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

Authors: A. Gasmi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

LetTm(f) be a multiplier operator defined in terms of Fourier transforms by Tm(f)=ℱ-1(mℱ(f)) for suitable functions f. The multiplier theorem of Hörmander [1] gives a sufficient condition on m for the operator Tm to be bounded on Lp(ℝn) whenever 1<p<∞, namely, that m is a bounded Cℓ-function on ℝn∖{0} satisfying the Hörmander condition M(2,ℓ) as follows:
(1)(∫R2R‍|m(s)(ξ)|2dξ)1/2≤CR(n+1)/2-s,∀R>0,
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 Tm is bounded on the Hardy spaces Hp(ℝn) with 0<p≤1.In [3], the authors considered the following multiplier operator which is associated with the Dunkl transform:
(2)Tmα(f)=ℱα-1(mℱα(f)),
where ℱα designs the Dunkl transform and using Hörmander’s technique proved the following theorem.Theorem 1.
Letℓ be the least integer greater than α+1 and let m be a bounded Cℓ-function on ℝ∖{0} which satisfies the Hörmander condition Mα(2,ℓ) as follows:
(3)(∫R2R‍|m(s)(ξ)|2dμα(ξ))1/2≤CRα+1-s,∀R>0,
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 Lp(μα) into itself for 1<p<∞, where Lp(μα) is the Lebesgue space on ℝ with respect to the following measure:
(4)μα(x)=(2α+1Γ(α+1))-1|x|2α+1,(α>-12).The Hardy spaces associated with Herz spaces can be regarded as the local version at the origin of the classical Hardy spacesHp and they are good substitutes for Hp when we study the boundedness of nontranslation invariant operators. To establish the boundedness of operators in hardy-type spaces on ℝn, one usually appeals to the atomic decomposition characterization of these spaces. In [4, 5], the authors studied the Herz-type Hardy spaces HK˙α,2β,p 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 HK˙α,2β,p.Theorem 2.
Let0<p≤1, β=(1/p)-(1/2), and ℓ be an integer greater than 2(α+1)β. If m satisfies the Hörmander condition Mα(2,ℓ), then the operator Tmα is bounded on HK˙α,2β,p.The paper is organized as follows. In Section2, 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 HK˙α,2β,p.Throughout this paper, letS(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].Letα>-1/2. We consider the differential-difference operator introduced in [9] as follows:
(5)Λα(f)(x)=dfdx(x)+2α+1x·f(x)-f(-x)2,f∈ℰ(ℝ),
and call it the Dunkl operator.Forλ∈ℂ, the following initial value problem:
(6)Λα(f)(x)=λf(x),f(0)=1,x∈ℝ,
has a unique solution Eα(λ·) (called the Dunkl kernel) given by
(7)Eα(z)=jα(iz)+z2(α+1)jα+1(iz),z∈ℂ,
where jα is the normalized Bessel function of the first kind (with order α) defined on ℂ by
(8)jα(z)=Γ(α+1)∑n=0∞‍(-1)n(z/2)2nn!Γ(n+α+1).
The integral representation of Eα is given by
(9)Eα(iλx)=Γ(α+1)πΓ(α+(1/2))∫-11‍(1-t)(1-t2)α-(1/2)e-iλxtdt.
From which, we get
(10)|∂xnEα(iλx)|≤|λ|n,λ,x∈ℝ,n∈ℕ.TheDunkl transformℱα, which was introduced by [10] and studied in [11], is defined for f∈L1(μα) by
(11)ℱα(f)(x)=∫ℝ‍Eα(-ixy)f(y)dμα(y),x∈ℝ.
This transform satisfies the following properties.(i)
For allf∈L1(μα), we have
(12)∥ℱα(f)∥∞,α≤∥f∥1,α.(ii)
For allf∈L1(μα) such that ℱα(f)∈L1(μα), we have the following inversion formula:
(13)ℱα-1(f)(x)=ℱα(f)(-x),a.e.x∈ℝ.(iii)
For allf∈𝒮(ℝ),
(14)ℱα(Λαf)(x)=ixℱα(f)(x).(iv)
ℱα is a topological isomorphism from 𝒮(ℝ) into itself.(v)
ℱα is an isometric isomorphism of L2(μα), and we have the following Parseval formula:
(15)∫ℝ‍f(x)g(x)¯dμα(x)=∫ℝ‍ℱα(f)(x)ℱα(g)(x)¯dμα(x),∥ℱα(f)∥2,α=∥f∥2,α.The following lemma can be proved, similar to Lemma 7.25, page 343, in [12].Lemma 3.
Letℓ be the least integer greater than α+1. If m satisfies the Hörmander condition Mα(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:
(16)∫R2R‍|m(s)(ξ)|2qdμα(ξ)≤CR2(α+1)-2qs,∀R>0.
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 allx,y,z∈ℝ, we put
(17)Wα(x,y,z)=[1-σx,y,z+σz,x,y+σz,y,x]Δα(|x|,|y|,|z|),
where
(18)σx,y,z={x2+y2-z22xy,ifx,y∈ℝ∖{0},0,otherwise,Δα(|x|,|y|,|z|)={dα[((|x|+|y|)2-z2)(z2-(|x|-|y|)2)]α-1/2|xyz|2α,if|z|∈Ax,y,0,otherwise,dα=21-α(Γ(α+1))2πΓ(α+1/2),Ax,y=[||x|-|y||,|x|+|y|].
The Dunkl translation operator τx, x∈ℝ is defined for a continuous function f on ℝ by
(19)τxf(y)=∫ℝ‍f(z)dγx,y(z),y∈ℝ,
where γx,y is the signed measures given by
(20)dγx,y(z)={Wα(x,y,z)dμα(z),ifx,y∈ℝ∖{0},dδx(z),ify=0,dδy(z),ifx=0.
The operator τx has the following properties. (i)
Forx,y∈ℝ and a continuous function f on ℝ, we have
(21)τx(f)(y)=τy(f)(x).(ii)
For allx∈ℝ, the operator τx can be extended to Lp(μα)(p≥1), and for f∈Lp(μα), we have
(22)∥τx(f)∥p,α≤3∥f∥p,α.(iii)
For allx,λ∈ℝ and f∈L1(μα), we have
(23)ℱα(τx(f))(λ)=Eα(iλx)ℱα(f)(λ).
Letp,q,r∈[1,∞] such that 1/p+1/q=1/r+1. The convolution product of f∈Lp(μα) and g∈Lq(μα) is defined by
(24)f*αg(x)=∫ℝ‍τx(f)(-y)g(y)dμα(y),a.e.x,
and we have
(25)∥f*αg∥r,α≤3∥f∥p,α∥g∥q,α.
If f,g∈L1(μα), then
(26)ℱα(f*αg)=ℱα(f)ℱα(g).Now, let us recall the definition of the Herz-type Hardy space and its atomic decomposition. ForN∈ℕ being sufficiently large, we denote by FN 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
(27)ρm,n(ϕ)=supx∈ℝ(1+|x|)m|Λαnϕ(x)|≤1.
Moreover, the system of seminorms {ρm,n}m,n∈ℕ generates the topology of S(ℝ).Letf∈S′(ℝ). We define the α-grand maximal function Gα(f) of f by
(28)Gα(f)(x)=supt>0,ϕ∈FN|ϕt*αf(x)|,x∈ℝ,
where ϕt is the dilation of ϕ given by
(29)ϕt(x)=t-2(α+1)ϕ(xt),x∈ℝ.Definition 4.
Letβ∈ℝ, p∈]0,∞[, and q∈[1,∞].(i)
The homogeneous weighted Herz spaceK˙α,qβ,p is the space constituted by all functions f∈Llocq(μα), such that
(30)∥f∥K˙α,qβ,p=[∑k=-∞∞‍22(α+1)βkp∥fχk∥q,αp]1/p<∞,
where χk is the characteristic function of Ak={x∈ℝ/2k-1≤|x|≤2k}.(ii)
The nonhomogeneous weighted Herz spaceKα,qβ,p is defined, as usual, by Kα,qβ,p=Lq(μα)∩K˙α,qβ,p. Moreover, ∥f∥Kq,αβ,p=∥f∥q,α+∥f∥K˙α,qβ,p.
Note thatK˙α,q0,q=Kα,q0,q=Lq(μα).Definition 5.
Letβ∈ℝ, p∈]0,∞], and q∈]1,∞]. The Herz-type Hardy space HK˙α,qβ,p is the space of distributions f∈S'(ℝ) such that Gα(f)∈K˙α,qβ,p. Moreover, we define
(31)∥f∥HK˙α,qβ,p=∥Gα(f)∥K˙α,qβ,p.
In the same way, we define the space HKα,qβ,p for the non-homogeneous case.Definition 6.
Letq∈]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)xkdμα(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.
Let0<p≤1<q≤∞ and β≥1-1/q. Then, f∈HK˙α,qβ,p if and only if, for all j∈ℕ∖{0}, there exist a (β,q,s)-atom aj and λj∈ℂ, such that ∑j=1∞‍|λj|p<∞ and f=∑j=1∞‍λjaj. Moreover,
(32)∥f∥HK˙α,qβ,p~inf(∑j=1∞‍|λj|p)1/p,
where the infimum is taking over all atomic decompositions of f.In the sequel, fixq=2 and β=1/p-1/2.Definition 8.
For0<p≤1. Set s≥[2(α+1)(1/p-1)], ε>s/2(α+1), a=1-1p+ε, and b=1/2+ε. A central (p,s,ε)-molecule is a function M∈L2(μα) satisfying the following: (i)
M(x)|x|2(α+1)b∈L2(μα),(ii)
∥M∥2,αa/b∥M(x)|x|2(α+1)b∥2,α1-a/b≡N(M)<∞,(iii)
∫ℝ‍M(x)xkdμα(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 HK˙α,2β,p and ∥M∥HK˙α,2β,p≤CN(M), where the constant C is independent of M.Proof.
LetM be a central (p,s,ε)-molecule and suppose that ∥M∥HK˙α,2β,p=1. In the general case, letting M~=M/∥M∥HK˙α,2β,p, we have ∥M~∥HK˙α,2β,p=1.
LetE0={|x|≤1}, Ek={2k-1<|x|≤2k}, and Mk=Mχk, k=1,2,3,…, where χk is the characteristic function of Ek. For each k, there exists a unique polynomial Qk, of degree at most s, such that if Pk=Qkχk; then
(33)∫ℝ‍(Mk-Pk)xjdμα(x)=0,j=0,1,…,s.
Using some ideas in [2], we can show that each (Mk-Pk) is a multiple of a central (β,2,s)-atom with a sequence of coefficients in lp. We also show that the sum ∑k=0+∞‍Pk can be written as an infinite linear combination of central (β,∞,s)-atom with a sequence of coefficients in lp. Since a (β,∞,s)-atom is also (β,2,s)-atom, hence,
(34)M=∑k=0+∞‍Mk=∑k=0+∞‍(Mk-Pk)+∑k=0+∞‍Pk=∑i=0+∞‍λiai,
where ai is a central (β,2,s)-atom and ∑k=0+∞‍|λi|p<∞. It follows from Theorem 7 that M∈HK˙α,2β,p and ∥M∥HK˙α,2β,p≤CN(M).The following Lemma plays an important role in the proof of the main result of this work.Lemma 10.
Leta 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
(35)(i)|(ℱα(a))(k)(y)|≤C|y|s+1-k∥a∥2,αA,A=1-1β(12+s+12(α+1))y∈ℝ,(ii)∥((ℱα(a))(k)(y))2∥u′,α≤C∥a∥2,α2-(1/β)((k/α+1)+(1/u)),1u+1u′=1y∈ℝ.Proof.
(i) Leta 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
(36)(ℱα(a))(k)(y)=Cα∫-11‍((1-t)(1-t2)α-1/2tk×∫-rr‍(ix)k[exp(ixyt)-∑n=0s-k‍(ixyt)nn!]×a(x)dμα(x)(1-t)(1-t2)α-1/2tk)dt≤C|y|s+1-k∫-rr‍|x|s+1|a(x)|dμα(x)≤C|y|s+1-k∥a∥2,α[∫-rr|x|2(s+1)dμα(x)‍]1/2≤C|y|s+1-k∥a∥2,αrs+α+2.
From (ii) of Definition (19), we obtain
(37)|(ℱα(a))(k)(y)|≤C|y|s+1-k∥a∥2,αA,A=1-1β(12+s+12(α+1)).
(ii) For u=1,
(38)(ℱα(a))(k)(y)=∫-rr‍∂ykEα(-iyx)a(x)dμα(x).
Using (10), we get the following for all y∈ℝ:
(39)|(ℱα(a))(k)(y)|≤C∫-rr‍|x|k|a(x)|dμα(x)≤C(∫-rr‍|a(x)|2dμα(x))1/2×(∫-rr‍|x|2kdμα(x))1/2≤C∥a∥2,αrk+α+1.
From (ii) of Definition (19), we obtain the following for all y∈ℝ:
(40)|(ℱα(a))(k)(y)|2≤C∥a∥2,α2-(1/β)(k/(α+1)+1).
For u=∞,
(41)∫ℝ‍|(ℱα(a))(k)(x)|2dμα(x)≤C∫-rr‍|x|2k|a(x)|2dμα(x)≤Cr2k∥a∥2,α2≤C∥a∥2,α2-(1/β)(k/(α+1)).
For 1<u<∞,
(42)∫ℝ‍|(ℱα(a))(k)(y)|2u′dμα(x)=∫ℝ‍|(ℱα(a))(k)(y)|2|(ℱα(a))(k)(y)|2u′-2dμα(x)≤C∥a∥2,α(u′-1)(2-(1/β)(k/(α+1))∫ℝ‍|(ℱα(a))(k)(x)|2dμα(x)≤C∥a∥2,αu′(2-(1/β)(k/(α+1)+(1/u′)).
Finally, we get the following for all y∈ℝ:
(43)∥((ℱα(a))(k)(y))2∥u′,α≤C∥a∥2,α2-(1/β)(k/(α+1)+(1/u)).


## 3. Proof of Theorem2

Let0<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 haveℓ-1≥s; then, according to Proposition 9 to prove Theorem 2 it suffices to prove that, for any (β,2,ℓ)-atom f, Tmαf is a central (p,s,ϵ)-molecule with N(Tmαf)<C for some constant C independent of f. In other words, we need to check that
(44)(i)Tmαf(ξ),Tmαf(ξ)|ξ|ℓ∈L2(μα),(ii)∥Tmαf∥2,αa/b∥Tmαf(ξ)|ξ|ℓ∥2,α1-a/b≡N(Tmαf)<C,(iii)∫ℝ‍Tmαf(ξ)ξjdμα(ξ)=0∀j=0,1,…,s.
Firstly, we prove (i) and (ii).m satisfies the Hörmander condition Mα(2,ℓ); then, by Theorem 1, there exists a constant C independent of f, such that
(45)∥Tmαf∥2,α≤C∥f∥2,α.
From (14) and (13), we have
(46)Λαℓ(ℱα(Tmαf))(ξ)=ℱα((-ix)ℓTmαf(x))(ξ).
Then, by Plancherel theorem to estimate ∥Tmαf(ξ)|ξ|ℓ∥2,α, it suffices to estimate ∥Λαℓ(mℱα(f))∥2,α, which turns out to prove that
(47)∥Λαℓ(mℱα(f))∥2,α≤C∥f∥2,α1-(ℓ/2(α+1)β).By induction, we have(48)Λαℓ(mℱα(f))(ξ)=∑r=0l‍arξr-ℓ(mℱα(f))(r)(ξ)+∑r=0l‍brξr-ℓ(mℱα(f))(r)(-ξ),
where ar and br are constants.But, using Leibniz formula, we have the following forr∈{0,1,…,ℓ}:
(49)(mℱα(f))(r)(ξ)=∑k=0r‍Crk(ℱα(f))(k)(ξ)(m)(r-k)(ξ).
So, to establish (47), it suffices to claim that
(50)∥ξr-ℓ(ℱα(f))(k)(ξ)(m)(r-k)(ξ)∥2,α≤C∥f∥2,α1-(ℓ/2(α+1)β)forallintegers0≤k≤r≤ℓ.
For the case k=ℓ, we use Lemma 10 (ii) with u=∞ and Lemma 3 to get the following:
(51)∥ξr-ℓ(ℱα(f))(k)(ξ)(m)(r-k)(ξ)∥2,α≤C∥(ℱα(f))(k)∥2,α≤C∥f∥2,α1-(ℓ/2(α+1)β).
For 0≤k<ℓ, we have
(52)∥ξr-ℓ(ℱα(f))(k)(ξ)(m)(r-k)(ξ)∥2,α2=∑j∈ℤ∫2j<|ξ|<2j+1|ξ|2(r-ℓ)|(ℱα(f))(k)(ξ)|2×|(m)(r-k)(ξ)|2dμα(ξ)=S1+S2,
where
(53)S1=∑j=-∞j0‍∫2j<|ξ|<2j+1‍|ξ|2(r-ℓ)|(ℱα(f))(k)(ξ)|2×|(m)(r-k)(ξ)|2dμα(ξ),S2=∑j=j0+1+∞‍∫2j<|ξ|<2j+1‍|ξ|2(r-ℓ)|(ℱα(f))(k)(ξ)|2×|(m)(r-k)(ξ)|2dμα(ξ),
and j0 is the integer, such that
(54)22(α+1)j0<∥f∥2,α1/β≤22(α+1)(j0+1).
Firstly, we estimate S1.Using (i) of Lemma10 and the fact that m satisfies the Hörmander condition Mα(2,ℓ), we get
(55)∫2j<|ξ|<2j+1‍|ξ|2(r-ℓ)|(ℱα(f))(k)(ξ)|2|(m)(r-k)(ξ)|2dμα(ξ)≤C∥f∥2,α2-(1/β)(ℓ/(α+1)+1)×∫2j<|ξ|<2j+1‍|ξ|2(r+1-k)|(m)(r-k)(ξ)|2dμα(ξ)≤C∥f∥2,α2-(1/β)(ℓ/(α+1)+1)22(α+1)j.
By (54), we obtain
(56)S1≤C∥f∥2,α2-(1/β)(ℓ/(α+1)+1)22(α+1)j0≤C∥f∥2,α2-(ℓ/(α+1)β).
Now, we estimate S2. By Holder’s inequality, we have
(57)∫2j<|ξ|<2j+1‍|ξ|2(r-ℓ)|(ℱα(f))(k)(ξ)|2|(m)(r-k)(ξ)|2dμα(ξ)≤22j(r-ℓ)(∫2j<|ξ|<2j+1‍|(ℱα(f))(k)(ξ)|2u′dμα(ξ))1/u′×(∫2j<|ξ|<2j+1‍|(m)(r-k)(ξ)|2udμα(ξ))1/u.
Using (ii) of Lemmas 10 and 3, we get
(58)S2≤C∥f∥2,α2-(1/β)((k/(α+1))+(1/u))∑j=j0+1+∞‍(2(2(α+1)/u)-2(ℓ-k))j.
To guarantee the convergence of this summation, we choose the pair (k,u) as follows:(a)
ifl-k>α+1, we choose u=1;(b)
if0<l-k≤α+1 and k>α+1, we choose u=∞;(c)
if0<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
(59)S2≤C∥f∥2,α2-(1/β)(k/(α+1)+(1/u))(2(2(α+1)/u)-2(ℓ-k))j0+1≤C∥f∥2,α2-(ℓ/(α+1)β).Finally, combining (56) and (59), we obtain (47). (i) and (ii) are hence proved.To prove (iii), it suffices to prove thatTmαf(ξ)ξj∈L1(μα) for all integer 0≤j≤s and Λαj(mℱα(f))(0)=0: indeed if Tmαf(ξ)ξj∈L1(μα) according to (14), which we have Λαj(ℱα(Tmαf))(x)=Cℱα(Tmαf(ξ)ξj)(x) is continuous, and hence ∫ℝ‍Tmαf(ξ)ξjdμα(ξ)=CΛαj(mℱα(f))(0).Now, we checkTmαf(ξ)ξj∈L1(μα). We write ∫ℝ‍|Tmαf(ξ)ξj|dμα(ξ)=I1+I2, where
(60)I1=∫|ξ|≤1‍|Tmαf(ξ)ξj|dμα(ξ),I2=∫|ξ|>1‍|Tmαf(ξ)ξj|dμα(ξ).
Using the fact that Tmαf∈L2(μα) and Schwarz’s inequality, we get
(61)I1≤∫|ξ|≤1‍|Tmαf(ξ)|dμα(ξ)≤(∫|ξ|≤1‍|Tmαf(ξ)|2dμα(ξ))1/2(∫|ξ|≤1‍dμα(ξ))1/2≤C∥Tmαf∥2,α≤∞.
For 0≤j≤s, we have
(62)I2≤∫|ξ|>1‍|Tmαf(ξ)ξs|dμα(ξ)≤(∫|ξ|>1‍|Tmαf(ξ)|2|ξ2ℓ|dμα(ξ))1/2×(∫|ξ|>1‍|ξ2(s-ℓ)|dμα(ξ))1/2=∥Tmαf(ξ)|ξ|l∥2,α(∫|ξ|>1‍|ξ2(s-ℓ)|dμα(ξ))1/2.
Using the fact that s-ℓ<α+1, we get I2≤C.Finally, we check(63)Λαj(mℱα(f))(0)=0,0≤j≤s.
We have
(64)Λαj(mℱα(f))(h)=∑r=0j‍arhr-j(mℱα(f))(r)(h)+∑r=0j‍brhr-j(mℱα(f))(r)(-h),
where ar and br are constants. Then, to prove (63), it suffices to prove that
(65)limh→0|hr-jm(r-k)(h)(ℱα(f))(k)(h)|=0,forallintegers0≤k≤r≤j≤s.
By (i) of Lemma 10, we have
(66)|hr-jm(r-k)(h)(ℱα(f))(k)(h)|≤C|h|s+1-j|h|r-k|m(r-k)(h)|∥f∥2,αA.
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
(67)limh→0|hr-jm(r-k)(h)(ℱα(f))(k)(h)|≤Climh→0|h|s+1-j=0,
where (63) is hence proved. This finishes the proof of Theorem 2.Corollary 11.
Let0<p≤1. Then, the generalized Hilbert transform Hα defined by
(68)Hα(f)=Γ(α+(3/2))πΓ(α+1)limε→0∫|y|>ε‍τx(f)(-y)ydy,
where τx is given by (19), is bounded on HK˙α,2β,p.Proof.
From Proposition 3.6 in [3], the generalized Hilbert transform Hα is a multiplier operator Tmα with m(ξ)=-sign(ξ); then the proof of the corollary follows from Theorem 2.



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*Source: 694714-2013-11-20.xml*