# Some Embeddings into the Morrey and Modified Morrey Spaces Associated with the Dunkl Operator

Authors: Emin V. Guliyev; Yagub Y. 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 some embeddings into the Morrey space (D-Morrey space) Lp,λ,α, 0≤λ<2α+2 and modified Morrey space (modified D-Morrey space) L̃p,λ,α associated with the Dunkl operator on ℝ. As applications we get boundedness of the fractional maximal operator Mβ, 0≤β<2α+2, associated with the Dunkl operator (fractional D-maximal operator) from the spaces Lp,λ,α to L∞(ℝ) for p=(2α+2-λ)/β and from the spaces L̃p,λ,α(ℝ) to L∞(ℝ) for (2α+2-λ)/β≤p≤(2α+2)/β.


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## Body

## 1. Introduction

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl [1] and are denoted by Λα, where α is a real parameter >-1/2. These operators are associated with the reflection group ℤ2 on ℝ. Rösler in [2] shows that the Dunkl kernel verifies a product formula. This allows us to define the Dunkl translations τx, x∈ℝ.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 [3]). Later, Morrey spaces found important applications to Navier-Stokes [4, 5] and Schrödinger [6–8] equations, elliptic problems with discontinuous coefficients [9, 10], and potential theory [11–13]. An exposition of the Morrey spaces can be found in the book [14].In the present work, we study some embeddings into theD-Morrey and modified D-Morrey spaces. As applications we give boundedness of the fractional D-maximal operator in the D-Morrey and modified D-Morrey spaces.The paper is organized as follows. In Section2, we present some definitions and auxiliary results. In Section 3, we give some embeddings into the D-Morrey and modified D-Morrey spaces. In Section 4, we prove the boundedness of the fractional D-maximal operator Mβ from the spaces Lp,λ,α to L∞(ℝ) for p=(2α+2-λ)/β and from the spaces L̃p,λ,α(ℝ) to L∞(ℝ) for (2α+2-λ)/β≤p≤(2α+2)/β.


## 2. Preliminaries

On the real line, we consider the first-order differential-difference operator defined by(2.1)Λα(f)(x):=ddxf(x)+2α+1x(f(x)-f(-x)2),α>-12,
which is called the Dunkl operator. For λ∈ℂ, the Dunkl kernel Eα(λ·) on ℝ was introduced by Dunkl in [1] (see also [15–17]) and is given by(2.2)Eα(λx)=jα(iλx)+λx2(α+1)jα+1(iλx),x∈ℝ,
where jα is the normalized Bessel function of the first kind and order α [18], defined by(2.3)jα(z):=2αΓ(α+1)Jα(z)zα=Γ(α+1)∑n=0∞(-1)n(z/2)2nn!Γ(n+α+1),z∈ℂ.
The Dunkl kernel Eα(λ·) is the unique analytic solution on ℝ of the initial problem for the Dunkl operator (see [1]).Letμα be the weighted Lebesgue measure on ℝ given by(2.4)dμα(x):=|x|2α+12α+1Γ(α+1)dx.For every1≤p≤∞, we denote by Lp,α(ℝ)=Lp(dμα) the spaces of complex-valued functions f, measurable on ℝ such that(2.5)∥f∥Lp,α:=(∫ℝ|f(x)|pdμα(x))1/p<∞ifp∈[1,∞),∥f∥L∞,α:=esssupx∈ℝ|f(x)|ifp=∞.For1≤p<∞, we denote by WLp,α(ℝ) the weak Lp,α spaces defined as the set of locally integrable functions f(x), x∈ℝ with the finite norm(2.6)∥f∥WLp,α:=supr>0r(μα{x∈ℝ:|f(x)|>r})1/p.Note that(2.7)Lp,α(ℝ)⊂WLp,α(ℝ),∥f∥WLp,α≤∥f∥Lp,α∀f∈Lp,α.For allx,y,z∈ℝ, we put(2.8)Wα(x,y,z):=(1-σx,y,z+σz,x,y+σz,y,x)Δα(x,y,z),
where(2.9)σx,y,z:={x2+y2-z22xyifx,y∈ℝ∖{0},0otherwise,
and Δα is the Bessel kernel given by(2.10)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(2.11)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(2.12)τxf(y):=∫ℝf(z)dνx,y(z).Using the change of variablez=Ψ(x,y,θ)=x2+y2-2xycosθ, we have also (see [2])(2.13)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 [16]).
For allx∈ℝ the operator τx extends to Lp,α(ℝ), p≥1 and we have for f∈Lp,α(ℝ),
(2.14)∥τxf∥Lp,α≤4∥f∥Lp,α.LetB(0,t)=]-t,t[, t>0 and μα(]-t,t[)=bαt2α+2, where bα=2-α-1((α+1)Γ(α+1))-1. For L1,αloc(ℝ) (the space of locally integrable functions on ℝ), we consider(2.15)Mf(x):=supr>0(μαB(0,r))-1∫B(0,r)τx|f|(y)dμα(y).Theorem 2.2 (see [19]).
(1) If f∈L1,α(ℝ), then for every β>0,(2.16)μα{x∈ℝ:Mf(x)>β}≤Cβ∥f∥L1,α,
where C>0 is independent of f.
(2) Iff∈Lp,α(ℝ), 1<p≤∞, then Mf∈Lp,α(ℝ) and
(2.17)∥Mf∥Lp,α≤Cp∥f∥Lp,α,
where Cp>0 is independent of f.Corollary 2.3.
Iff∈L1,αloc(ℝ), then
(2.18)limr→0(μαB(0,r))-1∫B(0,r)τxf(y)dμα(y)=f(x)
for a.e. x∈ℝ.


## 3. Some Embeddings into theD-Morrey and Modified D-Morrey Spaces

Definition 3.1 (see [20]).
Let1≤p<∞, 0≤λ≤2α+2, and [t]1=min{1,t}, t>0. We denote by Lp,λ,α(ℝ) Morrey space (≡D-Morrey space) and by L̃p,λ,α(ℝ) the modified Morrey space (≡ modified D-Morrey space), associated with the Dunkl operator as the set of locally integrable functions f(x), x∈ℝ, with the finite norms
(3.1)∥f∥Lp,λ,α:=supx∈ℝ,t>0(t-λ∫B(0,t)τx|f|p(y)dμα(y))1/p,∥f∥L̃p,λ,α:=supx∈ℝ,t>0([t]1-λ∫B(0,t)τx|f|p(y)dμα(y))1/p,
respectively.Ifλ<0 or λ>2α+2, then L̃p,λ,α(ℝ)=Θ, where Θ is the set of all functions equivalent to 0 on ℝ.Note that(3.2)Lp,α(ℝ)⊂≻L̃p,0,α(ℝ)=Lp,0,α(ℝ),∥f∥L̃p,0,α=∥f∥Lp,0,α≤4∥f∥Lp,α,(3.3)L̃p,λ,α(ℝ)⊂≻Lp,α(ℝ),∥f∥Lp,α≤∥f∥L̃p,λ,α,L̃p,λ,α(ℝ)⊂≻Lp,λ,α(ℝ),∥f∥Lp,λ,α≤∥f∥L̃p,λ,α.Definition 3.2 (see [19]).
Let1≤p<∞, 0≤λ≤2α+2. We denote by WLp,λ,α(ℝ) the weak D-Morrey space and by WL̃p,λ,α(ℝ) the modified weak D-Morrey space as the set of locally integrable functions f(x), x∈ℝ with finite norms
(3.4)∥f∥WLp,λ,α:=supr>0rsupx∈ℝ,t>0(t-λμα{y∈B(0,t):τx|f|(y)>r})1/p,∥f∥WL̃p,λ,α:=supr>0rsupx∈ℝ,t>0([t]1-λμα{y∈B(0,t):τx|f|(y)>r})1/p,
respectively.We note that(3.5)Lp,λ,α(ℝ)⊂WLp,λ,α(ℝ),∥f∥WLp,λ,α≤∥f∥Lp,λ,α,L̃p,λ,α(ℝ)⊂WL̃p,λ,α(ℝ),∥f∥WL̃p,λ,α≤∥f∥L̃p,λ,α.Lemma 3.3 (see [20]).
Let1≤p<∞. Then
(3.6)Lp,2α+2,α(ℝ)=L∞(ℝ),∥f∥Lp,2α+2,α=4bα1/p∥f∥L∞.Lemma 3.4.
Let1≤p<∞, 0≤λ≤2α+2. Then
(3.7)L̃p,λ,α(ℝ)=Lp,λ,α(ℝ)∩Lp,α(ℝ),max{∥f∥Lp,λ,α,∥f∥Lp,α}≤∥f∥L̃p,λ,α≤max{∥f∥Lp,λ,α,4∥f∥Lp,α}.Proof.
Letf∈L̃p,λ,α(ℝ). Then by (3.3) we have
(3.8)L̃p,λ,α(ℝ)⊂≻Lp,λ,α(ℝ)∩Lp,α(ℝ),max{∥f∥Lp,λ,α,∥f∥Lp,α}≤∥f∥L̃p,λ,α.
Letf∈Lp,λ,α(ℝ)∩Lp,α(ℝ). Then
(3.9)∥f∥L̃p,λ,α=supx∈ℝ,t>0([t]1-λ∫B(0,t)τx|f|p(y)dμα(y))1/p=max{supx∈ℝ,0<t≤1(t-λ∫B(0,t)τx|f|p(y)dμα(y))1/p,supx∈ℝ,t>1(∫B(0,t)τx|f|p(y)dμα(y))1/p}≤max{∥f∥Lp,λ,α,4∥f∥Lp,α}.
Therefore, f∈L̃p,λ,α(ℝ) and the embedding Lp,λ,α(ℝ)∩Lp,α(ℝ)⊂≻L̃p,λ,α(ℝ) is valid.
ThusL̃p,λ,α(ℝ)=Lp,λ,α(ℝ)∩Lp,α(ℝ).From Lemmas3.3 and 3.4 for 1≤p<∞, we have(3.10)L̃p,2α+2,α(ℝ)=L∞(ℝ)∩Lp,α(ℝ).Lemma 3.5.
Let0≤λ≤2α+2. Then
(3.11)L(2α+2)/(2α+2-λ),α(ℝ)⊂≻L1,λ,α(ℝ),∥f∥L1,λ,α≤4bαλ/(2α+2)∥f∥L(2α+2)/(2α+2-λ),α.Proof.
The embedding is a consequence of Hölder's inequality and Proposition2.1. Indeed,
(3.12)∥f∥L1,λ,α=supx∈ℝ,t>0t-λ∫B(0,t)τx|f|(y)dμα(y)≤supx∈ℝ,t>0t-λ(μαB(0,t))λ/(2α+2)(∫B(0,t)τx|f|(y)(2α+2)/(2α+2-λ)dμα(y))(2α+2-λ)/(2α+2)≤4bαλ/(2α+2)∥f∥L(2α+2)/(2α+2-λ),α.On theD-Morrey spaces, the following embedding is valid.Lemma 3.6 (see [20]).
Let0≤λ<2α+2 and 0≤β<2α+2-λ. Then for p=(2α+2-λ)/β,(3.13)Lp,λ,α(ℝ)⊂L1,2α+2-β,α(ℝ),∥f∥L1,2α+2-β,α≤bα1/p′∥f∥Lp,λ,α,
where 1/p+1/p'=1.On the modifiedD-Morrey spaces, the following embedding is valid.Lemma 3.7.
Let0≤λ<2α+2 and 0≤β<2α+2-λ. Then for (2α+2-λ)/β≤p≤(2α+2)/β,(3.14)L̃p,λ,α(ℝ)⊂≻L1,2α+2-β,α(ℝ),∥f∥L1,2α+2-β,α≤bα1/p′∥f∥L̃p,λ,α.Proof.
Let0<λ<2α+2, 0<β<2α+2-λ, f∈L̃p,λ,α(ℝ), and (2α+2-λ)/β≤p≤(2α+2)/β. By the Hölder's inequality, we have
(3.15)∥f∥L1,2α+2-β,α=supx∈ℝ,t>0[t]1β-2α-2∫B(0,t)τx|f|(y)dμα(y)≤bα1/p′supx∈ℝ,t>0([t]1t-1)-(2α+2)/p′[t]1β-(2α+2-λ)/p×([t]1-λ∫B(0,t)τx|f|p(y)dμα(y))1/p=bα1/p′supx∈ℝ,t>0([t]1t-1)2α+2-β([t]1t-1)-(2α+2)/p′[t]1β-(2α+2-λ)/p×([t]1-λ∫B(0,t)τx|f|p(y)dμα(y))1/p≤bα1/p′∥f∥L̃p,λ,αsupt>0([t]1t-1)(2α+2)/p-β[t]1β-(2α+2-λ)/p.
Note that
(3.16)supt>0([t]1t-1)(2α+2)/p-β[t]1β-(2α+2-λ)/p=max{sup0<t≤1tβ-(2α+2-λ)/p,supt>1tβ-(2α+2)/p}<∞iff2α+2-λβ≤p≤2α+2β.
Therefore,f∈L1,2α+2-β,α(ℝ) and
(3.17)∥f∥L1,2α+2-β,α≤bα1/p′∥f∥L̃p,λ,α.


## 4. Some Applications

In this section, using the results of Section3, we get the boundedness of the fractional D-maximal operator in the D-Morrey and modified D-Morrey spaces.For0≤β<2α+2, we define the fractional maximal functions(4.1)Mβf(x):=supt>0(μαB(0,t))-1+β/(2α+2)∫B(0,t)τx|f|(y)dμα(y),Mp,βf(x):=(Mβ|f|p)(1/p)(x).
In the case β=0, we denote Mp,0f by Mpf. Note that M1f=Mf.Lemma 4.1.
Let1≤p<∞, 0≤β<2α+2, and f∈Lp,2α+2-β,α(ℝ). Then Mp,βf∈L∞(ℝ) and the following equality
(4.2)∥Mp,βf∥L∞=bα(β/(2α+2)-1)(1/p)∥f∥Lp,2α+2-β,α
is valid.Proof.
(4.3)∥Mp,βf∥L∞=bα(β/(2α+2)-1)(1/p)supx∈ℝ,t>0(tβ-2α-2∫B(0,t)τx|f|p(y)dμα(y))1/p=bα(β/(2α+2)-1)(1/p)∥f∥Lp,2α+2-β,α.Takingβ=0 in Lemma 4.1 and using Lemma 3.3, we get for Mpf the following result.Corollary 4.2.
Let1≤p<∞. Then
(4.4)∥Mpf∥L∞=4∥f∥L∞.Lemma 4.3.
Let1≤p<∞, 0≤β<2α+2, and f∈L̃p,2α+2-β,α(ℝ). Then Mp,βf∈L∞(ℝ) and the following equality
(4.5)∥Mp,βf∥L∞=bα(β/(2α+2)-1)(1/p)∥f∥L̃p,2α+2-β,α
is valid.Corollary 4.4.
Let0≤λ<2α+2 and 0≤β<2α+2-λ. Then the operator Mβ is bounded from Lp,λ,α to L∞ for p=(2α+2-λ)/β. Moreover,
(4.6)∥Mβf∥L∞=bαβ/(2α+2)-1∥f∥L1,2α+2-β,α≤bαβ/(2α+2)-1/p∥f∥Lp,λ,α.Corollary 4.5.
1≤p<∞,0≤λ<2α+2,0≤β<2α+2-λ. Then the operator Mβ is bounded from L̃p,λ,α to L∞ for (2α+2-λ)/β≤p≤(2α+2)/β. Moreover,
(4.7)∥Mβf∥L∞=bαβ/(2α+2)-1∥f∥L̃1,2α+2-β,α≤bαβ/(2α+2)-1/p∥f∥L̃p,λ,α.



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