New characterizations of weights on dynamic inequalities involving a Hardy operator

In this paper, we establish some new characterizations of weighted functions of dynamic inequalities containing a Hardy operator on time scales. These inequalities contain the characterization of Ariňo and Muckenhoupt when $\mathbb{T}=\mathbb{R}$
 T
 =
 R
 , whereas they contain the characterizations of Bennett–Erdmann and Gao when $\mathbb{T}=\mathbb{N}$
 T
 =
 N
 .


Introduction
In [10], Muckenhoupt characterized the weights such that the inequality 1/k holds for all measurable ≥ 0 and the constant C is independent of (here 1 < k < ∞). The characterization reduces to the condition that the nonnegative functions and ω satisfy and K ≤ C ≤ Kk 1/k (k * ) 1/k * . In [7], Bradley gave new characterizations of weights such that the general inequality holds for all measurable ≥ 0 and the constant C is independent of (here 1 ≤ k ≤ q ≤ ∞). The characterization reduces to the condition that the nonnegative functions and ω satisfy and K ≤ C ≤ Kk 1/q (k * ) 1/k * for 1 < k < q < ∞ and K = C if k = 1 and q = ∞.
In [3], Ariňo and Muckenhoupt characterized the weight function such that the inequality holds for all nonnegative nonincreasing measurable function on (0, ∞) with a constant C > 0 independent on (here 1 ≤ k < ∞). The characterization reduces to the condition that the function ϕ satisfies ϕ(x) dx for all ς ∈ (0, ∞) and B > 0.
In [12], Sinnamon characterized the weights such that the inequality holds for all measurable ≥ 0 and the constant C is independent of (here 0 < q < 1 < k and 1 r = 1 q -1 k ). The characterization reduces to the condition that the nonnegative functions and ω satisfy b For the discrete case, however, Bennett and Erdmann [4] characterized the weights such that the inequality holds for all nonnegative nonincreasing sequence z n . The characterization reduces to the condition that the nonnegative sequence ϕ n satisfies ϕ k for all n ∈ N and B > 0.
In [8], Gao extended the results of Bennett and Erdmann and characterized the weights such that the inequality holds for all nonnegative and nonincreasing sequences z n and a n with a 1 > 0, where the constant C is independent of a n and z n . The characterization reduces to the condition that the nonnegative sequences a n and ϕ n satisfy ϕ k for all n ∈ N and B > 0, where A n = n k=1 a k . In this paper, we are concerned with proving some dynamic inequalities on time scales; see [1,2]. The general idea is to prove our results where the domain of the unknown function is a so-called time scale T, which is an arbitrary nonempty closed subset of the real numbers R. In [11], the authors characterized the weights such that the dynamic inequality on a time scale T holds for all nonnegative rd-continuous function on The characterization reduces to the condition that the nonnegative functions and υ satisfy Moreover, the estimate for the constant C in (4) is given by As a particular case of (4) if k = q, (ς) = ϕ(ς)(σ (ς)ς 0 ) -k and υ = ϕ, then we get the inequality and the characterization reduces to the condition that the nonnegative function ϕ satisfies Our aim in this paper is to establish some new characterizations of the weights for the dynamic inequalities of the form (5) and for the general inequalities of the form where ϒ(ς) = ς ς 0 ψ(x) x, 1 < k < ∞, and c > 0.
The paper is organized as follows. In Sect. 2, we present some definitions and basic concepts of time scales and prove essential lemmas needed in Sect. 3 where the main results are proved. Our findings significantly recover particular cases. Indeed, the proposed theorems contain the characterizations of inequalities (2) and (3) proved by Bennett and Erdmann and Gao when T = N, whereas they give the characterizations of inequality (1) proved by Ariňo and Muckenhoupt when T = R.

Preliminaries and basic lemmas
For completeness, we recall the following concepts related to the notion of time scales. We refer the reader to the two books by Bohner and Peterson [5,6]. A time scale T is an arbitrary nonempty closed subset of the real numbers R.
We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. The forward jump operator and the backward jump operator are defined by: σ (ς) := inf{s ∈ T : s > ς} and ρ(ς) := sup{s ∈ T : s < ς}, respectively. A point ς ∈ T is said to be left-dense if ρ(ς) = ς , right-dense if σ (ς) = ς , left-scattered if ρ(ς) < ς , and right-scattered if σ (ς) > ς . A function z : T → R is said to be right-dense continuous (rd-continuous) provided z is continuous at right-dense points and at left-dense points in T, left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by C rd (T, R).

and the integration by parts formula on time scales is given by
The time scales chain rule (see [5,Theorem 1.87]) is given by where it is assumed that z : R → R is continuously differentiable and δ : T → R is delta differentiable. A simple consequence of Keller's chain rule [5, Theorem 1.90] is given by The Hölder inequality, see [5,Theorem 6.13], on time scales is given by where ς 0 , b ∈ T, , z ∈ C rd (I, R), γ > 1, and 1 γ + 1 ν = 1. The special case γ = ν = 2 in (11) yields the time scales Cauchy-Schwarz inequality.
Throughout the paper, we assume (without mentioning) that the functions are nonnegative rd-continuous functions on [ς 0 , ∞) T and the integrals considered are assumed to exist (finite i.e. convergent). We define The following lemma is adopted from [11].
The following lemmas are needed in Sect. 3.

Lemma 2.2 Assume that ϕ, ψ are nonnegative rd-continuous functions defined on
Using ω(ς 0 ) = ϒ(∞) = 0 (recall all integrals are assumed to be convergent), we obtain that The proof is complete.
where z is a nonincreasing function.
Remark 2.1 Note in the above lemma that

Main results
This section is devoted to state and prove our main results.
Furthermore, assume that is nonincreasing and Proof Suppose that (18) holds. Apply Lemma 2.1 with z = , and we have that Substituting (20) into the left-hand side of (19), we get that Applying Lemma 2.2, with on the right hand side of (21), we see that Using the additive property of integrals [5, Theorem 1.77(iv)] on time scales, we have for Substituting (18) into the right-hand side of (23), we see that Applying integration by parts formula (8) on the term where υ(s) = s ς 0 (σ (x)ς 0 ) k-1 x. Using ϒ σ (ς) = 0 and υ(ς 0 ) = 0, we have that Substituting (25) into the right-hand side of (24), we get that Since inequality (26) becomes Applying Lemma 2.2 on the term From (27) and (29), we have Putting ψ(x) = 1 in Lemma 2.4, we see that the function is nonincreasing. Now, by applying Lemma 2.3, with we have from (28) and (30) that Substituting (31) into the right-hand side of (22), we see that Applying Hölder's inequality (11) on the term with indices k and k/(k -1), we get that Finally substituting (33) into (32), we have that This implies that which is (19). The proof is complete.

Thus we get the inequality
Remark 3.6 In the case when T = R and ς 0 = 0, then K = 1 in the previous remark and from (38) we have which is a Hardy-type inequality with constant (k 2 /(k -1)) k (see [9]).
Remark 3.7 In Theorem 3.2 we could replace is rd-continuous with is integrable.
Remark 3.8 Suppose that is integrable, and also assume that