Our aim in this paper is to obtain Hardy’s inequality in variable exponent Lebesgue spaces Lp(x)(0

http://jipam.vu.edu.au/

Volume 7, Issue 3, Article 106, 2006

ON HARDY’S INEQUALITY IN L^p(x)(0,∞)

RABIL A. MASHIYEV, BILAL ÇEKIÇ, AND SEZAI OGRAS UNIVERSITY OFDICLE, FACULTY OFSCIENCES ANDARTS

DEPARTMENT OFMATHEMATICS

21280- DIYARBAKIRTURKEY mrabil@dicle.edu.tr bilalc@dicle.edu.tr sezaio@dicle.edu.tr

Received 14 October, 2005; accepted 07 April, 2006 Communicated by L. Pick

ABSTRACT. Our aim in this paper is to obtain Hardy’s inequality in variable exponent Lebesgue spaces L^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.

Key words and phrases: Variable exponent, Hardy’s inequality.

2000 Mathematics Subject Classification. 46E35, 26D10.

1. INTRODUCTION

Over the last decades the variable exponent Lebesgue spaceL^p(·)(Ω) and the corresponding Sobolev space W^m,p(·)(Ω) have been a subject of active research stimulated by development of the studies of problems in elasticity, fluid dynamics, calculus of variations, and differential equations withp(x)−growth [10, 12]. These spaces are a special case of the Musielak-Orlicz spaces [8]. If p is the constant, then L^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,∞) and u(∞) = lim

x→∞u(x) = 0.

ISSN (electronic): 1443-5756 c

2006 Victoria University. All rights reserved.

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.

310-05

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 weightρ(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,`),

whereβ >−1,1< p⁻ ≤p⁺<∞, `is a positive finite number, anduis an absolutely continu- ous function on(0, `)in the Lebesgue space with variable exponent for bounded domains from Theorem E in [6].

Recently, Harjulehto, Hästö and Koskenoja [5] have obtained the norm version 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 variable p(x). Such inequalities with variable p(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 spaceL^p(x)(0,∞)using a distinct method, by considering relevant studies in [1] and [6].

2. PRELIMINARIES

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

x∈Ω

p(x)andp⁻ = essinf

x∈Ω p(x). We define 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. If p⁺ < ∞ then we call p a bounded exponent and we can introduce the norm on L^p(·)(Ω)by

(2.1) kfk_p(·),Ω := inf

λ >0 :A_p f

λ

≤1

andL^p(·)(Ω)becomes a Banach space. The normkfk_p(·),Ωis in close relation with the modular A_p(f).

Lemma 2.1 ([4]). Letp(x)be a measurable exponent such that1≤p⁻ ≤p(x)≤p⁺ <∞and letΩbe a measurable set inRⁿ. Then,

(i) kfk_p(x) =λ6= 0if and only ifA_p ^f_λ

= 1;

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

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

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

kfk^p_p(x)⁻ ≤A_p(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 > 0 such 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]). Let I = [0, M) forM < ∞, p : I → [1,∞) be bounded, p(0)>1and

lim sup

x→0+

(p(x)−p(0)) log1 x <∞ andp⁻_(0,x

0) =p(0)for somex₀ ∈(0,1). Ifa∈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 thatp(x)is a measurable function and use this notation kfk_p(x) :=kfk_p(x),(0,∞).

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

3. MAINRESULT

Theorem 3.1. Letβ >−1andp: (0,∞)→(1,∞)be such that1≤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 +1}u⁰(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 +1}u⁰(x) p(x)

= 1

for a monotone decreasing functionu. Using Hölder’s inequality, we obtain (3.4) u(a) = −

Z ∞

a

u⁰(t)dt ≤ckt^{p(t)β +1}u⁰(t)kp(t),(a,∞)kt^{−p(t)β −1}k_p⁰(t),(a,∞)≤c1,

wherep⁰(x) = _p(x)−1^p(x) , and the positive constantc₁depends only onp(x)andβ. Sinceu(x)≤c₁ for(0,∞), using Hardy’s inequality for the fixed exponentp(∞)we have

(3.5)

Z ∞

a

x^βu(x)^p(x)dx≤c^p₂⁺ Z ∞

a

x^βu(x)^p(∞)dx≤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

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)logt 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.

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) kt^p(t)β u(t)k_p(t)≤c.

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

t^{p(t)β +1}u⁰(t) p(t)

.

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