Best Constants between Equivalent Norms in Lorentz Sequence Spaces

We find the best constants in inequalities relating the standard norm, the dual norm, and the norm ‖x‖ p,s : inf{ ∑ k ‖x k ‖p,s}, where the infimum is taken over all finite representations x ∑ k x k 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.


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 1 < p < ∞ and 1 ≤ s ≤ ∞.For a sequence x x n n , the decreasing rearrangement x * x * n n is obtained by rearranging |x n | n in decreasing order.We recall the definition of Lorentz sequence spaces p,s ⎧ ⎨ ⎩ x : x p,s : with the usual modification if s ∞.Lorentz proved that • p,s is a norm if and only if 1 ≤ s ≤ p < ∞ and that the space p,s is always normable i.e., there exists a norm which is equivalent to • p,s when 1 < p < s ≤ ∞ for the remaining cases, it is known that p,s cannot be endowed with an equivalent norm .
The study of normability for p < s was carried out by means of the maximal norm: x * p,s x n n * p,s As a consequence of the fact that • p,s 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 C ps > 0 such that, for every finite collection {x k } k 1,2,...,N ⊂ p,s , it yields that 1.4 Also the converse result holds, namely that if 1.4 holds, then • p,s is equivalent with a norm and an alternative equivalent norm is given by means of the following decomposition norm: 1.5 One can prove that • p,s is always a norm and that • p,s is equivalent with • p,s see, e.g., 7, 8 .We also remark that the best constant C ps 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 K öthe duality, which we will call the dual norm: x n y n : y p ,s 1 .

1.7
The dual norm • p,s is a norm indeed, equivalent to • p,s and, moreover, if 1 ≤ s ≤ p, then x p,s x p,s x p,s . 1.8 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 n-terms.In 9 , the case of the classical Lorenz spaces L p,s was considered, while in 10 similar results were proved for the weighted Lorentz space Γ p w , where w 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 C ps in the triangle inequality 1.4 .This constant was found by a different approach in 13 where the best constants in q-convexity and q-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 p,s of x x n n and of its level sequence x • x • n n .Section 4 is dedicated to the study of the dual norm x p,s , 1 < p < s ≤ ∞, in relation to the other norms.For instance, one of the main results in Section 4 is that the dual norm of x x n n ∈ p,s coincides with the quasinorm of the level sequence x • p,s see Theorem 4.2 .Moreover, x p,s ≤ C ps x p,s , where C ps 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 x p,s .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 x x n n , y y n n , we denote that x ≺ y if and we use the notation x ≤ y if x n ≤ y n , ∀n ≥ 1.

1.10
The number p stands for the conjugate index of p; that is, 1/p 1/p 1 and N, N * stand for the sets of nonnegative, respectively, positive integers.

Some Lemmas
First, we recall the following well-known majorization lemma see, e.g., 14, page 9 .
Lemma 2.1.Let u, v, w be nonnegative sequences and suppose that w is decreasing.If We also need the following auxiliary statement related to the dual norm.
Proof.The proof is similar to that of Lemma 2.7 in 9 , so we omit the details.

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 L p,s R, μ , where R, μ is a totally σ-finite nonatomic measure space and 1 < p < s ≤ ∞, 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 ϕ ϕ n n be a sequence of positive numbers and 3.1 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 ϕ ϕ n n be a positive sequence and Φ n n i 1 ϕ i .Let x x n n be a positive sequence and suppose that Then, there exists a unique nonnegative sequence x • x • n n satisfying the following conditions: x n n with respect to ϕ ϕ n n .
Proof. a Let X n n i 1 x i , n ∈ N * .Condition 3.6 implies the existence of a sequence X X n n , which is a ϕ-concave majorant of X.We denote X • n : inf X n and from 3.5 we get that Thus, there exists a unique positive sequence x n n such that

3.8
According to 3.5 , we have that k , k ≥ 1.Now, we assume that there exists j ∈ N * such that X • j > X j and we denote n : max{k < j : 3.9 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: x n χ n,n 1 t .

3.11
Example 3.5.For f t χ 0,1 t , the level function is where α 1 − s /p so that If x x n n , where

3.15
We also need the following technical lemma.

3.17
Proof.We prove first that

3.18
The inequality is trivial.For the converse inequality, it is enough to prove that for all m ≥ n.
The above inequality can be obviously written in the form

3.21
Let us fix m ≥ 1. Observe that for m 1 we have equality, so we may suppose that m > 1.If we denote x k k s /p −1 , z k k m − 1 s /p −1 , and r −s/s < 0, we have to prove that

3.22
Observe that both x k k and z k k are decreasing sequences.Denote Applying Karamata's inequality see, e.g., 20 for the decreasing sequences x k k and y k Cz k and for the convex function Φ t t r , r < 0, we get that

3.25
Since m was arbitrary, we get the desired inequality and 3.18 is proved.Moreover, it is known that with C ps 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 ϕ ϕ n n with ϕ n n −1 s /p , n ≥ 1. Theorem 3.7.Let 1 < p < ∞, p < s ≤ ∞, and let x x n ∈ p,s be a nonnegative and decreasing sequence, and let x • x • n n be the level sequence with respect to the sequence ϕ ϕ n , where ϕ n n −α , α 1 − s /p .Then,

3.26
The constants in 3.26 are optimal.
Proof.Assume first that s < ∞ and consider the left-hand side inequality in 3.26 .By applying Theorem 3.3, we have that x , applying H ölder's inequality, we obtain that x s j j s/p−1 .

3.27
By the above estimate and Theorem 3.3, we get the left-hand side inequality in 3.26 .Let us now consider the sequence ψ ψ n n , ψ n x s−1 n n s/p−1 , n ≥ 1, and let ψ ψ n n be the level sequence of ψ ψ n n with respect to ϕ ϕ n n , ϕ n 1, n ≥ 1.By applying Theorem 3.3, Lemma 2.1, and Hölder's inequality, we obtain that

3.28
We note that to obtain the right-hand side inequality in 3.26 it is sufficient to prove that x s−1 p,s .

3.29
Let E : {n ≥ 1 : ψ n ψ n }.Then, we have that where {I k } are disjoint and such that

3.31
By H ölder's inequality, we obtain that

3.32
We also have that x s n n s/p−1 .

3.33
Hence, by 3.31 and 3.32 , it yields that

3.35
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 k 0 , we take x x n n , where

3.36
The right-hand side inequality 3.26 becomes equality for x x n n , where

3.37
We have that x p,s k 0 n 1 n s/p−1 1/s .It is easy to verify that if x • x • n n , where

3.39
Since k 0 is arbitrary, we get that also the constant on the right-hand side inequality 3.26 is optimal.
Let us now consider the case s ∞ and, hence, α 1 − 1/p 1/p.By using Hölder's inequality, we find that

3.40
Then, λ k ≤ x p,∞ for every k, which implies that x • p,∞ ≤ x p,∞ .On the other hand, for any n ∈ I k , and using again H ölder's inequality, we obtain that

3.41
It follows that sup

3.43
The constants are optimal also in this case.This can easily be proved if we consider the same sequences as in the case s < ∞.The proof is complete.

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 x x n n ∈ p,s .and x x n n ∈ p,s .Then, the following statements hold.
a One has x p,s ≤ x p,s . 4 x * n ψ n .

4.4
The last two equalities imply that x p,s ≥ x p,s 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 z z n n ∈ p,s such that x ≺ z and where y y n n ∈ p ,s 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 0, 1 and that the infimum is attained.For p < s ≤ ∞, a complete proof for Lorentz spaces over σ-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 R, μ is a totally σ-finite measure space, completely atomic, with all atoms having the same measure.For completeness, we prove this result in the following theorem.

4.13
This implies 4.7 , and the proof is complete.where C ps is defined by 3.17 .The constant is optimal.
Proof.The proof follows immediately from Theorems 4.2 and 3.7.

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 .
Our main result in this section reads the following.Then, by H ölder's inequality, we have that p,s y p ,s .

5.8
By taking the infimum over all representation 5.7 , we obtain 5.6 .It remains only to prove that x p,s ≤ x p,s .5.9 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 n 0 such that x i 0, for i > n 0 , n 0 ∈ N * .From Theorem 3.3, we have also that x

Further Results and Remarks
From Theorems 4. where C ps is given by 3.17 and the constant is optimal.Remark 6.2.This result can also be derived from a recent result of Kami ńska and Parrish 13 .They solved the problem in a completely different way.
Proof.We note that 6. 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 p,s 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 R, μ .For example, Theorem 5.2 can be generalized as follows.Theorem 6.4.Let 1 < p < ∞ and 1 ≤ s ≤ ∞.Then, for any function f ∈ L p,s R, μ , where R, μ is a totally σ-finite resonant measure space (see, e.g., [15,Definition 2.3,page 45]), one has that f p,s f p,s .6.6 Proof.Indeed, according to Theorem 5.2, we conclude that equality 6.6 holds for f ∈ L p,s R, μ , where R, μ is a totally σ-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.

Theorem 6 . 1 .
4 and 5.2, we can obtain the following sharp version of the "triangle inequality."Let 1 < p < s ≤ ∞, and suppose that x k x k n n ∈ p,s , k 1, . . ., N.Then, one has that n is decreasing and this proves a .b This statement follows directly from 3.8 and the definition of X • n .c Let us first note that two consecutive terms X • k−1 and X • k are equal to X k−1 and X k , respectively, if and only if x k x • Remark 3.8.Let 1 < p < s ≤ ∞.Let xx n n ∈ p,s be a nonnegative and nonincreasing sequence, and let x •x • n n be the level sequence of x with respect to the sequence According to Theorem 3.3 it yields that N * − E ∪ k I k , where I k are disjoint finite subsets of positive integers such that We consider only the nontrivial case I 1 {1, 2, . . ., m}.Let y y n n , where Let 1 < p < ∞ and p < s ≤ ∞.Then, for any sequence x x n n ∈ p,s , it yields that x p,s ≤ C ps x p,s , 4.15 n 1 }.According to Theorem 4.2, it yields that we have that there exists a permutation of {η jk }, { η jk } such that Since ε was arbitrary, it follows that 5.9 holds.The proof is complete.Let x x n n ∈ p,s , 1 ≤ p < ∞, 1 ≤ s ≤ ∞.Then,Proof.Equality 5.16 follows immediately from 5.4 and Lemma 2.2.
1 is equivalent to the inequality x p,s ≤ C ps x p,s , 6.2 where x x n n is any sequence from p,s .Inequality 6.2 follows directly from Theorems 4.4 and 5.2.Moreover, for x x n n with