Abstract
In this paper we characterize the validity of the Hardy-type inequality
where , , u, w and v are weight functions on . 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.
MSC:26D10, 46E20.
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1 Introduction
Throughout the paper, we assume that . By we denote the set of all measurable functions on I. The symbol stands for the collection of all which are non-negative on I, while 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 , is denoted by .
For and , we define the functional on by
If, in addition, , then the weighted Lebesgue space is given by
and it is equipped with the quasi-norm .
When on I, we write simply and instead of and , respectively.
Everywhere in the paper, u, v and w are weights. We denote by
and assume that for every .
In this paper we characterize the validity of the inequality
where , , , u, w and v are weight functions on . Note that inequality (1.1) was considered in the case in [1] (see also [2]), where the result was presented without proof, in the case in [3] and in the case in [4] and [5], where the special type of a weight function v was considered, and recently in [6] in the case , , .
We pronounce that the characterization of 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 of the obtained results in Section 5. 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 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 [4, 7, 8] and [6]. Some basic facts concerning these methods and other preliminaries are presented in Section 2. In Section 3 discretizations of inequalities (1.1) are given. Anti-discretization 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.
2 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 does not change in different occurrences. By () we mean that , where depends only on inessential parameters. If and , we write and say that a and b are equivalent. Throughout the paper, we use the abbreviation () for the left (right) hand side of the relation . By 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.
Convention 2.1 (i) Throughout the paper, we put , , , , , and if .
(ii) If , we define by . Moreover, we put if and if .
(iii) If and g is a monotone function on I, then by and we mean the limits and , respectively.
In this paper we shall use the Lebesgue-Stieltjes integral. To this end, we recall some basic facts.
Let φ be a non-decreasing and finite function on the interval . 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. [9], Chapter 10), it admits a unique extension to a non-negative Borel measure λ on I.
Formula (2.2) implies that
Note also that the associated Borel measure can be determined, e.g., only by putting
(since the Borel subsets of I can be generated by subintervals ).
If , then the Lebesgue-Stieltjes integral is defined as . We shall also use the Lebesgue-Stieltjes integral when φ is non-increasing and finite on the interval I. In such a case, we put
We conclude this section by recalling an integration by parts formula for Lebesgue-Stieltjes integrals. For any non-decreasing function f and a continuous function g on ℝ, the following formula is valid for :
Remark 2.1 Let . If and φ is a non-decreasing, right-continuous and finite function on I, then it is possible to show that for any , the Riemann-Stieltjes integral (written usually as ) coincides with the Lebesgue-Stieltjes integral . In particular, if and φ is non-decreasing on I, then the Riemann-Stieltjes integral coincides with the Lebesgue-Stieltjes integral for any .
Let us now recall some definitions and basic facts concerning discretization and anti-discretization which can be found in [7, 8] and [4].
Definition 2.1 Let be a sequence of positive real numbers. We say that is geometrically increasing or geometrically decreasing and write or when
respectively.
Definition 2.2 Let U be a continuous strictly increasing function on such that and . Then we say 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 and is equivalent to a decreasing function on . We say that a U-quasiconcave function φ is non-degenerate if
The family of non-degenerate U-quasiconcave functions is denoted by . We say that φ is quasiconcave when with . 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 of a decreasing function, the Peetre K-functional in interpolation theory and the fundamental function , X is a rearrangement invariant space, all are quasiconcave.
Definition 2.3 Assume that U is admissible and . We say that is a discretizing sequence for φ with respect to U if
(i) and ;
(ii) and ;
(iii) there is a decomposition such that and for every ,
Let us recall [[7], Lemma 2.7] that if , 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 . We say that the function φ defined by
is the fundamental function of the measure ν with respect to U. We also say that ν is a representation measure of φ with respect to U.
We say that ν is non-degenerate with respect to U if the following conditions are satisfied for every :
We recall from Remark 2.10 of [7] that
Lemma 2.1 ([[8], Lemma 1.5])
Let , u, w be weights and φ be defined by
Then φ is the least -quasiconcave majorant of w, and
for any non-negative measurable h on . Further, for ,
Theorem 2.1 ([[7], Theorem 2.11])
Let . Assume that U is an admissible function, ν is a non-negative non-degenerate Borel measure on , and φ is the fundamental function of ν with respect to and . If is a discretizing sequence for φ with respect to , then
Lemma 2.2 ([[7], Corollary 2.13])
Let . Assume that U is an admissible function, , ν is a non-negative non-degenerate Borel measure on and φ is the fundamental function of ν with respect to . If is a discretizing sequence for φ with respect to , then
Lemma 2.3 ([[7], Lemma 3.5])
Let . Assume that U is an admissible function, and . If is a discretizing sequence for φ with respect to , then
We shall use some Hardy-type inequalities in this paper. Define
Lemma 2.4 We have the following Hardy-type inequalities:
(a) Let . Then the inequality
holds for all if and only if
and the best constant in (2.9) satisfies
(b) Let . Then inequality (2.9) holds for all if and only if
and
These well-known results can be found in Maz’ya and Rozin [10], Sinnamon [11], Sinnamon and Stepanov [5] (cf. also [12] and [13]).
We shall also use the following fact (cf. [[14], p.188]):
Finally, if and is a sequence of positive numbers, we denote by the following discrete analogue of a weighted Lebesgue space: if , then
and
If for all , we write simply instead of .
We quote some known results. Proofs can be found in [15] and [16].
Lemma 2.5 Let . If is a geometrically decreasing sequence, then
and
for all non-negative sequences .
Let be a geometrically increasing sequence. Then
and
for all non-negative sequences .
We shall use the following inequality, which is a simple consequence of the discrete Hölder inequality:
where .a
Given two (quasi-)Banach spaces X and Y, we write if and if the natural embedding of X in Y is continuous.
The following two lemmas are discrete versions of the classical Landau resonance theorems. Proofs can be found, for example, in [7].
Proposition 2.1 ([[7], Proposition 4.1])
Let , and let and be two sequences of positive numbers. Assume that
(i) If , then
where C stands for the norm of inequality (2.13).
(ii) If , then
where and C stands for the norm of inequality (2.13).
3 Discretization of inequalities
In this section we discretize the inequalities
and
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 with respect to , that is,
where
Assume that is non-degenerate with respect to . Then , and therefore there exists a discretizing sequence for φ with respect to . Let be one such sequence. Then and . Furthermore, there is a decomposition , such that for every and , and for every and , .
Next, we state a necessary lemma which is also of independent interest.
Lemma 3.1 Let , , , and let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to . Let be any discretizing sequence for φ defined by (3.3). Then inequality (3.1) holds for every if and only if
and the best constant in inequality (3.1) satisfies
Proof By using Lemma 2.2 with
we get that
Moreover, by using Lemma 2.5,
where , . By now, using the fact that
we find 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 ) 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 and .
Next we prove that condition (3.4) is also necessary for the validity of inequality (3.1). Assume that inequality (3.1) holds with . By (2.8), there are , , such that
and
Define , , as the extension of by 0 to the whole interval and put
where is any sequence of positive numbers. We obtain that
Moreover,
Therefore, by (3.1), (3.13) and (3.14), we arrive at
and Proposition 2.1 implies that
On the other hand, there are , , such that
and
Define , , as the extension of by 0 to the whole interval and put
where is any sequence of positive numbers. We obtain that
Note that
Then, by (3.1) and previous two inequalities, we have that
Proposition 2.1 implies that
Inequalities (3.16) and (3.20) prove that . □
Before we proceed to inequality (3.2), we make the following remark.
Remark 3.2 Suppose that for all , where φ is defined by (2.7). Let φ be non-degenerate with respect to . Then, by Lemma 2.1, , and therefore there exists a discretizing sequence for φ with respect to . Let be one such sequence. Then and . Furthermore, there is a decomposition , such that for every and , and for every and , .
The following lemma is proved analogously, and for the sake of completeness, we give the full proof.
Lemma 3.2 Let , and let u, v, w be weights. Assume that u is such that is admissible. Let φ, defined by (2.7), be non-degenerate with respect to . Let be any discretizing sequence for φ. Then inequality (3.2) holds for every if and only if
and the best constant in inequality (3.2) satisfies .
Proof Using Lemma 2.1, Lemma 2.3, Lemma 2.5, we obtain for the left-hand side of (3.2) that
To find a sufficient condition for the validity of inequality (3.2), we apply to III locally 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 and .
Next we prove that condition (3.21) is also necessary for the validity of inequality (3.2). Assume that inequality (3.2) holds with . By (3.10), (3.11) and (3.12), we obtain that
Moreover,
Therefore, by (3.2), (3.25) and (3.26),
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 . □
Remark 3.3 In view of (2.11) and Lemma 2.5, it is evident now that
Monotonicity of implies that
Since is geometrically increasing, we obtain that
This inequality shows that must be equal to 0, because 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
4 Anti-dicretization of conditions
In this section, we anti-discretize the conditions obtained in Lemmas 3.1 and 3.2. We distinguish several cases.
The case , . We need the following lemma.
Lemma 4.1 Let , , , and let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to . Let be any discretizing sequence for φ defined by (3.3). Then
where
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
where
Proof Evidently, . Using integrating by parts formula (2.6), we have that
Again integrating by parts, we have that
Since
we obtain that
□
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
where
Proof By Lemma 2.5, we have that
Hence,
□
We are now in a position to state and prove our first main theorem.
Theorem 4.1 Let , , and let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to .
(i) Let . Then inequality (3.1) holds for every if and only if
Moreover, the best constant c in (3.1) satisfies .
(ii) Let . Then inequality (3.1) holds for every if and only if
Moreover, the best constant c in (3.1) satisfies .
Proof (i) The proof of the statement follows by using Lemmas 3.1, 4.1-4.5 and 2.3.
(ii) The proof of the statement follows by combining Lemmas 3.1, 4.1-4.5 and Theorem 2.1. □
The case , . The following lemma is true.
Lemma 4.6 Let , , and let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to . Let be any discretizing sequence for φ defined by (3.3). Then
where
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,
Since
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
where
Proof By Lemma 2.5, we get that
□
Our next main result reads as follows.
Theorem 4.2 Let , , and let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to .
(i) Let . Then inequality (3.1) holds for every if and only if
Moreover, the best constant c in (3.1) satisfies that .
(ii) Let . Then inequality (3.1) holds for every if and only if
Moreover, the best constant c in (3.1) satisfies that .
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. □
The case , . The following lemma is true.
Lemma 4.10 Let , 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 . Let be any discretizing sequence for φ.
(i) If , then
(ii) If , then
Proof (i) The proof of the statement follows by using Lemmas 3.2, 2.4 and 4.1-4.5.
(ii) The proof of the statement follows by combining Lemmas 3.2, 2.4 and 4.6-4.9. □
Now we are in a position to formulate our last main result.
Theorem 4.3 Let , 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 .
(i) Let . Then inequality (3.2) holds for every if and only if
Moreover, the best constant c in (3.2) satisfies that .
(ii) Let . Then inequality (3.2) holds for every if and only if
Moreover, the best constant c in (3.2) satisfies that .
Proof Both statements of the theorem follow by using Lemmas 3.2, 4.10, 2.3 and 2.1. □
5 Some applications
In this section, we give some applications of the obtained results. We start with the weighted Hardy inequality on the cone of non-increasing functions. Denote by the weighted Hardy operator
Note that the characterization of the weighted Hardy inequality on the cone of non-increasing functions
has been obtained in [17] and [4].
The following reduction theorem is true.
Theorem 5.1 Let , and let u, v, w be weights. Then inequality (5.1) holds for every if and only if the inequality
holds for all . Moreover, the best constants c and C in (5.1) and (5.2), respectively, satisfy .
Proof It is well known that every non-negative, non-increasing function f is the pointwise limit of an increasing sequence of functions of the form for (cf. [[5], p.97]). Since f is non-increasing if and only if is non-increasing, by the monotone convergence theorem, (5.1) is equivalent to
which, by Fubini’s theorem, is equivalent to
□
Analogously, the following theorem can be proved.
Theorem 5.2 Let , and let u, v, w be weights. Then the inequality
holds for every if and only if the inequality
holds for all . Moreover, for the best constants c and C in (5.3) and (5.4), respectively, it yields that .
Combining Theorem 5.1 with Theorems 4.1 and 4.2, we obtain the following statement.
Theorem 5.3 Let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to .
(i) Let , . Then inequality (5.1) holds for every if and only if
Moreover, the best constant c in (5.1) satisfies that .
(ii) Let , . Then inequality (5.1) holds for every if and only if
Moreover, the best constant c in (5.1) satisfies that .
(iii) Let . Then inequality (5.1) holds for every if and only if
Moreover, the best constant c in (5.1) satisfies that .
(iv) Let , . Then inequality (5.1) holds for every if and only if
Moreover, the best constant c in the (5.1) satisfies that .
Combining Theorems 5.2 and 4.3, we arrive at the following statement.
Theorem 5.4 Let u, v, w be weights. Assume that u is such that U is admissible. Let φ, defined by
be non-degenerate with respect to U.
(i) Let . Then inequality (5.3) holds for every if and only if
Moreover, the best constant c in (5.3) satisfies that .
(ii) Let . Then inequality (5.3) holds for every if and only if
Moreover, the best constant c in (5.3) satisfies that .
Now we consider the generalized Stieltjes transform S defined by
for all ; the usual Stieltjes transform is obtained by putting . In the case , , the boundedness of the operator S between weighted and spaces was investigated in [18] (when ) and in [19, 20] (when ). This problem was also considered in [21] and [22], where a completely different approach was used, based on the so call ‘gluing lemma’ (see also [23]).
The following reduction theorem is true.
Theorem 5.5 Let , , and let u, v, w be weights. Then the inequality
holds if and only if
holds.
Proof Evidently, inequality (5.5) is equivalent to the following inequality:
It is easy to see that
Indeed, by Fubini’s theorem, we have that
that is,
Hence, we see that inequality (5.5) is equivalent to inequality (5.6). □
Combining Theorem 5.5 with Theorems 4.2, 4.3 and Theorems 3.1, 3.2 in [6], we obtain the following statements.
Theorem 5.6 Let u, v, w be weights. Assume that u is such that U is admissible and the measure is non-degenerate with respect to . Let . When , we set .
(i) Let , . Then inequality (5.5) holds for every if and only if
Moreover, the best constant c in (5.5) satisfies that .
(ii) Let , . Then inequality (5.5) holds for every if and only if
Moreover, the best constant c in (5.5) satisfies that .
(iii) Let . Then inequality (5.5) holds for every if and only if
Moreover, the best constant c in (5.5) satisfies that .
(iv) Let , . Then inequality (5.5) holds for every if and only if
Moreover, the best constant c in (5.5) satisfies that .
(v) Let , . Then inequality (5.5) holds for every if and only if
Moreover, the best constant c in (5.5) satisfies that .
Theorem 5.7 Let u, v, w be weights. Assume that u is such that U is admissible. Let φ, defined by
be non-degenerate with respect to U.
(i) Let . Then the inequality
holds for every if and only if
Moreover, the best constant c in (5.7) satisfies that .
(ii) Let . Then inequality (5.7) holds for every if and only if
Moreover, the best constant c in (5.7) satisfies that .
(iii) Let . Then inequality (5.7) holds for every if and only if
Moreover, the best constant c in (5.7) satisfies that .
Endnote
For any , define when and when .
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Acknowledgements
We thank the anonymous referees for his/her remarks, which have improved the final version of this paper. The research of the first author was partly supported by the grants 201/08/0383 and P201-13-14743S of the Grant Agency of the Czech Republic and RVO: 67985840. The research of the first and second authors was partly supported by the joint project between Academy of Sciences of Czech Republic and the Scientific and Technological Research Council of Turkey.
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Gogatishvili, A., Mustafayev, R. & Persson, LE. Some new iterated Hardy-type inequalities: the case . J Inequal Appl 2013, 515 (2013). https://doi.org/10.1186/1029-242X-2013-515
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DOI: https://doi.org/10.1186/1029-242X-2013-515