DOI: 10.2478/auom-2013-0026
An. Şt. Univ. Ovidius Constanţa
Vol. 21(2),2013, 111–130
Necessary and sufficient conditions for the
boundedness of the anisotropic Riesz potential
in anisotropic modified Morrey spaces
Malik S. Dzhabrailov and Sevinc Z. Khaligova
Abstract
We prove that the anisotropic fractional maximal operator Mα,σ
and the anisotropic Riesz potential operator Iα,σ , 0 < α < |σ| are
e 1,b,σ (Rn ) to
bounded from the anisotropic modified Morrey space L
n
e q,b,σ (R ) if and only
the weak anisotropic modified Morrey space W L
e p,b,σ (Rn ) to L
e q,b,σ (Rn )
if, α/|σ| ≤ 1−1/q ≤ α/(|σ|(1−b)) and from L
if and only if, α/|σ| ≤ 1/p − 1/q ≤ α/((1 − b)|σ|) . In the limiting case
|σ|(1−b)
≤ p ≤ |σ|
we prove that the operator Mα,σ is bounded from
α
α
e p,b,σ (Rn ) to L∞ (Rn ) and the modified anisotropic Riesz potential
L
e p,b,σ (Rn ) to BM Oσ (Rn ) .
operator Ieα,σ is bounded from L
1
Introduction
For x ∈ Rn and t > 0 , let B(x, t) denote the open ball centered at x of
{
radius t and B(x, t) = Rn \B(x, t) . Let 0 ≤ b ≤ 1 , σ = (σ1 , · · · , σn ) with
σ
σi > 0 for i = 1, · · · , n , |σ| = σ1 + · · · + σn and tQ
x ≡ (tσ1 x1 , . . . , tσn xn )
n
n
for t > 0 . For x ∈ R and t > 0, let Eσ (x, t) = i=1 (xi − tσi , xi + tσi )
denote the open parallelepiped centered at x of side length 2tσi for i =
1, · · · , n .
Key Words: anisotropic Riesz potential, anisotropic fractional maximal function,
anisotropic modified Morrey space, anisotropic BMO space
2010 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35
Received: September 2011.
Revised: December 2011.
Accepted: June 2012.
111
112
M.S. Dzhabrailov, S.Z. Khaligova
Pn
By [3, 10], the function F (x, ρ) = i=1 x2i ρ−2σi , considered for any fixed
x ∈ Rn , is a decreasing one with respect to ρ > 0 and the equation F (x, ρ) =
1 is uniquely solvable. This unique solution will be denoted by ρ(x) . Define
ρ(x) = ρ and ρ(0) = 0 . It is a simple matter to check that ρ(x − y) defines
a distance between any two points x, y ∈ Rn . Thus Rn , endowed with the
metric ρ , defines a homogeneous metric space ([3, 5, 10]). Note that ρ(x) is
1
equivalent to |x|σ = max |xi | σi .
1≤i≤n
One of the most important variants of the anisotropic maximal function is
the so-called anisotropic fractional maximal function defined by the formula
Z
Mα,σ f (x) = sup |Eσ (x, t)|−1+α/|σ|
|f (y)|dy, 0 ≤ α < |σ|,
t>0
Eσ (x,t)
where |Eσ (x, t)| = 2n t|σ| is the Lebesgue measure of the parallelepiped
Eσ (x, t) .
It coincides with the anisotropic maximal function Mσ f ≡ M0,σ f and is
intimately related to the anisotropic Riesz potential operator
Z
f (y)dy
Iα,σ f (x) =
,
0 < α < |σ|.
|σ|−α
Rn |x − y|σ
If σ = 1 , then Mα ≡ Mα,1 and Iα ≡ Iα,1 is the fractional maximal
operator and Riesz potential, respectively. The operators Mα , Mα,σ , Iα
and Iα,σ play important role in real and harmonic analysis (see, for example
[4] and [35]).
Definition 1.1. Let 0 ≤ b ≤ 1 , 1 ≤ p < ∞ and [t]1 = min{1, t} . We
e p,b,σ (Rn ) the moddenote by Lp,b,σ (Rn ) anisotropic Morrey space, and by L
ified anisotropic Morrey space, the set of locally integrable functions f (x) ,
x ∈ Rn , with the finite norms
!1/p
Z
kf kLp,b,σ =
kf kLep,b,σ =
sup
t−b|σ|
x∈Rn , t>0
sup
x∈Rn , t>0
|f (y)|p dy
,
Eσ (x,t)
−b|σ|
[t]1
Z
!1/p
p
|f (y)| dy
Eσ (x,t)
respectively.
Remark 1.1. Note that Lp,0,σ = Lp (Rn ) and Lp,1,σ = L∞ (Rn ) . If b < 0
or b > 1 , then Lp,b,σ = Θ , where Θ is the set of all functions equivalent to
0 on Rn . In the case σ ≡ 1 = (1, . . . , 1) and b = nλ we get the classical
Morrey space Lp,λ (Rn ) = Lp, λ ,1 (Rn ) , 0 ≤ λ ≤ n .
n
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
113
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 C. B. Morrey in 1938 in connection with certain
problems in elliptic partial differential equations and calculus of variations (see
[25]). Later, Morrey spaces found important applications to Navier-Stokes
([22], [36]) and Schrödinger ([26], [28], [29], [31], [32]) equations, elliptic problems with discontinuous coefficients ([8], [11]), and potential theory ([1], [2]).
An exposition of the Morrey spaces can be found in the book [20].
e p,b,σ (Rn ) firstly was defined and investigated
The modified Morrey space L
by [19] (see also [4]).
Note that
e p,0,σ (Rn ) = Lp,0,σ (Rn ) = Lp (Rn ),
L
e p,b,σ (Rn ) ⊂ Lp,b,σ (Rn ) ∩ Lp (Rn )
L
and
max{kf kLp,b,σ , kf kLp } ≤ kf kLep,b,σ
(1.1)
e p,b,σ (Rn ) = Θ .
and if b < 0 or b > 1 , then Lp,b,σ (Rn ) = L
Definition 1.2. [6] Let 1 ≤ p < ∞ , 0 ≤ b ≤ 1 . We denote by W Lp,b,σ (Rn )
e p,b,σ (Rn ) the weak modified
the weak anisotropic Morrey space and by W L
anisotropic Morrey space as the set of locally integrable functions f (x) , x ∈
Rn with finite norms
kf kW Lp,b,σ = sup r
r>0
kf kW Lep,b,σ = sup r
r>0
sup
t−b|σ| |{y ∈ Eσ (x, t) : |f (y)| > r}|
1/p
,
x∈Rn , t>0
sup
x∈Rn , t>0
−b|σ|
[t]1
1/p
|{y ∈ Eσ (x, t) : |f (y)| > r}|
respectively.
Note that
e p,0,σ (Rn ),
W Lp (Rn ) = W Lp,0,σ (Rn ) = W L
Lp,b,σ (Rn ) ⊂ W Lp,b,σ (Rn ) and kf kW Lp,b,σ ≤ kf kLp,b,σ ,
e p,b,σ (Rn ) ⊂ W L
e p,b,σ (Rn ) and kf k e
L
e p,b,σ .
W Lp,b,σ ≤ kf kL
The anisotropic result by Hardy-Littlewood-Sobolev states that if 1 <
n
n
p < q <∞ , then
Iα,σ is bounded from Lp (R ) to Lq (R ) if and only if
1
p
1
q
and for p = 1 < q < ∞ , Iα,σ is bounded from L1 (Rn )
to W Lq (Rn ) if and only if α = |σ| 1 − 1q . Spanne (see [33]) and Adams
α = |σ|
−
114
M.S. Dzhabrailov, S.Z. Khaligova
[1] studied boundedness of the Riesz potential Iα for 0 < α < n in Morrey
spaces Lp,λ . Later on Chiarenza and Frasca [9] was reproved boundedness
of the Riesz potential Iα in these spaces. By more general results of Guliyev
[13] (see also [14, 17]) one can obtain the following generalization of the results
in [1, 9, 33] to the anisotropic case.
.
Theorem A. Let 0 < α < |σ| and 0 ≤ b < 1 , 1 ≤ p < (1−b)|σ|
α
α
, then condition p1 − 1q = (1−b)|σ|
is necessary
1) If 1 < p < (1−b)|σ|
α
and sufficient for the boundedness of the operator Iα,σ from Lp,b,σ (Rn ) to
Lq,b,σ (Rn ) .
α
2) If p = 1 , then condition 1 − 1q = (1−b)|σ|
is necessary and sufficient
for the boundedness of the operator Iα,σ from L1,b,σ (Rn ) to W Lq,b,σ (Rn ) .
If α = (1 − b)|σ| p1 − 1q , then b = 0 and the statement of Theorem A reduces to the aforementioned anisotropic result by Hardy-Littlewood-Sobolev.
Recall that, for 0 < α < |σ| ,
α
Mα,σ f (x) ≤ 2n( |σ| −1) Iα,σ (|f |)(x),
(1.2)
hence Theorem A also implies the boundedness of the fractional maximal
operator Mα,σ . It is known that the anisotropic maximal operator Mσ is
also bounded from Lp,b,σ to Lp,b,σ for all 1 < p < ∞ and 0 < b < 1 ,
which isotropic case proved by F. Chiarenza and M. Frasca [9].
In this paper we study the fractional maximal integral and the Riesz potential in the modified Morrey space. In the case p = 1 we prove that the
e 1,b,σ (Rn ) to W L
e q,b,σ (Rn ) if
operators Mα,σ and Iα,σ are bounded from L
we
and only if, α/|σ| ≤ 1−1/q ≤ α/((1−b)|σ|) . In the case 1 < p < (1−b)|σ|
α
n
e
prove that the operators Mα,σ and Iα,σ are bounded from Lp,b,σ (R ) to
e q,b,σ (Rn ) if and only if, α/|σ| ≤ 1/p − 1/q ≤ α/((1 − b)|σ|) . In the limiting
L
case |σ|(1−b)
≤ p ≤ |σ|
α
α we prove that the operator Mα,σ is bounded from
n
e p,b,σ (R ) to L∞ (Rn ) and the modified anisotropic Riesz potential operator
L
e p,b,σ (Rn ) to BM Oσ (Rn ) .
Ieα,σ is bounded from L
The structure of the paper is as follows. In section 2 the boundedness of
e p,b,σ
the anisotropic maximal operator in anisotropic modified Morrey space L
is proved. The main result of the paper is the Hardy-Littlewood-Sobolev
inequality in anisotropic modified Morrey space for the anisotropic Riesz potential, established in section 3. In section 4 we prove that the operator Ieα,σ
e p,b,σ (Rn ) to BM Oσ (Rn ) for |σ|(1−b) ≤ p ≤ |σ| .
is bounded from L
α
α
By A . B we mean that A ≤ CB with some positive constant C
independent of appropriate quantities. If A . B and B . A , we write
A ≈ B and say that A and B are equivalent.
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
2
115
ep,b,σ -boundedness of the maximal operator
L
Define ftσ (x) =: f (tσ x) and [t]1,+ = max{1, t} . Then
kftσ kLp = t−
kftσ kLp,b,σ = t
=t
|σ|
p
kf kLp ,
|σ|
− p
sup
r−b|σ|
x∈Rn , r>0
(b−1)|σ|
p
!1/p
Z
|f (y)|p dy
Eσ (tσ x,tr)
kf kLp,b,σ ,
and
kftσ kLep,b,σ =
−b|σ|
[r]1
sup
x∈Rn , r>0
=t
−
|σ|
p
=t
sup
sup
r>0
= t−
|σ|
p
p
|ftσ (y)| dy
Eσ (x,r)
x∈Rn , r>0
|σ|
− p
!1/p
Z
[tr]1
[r]1
−b|σ|
[r]1
!1/p
Z
p
|f (y)| dy
Eσ (tσ x,tr)
b|σ|/p
sup
x∈Rn , r>0
−b|σ|
[tr]1
Z
!1/p
p
|f (y)| dy
Eσ (tσ x,tr)
b|σ|
p
[t]1,+
kf kLep,b,σ .
(2.1)
e p,b,σ -boundedness of the maximal operator
In this section we study the L
Mσ .
Lemma 2.1. Let 1 ≤ p < ∞ , 0 ≤ b ≤ 1 . Then
e p,b,σ (Rn ) = Lp,b,σ (Rn ) ∩ Lp (Rn )
L
and
n
o
kf kLep,b,σ = max kf kLp,b,σ , kf kLp .
e p,b,σ (Rn ) . Then from (1.1) we have that f ∈ Lp,b,σ (Rn ) ∩
Proof. Let f ∈ L
n
o
n
Lp (R ) and max kf kLp,b,σ , kf kLp ≤ kf kLep,b,σ .
116
M.S. Dzhabrailov, S.Z. Khaligova
Let now f ∈ Lp,b,σ (Rn ) ∩ Lp (Rn ) . Then
kf kLep,b,σ =


sup
x∈Rn ,t>0
−b|σ|
[t]1
p
|f (y)| dy
Eσ (x,t)
Z
sup
t−b|σ|
x∈Rn ,0<t≤1
n
o
≤ max kf kLp,b,σ , kf kLp .
= max
!1/p
Z
!1/p
|f (y)|p dy
Eσ (x,t)
Z
, sup
x∈Rn ,t>1
|f (y)|p dy
Eσ (x,t)
!1/p 


e p,b,σ (Rn ) and the embedding
Therefore, f ∈ L
e p,b,σ (Rn )
Lp,b,σ (Rn ) ∩ Lp (Rn ) ⊂ L
is valid.
Thus
e p,b,σ (Rn ) = Lp,b,σ (Rn ) ∩ Lp (Rn ) and
L
n
o
kf kLep,b,σ = max kf kLp,b,σ , kf kLp .
Analogously proved the following statement.
Lemma 2.2. Let 1 ≤ p < ∞ , 0 ≤ b ≤ 1 . Then
e p,b,σ (Rn ) = W Lp,b,σ (Rn ) ∩ W Lp (Rn )
WL
and
n
o
kf kW Lep,b,σ = max kf kW Lp,b,σ , kf kW Lp .
To prove our main result in this section we need the following statement.
Theorem 2.1. [23] 1. If f ∈ L1,b,σ (Rn ) , 0 ≤ b < 1 , then Mσ f ∈
W L1,b,σ (Rn ) and
kMσ f kW L1,b,σ ≤ Cb,σ kf kL1,b,σ ,
where Cb,σ depends only on n , b and σ .
2. If f ∈ Lp,b,σ (Rn ) , 1 < p < ∞ , 0 ≤ b < 1 , then Mσ f ∈ Lp,b,σ (Rn )
and
kMσ f kLp,b,σ ≤ Cp,b,σ kf kLp,b,σ ,
where Cp,b,σ depends only on n , p , b and σ .
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
117
Our main theorem in this section is the following statement:
e 1,b,σ (Rn ) , 0 ≤ b < 1 , then Mσ f ∈ W L
e 1,b,σ (Rn )
Theorem 2.2. 1. If f ∈ L
and
kMσ f kW Le1,b,σ ≤ C 1,b,σ kf kLe1,b,σ ,
where C1,b,σ depends only on b and σ .
e p,b,σ (Rn ) , 1 < p < ∞ , 0 ≤ b < 1 , then Mσ f ∈ L
e p,b,σ (Rn )
2. If f ∈ L
and
kMσ f kLep,b,σ ≤ C p,b,σ kf kLep,b,σ ,
where Cp,b,σ depends only on p , b and σ .
Proof. It is obvious that (see Lemmas 2.1 and 2.2)
n
o
kMσ f kLep,b,σ = max kMσ f kLp,b,σ , kMσ f kLp
for 1 < p < ∞ and
n
o
kMσ f kW Le1,b,σ = max kMσ f kW L1,b,σ , kMσ f kW L1
for p = 1 .
Let 1 < p < ∞ . By the boundedness of Mσ on Lp (Rn ) and from
Theorem 2.1 we have
kMσ f kLep,b,σ ≤ max {Cp,σ , Cp,b,σ } kf kLep,b,σ .
Let p = 1 . By the boundedness of Mσ from L1 (Rn ) to W L1 (Rn ) and
from Theorem 2.1 we have
kMσ f kW Le1,b,σ ≤ max {C1,σ , C1,b,σ } kf kLe1,b,σ .
3
Hardy-Littlewood-Sobolev inequality in modified Morrey spaces
The following Hardy-Littlewood-Sobolev inequality in modified Morrey spaces
is valid.
118
M.S. Dzhabrailov, S.Z. Khaligova
(1−b)|σ|
α
.
|σ| and 1 ≤ p <
α
α
1
1
α
|σ| ≤ p − q ≤ (1−b)|σ| is necessary
e p,b,σ (Rn ) to
operator Iα,σ from L
Theorem 3.3. Let 0 < α < |σ| , 0 ≤ b < 1 −
(1−b)|σ|
α
1) If 1 < p <
, then condition
and sufficient for the boundedness of the
e q,b,σ (Rn ) .
L
2) If p = 1 < (1−b)|σ|
, then condition
α
α
|σ|
≤ 1 − 1q ≤
α
(1−b)|σ|
is necessary
e 1,b,σ (Rn ) to
and sufficient for the boundedness of the operator Iα,σ from L
e q,b,σ (Rn ) .
WL
Proof. 1) Sufficiency. Let 0 < α < |σ| , 0 < b < 1 −
and 1 < p <
(1−b)|σ|
α
e p,b,σ (Rn )
, f ∈L
. Then
Z
!
Z
Iα,σ f (x) =
f (y)|x − y|α−|σ|
dy ≡ A(x, t) + C(x, t).
σ
+
{
Eσ (x,t)
Eσ (x,t)
For A(x, t) we have
Z
|A(x, t)| ≤
≤
α
|σ|
|x − y|α−|σ|
|f (y)| dy
σ
Eσ (x,t)
∞
X
−j
2
t
α−|σ|
Z
|f (y)| dy.
Eσ (x,2−j+1 t)\Eσ (x,2−j t)
j=1
Hence
|A(x, t)| . tα M f (x).
(3.1)
In the second integral by the Hölder’s inequality we have
!1/p
Z
|C(x, t)| ≤
|x −
{
p
y|−β
σ |f (y)| dy
Eσ (x,t)
Z
( β +α−|σ|)p0
dy
|x − y|σ p
×
{
!1/p0
= J1 · J2 .
Eσ (x,t)
For J2 we obtain
Z
J2 .
t
where β < |σ| − αp .
∞
β
0
r|σ|−1+( p +α−|σ|)p dr
10
p
β
≈ t p +α−
|σ|
p
,
(3.2)
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
119
Let b|σ| < β < |σ| − αp . For J1 we get
∞ Z
X
1/p
p
J1 =
|x − y|−β
|f
(y)|
dy
σ
j=0
Eσ (x,2j+1 t)\Eσ (x,2j t)
β
≤ t− p kf kLep,b,σ
∞
X
b|σ|
2−βj [2j+1 t]1
1/p
j=0
β
= t− p kf kLep,b,σ

1
1/p
[log2 2t
∞

P ] (b|σ|−β)j
P

b|σ| b|σ|

2
+
2−βj
, 0 < t < 12 ,
t
 2
j=0
kf kLep,b,σ
≈t
(
≈ kf kLep,b,σ
b|σ|
p
= [2t]1
t
2−βj
1/p
1
2t ]+1
t≥
,
j=0
( −β
p
j=[log2
P
∞




tb|σ| + tβ
1,
1/p
1
2
, 0 < t < 12 ,
t ≥ 12
b|σ|−β
p
−β
p
t
, 0 < t < 12 ,
,
t ≥ 12
β
t− p kf kLep,b,σ .
(3.3)
From (3.2) and (3.3) we have
b|σ|
|C(x, t)| . [t]1 p tα−
|σ|
p
kf kLep,b,σ .
(3.4)
Thus for all t > 0 we get
b|σ|
|Iα,σ f (x)| . tα Mσ f (x) + [t]1 p tα−
|σ|
p
kf kLep,b,σ
n
o
|σ|
(1−b)|σ|
≤ min tα Mσ f (x) + tα− p kf kLep,b,σ , tα Mσ f (x) + tα− p
kf kLep,b,σ .
Minimizing with respect to t , at
h
ip/((1−b)|σ|)
−1
t = (Mσ f (x)) kf kLep,b,σ
and
h
ip/|σ|
−1
t = (Mσ f (x)) kf kLep,b,σ
we get


Mσ f (x)
|Iα,σ f (x)| . min
 kf kLep,b,σ
pα
!1− (1−b)|σ|
,
Mσ f (x)
kf kLep,b,σ
pα 
!1− |σ|


kf kLep,b,σ .
120
M.S. Dzhabrailov, S.Z. Khaligova
Then
|Iα,σ f (x)| . (Mσ f (x))
p/q
1−p/q
kf kLe
Hence, by Theorem 2.2, we have
Z
Z
q
q−p
|Iα,σ f (y)| dy . kf kLe
p,b,σ
Eσ (x,t)
.
b|σ|
[t]1
.
p,b,σ
p
(Mσ f (y)) dy
Eσ (x,t)
q
kf kLe
p,b,σ
,
e p,b,σ (Rn ) to L
e q,b,σ (Rn ) .
which implies that Iα,σ is bounded from L
(1−b)|σ|
e p,b,σ (Rn ) and Iα,σ bounded from
, f ∈L
Necessity. Let 1 < p <
α
n
n
e p,b,σ (R ) to L
e q,b,σ (R ) . Then from (2.1) we have
L
kftσ kLep,b,σ = t−
|σ|
p
b|σ|
p
[t]1,+
kf kLep,b,σ ,
and
Iα,σ ftσ (x) = t−α Iα,σ f (tσ x),
kIα,σ ftσ kLeq,b,σ = t
=t
−α−
|σ|
q
sup
r>0
= t−α−
|σ|
q
−α
sup
x∈Rn , r>0
[tr]1
[r]1
−b|σ|
[r]1
(3.5)
!1/q
Z
σ
q
|Iα,σ f (t y)| dy
Eσ (x,r)
b|σ|/q
sup
x∈Rn , r>0
−b|σ|
[tr]1
!1/q
Z
q
|Iα,σ f (y)| dy
Eσ (tσ x,tr)
b|σ|
q
[t]1,+
kIα,σ f kLeq,b,σ .
e p,b,σ (Rn ) to L
e q,b,σ (Rn )
By the boundedness of Iα,σ from L
−
b|σ|
−
b|σ|
kIα,σ f kLeq,b,σ = tα+
|σ|
q
≤ tα+
|σ|
q
= tα+
|σ|
|σ|
q − p
[t]1,+q
[t]1,+q
kIα,σ ftσ kLeq,b,σ
kftσ kLep,b,σ
b|σ|
p
[t]1,+
−
b|σ|
q
kf kLep,b,σ ,
where Cp,q,b,σ depends only on p , q , b and σ .
α
If p1 < 1q + |σ|
, then in the case t → 0 we have kIα,σ f kLeq,b,σ = 0 for all
n
e
f ∈ Lp,b,σ (R ) .
α
As well as if p1 > 1q + (1−b)|σ|
, then at t → ∞ we obtain kIα,σ f kLeq,b,σ = 0
n
e
for all f ∈ Lp,b,σ (R ) .
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
α
|σ|
α
≤ (1−b)|σ|
.
n
e 1,b,σ (R ) . We have
2) Sufficiency. Let f ∈ L
Therefore
≤
1
p
121
−
1
q
|{y ∈ Eσ (x, t) : |Iα,σ f (y)| > 2β}|
≤ |{y ∈ Eσ (x, t) : |A(y, t)| > β}|
+ |{y ∈ Eσ (x, t) : |C(y, t)| > β}| .
Then
C(y, t) =
∞ Z
X
Eσ (y,2j+1 t)\Eσ (y,2j t)
j=0
≤ tα−|σ| kf kLe1,b,σ
∞
X
|f (z)||y − z|α−|σ|
dz
σ
b|σ|
2−(|σ|−α)j [2j+1 t]1
= tα−|σ| kf kLe1,b,σ
j=0
×

1
[log2 2t

P ] (b|σ|−|σ|+α)j

b|σ| b|σ|

2
+
2
t

j=0
∞
P




∞
P
j=[log2
1
2t ]+1
2−(|σ|−α)j , 0 < t < 12 ,
2−(|σ|−α)j ,
t≥
j=0
1
2
tb|σ| + t|σ|−α , 0 < t < 21 ,
kf kLe1,b,σ
1,
t ≥ 12
b|σ|+α−|σ|
t
, 0 < t < 21 ,
≈ kf kLe1,b,σ
tα−|σ| ,
t ≥ 12
≈t
α−|σ|
b|σ|
= [2t]1
tα−|σ| kf kLe1,b,σ .
Taking into account inequality (3.1) and Theorem 2.2, we have
|{y ∈ Eσ (x, t) : |A(y, t)| > β}|
β
≤ y ∈ Eσ (x, t) : M f (y) >
α
C1 t ≤
C 2 tα
b|σ|
· [t]1 kf kLe1,b,σ ,
β
b|σ|
where C2 = C1 · C1,b,σ and thus if C2 [2t]1 tα−|σ| kf kLe1,b,σ = β , then
|C(y, t)| ≤ β and consequently, | {y ∈ Eσ (x, t) : |C(y, t)| > β} | = 0 .
Then
1 b|σ|
|{y ∈ Eσ (x, t) : |Iα,σ f (y)| > 2β}| . [t]1 tα kf kLe1,b,σ
β
! (1−b)|σ|
kf kLe1,b,σ (1−b)|σ|−α
b|σ|
. [t]1
, if 2t < 1
β
122
M.S. Dzhabrailov, S.Z. Khaligova
and
1 b|σ| α
[t]
t kf kLe1,b,σ
β 1
! |σ|
kf kLe1,b,σ |σ|−α
b|σ|
. [t]1
, if 2t ≥ 1.
β
|{y ∈ Eσ (x, t) : |Iα,σ f (y)| > 2β}| .
Finally we have
|{y ∈ Eσ (x, t) : |Iα,σ f (y)| > 2β}|

(1−b)|σ|
! (1−b)|σ|−α
 kf k e
L1,b,σ
b|σ|
. [t]1 min
,

β
q
1
b|σ|
≤ [t]1
kf kLe1,b,σ .
β
kf kLe1,b,σ
|σ| 
! |σ|−α

β

e 1,b,σ (Rn ) to W L
e q,b,σ (Rn ) . From
Necessity. Let Iα,σ is bounded from L
(3.5) we have
!1/q
Z
kIα,σ ftσ kW Leq,b,σ = sup r
r>0
= sup r
sup
r>0
x∈Rn , τ >0
= t−α−
|σ|
q
sup
τ >0
×
sup
x∈Rn , τ >0
= t−α−
|σ|
q
−b|σ|
sup
[τ ]1
x∈Rn , τ >0
−b|σ|
[τ ]1
[tτ ]1
[τ ]1
!1/q
Z
b|σ|/q
−b|σ|
[tτ ]1
dy
{y∈Eσ (x,τ ) : |Iα,σ ftσ (y)|>r}
dy
{y∈Eσ (x,τ ) : |Iα,σ f (tσ y)|>rtα }
sup rtα
r>0
!1/q
Z
dy
{y∈Eσ (tσ x,tτ ) : |Iα,σ f (y)|>rtα }
b|σ|
q
[t]1,+
kIα,σ f kW Leq,b,σ .
e 1,b,σ (Rn ) to W L
e q,b,σ (Rn ) and from
By the boundedness of Iα,σ from L
(2.1) we get
−
b|σ|
−
b|σ|
kIα,σ f kW Leq,b,σ = tα+
|σ|
q
. tα+
|σ|
q
. tα+
|σ|
q −|σ|
[t]1,+q kIα,σ ftσ kW Leq,b,σ
[t]1,+q kftσ kLe1,b,σ
b|σ|−
[t]1,+
b|σ|
q
kf kLe1,b,σ .
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
123
α
If 1 < 1q + |σ|
, then in the case t → 0 we have kIα,σ f kW Leq,b,σ = 0 for
n
e 1,b,σ (R ) .
all f ∈ L
α
, then for t → ∞ we obtain kIα,σ f kW Leq,b,σ =
Similarly, if 1 > 1q + (1−b)|σ|
n
e
0 for all f ∈ L1,b,σ (R ) .
Therefore
1
p
−
1
q
=
α
(1−b)|σ|
.
Corollary 3.1. Let 0 < α < |σ| , 0 ≤ b < 1 −
1) If 1 < p <
(1−b)|σ|
α
α
|σ|
, then condition
≤
α
|σ|
and 1 ≤ p ≤
1
p
1
q
−
≤
and sufficient for the boundedness of the operator Mα,σ
e q,b,σ (Rn ) .
L
2) If p = 1 <
(1−b)|σ|
α
, then condition
α
|σ|
≤ 1 − 1q ≤
|σ|
α
.
α
(1−b)|σ|
is necessary
e
from Lp,b,σ (Rn ) to
α
(1−b)|σ|
is necessary
e 1,b,σ (Rn ) to
and sufficient for the boundedness of the operator Mα,σ from L
n
e
W Lq,b,σ (R ) .
3) If (1−b)|σ|
≤ p ≤
α
e p,b,σ (Rn ) to L∞ (Rn ) .
L
|σ|
α
, then the operator Mα,σ is bounded from
Proof. Sufficiency of Corollary 3.1 follows from Theorem 3.3 and inequality
(1.2).
e p,b,σ (Rn ) to L
e q,b,σ (Rn ) for
Necessity. (1) Let Mα,σ be bounded from L
(1−b)|σ|
1<p<
. Then we have
α
Mα,σ ftσ (x) = t−α Mα,σ f (tσ x),
and
kMα,σ ftσ kLeq,b,σ = t−α−
|σ|
q
b|σ|
q
[t]1,+
kMα,σ f kLeq,b,σ .
By the same argument in Theorem 3.3 we obtain
α
|σ|
≤
1
p
−
1
q
≤
α
(1−b)|σ|
e 1,b,σ (Rn ) to W L
e q,b,σ (Rn ) . Then
(2) Let Mα,σ be bounded from L
kMα,σ ftσ kW Leq,b,σ = t−α−
Hence we obtain
α
|σ|
≤1−
1
q
≤
|σ|
q
b|σ|
q
[t]1,+
kMα,σ f kW Leq,b,σ .
α
(1−b)|σ|
.
.
124
M.S. Dzhabrailov, S.Z. Khaligova
(3) Let
(1−b)|σ|
α
≤p≤
kMα,σ f kL∞ = 2−n
≤2
|σ|
α
. Then by the Hölder’s inequality we have
Z
sup tα−|σ|
|f (y)| dy
x∈Rn , t>0
−n
p
sup
Eσ (x,t)
α−
t
x∈Rn , t>0
|σ|
p
b|σ|
p
[t]1
!1/p
Z
p
|f (y)| dy
Eσ (x,t)
|σ|
n
−b|σ|
[t]1
b|σ|
≤ 2− p kf kLep,b,σ sup tα− p [t]1 p
t>0
o
n
|σ|
|σ|(1−b)
−n
p
kf kLep,b,σ max sup tα− p , sup tα− p
=2
t>1
0<t≤1
−n
p
=2
4
kf kLep,b,σ .
The modified anisotropic Riesz potential in the spaces
ep,b,σ (Rn )
L
The examples show that the anisotropic Riesz potential Iα,σ are not defined
α
for all functions f ∈ Lp,b,σ (Rn ) , 0 ≤ b < 1 − |σ|
, if p ≥ |σ|(1−b)
, and
α
α
n
e
, if
Iα,σ are not defined for all functions f ∈ Lp,b,σ (R ) , 0 ≤ b < 1 −
|σ|
.
p ≥ |σ|(1−b)
α
We consider the modified Riesz potential
Z e
Iα,σ f (x) =
|x − y|α−|σ|
− |y|α−|σ|
χ {E
σ
σ
Rn
σ
(y)
f (y) dy.
(0,1)
Note that in the limiting case |σ|(1−b)
≤ p ≤ |σ|
α
α statement 1) in Theorem
e
A does not hold. Moreover, there exists f ∈ Lp,b,σ (Rn ) such that Iα,σ f (x) =
∞ for all x ∈ Rn . However, as will be proved, statement 1) holds for the
modified anisotropic Riesz potential Ieα,σ if the space L∞ (Rn ) is replaced
by a wider space BM Oσ (Rn ) .
The following theorem is our main result in which we obtain conditions
ensuring that the modified anisotropic Riesz potential Ieα,σ is bounded from
e p,b,σ (Rn ) to BM Oσ (Rn ) .
the space L
α
Theorem 4.4. Let 0 < α < |σ| , 0 ≤ b < 1 − |σ|
, and |σ|(1−b)
≤ p ≤
α
|σ|
n
n
e
e
α , then the operator Iα,σ is bounded from Lp,b,σ (R ) to BM Oσ (R ) .
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
125
e p,b,σ (Rn ) ,
Moreover, if the integral Iα,σ f exists almost everywhere for f ∈ L
|σ|(1−b)
|σ|
≤ p ≤ α , then Iα,σ f ∈ BM Oσ (Rn ) and the following inequality is
α
valid
kIα,σ f kBM Oσ ≤ Ckf kLep,b,σ ,
where C > 0 is independent of f .
Proof. For given t > 0 we denote
f1 (x) = f (x)χEσ (0,2t) (y),
f2 (x) = f (x) − f1 (x),
(4.1)
where χEσ (0,2t) is the characteristic function of the set Eσ (0, 2t) . Then
Ieα,σ f (x) = Ieα,σ f1 (x) + Ieα,σ f2 (x) = F1 (x) + F2 (x),
(4.2)
where
Z
F1 (x) =
Eσ (0,2t)
Z
F2 (x) =
|x − y|α−|σ|
− |y|α−|σ|
χ {E
σ
σ
{
Eσ (0,2t)
σ
|x − y|α−|σ|
− |y|α−|σ|
χ {E
σ
σ
(y)
f (y) dy,
(0,1)
σ
(y)
f (y) dy.
(0,1)
Note that the function f1 has compact (bounded) support and thus
Z
a1 = −
|y|α−|σ|
f (y) dy
σ
Eσ (0,2t)\Eσ (0,min{1,2t})
is finite.
Note also that
Z
F1 (x) − a1 =
|x − y|α−|σ|
f (y) dy
σ
Eσ (0,2t)
Z
|y|α−|σ|
f (y) dy
σ
−
Eσ (0,2t)\Eσ (0,min{1,2t})
Z
|y|α−|σ|
f (y) dy
σ
+
Eσ (0,2t)\Eσ (0,min{1,2t})
Z
=
Rn
|x − y|α−|σ|
f1 (y) dy = Iα,σ f1 (x).
σ
Therefore
Z
|F1 (x) − a1 | ≤
Rn
|y|σα−|σ| |f1 (x − y)| dy =
Z
Eσ (0,2t)
|y|α−|σ|
|f (x − y)| dy.
σ
126
M.S. Dzhabrailov, S.Z. Khaligova
Then
Z
−1
|Eσ (x, t)|
|F1 (y) − a1 | dy
Eσ (x,t)
= |Eσ (0, t)|
−1
Z
|F1 (x − y) − a1 | dy
Eσ (0,t)
−1
Z
!
Z
|z|α−|σ|
|f (x
σ
≤ |Eσ (0, t)|
Eσ (0,t)
−1
Z
|f (x − y − z)|dy |z|α−|σ|
dz
σ
= |Eσ (0, t)|
Eσ (0,2t)
≈ kf kL1,1−
Eσ (0,t)
Z
α ,σ
|σ|
|z|α−|σ|
dz
σ
Eσ (0,2t)
.
α ,σ
|σ|
Denote
dy
!
Z
. t−|σ| t|σ|−α kf kL1,1−
− y − z)| dz
Eσ (0,2t)
(4.3)
Z
|y|α−|σ|
f (y) dy.
σ
a2 =
Eσ (0,max{1,2t})\Eσ (0,2t)
If 2|x|σ ≤ |y|σ , then
||x − y|α−|σ|
− |y|α−|σ|
| ≤ C|x|σ |y|α−|σ|−1
.
σ
σ
σ
By the Hölder’s inequality we have
Z
|F2 (x) − a2 | ≤ C|x|σ
|y|α−|σ|−1
|f (y)| dy
σ
. |x|σ
{
Eσ (0,2t)
∞
XZ
j=0
. |x|σ
∞
X
2j+1 t≤|y|σ ≤2j+2 t
(2j+1 t)α−|σ|−1
. |x|σ kf kL1,1−
≈ |x|σ t
Z
|f (y)| dy
|y|σ ≤2j+2 t
j=0
−1
|y|α−|σ|−1
|f (y)| dy
σ
α ,σ
|σ|
kf kL1,1−
∞
X
(2j+2 t)α−|σ|−1 (2j+2 t)|σ|−α
j=0
α ,σ
|σ|
.
Therefore, from (4.3) and (4.4) we have
Z
1
e
sup
Iα,σ f (x − y) − af dy . kf kL1,1− α ,σ .
|σ|
x,t |Eσ (0, t)| Eσ (0,t)
(4.4)
(4.5)
Necessary and sufficient conditions for the boundedness of the
anisotropic Riesz potential in anisotropic modified Morrey spaces
By the Hölder’s inequality we have
Z
kf kL1,1− α ,σ = sup tα−|σ|
|σ|
x∈Rn , t>0
≤2
n
p0
|f (y)|dy
Eσ (x,t)
sup
t
|σ|
α− p
x∈Rn , t>0
b|σ|
p
[t]1
−b|σ|
[t]1
!1/p
Z
|f (y)|p dy
Eσ (x,t)
b|σ|
|σ|
n
≤ 2 p0 kf kLep,b,σ sup tα− p [t]1 p
t>0
o
n
|σ|
|σ|(1−b)
n
= 2 p0 kf kLep,b,σ max sup tα− p , sup tα− p
0<t≤1
=2
Finally let
(4.6) we get
e
Iα,σ f BM Oσ
n
p0
127
(4.6)
t>1
kf kLep,b,σ .
|σ|(1−b)
α
≤p≤
|σ|
α
1
≤ 2 sup
|E
(0,
t)|
x,t
σ
e p,b,σ (Rn ) , then from (4.5) and
and f ∈ L
Z
Eσ (0,t)
e
Iα,σ f (x − y) − af dy . kf kLep,b,σ .
The Theorem 4.4 is proved.
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Malik S. Dzhabrailov
Department of Mathematics
Azerbaijan State Pedagogical University, Baku, Azerbaijan
E-mail: vagif@guliyev.com
Sevinc Z. Khaligova
Department of Mathematics
Azerbaijan State Pedagogical University, Baku, Azerbaijan
Email: seva.xaligova@hotmail.com