# Necessary and Sufficient Conditions for the Boundedness of Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

Authors: Emin Guliyev; Ahmet Eroglu; Yagub Mammadov
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We consider the generalized shift operator, associated with the Dunkl operatorΛα(f)(x)=(d/dx)f(x)+((2α+1)/x)((f(x)-f(-x))/2), α>-1/2. We study the boundedness of the Dunkl-type fractional maximal operator Mβ in the Dunkl-type Morrey space Lp,λ,α(ℝ), 0≤λ<2α+2. We obtain necessary and sufficient conditions on the parameters for the boundedness Mβ, 0≤β<2α+2 from the spaces Lp,λ,α(ℝ) to the spaces Lq,λ,α(ℝ), 1<p≤q<∞, and from the spaces L1,λ,α(ℝ) to the weak spaces WLq,λ,α(ℝ), 1<q<∞. As an application of this result, we get the boundedness of Mβ from the Dunkl-type Besov-Morrey spaces Bpθ,λ,αs(ℝ) to the spaces Bqθ,λ,αs(ℝ), 1<p≤q<∞, 0≤λ<2α+2, 1/p-1/q=β/(2α+2-λ), 1≤θ≤∞, and 0<s<1.


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## Body

## 1. Introduction

On the real line, the Dunkl operatorsΛα are differential-difference operators introduced in 1989 by Dunkl [1]. For a real parameter α>-1/2, we consider the Dunkl operator, associated with the reflection group ℤ2 on ℝ: (1.1)Λα(f)(x):=ddxf(x)+2α+1x(f(x)-f(-x)2).In the theory of partial differential equations, together with weightedLp,w(ℝn) spaces, Morrey spaces Lp,λ(ℝn) play an important role. Morrey spaces were introduced by Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variations (see [2]).The Hardy-Littlewood maximal function, fractional maximal function, and fractional integrals are important technical tools in harmonic analysis, theory of functions, and partial differential equations. In the works [3–5], the maximal operator and in [6, 7] the fractional maximal operator associated with the Dunkl operator on ℝ were studied. In this work, we study the boundedness of the fractional maximal operator Mβ (Dunkl-type fractional maximal operator) in Morrey spaces Lp,λ,α(ℝ) (Dunkl-type Morrey spaces) associated with the Dunkl operator on ℝ. We obtain the necessary and sufficient conditions for the boundedness of the operator Mβ from the spaces Lp,λ,α(ℝ) to Lq,λ,α(ℝ), 1<p≤q<∞, and from the spaces L1,λ,α(ℝ) to the weak spaces WLq,λ,α(ℝ), 1<q<∞.The paper is organized as follows. In Section2, we present some definitions and auxiliary results. In Section 3, we give our main result on the boundedness of the operator Mβ in Lp,λ,α(ℝ). We obtain necessary and sufficient conditions on the parameters for the boundedness of the operator Mβ from the spaces Lp,λ,α(ℝ) to the spaces Lq,λ,α(ℝ), 1<p≤q<∞, and from the spaces L1,λ,α(ℝ) to the weak spaces WLq,λ,α(ℝ), 1<q<∞. As an application of this result, in Section 4 we prove the boundedness of the operator Mβ from the Dunkl-type Besov-Morrey spaces Bpθ,λ,αs(ℝ) to the spaces Bqθ,λ,αs(ℝ), 1<p≤q<∞, 0≤λ<2α+2, 1/p-1/q=β/(2α+2-λ), 1≤θ≤∞, and 0<s<1.Finally, we mention that,C will be always used to denote a suitable positive constant that is not necessarily the same in each occurrence.


## 2. Preliminaries

Letα>-1/2 be a fixed number and μα be the weighted Lebesgue measure on ℝ, given by (2.1)dμα(x):=(2α+1Γ(α+1))-1|x|2α+1dx.For every1≤p≤∞, we denote by Lp,α(ℝ)=Lp(dμα)(ℝ) the spaces of complex-valued functions f, measurable on ℝ such that (2.2)∥f∥p,α:=(∫ℝ|f(x)|pdμα(x))1/p<∞ifp∈[1,∞),∥f∥∞,α:=esssupx∈ℝ|f(x)|ifp=∞.For1≤p<∞ we denote by WLp,α(ℝ), the weak Lp,α(ℝ) spaces defined as the set of locally integrable functions f(x), x∈ℝ with the finite norm (2.3)∥f∥WLp,α:=supr>0r(μα{x∈ℝ:|f(x)|>r})1/p.Note that(2.4)Lp,α⊂WLp,α,∥f∥WLp,α≤∥f∥p,α∀f∈Lp,α(ℝ).For allx,y,z∈ℝ, we put (2.5)Wα(x,y,z):=(1-σx,y,z+σz,x,y+σz,y,x)Δα(x,y,z),
where (2.6)σx,y,z:={x2+y2-z22xyifx,y∈ℝ∖0,0otherwise
and Δα is the Bessel kernel given by (2.7)Δα(x,y,z):={dα([(|x|+|y|)2-z2][z2-(|x|-|y|)2])α-1/2|xyz|2αif|z|∈Ax,y,0otherwise,
where dα=(Γ(α+1))2/(2α-1πΓ(α+1/2)) and Ax,y=[||x|-|y||,|x|+|y|].In the sequel we consider the signed measureνx,y, on ℝ, given by (2.8)νx,y:={Wα(x,y,z)dμα(z)ifx,y∈ℝ∖0,dδx(z)ify=0,dδy(z)ifx=0.Forx,y∈ℝ and f being a continuous function on ℝ, the Dunkl translation operator τx is given by (2.9)τxf(y):=∫ℝf(z)dνx,y(z).Using the change of variablez=Ψ(x,y,θ)=x2+y2-2xycosθ, we have also (see [8]) (2.10)τxf(y)=Cα∫0π[f(Ψ)+f(-Ψ)+x+yΨ(f(Ψ)-f(-Ψ))]dνα(θ),
where dνα(θ)=(1-cosθ)sin2αθdθ and Cα=Γ(α+1)/2πΓ(α+1/2).Proposition 2.1 (see Soltani [9]).
For allx∈ℝ the operator τx extends to Lp,α(ℝ), p≥1 and we have for f∈Lp,α(ℝ),
(2.11)∥τxf∥Lp,α≤4∥f∥Lp,α.LetB(x,r)={y∈ℝ:|y|∈]max{0,|x|-r}, |x|+r[},r>0, and bα=[2α+1(α+1)Γ(α+1)]-1. Then B(0,r)=]-r,r[ and μαB(0,r)=bαr2α+2.Now we define the Dunkl-type fractional maximal function (see [3–5]) by (2.12)Mβf(x)=supr>0(μαB(0,r))-1+β/(2α+2)∫B(0,r)τx|f|(y)dμα(y),0≤β<2α+2.
If β=0, then M=M0 is the Dunkl-type maximal operator.In [3–5] was proved the following theorem (see also [10]).Theorem 2.2.
(1) If f∈L1,α(ℝ), then for every β>0(2.13)μα{x∈ℝ:Mf(x)>β}≤Cβ∥f∥L1,α,
where C>0 is independent of f.
(2) If f∈Lp,α(ℝ),1<p≤∞, then Mf∈Lp,α(ℝ) and(2.14)∥Mf∥Lp,α≤Cp∥f∥Lp,α,
where Cp>0 is independent of f.Definition 2.3.
Let1≤p<∞, 0≤λ≤2α+2. We denote by Lp,λ,α(ℝ) Morrey space (≡ Dunkl-type Morrey space), associated with the Dunkl operator as the set of locally integrable functions f(x), x∈ℝ, with the finite norm
(2.15)∥f∥p,λ,α=supx∈ℝ,r>0(r-λ∫B(0,r)τx|f(y)|pdμα(y))1/p.Note thatLp,0,α(ℝ)=Lp,α(ℝ), and if λ<0 or λ>2α+2, then Lp,λ,α(ℝ)=Θ, where Θ is the set of all functions equivalent to 0 on ℝ (see also [7]).Definition 2.4.
Let1≤p<∞and 0≤λ≤2α+2. We denote by WLp,λ,α(ℝ) a weak Dunkl-type Morrey space as the set of locally integrable functions f(x), x∈ℝ with finite norm
(2.16)∥f∥WLp,λ,α=supt>0tsupx∈ℝ,r>0(r-λ∫{y∈B(0,r):τx|f(y)|>t}dμα(y))1/p.We note that(2.17)Lp,λ,α(ℝ)⊂WLp,λ,α(ℝ),∥f∥WLp,λ,α≤∥f∥p,λ,α.


## 3. Main Results

The following theorem is our main result in which we obtain the necessary and sufficient conditions for the Dunkl-type fractional maximal operatorMβ to be bounded from the spaces Lp,λ,α(ℝ) to Lq,λ,α(ℝ), 1<p<q<∞ and from the spaces L1,λ,α(ℝ) to the weak spaces WLq,λ,α(ℝ), 1<q<∞.Theorem 3.1.
Let0≤β<2α+2, 0≤λ<2α+2, and 1≤p≤(2α+2-λ)/β.(1)
Ifp=1, then the condition 1-1/q=β/(2α+2-λ) is necessary and sufficient for the boundedness of Mβ from L1,λ,α(ℝ) to WLq,λ,α(ℝ).(2)
If1<p<(2α+2-λ)/β, then the condition (1/p)-(1/q)=β/(2α+2-λ) is necessary and sufficient for the boundedness of Mβ from Lp,λ,α(ℝ) to Lq,λ,α(ℝ).(3)
Ifp=(2α+2-λ)/β, then Mβ is bounded from Lp,λ,α(ℝ) to L∞(ℝ).For1≤p,θ≤∞, 0≤λ<2α+2, and 0<s<2, the Dunkl-type Besov-Morrey Bpθ,λ,αs(ℝ) consists of all functions f in Lp,λ,α(ℝ) so that (3.1)∥f∥Bpθ,λ,αs=∥f∥Lp,λ,α+(∫ℝ∥τxf(·)-f(·)∥Lp,λ,αθ|x|2α+2+sθdμα(x))1/θ<∞.Besov spaces in the setting of the Dunkl operators were studied by Abdelkefi and Sifi [11], Bouguila et al. [12], Guliyev and Mammadov [10], and Kamoun [13]. In the following theorem, we prove the boundedness of the Dunkl-type fractional maximal operator in the Dunkl-type Besov-Morrey spaces.Theorem 3.2.
For1<p≤q<∞,0≤λ<2α+2, (1/p)-(1/q)=β/(2α+2-λ), 1≤θ≤∞, and 0<s<1, the Dunkl-type fractional maximal operator Mβ is bounded from Bpθ,λ,αs(ℝ) to Bqθ,λ,αs(ℝ). More precisely, there is a constant C>0 such that
(3.2)∥Mβf∥Bqθ,λ,αs≤C∥f∥Bpθ,λ,αs
hold for all f∈Bpθ,λ,αs(ℝ).Remark 3.3.
Note that Theorem3.2 in the case λ=0 was proved in [10].


## 4. Boundedness of the Dunkl-Type Fractional Maximal Operator in the Dunkl-Type Morrey Spaces

In the following theorem, we obtain the boundedness of the Dunkl-type fractional maximal operatorMβ in the Dunkl-type Morrey spaces Lp,λ,α(ℝ).Theorem 4.1.
Let0≤β<2α+2, 0≤λ<2α+2, f∈Lp,λ,α(ℝ), and 1≤p≤(2α+2-λ)/β.(1)
Ifp=1 and 1-1/q=β/(2α+2-λ), then Mβf∈WLq,λ,α(ℝ) and
(4.1)∥Mβf∥WLq,λ,α≤C∥f∥1,λ,α,
where C>0 is independent of f.(2)
If1<p<(2α+2-λ)/β and (1/p)-(1/q)=β/(2α+2-λ), then Mβf∈Lq,λ,α(ℝ) and
(4.2)∥Mβf∥q,λ,α≤C∥f∥p,λ,α,
where C>0 is independent of f.(3)
Ifp=(2α+2-λ)/β and q=∞, then Mβf∈L∞(ℝ) and
(4.3)∥Mβf∥∞≤bα-1/p(2α+2)∥f∥p,λ,α.Proof.
The maximal functionMf(x) may be interpreted as a maximal function defined on a space of homogeneous type. By this we mean a topological space X equipped with a continuous pseudometric ρ and a positive measure μ satisfying
(4.4)μE(x,2r)≤C0μE(x,r)
with a constant C0 being independent of x and r>0. Here E(x,r)={y∈X:ρ(x,y)<r},ρ(x,y)=|x-y|. Let (X,ρ,μ) be a space of homogeneous type, where X=ℝ, ρ(x,y)=|x-y|, and dμ(x)=dμα(x). It is clear that this measure satisfies the doubling condition (4.4). Define
(4.5)Mμf(x)=supr>0(μE(x,r))-1∫E(x,r)|f(y)|dμ(y).
It is well known that the maximal operatorMμ is bounded from L1,λ(X,μ) to WL1,λ(X,μ) and is bounded on Lp,λ(X,μ) for 1<p<∞, 0≤λ<2α+2 (see [14, 15]).
The following inequality was proved in [6](4.6)Mf(x)≤CMμf(x),
where C>0 is independent of f.
Then from (4.6) we get the boundedness of the operator M from L1,λ,α(ℝ) to WL1,λ,α(ℝ) and on Lp,λ,α(ℝ), 1<p<∞. Thus in the case β=0 we complete the proof of (1) and (2).
Lett>0, 0<β<2α+2, f∈Lp,λ,α(ℝ), 1≤p≤(2α+2-λ)/β and (1/p)-(1/q)=β/(2α+2-λ). Applying the Hölders inequality we have(4.7)Mβf(x)=max{supr≥t(μαB(0,r))β/(2α+2)-1∫B(0,r)τx|f(y)|dμα(y),supr<t(μαB(0,r))β/(2α+2)-1∫B(0,r)τx|f(y)|dμα(y)}≤bαβ/(2α+2)max{bα-1/ptβ-(2α+2-λ)/p∥f∥p,λ,α,tβMf(x)}.
Therefore, for allt>0, we get(4.8)Mβf(x)≤bαβ/(2α+2)(bα-1/ptβ-(2α+2-λ)/p+∥f∥p,λ,α,tβMf(x)).
The minimum value of the right-hand side (4.8) is attained at
(4.9)t=(2α+2-λpbα-1/p∥f∥p,λ,αMf(x))p/(2α+2-λ)
and hence
(4.10)Mβf(x)≤bαβ/(2α+2)-β/(2α+2-λ)∥f∥p,λ,α1-p/q(Mf(x))p/q.
Then for1<p≤(2α+2-λ)/β from (4.10), we have(4.11)∥Mβf∥q,λ,α=supr>0(r-λ∫B(0,r)τx(Mβf(y))qdμα(y))1/q≤bαβ/(2α+2)-β/(2α+2-λ)∥f∥p,λ,α1-p/q(r-λ∫B(0,r)τx(Mf(y))pdμα(y))1/q≤bαβ/(2α+2)-β/(2α+2-λ)∥f∥p,λ,α1-p/q∥Mf∥p,λ,αp/q≤C∥f∥p,λ,α,
where C>0 is independent of f.
Also forp=1 from (4.10) we have(4.12)∥Mβf∥WLq,λ,α=supt>0tsupx∈ℝ,r>0(r-λ∫{y∈B(0,r):τxMβf(y)>t}dμα(y))1/q≤supt>0tsupx∈ℝ,r>0(r-λ∫{y∈B(0,r):τxMf(y)>bα-βq/(2α+2-λ)+βq/(2α+2)∥f∥1,λ,α1-qtq}dμα(y))1/q≤bαβ/(2α+2-λ)-β/(2α+2)∥f∥1,λ,α1-1/q∥Mf∥WL1,λ,α1/q≤C∥f∥1,λ,α,
where C>0 is independent of f.
Therefore, the caseβ>0 complete the proof of (1) and (2).
(3) Let p=(2α+2-λ)/β, f∈Lp,λ,α(ℝ); then applying Hölders inequality, we obtain(4.13)(μαB(0,r))-1+β/(2α+2)∫B(0,r)τx|f|(y)dμα(y)≤(μαB(0,r))-1+β/(2α+2)+1/p(∫B(0,r)τx|f(y)|pdμα(y))1/p=bα-λ/p(2α+2)(r-λ∫B(0,r)τx|f(y)|pdμα(y))1/p≤bα-λ/p(2α+2)∥f∥p,λ,α.
Thus the caseβ>0 completes the proof of (3).
Theorem4.1 has been proved.Proof of Theorem3.1.
Sufficiency part of the proof follows from Theorem4.1.Necessity.
(1) Let 1<p≤(2α+2-λ)/α and Mβ be bounded from Lp,λ,α(ℝ) to Lq,λ,α(ℝ).
Defineft(x):=f(tx), t>0. Then(4.14)∥ft∥p,λ,α=t-(2α+2)/psupx∈ℝ,r>0(r-λ∫B(0,tr)τtx|f(y)|pdμα(y))1/p=t-(2α+2-λ)/p∥f∥p,λ,α
and Mβft(x)=t-βMβf(tx),
(4.15)∥Mβft∥Lq,λ,α=t-βsupx∈ℝ,r>0(r-λ∫B(0,r)τtx|Mβf(y)|qdμα(y))1/q=t-β-(2α+2)/qsupx∈ℝ,r>0(r-λ∫B(0,tr)τx|Mβf(y)|qdμα(y))1/q=t-β-(2α+2-λ)/q∥Mβf∥Lq,λ,α.
By the boundedness ofMβ from Lp,λ,α(ℝ) to Lq,λ,α(ℝ),(4.16)∥Mβf∥Lq,λ,α=rβ+(2α+2-λ)/q∥Mβfr∥Lq,λ,α≤Crβ+(2α+2-λ)/q∥fr∥p,λ,α=Crβ+(2α+2-λ)/q-(2α+2-λ)/p∥f∥p,λ,α,
where C depends only on p, β, λ, and α.
If1/p>1/q+β/(2α+2-λ), then for all f∈Lp,λ,α(ℝ) we have ∥Mβf∥q,λ,α=0 as r→0, which is impossible. Similarly, if 1/p<1/q+β/(2α+2-λ), then for all f∈Lp,λ,α(ℝ) we obtain ∥Mβf∥q,λ,α=0 as r→∞, which is also impossible.
Therefore, we get1/p=1/q+β/(2α+2-λ).Necessity.
LetMβ be bounded from L1,λ,α(ℝ) to WLq,λ,α(ℝ). We have
(4.17)∥Mβfr∥WLq,λ,α=r-β-(2α+2-λ)/q∥Mβf∥WLq,λ,α.
By the boundedness ofMβ from L1,λ,α(ℝ) to WLq,λ,α(ℝ) it follows that(4.18)∥Mβf∥WLq,λ,α=rβ+(2α+2-λ)/q∥Mβfr∥WLq,λ,α≤Crβ+(2α+2-λ)/q∥fr∥1,λ,α=Crβ+(2α+2-λ)/q-(2α+2)∥f∥1,λ,α,
where C depends only on β, λ, and α.
If1<1/q+β/(2α+2-λ), then for all f∈L1,λ,α(ℝ) we have ∥Mβf∥WLq,λ,α=0 as r→0. Similarly, if 1>1/q+β/(2α+2-λ), then for all f∈L1,λ,α(ℝ) we obtain ∥Mβf∥WLq,λ,α=0 as r→∞.
Hence we get1=1/q+β/(2α+2-λ). Thus the proof of Theorem 3.1 is completed.Proof of Theorem3.2.
Forx∈ℝ, let τx be the generalized translation by x. By definition of the Besov spaces, it suffices to show that
(4.19)∥τxMβf-Mβf∥Lq,λ,α≤C2∥τxf-f∥Lp,λ,α.
It is easy to see that τx commutes with Mβ, that is, τxMβf=Mβ(τxf). Hence we have
(4.20)|τxMβf-Mβf|=|Mβ(τxf)-Mβf|≤Mβ(|τxf-f|).
Taking Lp,λ,α(ℝ) norm on both ends of the above inequality, by the boundedness of Mβ from Lp,λ,α(ℝ) to Lq,λ,α(ℝ), we obtain the desired result. Theorem 3.2 is proved.



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