Some New Iterated Hardy-Type Inequalities

1 Institute of Mathematics, Academy of Sciences of Czech Republic, Zitna 25, 115 67 Prague 1, Czech Republic 2 Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, F. Agayev Street 9, 1141 Baku, Azerbaijan 3 Department of Mathematics, Faculty of Science and Arts, Kirikkale University, Yahsihan, 71450 Kirikkale, Turkey 4 Department of Mathematics, Luleå University of Technology, 971 87 Luleå, Sweden 5 Narvik University College, P.O. Box 385, 8505 Narvik, Norway


Introduction
Everywhere in the paper, u, v, and w are weights, that is, locally integrable nonnegative functions on 0, ∞ , and we denote

1.2
We assume that u is such that U t > 0 for every t ∈ 0, ∞ .For 0 < p < ∞ and w, a weight function on a, b ⊆ 0, ∞ , let us denote by L p,w a, b the weighted Lebesgue space defined as the set of all measurable functions u on a, b for which the quantity In this paper we characterize the validity of the inequality ∞ s h z dz p,u, 0,t q,w, 0,∞ ≤ c h θ,v, 0,∞ , where 0 < p < ∞, 0 < q ≤ ∞, 1 < θ ≤ ∞, u, w, and v are weight functions on 0, ∞ .Note that inequality 1.4 has been considered in the case p 1 in 1 see also 2 , where the result is presented without proof, in the case p ∞ in 3 and in the case θ 1 in 4, 5 , where weight functions v of special type were considered.For general weight functions v, the characterization of the inequality 1.4 in the case θ 1 does not follow directly by this method there are some technical problems and we are working on it.
It is worth to mention that, by Fubini's theorem, Hence, we see that the inequality 1.4 with p 1 is equivalent with the following inequality: where the operator S defined by for all nonnegative measurable functions h on 0, ∞ .We call this operator the generalized Stieltjes transform; 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 6 when 1 ≤ p ≤ q ≤ ∞ and in 7, 8 when 1 ≤ q < p ≤ ∞ .
Our approach is based on discretization and antidiscretization methods developed in 4, 9, 10 .Some basic facts concerning these methods and other preliminaries are presented in Section 2. The main results Theorems 3.1 and 3.2 are stated and proved in Section 3.
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 on inessential parameters.If a b and b a, we write a ≈ b and say that a and b are equivalent.We put 1/∞ 0, 0 • ∞ 0, 0/0 0, and ∞/∞ 0.

Preliminaries
Let us now recall some definitions and basic facts concerning discretization and antidiscretization which can be found in 4, 9, 10 .Definition 2.1.Let {a k } be a sequence of positive real numbers.One says that {a k } is strongly increasing or strongly decreasing and write a k ↑↑ or a k ↓↓ when Definition 2.2.Let U be a continuous strictly increasing function on 0, ∞ such that U 0 0 and lim t → ∞ U t ∞.Then One says that U is admissible.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 nondegenerate if The family of nondegenerate 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, for example, 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 .One says that {x k } k∈Z is a discretizing sequence for ϕ with respect to U if i x 0 1 and U x k ↑↑; ii ϕ x k ↑↑ and ϕ x k /U x k ↓↓; 4

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iii there is a decomposition Z Z 1 ∪Z 2 such that Z 1 ∩Z 2 ∅ and for every t

2.3
Let us recall see 9, 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 nonnegative Borel measure on 0, ∞ .We say that the function ϕ defined by is the fundamental function of the measure ν with respect to U. One will also say that ν is a representation measure of ϕ with respect to U. We say that ν is nondegenerate if the following conditions are satisfied for every t ∈ 0, ∞ :

2.5
We recall from 9, Remark 2.10 that Corollary 2.5 see 10, Lemma 1.5 .Let u, w be weights, and let ϕ be defined by Then ϕ is the least U

2.9
Theorem 2.6 see 9, Theorem 2.11 .Let p, q, r ∈ 0, ∞ .Assume that U is an admissible function, ν is a nonnegative nondegenerate 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

2.10
Lemma 2.7 see 9, Corollary 2.13 .Let q ∈ 0, ∞ .Assume that U is an admissible function, f ∈ Ω U , ν is a nonnegative nondegenerate 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 2.8 see 9, Lemma 3.5 .Let p, q, r ∈ 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 and {λ k } is a discretizing sequence of g with respect to U p , then

2.13
Lemma 2.10 see 9, Lemma 3.6 .Let q ∈ 0, ∞ .Assume that U is an admissible function, ν is a nondegenerate nonnegative Borel measure on 0, ∞ , ϕ is the fundamental function of ν with respect to U q , and f is a measurable function on 0,

2.14
Lemma 2.11 see 9, Lemma 3.7 .Let q ∈ 0, ∞ .Assume that U is an admissible function, ν is a nondegenerate nonnegative Borel measure on 0, ∞ , ϕ is the fundamental function of ν with respect to U q , and f is a measurable function on 0, ∞ .If {x k } is a discretizing sequence for ϕ with respect to U q , then 0,∞ ess sup f y q U −q y ϕ y .

2.15
Lemma 2.12 see 9, Lemma 3.8 .Let q ∈ 0, ∞ .Assume that U is an admissible function, ϕ ∈ Ω U q , {x k } is a discretizing sequence for ϕ with respect to U q , and f is a measurable function on 0, ∞ .Then

2.16
Lemma 2.13 see 9, Lemma 3.9 .Let U be an admissible function, ϕ ∈ Ω U , {x k } be a discretizing sequence for ϕ with respect to U, and f be a measurable function on 0, ∞ .Then

2.17
Proposition 2.14 see 9, Proposition 4.1 .Let {ω k } and {υ k }, k ∈ Z, be two sequences of positive real numbers.Let p, q ∈ 0, ∞ , and assume that the inequality is satisfied for every sequence {a k } of positive real numbers.
ii If p > q and r pq/ p − q , then
Then the inequality

holds for all nonnegative measurable h if and only if
A : sup and the best constant in 2.21 satisfies c ≈ A.
b Let 1 < θ < ∞, p ∞. Then the inequality 2.21 holds if and only if B : sup and the best constant in 2.21 satisfies c ≈ C.
and the best constant in 2.21 satisfies c ≈ D.
e Let θ ∞, 0 < p < ∞.Then the inequality 2.21 holds if and only if and the best constant in 2.21 satisfies c ≈ E.
These results are just classical results of Maz'ja 11 and Sinnamon 12 cf.also 13, 14 .

The Main Results
In this section we characterize the validity of the inequalities First we characterize 3.1 as follows.
Theorem 3.1.Let 0 < q < ∞, 0 < p < ∞, 1 < θ ≤ ∞, and let u, v, w be weights.Assume that u is such that U q/p is admissible and the measure w t dt is nondenerate with respect to U q/p .Then the inequality 3.1 holds for every measurable function f on 0, ∞ if and only if Moreover, the best constant c in 3.

3.5
Moreover, the best constant c in 3.
Moreover, the best constant c in 3.

3.7
Moreover, the best constant c in 3.
Moreover, the best constant c in 3.1 satisfies c ≈ A 5 .
Proof.Define ϕ x ∞ 0 U x, s q/p w s ds.

3.9
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.Then ϕ x k ↑↑ and ϕ x k U −q/p ↓↓.Furthermore, there is a decomposition For the left-hand side of 3.
Moreover, by using Lemma 2.9, we get that 3.12 By now using the fact that is, by using Lemma 2.9 on the second term,

3.14
Now we will distinguish several cases.We start with the case 1 < θ ≤ p < ∞.Then, by using Lemma 2.15, we get that Moreover, by applying H ölder's inequality for II, we find that 3.16 i In the case q/θ ≥ 1, according to 3.15 , we have that 3.17 Similarly, if q/θ ≥ 1, then, according to 3.16 , we obtain that and, finally, by using 3.9 , Lemma 2.13, and 3.14 , we get that

3.19
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ii For the case 0 < q < θ < ∞, l θq/ θ − q , by applying H ölder's inequality for sums to the right-hand side of 3.15 and 3.16 with exponents θ/q and l/q, we find that

3.20
Therefore, we get that 3.21 so that, in view of Lemma 2.11, Theorem 2.6, and 3.14 , x, t q/p w t dt l−q /q w x U x −l/p sup t∈ 0,x U t l/p V θ t l/θ dx 1/l h θ,v, 0,∞ .

3.25
Hence, using Lemmas 2.9 and 2.12, and 3.14 , we get that iv Next, we consider the case 0 < q < θ, 1/l 1/q−1/θ.By using Hölder's inequality for sums to the right-hand side of 3.23 and 3.24 with exponents θ/q and l/q, we get that

3.27
Journal of Function Spaces and Applications 15 Therefore, using Lemmas 2.9 and 2.10, Theorem 2.6, and 3.14 , we find that According to Lemma 2.15, we have that

3.29
Moreover, it yields that

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Hence, by integrating by parts, using Lemmas 2.9 and 2.10, and 3.14 , we get that

3.31
Now we prove the lower bounds necessity .Let 0 < q < ∞ and {x k } be a discretizing sequence for ϕ from 3.9 .Then, by 3.14 , we find that

3.32
Let 1 < θ ≤ p < ∞.For k ∈ Z, let h k be functions that saturate the Hardy inequality 2.21 and H ölder's inequality, that is, functions

3.33
Now we define the test function where {a k } is a sequence of positive real numbers.Thus, using test function 3.34 in 3.32 , we get that

3.35
Now using Proposition 2.14 for the case θ ≤ q, we obtain that

3.36
On the other hand, using Lemma 2.9, we get that

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Let 0 < q < θ < ∞.From 3.35 and Proposition 2.14, we obtain that

3.38
Since by Lemma 2.9, we arrive at

3.41
Now we define the test function where {a k } is a sequence of positive real numbers.Thus, using test function 3.42 in 3.32 , we get that

3.43
Now using Proposition 2.14 for the case θ ≤ q, we obtain that

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Since

3.46
Again integrating by parts, we arrive at

3.48
Since integrating by parts, we find that

3.50
Again integrating by parts, we arrive at

3.52
Now we define the test function where {a k } is a sequence of positive real numbers.Thus, using test function 3.53 in 3.32 , we get

3.54
Hence, by Proposition 2.14, we have that

3.55
On the other hand,

3.56
Integrating by part and using Lemma 2.9, we get that

3.57
The proof is complete.
We now state the announced characterization of 3.2 .
Theorem 3.2.Let 0 < p < ∞, 1 < θ ≤ ∞, and let u, v, w be weights.Assume that u is such that U 1/p is admissible and the measure w t dt is nondenerate with respect to U 1/p .Then the inequality 3.2 holds for every measurable function f on 0, ∞ if and only if

3.58
Moreover, the best constant c in 3.2 satisfies c ≈ B 1 .

3.59
Moreover, the best constant c in 3.2 satisfies c ≈ B 2 .

3.60
Moreover, the best constant c in 3.2 satisfies c ≈ B 3 .
Proof.Using Corollary 2.5, Lemmas 2.8 and 2.9, we obtain for the left-hand side J 0 of 3.2 that ϕ is defined by 2.7 x k h z dz : III IV.

3.61
i For the case 1 < θ ≤ p < ∞, by using Lemma 2.15 for III and applying H ölder's inequality for IV , we arrive at so that, by Lemma 2.13 and 3.61 , we obtain that

3.65
iii Now let 0 < p < ∞, θ ∞.By using Lemma 2.15 for III, we deduce that Moreover, for IV , it yields that

3.67
Therefore, by using integration by parts, Lemma 2.12, and 3.61 , we get that

3.68
Now we prove the lower bounds necessity .Let {x k } be a discretizing sequence for ϕ defined by 2.7 .Then, by 3.61 , we have

3.70
Therefore, by Proposition 2.14, we have that

3.71
Since    x k dz v z < ∞.

3.82
The proof is complete.
we use in 3.69 the test function defined by 3.34 , we obtain that θ ds