Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 291345, 10 pages
doi:10.1155/2010/291345
Research Article
Some Embeddings into the Morrey and Modified
Morrey Spaces Associated with the Dunkl Operator
Emin V. Guliyev1 and Yagub Y. Mammadov1, 2
1
2
Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan
Nakhchivan Teacher-Training Institute, Azerbaijan
Correspondence should be addressed to Yagub Y. Mammadov, yagubmammadov@yahoo.com
Received 29 October 2009; Revised 5 February 2010; Accepted 2 March 2010
Academic Editor: Ağacik Zafer
Copyright q 2010 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 Λα fx d/dxfx2α1/xfx−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
p,λ,α associated with the Dunkl operator on R. As applications we get boundedness of the
space L
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
p,λ,α R to L∞ R for 2α 2 − λ/β ≤ p ≤ 2α 2/β.
L
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 weighted Lp,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 the D-Morrey and modified DMorrey spaces. As applications we give boundedness of the fractional D-maximal operator
in the D-Morrey and modified D-Morrey spaces.
2
Abstract and Applied Analysis
The paper is organized as follows. In Section 2, we present some definitions and
auxiliary results. In Section 3, we give some embeddings into the D-Morrey and modified DMorrey spaces. In Section 4, we prove the boundedness of the fractional D-maximal operator
p,λ,α R to
Mβ from the spaces Lp,λ,α to L∞ R for p 2α 2 − λ/β and from the spaces L
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
2α 1 fx − f−x
d
fx ,
Λα f x :
dx
x
2
1
α>− ,
2
2.1
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
Eα λx jα iλx λx
jα1 iλx,
2α 1
x ∈ R,
2.2
where jα is the normalized Bessel function of the first kind and order α 18, defined by
jα z : 2α Γα 1
∞
Jα z
−1n z/22n
,
Γα
1
zα
n!Γn α 1
n0
z ∈ C.
2.3
The Dunkl kernel Eα λ· is the unique analytic solution on R of the initial problem for the
Dunkl operator see 1.
Let μα be the weighted Lebesgue measure on R given by
dμα x :
|x|2α1
dx.
2α1 Γα 1
2.4
For every 1 ≤ p ≤ ∞, we denote by Lp,α R Lp dμα the spaces of complex-valued
functions f, measurable on R such that
f :
Lp,α
R
fx
p dμα x
1/p
f : ess sup
fx
L∞,α
< ∞ if p ∈ 1, ∞,
2.5
if p ∞.
x∈R
For 1 ≤ p < ∞, we denote by WLp,α R the weak Lp,α spaces defined as the set of
locally integrable functions fx, x ∈ R with the finite norm
1/p
f : sup r μα x ∈ R : fx
> r
.
WLp,α
r>0
2.6
Abstract and Applied Analysis
3
Note that
f ≤ f Lp,α
WLp,α
∀f ∈ Lp,α .
2.7
Wα x, y, z : 1 − σx,y,z σz,x,y σz,y,x Δα x, y, z ,
2.8
Lp,α R ⊂ WLp,α R,
For all x, y, z ∈ R, we put
where
σx,y,z
⎧ 2
x y2 − z2
⎪
⎪
⎨
2xy
:
⎪
⎪
⎩
0
if x, y ∈ R \ {0},
2.9
otherwise,
and Δα is the Bessel kernel given by
Δα x, y, z :
⎧
⎪
⎪
⎪
⎨
dα
⎪
⎪
⎪
⎩
2 α−1/2
2
|x| y
− z2 z2 − |x| − y
xyz
2α
0
if |z| ∈ Ax,y ,
2.10
otherwise,
√
where dα Γα 12 /2α−1 π Γα 1/2 and Ax,y x| − |y, |x| |y|.
In the sequel we consider the signed measure νx,y , on R, given by
νx,y
⎧ ⎪
Wα x, y, z dμα z
⎪
⎪
⎨
: dδx z
⎪
⎪
⎪
⎩
dδy z
if x, y ∈ R \ {0},
if y 0,
2.11
if x 0.
For x, y ∈ R and f being a continuous function on R, the Dunkl translation operator
τx is given by
τx f y :
R
fzdνx,y z.
Using the change of variable z Ψx, y, θ 2.12
x2 y2 − 2xy cos θ, we have also see
2
τx f y Cα
π
0
x y
fΨ f−Ψ fΨ − f−Ψ dνα θ,
Ψ
√
where dνα θ 1 − cos θsin2α θdθ and Cα Γα 1/2 πΓα 1/2.
2.13
4
Abstract and Applied Analysis
Proposition 2.1 see Soltani 16. For all x ∈ R the operator τx extends to Lp,α R, p ≥ 1 and we
have for f ∈ Lp,α R,
τx f ≤ 4f Lp,α .
Lp,α
For
2.14
Let B0, t − t, t, t > 0 and μα − t, t bα t2α2 , where bα 2−α−1 α 1Γα 1−1 .
the space of locally integrable functions on R, we consider
Lloc
1,α R
Mfx : sup μα B0, r
−1
r>0
τx f y dμα y .
2.15
B0,r
Theorem 2.2 see 19. 1 If f ∈ L1,α R, then for every β > 0,
C μα x ∈ R : Mfx > β ≤ f L1,α ,
β
2.16
where C > 0 is independent of f.
2 If f ∈ Lp,α R, 1 < p ≤ ∞, then Mf ∈ Lp,α R and
Mf ≤ Cp f Lp,α ,
Lp,α
2.17
where Cp > 0 is independent of f.
Corollary 2.3. If f ∈ Lloc
1,α R, then
−1
lim μα B0, r
r →0
τx f y dμα y fx
2.18
B0,r
for a.e. x ∈ R.
3. Some Embeddings into the D-Morrey and Modified
D-Morrey Spaces
Definition 3.1 see 20. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 2α 2, and t1 min{1, t}, t > 0. We denote
p,λ,α R the modified Morrey space ≡
by Lp,λ,α R Morrey space ≡ D-Morrey space and by L
modified D-Morrey space, associated with the Dunkl operator as the set of locally integrable
functions fx, x ∈ R, with the finite norms
1/p
p
−λ
f : sup t
τx f ydμα y
,
Lp,λ,α
f : sup
Lp,λ,α
x∈R,t>0
respectively.
B0,t
x∈R,t>0
t−λ
1
p
τx f ydμα y
B0,t
3.1
1/p
,
Abstract and Applied Analysis
5
p,λ,α R Θ, where Θ is the set of all functions equivalent
If λ < 0 or λ > 2α 2, then L
to 0 on R.
Note that
p,0,α R Lp,0,α R,
Lp,α R ⊂ L
f f Lp,0,α ≤ 4f Lp,α ,
Lp,0,α
p,λ,α R ⊂ Lp,α R,
L
p,λ,α R ⊂ Lp,λ,α R,
L
f Lp,α
≤ f L p,λ,α ,
f ≤ f L p,λ,α .
Lp,λ,α
3.2
3.3
Definition 3.2 see 19. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 2α 2. We denote by WLp,λ,α R the weak
p,λ,α R the modified weak D-Morrey space as the set of locally
D-Morrey space and by W L
integrable functions fx, x ∈ R with finite norms
1/p
f : sup r sup t−λ μα y ∈ B0, t : τx f y > r
,
WLp,λ,α
r>0
f p,λ,α
WL
x∈R,t>0
: sup r sup
r>0
x∈R,t>0
t−λ
1 μα y
1/p
,
∈ B0, t : τx f y > r
3.4
respectively.
We note that
Lp,λ,α R ⊂ WLp,λ,α R,
f ≤ f Lp,λ,α ,
WLp,λ,α
p,λ,α R,
p,λ,α R ⊂ W L
L
f ≤ f L p,λ,α .
W Lp,λ,α
3.5
Lemma 3.3 see 20. Let 1 ≤ p < ∞. Then
Lp,2α2,α R L∞ R,
1/p f 4bα f L∞ .
Lp,2α2,α
3.6
Lemma 3.4. Let 1 ≤ p < ∞, 0 ≤ λ ≤ 2α 2. Then
p,λ,α R Lp,λ,α R ∩ Lp,α R,
L
max f Lp,λ,α , f Lp,α ≤ f L p,λ,α ≤ max f Lp,λ,α , 4f Lp,α .
3.7
6
Abstract and Applied Analysis
p,λ,α R. Then by 3.3 we have
Proof. Let f ∈ L
p,λ,α R⊂ Lp,λ,α R ∩ Lp,α R,
L
max f Lp,λ,α , f Lp,α ≤ f L p,λ,α .
3.8
Let f ∈ Lp,λ,α R ∩ Lp,α R. Then
f sup
Lp,λ,α
x∈R,t>0
max
t−λ
1
p
τx f ydμα y
1/p
B0,t
⎧
⎨
sup
t
⎩x∈R,0<t≤1
−λ
1/p
3.9
,
B0,t
p
τx f ydμα y
sup
x∈R,t>1
p
τx f ydμα y
B0,t
1/p ⎫
⎬
⎭
≤ max f Lp,λ,α , 4f Lp,α .
p,λ,α R and the embedding Lp,λ,α R ∩ Lp,α R ⊂ L
p,λ,α R is valid.
Therefore, f ∈ L
Thus Lp,λ,α R Lp,λ,α R ∩ Lp,α R.
From Lemmas 3.3 and 3.4 for 1 ≤ p < ∞, we have
p,2α2,α R L∞ R ∩ Lp,α R.
L
3.10
Lemma 3.5. Let 0 ≤ λ ≤ 2α 2. Then
λ/2α2 f f ≤ 4bα
.
L1,λ,α
L2α2/2α2−λ,α
L2α2/2α2−λ,α R ⊂ L1,λ,α R,
3.11
Proof. The embedding is a consequence of Hölder’s inequality and Proposition 2.1. Indeed,
f L1,λ,α
sup t
−λ
B0,t
x∈R,t>0
≤ sup t
τx f y dμα y
−λ
μα B0, t
λ/2α2
x∈R,t>0
τx f y2α2/2α2−λ dμα y
2α2−λ/2α2
B0,t
λ/2α2 f ≤ 4bα
.
L2α2/2α2−λ,α
On the D-Morrey spaces, the following embedding is valid.
3.12
Abstract and Applied Analysis
7
Lemma 3.6 see 20. Let 0 ≤ λ < 2α 2 and 0 ≤ β < 2α 2 − λ. Then for p 2α 2 − λ/β,
1/p f ≤ bα f Lp,λ,α ,
L1,2α2−β,α
Lp,λ,α R ⊂ L1,2α2−β,α R,
3.13
where 1/p 1/p 1.
On the modified D-Morrey spaces, the following embedding is valid.
Lemma 3.7. Let 0 ≤ λ < 2α 2 and 0 ≤ β < 2α 2 − λ. Then for 2α 2 − λ/β ≤ p ≤ 2α 2/β,
1/p f ≤ bα f L p,λ,α .
L1,2α2−β,α
p,λ,α R ⊂ L1,2α2−β,α R,
L
3.14
p,λ,α R, and 2α 2 − λ/β ≤ p ≤ 2α 2/β.
Proof. Let 0 < λ < 2α 2, 0 < β < 2α 2 − λ, f ∈ L
By the Hölder’s inequality, we have
β−2α−2
f sup t1
L1,2α2−β,α
B0,t
x∈R,t>0
1/p
≤ bα
sup
t−λ
1
×
1/p
bα
t1 t−1
sup
t−λ
1
−2α2/p
β−2α2−λ/p
t1
p
τx f ydμα y
1/p
B0,t
x∈R,t>0
×
x∈R,t>0
τx f y dμα y
t1 t−1
2α2−β t1 t−1
p
τx f ydμα y
−2α2/p
3.15
β−2α2−λ/p
t1
1/p
B0,t
2α2/p−β β−2α2−λ/p
1/p ≤ bα f L p,λ,α sup t1 t−1
.
t1
t>0
Note that
sup t1 t
t>0
−1
2α2/p−β
β−2α2−λ/p
t1
max sup tβ−2α2−λ/p , suptβ−2α2/p
0<t≤1
<∞
t>1
iff
2α 2
2α 2 − λ
≤p≤
.
β
β
3.16
8
Abstract and Applied Analysis
Therefore, f ∈ L1,2α2−β,α R and
1/p f ≤ bα f L p,λ,α .
L1,2α2−β,α
3.17
4. Some Applications
In this section, using the results of Section 3, we get the boundedness of the fractional Dmaximal operator in the D-Morrey and modified D-Morrey spaces.
For 0 ≤ β < 2α 2, we define the fractional maximal functions
−1β/2α2
Mβ fx : sup μα B0, t
t>0
τx f y dμα y ,
B0,t
4.1
p 1/p
Mp,β fx : Mβ f x.
In the case β 0, we denote Mp,0 f by Mp f. Note that M1 f Mf.
Lemma 4.1. Let 1 ≤ p < ∞, 0 ≤ β < 2α 2, and f ∈ Lp,2α2−β,α R. Then Mp,β f ∈ L∞ R and the
following equality
Mp,β f bαβ/2α2−11/p f Lp,2α2−β,α
L∞
4.2
is valid.
Proof.
Mp,β f bαβ/2α2−11/p sup
L∞
t
β−2α−2
x∈R,t>0
p
τx f ydμα y
B0,t
1/p
4.3
β/2α2−11/p f bα
.
Lp,2α2−β,α
Taking β 0 in Lemma 4.1 and using Lemma 3.3, we get for Mp f the following result.
Corollary 4.2. Let 1 ≤ p < ∞. Then
Mp f 4f .
L∞
L∞
4.4
p,2α2−β,α R. Then Mp,β f ∈ L∞ R and the
Lemma 4.3. Let 1 ≤ p < ∞, 0 ≤ β < 2α 2, and f ∈ L
following equality
Mp,β f bαβ/2α2−11/p f Lp,2α2−β,α
L∞
is valid.
4.5
Abstract and Applied Analysis
9
Corollary 4.4. Let 0 ≤ 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,
β/2α2−1/p Mβ f bαβ/2α2−1 f f ≤ bα
L1,2α2−β,α
Lp,λ,α .
L∞
4.6
Corollary 4.5. 1 ≤ p < ∞, 0 ≤ λ < 2α 2, 0 ≤ β < 2α 2 − λ. Then the operator Mβ is bounded
p,λ,α to L∞ for 2α 2 − λ/β ≤ p ≤ 2α 2/β. Moreover,
from L
β/2α2−1/p Mβ f bαβ/2α2−1 f f .
≤ bα
L1,2α2−β,α
Lp,λ,α
L∞
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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