On the Logarithmic Regularity Conditions for the Variable Exponent Hardy Type Inequality

We discuss a logarithmic regularity condition in a neighborhood of the origin and infinity on the exponent functions and for the variable exponent Hardy inequality to hold.

In this note, we will focus on the results of sufficiency and necessity of regularity conditions Λ 0 , Λ ∞ , and Λ 1 below for inequality (2) to hold.

Main Results
We will state some sufficiency and necessity assertions concerning inequality (2). Along the way, it will be given a proof for two elementary estimates that we had used. Let us introduce the following classes of measurable functions. We say, is in the class Λ ∞ if and is in the class Λ 1 if Theorem 1 (see [8]). Suppose ∈ R and : (0, ) → [1, ∞) is an increasing function on (0, ) such that ( ) is continuous at = 0 and < 1 − (1/ (0)), − > 1; then for the inequality to hold it is necessary that (⋅) ∈ Λ 1 .

Proof of Main Results
For the proof of Theorems 1 and 2 we refer to [8]. Other proofs of these theorems are given in [10]. The proof of Theorem 3 also is given in [8]. Here we derive an alternative proof of that theorem using the general results of [2,4].
In the proof of main results we use the following elementary Lemma. Lemma 6. Suppose : R → (0, ∞) is a measurable function such that ∈ Λ 0 ∩ Λ ∞ and 0 < − , + < ∞; then it holds the estimate for 0 < < 1 and the estimate for ≥ 2.
Therefore, for 0 < ≤ we have the estimation ) .

(23)
By using these inequalities and by the representation we have estimate (14).
To show estimate (15) note that for ≥ and ( ) ≤ is a certain number. If ≥ and ( ) ≥ (∞) then by the condition ∈ Λ ∞ we have Combining the estimates for the functions ( )− (∞) , (∞)− ( ) for ≥ by the presentation we get estimate (15). To complete the proof of Lemma 6, note that the condition ∈ Λ 0 is equivalent to and the ∈ Λ ∞ is equivalent to respectively.
Proof of Theorem 5. Let us assume that We define the step function as where ⋅ ln(1/ ) → ∞. Then, as → ∞. The last relation contradicts the validity of inequality (2).

Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.