On Boundedness of Weighted Hardy Operator in L

Volume 2010, Article ID 837951,14pages doi:10.1155/2010/837951

Research Article

On Boundedness of Weighted Hardy Operator in L

and Regularity Condition

Aziz Harman

and Farman Imran Mamedov

1Education 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 spacesL^p·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|fydyfromL^p·

|x|^β·Rⁿinto

L^q·

|x|^{β·−n/p·−n/q·}R^N. 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

L^q·Rⁿ≤C|x|^β·f

L^p·Rⁿ, Hfx

|^y|^≤|x|f y

dy 1.1

in the norms of generalized Lebesgue spacesL^p·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 function|x|⁻ⁿHfx≤CMfx.

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 spacesL^p·, 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 sup_x∈Rnp <∞. The space of functionsL^p·Ωis introduced as the class of measurable functionsfxinΩ, which have a finiteI_pf :

Ω|fx|^pxdx- modular. A norm inL^p·Ωis given in the form

f

L^p·Ωinf

λ >0 :Ip

f λ ≤1

. 2.1

Forp⁻>1,p<∞the spaceL^p·Ωis a reflexive Banach space.

Denote byΛ a class of measurable functionsf : Rⁿ → Rsatisfying the following conditions:

∃m∈

0,1

2 , ∃f0∈R, sup

x∈B0,m

fx−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.

We will many times use the following statement in the proof of main results.

Lemma 2.1. Lets ∈ Λbe a measurable function such that−∞ < s⁻, s < ∞. Then the condition 2.2for the functionsxis equivalent to the estimate

C⁻¹₃ |x|^s0 ≤ |x|^sx≤C₃|x|^s0 2.4

when|x| ≤mand the condition2.3forsxis equivalent to the estimate

C₄⁻¹|x|^s∞ ≤ |x|^sx≤C₄|x|^s∞ 2.5

when|x| ≥M. Where the constantsC₃, C₄>1 depend 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

|y|≥|x|fydy.

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

L^p·Rⁿ ≤C|x|^β·f

L^p·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

L^p·Rⁿ≤1. 4.1

We will prove that

|x|^{β·−n/p·−n/q·}Hf

L^q·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

L^q·Rⁿ≤|x|^{β·−n/p·−n/q·}Hf

L^q·B0,δ

|x|^{β·−n/p·−n/q·}Hf

L^q·B0,N\B0,δ

|x|^{β·−n/p·−n/q·}

{t:|t|<N}ftdt

L^q·Rⁿ\B0,N

|x|^{β·−n/p·−n/q·}

{t:N<|t|<|x|}ftdt

L^q·R^N\B0,N

:i1i2i3i4.

4.3

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

L^q·B0,δ

≤^∞

k0

|x|^{β·−n/p·−n/q·}

{t:2^−k−1|x|<|t|<2^−k|x|}ftdt

L^q·B0,δ

.

4.4

DenoteB_x,k {y ∈ Rⁿ : 2^−k−1|x| < |y| < 2^−k|x|}andp⁻_x,k minpx,inf_y∈Bx,kpy.

By2.2andLemma 2.1, forx ∈ B0, δ,t ∈ B_x,k, we have|x|^βx ∼ 2^kβ0t^βt. 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

≤C₆2^kβ0|x|^{−n/px−n/qx}

Bx,k

|t|^βtftdt

≤C62^kβ0|x|^{−n/p0−n/qx}

Bx,k

|t|^βtftp_x,k⁻

dt _1/p⁻

x,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

∼t^n/pt ∼t^n/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

∼2^−kn/px|x|^n/px≤2^−kn/p0εk|x|^n/px; x∈B0, δ. 4.8

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|^βtft_pt dt

Bx,k

dt≤12^−knδⁿC8.

4.9

By4.5and4.9, we have

I_q;B0,δ

|x|^{β·−n/p·−n/q·}

Bx,k

ftdt

≤C92^−kεq−

B0,δ|x|⁻ⁿ

Bx,k

|t|^βtft_p⁻

x,kdt

_qx/px,k⁻ dx

≤C₉C₈^q/p−−12^−kεq−

B0,δ

Bx,k

|t|^βtft_pt 1 dt

|x|⁻ⁿdx

4.10

which, due to Fubini’s theorem, yields

≤C₉C^q₈^/p−−12^−kεq−

{t:|t|<2^−kδ}

ft|t|^βt_pt

B^{0,2k1|t|\B0,2k|t|}|x|⁻ⁿdx

dt C₁₀2^−kεq−ln 2

{t:|t|<2^−kδ}

ft|t|^βt_pt

1 dt≤C₁₁2^−kεq−.

4.11

Therefore,

|x|^{β·−n/p·−n/q·}

Bx,k

ftdt

L^q·B0,δ

≤C122^−kεq−/q. 4.12

By4.12and4.4, we get

i₁≤C₁₂ ∞ k0

2^−kεq−/qC₁₃<∞. 4.13

The estimate at infinityi4.

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

L^q·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∞∼2^kβ∞t^β∞∼2^kβ∞t^βt. 4.15

Therefore, by virtue of Holder’s inequality,

|x|^{βx−n/px−n/qx}

Bx,k

fNtdt

≤C142^kβ∞|x|^{−n/px−n/qx}

Bx,k

|t|^βtfNtdt

≤C142^kβ∞|x|^{−n/px−n/qx}

Bx,k

|t|^βtfNt_p⁻

x,kdt _1/p⁻

x,k

2^−k|x|_n/p⁻

x,k

.

4.16

iIfp⁻_x,k/pxandt∈B_x,k, by2.3andLemma 2.1, we have

2^−k|x|_n/p⁻

x,k

∼t^n/pt∼t^n/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|^βtft_p⁻

x,kdt≤

Bx,k∩{t∈Rⁿ:|t|^βtft≥Gt}

|t|^βtft Gt

_p⁻

x,k

Gt^p−x,kdt

Bx,k

Gt^p−dt, 4.20

whereGt 1/1t². Hence,

≤

Bx,k

fNt|t|^βt_pt

Gt^p−x,k−pt

Bx,k

Gtdt. 4.21

By2.3, fort∈Bx,k, we have

Gt^p−x,k−pt≤

1t²_pt−p⁻

x,k ≤C16. 4.22

Then4.21implies

Bx,k

|t|^βtf_Nt_p⁻

x,kdt≤C₁₇. 4.23

Therefore,

I_q;Rⁿ_\B0,N

|x|^{βx−n/px−n/qx}

Bx,k

fNtdt

≤C^q₁₇^/p−2^−kδ1q−

Rⁿ\B0,N|x|⁻ⁿ

Bx,k

|t|^βtfNt_pt dt

dx,

4.24

by Fubini’s theorem,

≤C^q₁₇^/p−−12^−kδ1q−ln 2

{t:|t|>2^−kN}

fNt|t|^βt_pt

dt≤C182^−kδ1q−. 4.25

From4.25and expansion4.14, we get

i₄≤C₁₈ ∞ k0

2^{−kq−δ1/q}C₁₉. 4.26

The estimate in the middlei2, i3. We have

i2

|x|^{β·−n/p·−n/q·}

{t∈Rⁿ:|t|<|x|}ftdt

L^q·B0,N\B0,δ

≤

B0,Nftdt

|x|^{β·−n/p·−n/q·}

L^q·B0,N\B0,δ

≤C₂₀

B0,Nftdt,

4.27

from which, by virtue of Holder’s inequality, forpx-norms, we obtain the estimate

B0,Nftdt≤|t|^β·ft

L^p·B0,N

|t|^−β·

L^p·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|^−βtptdt≤C21

B0,N|t|^−β0p0dtC22. 4.28 From4.27and4.28, it follows that

i2≤C23. 4.29

Furthermore, we have

i3≤

B0,Nftdt

|x|^{βx−n/px−n/qx}

L^q·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

I_q;Rⁿ_/B0,N

|x|^{β·−n/p·−n/q·}

≤C₂₄

Rⁿ\B0,N|x|^−n−2δ1dxC₂₅. 4.32

Then

i3 ≤C₂₅^1/p−. 4.33

Necessity. Letβ0 > n/p0. Fix a sufficiently largeτ >0 and apply inequality1.1by the test function

fτt t^{−n/pt−βt}χB0,δ/τ\B0,δ/2τt. 4.34

We come to a contradiction

I_p

|t|^β·f_τ

B0,δ/τ\B0,δ/2τ|x|⁻ⁿdxC₀ln 2<∞, Iq

|t|^{β·−n/p·−n/q·}

B0,tfτ

y dy

≥

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−n}dt−→ ∞

4.35

asτ → ∞.

If 0< p0≤1, then by virtue of inequalities4.35and3.2we obtain

Iq

|t|^{βt−n/pt−n/qt}

B0,tfτ

y dy

−→ ∞, asτ −→ ∞. 4.36

Also,

I_p

|t|^βtf_τt

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,

I_q

|t|^{βt−n/pt−n/qt}

B0,tf_τtdy

≥

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−n}dt−→ ∞

4.38

asτ → ∞. Ifβ∞ n/p∞, then from4.38we have

I_q

|t|^{βt−n/pt−n/qt}

B0,tf_τtdy

∞. 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/qx}Hfx

L^q·Rⁿ≤C|x|^βxfx

L^p·Rⁿ, |z|^{n−βz−2n/qz}Hfx

L^q·Rⁿ≤C|z|^{−βz−2n/pz}fz

L^p·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

L^p·Rⁿ|z|^−2n/pzg

L^p·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. Assume δk 1/4^k,k ∈ N,fkx |x|^{−n/px−βx}χ_{B0,2δk\B0,δk}x, 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

kαk−→ ∞ ask−→ ∞. 5.2

Thenαkln1/δk → ∞, and condition2.2does not hold for the functionpx. Since

I_p

|x|^βxf_kx

B0,2δk\B0,δk

|t|^β0· |t|^{−n/p0−β0}p0

dt

B0,2δk\B0,δk|t|⁻ⁿdtC_{0 2δk}

δk

dt

t ω_n−1ln 2, Ip

H

|·|^β·−nfk·

≥

B0,4δk\B0,2δk

B0,2δk\B0,δk|t|^{−n/pt−β0}dt pk

|x|^β0−npkdx

≥C

B0,3δk\B0,2δkδkn−n/p0−β0pk|x|^β0−np0αkdx

≥Cδ_k^−nαk/p0 e^{nαk/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 equalityδk4^k, k ∈N. Letfkx |x|^−n/px−βχ_{B0,2δk\B0,δk}x,βx β_∞, and x∈Rⁿ. We define the functionpas

px

⎧⎨

⎩

p_∞, x∈B0,2δ_k\B0, δk,

p_k, x∈B0,4δk\B0,2δk, k∈N 5.4

wherep_∞> 1,β_∞ ∈R,p_k p_∞−αk, and{αk}is an arbitrary sequence of positive numbers satisfying the conditionkα_k → ∞ ask → ∞. Thenα_klnδ_k → ∞; 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∞

dtω_n−1ln 2,

I_p

|x|^βx−nf_kx

≥

B0,4δk\B0,2δk

B0,2δk\B0,δk|t|^{−n/pt−β∞}dt _pk

|x|^β∞−npkdx

≥C

B0,3δk\B0,2δkδ_k^{n−n/p∞−β∞pk}|x|^{β∞−np∞−αk}dx

≥Cδ^nα_k ^k/p∞Ce^{nαk/p∞lnδk} −→ ∞

5.5

ask → ∞, which contradicts inequality3.4.

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

⎧⎨

⎩

β₀α_k, x∈B0,2δ_k\B0, δk,

β₀, x∈B0,4δ_k\B0,2δ_kk∈N. 5.6 Then

I_p

|x|^βx−nf_kx

≥Cδ^−p_k ^0αk −→ ∞ ask−→ ∞, I_p

|x|^βxf_kx

≤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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