SOME NEW SCALES OF CHARACTERIZATION OF HARDY’S INEQUALITY

Let 1 < pq < ¥: Inspired by some recent results concerning Hardy-type inequalities where the equivalence of four scales of integral conditions was proved, we use related ideas to find ten new equivalence scales of integral conditions. By applying this result to the original Hardy-type inequality, we obtain a new proof of a number of characterizations of the Hardy inequality and also some new weight characterizations.


Introduction
We consider the general one-dimensional Hardy inequality with a fixed b, 0 < b ≤ ∞, for measurable functions f ≥ 0,weights u and v and for the parameters p, q satisfying 1 < p ≤ q < ∞. and assume that U (x) < ∞, V (x) < ∞ for every x ∈ (0, b).
In [2] the equivalence of four scales of integral conditions that characterize the inequality (1.1) (with the usual Muckenhoupt condition as a special case) was proved.The proof was carried out by first proving an equivalence theorem of independent interest.We will here extend this theorem by finding some additional new scales of conditions.
As it was shown in [6], [2], [8] and [3], the validity of Hardy's inequality (1.1) for all functions f ≥ 0 in fact can be characterized e.g. by prescribing that any of the following expressions is finite: Here, we will extend this list.
The paper is organized as follows: In Section 2 we formulate an equivalence theorem of independent interest, and in Section 3 we use this equivalence theorem to describe some new scales of weight characterization of the Hardy inequality.The main result is formulated in Theorem 3.1, which includes the results mentioned in (1.5) but gives also ten new weight characterizations.In Section 4 we give some outlines of the proof of the equivalence theorem ( Theorem 2.1), whose detailed proof can be found in the research note [1].

The equivalence theorem
are mutually equivalent.The constants in the equivalence relations can depend on α, β and s.
Remark 2.1.The proof of Theorem 2.1 (see [1] and Section 4)is carried out by determining positive constants c i and d i so that

The main result
Theorem 3.1.Let 1 < p ≤ q < ∞ , 0 < s < ∞, and define, for the weight functions u, v, the functions U and V by (1.4), and the functions A i (s), i = 1, 2, . . ., 15, as follows (3.1) Then the Hardy inequality (1.1) holds for all measurable functions f ≥ 0 if and only if any of the quantities A i (s) is finite.Moreover, for the best constant C in (1.1) we have C ≈ A i (s), i = 1, 2, 3, . . ., 14.
Remark 3.1.The conditions in (1.5) can be described in the following way: Hence, Theorem 3.1 generalizes the corresponding results in [2], [6] and also all previous results of this type.

Outlines of the proof of the equivalence theorem
In the proof, which is rather technical, we use -among other tools -the fact that the function F from (2.1) is decreasing and the function of G from (2.1) is increasing, and that Moreover, the equivalences have been proved in [2, Theorem 2.1], so that it is remains to prove the other 10 equivalences.
Here, we will give a detailed proof only for some equivalence, in order to show the typical steps used.As mentioned, full proofs can be found in [1].
(we have used the fact that G is increasing).Now we take the suprema for x ∈ (a, b) and have that (we have used the fact that B 1 (t, α, β) ≤ B 1 (α, β), and formula (4.1) for F with λ = − α α+s ).Taking the supremum on the left-hand side, we have that B 6 (α, β, s) ≤ 1 Then and (we have used the fact that G is increasing).Taking the supremum with respect to y (right) and x (left), we have that