Some new iterated hardy-type inequalities: The case $\theta = 1$

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p<\infty$, $0<q\leq +\infty$, $u$, $w$ and $v$ are weight functions on $(0,\infty)$. It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator.


Introduction
Throughout the paper we assume that I := (a, b) ⊆ (0, ∞). By M(I) we denote the set of all measurable functions on I. The symbol M + (I) stands for the collection of all f ∈ M(I) which are non-negative on I, while M + (I, ; ↓) is used to denote the subset of those functions which are non-increasing on I. The family of all weight functions (also called just weights) on I, that is, locally integrable non-negative functions on (0, ∞), is denoted by W(I).
For p ∈ (0, +∞] and w ∈ M + (I), we define the functional · p,w,I on M(I) by If, in addition, w ∈ W(I), then the weighted Lebesgue space L p (w, I) is given by L p (w, I) = {f ∈ M(I) : f p,w,I < +∞} and it is equipped with the quasi-norm · p,w,I . When w ≡ 1 on I, we write simply L p (I) and · p,I instead of L p (w, I) and · p,w,I , respectively.
Everywhere in the paper, u, v and w are weights. We denote by and assume that U (t) > 0 for every t ∈ (0, ∞).
We pronounce that the characterization of the inequality (1.1) is important because many inequalities for classical operators can be reduced to this form. Just to illustrate this important fact we give two applications in Section 5 of the obtained results. Firstly, we present some new characterizations of weighted Hardy-type inequalities restricted to the cone of monotone functions (see Theorems 5.3 and 5.4). Secondly, we point out boundedness results in weighted Lebesgue spaces concerning the weighted Stieltjes's transform (see Theorems 5.6 and 5.7). Here we also need to prove some reduction theorems of independent interest (see Theorems 5.1, 5.2 and 5.5).
Our approach is based on discretization and anti-discretization methods developed in [8], [9], [11] and [13]. Some basic facts concerning these methods and other preliminaries are presented in Section 2. In Section 3 discretizations of the inequalities (1.1) are given. Antidiscretization of the obtained conditions in Section 3 and the main results (Theorems 4.1, 4.2 and 4.3) are stated and proved in Section 4. Finally, the described applications can be found in Section 5.

Notations and Preliminaries
Throughout the paper, we always denote by c or C a positive constant, which is independent of the main parameters but it may vary from line to line. However a constant with subscript such as c 1 does not change in different occurrences. By a b, (b a) we mean that a ≤ λb, where λ > 0 depends only on inessential parameters. If a b and b a, we write a ≈ b and say that a and b are equivalent. Throughout the paper we use the abbreviation LHS( * ) (RHS( * )) for the left (right) hand side of the relation ( * ). By χ Q we denote the characteristic function of a set Q. Unless a special remark is made, the differential element dx is omitted when the integrals under consideration are the Lebesgue integrals. (ii) If p ∈ [1, +∞], we define p ′ by 1/p + 1/p ′ = 1. Moreover, we put p * = p 1−p if p ∈ (0, 1) and p * = +∞ if p = 1.
(iii) If I = (a, b) ⊆ R and g is a monotone function on I, then by g(a) and g(b) we mean the limits lim x→a+ g(x) and lim x→b− g(x), respectively.
In the paper we shall use the Lebesgue-Stieltjes integral. To this end, we recall some basic facts.
Let ϕ be non-decreasing and finite function on the interval I := (a, b) ⊆ R. We assign to ϕ the function λ defined on subintervals of I by The function λ is a non-negative, additive and regular function of intervals. Thus (cf. [23], Chapter 10), it admits a unique extension to a non-negative Borel measure λ on I. The formula (2.2) imply that Note also that the associated Borel measure can be determined, e.g., only by putting for any [y, z] ⊂ I z y f dϕ) coincides with the Lebesgue-Stieltjes integral (y,z] f dϕ. In particular, if f, g ∈ C(I) and ϕ is non-decreasing on I, then the Riemann-Stieltjes integral [y,z] f dϕ coincides with the Lebesgue-Stieltjes integral (y,z] f dϕ for any [y, z] ⊂ I. Let us now recall some definitions and basic facts concerning discretization and antidiscretization which can be found in [8], [9] and [11]. Let U be an admissible function. We say that a function ϕ is U -quasiconcave if ϕ is equivalent to an increasing function on (0, ∞) and ϕ U is equivalent to a decreasing function on (0, ∞). We say that a U -quasiconcave function ϕ is non-degenerate if The family of non-degenerate U -quasiconcave functions will be denoted by Ω U . We say that ϕ is quasiconcave when ϕ ∈ Ω U with U (t) = t. A quasiconcave function is equivalent to a concave function. Such functions are very important in various parts of analysis. Let us just mention that e.g. the Hardy operator Hf (x) = x 0 f (t)dt of a decreasing function, the Peetre K-functional in interpolation theory and the fundamental function χ E X , X is a rearrangement invariant space, all are quasiconcave. Definition 2.3. Assume that U is admissible and ϕ ∈ Ω U . We say that {x k } k∈Z is a discretizing sequence for ϕ with respect to U if (i) x 0 = 1 and U (x k ) ↑↑; Let us recall ( [8], Lemma 2.7) that if ϕ ∈ Ω U , then there always exists a discretizing sequence for ϕ with respect to U . Definition 2.4. Let U be an admissible function and let ν be a non-negative Borel measure on [0, ∞). We say that the function ϕ defined by is the fundamental function of the measure ν with respect to U . We will also say that ν is a representation measure of ϕ with rspect to U . We say that ν is non-degenerate with respect to U if the following conditions are satisfied for every t ∈ (0, ∞): We recall from Remark 2.10 of [8] that , Lemma 1.5). Let p ∈ (0, ∞). Let u, w be weights and let ϕ be defined by Then ϕ is the least U for any non-negative measurable h on (0, ∞). Further, for t ∈ (0, ∞) Theorem 2.11). Let p, q, r ∈ (0, ∞). Assume that U is an admissible function, ν is a non-negative non-degenerate Borel measure on [0, ∞), and ϕ is the fundamental function of ν with respect to U q and σ ∈ Ω U p . If {x k } is a discretizing sequence for ϕ with respect to U q , then Corollary 2.13). Let q ∈ (0, ∞). Assume that U is an admissible function, f ∈ Ω U , ν is a non-negative non-degenerate Borel measure on [0, ∞) and ϕ is the fundamental function of ν with respect to U q . If {x k } is a discretizing sequence for ϕ with respect to U q , then Lemma 3.5). Let p, q ∈ (0, ∞). Assume that U is an admissible function, ϕ ∈ Ω U q and g ∈ Ω U p . If {x k } is a discretizing sequence for ϕ with respect to U q , then sup t∈(0,∞) We shall use some Hardy-type inequalities in this paper. Denote by These well-known results can be found in Maz'ya and Rozin [17], Sinnamon [21], Sinnamon and Stepanov [22] (cf. also [18] and [14]).
We shall also use the following fact (cf. [3], p. 188): Finally, if q ∈ (0, +∞] and {w k } = {w k } k∈Z is a sequence of positive numbers, we denote by ℓ q ({w k }, Z) the following discrete analogue of a weighted Lebesgue space: if 0 < q < +∞, then If w k = 1 for all k ∈ Z, we write simply ℓ q (Z) instead of ℓ q ({w k }, Z). We quote some known results. Proofs can be found in [15] and [16]. for all non-negative sequences {a k } k∈Z .
We shall use the following inequality, which is a simple consequence of the discrete Hölder inequality: Given two (quasi-)Banach spaces X and Y , we write X ֒→ Y if X ⊂ Y and if the natural embedding of X in Y is continuous.
The following two lemmas are discrete version of the classical Landau resonance theorems. Proofs can be found, for example, in [8].
where 1/r := 1/q − 1/p and C stands for the norm of the inequality (2.13).

Discretization of Inequalities
In this section we discretize the inequalities We start with inequality (3.1). At first we do the following remark.
Remark 3.1. Let ϕ be the fundamental function of the measure w(t)dt with respect to U q p , that is, Assume that w(t)dt is non-degenerate with respect to U q p . Then ϕ ∈ Ω U q p , and therefore there exists a discretizing sequence for ϕ with respect to U q p . Let {x k } be one such sequence.
Next, we state a necessary lemma which is also of independent interest.
Lemma 3.1. Let 0 < q < ∞, 0 < p < ∞, 1/ρ = (1/q − 1) + , and let u, v, w be weights. Assume that u is such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q p . Let {x k } be any discretizing sequence for ϕ defined by (3.3). Then inequality

4)
and the best constant in inequality (3.1) satisfies Proof. By using Lemma 2.2 with .
Moreover, by using Lemma 2.5, By now using the fact that .
Consequently, by using Lemma 2.5 on the second term, To find a sufficient condition for the validity of inequality (3.1), we apply to I locally (that is, for any k ∈ Z) the Hardy-type inequality Thus, in view of inequality (2.12), we have that For II, by inequalities (2.11) and (2.12), we get that Combining (3.7) and (3.8), in view of (3.5), we obtain that Consequently, (3.1) holds provided that A < ∞ and c ≤ A.
Next we prove that condition (3.4) is also necessary for the validity of inequality (3.1). Assume that inequality (3.1) holds with c < ∞. By (2.8) Define g k , k ∈ Z, as the extension of h k by 0 to the whole interval (0, ∞) and put where {a k } k∈Z is any sequence of positive numbers. We obtain that Therefore, by (3.1), (3.13) and (3.14), we arrive at and Proposition 2.1 implies that On the other hand, there are ψ k ∈ M + (I k ), k ∈ Z, such that and Define f k , k ∈ Z, as the extension of ψ k by 0 to the whole interval (0, ∞) and put where {b k } k∈Z is any sequence of positive numbers. We obtain that .
Note that Then, by (3.1) and previous two inequalities, we have that Inequalities (3.16) and (3.20) prove that A c.
Before we proceed to inequality (3.2) we make the following remark.
The following lemma is proved analogously, and for the sake of completeness we give the full proof.
To find a sufficient condition for the validity of inequality (3.2), we apply to III locally the Hardy-type inequality (3.6). Thus For IV we have that Combining (3.23) and (3.24), in view of (3.22), we get that Consequently, inequality (3.2) holds provided that D < ∞, and c D.

27)
and Proposition 2.1 implies that On the other hand, accordingly to (3.17), (3.18) and (3.19), we obtain that Since, in view of (3.2) and previous two inequalities, we have that Proposition 2.1 implies that Finally, inequalities (3.28) and (3.29) imply that D c.
Remark 3.3. In view of (2.11) and Lemma 2.5, it is evident now that .
Monotonicity of v(t, ∞) implies that is geometrically increasing, we obtain that This inequality shows that lim t→∞ v(t, ∞) must be equal to 0, because ϕ(∞) is always equal to ∞ by our assumptions on the function ϕ. Therefore, in the remaining part of the paper we consider weight functions v such that lim t→∞ v(t, ∞) = 0.

Anti-dicretization of Conditions
In this section we anti-discretize the conditions obtained in Lemmas 3.1 and 3.2. We distinguish several cases.
Lemma 4.1. Let 0 < q < ∞, 0 < p < 1, 1/ρ = (1/q − 1) + , and let u, v, w be weights. Assume that u be such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q p . Let {x k } be any discretizing sequence for ϕ defined by (3.3). Then .

Proof. By Lemma 2.4, in this case it yields that
Therefore, in view of (2.11), Lemma 3.1, we have that .

It is easy to see that
On the other hand,

Lemma 4.2. Assume that the conditions of Lemma 4.1 are fulfilled. Then
Proof. Evidently, A 1 ≤ A 2 . Using integrating by parts formula (2.6), we have that .

Again integrating by parts we have that
. Since we obtain that A 2 A 1 .

Lemma 4.3. Assume that the conditions of Lemma 4.1 are fulfilled. Then
where Proof. Integrating by parts, in view of inequality (4.1) and Lemma 4.2, we have that On the other hand, again integrating by parts, we get that

Lemma 4.4. Assume that the conditions of Lemma 4.1 are fulfilled. Then
where .
Proof. By Lemma 2.5, in view of Remark 3.3, we have that .

Lemma 4.5. Assume that the conditions of Lemma 4.1 are fulfilled. Then
.
Proof. By Lemma 2.5, we have that . Hence We are now in position to state and prove our first main theorem.
Theorem 4.1. Let 0 < p < 1, 0 < q < ∞, and let u, v, w be weights. Assume that u is such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q p . (i) Let 1 ≤ q < ∞. Then inequality (3.1) holds for every h ∈ M + (0, ∞) if and only if Moreover, the best constant c in (3.1) satisfies c ≈ I 1 .
Proof. The case 1 ≤ p < ∞, 0 < q < ∞. The following lemma is true. Lemma 4.6. Let 1 ≤ p < ∞, 0 < q < ∞ and let u, v, w be weights. Assume that u is such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q p . Let {x k } be any discretizing sequence for ϕ defined by (3.3). Then Proof. By Lemma 2.4, in this case we find that By using (2.11), in view of Lemma 3.1, we have that . Obviously, On the other hand, Lemma 4.7. Assume that the conditions of Lemma 4.6 are fulfilled. Then where .
Proof. Obviously, we obtain that Lemma 4.8. Assume that the conditions of Lemma 4.6 are fulfilled. Then where .
Proof. Obviously, On the other hand, by (4.2), we get that Thus Lemma 4.9. Assume that the conditions of Lemma 4.6 are fulfilled. Then .
Proof. By Lemma 2.5, we get that Our next main result reads: Theorem 4.2. Let 1 ≤ p < ∞, 0 < q < ∞, and let u, v, w be weights. Assume that u is such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q p . (i) Let 1 ≤ q < ∞. Then inequality (3.1) holds for every h ∈ M + (0, ∞) if and only if Moreover, the best constant c in (3.1) satisfies that c ≈ I 3 .
Proof. (i) The proof of the statement follows by combining Lemmas 4.6 -4.9, 2.3 and 2.1.
(ii) The proof of the statement follows by using Lemmas 4.6 -4.9, 2.1 and Theorem 2.1.
Lemma 4.10. Let 0 < p < ∞ and let u, v, w be weights. Assume that u is such that U is admissible. Let ϕ, defined by (2.7), be non-degenerate with respect to U 1 p . Let {x k } be any discretizing sequence for ϕ.
Proof. (i) The proof of the statement follows by using Lemmas 3.
holds for all h ∈ M + (0, ∞). Moreover, for the best constants c and C in ( Moreover, the best constant c in (5.1) satisfies that c ≈ C 1 .
Combining Theorems 5.2 and 4.3, we arrive at the following statement.
Moreover, the best constant c in (5.3) satisfies that c ≈ C 5 .
Now we consider the generalized Stieltjes transform S defined by for all h ∈ M + (0, ∞); the usual Stieltjes transform is obtained on putting U (x) ≡ x. In the case U (x) ≡ x λ , λ > 0, the boundedness of the operator S between weighted L p and L q spaces was investigated in [1] (when 1 ≤ p ≤ q ≤ ∞) and in [19], [20] (when 1 ≤ q < p ≤ ∞). This problem also was considered in [6] and [7], where completely different approach was used, based on the so call "gluing lemma" (see also [12]).
It is easy to see that Indeed, by Fubini's Theorem, we have that x ∈ (0, ∞).
Hence, we see that the inequality (5.5) is equivalent to the inequality (5.6).
Theorem 5.6. Let u, v, w be weights. Assume that u is such that U is admissible and the measure w(t)dt is non-degenerate with respect to U q . Let p, q ∈ (0, ∞]. When q < p < ∞, we set r = pq p−q . Moreover, the best constant c in (5.5) satisfies that c ≈ S 1 .
Moreover, the best constant c in (5.7) satisfies that c ≈ S 7 .
Moreover, the best constant c in (5.7) satisfies that c ≈ S 8 .