#### Abstract

We characterize the validity of the Hardy-type inequality , where , , , , , and are weight functions on . Some fairly new discretizing and antidiscretizing techniques of independent interest are used.

#### 1. Introduction

Everywhere in the paper, , , and are weights, that is, locally integrable nonnegative functions on , and we denote where We assume that is such that for every .

For and , a weight function on , let us denote by the weighted Lebesgue space defined as the set of all measurable functions on for which the quantity is finite.

In this paper we characterize the validity of the inequality where , , , , , and are weight functions on . Note that inequality (1.4) has been considered in the case in [1] (see also [2]), where the result is presented without proof, in the case in [3] and in the case in [4, 5], where weight functions of special type were considered. For general weight functions , the characterization of the inequality (1.4) in the case 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 ) is equivalent with the following inequality: where the operator defined by for all nonnegative measurable functions on . We call this operator the generalized Stieltjes transform; the usual Stieltjes transform is obtained on putting .

In the case , , the boundedness of the operator between weighted and spaces was investigated in [6] (when ) and in [7, 8] (when ).

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 or 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 on inessential parameters. If and , we write and say that and are equivalent. We put , , , and .

#### 2. 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 be a sequence of positive real numbers. One says that is strongly increasing or strongly decreasing and write or when
respectively.

*Definition 2.2. *Let be a continuous strictly increasing function on such that and . Then One says that is admissible.

Let be an admissible function. We say that a function is -quasiconcave if is equivalent to an increasing function on and is equivalent to a decreasing function on . We say that a -quasiconcave function is nondegenerate if
The family of nondegenerate -quasiconcave functions will be denoted by . We say that 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, for example, the Hardy operator of a decreasing function, the Peetre -functional in interpolation theory, and the fundamental function , is a rearrangement invariant space, all are quasiconcave.

*Definition 2.3. *Assume that is admissible and . One says that is a discretizing sequence for with respect to if(i) and ;(ii) and ;(iii) there is a decomposition such that and for every

Let us recall (see [9, Lemma 2.7]) that if , then there always exists a discretizing sequence for with respect to .

*Definition 2.4. * Let be an admissible function, and let be a nonnegative Borel measure on . We say that the function defined by
is the fundamental function of the measure with respect to . One will also say that is a representation measure of with respect to .

We say that is nondegenerate if the following conditions are satisfied for every :

We recall from [9, Remark 2.10] that

Corollary 2.5 (see [10, Lemma 1.5]). *Let , be weights, and let be defined by
**
Then is the least -quasiconcave majorant of , and
**
for any nonnegative measurable on . Further, for ,
*

Theorem 2.6 (see [9, Theorem 2.11]). *Let . Assume that is an admissible function, is a nonnegative nondegenerate Borel measure on , and is the fundamental function of with respect to and . If is a discretizing sequence for with respect to , then
*

Lemma 2.7 (see [9, Corollary 2.13]). *Let . Assume that is an admissible function, , is a nonnegative nondegenerate Borel measure on , and is the fundamental function of with respect to . If is a discretizing sequence for with respect to , then
*

Lemma 2.8 (see [9, Lemma 3.5]). *Let . Assume that is an admissible function, , and . If is a discretizing sequence for with respect to and is a discretizing sequence of with respect to , then
*

Lemma 2.9 (see [4, Lemma 2.5]). *If , then
*

Lemma 2.10 (see [9, Lemma 3.6]). *Let . Assume that is an admissible function, is a nondegenerate nonnegative Borel measure on , is the fundamental function of with respect to , and is a measurable function on . If is a discretizing sequence for with respect to , then
*

Lemma 2.11 (see [9, Lemma 3.7]). * Let . Assume that is an admissible function, is a nondegenerate nonnegative Borel measure on , is the fundamental function of with respect to , and is a measurable function on . If is a discretizing sequence for with respect to , then
*

Lemma 2.12 (see [9, Lemma 3.8]). *Let . Assume that is an admissible function, , is a discretizing sequence for with respect to , and is a measurable function on . Then
*

Lemma 2.13 (see [9, Lemma 3.9]). *Let be an admissible function, , be a discretizing sequence for with respect to , and be a measurable function on . Then
*

Proposition 2.14 (see [9, Proposition 4.1]). * Let and , , be two sequences of positive real numbers. Let , and assume that the inequality
**
is satisfied for every sequence of positive real numbers.*(i)*If **, then *(ii)*If ** and **, then*

Lemma 2.15. * One has the following Hardy-type inequalities.*(a)* Let . Then the inequality
* *holds for all nonnegative measurable if and only if
* *and the best constant in (2.21) satisfies .*(b)* Let , . Then the inequality (2.21) holds if and only if
* *and the best constant in (2.21) satisfies .*(c)* Let . Then the inequality (2.21) holds if and only if
* *and the best constant in (2.21) satisfies .*(d)* Let , , and . Then the inequality (2.21) holds if and only if
* *and the best constant in (2.21) satisfies .*(e)* Let , . Then the inequality (2.21) holds if and only if
* *and the best constant in (2.21) satisfies . *

These results are just classical results of Maz’ja [11] and Sinnamon [12] (cf. also [13, 14]).

#### 3. The Main Results

In this section we characterize the validity of the inequalities

Denote

First we characterize (3.1) as follows.

Theorem 3.1. *Let , , , and let be weights. Assume that is such that is admissible and the measure is nondenerate with respect to . Then the inequality (3.1) holds for every measurable function on if and only if*(i)*, * *Moreover, the best constant in (3.1) satisfies .*(ii)*, , * *Moreover, the best constant in (3.1) satisfies .*(iii)*, , * *Moreover, the best constant in (3.1) satisfies .*(iv)*, , , , ,
* *Moreover, the best constant in (3.1) satisfies .*(v)* Let , , ,
* *Moreover, the best constant in (3.1) satisfies . *

*Proof. *Define
Then , and therefore there exists a discretizing sequence for with respect to . Let be one such sequence. Then and . Furthermore, there is a decomposition , such that for every and , and for every and , .

For the left-hand side of (3.1), by using Lemma 2.7 with
we get that
Moreover, by using Lemma 2.9, we get that

By now using the fact that , we find that
is, by using Lemma 2.9 on the second term,
Now we will distinguish several cases. We start with the case . Then, by using Lemma 2.15, we get that
Moreover, by applying Hölder’s inequality for , we find that
(i) In the case , according to (3.15), we have that
Similarly, if , then, according to (3.16), we obtain that
and, finally, by using (3.9), Lemma 2.13, and (3.14), we get that
(ii) For the case , , by applying Hölder’s inequality for sums to the right-hand side of (3.15) and (3.16) with exponents and , we find that
Therefore, we get that
so that, in view of Lemma 2.11, Theorem 2.6, and (3.14),
Now let us assume that , , . By Lemma 2.15, we have that
Moreover, by applying Hölder’s inequality for , we find that
(iii) Now, we assume that . Then, according to (3.7) and (3.24), we obtain that
Hence, using Lemmas 2.9 and 2.12, and (3.14), we get that
(iv) Next, we consider the case , . By using Hölder’s inequality for sums to the right-hand side of (3.7) and (3.24) with exponents and , we get that
Therefore, using Lemmas 2.9 and 2.10, Theorem 2.6, and (3.14), we find that
(v) Let , , . According to Lemma 2.15, we have that
Moreover, it yields that
Hence, by integrating by parts, using Lemmas 2.9 and 2.10, and (3.14), we get that

Now we prove the lower bounds (necessity). Let and be a discretizing sequence for from (3.9). Then, by (3.14), we find that
Let . For , let be functions that saturate the Hardy inequality (2.21) and Hölder’s inequality, that is, functions satisfying
Now we define the test function
where is a sequence of positive real numbers. Thus, using test function (3.34) in (3.32), we get that
Now using Proposition 2.14 for the case , we obtain that