Volume 2010, Article ID 837951,14pages doi:10.1155/2010/837951
Research Article
On Boundedness of Weighted Hardy Operator in
L
p
·
and Regularity Condition
Aziz Harman
1and Farman Imran Mamedov
1, 21Education Faculty, Dicle University, 21280 Diyarbakir, Turkey
2Institute 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
Copyrightq2010 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 spacesLp·Rn. Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponentspx≤qx, βx, necessary and sufficient conditions are proved for the boundedness of the Hardy operatorHfx |y|≤|x|fydyfromLp·
|x|β·Rninto Lq·
|x|β·−n/p·−n/q·RN. Also a separate statement on the exactness of logarithmic conditions at zero
and at infinity is given. This shows that logarithmic regularity conditions for the functionsβ, pat the origin and infinity are essentially one.
1. Introduction
The object of this investigation is the Hardy-type weighted inequality
|x|β·−n/p·−n/q·Hf
Lq·Rn≤C
|x|β·f
Lp·Rn, Hfx
|y|≤|x|f
ydy 1.1
in the norms of generalized Lebesgue spacesLp·Rn. This subject was investigated in the
papers1–7. For the one-dimensional Hardy operator in 1, the necessary and sufficient
condition was obtained for the exponents β, p, q. We give a new proof for this result in
more general settings for the multidimensional Hardy operator. Also we prove that the logarithmic regularity conditions are essential one for such kind of inequalities to hold. In
that proposal, we improve a result sort of8 since, there is an estimation by the maximal
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≥p0was imposed on the exponent. In3,9it was shown that it is sufficient to assume
the logarithmic condition only at the zero point. In10the case of an entire semiaxis was
considered without using the conditionpx ≥ p0. However, a more rigid conditionβ <
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 in10, based on the property of triangles forpx-norms and
binary decomposition near the origin and infinity. We consider the multidimensional case,
and the conditionβx const is not obligatory, while the necessary and sufficient condition
is obtained by a set of exponentsp, q, βwithout imposing any preliminary restrictions on their
valuesTheorems3.1and3.2. 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 caseqp.
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 authorssee
11–18.
2. Lebesgue Spaces with a Variable Exponent
As to the basic properties of spacesLp·, we refer to19. Throughout this paper, it is assumed
thatpxis a measurable function inΩ, whereΩ∈Rnis an open domain, taking its values
from the interval1,∞withp supx∈Rnp <∞. The space of functionsLp·Ωis introduced
as the class of measurable functionsfxinΩ, which have a finiteIpf : Ω|fx|pxdx
-modular. A norm inLp·Ωis given in the form
f
Lp·Ωinf
λ >0 :Ip
f
λ ≤1
. 2.1
Forp−>1,p<∞the spaceLp·Ωis a reflexive Banach space.
Denote byΛ a class of measurable functionsf : Rn → Rsatisfying the following
conditions:
∃m∈
0,1
2 , ∃f0∈R, x∈supB0,mfx−f0ln
1
|x| <∞, 2.2
∃M >1, ∃f∞∈R, sup
x∈Rn\B0,M
fx−f∞ln|x|<∞.
2.3
For the exponential functionsβx, px, andqx, we further assumeβ, p, q∈Λ.
Lemma 2.1. Lets ∈ Λbe a measurable function such that−∞ < s−, s < ∞. Then the condition 2.2for the functionsxis equivalent to the estimate
C−31|x|s0 ≤ |x|sx≤C3|x|s0 2.4
when|x| ≤mand the condition2.3forsxis equivalent to the estimate
C−1
4 |x|s∞ ≤ |x|sx≤C4|x|s∞ 2.5
when|x| ≥M. Where the constantsC3, C4>1depend ons0,s∞,s−,s,s0,s∞,m,M,C1,
C2.
To proveLemma 2.1, for example2.4, it suffices to rewrite the inequality2.4in the
form
C3−1≤ |x|sx−s0≤C3 2.6
and pass to logarithmic in this inequalitysee also,1,7,17.
For 1 < p < ∞, p denotes the conjugate number of p,p p/p−1. It is further
assumed thatp∞forp1, andp1 forp∞, 1/∞0, 1/0∞. We denote byC, C1, C2
various positive constants whose values may vary at each appearance.Bx, rdenotes a ball
with center atxand radiusr >0. We writeu∼vif there exist positive constantsC3, C4such
thatC3ux≤vx≤C4ux. ByχE, we denote the characteristic function of the setE.
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. Letqx ≥ pxand βxbe measurable functions taken from the classΛ. Let the following conditions be fulfilled:
0< p−≤px, qx≤q <∞, −∞< β−≤βx≤β<∞. 3.1
Then the inequality1.1for any positive measurable functionfis fulfilled if and only if
p0>1, p∞>1, β0< n
1− 1
p0 , β∞< n
1− 1
p∞ . 3.2
We have the following analogous result for the conjugate Hardy operatorHfx
Theorem 3.2. Letqx ≥ pxand βxbe measurable functions taken from the classΛ. Let the conditions3.1be fulfilled. Then the inequality 1.1 for any positive measurable functionf and operatorHfis fulfilled if and only if
p0>1, p∞>1, β0> n
1− 1
p0 , β∞> n
1− 1
p∞ . 3.3
In the next theorem, we prove that the logarithmic conditions near zero and at infinity are essentially one.
Theorem 3.3. If condition2.2or2.3does not hold, then there exists an example of functionsp, β, and a sequencefbelow indexkviolating the inequality
|x|β·−nHf
Lp·Rn ≤C
|x|β·f
Lp·Rn. 3.4
4. Proofs of the Main Results
Proof ofTheorem 3.1.
Sufficiency. Letfx≥0 be a measurable function such that
|x|β·f
Lp·Rn≤1. 4.1
We will prove that
|x|β·−n/p·−n/q·Hf
Lq·Rn≤C5. 4.2
Assume that 0< δ < mis a sufficiently small number such thatn/px> n/p0−ε
for allx∈B0, δ, whereε n/p0−β0/2. Let, furthermore,M < N <∞be a sufficiently
large number such thatn/px> n/p∞−δ1for allx∈Rn\B0, N, whereδ1 n/p∞−
β∞/2.
By Minkowski inequality, forpx-norms, we have
|x|β·−n/p·−n/q·Hf Lq·Rn≤
|x|β·−n/p·−n/q·Hf
Lq·B0,δ
|x|β·−n/p·−n/q·Hf
Lq·B0,N\B0,δ
|x|β·−n/p·−n/q·
{t:|t|<N}ftdt
Lq·Rn\B0,N
|x|β·−n/p·−n/q·
{t:N<|t|<|x|}ftdt
Lq·RN\B0,N
:i1i2i3i4.
The estimate near zeroi1.
By Minkowski inequality, we have the inequalities
i1≤
|x|β·−n/p
·−n/q·∞
k0 {t:2−k−1|x|<|t|<2−k|x|}
ftdt
Lq·B0,δ
≤∞
k0
|x|β·−n/p
·−n/q·
{t:2−k−1|x|<|t|<2−k|x|}
ftdt
Lq·B0,δ .
4.4
DenoteBx,k {y ∈ Rn : 2−k−1|x| < |y| < 2−k|x|}andp−x,k minpx,infy∈Bx,kpy.
By2.2andLemma 2.1, forx ∈ B0, δ,t ∈ Bx,k, we have|x|βx ∼ 2kβ0tβt. To prove this
equivalence, we use that|t| ∼ |x|2−k,|x|βx ∼ |x|β0and|t|βt ∼ |t|β0. Therefore, and due to
Holder’s inequality, forx∈B0, δ, we get
|x|βx−n/px−n/qx
Bx,k ftdt
≤C62kβ0|x|−n/p
x−n/qx
Bx,k
|t|βtftdt
≤C62kβ0|x|−n/p
0−n/qx
Bx,k
|t|βtftp −
x,k dt
1/px,k−
2−k|x|n/p
−
x,k .
4.5
aIfp−x,k/px, then by2.2andLemma 2.1,
2−k|x|n/p
−
x,k
∼tn/pt ∼tn/p0∼2−kn/p0|x|n/p0∼2−kn/p0|x|n/px. 4.6
Demonstrate details in proof of4.6. Fort∈Bx,kandx∈B0, δ, we have 2−k−1|x|<
|t| ≤2−k|x|. Then
2−k|x|n/p
−
x,k
∼ |t|n/p−x,k
. 4.7
By hypothesisa,p−x,kattains in the intervalBx,k, because there exists a pointy∈Bx,kwhere
p−x,k ∼ py. Obviously, the pointy depends onx, k. Then|t|n/p−x,k
∼ |t|n/py. By virtue of
2−k−1|x|<|y| ≤2−k−1|x|, we have|t|/2 <|y| ≤2|t|. Hence,|t|n/py ∼ |y|n/py, byLemma 2.1,
|y|n/py∼ |y|n/p0∼ |t|n/p0.
bIfp−x,kpx, then by choice ofδ,
2−k|x|n/p−x,k
Applying estimate4.8to both hypothesesaandb, by choosing ofεandδ, the right-hand
part of4.5is less than
C7|x|−n/qx2−kε
Bx,k
|t|βtftp −
x,k dt
1/p−x,k
. 4.9
Simultaneously,
Bx,k
|t|βtftp −
x,k dt
≤
Bx,k∩{t∈Rn:|t|βtft≥1}
|t|βtftptdt
Bx,k
dt≤12−knδnC8.
4.9
By4.5and4.9, we have
Iq;B0,δ
|x|β·−n/p·−n/q·
Bx,k ftdt
≤C92−kεq −
B0,δ|x| −n
Bx,k
|t|βtftp − x,k dt qx/p− x,k dx
≤C9Cq
/p−−1 8 2−kεq
−
B0,δ
Bx,k
|t|βtftpt1 dt
|x|−ndx
4.10
which, due to Fubini’s theorem, yields
≤C9Cq
/p−−1 8 2−kεq
−
{t:|t|<2−kδ}
ft|t|βtpt
B0,2k1|t|\B0,2k|t|| x|−ndx
dt
C102−kεq −
ln 2
{t:|t|<2−kδ}
ft|t|βtpt1 dt≤C112−kεq − . 4.11 Therefore,
|x|β·−n/p
·−n/q·
Bx,k
ftdt
Lq·B0,δ
≤C122−kεq −/q
. 4.12
By4.12and4.4, we get
i1≤C12
∞
k0
2−kεq−/qC13<∞. 4.13
PutfNt ftχ|t|>N. Analogously to the case of4.4, we have
i4≤
∞
k0
|x|β·−n/p
·−n/q·
{t:2−k−1|x|<|t|<2−k|x|}
fNtdt
Lq·Rn\B0,N
. 4.14
By|t| ∼ |x|2−k, condition2.3andLemma 2.1forx∈Rn\B0, N,t∈B
x,k, we have
|x|βx ∼ |x|β∞∼2kβ∞tβ∞∼2kβ∞tβt. 4.15
Therefore, by virtue of Holder’s inequality,
|x|βx−n/px−n/qx
Bx,k
fNtdt
≤C142kβ∞|x|−n/p
x−n/qx
Bx,k
|t|βtfNtdt
≤C142kβ∞|x|−n/p
x−n/qx
Bx,k
|t|βtfNtp −
x,k dt
1/p−x,k
2−k|x|n/p
−
x,k
.
4.16
iIfp−x,k/pxandt∈Bx,k, by2.3andLemma 2.1, we have
2−k|x|n/p
−
x,k
∼tn/pt∼tn/p∞ ∼2−kn/p∞|x|n/p∞∼2−kn/p∞|x|n/px. 4.17
iiIfp−x,kpx, then by choice ofδ1,
2−k|x|n/p
−
x,k
∼2−kn/px|x|n/px≤2−kn/p∞δ1k|x|n/px. 4.18
In both hypothesesiandiiby choosing ofδ1, we have
|x|βx−n/px−n/qx
Bx,k
fNtdt≤C15|x|−n/qx2−kδ1
Bx,k
|t|βtfNtp −
x,k dt
1/p−x,k
. 4.19
On the other hand,
Bx,k
|t|βtftp −
x,k dt≤
Bx,k∩{t∈Rn:|t|βtft≥Gt}
|t|βtft Gt
p−x,k
Gtp−x,kdt
Bx,k
Gtp−dt,
whereGt 1/1t2. Hence,
≤
Bx,k
fNt|t|βtptGtp−x,k−pt
Bx,k
Gtdt. 4.21
By2.3, fort∈Bx,k, we have
Gtp−x,k−pt≤
1t2pt−p
−
x,k
≤C16. 4.22
Then4.21implies
Bx,k
|t|βtfNtp −
x,k
dt≤C17. 4.23
Therefore,
Iq;Rn\B0,N
|x|βx−n/px−n/qx
Bx,k
fNtdt
≤Cq17/p−2−kδ1q−
Rn\B0,N| x|−n
Bx,k
|t|βtfNtptdt
dx,
4.24
by Fubini’s theorem,
≤Cq17/p−−12−kδ1q−ln 2
{t:|t|>2−kN}
fNt|t|βtptdt≤C182−kδ1q −
. 4.25
From4.25and expansion4.14, we get
i4≤C18
∞
k0
2−kq−δ1/qC
19. 4.26
The estimate in the middlei2, i3.
We have
i2
|x|β·−n/p
·−n/q·
{t∈Rn:|t|<|x|}
ftdt
Lq·B0,N\B0,δ
≤
B0,Nftdt
|x|β·−n/p·−n/q·
Lq·B0,N\B0,δ
≤C20
B0,Nftdt,
from which, by virtue of Holder’s inequality, forpx-norms, we obtain the estimate
B0,Nftdt≤
|t|β·ft
Lp·B0,N
|t|−β·
Lp·B0,N. 4.27
Using t−βtpt ∼ t−β0p0
by Lemma 2.1for t ∈ B0, N and taking the condition β0 <
n/p0into account, we find
Ip;B0,N
|t|−β·
B0,N|t| −βtpt
dt≤C21
B0,N|t| −β0p0
dtC22. 4.28
From4.27and4.28, it follows that
i2≤C23. 4.29
Furthermore, we have
i3≤
B0,Nftdt
|x|βx−n/px−n/qx
Lq·Rn\B0,δ
. 4.30
The boundedness of the first term follows by4.27. Due to2.3andLemma 2.1, forx ∈
Rn\B0, N, we have
|x|βx−n/pxqx−n∼ |x|β∞−n/p∞qx−n. 4.31
Applying condition4.31, we get
Iq;Rn/B0,N
|x|β·−n/p·−n/q·≤C24
Rn\B0,N
|x|−n−2δ1dxC
25. 4.32
Then
i3 ≤C1/p −
25 . 4.33
Necessity. Letβ0 > n/p0. Fix a sufficiently largeτ >0 and apply inequality1.1by the test function
We come to a contradiction
Ip
|t|β·fτ
B0,δ/τ\B0,δ/2τ|x|
−ndxC
0ln 2<∞,
Iq
|t|β·−n/p·−n/q·
B0,tfτ
ydy
≥
B0,1\B0,δ/τ|t|
βt−n/pt−n/qtqt
B0,δ/τ\B0,δ/2τ
y−n/p0−β0
dy
qt
dt
≥
δ
2τ
n/p0−β0q−
B0,1\B0,δ/τ|
t|β0−n/p0qt−ndt−→ ∞
4.35
asτ → ∞.
If 0< p0≤1, then by virtue of inequalities4.35and3.2we obtain
Iq
|t|βt−n/pt−n/qt
B0,tfτ
ydy
−→ ∞, asτ −→ ∞. 4.36
Also,
Ip
|t|βtfτt
C0ln 2, 4.37
and we come to a contradiction.
Ifβ∞≥n/p∞, then, using condition2.3andLemma 2.1assuming 0< τ <1, we
again obtain
Ip
|t|βtfτt
C0ln 2,
Iq
|t|βt−n/pt−n/qt
B0,t
fτtdy
≥
Rn\B0,δ/τ
|t|βt−n/ptqt−n
B0,δ/τ\B0,δ/2τ
y−n/p∞−β∞dy
dt
≥
δ
2τ
n/p∞−β∞q
Rn\B0,δ/τ
|t|β∞−n/p∞qt−ndt−→ ∞
asτ → ∞. Ifβ∞ n/p∞, then from4.38we have
Iq
|t|βt−n/pt−n/qt
B0,tfτtdy
∞. 4.39
From4.38and3.2, we derive, as above, the necessity of the conditionp∞>1.
This completes the proof ofTheorem 3.1.
The proof ofTheorem 3.2easily follows fromTheorem 3.1by using the equivalence of
inequalities
|x|βx−n/px−n/qxHfx
Lq·Rn≤C
|x|βxfx Lp·Rn,
|z|n−βz−2n/qzHfx Lq·Rn
≤C|z|−βz−2n/pzfz
Lp·Rn,
4.40
where px, qx, and βx stand for the functions px/|x|2, qx/|x|2, and βx/|x|2,
respectively. The equivalence readily follows from the equality
g
Lp·Rn|z|−2n/pzg
Lp·Rn 4.41
for any functiong:Rn → R, wheregz gz/|z|2, which easily can be proved by changing
of variablexz/|z|2in the definition ofpx-norm.
5. Exactness of the Logarithmic Conditions
Proof ofTheorem 3.3. Assumeδk 1/4k,k ∈ N,fkx |x|−n/px−βxχB0,2δk\B0,δkx, and
βx β0. Define the functionp:0,∞ → 1,∞as
px
⎧ ⎨ ⎩
p0, x∈B0,2δk\B0, δk,
pk, x∈B0,4δk\B0,2δk, k∈N
5.1
wherep0 > 1,pk p0αk,β0 ∈ R, and{αk}is an arbitrary sequence of positive numbers
satisfying the condition
Thenαkln1/δk → ∞, and condition2.2does not hold for the functionpx. Since
Ip
|x|βxfkx
B0,2δk\B0,δk
|t|β0· |t|−n/p0−β0p0dt
B0,2δk\B0,δk
|t|−ndtC0
2δk
δk dt
t ωn−1ln 2,
Ip
H|·|β·−nfk·≥
B0,4δk\B0,2δk
B0,2δk\B0,δk
|t|−n/pt−β0dt
pk
|x|β0−npkdx
≥C
B0,3δk\B0,2δk
δkn−n/p0−β0pk|x|β0−np0αkdx
≥Cδ−nαk/p0
k enαk/p0ln1/δk−→ ∞
5.3
ask → ∞, we see that this contradicts inequality3.4.
The given functionfkxand the exponential functionspxandβxare also suitable
for proving the necessity of condition2.3for the functionp. For this we define the numbers
δkfrom the equalityδk4k, k ∈N. Letfkx |x|−n/px−βχB0,2δk\B0,δkx,βx β∞, and
x∈Rn. We define the functionpas
px
⎧ ⎨ ⎩
p∞, x∈B0,2δk\B0, δk,
pk, x∈B0,4δk\B0,2δk, k∈N
5.4
wherep∞> 1,β∞ ∈R,pk p∞−αk, and{αk}is an arbitrary sequence of positive numbers
satisfying the conditionkαk → ∞ ask → ∞. Thenαklnδk → ∞; hence, condition2.3
does not hold for the functionpx. Furthermore, we have
Ip
|x|βxfkx
B0,2δk\B0,δk
|t|β∞· |t|−n/p∞−β∞p∞dtωn−1ln 2,
Ip
|x|βx−nfkx≥
B0,4δk\B0,2δk
B0,2δk\B0,δk
|t|−n/pt−β∞dt
pk
|x|β∞−npkdx
≥C
B0,3δk\B0,2δk
δkn−n/p∞−β∞pk|x|β∞−np∞−αkdx
≥Cδnαk/p∞
k Cenαk/p∞
lnδk −→ ∞
5.5
The same reasoning brings us to the proof of the exactness of conditions2.2and
2.3for the functionβxalso. For instance, to show the necessity of condition2.2, it can
be assumed thatpx≡p0>1,x∈Rn,
βx
⎧ ⎨ ⎩
β0αk, x∈B0,2δk\B0, δk,
β0, x∈B0,4δk\B0,2δkk∈N.
5.6
Then
Ip
|x|βx−nfkx≥Cδ−p0αk
k −→ ∞ ask−→ ∞,
Ip
|x|βxfkx≤C0ln 2.
5.7
This completes the proof ofTheorem 3.3.
Acknowledgment
F. I. Mamedov was supported partially by the INTAS Grant for the South-Caucasian Republics, no. 8792.
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