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

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Volume 2010, Article ID 291345,10pages doi:10.1155/2010/291345

Research Article

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

Emin V. Guliyev

¹

and Yagub Y. Mammadov

^{1, 2}

1Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

2Nakhchivan Teacher-Training Institute, Azerbaijan

Correspondence should be addressed to Yagub Y. Mammadov,[email protected] Received 29 October 2009; Revised 5 February 2010; Accepted 2 March 2010

Academic Editor: A ˘gacik Zafer

Copyrightq2010 E. V. Guliyev and Y. Y. Mammadov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the generalized shift operator, associated with the Dunkl operator Λαfx d/dxfx2α1/xfx−f−x/2,α >−1/2. We study some embeddings into the Morrey spaceD-Morrey spaceLp,λ,α, 0 ≤λ < 2α2 and modified Morrey spacemodifiedD-Morrey spaceLp,λ,αassociated with the Dunkl operator onR. As applications we get boundedness of the fractional maximal operatorMβ, 0 ≤β < 2α2, associated with the Dunkl operatorfractional D-maximal operatorfrom the spacesLp,λ,αtoL∞Rforp 2α2−λ/βand from the spaces Lp,λ,αRtoL∞Rfor2α2−λ/β≤p≤2α2/β.

1. Introduction

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl1 and are denoted byΛα, whereαis a real parameter> −1/2. These operators are associated with the reflection groupZ2 onR. R ösler in2 shows that the Dunkl kernel verifies a product formula. This allows us to define the Dunkl translationsτx,x∈R.

In the theory of partial differential equations, together with weightedLp,wRⁿspaces, Morrey spacesLp,λRⁿplay an important role. Morrey spaces were introduced by Morrey in 1938 in connection with certain problems in elliptic partial differential equations and calculus of variationssee3 . Later, Morrey spaces found important applications to Navier-Stokes 4,5 and Schr ödinger6–8 equations, elliptic problems with discontinuous coefficients9, 10 , and potential theory11–13 . An exposition of the Morrey spaces can be found in the book14 .

In the present work, we study some embeddings into theD-Morrey and modifiedD- Morrey spaces. As applications we give boundedness of the fractionalD-maximal operator in theD-Morrey and modifiedD-Morrey spaces.

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The paper is organized as follows. In Section 2, we present some definitions and auxiliary results. InSection 3, we give some embeddings into theD-Morrey and modifiedD- Morrey spaces. InSection 4, we prove the boundedness of the fractionalD-maximal operator Mβfrom the spacesLp,λ,αtoL_∞Rforp 2α2−λ/βand from the spacesLp,λ,αRto L∞Rfor2α2−λ/β≤p≤2α2/β.

2. Preliminaries

On the real line, we consider the first-order diﬀerential-diﬀerence operator defined by

Λα

f

x: d

dxfx 2α1 x

fx−f−x 2

, α >−1

2, 2.1

which is called the Dunkl operator. Forλ∈C, the Dunkl kernelEαλ·onRwas introduced by Dunkl in1 see also15–17 and is given by

Eαλx jαiλx λx

2α1jα1iλx, x∈R, 2.2

wherejαis the normalized Bessel function of the first kind and orderα18 , defined by

jαz:2^αΓα1Jαz

z^α Γα1^∞

n0

−1ⁿz/2²ⁿ

n!Γnα1, z∈C. 2.3

The Dunkl kernelEαλ·is the unique analytic solution on Rof the initial problem for the Dunkl operatorsee1 .

Letμαbe the weighted Lebesgue measure onRgiven by

dμαx: |x|^2α1

2^α1Γα1dx. 2.4

For every 1≤ p ≤ ∞, we denote byLp,αR Lpdμαthe spaces of complex-valued functionsf, measurable onRsuch that

f

Lp,α :

R

fx^pdμαx _1/p

<∞ ifp∈1,∞, f

L∞,α :ess sup

x∈R

fx ifp∞.

2.5

For 1 ≤ p < ∞,we denote by WLp,αRthe weakLp,α spaces defined as the set of locally integrable functionsfx,x∈Rwith the finite norm

f

WLp,α :sup

r>0

r μα

x∈R:fx> r_1/p

. 2.6

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Note that

Lp,αR⊂WLp,αR, f

WLp,α ≤f

Lp,α ∀f∈Lp,α. 2.7

For allx, y, z∈R, we put Wα

x, y, z :

1−σx,y,zσz,x,yσz,y,x

Δα

x, y, z

, 2.8

where

σx,y,z:

⎧⎪

⎪⎨

⎪⎪

⎩

x²y²−z²

2xy if x, y∈R\ {0},

0 otherwise,

2.9

andΔαis the Bessel kernel given by

Δα

x, y, z :

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ dα

|x|y²−z² z²−

|x| −y²_α−1/2

xyz^2α if|z| ∈Ax,y,

0 otherwise,

2.10

wheredα Γα1²/2^α−1√

π Γα1/2andAx,y x| − |y,|x||y| . In the sequel we consider the signed measureνx,y, onR, given by

νx,y :

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ Wα

x, y, z

dμαz ifx, y∈R\ {0},

dδxz ify0,

dδyz ifx0.

2.11

Forx, y ∈Randfbeing a continuous function onR, the Dunkl translation operator τxis given by

τxf y

: Rfzdνx,yz. 2.12

Using the change of variablez Ψx, y, θ

x²y²−2xycosθ, we have alsosee 2

τxf y

Cα π 0

fΨ f−Ψ xy Ψ

fΨ−f−Ψ

dναθ, 2.13

wheredναθ 1−cosθsin^2αθdθandCα Γα1/2√πΓα1/2.

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Proposition 2.1see Soltani16 . For allx∈Rthe operatorτxextends toLp,αR,p≥1 and we have forf ∈Lp,αR,

τxf

Lp,α ≤4f

Lp,α. 2.14

LetB0, t −t, t,t >0 andμα −t, t bαt^2α2, wherebα2^−α−1α1Γα1⁻¹. ForL^loc_1,αR the space of locally integrable functions onR, we consider

Mfx:sup

r>0

μαB0, r₋₁

B0,rτxfy dμα

y

. 2.15

Theorem 2.2see19 . 1Iff ∈L1,αR, then for everyβ >0,

μα

x∈R:Mfx> β

≤ C βf

L1,α, 2.16

whereC >0 is independent off.

2Iff∈Lp,αR, 1< p≤ ∞, thenMf ∈Lp,αRand Mf

Lp,α ≤Cpf

Lp,α, 2.17

whereCp>0 is independent off. Corollary 2.3. Iff∈L^loc_1,αR, then

rlim→0

μαB0, r₋₁

B0,rτxf y

dμα

y

fx 2.18

for a.e.x∈R.

3. Some Embeddings into the D -Morrey and Modified D -Morrey Spaces

Definition 3.1see20 . Let 1≤p <∞, 0≤λ≤2α2, andt ₁ min{1, t},t >0. We denote byLp,λ,αRMorrey space≡D-Morrey spaceand byLp,λ,αRthe modified Morrey space≡ modifiedD-Morrey space, associated with the Dunkl operator as the set of locally integrable functionsfx,x∈R,with the finite norms

f

Lp,λ,α : sup

x∈R,t>0

t^−λ

B0,tτxf^pydμαy 1/p

,

f

Lp,λ,α : sup

x∈R,t>0

t ^−λ₁

B0,tτxf^pydμαy _1/p

,

3.1

respectively.

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Ifλ <0 orλ >2α2, thenLp,λ,αR Θ, whereΘis the set of all functions equivalent to 0 onR.

Note that

Lp,αR⊂Lp,0,αR Lp,0,αR, f

Lp,0,α f

Lp,0,α ≤4f

Lp,α, 3.2

Lp,λ,αR⊂Lp,αR, f

Lp,α ≤f

Lp,λ,α, Lp,λ,αR⊂Lp,λ,αR, f

Lp,λ,α ≤f

Lp,λ,α.

3.3

Definition 3.2see19 . Let 1≤ p < ∞, 0 ≤ λ ≤ 2α2. We denote byWLp,λ,αRthe weak D-Morrey space and byWLp,λ,αRthe modified weakD-Morrey space as the set of locally integrable functionsfx,x∈Rwith finite norms

f

WLp,λ,α :sup

r>0

r sup

x∈R,t>0

t^−λμα

y∈B0, t: τxfy> r^1/p , f

WLp,λ,α :sup

r>0

r sup

x∈R,t>0

t ^−λ₁ μα

y∈B0, t: τxfy> r^1/p ,

3.4

respectively.

We note that

Lp,λ,αR ⊂WLp,λ,αR, f

WLp,λ,α ≤f

Lp,λ,α, Lp,λ,αR⊂WLp,λ,αR, f

WLp,λ,α ≤f

Lp,λ,α.

3.5

Lemma 3.3see20 . Let 1≤p <∞. Then

Lp,2α2,αR L∞R, f

Lp,2α2,α 4b^1/pα f

L∞. 3.6

Lemma 3.4. Let 1≤p <∞, 0≤λ≤2α2. Then

Lp,λ,αR Lp,λ,αR∩Lp,αR, maxf

Lp,λ,α,f

Lp,α

≤f

Lp,λ,α ≤maxf

Lp,λ,α,4f

Lp,α

.

3.7

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Proof. Letf ∈Lp,λ,αR. Then by3.3we have

Lp,λ,αR⊂Lp,λ,αR∩Lp,αR, maxf

Lp,λ,α,f

Lp,α

≤f

Lp,λ,α.

3.8

Letf∈Lp,λ,αR∩Lp,αR. Then

f

Lp,λ,α sup

x∈R,t>0

t ^−λ₁

max

⎧⎨

⎩ sup

x∈R,0<t≤1

t^−λ

,

sup

x∈R,t>1 B0,tτxf^pydμαy _1/p⎫

⎬

⎭≤maxf

Lp,λ,α,4f

Lp,α

.

3.9

Therefore,f ∈Lp,λ,αRand the embeddingLp,λ,αR∩Lp,αR⊂Lp,λ,αRis valid.

ThusLp,λ,αR Lp,λ,αR∩Lp,αR.

From Lemmas3.3and3.4for 1≤p <∞, we have

L_p,2α2,αR L_∞R∩Lp,αR. 3.10

Lemma 3.5. Let 0≤λ≤2α2. Then

L2α2/2α2−λ,αR⊂L1,λ,αR, f

L1,λ,α ≤4bα^λ/2α2f

L2α2/2α2−λ,α. 3.11

Proof. The embedding is a consequence of H ¨older’s inequality andProposition 2.1. Indeed, f

L1,λ,α sup

x∈R,t>0t^−λ

B0,tτxfy dμα

y

≤ sup

x∈R,t>0t^−λ

μαB0, t_λ/2α2

B0,tτxfy^{2α2/2α2−λ}dμαy

_{2α2−λ/2α2}

≤4b^λ/2α2α f

L2α2/2α2−λ,α.

3.12 On theD-Morrey spaces, the following embedding is valid.

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Lemma 3.6see20 . Let 0≤λ <2α2 and 0≤β <2α2−λ. Then forp 2α2−λ/β,

Lp,λ,αR⊂L1,2α2−β,αR, f

L1,2α2−β,α ≤b^1/pα f

Lp,λ,α, 3.13

where 1/p1/p1.

On the modifiedD-Morrey spaces, the following embedding is valid.

Lemma 3.7. Let 0≤λ <2α2 and 0≤β <2α2−λ. Then for2α2−λ/β≤p≤2α2/β,

Lp,λ,αR⊂L1,2α2−β,αR, f

L1,2α2−β,α ≤b^1/pα f

Lp,λ,α. 3.14

Proof. Let 0< λ <2α2, 0< β <2α2−λ,f∈Lp,λ,αR,and2α2−λ/β≤p≤2α2/β.

By the H ¨older’s inequality, we have

f

L1,2α2−β,α sup

x∈R,t>0t ^β−2α−2₁

B0,tτxfy dμα

y

≤b^1/pα sup

x∈R,t>0

t ₁t⁻¹_−2α2/p

t ^{β−2α2−λ/p}₁

×

t ^−λ₁

B0,tτxf^pydμαy 1/p

b^1/pα sup

x∈R,t>0

t ₁t⁻¹_2α2−β

t ₁t⁻¹_−2α2/p

t ^{β−2α2−λ/p}₁

×

t ^−λ₁

≤b^1/pα f

Lp,λ,αsup

t>0

t ₁t⁻¹_2α2/p−β

t ^{β−2α2−λ/p}₁ .

3.15

Note that

sup

t>0

t ₁t⁻¹_2α2/p−β

t ^{β−2α2−λ/p}₁ max

sup

0<t≤1t^{β−2α2−λ/p},sup

t>1

t^β−2α2/p <∞ iﬀ 2α2−λ

β ≤p≤ 2α2 β .

3.16

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Therefore,f∈L1,2α2−β,αRand f

L_{1,2α2−β,α} ≤b^1/pα f

Lp,λ,α. 3.17

4. Some Applications

In this section, using the results ofSection 3, we get the boundedness of the fractional D- maximal operator in theD-Morrey and modifiedD-Morrey spaces.

For 0≤β <2α2,we define the fractional maximal functions

Mβfx:sup

t>0

μαB0, t_−1β/2α2

B0,tτxfy dμα

y ,

Mp,βfx:

Mβf^p1/px.

4.1

In the caseβ0, we denoteMp,0fbyMpf. Note thatM1f Mf.

Lemma 4.1. Let 1≤p <∞, 0≤β <2α2,andf∈Lp,2α2−β,αR. ThenMp,βf∈L∞Rand the following equality

Mp,βf

L∞b^{β/2α2−11/p}α f

Lp,2α2−β,α 4.2

is valid.

Proof.

Mp,βf

L∞ b^{β/2α2−11/p}α sup

x∈R,t>0

t^β−2α−2

b^{β/2α2−11/p}α f

Lp,2α2−β,α.

4.3

Takingβ0 inLemma 4.1and usingLemma 3.3, we get forMpfthe following result.

Corollary 4.2. Let 1≤p <∞. Then

Mpf

L∞4f

L∞. 4.4

Lemma 4.3. Let 1≤p <∞, 0≤β <2α2,andf∈Lp,2α2−β,αR. ThenMp,βf∈L∞Rand the following equality

Mp,βf

L∞b^{β/2α2−11/p}α f

L_{p,2α2−β,α} 4.5

is valid.

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Corollary 4.4. Let 0≤λ <2α2 and 0≤β < 2α2−λ. Then the operatorMβis bounded from Lp,λ,αtoL_∞forp 2α2−λ/β. Moreover,

Mβf

L∞b^β/2α2−1α f

L1,2α2−β,α ≤b^{β/2α2−1/p}α f

Lp,λ,α. 4.6

Corollary 4.5. 1≤ p <∞,0 ≤λ <2α2,0 ≤β <2α2−λ. Then the operatorMβis bounded fromLp,λ,αtoL∞for2α2−λ/β≤p≤2α2/β. Moreover,

Mβf

L∞b^β/2α2−1α f

L1,2α2−β,α ≤b^{β/2α2−1/p}α f

Lp,λ,α. 4.7

Acknowledgment

The authors express their thank to the referees for careful reading, helpful comments and suggestions on the manuscript of this paper.

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