Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 713534, 19 pagesdoi:10.1155/2012/713534

Research Article

Best Constants between Equivalent Norms in Lorentz Sequence Spaces

S. Barza,¹ A. N. Marcoci,² and L. E. Persson³

¹Department of Mathematics, Karlstad University, 65188 Karlstad, Sweden²Department of Mathematics and Computer Science, Technical University of Civil Engineering Bucharest, 020 396 Bucharest, Romania³Department of Mathematics, Luleå University of Technology, 97 187 Luleå, Sweden

Received 19 March 2010; Accepted 24 May 2010

Academic Editor: Nicolae Popa

Copyright © 2012 S. Barza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm , where the infimum is taken over all finite representations in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.

1. Introduction

The study of Lorentz spaces goes back to the work of Lorentz [1, 2] (see also [3, 4] for more recent results concerning functional properties of Lorentz spaces). They play an important role in the theory of Banach function spaces, in particular in the interpolation theory of linear operators.

Let and . For a sequence , the decreasing rearrangement is obtained by rearranging in decreasing order. We recall the definition of Lorentz sequence spaces with the usual modification if . Lorentz proved that is a norm if and only if and that the space is always normable (i.e., there exists a norm which is equivalent to ) when (for the remaining cases, it is known that cannot be endowed with an equivalent norm).

The study of normability for was carried out by means of the maximal norm: defined in terms of the discrete Hardy operator. It is well known that is always a norm. Moreover, one can prove that is equivalent to with the following estimates: see, for example, [5, 6].

As a consequence of the fact that is equivalent to a norm, it is easy to see that it is a quasinorm satisfying the triangle inequality uniformly in the numbers of terms; that is, there exists a constant such that, for every finite collection , it yields that

Also the converse result holds, namely that if (1.4) holds, then is equivalent with a norm and an alternative equivalent norm is given by means of the following decomposition norm:

One can prove that is always a norm and that is equivalent with , . If , then (see, e.g., [7, 8]). We also remark that the best constant in the inequality is the same as the optimal one in (1.4).

It is natural to consider also the following norm defined in terms of Köthe duality, which we will call the dual norm: The dual norm is a norm indeed, equivalent to and, moreover, if , then

In the recent papers [9, 10], the authors considered estimates between the dual norm, decomposition norm, and the norm which defines the Lorentz spaces over nonatomic resonant measure spaces. The main reason for these consideration was that the technique based on the norm defined in terms of the maximal function (or Hardy operator) does not give the best constant in the triangle inequality with -terms. In [9], the case of the classical Lorenz spaces was considered, while in [10] similar results were proved for the weighted Lorentz space , where is an increasing weight function.

The aim of this paper is to treat similar problems in the context of Lorentz spaces of sequences. In particular, in this paper, we introduce the notion of level sequence, which corresponds to the notion of level function introduced by Halperin in [11] and Lorentz in [12]. We would like to pronounce that our results are not corollaries of the results proved in [9] although some of the techniques are similar (cf. our Example 3.5). One of the main applications of our results is that we obtain the best constant in the triangle inequality (1.4). This constant was found by a different approach in [13] where the best constants in -convexity and -concavity inequalities were found for Lorentz and Marcinkiewicz spaces of functions and sequences.

The paper is organized as follows. In Section 2, we state and discuss several technical lemmas, which will be used in the proofs of our theorems. In Section 3, we introduce and discuss one of the key tools used in this paper, the so-called level sequence. In Theorem 3.7 of Section 3, we prove optimal estimates between the quasinorm in of and of its level sequence . Section 4 is dedicated to the study of the dual norm , , in relation to the other norms. For instance, one of the main results in Section 4 is that the dual norm of coincides with the quasinorm of the level sequence (see Theorem 4.2). Moreover, , where is the optimal constant which appears in Theorem 4.4. The main result of Section 5, that is, that the dual norm coincides with the decomposition norm is given in Theorem 5.2. Finally, in the last section, we give some complementary results and remarks, as, for example, Theorem 6.1, which gives us the best constant in the “triangle inequality” for the quasinorm . Moreover, we point out that our results can be given in a somewhat more general setting (see Remark 6.3 and Theorem 6.4).

Throughout this paper, given two sequences , , we denote that if and we use the notation if The number stands for the conjugate index of ; that is, and , stand for the sets of nonnegative, respectively, positive integers.

2. Some Lemmas

First, we recall the following well-known majorization lemma (see, e.g., [14, page 9]).

Lemma 2.1. Let be nonnegative sequences and suppose that is decreasing. If then

We also need the following auxiliary statement related to the dual norm.

Lemma 2.2. Let , , . Then,

A proof can be found, for example, in [15, pages 45–49].

The next lemmas are crucial in the proofs of some of our main results and are also of independent interest.

Lemma 2.3. Let , . Then, the following statements hold. (a)The equality holds, where the infimum is taken over all finite nonnegative sequences .(b) If , then .(c) If and as , then as .

Proof. The proof is similar to that of Lemma  2.7 in [9], so we omit the details.

3. Level Sequences

The notion of level function was introduced in the early 1950s by Halperin [11] and Lorentz [12] and generalized more recently by Sinnamon in, for example, [16–18] and Mastylo and Sinnamon in [19]. Based on the extension given by Lorentz, the optimal constant in the triangle inequality in Lorentz spaces , where is a totally -finite nonatomic measure space and , was found.

Inspired by these results, we will use in this paper the concept of level sequence with respect to a given sequence and study Lorentz sequence spaces in this frame.

Definition 3.1. Let be a sequence of positive numbers and . A sequence is called -concave if, for arbitrary , we have that , where , and are constants, such that “interpolate” in , ; that is,

Proposition 3.2. The following statements are equivalent:

Proof. The proof of this proposition is literally the same as the proof of the equivalence of the corresponding statements in the case of functions, so we omit the details (see, e.g., [12]).

In the next theorem, we prove the existence of the level sequence and derive its properties. Since, as far as we know, this theorem has not been proved earlier in this context, we give the entire proof here.

Theorem 3.3. Let be a positive sequence and . Let be a positive sequence and suppose that Then, there exists a unique nonnegative sequence satisfying the following conditions: (a) is decreasing;(b);(c) the set , where are disjoint sets of positive integers, such that and , for all .

Remark 3.4. The unique sequence in Theorem 3.3 is called the level sequence of with respect to .

Proof. (a) Let , . Condition (3.6) implies the existence of a sequence , which is a -concave majorant of . We denote and from (3.5) we get that that is, is increasing. Thus, there exists a unique positive sequence such that , with and According to (3.5), we have that , which means that is decreasing and this proves (a).(b) This statement follows directly from (3.8) and the definition of .(c) Let us first note that two consecutive terms and are equal to and , respectively, if and only if , .Now, we assume that there exists such that and we denote and . Now, we take to “interpolate” in and . Then, and . It follows that for .Thus, , , and . Since we obtain that and the proof is complete.

The following example shows that the results for sequences do not follow from the results for function simply by taking step functions:

Example 3.5. For , the level function is where so that If , where then

We also need the following technical lemma.

Lemma 3.6. Let and Then,

Proof. We prove first that The inequality is trivial. For the converse inequality, it is enough to prove that for all .The above inequality can be obviously written in the form Let us fix . Observe that for we have equality, so we may suppose that . If we denote , , and , we have to prove that Observe that both and are decreasing sequences. Denote , , and .Since is decreasing, so is also , and hence for . Applying Karamata’s inequality (see, e.g., [20]) for the decreasing sequences and and for the convex function , , we get that or The last inequality implies Since was arbitrary, we get the desired inequality and (3.18) is proved. Moreover, it is known that with defined by (3.18) it yields that (3.17) holds (see Theorem  15 in [13]). The proof is complete.

The main result of this section contains optimal estimates between the norm of a sequence and of its level sequence with respect to the sequence with , .

Theorem 3.7. Let , , and let be a nonnegative and decreasing sequence, and let be the level sequence with respect to the sequence , where , . Then, The constants in (3.26) are optimal.

Proof. Assume first that and consider the left-hand side inequality in (3.26). By applying Theorem 3.3, we have that , . Since , , applying Hölder’s inequality, we obtain that By the above estimate and Theorem 3.3, we get the left-hand side inequality in (3.26). Let us now consider the sequence , , and let be the level sequence of with respect to , , . By applying Theorem 3.3, Lemma 2.1, and Hölder’s inequality, we obtain that We note that to obtain the right-hand side inequality in (3.26) it is sufficient to prove that Let . Then, we have that where are disjoint and such that By Hölder’s inequality, we obtain that We also have that Hence, by (3.31) and (3.32), it yields that where Hence, according to Lemma 3.6, it follows that (3.29) holds, which means that the right-hand side inequality in (3.26) is proved. It only remains to prove the sharpness of the obtained inequalities.We note that the left-hand side inequality in (3.26) becomes equality if, for a fixed , we take , where The right-hand side inequality (3.26) becomes equality for , where We have that . It is easy to verify that if , where then Since is arbitrary, we get that also the constant on the right-hand side inequality (3.26) is optimal.Let us now consider the case and, hence, . By using Hölder’s inequality, we find that Then, for every , which implies that .On the other hand, for any , and using again Hölder’s inequality, we obtain that It follows that where The constants are optimal also in this case. This can easily be proved if we consider the same sequences as in the case . The proof is complete.

Remark 3.8. Let . Let be a nonnegative and nonincreasing sequence, and let be the level sequence of with respect to the sequence Then, the equality holds if and only if for all .Indeed, inequality (3.27) becomes equality if and only if is constant on . Therefore, (3.45) holds if and only if , for all , hence if and only if is decreasing.

4. Level Sequences and the Dual Norm

This section is devoted to investigate the dual norm of a sequence defined in (1.7).

In the next proposition, we summarize some well-known properties of the dual norm of a sequence .

Proposition 4.1. Let , , and . Then, the following statements hold. (a) One has (b) If , then (c) If and if the sequence is nonincreasing, then (4.2)holds.(d) If , then one has that

Proof. Indeed, (a) follows from Hölder’s inequality.(b) Take with , . The sequence is nonincreasing and we have that The last two equalities imply that and, by (a), we get equality (4.2).(c) The proof is similar to that of (b), so we omit the details.(d) In view of Lemma 2.1, we have that where the infimum is taken over all nonnegative sequences such that and where is a nonnegative and nonincreasing sequence. This implies that (d) holds.

It was proved by Halperin in [11] (see also [12, Theorem  3.6.5]) that we have equality in (4.3) in the case of real functions defined on and that the infimum is attained. For , a complete proof for Lorentz spaces over -finite nonatomic measure spaces was given in the recent paper [9]. We remark here that the proofs given in [9, 11, 12] do not cover the result in the case when is a totally -finite measure space, completely atomic, with all atoms having the same measure. For completeness, we prove this result in the following theorem.

Theorem 4.2. Let , and let be a nonnegative and nonincreasing sequence. Let and , where . Then, one has that where is the level sequence of with respect to the sequence .

Proof. By Proposition 4.1 and Theorem 3.3(b), it is sufficient to prove that We denote . According to Theorem 3.3 it yields that , where are disjoint finite subsets of positive integers such that Assume first that and let , where , . As before, we have that Choose , where . Then, .From Theorem 3.3, it follows that, for each , , . Thus, and Moreover, we have that Thus, we obtain that . This implies (4.7).Let now . We consider only the nontrivial case . Let , where and . Then, This implies (4.7), and the proof is complete.

Remark 4.3. Let and let be a nonnegative and nonincreasing sequence. Then, by Remark 3.8, the equality (see (4.2)) holds if and only if is decreasing. Note also that (4.2) holds in general if (see Proposition 4.1(b)).The final result in this section gives the sharp estimate of the standard norm via the dual norm.

Theorem 4.4. Let and . Then, for any sequence , it yields that where is defined by (3.17). The constant is optimal.

Proof. The proof follows immediately from Theorems 4.2 and 3.7.

5. The Decomposition Norm

In this section, we prove that the dual norm and the decomposition norm coincide. The following lemma plays an important role in the proof of the main result, and it was proved in the recent paper [9].

Lemma 5.1. Let be positive numbers, and let be a -matrix of positive numbers, . Set Assume that for any . Let . Then, for any , there exists a permutation of the -tuple such that for any .

Our main result in this section reads the following.

Theorem 5.2. Let and . Then, for any sequence ,

Proof. If , then it is well known that (see, e.g., [7, 15]) Let . We prove first that Let , and let Then, by Hölder’s inequality, we have that By taking the infimum over all representation (5.7), we obtain (5.6). It remains only to prove that In view of Lemma 2.3, it is sufficient to prove (5.9) in the case of positive nonincreasing sequences. We can assume that there exists such that , for , . From Theorem 3.3, we have also that for , . Denote . According to Theorem 4.2, it yields that where is the level sequence of with respect to the sequence , , .Let . Choose such that , with , where Choose such that By using Lemma 5.1 with , , , , , we have that there exists a permutation of , such that We define now , where . Since is equimeasurable with , we have that which, in its turn, implies that Since was arbitrary, it follows that (5.9) holds. The proof is complete.

Corollary 5.3. Let , , . Then,

Proof. Equality (5.16) follows immediately from (5.4) and Lemma 2.2.

6. Further Results and Remarks

From Theorems 4.4 and 5.2, we can obtain the following sharp version of the “triangle inequality.”

Theorem 6.1. Let , and suppose that , . Then, one has that where is given by (3.17) and the constant is optimal.

Remark 6.2. This result can also be derived from a recent result of Kamińska and Parrish [13]. They solved the problem in a completely different way.

Proof. We note that (6.1) is equivalent to the inequality where is any sequence from . Inequality (6.2) follows directly from Theorems 4.4 and 5.2. Moreover, for with for a fixed , we obtain that From Theorems 4.2 and 5.2, we have that and therefore we obtain equality in (6.2), and the proof is complete.

Remark 6.3. We want to pronounce that in this paper we have formulated our results only for spaces, but our proofs show that some results are true for much more general spaces. For example, Theorems 3.7, 4.4, 5.2, and 6.1 are true in the more general case of resonant measure space . For example, Theorem 5.2 can be generalized as follows.

Theorem 6.4. Let and . Then, for any function , where is a totally -finite resonant measure space (see, e.g., [15, Definition  2.3, page 45]), one has that

Proof. Indeed, according to Theorem 5.2, we conclude that equality (6.6) holds for , where is a totally -finite measure space, completely atomic, with all atoms having the same measure. Hence, together with Theorem  5.2 in [9] and by applying [15, Theorem  2.7, page 51], we obtain the desired result.

Acknowledgments

The authors would like to thank Professor Nicolae Popa for careful reading of the manuscript and for valuable comments, to Professor A. Kamińska who made us aware of the information given in Remark 6.2 and to Professor G. Sinnamon for giving us some information regarding level functions and for his remarks about some of our Theorems. Moreover the second named author wants to thank the Department of Mathematics at Luleå University of Technology for hospitality and for financial support. A. N. Marcoci was partially supported by CNCSIS-UEFISCSU, project number 538/2009 PNII-IDEI code 1905/2008.

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