Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 837951, 14 pages
doi:10.1155/2010/837951
Research Article
On Boundedness of Weighted Hardy Operator in
L
p·
and Regularity Condition
Aziz Harman
1
and Farman Imran Mamedov
1, 2
1
Education Faculty, Dicle University, 21280 Diyarbakir, Turkey
2
Institute of Mathematics and Mechanics of National Academy of Science, Azerbaijan
Correspondence should be addressed to Farman Imran Mamedov, [email protected]
Received 22 September 2010; Accepted 26 November 2010
Academic Editor: P. J. Y. Wong
Copyright q 2010 A. Harman and F. I. Mamedov. 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 give a new proof for power-type weighted Hardy inequality in the norms of generalized
Lebesgue spaces L
p·
R
n
. Assuming the logarithmic conditions of regularity in a neighborhood
of zero and at infinity for the exponents px ≤ qx,βx, necessary and sufficient conditions are
proved for the boundedness of the Hardy operator Hfx
·−n/q·
Hf
L
q·
R
n
≤ C
|
x
|
β·
f
L
p·
R
n
,Hf
x
At the beginning, a one-dimensional Hardy inequality was considered assuming the
the local log condition at the finite interval 0,l. Subsequently, the logarithmic condition
was assumed in an arbitrarily small neighborhood of zero, where an additional restriction
px ≥ p0 was imposed on the exponent. In 3, 9 it was shown that it is sufficient to assume
the logarithmic condition only at the zero point. In 10 the case of an entire semiaxis was
considered without using the condition px ≥ p0. However, a more rigid condition β
<
1 − 1/p
−
was introduced for a range of exponents. The exact condition was found in 1.
They proved this result by using of interpolation approaches. In this paper, we use other
approaches, analogous to those in 10, based on the property of triangles for px-norms and
binary decomposition near the origin and infinity. We consider the multidimensional case,
and the condition βxconst is not obligatory, while the necessary and sufficient condition
is obtained by a set of exponents p, q, β without imposing any preliminary restrictions on their
values Theorems 3.1 and 3.2.InTheorem 3.3, it has been proved that logarithmic conditions
at zero and at infinity are exact for the Hardy inequality to be valid in the case q p.
Problems of the boundedness of classical integral operators such as maximal and
singular operators, the Riesz potential, and others in Lebesgue spaces with variable exponent,
as well as the investigation of problems of regularity of nonlinear equations with nonstandard
growth condition have become of late the arena of an intensive attack of many authors see
11–18.
2. Lebesgue Spaces with a Variable Exponent
As to the basic properties of spaces L
p·
, we refer to 19. Throughout this paper, it is assumed
that px is a measurable function in Ω, where Ω ∈ R
n
is an open domain, taking its values
p
f
λ
≤ 1
.
2.1
For p
−
> 1, p
< ∞ the space L
p·
Ω is a reflexive Banach space.
Denote by Λ a class of measurable functions f : R
n
→ R satisfying the following
conditions:
∃m ∈
0,
1
2
, ∃f
0
n
\B
0,M
f
x
− f
∞
ln
|
x
|
< ∞.
2.3
For the exponential functions βx,px,andqx, we further assume β, p, q ∈ Λ.
We will many times use the following statement in the proof of main results.
Journal of Inequalities and Applications 3
Lemma 2.1. Let s ∈ Λ be a measurable function such that −∞ <s
−
,s
|
x
|
sx
≤ C
4
|
x
|
s∞
2.5
when |x|≥M. Where the constants C
3
,C
4
> 1 depend on s0, s∞, s
−
, s
, s0, s∞, m, M, C
1
,
C
2
.
To prove Lemma 2.1, for example 2.4,itsuffices to rewrite the inequality 2.4 in the
form
C
−1
3
ux ≤ vx ≤ C
4
ux.Byχ
E
, we denote the characteristic function of the set E.
3. The Main Results
The main results of the paper are contained in the next statements. The theorem below gives
a solution of the two-weighted problem for the multidimensional Hardy operator in the case
of power-type weights.
Theorem 3.1. Let qx ≥ px and βx be measurable functions taken from the class Λ. Let the
following conditions be fulfilled:
0 <p
−
≤ p
x
,q
x
≤ q
< ∞, −∞ <β
−
≤ β
x
≤ β
1 −
1
p
∞
.
3.2
We have the following analogous result for the conjugate Hardy operator
Hfx
|y|≥|x|
fydy.
4 Journal of Inequalities and Applications
Theorem 3.2. Let qx ≥ px and βx be measurable functions taken from the class Λ. Let the
conditions 3.1 be fulfilled. Then the inequality 1.1 for any positive measurable function f and
operator
Hf is fulfilled if and only if
p
0
> 1,p
∞
> 1,β
0
|
x
|
β·−n
Hf
L
p·
R
n
≤ C
|
x
|
β·
f
L
p·
R
x
|
β·−n/p
·−n/q·
Hf
L
q·
R
n
≤ C
5
.
4.2
Assume that 0 <δ<mis a sufficiently small number such that n/p
x >n/p
0 − ε
for all x ∈ B0,δ, where ε n/p
0−β0/2. Let, furthermore, M<N<∞ be a sufficiently
large number such that n/p
x >n/p
|
x
|
β·−n/p
·−n/q·
Hf
L
q·
B0,δ
|
x
|
β·−n/p
·−n/q·
Hf
L
q·
R
n
\B0,N
|
x
|
β·−n/p
·−n/q·
{
t:N<
|
t
|
<
|
x
|}
i
1
≤
|
x
|
β·−n/p
·−n/q·
∞
k0
{
t:2
−k−1
|
x
|
<
|
t
|
<2
−k
·−n/q·
{t:2
−k−1
|
x
|
<
|
t
|
<2
−k
|
x
|
}
f
t
dt
L
q·
βx
∼|x|
β0
and |t|
βt
∼|t|
β0
. Therefore, and due to
Holder’s inequality, for x ∈ B0,δ,weget
|
x
|
βx−n/p
x−n/qx
B
x,k
f
t
dt
≤ C
6
2
kβ0
|
x
|
|
t
|
βt
f
t
p
−
x,k
dt
1/p
−
x,k
2
−k
|
x
|
n/p
−
x,k
.
|
x
|
n/p
0
∼ 2
−kn/p
0
|
x
|
n/p
x
.
4.6
Demonstrate details in proof of 4.6. For t ∈ B
x,k
and x ∈ B0,δ, we have 2
−k−1
|x| <
|t|≤2
−k
|x|. Then
2
−k
|
∼ py. Obviously, the point y depends on x, k. Then |t|
n/p
−
x,k
∼|t|
n/p
y
.Byvirtueof
2
−k−1
|x| < |y|≤2
−k−1
|x|, we have |t|/2 < |y|≤2|t|. Hence, |t|
n/p
y
∼|y|
n/p
y
,byLemma 2.1,
|y|
n/p
y
∼|y|
n/p
n/p
x
≤ 2
−kn/p
0εk
|
x
|
n/p
x
; x ∈ B
0,δ
.
4.8
6 Journal of Inequalities and Applications
Applying estimate 4.8 to both hypotheses a and b, by choosing of ε and δ, the right-hand
part of 4.5 is less than
C
7
|
x
|
−n/qx
2
−kε
t
|
βt
f
t
p
−
x,k
dt
≤
B
x,k
∩{t∈R
n
:
|
t
|
βt
f
t
≥1}
|
x
|
β·−n/p
·−n/q·
B
x,k
f
t
dt
≤ C
9
2
−kεq
−
B
0,δ
|
x
|
−n
2
−kεq
−
B
0,δ
B
x,k
|
t
|
βt
f
t
pt
1
dt
|
x
|
t
|
βt
pt
B
0,2
k1
|
t
|
\B
0,2
k
|
t
|
|
x
|
−n
dx
11
2
−kεq
−
.
4.11
Therefore,
|
x
|
β·−n/p
·−n/q·
B
x,k
f
t
dt
The estimate at infinity i
4
.
Journal of Inequalities and Applications 7
Put f
N
tftχ
|t|>N
. Analogously to the case of 4.4, we have
i
4
≤
∞
k0
|
x
|
β·−n/p
·−n/q·
{t:2
−k−1
|
By |t|∼|x|2
−k
, condition 2.3 and Lemma 2.1 for x ∈ R
n
\ B0,N, t ∈ B
x,k
, we have
|
x
|
βx
∼
|
x
|
β∞
∼ 2
kβ∞
t
β∞
∼ 2
kβ∞
t
βt
.
4.15
Therefore, by virtue of Holder’s inequality,
|
x
|
N
t
dt
≤ C
14
2
kβ∞
|
x
|
−n/p
x−n/qx
B
x,k
|
t
|
βt
f
N
t
2
−k
|
x
|
n/p
−
x,k
∼ t
n/p
t
∼ t
n/p
∞
∼ 2
−kn/p
∞
|
x
|
n/p
∞
∼ 2
−kn/p
x
|
x
|
n/p
x
≤ 2
−kn/p
∞δ
1
k
|
x
|
n/p
x
.
4.18
In both hypotheses i and ii by choosing of δ
1
, we have
|
x
|
βx−n/p
t
p
−
x,k
dt
1/p
−
x,k
.
4.19
On the other hand,
B
x,k
|
t
|
βt
f
t
p
−
G
t
p
−
x,k
G
t
p
−
x,k
dt
B
x,k
G
t
p
−
dt,
4.20
8 Journal of Inequalities and Applications
where Gt1/1 t
2
t
dt.
4.21
By 2.3,fort ∈ B
x,k
, we have
G
t
p
−
x,k
−pt
≤
1 t
2
pt−p
−
x,k
≤ C
16
.
4.22
Then 4.21 implies
x−n/qx
B
x,k
f
N
t
dt
≤ C
q
/p
−
17
2
−kδ
1
q
−
R
n
\B
0,N
−1
17
2
−kδ
1
q
−
ln 2
{t:
|
t
|
>2
−k
N}
f
N
t
|
t
|
βt
pt
dt ≤ C
18
.
We have
i
2
|
x
|
β·−n/p
·−n/q·
{
t∈R
n
:
|
t
|
<
|
x
|}
f
β·−n/p
·−n/q·
L
q·
B0,N\B0,δ
≤ C
20
B
0,N
f
t
dt,
4.27
Journal of Inequalities and Applications 9
from which, by virtue of Holder’s inequality, for px-norms, we obtain the estimate
B
0,N
f
L
p
·
B0,N
.
4.27
Using t
−βtp
t
∼ t
−β0p
0
by Lemma 2.1 for t ∈ B0,N and taking the condition β0 <
n/p
0 into account, we find
I
p
;B0,N
|
t
|
4.28
From 4.27
and 4.28, it follows that
i
2
≤ C
23
. 4.29
Furthermore, we have
i
3
≤
B
0,N
f
t
dt
|
x
β∞−n/p
∞qx−n
.
4.31
Applying condition 4.31,weget
I
q;R
n
/B0,N
|
x
|
β·−n/p
·−n/q·
≤ C
24
R
n
\B
0,N
|
x
|
.
4.34
10 Journal of Inequalities and Applications
We come to a contradiction
I
p
|
t
|
β·
f
τ
B
0,δ/τ
\B
0,δ/
2τ
|
x
|
\B
0,δ/τ
|
t
|
βt−n/p
t−n/qtqt
B
0,δ/τ
\B
0,δ/
2τ
y
−n/p0−β0
dy
If 0 <p0 ≤ 1, then by virtue of inequalities 4.35 and 3.2 we obtain
I
q
|
t
|
βt−n/p
t−n/qt
B
0,t
f
τ
y
dy
−→ ∞ , as τ −→ ∞ . 4.36
Also,
I
p
|
t
|
ln 2,
I
q
|
t
|
βt−n/p
t−n/qt
B
0,t
f
τ
t
dy
≥
R
n
\B
0,δ/τ
2τ
n/p
∞−β∞q
R
n
\B
0,δ/τ
|
t
|
β∞−n/p
∞qt−n
dt −→ ∞
4.38
Journal of Inequalities and Applications 11
as τ →∞.Ifβ∞n/p
∞, then from 4.38 we have
I
q
|
t
Hf
x
L
q·
R
n
≤ C
|
x
|
βx
f
x
L
p·
R
z
|
−βz−2n/pz
f
z
L
p·
R
n
,
4.40
where
px, qx,andβx stand for the functions px/|x|
2
,qx/|x|
2
,andβx/|x|
2
,
respectively. The equivalence readily follows from the equality
g
2
in the definition of px-norm.
5. Exactness of the Logarithmic Conditions
Proof of Theorem 3.3. Assume δ
k
1/4
k
, k ∈ N, f
k
x|x|
−n/px−βx
χ
B0,2δ
k
\B0,δ
k
x,and
βxβ
0
. Define the function p : 0, ∞ → 1, ∞ as
p
x
⎧
⎨
⎩
p
p
0
α
k
, β
0
∈ R,and{α
k
} is an arbitrary sequence of positive numbers
satisfying the condition
kα
k
−→ ∞ as k −→ ∞ . 5.2
12 Journal of Inequalities and Applications
Then α
k
ln1/δ
k
→∞, and condition 2.2 does not hold for the function px. Since
I
p
|
x
|
βx
f
k
x
0
dt
B
0,2δ
k
\B
0,δ
k
|
t
|
−n
dt C
0
2δ
k
δ
k
dt
t
ω
n−1
ln 2,
0,2δ
k
\B
0,δ
k
|
t
|
−n/pt−β
0
dt
p
k
|
x
|
β
0
−np
k
dx
≥ C
B
/p
0
k
e
nα
k
/p
0
ln1/δ
k
−→ ∞
5.3
as k →∞, we see that this contradicts inequality 3.4.
The given function f
k
x and the exponential functions px and βx are also suitable
for proving the necessity of condition 2.3 for the function p. For this we define the numbers
δ
k
from the equality δ
k
4
k
,k∈ N.Letf
k
x|x|
−n/px−β
χ
B0,2δ
,
p
k
,x∈ B
0, 4δ
k
\ B
0, 2δ
k
,k∈ N
5.4
where p
∞
> 1, β
∞
∈ R, p
k
p
∞
− α
k
,and{α
k
} is an arbitrary sequence of positive numbers
satisfying the condition kα
k
|
t
|
β
∞
·
|
t
|
−n/p
∞
−β
∞
p
∞
dt ω
n−1
ln 2,
I
p
|
x
|
βx−n
f
k
|
−n/pt−β
∞
dt
p
k
|
x
|
β
∞
−np
k
dx
≥ C
B
0,3δ
k
\B
0,2δ
k
δ
k
n−n/p
5.5
as k →∞, which contradicts inequality 3.4.
Journal of Inequalities and Applications 13
The same reasoning brings us to the proof of the exactness of conditions 2.2 and
2.3 for the function βx also. For instance, to show the necessity of condition 2.2, it can
be assumed that px ≡ p
0
> 1, x ∈ R
n
,
β
x
⎧
⎨
⎩
β
0
α
k
,x∈ B
0, 2δ
k
\ B
0,δ
≥ Cδ
−p
0
α
k
k
−→ ∞ as k −→ ∞ ,
I
p
|
x
|
βx
f
k
x
≤ C
0
ln 2.
5.7
This completes the proof of Theorem 3.3.
Acknowledgment
F. I. Mamedov was supported partially by the INTAS Grant for the South-Caucasian
Republics, no. 8792.
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px
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