝑓^{2−(1/𝛽)(ℓ/(𝛼+1)+1)}_2,𝛼×_{2𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟+1−𝑘)}(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉)≤𝐶𝑓^{2−(1/𝛽)(ℓ/(𝛼+1)+1)}_2,𝛼2^{2(𝛼+1)𝑗}. (55) By (54), we obtain 𝑆₁≤𝐶𝑓^{2−(1/𝛽)(ℓ/(𝛼+1)+1)}_2,𝛼2^{2(𝛼+1)𝑗0}≤𝐶𝑓^{2−(ℓ/(𝛼+1)𝛽)}_2,𝛼. (56) Now, we estimate 𝑆₂. By Holder’s inequality, we have _{2𝑗<|𝜉|<2𝑗+1}𝜉^{2(𝑟−ℓ)}ℱ_𝛼𝑓^(𝑘)(𝜉)²(𝑚)^{(𝑟−𝑘)}(𝜉)²𝑑𝜇_𝛼(𝜉)≤2^{2𝑗(𝑟−ℓ)}_{2𝑗<|𝜉|<2𝑗+1}ℱ_𝛼𝑓^(𝑘)(𝜉)^2𝑢′𝑑𝜇_𝛼(𝜉)^1/𝑢′×_{2𝑗<|𝜉|<2𝑗+1}(𝑚)^{(𝑟−𝑘)}(𝜉)^2𝑢𝑑𝜇_𝛼(𝜉)^1/𝑢. (57) Using (ii) of Lemmas 10 and 3, we get 𝑆₂≤𝐶𝑓^{2−(1/𝛽)((𝑘/(𝛼+1))+(1/𝑢))}_2,𝛼^+∞_𝑗=𝑗0+12^{(2(𝛼+1)/𝑢)−2(ℓ−𝑘)𝑗}. (58) To guarantee the convergence of this summation, we choose the pair (𝑘,𝑢) as follows:(a)if 𝑙−𝑘>𝛼+1, we choose 𝑢=1;(b)if 0<𝑙−𝑘≤𝛼+1 and 𝑘>𝛼+1, we choose 𝑢=∞;(c)if 0<𝑙−𝑘≤𝛼+1 and 𝑘≤𝛼+1, we choose 0<𝑢<∞such that 𝑘>(𝛼+1)(1−(1/𝑢)).

Furthermore, by (54), we get 𝑆₂≤𝐶𝑓^{2−(1/𝛽)(𝑘/(𝛼+1)+(1/𝑢))}_2,𝛼2^{(2(𝛼+1)/𝑢)−2(ℓ−𝑘)𝑗0+1}≤𝐶𝑓^{2−(ℓ/(𝛼+1)𝛽)}_2,𝛼. (59)

Finally, combining (56) and (59), we obtain (47). (i) and (ii) are hence proved.

To prove (iii), it suffices to prove that 𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗∈𝐿¹(𝜇_𝛼) for all integer 0≤𝑗≤𝑠 and Λ^𝑗_𝛼(𝑚ℱ_𝛼(𝑓))(0)=0: indeed if 𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗∈𝐿¹(𝜇_𝛼) according to (14), which we have Λ^𝑗_𝛼(ℱ_𝛼(𝑇^𝛼_𝑚𝑓))(𝑥)=𝐶ℱ_𝛼(𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗)(𝑥) is continuous, and hence ∫_ℝ𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗𝑑𝜇_𝛼(𝜉)=𝐶Λ^𝑗_𝛼(𝑚ℱ_𝛼(𝑓))(0).

Now, we check 𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗∈𝐿¹(𝜇_𝛼). We write ∫_ℝ|𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗|𝑑𝜇_𝛼(𝜉)=𝐼₁+𝐼₂, where 𝐼₁=_|𝜉|≤1𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗𝑑𝜇_𝛼(𝜉),𝐼₂=_|𝜉|>1𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑗𝑑𝜇_𝛼(𝜉). (60) Using the fact that 𝑇^𝛼_𝑚𝑓∈𝐿²(𝜇_𝛼) and Schwarz’s inequality, we get 𝐼₁≤_|𝜉|≤1𝑇^𝛼_𝑚𝑓(𝜉)𝑑𝜇_𝛼(𝜉)≤_|𝜉|≤1𝑇^𝛼_𝑚𝑓(𝜉)²𝑑𝜇_𝛼(𝜉)^1/2_|𝜉|≤1𝑑𝜇_𝛼(𝜉)^1/2≤𝐶𝑇^𝛼_𝑚𝑓_2,𝛼≤∞. (61) For 0≤𝑗≤𝑠, we have 𝐼₂≤_|𝜉|>1𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑠𝑑𝜇_𝛼(𝜉)≤_|𝜉|>1𝑇^𝛼_𝑚𝑓(𝜉)²𝜉^2ℓ𝑑𝜇_𝛼(𝜉)^1/2×_|𝜉|>1𝜉^{2(𝑠−ℓ)}𝑑𝜇_𝛼(𝜉)^1/2=𝑇^𝛼_𝑚𝑓(𝜉)𝜉^𝑙_{2,𝛼|𝜉|>1}𝜉^{2(𝑠−ℓ)}𝑑𝜇_𝛼(𝜉)^1/2. (62) Using the fact that 𝑠−ℓ<𝛼+1, we get 𝐼₂≤𝐶.

Finally, we check Λ^𝑗_𝛼𝑚ℱ_𝛼𝑓(0)=0,0≤𝑗≤𝑠. (63) We have Λ^𝑗_𝛼𝑚ℱ_𝛼𝑓(ℎ)=^𝑗_𝑟=0𝑎_𝑟ℎ^𝑟−𝑗𝑚ℱ_𝛼𝑓^(𝑟)(ℎ)+^𝑗_𝑟=0𝑏_𝑟ℎ^𝑟−𝑗𝑚ℱ_𝛼𝑓^(𝑟)(−ℎ), (64) where 𝑎_𝑟 and 𝑏_𝑟 are constants. Then, to prove (63), it suffices to prove that lim_ℎ→0ℎ^𝑟−𝑗𝑚^{(𝑟−𝑘)}(ℎ)ℱ_𝛼𝑓^(𝑘)(ℎ)=0,forallintegers0≤𝑘≤𝑟≤𝑗≤𝑠. (65) By (i) of Lemma 10, we have ℎ^𝑟−𝑗𝑚^{(𝑟−𝑘)}(ℎ)ℱ_𝛼𝑓^(𝑘)(ℎ|ℎ|)≤𝐶^{𝑠+1−𝑗}|ℎ|^𝑟−𝑘𝑚^{(𝑟−𝑘)}(ℎ)𝑓^𝐴_2,𝛼. (66) According to Lemma 3, we have |ℎ|^𝑟−𝑘|𝑚^{(𝑟−𝑘)}(ℎ)|≤𝐶; indeed 2(𝑟−𝑘−ℓ)+𝛼+1<0; then, we obtain lim_ℎ→0ℎ^𝑟−𝑗𝑚^{(𝑟−𝑘)}(ℎ)ℱ_𝛼𝑓^(𝑘)(ℎ)≤𝐶lim_ℎ→0|ℎ|^{𝑠+1−𝑗}=0, (67) where (63) is hence proved. This finishes the proof of Theorem 2.

Corollary 11. Let 0<𝑝≤1. Then, the generalized Hilbert transform 𝐻_𝛼 defined by 𝐻_𝛼𝑓=Γ(𝛼+(3/2))√𝜋Γ(𝛼+1)lim_{𝜀→0|𝑦|>𝜀}𝜏_𝑥𝑓−𝑦𝑦𝑑𝑦, (68) where 𝜏_𝑥 is given by (19), is bounded on 𝐻𝐾̇^𝛽,𝑝_𝛼,2.

Proof. From Proposition 3.6 in [3], the generalized Hilbert transform 𝐻_𝛼 is a multiplier operator 𝑇^𝛼_𝑚 with 𝑚(𝜉)=−sign(𝜉); then the proof of the corollary follows from Theorem 2.

Acknowledgment

This paper was supported by a generous grant from Taibah University Research Project.