On Necessary Condition for the Variable Exponent Hardy Inequality

Abstract

We derive a necessary condition for exponent functions 𝑝,𝛽 such that the variable exponent Hardy inequality ‖𝑥^{𝛽(𝑥)−1}∫^𝑥0𝑓(𝑡)𝑑𝑡‖_{𝐿𝑝(.)(0,𝑙)}≤𝐶‖𝑥^𝛽(𝑥)𝑓‖_𝐿^𝑝(.)_(0,𝑙) holds.

1. Introduction

A sufficient condition on measurable functions 𝑝∶(0,𝑙)→[1,∞), 𝛽∶(0,𝑙)→(−∞,∞) for which the variable exponent Hardy inequality‖‖|𝑥|^{𝛽(⋅)−1}𝐻𝑓‖‖_{𝐿𝑝(⋅)(0,𝑙)}≤𝐶‖‖|𝑥|^𝛽(⋅)𝑓‖‖_{𝐿𝑝(⋅)(0,𝑙)},𝐻𝑓(𝑥)=^𝑥0𝑓(𝑡)𝑑𝑡(1.1) holds for all 𝑓≥0 have been known (see [1–3]). According to mentioned works, if 𝛽(0)<1−(1/𝑝(0)), 𝑝⁻∶=inf{𝑝(𝑥)∶𝑥∈(0,𝑙)}>1, then a sufficient condition is 𝑝,𝛽∈Λ, where Λ means a class of measurable functions 𝑔∶(0,𝑙)→(−∞,∞) such that ∃𝑔(0),𝐶>0||𝑔(𝑥)−𝑔(0)||ln1|𝑥|≤𝐶,0<𝑥<12.(1.2)

The purpose of this paper is to prove that a weaker continuity condition on 𝑝(⋅) and 𝛽(⋅) is necessary for the norm inequality to hold provided that 𝑝(⋅) and 𝛽(⋅) are monotone (see Theorems 2.1 and 2.2 for a precise statement), which is the following condition:||𝑔(2𝑥)−𝑔(𝑥)||ln1𝑥≤𝐶,0<𝑥<12.(1.3)

Note that condition (1.3) is strictly weaker than (1.2). For example, it is satisfied by 𝑝(𝑥)=𝐶/|log(𝑥)|^1/2. This condition is new and somewhat surprising. Since in the corresponding theorem for the maximal operator, it is known that 𝑝(⋅) need not be continuous, and the problem of determining which exponent conditions are necessary and/or sufficient is an open one.

If the powers are not monotone, it follows from the results of the paper [2] that condition (1.2) is close to be sharp. Also in [2], the necessity of conditions 𝑝⁻>1 and 𝛽(0)<1−1/𝑝(0) was proved. Recently, there have been quite a number of papers discussing the Hardy inequality in norms of the variable exponent Lebesgue spaces [3–11].

For problems of boundedness of classical integral operators in variable exponent Lebesgue spaces and regularity results for nonlinear equations with nonstandard growth condition, see monograph [12] and references therein.

2. Main Results

As to the basic properties of spaces 𝐿^𝑝(⋅), we refer to [13]. Throughout this paper it is assumed that 𝑝(𝑥) is a measurable function in (0,𝑙) taking its values from the interval [1,∞) with 𝑝⁺∶=sup{𝑝(x)∶𝑥∈(0,𝑙)}<∞. The space of functions 𝐿^𝑝(⋅)(0,𝑙) is introduced as the class of measurable functions 𝑓(𝑥) in (0,𝑙), which have a finite 𝐼_𝑝(𝑓)∶=∫^𝑙0|𝑓(𝑥)|^𝑝(𝑥)𝑑𝑥-modular. A norm in 𝐿^𝑝(⋅)(0,𝑙) is given in the form‖𝑓‖_𝐿^𝑝(⋅)_(0,𝑙)=inf𝜆>0∶𝐼_𝑝𝑓𝜆≤1.(2.1) There exists a relation between modular and norm, which is expressed by the following inequalities:‖𝑓‖^𝑝+_{𝐿𝑝(⋅)(0,𝑙)}≤𝐼_𝑝(𝑓)≤‖𝑓‖^𝑝−_{𝐿𝑝(⋅)(0,𝑙)},1≥‖𝑓‖_𝑝(⋅),(2.2)‖𝑓‖^𝑝−_{𝐿𝑝(⋅)(0,𝑙)}≤𝐼_𝑝(𝑓)≤‖𝑓‖^𝑝+_{𝐿𝑝(⋅)(0,𝑙)},1≤‖𝑓‖_𝑝(⋅).(2.3) Such estimates allow us to perform our estimates in terms of a modular. In the following two theorems, we show that if functions 𝑝,𝛽 are monotone, then condition (1.3) for them is necessary for inequality (1.1) to hold.

Theorem 2.1. Let 𝛽∈ℝ a function 𝑝∶(0,𝑙)→[1,∞) be increasing on (0,𝜀) and such that 𝑝(0)=lim_𝑥→0𝑝(𝑥) exists, 𝛽(0)<1−1/𝑝(0), 𝑝⁻>1. Then for inequality (1.1) to hold, it is necessary that for the function 𝑝(⋅) condition (1.3) is satisfied.

Theorem 2.2. Let 𝑝∈ℝ, let 𝛽∶(0,𝑙)→[−∞,∞) be a function decreasing on (0,𝜀) such that 𝛽(0)=lim_𝑥→0𝛽(𝑥) exists, and let the conditions 𝛽(0)<1−1/𝑝(0), 𝑝⁻>1 be satisfied. Then for inequality (1.1) to hold, it is necessary that for the function 𝛽(⋅) condition (1.3) is satisfied.

The following two theorems show that the logarithmic regularity conditions (1.2) for the functions 𝑝,𝛽 are essential for inequality (1.1) to hold.

Theorem 2.3. Let 𝛽∈ℝ, and 𝛿_𝑛=4^−𝑛,𝑛∈ℕ. There exist a sequence of functions {𝑓_𝑛} and a function 𝑝∶(0,𝑙)→[1,∞) satisfying the conditions 𝛽<1−1/𝑝(0), 𝑝⁻>1 such that lim_𝑛→∞||𝑝𝛿_𝑛−𝑝(0)||ln1𝛿_𝑛=∞,(2.4) and inequality (1.1) is violated.

Theorem 2.4. Let 𝑝>1, 𝛿_𝑛=4^−𝑛,𝑛∈ℕ. Then there exist a sequence of functions {𝑓_𝑛} and a function 𝛽∶(0,𝑙)→(−∞,∞) satisfying the conditions 𝛽(0)<1−1/𝑝, lim_𝑛→∞||𝛽𝛿_𝑛−𝛽(0)||ln1𝛿_𝑛=∞,(2.5) such that inequality (1.1) is violated.

3. Proofs of Main Results

Proof of Theorem 2.1. Denote 𝐼_𝑝(⋅)(𝑓)=∫^𝑙0|𝑓(𝑡)|^𝑝(𝑡)𝑑𝑡. By (2.2) note that the condition 𝐼_𝑝(𝑓)≤1 is equivalent to ‖𝑓‖_𝐿^𝑝(.)_(0,𝑙)≤1.
Put 𝛿_𝑘=𝜀4^−𝑘,𝑘∈ℕ, and 𝑓_𝑘(𝑥)=𝑥^{−1/𝑝(𝑥)−𝛽}𝜒_{(𝛿𝑘,2𝛿𝑘)}(𝑥),𝑥∈(0,𝑙). Then for sufficiently large 𝑘, 𝐼_𝑝(⋅)𝑥^𝛽(𝑥)𝑓_𝑘(𝑥)==^2𝛿𝑘_𝛿𝑘𝑡^𝛽𝑡^{−1/𝑝(𝑡)−𝛽}^𝑝(𝑡)𝑑𝑡_{2𝛿𝑘𝛿𝑘}𝑡⁻¹𝑑𝑡=ln2.(3.1) Also 𝐼_𝑝(⋅)𝑥^{𝛽(𝑥)−1}𝐻𝑓_𝑘(𝑥)≥^4𝛿𝑘_3𝛿𝑘𝑥^(𝛽−1)^2𝛿𝑘_𝛿𝑘≥𝐶_{4𝛿𝑘3𝛿𝑘}𝑡^{−(1/𝑝(𝑡))−𝛽}𝑑𝑡^𝑝(𝑥)𝑑𝑥≥𝐶𝛿_{1−𝑝(3𝛿𝑘)/𝑝(2𝛿𝑘)𝑘}𝛿^{(1−1/𝑝(2𝛿𝑘𝑘)−𝛽)}𝑥^{(𝛽−1)𝑝(2𝛿𝑘)}𝑑𝑥=𝐶𝑒^{(1/𝑝+)[𝑝(3𝛿𝑘)−𝑝(2𝛿k)]ln(1/2𝛿𝑘)}.(3.2) Applying inequality (1.1), we have ||𝑝3𝛿_𝑘−𝑝2𝛿_𝑘||ln12𝛿_𝑘≤𝐶,𝑘∈ℕ,(3.3) which by using of monotony of 𝑝 and its boundedness implies (1.3).
This completes the proof of Theorem 2.1.

Proof of Theorem 2.2. Put 𝛿_𝑘=𝜀4^−𝑘,𝑘∈ℕ and 𝑓_𝑘(𝑥)=𝑥^{−1/𝑝−𝛽(𝑥)}𝜒_{(𝛿𝑘,2𝛿𝑘)}(𝑥),𝑥∈(0,𝑙). Then 𝐼_𝑝(⋅)𝑥^𝛽(𝑥)𝑓_𝑘(𝑥)==^2𝛿𝑘_𝛿𝑘𝑡^𝛽(𝑡)𝑡^{−1/𝑝−𝛽(𝑡)}^𝑝(𝑡)𝑑𝑡_{2𝛿𝑘𝛿𝑘}𝑡⁻¹𝑑𝑡=ln2.(3.4) Also 𝐼_𝑝(⋅)𝑥^{𝛽(𝑥)−1}𝐻𝑓_𝑘(𝑥)≥^4𝛿𝑘_3𝛿𝑘𝑥^{𝛽(𝑥)−1}^2𝛿𝑘_𝛿𝑘≥𝐶𝛿_{[𝛽(3𝛿𝑘)−𝛽(2𝛿𝑘𝑘)]𝑝}𝑡^{−1/𝑝−𝛽(𝑡)}𝑑𝑡^𝑝(𝑥)𝑑𝑥≥𝐶𝑒^{𝑝[𝛽(3𝛿𝑘)−𝛽(2𝛿𝑘)]ln(1/𝛿𝑘)}.(3.5) Applying inequality (1.1), we have ||𝛽2𝛿_𝑘−𝛽3𝛿_𝑘||ln1𝛿_𝑘≤𝐶,𝑘∈ℕ(3.6) which by using monotony of 𝛽 implies (1.3).
This completes the proof of Theorem 2.2.

Proof of Theorem 2.3. Let us assume that 𝑓_𝑘(𝑥)=𝑥^{−1/𝑝(𝑥)−𝛽}𝜒_{(𝛿𝑘,2𝛿𝑘)}(𝑥), 𝑥∈(0,𝑙). Fix 𝑘∈ℕ. We define the step function 𝑝(𝑥)=𝑝⁰𝑝₀+𝛼_𝑛if𝑥∈2𝛿_𝑛,4𝛿_𝑛,if𝑥∈𝛿_𝑛,2𝛿_𝑛,𝑛∈ℕ.(3.7) Here {𝛼_𝑛}is a sequence of positive numbers that satisfies the condition 𝑛𝛼_𝑛⟶∞as𝑛⟶∞.(3.8) Then 𝛼_𝑛ln(1/𝛿_𝑛)→∞ as 𝑛→∞; that is, condition (1.2) is not satisfied for the function 𝑝(𝑥). Also note that this function 𝑝(⋅) is not monotone. We have 𝐼_𝑝(⋅)𝑥^𝛽𝑓_𝑘(𝑥)==^2𝛿𝑘_𝛿𝑘𝑡^𝛽𝑡^{−1/𝑝(𝑡)−𝛽}^𝑝(𝑡)𝑑𝑡_{2𝛿𝑘𝛿𝑘}𝐼𝑡⁻¹𝑑𝑡=ln2,(3.9)_𝑝(⋅)𝑥^𝛽−1𝐻𝑓_𝑘(𝑥)≥≥𝐶^4𝛿𝑘_2𝛿𝑘_{4𝛿𝑘2𝛿𝑘}^2𝛿𝑘_𝛿𝑘𝑡^{−1/𝑝(𝑡)−𝛽}𝑑𝑡^{(𝑝0+𝛼𝑘)}𝑥^{(𝛽−1)(𝑝0+𝛼𝑘)}𝑑𝑥≥𝐶𝛿_{−𝛼𝑘/𝑝0𝑘}𝛿^{(1−1/𝑝0−𝛽)(𝑝0+𝛼𝑘)𝑘}𝑥^{(𝛽−1)(𝑝0+𝛼𝑘)}𝑑𝑥=𝐶𝑒^{𝛼𝑘/𝑝0ln(1/𝛿𝑘)}⟶∞as𝑘⟶∞.(3.10) The last relation shows violating of inequality (1.1) for sufficiently large 𝑘.

Proof of Theorem 2.4. Let us assume that 𝑓_𝑘(𝑥)=𝑥^{−1/𝑝−𝛽(𝑥)}𝜒_{(𝛿𝑛,2𝛿𝑛)}(𝑥), 𝑥∈(0,𝑙), 𝑛∈ℕ. Fix 𝑛∈ℕ. We define the step function 𝛽 as 𝛽(𝑥)=𝛽⁰𝛽₀+𝛼_𝑛if𝑥∈𝛿_𝑛,2𝛿_𝑛,if𝑥∈2𝛿_𝑛,4𝛿_𝑛,𝑛∈ℕ,(3.11) where 𝛼_𝑛ln(1/𝛿_𝑛)→∞; that is, condition (1.2) is not satisfied for the function 𝛽. Note that this function 𝛽 is not monotone.
We have 𝐼_𝑝(⋅)𝑥^{𝛽(𝑥)−1}𝐼𝑓_𝑛(𝑥)≥𝐶𝛿^{−𝑝𝛼𝑛}_𝑛⟶∞as𝑛⟶∞,(3.12)_𝑝(⋅)𝑥^𝛽(𝑥)𝑓_𝑛(𝑥)≤𝐶₀ln2.(3.13) The last relation contradicts to inequality (1.1) for sufficiently large 𝑘.

Acknowledgment

The author thanks Professor Mamedov F.I. for his valuable discussion of the subject. The author also expresses his acknowledgement to the referee for drawing attention to the log example and comments of stylistic character.