New characterization of weighted inequalities involving superposition of Hardy integral operators

. Let 1 ≤ p < ∞ and 0 < q, r < ∞ . We characterize the validity of the inequality for the composition of the Hardy operator,


Introduction and the main results
In operator theory, weighted inequalities involving operator composition may be found in various topics.Let 0 < q, r < ∞ and 1 ≤ p < ∞.The validity of inequalities w(x)dx for all h ∈ M + (0, ∞) are crucial because many classical inequalities can be reduced to them.For example, duality techniques reduce the embeddings between Lorentz-type spaces, Morreytype spaces, and Cesáro-type spaces to the weighted iterated inequalities (see, e.g., [4,5,7,9,34]).On the other hand, characterizations of weighted bilinear Hardy and Copson inequalities reduce to the characterizations of iterated Hardy inequalities (see, e.g., [1,23,33]).
Various approaches have been used to handle inequalities (1.1) and (1.2) resulting in conditions of a different nature.Inequality (1.1) is investigated thoroughly.The most recent results are presented in the paper [20].Alternative characterizations of inequalities (1.1) and (1.2) are found in [19].Detailed information on the development and history of this inequality may be found in the recent paper [6].
(1.3) Inequality (1.3) was completely characterized in [26,3,28,32] when 1 ≤ r, p < ∞.However, for a long period, there was no adequate characterization in the case when 0 < r < 1 ≤ p < ∞.Several attempts have been made to tackle this case (see, e.g., [32,25,29,13]); in some works, necessary and sufficient conditions did not match, while in others, characterization had a discrete form or involved auxiliary functions.Hence, it was not easily verifiable.Finally, in [22], the missing integral conditions were provided.
In the general cases, (1.2) is characterized in [8], but the conditions are in a non-standard form.It was also considered in [30], but the conditions are not applicable because they involve auxiliary functions.Recently, in [24], a more complicated discretization method is used to establish a characterization of the same inequality that involves iteration of the Copson operators and is restricted to non-degenerate weights, and the case p = 1 is presented without a proof.In our approach, the case p > 1 is not separated from p = 1.
Recently, in [6], with a new and simpler discretization technique that requires neither parameter restrictions nor non-degeneracy conditions, characterization of (1.1) is given.We adapt this approach to the specific demands of the inequality considered in this paper.Our technique allows us to obtain a scale of characterization which was not possible before; moreover, we obtain the previous characterizations of inequality (1.2) as a special case (more, precisely for β = 0, see Theorem 1.1).
We would like to point out that the characterization of Hardy inequality involving nondecreasing functions, that is, is obtained directly from inequality (1.2) without any further work (see the proof of Theorem 1.2) as a direct outcome of our main theorem (see Theorem 1.1).We should also mention that in [12,13], using reduction techniques, (1.4) is reduced to inequality (1.3).However, as we have already mentioned, at that point of time the characterizations of the reduced inequalities were not known.Corollary 3.2 from [13] provides a characterization of (1.4) but the result is non-standard.The earlier works on inequality (1.4) can be found in [17,27,14,15].
We should note that Theorem 1.1 allowed us to give a scale of characterizations of (1.4), and we would like to provide the characterization here to integrate all relevant parameter choices into a single theorem for the reader's convenience (see, Theorem 1.2).
As one can see in Section 3, the discretization method transforms the inequality at hand equivalently to discrete inequalities that involve local characterizations of inequalities having low-order iterations.For this very reason, our aim in this paper is to revisit inequality Let −∞ ≤ a < b ≤ ∞ and a weight be a positive measurable function on (a, b).The principal goal of this study is to determine the scale of necessary and sufficient conditions on weights u, v, w on (a, b) for which It is worth noting that if p < 1, inequality (1.5)only holds for trivial functions.
Let us first review the essential notations and conventions before presenting our main results.The left and right sides of the inequality numbered by ( * ) are denoted by LHS( * ) and RHS( * ), respectively.We put 0.∞ = ∞/∞ = 0/0 = 0.The symbol A B means a constant c > 0 exists such that A ≤ cB where c depends only on the parameters p, q, r.If both A B and B A, then we write A ≈ B.
For 1 ≤ p < ∞, non-negative measurable functions v and w on (a, b), and x, y ∈ [a, b], denote by and Now, we are ready to formulate our main results.
where C 1 is defined in (1.7).Moreover, the best constant C in inequality (1.5) satisfies (iv) r < p, q < p, C 3 < ∞ and where C 3 is defined in (1.8).Moreover, the best constant C in inequality (1.5) satisfies holds for all f ∈ M ↑ (a, b) if and only if (i) p ≤ q, p ≤ 1 and Moreover, the best constant C in inequality (1.9) satisfies C ≈ C 2 + C 3 .
(iii) 1 < p ≤ q, C 1 < ∞ and where C 1 is defined in (1.10).Moreover, the best constant C in inequality (1.9) satisfies (iv) q < p, 1 < p, C 3 < ∞ and where C 3 is defined in (1.11).Moreover, the best constant C in inequality (1.9) satisfies Proofs of Theorem 1.1 and Theorem 1.2 will be given in Section 4.

Preliminary Results
In this section, we cover the foundations of discretization and several new results that will be employed frequently throughout the proof of the main theorem.
Definition 2.1.Let N ∈ Z ∪ {−∞}, M ∈ Z ∪ {+∞}, N < M , and {a k } M k=N be a sequence of positive numbers.We say that {a k } M k=N is geometrically decreasing if and for all non-negative sequences {a k } ∞ k=n .
k=n+1 is a geometrically decreasing sequence, then, and for all non-negative measurable g on (a, b).
Proof.For each n ∈ Z ∪ {−∞}, we can write Then (2.4) k=n+1 is a geometrically decreasing sequence, and {σ k } ∞ k=n is a positive non-decreasing sequence.Then and hold for all non-negative measurable g on (a, b).
Proof.Let us start with the equivalency (2.7).Since {τ k } ∞ k=n+1 is a geometrically decreasing sequence, interchanging supremum and (2.4) give Interchanging supremum once again and monotonicity of {σ k } ∞ k=n results in

7).
Let us now tackle (2.8).Monotonicity of {σ k } ∞ k=n gives that Then, using (2.2), we have the following upper estimate On the other hand, the reverse estimate is clear, and the proof is complete.
Definition 2.5.Let w be a non-negative measurable function on (a, b) Remark 2.6.Let w be a non-negative measurable function on (a, b) such that 0 < W(t) < ∞ for all t ∈ (a, b).The Darboux property of continuous functions implies that the discretizing sequence exists for W. We can define the discretizing sequence in the following way: k=N is a discretizing sequence of the function W. Then for any n : and ess sup hold for all non-negative and non-decreasing h on (a, b).

Proof. Monotonicity of h and properties of the discretizing sequence {x
and, conversely Thus, (2.9) holds.
On the other hand, similarly, k=N and {v k } M k=N be two sequences of positive real numbers.Then inequality holds for every sequence {a k } M k=N of non-negative real numbers if and only if either Moreover, if we denote by C the optimal embedding constant in (2.11), then (2.12) holds for every sequence {x k } M k=N of non-negative real numbers if and only if (i) either p ≤ 1, p ≤ q and (iii) or 1 < p, q < p and Moreover, if we denote by C the best constant in (2.12), then Moreover, the multiplicative constants in all the equivalences above depend only on p, q.

Discrete Characterization
We begin this section by observing that inequality (1.5) is equivalent to two other discrete inequalities, and we present the characterization in discrete form, which is noteworthy on its own.Let us start with the discretization of inequality (1.5).To this end we need the following notation, denote by M (x k−1 , x k ) the best constant of weighted Hardy inequality, that is, and using the classical characterizations of weighted Hardy inequalities (see, [21,31]), we have (1.6) can be expressed as ] is a discretizing sequence of the function W.Then, there exists a positive constant C such that inequality (1.5) holds for all f ∈ M + (a, b) if and only if there exist positive constants C ′ and C ′′ such that Proof.Applying (2.9) with α = 1 and h(x) = x a t a f q u(t)dt r q , we have that .
Then, it is clear that there exists a positive constant C such that inequality (1.5) holds for all f ∈ M + (a, b) if and only if there exist positive constants C ′ and C ′′ such that (3.2) and (3. Then there exists a positive constant C ′ such that inequality (3.2) holds for all f ∈ M + (a, b) if and only if there exists a positive constant C ′ such that holds for every sequence of non-negative numbers {a k } ∞ k=N +1 .Moreover the best constants C ′ and C ′ , respectively, in (3.2) and (3.4) satisfy C ′ ≈ C ′ .
Proof.Assume that (3.2) holds.By the definition of B(x k−1 , x k ), there exist non-negative measurable functions Conversely, observe first that for each Then (3.2) follows by, inserting a k = ( Then there exists a positive constant C ′′ such that inequality (3.3) holds for all f ∈ M + (a, b) if and only if there exists a positive constant C ′′ such that Proof.Suppose that (3.3) holds for all f ∈ M + (a, b).Using (3.5), there exist non-negative measurable functions Thus, inserting f = ∞ m=N +1 a m g m , where {a m } ∞ m=N +1 is any sequence of non-negative numbers, into (3.3),(3.5) follows.Moreover, C ′′ C ′′ holds.
Conversely, taking a k = ( p in (3.5) and using the estimate Theorem 3.4.Let 1 ≤ p < ∞, 0 < q, r < ∞, −∞ < β < 1 and let u, v, w be weights on (a, b) such that 0 < W(t) < ∞ for all t ∈ (a, b), and {x k } ∞ k=N +1 be the discretizing sequence of W. Then inequality (3.4) holds for every sequence of non-negative numbers {a k } ∞ k=N +1 if and only if (i) p ≤ r, p ≤ q and (iii) q < p ≤ r and (iv) r < p, q < p and Moreover, the best constant C ′ in inequality (3.4) satisfies A 2 in the case (ii), A 3 in the case (iii), A 4 in the case (iv).
Proof.The result follows easily by combining Theorem 2.8 for suitable parameters and weights with (3.1) and the fact that Theorem 3.5.Let 1 ≤ p < ∞, 0 < q, r < ∞, −∞ < β < 1 and let u, v, w be weights on (a, b) such that 0 < W(t) < ∞ for all t ∈ (a, b), and {x k } ∞ k=N +1 be the discretizing sequence of W. Then inequality (3.5) holds for every sequence of non-negative numbers {a k } ∞ k=N +1 if and only if (i) p ≤ r and (ii) r < p and Moreover, the best constant C ′′ in inequality (3.5) satisfies Proof.We will apply Theorem 2.9 with suitable parameters and weights a k = 2 −k x k+1 x k u r q and b k = V p (x k−1 , x k ).We need to treat the cases when p > 1 and p = 1, separately.Note that when p > 1, since {x k } ∞ k=N +1 is the discretizing sequence of W, we have for each k : Thus, applying [Theorem 2.9, (iv)] when 1 < p ≤ r and [Theorem 2.9, (iii)] when 1 < p, r < p, the result follows.
On the other hand, if p = 1, for each k : Therefore, applying [Theorem 2.9, (ii)] when r < p = 1, we have Lastly, if p = 1 ≤ r, then applying [Theorem 2.9, (i)], and the fact that W(t) ≈ 2 −i , for every t ∈ [x i , x i+1 ] we have Finally, interchanging supremum yields that We can formulate the discrete characterization of inequality (1.5).
Moreover the best constant C in (1.5) satisfies

Proofs
Proof of Theorem 1.1 (i) Let 1 ≤ p ≤ min{r, q}.We have from [Theorem 3.6, (i)] that C ≈ A 1 + B 1 .We will prove that C 1 ≈ A 1 + B 1 .First, we will show that A 1 + B 1 ≈ A 1 + B 1 , where It is clear that A 1 ≤ A 1 .On the other hand, observe that Then, interchanging the supremum in the first term and applying (2.7) with n = N + 2, for the second term, we have that Note that, for any k ≥ N + 2, we have Then, in view of (4.1), V p (a, x k−1 ) Then we have that

It remains to show that
holds, and interchanging supremum gives that On the other hand, applying (2.9) with α = 1 − β, then using (2.5) with n = k, we obtain for Therefore, in view of (4.3), Thus, combining (4.2) with (4.4), we have that Conversely, using (4.1), we have V p (a, x k−1 ).
Then, using (4.3), we arrive at As a result, we arrive at the conclusion that the best constant C in (1.5) satisfies C ≈ C 1 .
(ii) Let r < p ≤ q.Then, we have from [Theorem 3.6, (ii)] that the best constant in (1.5) satisfies C ≈ A 2 + B 2 .We will start by showing that

.6)
We have A 2 ≤ A 2 by the definitions of A 2 and A 2 .On the other hand, Next, we will show that 3) for the first term and (2.8) for the second term, we obtain that

Using (4.1) we arrive at
Furthermore, it is clear from the definitions of B 2 and B 2 that B 2 ≤ B 2 .Then, we have Next, we will prove that On the other hand, using (4.3), Combination of (4.8) and (4.9) yield that Conversely, In view of (4.10), we get Furthermore, using (4.10) once more, Since, the sequence {a k } ∞ k=N +1 , with Then, we have Thus, (4.3) ensures that As a result, {y k } ∞ k=N +1 is a discretizing sequence of W, as well.This fact together with (4.7) yield On the other hand, applying (2.8) with Lastly, using (4.3) and (4.7), we obtain that Therefore, (iii) Let q < p ≤ r.According to [Theorem 3.6, (iii)], the best constant in (1.5) satisfies C ≈ A 3 + B 1 .We will begin our proof by showing that A 3 + B 1 ≈ A 3 + B 1 , where It is clear that A 3 ≤ A 3 , the proof of this part is complete if we show that A 3 . Therefore, we have (4.12) In that case, integration by parts gives Moreover, Minkowski's inequality with p p−q > 1 yields that .
Then, we arrive at Now, we are in a position to find the upper estimate for A 3 .Using (4.13), we have that Further, (2.1) yields, Integrating by parts again, we have that Lastly, we will find a suitable upper estimate for A 3,3 .To this end, we will treat the cases Then, then using (4.16) together with (2.7), we get V p (a, t) V p (a, x N +1 ).V p (a, t) V p (a, x i ).
Then, using (4.18) for the first term and applying (2.7) for the second term, we get V p (a, x k−1 ).
Finally, using (4.1), we arrive at V p (a, x k−1 ) A 3 + B 1 .(iv) Let max{r, q} < p.Then, using [Theorem 3.6, (iv)], we have that C ≈ B 2 + A 4 .First of all, we will show that B 2 + A 4 ≈ B 2 + A 4 , where It is clear that A 4 ≤ A 4 .We have already shown in (4.7) that B 2 A 2 + B 2 .Moreover, analogously as in the previous proof, using (4.18), one can easily see that A 2 A 4 .Thus, B 2 + A 4 B 2 + A 4 follows.

Consequently
It remains to prove that B 2 + A 4 B 2 + A 4 .Assume that max{A 4 , B 2 } < ∞.Then, using the same steps as in the previous case, we can see that (4.12) holds; therefore, (4.13) is true in this case, as well.
Applying (4.13) combined with (2.5), we obtain that As in the proof of the previous case, using integration by parts in combination with (4.1), we have that Assume that (1.9) holds for all f ∈ M ↑ (a, b).Substituting f (x) = Conversely, assume that (4.21) holds for all h ∈ M + (a, b).Since any f ∈ M ↑ (a, b), even if f (0) > 0, can be approximated pointwise from below by a function of the form f (x) p = x a h, x ∈ (a, b), then the validity of (4.21) yields (1.9).Therefore, the result follows from Theorem 1.1.
Let −∞ ≤ a < b ≤ ∞.Denote by M + (a, b) the set of all non-negative measurable functions on (a, b) and M ↑ (a, b) is the class of non-decreasing elements of M + (a, b).

Proof.
According to Lemmas [3.1-3.3],inequality (1.5) holds if and only if inequalities (3.4) and (3.5) hold.Moreover, the best constant C in (1.5) satisfies C ≈ C ′ + C ′′ , where C ′ and C ′′ are the best constants in inequalities (3.4) and (3.5), respectively.The result follows from the combination of Theorem 3.4 and Theorem 3.5.

and B 2
is defined in (4.6).

A 4
t)V p (a, t) pq p−q dt r(p−q) q(p−r) p−r prProof of Theorem 1.2 We will prove that inequality (1.9) holds for all f ∈ M ↑ (a, b) if and only if inequality h ∈ M + (a, b).

x a h 1 p
, x ∈ (a, b) for h ∈ M + (a, b) in (1.9) and applying Fubini's theorem on the right-hand side, (4.21) follows.