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volume 7, issue 3, article 106, 2006.

Received 14 October, 2005;

accepted 07 April, 2006.

Communicated by:L. Pick

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Journal of Inequalities in Pure and Applied Mathematics

ON HARDY’S INEQUALITY INL^p^(x)(0,∞)

RABIL A. MASHIYEV, BILAL ÇEKIÇ AND SEZAI OGRAS

University of Dicle, Faculty of Sciences and Arts Department of Mathematics

21280- Diyarbakır TURKEY EMail:mrabil@dicle.edu.tr EMail:bilalc@dicle.edu.tr EMail:sezaio@dicle.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 310-05

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On Hardy’s Inequality in L^p^(x)(0,∞)

Rabil A. Mashiyev Bilal Çekiç and Sezai Ogras

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J. Ineq. Pure and Appl. Math. 7(3) Art. 106, 2006

http://jipam.vu.edu.au

Abstract

Our aim in this paper is to obtain Hardy’s inequality in variable exponent Lebesgue spacesL^p(x)(0,∞), where the test functionu(x)vanishes at infinity. We use a local Dini-Lipschitz condition and its the natural analogue at infinity, which play a central role in our proof.

2000 Mathematics Subject Classification:46E35, 26D10.

Key words: Variable exponent, Hardy’s inequality.

The authors are thankful to the referees and Peter Hästö for their helpful suggestions and valuable contributions. This research was supported by DUAPK grant No. 04 FF 40.

1. Introduction

Over the last decades the variable exponent Lebesgue space L^p(·)(Ω) and the corresponding Sobolev spaceW^m,p(·)(Ω)have been a subject of active research stimulated by development of the studies of problems in elasticity, fluid dy- namics, calculus of variations, and differential equations with p(x)− growth [10,12]. These spaces are a special case of the Musielak-Orlicz spaces [8]. Ifp is the constant, thenL^p(·)(Ω)coincides with the classical Lebesgue spaces. We refer to [4, 7] for fundamental properties of these spaces and to [5, 6, 11] for Hardy type inequalities.

The classical Hardy inequality [9] is (1.1)

Z ∞

0

|u(x)|^px^βdx ≤ p

β+ 1

pZ ∞

0

|u⁰(x)|^px^β+pdx,

where 1 < p < ∞,−1 < β < ∞, u is an absolutely continuous function on (0,∞)andu(∞) = lim

x→∞u(x) = 0.

Kokilashvili and Samko [6] gave the boundedness of Hardy operators with fixed singularity in the spacesL^p(·)(ρ,Ω)over a bounded open set inRⁿwith a power weight ρ(x) = |x−x₀|^β, x₀ ∈ Ω¯ and an exponentp(x)satisfying the Dini-Lipschitz condition. The Hardy type inequality can be derived

(1.2)

x

β p(x)u

p(x),(0,`) ≤C(p(x), `) x

β p(x)+1

u⁰

p(x),(0,`),

where β > −1,1 < p⁻ ≤ p⁺ < ∞, ` is a positive finite number, and u is an absolutely continuous function on(0, `)in the Lebesgue space with variable exponent for bounded domains from Theorem E in [6].

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Recently, Harjulehto, Hästö and Koskenoja [5] have obtained the norm ver- sion of Hardy’s inequality using Diening’s corollaries in the variable exponent Sobolev space. Also they have given a necessary and sufficient condition for Hardy’s inequality to hold.

We consider the problem of the extension of Hardy’s inequality to the case of variablep(x). Such inequalities with variablep(x)are already known for a finite interval (0, `)in the one-dimensional case. Our aim in this paper is to obtain a Hardy type inequality in a one-dimensional Lebesgue space L^p(x)(0,∞)using a distinct method, by considering relevant studies in [1] and [6].

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2. Preliminaries

Let Ω ⊂ Rⁿ be an open set, p(·) : Ω → [1,∞) be a measurable bounded function and be denoted as p⁺ = esssup

x∈Ω

p(x) and p⁻ = essinf

x∈Ω p(x). We de- fine the variable exponent Lebesgue spaceL^p(·)(Ω)consisting of all measurable functionsf : Ω→Rsuch that the modular

Ap(f) :=

Z

Ω

|f(x)|^p(x)dx

is finite. Ifp⁺ < ∞then we call pa bounded exponent and we can introduce the norm onL^p(·)(Ω)by

(2.1) kfk_p(·),Ω := inf

λ >0 :A_p f

λ

≤1

and L^p(·)(Ω) becomes a Banach space. The normkfk_p(·),Ω is in close relation with the modularA_p(f).

Lemma 2.1 ([4]). Let p(x) be a measurable exponent such that 1 ≤ p⁻ ≤ p(x)≤p⁺<∞and letΩbe a measurable set inRⁿ. Then,

(i) kfk_p(x)=λ6= 0if and only ifAp f λ

= 1;

(ii) kfk_p(x)<1(= 1;>1)⇔A_p(f)<1 (= 1;>1);

(iii) For anyp(x), the following inequalities

kfk^p_p(x)⁺ ≤Ap(f)≤ kfk^p_p(x)⁻ , kfk_p(x) ≤1

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and

kfk^p_p(x)⁻ ≤Ap(f)≤ kfk^p_p(x)⁺ , kfk_p(x) ≥1 hold.

Lemma 2.2 ([4,7]). The generalization of Hölder’s inequality

Z

Ω

f(x)ϕ(x)dx

≤ckfk_p(x)kϕk_p0(x)

holds, wherep⁰(x) = _p(x)−1^p(x) and the constantc >0depends only onp(x).

We say that the exponentp(·) : Ω →[1,∞)is Dini-Lipschitz if there exists a constantc >0such that

(2.2) |p(x)−p(y)| ≤ c

−log|x−y|,

for everyx, y ∈Ωwith|x−y| ≤ ¹₂.The natural analogue of (2.2) is

(2.3) |p(x)−p(y)| ≤ c

log (e+|x|)

for every x, y ∈ Ω, |y| ≥ |x| at infinity. Under these conditions, most of the properties of the classical Lebesgue space can be readily generalized to the Lebesgue space with variable exponent.

Theorem 2.3 ([5, Theorem 5.2]). LetI = [0, M)forM <∞,p:I →[1,∞) be bounded,p(0) >1and

lim sup

x→0+

(p(x)−p(0)) log 1 x <∞

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and p⁻_(0,x

0) = p(0) for some x0 ∈ (0,1). If a ∈ h

0,1− _p(0)¹

, then Hardy’s inequality

(2.4)

u(x) x^1−a p(x)

≤Cku⁰(x)x^ak_p(x)

holds for everyu∈W^1,p(x)(I)withu(0) = 0.

Throughout this paper, we will assume that p(x) is a measurable function and use this notation

kfk_p(x) :=kfk_p(x),(0,∞).

Moreover, we will usecandc_i as generic constants, i.e. its value may change from line to line.

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3. Main Result

Theorem 3.1. Letβ > −1 andp : (0,∞) → (1,∞)be such that 1 ≤ p⁻ ≤ p⁺ <∞and

(3.1) |p(x)−p(y)| ≤ c

−log|x−y|, |x−y| ≤ 1

2, x, y ∈R⁺. Assume that there exists a numberp(∞)∈[1,∞)anda≥1such that

(3.2) 0≤p(x)−p(∞)≤ c

log(e+x), x≥a.

Then, we have (3.3)

x^p(x)^β u(x) p(x)

≤c

x^p(x)^β ⁺¹u⁰(x) p(x)

for every absolutely continuous functionu: (0,∞)→Rwithu(∞) = 0.

Proof. To prove this inequality it suffices to consider the case

x^p(x)^β ⁺¹u⁰(x) p(x)

= 1

for a monotone decreasing functionu. Using Hölder’s inequality, we obtain u(a) =−

Z ∞

a

u⁰(t)dt (3.4)

≤c

t^p(t)^β ⁺¹u⁰(t)

_p(t),(a,∞)

t⁻^p(t)^β ⁻¹

p⁰(t),(a,∞) ≤c₁,

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where p⁰(x) = _p(x)−1^p(x) , and the positive constantc₁ depends only on p(x) and β. Sinceu(x) ≤c1 for(0,∞), using Hardy’s inequality for the fixed exponent p(∞)we have

Z ∞

a

x^βu(x)^p(x)dx≤c^p₂⁺ Z ∞

a

x^βu(x)^p(∞)dx (3.5)

≤c₃ Z ∞

a

x^β(−xu⁰(x))^p(∞)dx.

If we divide the interval(a,∞)into three sets such that A={t∈(a,∞) :t|u⁰(t)|>1},

B ={t∈(a,∞) :t^−β−2 < t|u⁰(t)| ≤1}, C ={t∈(a,∞) :t|u⁰(t)| ≤t^−β−2}, then we can write

Z ∞

a

t^β|tu⁰(t)|^p(∞)dt

= Z

A

t^β|tu⁰(t)|^p(∞)dt+ Z

B

t^β|tu⁰(t)|^p(∞)dt+ Z

C

t^β|tu⁰(t)|^p(∞)dt.

Now, let us estimate each integral. It is easy to see that Z

A

t^β|tu⁰(t)|^p(∞)dt≤ Z ∞

a

t^β|tu⁰(t)|^p(t)dt ≤1

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and Z

C

t^β|tu⁰(t)|^p(∞)dt ≤ Z

C

t^βt^−β−2dt≤ Z ∞

a

t^βt^−β−2dt ≤c.

Since

t(β+2)(p(t)−p(∞))

= (t^{p(t)−p(∞)})^β+2

≤

t^log(e+t)¹ β+2

≤

e^log(e+t)^log^t β+2

≤e^β+2, we have

Z

B

t^β|tu⁰(t)|^p(∞)dt ≤ Z

B

t^β t^β+2|tu⁰(t)|p(t)−p(∞)

|tu⁰(t)|^p(∞)dt

≤ Z ∞

a

t(β+2)(p(t)−p(∞))

t^β|tu⁰(t)|^p(t)dt

≤e^β+2 Z ∞

a

t^β|tu⁰(t)|^p(t)dt

≤e^β+2.

Hence, we obtain (3.6)

Z ∞

a

t^β|u(t)|^p(t)dt ≤c.

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On the other hand, by using inequality (1.2) and the assumption (3.1) for the interval(0, a), we can write

(3.7)

Z a

0

t^β|u(t)|^p(t)dt ≤c.

Combining inequalities (3.6) and (3.7), we get Z ∞

0

t^β|u(t)|^p(t)dt≤c

and hence from the relation between norm and modular we have (3.8)

t

β p(t)u(t)

p(t) ≤c.

Consequently, we have the required result from (3.8) for u(t)

t^p(t)^β ⁺¹u⁰(t) p(t)

.

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References

[1] C. CAPONE, D. CRUZ-URIBEANDA. FIORENZA, The fractional maximal operator on variableL^pspaces, Rapporto Tecnico 281/04, May 2004 (preprint). [ONLINE:http://www.na.iac.cnr.it/rapporti] [2] L. DIENING, Maximal functions on generalized Lebesgue spaces L^p(·),

Math. Inequal. Appl., 7(2) (2004), 245–254.

[3] L. DIENING, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces L^p(·) and W^k,p(·), Math. Nachr., 263(1) (2004), 31–43.

[4] X.-L. FANANDD. ZHAO, On the spacesL^p(x)(Ω)andW^k,p(x)Ω, J. Math.

Anal. Appl., 263 (2001), 424–446.

[5] P. HARJULEHTO, P. HÄSTÖANDM. KOSKENOJA, Hardy’s inequality in variable exponent Sobolev spaces, Georgian Math. J., 12(3) (2005), 431–442.

[6] V. KOKILASHVILI AND S. SAMKO, Maximal and fractional operators in weightedL^p(x) Spaces, Rev. Mat. Iberoamericana., 20(2) (2004), 493–

515.

[7] O. KOVÁ ˇCIK AND J. RÁKOSNÍK, On the spaces L^p(x)(Ω) and W^1,p(x)(Ω), Czech. Math. J., 41 (116) (1991), 592–618.

[8] J. MUSIELAK, Orlicz Spaces and Modular Spaces, Springer-Verlag , Berlin, 1983.

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[9] B. OPIC ANDA. KUFNER, Hardy-type Inequalities, Longman Scientific

& Technical, Harlow, (1990).

[10] M. R ˚UŽI ˇCKA, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.

[11] S. SAMKO, Hardy inequality in the generalized Lebesgue spaces, Fract.

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[12] V.V. ZHIKOV, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR-Izv., 29(1), 33–66. [Translation of Izv. Akad.

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