Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 976493, 10 pages
doi:10.1155/2010/976493
Research Article
Necessary and Sufficient Conditions for the
Boundedness of Dunkl-Type Fractional Maximal
Operator in the Dunkl-Type Morrey Spaces
Emin Guliyev,1 Ahmet Eroglu,2 and Yagub Mammadov1, 3
1
Institute of Mathematics and Mechanics, AZ 1141 Baku, Azerbaijan
Department of Mathematics, Nigde University, 51100 Nigde, Turkey
3
Nakhchivan Teacher-Training Institute, AZ 7003 Nakhchivan, Azerbaijan
2
Correspondence should be addressed to Yagub Mammadov, yagubmammadov@yahoo.com
Received 15 April 2010; Revised 13 June 2010; Accepted 23 June 2010
Academic Editor: Dumitru Baleanu
Copyright q 2010 Emin Guliyev et al. 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/dxfx 2α 1/xfx − f−x/2, α > −1/2. We study the boundedness of the Dunkltype 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
s
s
of Mβ from the Dunkl-type Besov-Morrey spaces Bpθ,λ,α
R to the spaces Bqθ,λ,α
R, 1 < p ≤ q < ∞,
0 ≤ λ < 2α 2, 1/p − 1/q β/2α 2 − λ, 1 ≤ θ ≤ ∞, and 0 < s < 1.
1. Introduction
On the real line, the Dunkl operators Λα 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:
2α 1 fx − f−x
d
Λα f x :
fx .
dx
x
2
1.1
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 2
.
2
Abstract and Applied Analysis
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 Section 2, 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 Dunkls
s
R to the spaces Bqθ,λ,α
R, 1 < p ≤ q < ∞, 0 ≤ λ < 2α 2,
type Besov-Morrey spaces Bpθ,λ,α
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
Let α > −1/2 be a fixed number and μα be the weighted Lebesgue measure on R, given by
−1
dμα x : 2α1 Γα 1
|x|2α1 dx.
2.1
For every 1 ≤ p ≤ ∞, we denote by Lp,α R Lp dμα R the spaces of complex-valued
functions f, measurable on R such that
f :
p,α
R
f fx
p dμα x
∞,α
1/p
: ess sup
fx
<∞
if p ∈ 1, ∞,
2.2
if p ∞.
x∈R
For 1 ≤ p < ∞ we denote by WLp,α R, the weak Lp,α R 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
2.3
Note that
Lp,α ⊂ WLp,α ,
f ≤ f p,α
WLp,α
∀f ∈ Lp,α R.
2.4
Abstract and Applied Analysis
3
For all x, y, z ∈ R, we put
Wα x, y, z : 1 − σx,y,z σz,x,y σz,y,x Δα x, y, z ,
2.5
where
σx,y,z
⎧ 2
2
2
⎪
⎨x y − z
2xy
:
⎪
⎩0
if x, y ∈ R \ 0,
2.6
otherwise
and Δα is the Bessel kernel given by
Δα x, y, z :
⎧
⎪
⎪
⎪
⎨
2 α−1/2
2
|x| y
− z2 z2 − |x| − y
xyz
2α
dα
⎪
⎪
⎪
⎩0
if |z| ∈ Ax,y ,
2.7
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,
2.8
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 2.9
x2 y2 − 2xy cos θ, we have also see
8
τx f y Cα
π
0
fΨ f−Ψ x y
fΨ − f−Ψ dνα θ,
Ψ
2.10
√
where dνα θ 1 − cos θ sin2α θ dθ and Cα Γα 1/2 π Γα 1/2.
Proposition 2.1 see Soltani 9
. 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
2.11
4
Abstract and Applied Analysis
2
α1
Let Bx, r {y ∈ R : |y| ∈ max{0, |x| − r}, |x| r},r > 0, and bα −1
α 1 Γα 1
. Then B0, r − r, r and μα B0, r bα r 2α2 .
Now we define the Dunkl-type fractional maximal function see 3–5
by
Mβ fx sup μα B0, r
−1β/2α2
r>0
τx f y dμα y ,
0 ≤ β < 2α 2.
B0,r
2.12
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
C μα x ∈ R : Mfx > β ≤ f L1,α ,
β
2.13
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
2.14
where Cp > 0 is independent of f.
Definition 2.3. Let 1 ≤ 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 fx, x ∈ R, with the finite norm
f sup
p,λ,α
r
−λ
p
τx f y dμα y
1/p
2.15
.
B0,r
x∈R,r>0
Note that Lp,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. Let 1 ≤ 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 fx, x ∈ R with finite norm
f sup t sup
WLp,λ,α
t>0
r
−λ
x∈R,r>0
1/p
{y∈B0,r: τx |fy|>t}
dμα y
.
2.16
We note that
Lp,λ,α R ⊂ WLp,λ,α R,
f ≤ f p,λ,α .
WLp,λ,α
2.17
Abstract and Applied Analysis
5
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 operator Mβ 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. Let 0 ≤ β < 2α 2, 0 ≤ λ < 2α 2, and 1 ≤ p ≤ 2α 2 − λ/β.
1 If p 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 If 1 < 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 If p 2α 2 − λ/β, then Mβ is bounded from Lp,λ,α R to L∞ R.
s
For 1 ≤ p, θ ≤ ∞, 0 ≤ λ < 2α 2, and 0 < s < 2, the Dunkl-type Besov-Morrey Bpθ,λ,α
R
consists of all functions f in Lp,λ,α R so that
f s
B
pθ,λ,α
⎛ ⎞1/θ
τx f· − f·
θ
Lp,λ,α
f Lp,λ,α ⎝
dμα x⎠ < ∞.
|x|2α2sθ
R
3.1
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. For 1 < p ≤ q < ∞, 0 ≤ λ < 2α 2, 1/p − 1/q β/2α 2 − λ, 1 ≤ θ ≤ ∞, and
s
s
R to Bqθ,λ,α
R.
0 < s < 1, the Dunkl-type fractional maximal operator Mβ is bounded from Bpθ,λ,α
More precisely, there is a constant C > 0 such that
Mβ f s
Bqθ,λ,α
≤ Cf Bs
pθ,λ,α
3.2
s
hold for all f ∈ Bpθ,λ,α
R.
Remark 3.3. Note that Theorem 3.2 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
operator Mβ in the Dunkl-type Morrey spaces Lp,λ,α R.
6
Abstract and Applied Analysis
Theorem 4.1. Let 0 ≤ β < 2α 2, 0 ≤ λ < 2α 2, f ∈ Lp,λ,α R, and 1 ≤ p ≤ 2α 2 − λ/β.
1 If p 1 and 1 − 1/q β/2α 2 − λ, then Mβ f ∈ WLq,λ,α R and
Mβ f ≤ Cf 1,λ,α ,
WLq,λ,α
4.1
where C > 0 is independent of f.
2 If 1 < p < 2α 2 − λ/β and 1/p − 1/q β/2α 2 − λ, then Mβ f ∈ Lq,λ,α R and
Mβ f q,λ,α
≤ Cf p,λ,α ,
4.2
where C > 0 is independent of f.
3 If p 2α 2 − λ/β and q ∞, then Mβ f ∈ L∞ R and
Mβ f ≤ bα−1/p2α2 f .
p,λ,α
∞
4.3
Proof. The maximal function Mfx 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
μEx, 2r ≤ C0 μEx, r
4.4
with a constant C0 being independent of x and r > 0. Here Ex, r {y ∈ X : ρx, y <
r}, ρx, y |x − y|. Let X, ρ, μ be a space of homogeneous type, where X R, ρx, y |x − y|, and dμx dμα x. It is clear that this measure satisfies the doubling condition 4.4.
Define
−1
Mμ fx sup μEx, r
r>0
f y dμ y .
Ex,r
4.5
It is well known that the maximal operator Mμ 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
Mfx ≤ CMμ fx,
4.6
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.
Abstract and Applied Analysis
7
Let t > 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
Mβ fx max sup μα B0, r
β/2α2−1
r≥t
τx f y dμα y ,
B0,r
sup μα B0, r
β/2α2−1
r<t
τx f y dμα y
4.7
B0,r
−1/p
max bα
tβ−2α2−λ/p f p,λ,α , tβ Mfx .
β/2α2
≤ bα
Therefore, for all t > 0, we get
β/2α2
Mβ fx ≤ bα
−1/p β−2α2−λ/p
bα
t
f p,λ,α , tβ Mfx .
4.8
The minimum value of the right-hand side 4.8 is attained at
t
p/2α2−λ
2α 2 − λ −1/p f p,λ,α
bα
p
Mfx
4.9
and hence
1−p/q p/q
f p,λ,α Mfx
.
β/2α2−β/2α2−λ Mβ fx ≤ bα
4.10
Then for 1 < p ≤ 2α 2 − λ/β from 4.10, we have
−λ
Mβ f r
sup
q,λ,α
r>0
≤
1/q
q
τx Mβ fy dμα y
B0,r
β/2α2−β/2α2−λ f 1−p/q
bα
p,λ,α
β/2α2−β/2α2−λ ≤ bα
≤ Cf p,λ,α ,
where C > 0 is independent of f.
r
−λ
p
τx Mf y dμα y
B0,r
1−p/q p/q
f p,λ,α Mf p,λ,α
1/q
4.11
8
Abstract and Applied Analysis
Also for p 1 from 4.10 we have
Mβ f sup t sup
WLq,λ,α
t>0
r
{y∈B0,r : τx Mβ fy>t}
x∈R,r>0
⎛
≤ sup t sup ⎝r −λ
t>0
1/q
−λ
dμα y
x∈R,r>0
−βq/2α2−λβq/2α2
y∈B0,r: τx Mf y >bα
f 1,λ,α tq
1−q
⎞1/q
⎠
dμα y
1−1/q 1/q
f 1,λ,α Mf WL1,λ,α
β/2α2−λ−β/2α2 ≤ bα
≤ Cf 1,λ,α ,
4.12
where C > 0 is independent of f.
Therefore, the case β > 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
μα B0, r
−1β/2α2
τx f y dμα y
B0,r
≤ μα B0, r
−1β/2α21/p
p
τx f y dμα y
1/p
B0,r
−λ/p2α2
bα
p
τx f y dμα y
4.13
1/p
B0,r
−λ/p2α2 ≤ bα
r −λ
f p,λ,α .
Thus the case β > 0 completes the proof of 3.
Theorem 4.1 has been proved.
Proof of Theorem 3.1. Sufficiency part of the proof follows from Theorem 4.1.
Necessity. 1 Let 1 < p ≤ 2α 2 − λ/α and Mβ be bounded from Lp,λ,α R to Lq,λ,α R.
Define ft x : ftx, t > 0. Then
ft t−2α2/p sup
p,λ,α
r
−λ
x∈R,r>0
t−2α2−λ/p
f p,λ,α
B0,tr
p
τtx f y dμα y
1/p
4.14
Abstract and Applied Analysis
9
and Mβ ft x t−β Mβ ftx,
Mβ ft t−β sup
Lq,λ,α
r
−λ
q
τtx Mβ fy
dμα y
B0,r
x∈R,r>0
t
1/q
−β−2α2/q
sup
x∈R,r>0
r
−λ
q
τx Mβ fy
dμα y
1/q
4.15
B0,tr
t−β−2α2−λ/q Mβ f Lq,λ,α .
By the boundedness of Mβ from Lp,λ,α R to Lq,λ,α R,
Mβ f Lq,λ,α
r β2α2−λ/q Mβ fr Lq,λ,α
≤ Cr β2α2−λ/q fr p,λ,α
4.16
Cr β2α2−λ/q−2α2−λ/p f p,λ,α ,
where C depends only on p, β, λ, and α.
If 1/p > 1/q β/2α 2 − λ, then for all f ∈ Lp,λ,α R we have Mβ f
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
Mβ f
q,λ,α 0 as r → ∞, which is also impossible.
Therefore, we get 1/p 1/q β/2α 2 − λ.
Necessity. Let Mβ be bounded from L1,λ,α R to WLq,λ,α R. We have
Mβ fr r −β−2α2−λ/q Mβ f WLq,λ,α .
WLq,λ,α
4.17
By the boundedness of Mβ from L1,λ,α R to WLq,λ,α R it follows that
Mβ f r β2α2−λ/q Mβ fr WLq,λ,α
WLq,λ,α
≤ Cr β2α2−λ/q fr 1,λ,α
Cr β2α2−λ/q−2α2 f 1,λ,α ,
4.18
where C depends only on β, λ, and α.
If 1 < 1/q β/2α 2 − λ, then for all f ∈ L1,λ,α R we have Mβ f
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 Mβ f
WLq,λ,α 0 as
r → ∞.
Hence we get 1 1/q β/2α 2 − λ. Thus the proof of Theorem 3.1 is completed.
10
Abstract and Applied Analysis
Proof of Theorem 3.2. For x ∈ R, let τx be the generalized translation by x. By definition of the
Besov spaces, it suffices to show that
τx Mβ f − Mβ f ≤ C2 τx f − f Lp,λ,α .
Lq,λ,α
4.19
It is easy to see that τx commutes with Mβ , that is, τx Mβ f Mβ τx f. Hence we have
τx Mβ f − Mβ f Mβ τx f − Mβ f ≤ Mβ τx f − f .
4.20
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.
Acknowledgment
The authors express their thanks to the referee for careful reading, and helpful comments and
suggestions on the manuscript of this paper.
References
1
C. F. Dunkl, “Differential-difference operators associated to reflection groups,” Transactions of the
American Mathematical Society, vol. 311, no. 1, pp. 167–183, 1989.
2
C. B. Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Transactions
of the American Mathematical Society, vol. 43, no. 1, pp. 126–166, 1938.
3
C. Abdelkefi and M. Sifi, “Dunkl translation and uncentered maximal operator on the real line,”
International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 87808, 9 pages, 2007.
4
Y. Y. Mammadov, “On maximal operator associated with the Dunkl operator on R,” Khazar Journal of
Mathematics, vol. 2, no. 4, pp. 59–70, 2006.
5
F. Soltani, “Littlewood-Paley operators associated with the Dunkl operator on R,” Journal of Functional
Analysis, vol. 221, no. 1, pp. 205–225, 2005.
6
V. S. Guliyev and Y. Y. Mammadov, “Lp , Lq boundedness of the fractional maximal operator
associated with the Dunkl operator on the real line,” Integral Transforms and Special Functions, pp.
1–11, 2010.
7
E. V. Guliyev and Y. Y. Mammadov, “Some embeddings into the Morrey spaces associated with the
Dunkl operator,” Abstract and Applied Analysis, vol. 2010, Article ID 291345, 10 pages, 2010.
8
M. Rösler, “Bessel-type signed hypergroups on R,” in Probability Measures on Groups and Related
Structures, XI (Oberwolfach, 1994), pp. 292–304, World Scientific, River edge, NJ, USA, 1995.
9
F. Soltani, “Lp-Fourier multipliers for the Dunkl operator on the real line,” Journal of Functional
Analysis, vol. 209, no. 1, pp. 16–35, 2004.
10
V. S. Guliyev and Y. Y. Mammadov, “On fractional maximal function and fractional integrals
associated with the Dunkl operator on the real line,” Journal of Mathematical Analysis and Applications,
vol. 353, no. 1, pp. 449–459, 2009.
11
C. Abdelkefi and M. Sifi, “Characterization of Besov spaces for the Dunkl operator on the real line,”
Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 3, article no. 73, p. 11, 2007.
12
R. Bouguila, M. N. Lazhari, and M. Assal, “Besov spaces associated with Dunkl’s operator,” Integral
Transforms and Special Functions, vol. 18, no. 7-8, pp. 545–557, 2007.
13
L. Kamoun, “Besov-type spaces for the Dunkl operator on the real line,” Journal of Computational and
Applied Mathematics, vol. 199, no. 1, pp. 56–67, 2007.
14
G. Pradolini and O. Salinas, “Maximal operators on spaces of homogeneous type,” Proceedings of the
American Mathematical Society, vol. 132, no. 2, pp. 435–441, 2004.
15
N. Samko, “Weighted Hardy and singular operators in Morrey spaces,” Journal of Mathematical
Analysis and Applications, vol. 350, no. 1, pp. 56–72, 2009.