Best Constants between Equivalent Norms in Lorentz Sequence Spaces

Abstract

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=∑_𝑘𝑥^(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.

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 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 spacesℓ^𝑝,𝑠=⎧⎪⎨⎪⎩𝑥∶‖𝑥‖_𝑝,𝑠∶=^∞_𝑛=1𝑥^∗𝑛^𝑠𝑛^{𝑠/𝑝−1}^1/𝑠<∞⎫⎪⎬⎪⎭,(1.1) 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 ℓ^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 ℓ^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=^∞_𝑛=1𝑛^{𝑠/𝑝−1}1𝑛^𝑛_𝑖=1𝑥^∗𝑖^𝑠^1/𝑠(1.2) defined in terms of the discrete Hardy operator. It is well known that ‖⋅‖^∗𝑝,𝑠 is always a norm. Moreover, one can prove that ‖⋅‖^∗𝑝,𝑠 is equivalent to ‖⋅‖_𝑝,𝑠 with the following estimates:‖𝑥‖_𝑝,𝑠≤‖𝑥‖^∗𝑝,𝑠≤𝑝^‖𝑥‖_𝑝,𝑠,(1.3) see, for example, [5, 6].

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}⊂ℓ^𝑝,𝑠, it yields that‖‖‖‖^𝑁_𝑘=1𝑥^(𝑘)‖‖‖‖_𝑝,𝑠≤𝐶_𝑁𝑝𝑠_𝑘=1‖‖𝑥^(𝑘)‖‖_𝑝,𝑠.(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:‖𝑥‖_(𝑝,𝑠)∶=inf^𝑁_𝑘=1‖‖𝑥^(𝑘)‖‖_𝑝,𝑠∶𝑥=^𝑁_𝑘=1𝑥^(𝑘).(1.5)

One can prove that ‖⋅‖_(𝑝,𝑠) is always a norm and that ‖⋅‖_(𝑝,𝑠) is equivalent with ‖⋅‖_𝑝,𝑠, 1≤𝑝,𝑠≤∞. If 1≤𝑠≤𝑝, then ‖⋅‖_(𝑝,𝑠)=‖⋅‖_𝑝,𝑠 (see, e.g., [7, 8]). We also remark that the best constant 𝐶_𝑝𝑠 in the inequality‖𝑥‖_𝑝,𝑠≤𝐶_𝑝𝑠‖𝑥‖_(𝑝,𝑠)(1.6) 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: ‖𝑥‖^𝑝,𝑠=sup^∞_𝑛=1𝑥_𝑛𝑦_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 ℓ^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∈ℓ^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^𝑛_𝑖=1𝑥_𝑖≤^𝑛_𝑖=1𝑦_𝑖,∀𝑛≥1(1.9) and we use the notation 𝑥≤𝑦 if𝑥_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.

2. 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 ^𝑛_𝑘=1𝑢_𝑘≤^𝑛_𝑘=1𝑣_𝑘(𝑛=1,2,…),(2.1) then ^𝑛_𝑘=1𝑢_𝑘𝑤_𝑘≤^𝑛_𝑘=1𝑣_𝑘𝑤_𝑘(𝑛=1,2,…).(2.2)

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

Lemma 2.2. Let 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠, 1<𝑝<∞, 1≤𝑠≤∞. Then, ‖𝑥‖^𝑝,𝑠=‖𝑥^∗‖^𝑝,𝑠.(2.3)

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 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,s, 1<𝑝<∞,1≤𝑠≤∞. Then, the following statements hold.(a)The equality ‖𝑥‖_(𝑝,𝑠)=inf_𝑘‖‖𝑥^(𝑘)‖‖_𝑝,𝑠(2.4) holds, where the infimum is taken over all finite nonnegative sequences |𝑥_𝑛|=∑_𝑘𝑥^𝑛(𝑘).(b) If 0≤𝑦≤𝑥, then ‖𝑦‖_(𝑝,𝑠)≤‖𝑥‖_(𝑝,𝑠).(c) If 0≤𝑦^(𝑘)≤𝑥 and 𝑦^(𝑘)↗𝑥 as 𝑘→∞, then ‖𝑦^(𝑘)‖_(𝑝,𝑠)→‖𝑥‖_(𝑝,𝑠) as 𝑘→∞.

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 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 Φ_𝑛=∑^𝑛𝑖=1𝜑_𝑖. A sequence 𝑋=(𝑋_𝑛)_𝑛 is called 𝜑-concave if, for arbitrary 𝑛₁,𝑛₂∈ℕ, we have that 𝑋_𝑛≥𝐿_𝑛, where 𝐿_𝑛=𝐶Φ_𝑛+𝐷, 𝐶 and 𝐷 are constants, such that 𝐿 “interpolate” 𝑋 in 𝑛₁, 𝑛₂; that is, 𝐿𝐿_𝑛1𝑛2=𝑋_𝑛1,=𝑋_𝑛2.(3.1)

Proposition 3.2. The following statements are equivalent: 𝑋=𝑋_𝑛𝑋_𝑛is𝜑-concave,(3.2)_𝑛−𝑋_𝑛1Φ_𝑛−Φ_𝑛1≥𝑋^𝑛2−𝑋_𝑛1Φ_𝑛2−Φ_𝑛1𝑋,1≤𝑛₁<𝑛≤𝑛₂,(3.3)_𝑛−𝑋_𝑛1Φ_𝑛−Φ_𝑛1≥𝑋^𝑛2−𝑋_𝑛Φ_𝑛2−Φ_𝑛𝑋,1≤𝑛₁≤𝑛<𝑛₂,(3.4)_𝑛1−𝑋_𝑚1Φ_𝑛1−Φ_𝑚1≥𝑋^𝑛2−𝑋_𝑚2Φ_𝑛2−Φ_𝑚2,𝑚₁≤𝑚₂,𝑛₁≤𝑛₂,𝑚₁≠𝑛₁,𝑚₂≠𝑛₂.(3.5)

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=∑^𝑛𝑖=1𝜑_𝑖. Let 𝑥=(𝑥_𝑛)_𝑛 be a positive sequence and suppose that lim_𝑛→∞∑^𝑛𝑖=1𝑥_𝑖∑^𝑛𝑖=1𝜑_𝑖=0.(3.6)
Then, there exists a unique nonnegative sequence 𝑥^∘=(𝑥^∘𝑛)_𝑛 satisfying the following conditions:(a)(𝑥^∘𝑛/𝜑_𝑛)_𝑛 is decreasing;(b)𝑥≺𝑥^∘;(c) the set {𝑛∶𝑥^∘𝑛≠𝑥_𝑛}=∪^∞𝑘=1𝐼_𝑘, where {𝐼_𝑘} are disjoint sets of positive integers, 𝐼_𝑘={𝑛_𝑘,…,𝑛_𝑘+𝑚_𝑘} such that ∑_{𝑖∈𝐼𝑘}𝑥_𝑖=∑_{𝑖∈𝐼𝑘}𝑥^∘𝑖 and 𝑥^∘𝑖/𝜑_𝑖=𝜆_𝑘, for all 𝑖∈𝐼_𝑘.

Remark 3.4. The unique sequence 𝑥^∘=(𝑥^∘𝑛)_𝑛 in Theorem 3.3 is called the level sequence of 𝑥=(𝑥_𝑛)_𝑛 with respect to 𝜑=(𝜑_𝑛)_𝑛.

Proof. (a) Let 𝑋_𝑛=∑^𝑛𝑖=1𝑥_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∶=inf𝑋_𝑛 and from (3.5) we get that 𝑋_𝑘1𝑋_𝑘2≤Φ^𝑘1Φ_𝑘2≤1,for𝑘₁≤𝑘₂,(3.7) that is, 𝑋^∘=(𝑋^∘𝑘)_𝑘 is increasing. Thus, there exists a unique positive sequence (𝑥_𝑛)_𝑛 such that 𝑋^∘𝑛=∑^𝑛𝑖=1𝑥^∘𝑖, with 𝑥^∘1=𝑋^∘1 and 𝑥^∘𝑛∶=𝑋^∘𝑛−𝑋^∘𝑛−1.(3.8)
According to (3.5), we have that (𝑋^∘𝑛−𝑋^∘𝑛−1)/(Φ_𝑛−Φ_𝑛−1)≥(𝑋^∘𝑛+1−𝑋^∘𝑛)/(Φ_𝑛+1−Φ_𝑛), which means that (𝑥^∘𝑛/𝜑_𝑛)_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^∘𝑘, 𝑘≥1.
Now, we assume that there exists 𝑗∈ℕ^∗ such that 𝑋^∘𝑗>𝑋_𝑗 and we denote 𝑛∶=max{𝑘<𝑗∶𝑋^∘𝑘=𝑋_𝑘and𝑋^∘𝑘−1=𝑋_𝑘−1}+1 and 𝑚∶=min{𝑘>𝑗∶𝑋^∘𝑘=𝑋_𝑘and𝑋^∘𝑘+1=𝑋_𝑘+1}−𝑛+1. Now, we take 𝐿 to “interpolate” 𝑋^∘ in 𝑛−1 and 𝑛+𝑚−1. Then, 𝐿≥𝑋 and 𝑋^∘≥𝐿. It follows that 𝐿_𝑗=𝑋^∘𝑗=𝐶Φ_𝑗+𝐷 for 𝑗∈{𝑛−1,…,𝑛+𝑚−1}.
Thus, 𝑥^∘𝑗=𝐶𝜑_𝑗, 𝑗∈{𝑛,…,𝑛−1+𝑚}, 𝑥^∘𝑛+𝑚=𝑥_𝑛+𝑚 and 𝑥^∘𝑛−1=𝑥_𝑛−1. Since 𝑋𝑋^∘𝑛−1_{∘𝑛+𝑚−1}=𝑋_𝑛−1,=𝑋_{𝑛+𝑚−1},(3.9) we obtain that ^𝑛+^𝑚−1_𝑖=𝑛𝑥^∘𝑖=^{𝑛+𝑚−1}_𝑖=𝑛𝑥_𝑖(3.10) 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:𝑓(𝑡)=^∞_𝑛=0𝑥_𝑛𝜒_{[𝑛,𝑛+1)}(𝑡).(3.11)

Example 3.5. For 𝑓(𝑡)=𝜒_[0,1](𝑡), the level function is 𝑓^∘(𝑡)=(1−𝛼)𝑡^−𝛼𝜒_[0,1](𝑡),(3.12) where 𝛼=1−𝑠^′/𝑝^ so that ‖𝑓^∘‖_𝑝,𝑠=𝑠′𝑝′^1/𝑠′.(3.13)
If 𝑥=(𝑥_𝑛)_𝑛, where 𝑥_𝑛=10iiff𝑛𝑛≤>𝑘𝑘,,(3.14) then 𝑥^∘𝑛=⎧⎪⎨⎪⎩∑𝑘^𝑘𝑖=1𝑖^−𝛼‖𝑥_∘0,𝑛^−𝛼iofth𝑛e≤rw𝑘is,e,‖_𝑝,𝑠=𝑘^𝑘_𝑛=1𝑛^−𝛼^−1/𝑠′.(3.15)

We also need the following technical lemma.

Lemma 3.6. Let 1≤𝑝<𝑠<∞ and 𝐶_𝑝𝑠=sup_{𝑚,𝑛∈ℕ∗}1𝑛^{𝑚+𝑛−1}_𝑘=𝑚𝑘^{𝑠/𝑝−1}^1/𝑠1𝑛^{𝑚+𝑛−1}_𝑘=𝑚𝑘^{𝑠′/𝑝′−1}^1/𝑠′.(3.16) Then, 𝐶_𝑝𝑠=𝑝𝑠^1/𝑠𝑝𝑠^^1/𝑠′.(3.17)

Proof. We prove first that 𝐶_𝑝𝑠=sup_{𝑛∈ℕ∗}1𝑛^𝑛_𝑘=1𝑘^{𝑠/𝑝−1}^1/𝑠1𝑛^𝑛_𝑘=1𝑘^{𝑠′/𝑝′−1}^1/𝑠′.(3.18) The inequality 𝐶_𝑝𝑠≥sup_{𝑛∈ℕ∗}1𝑛^𝑛_𝑘=1𝑘^{𝑠/𝑝−1}^1/𝑠1𝑛^𝑛_𝑘=1𝑘^{𝑠′/𝑝′−1}^1/𝑠′(3.19) is trivial. For the converse inequality, it is enough to prove that ^𝑛_𝑘=1𝑘^{𝑠/𝑝−1}^1/𝑠^𝑛_𝑘=1𝑘^{𝑠′/𝑝′−1}^1/𝑠′≥^{𝑚+𝑛−1}_𝑘_=𝑚𝑘^{𝑠/𝑝−1}^1/𝑠^{𝑚+𝑛−1}_𝑘=𝑚𝑘^{𝑠′/𝑝′−1}^1/𝑠′(3.20) for all 𝑚≥𝑛.
The above inequality can be obviously written in the form ^𝑛_𝑘=1𝑘^{𝑠/𝑝−1}^𝑠′/𝑠^𝑛_𝑘=1𝑘^{𝑠′/𝑝′−1}≥^𝑛_𝑘=1(𝑘+𝑚−1)^{𝑠/𝑝−1}^𝑠′/𝑠^𝑛_𝑘=1(𝑘+𝑚−1)^{𝑠′/𝑝′−1}.(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 ∑^𝑛𝑘=1𝑥_𝑘∑^𝑛𝑘=1𝑥^𝑟𝑘^1/𝑟≥∑^𝑛𝑘=1𝑧_𝑘∑^𝑛𝑘=1𝑧^𝑟𝑘^1/𝑟.(3.22)
Observe that both (𝑥_𝑘)_𝑘 and (𝑧_𝑘)_𝑘 are decreasing sequences. Denote 𝑋_𝑛=∑^𝑛𝑘=1𝑥_𝑛, 𝑍_𝑛=∑^𝑛𝑘=1𝑧_𝑛, and 𝐶=𝑋_𝑛/𝑍_𝑛.
Since (𝑥_𝑘/𝑧_𝑘)_𝑘 is decreasing, so is also (𝑋_𝑘/𝑍_𝑘)_𝑘, and hence 𝑋_𝑘/𝑍_𝑘≥𝑋_𝑛/𝑍_𝑛=𝐶 for 𝑘≤𝑛−1. 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 ^𝑛_𝑘=1𝑥^𝑟𝑘≥^𝑛_𝑘=1𝐶𝑧_𝑘^𝑟(3.23) or ^𝑛_𝑘=1𝑥^𝑟𝑘≥𝐶^𝑟𝑛_𝑘=1𝑧^𝑟𝑘.(3.24) The last inequality implies ^𝑛_𝑘=1𝑥^𝑟𝑘^1/𝑟≤∑^𝑛𝑘=1𝑥_𝑘∑^𝑛𝑘=1𝑧_𝑘^𝑛_𝑘=1𝑧^𝑟𝑘^1/𝑟,since𝑟<0.(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)∈ℓ^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, ‖𝑥^∘‖_𝑝,𝑠≤‖𝑥‖_𝑝,𝑠≤𝑝𝑠^1/𝑠𝑝𝑠^^1/𝑠′‖𝑥^∘‖_𝑝,𝑠.(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^∘𝑗/𝑗^−𝛼=𝜆_𝑘, 𝑗∈𝐼_𝑘. Since 𝛼=(𝑠/𝑝−1)/(𝑠−1), (𝑥^∘𝑗)^𝑠−1𝑗^{𝑠/𝑝−1}=𝜆^𝑘𝑠−1, applying Hölder’s inequality, we obtain that _𝑗_∈𝐼𝑘𝑥^∘𝑗^𝑠𝑗^{𝑠/𝑝−1}=𝜆^𝑘𝑠−1_𝑗_∈𝐼𝑘𝑥^∘𝑗=⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑗^−𝛼⎞⎟⎟⎠^1−𝑠⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑥_𝑗⎞⎟⎟⎠^𝑠−1⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑥^∘𝑗=⎛⎜⎜⎝_{𝑗∈𝐼𝑘}⎞⎟⎟⎠𝑗^−𝛼⎞⎟⎟⎠^1−𝑠⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑥_𝑗⎞⎟⎟⎠^𝑠≤_{𝑗∈𝐼𝑘}𝑥^𝑠𝑗𝑗^{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^𝑛𝑠−1𝑛^{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 ‖𝑥‖^{𝑠𝑝,𝑠}=^∞_𝑛=1𝑥_𝑛𝜓_𝑛≤^∞_𝑛=1𝑥_𝑛𝜓_𝑛≤^∞_𝑛=1𝑥^∘𝑛𝜓_𝑛≤‖𝑥^∘‖_𝑝,𝑠‖‖𝜓‖‖_{𝑝′,𝑠′}.(3.28)
We note that to obtain the right-hand side inequality in (3.26) it is sufficient to prove that ‖‖𝜓‖‖_𝑝′,s′≤𝑝𝑠^1/𝑠𝑝𝑠^^1/𝑠′‖𝑥‖^{𝑠−1𝑝,𝑠}.(3.29)
Let 𝐸∶={𝑛≥1∶𝜓_𝑛=𝜓_𝑛}. Then, we have that ℕ^∗−𝐸=_𝑘𝐼_𝑘,(3.30) where {𝐼_𝑘} are disjoint and such that 𝜓_𝑛=1||𝐼_𝑘||+1_{𝑗∈𝐼𝑘}𝜓_𝑗,∀𝑛∈𝐼_𝑘.(3.31) By Hölder’s inequality, we obtain that _𝑗_∈𝐼𝑘𝜓_𝑗≤⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑗^{𝑠/𝑝−1}⎞⎟⎟⎠^1/𝑠⎛⎜⎜⎝_{𝑗∈𝐼𝑘}𝑥^𝑠𝑗𝑗^{𝑠/𝑝−1}⎞⎟⎟⎠^1/𝑠′.(3.32)
We also have that _𝑛_∈𝐸𝜓_𝑛^𝑠′𝑛^{𝑠′/𝑝′−1}=_𝑛∈𝐸𝜓_𝑛^𝑠′𝑛^{𝑠′/𝑝′−1}=_𝑛∈𝐸𝑥^𝑠𝑛𝑛^{𝑠/𝑝−1}.(3.33)
Hence, by (3.31) and (3.32), it yields that _𝑛_∈𝐼𝑘𝜓_𝑛^𝑠𝑛^{𝑠′/𝑝′−1}≤1||𝐼_𝑘||+1^𝑠′⎛⎜⎜⎝_{𝑚∈𝐼𝑘}𝑚^{𝑠/𝑝−1}⎞⎟⎟⎠^𝑠′/𝑠⎛⎜⎜⎝_{𝑚∈𝐼𝑘}𝑥^𝑠𝑚𝑚^{𝑠/𝑝−1}⎛⎜⎜⎝_{𝑛∈𝐼𝑘}⎞⎟⎟⎠,𝑛^{𝑠′/𝑝′−1}⎞⎟⎟⎠=𝐶_{𝑠′𝑝𝑠,𝑘𝑚}_∈𝐼𝑘𝑥^𝑠𝑚𝑚^{𝑠/𝑝−1},(3.34) where 𝐶_{𝑝𝑠,𝑘}=1||𝐼_𝑘||+1⎛⎜⎜⎝_{𝑛∈𝐼𝑘}𝑛^{𝑠/𝑝−1}⎞⎟⎟⎠^1/𝑠⎛⎜⎜⎝_{𝑛∈𝐼𝑘}𝑛^{𝑠′/𝑝′−1}⎞⎟⎟⎠^1/𝑠′.(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 𝑘₀, we take 𝑥=(𝑥_𝑛)_𝑛, where 𝑥_𝑛=⎧⎪⎨⎪⎩∑𝑘^0𝑘0_𝑖=1𝑖^−𝛼0,𝑛^−𝛼iofth𝑛e≤rw𝑘is₀e,.(3.36)
The right-hand side inequality (3.26) becomes equality for 𝑥=(𝑥_𝑛)_𝑛, where 𝑥_𝑛=1if𝑛≤𝑘⁰0,otherwise,.(3.37) We have that ‖𝑥‖_𝑝,𝑠=(∑^𝑘0_𝑛=1𝑛^{𝑠/𝑝−1})^1/𝑠. It is easy to verify that if 𝑥^∘=(𝑥^∘𝑛)_𝑛, where 𝑥^∘𝑛=⎧⎪⎨⎪⎩∑𝑘^0𝑘0_𝑖=1𝑖^−𝛼0,𝑛^−𝛼iofth𝑛e≤rw𝑘is₀e,,(3.38) then ‖𝑥^∘‖_𝑝,𝑠=𝑘₀⎛⎜⎜⎝^𝑘0_𝑛=1𝑛^−𝛼⎞⎟⎟⎠^−1/𝑠′.(3.39) Since 𝑘₀ 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 _𝑛_∈𝐼𝑘𝜆_𝑘𝑛^−1/𝑝=_{𝑛∈𝐼𝑘}𝑥^∘𝑛=_{𝑛∈𝐼𝑘}𝑥_𝑘≤‖𝑥‖_𝑝,∞𝑛_∈𝐼𝑘𝑛^−1/𝑝.(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 Hölder’s inequality, we obtain that 𝑥_𝑛𝑛^1/𝑝≤^𝑛_𝑖=1𝑥_𝑖𝑛^1/𝑝−1≤^𝑛_𝑖=1𝑥^∘𝑖𝑛^1/𝑝−1≤𝑛^1/𝑝^𝑛_𝑖=11𝑛𝑖^−1/𝑝sup_{1≤𝑖≤𝑛}𝑖^1/𝑝𝑥^∘𝑖.(3.41) It follows that sup_𝑛≥1𝑛^1/𝑝𝑥_𝑛≤sup_𝑛≥1𝑛^1/𝑝∑^𝑛𝑖=1𝑖^−1/𝑝𝑛sup_{1≤𝑖≤𝑛}𝑖^1/𝑝=‖𝑥_∘𝑥^∘𝑖‖_𝑝,∞𝐶_𝑝,(3.42) where 𝐶_𝑝=sup_𝑛≥1sup^𝑘=1,𝑛𝑘^1/𝑝∑^𝑛𝑘=1𝑘^−1/𝑝=sup_𝑛≥1𝑛𝑛^1/𝑝∑^𝑛𝑘=1𝑘^−1/𝑝𝑛=𝑝^.(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.

Remark 3.8. Let 1<𝑝<𝑠≤∞. Let 𝑥=(𝑥_𝑛)_𝑛∈ℓ^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 𝜑_𝑛=𝑛^−𝛼,𝛼=1−𝑠^𝑝^,𝑛≥1.(3.44) Then, the equality ‖𝑥^∘‖_𝑝,𝑠=‖𝑥‖_𝑝,𝑠(3.45) holds if and only if 𝑥^∘𝑛=𝑥_𝑛 for all 𝑛≥1.
Indeed, inequality (3.27) becomes equality if and only if (𝑥_𝑛𝑛^𝛼)_𝑛 is constant on 𝐼_𝑘. Therefore, (3.45) holds if and only if 𝑥^∘𝑛=𝑥_𝑛, for all 𝑛≥1, hence if and only if (𝑥_𝑛𝑛^𝛼)_𝑛 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 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠.

Proposition 4.1. Let 1<𝑝<∞, 1≤𝑠≤∞, and 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠. Then, the following statements hold.(a) One has ‖𝑥‖^𝑝,𝑠≤‖𝑥‖_𝑝,𝑠.(4.1)(b) If 𝑠≤𝑝, then ‖𝑥‖^𝑝,𝑠=‖𝑥‖_𝑝,𝑠.(4.2)(c) If 𝑝<𝑠≤∞ and if the sequence (𝑥^∗𝑛𝑛^{1−𝑠′/𝑝′})_𝑛 is nonincreasing, then (4.2)holds.(d) If 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠, then one has that ‖𝑥‖^𝑝,𝑠≤inf_𝑥≺𝑧‖𝑧‖_𝑝,𝑠.(4.3)

Proof. Indeed, (a) follows from Hölder’s inequality.
(b) Take 𝜓=(𝜓_𝑛)_𝑛 with 𝜓_𝑛=(𝑥^∗𝑛)^𝑠−1𝑛^{𝑠/𝑝−1}, 𝑛≥1. The sequence 𝜓=(𝜓_𝑛)_𝑛 is nonincreasing and we have that ‖𝜓‖^𝑠′_{𝑝′,𝑠′}=^∞_𝑛=1‖𝑥‖_{𝑠𝑝,𝑠}𝜓^𝑠′_𝑛𝑛^{𝑠′/𝑝′−1}=‖𝑥‖^{𝑠𝑝,𝑠},=^∞_𝑛=1𝑥^∗𝑛𝜓_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 ^∞_𝑛=1𝑥_𝑛𝑦_𝑛≤inf_𝑥≺𝑧‖𝑧‖_𝑝,𝑠‖𝑦‖_{𝑝,𝑠},(4.5) where the infimum is taken over all nonnegative sequences 𝑧=(𝑧_𝑛)_𝑛∈ℓ^𝑝,𝑠 such that 𝑥≺𝑧 and where 𝑦=(𝑦_𝑛)_𝑛∈ℓ^{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.

Theorem 4.2. Let 1<𝑝<𝑠≤∞, and let 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠 be a nonnegative and nonincreasing sequence. Let 𝛼=1−𝑠^/𝑝^ and 𝜑^(𝛼)=(𝜑^𝑛(𝛼))_𝑛, where 𝜑^𝑛(𝛼)=𝑛^−𝛼. Then, one has that ‖𝑥‖^𝑝,𝑠=inf_𝑥≺𝑧‖𝑧‖_𝑝,𝑠=‖𝑥^∘‖_𝑝,𝑠,(4.6) where 𝑥^∘=(𝑥^∘𝑛)_𝑛 is the level sequence of 𝑥=(𝑥_𝑛)_𝑛 with respect to the sequence 𝜑^(𝛼)=(𝜑^𝑛(𝛼))_𝑛.

Proof. By Proposition 4.1 and Theorem 3.3(b), it is sufficient to prove that ‖𝑥‖^𝑝,𝑠≥‖𝑥^∘‖_𝑝,𝑠.(4.7)
We denote 𝐸={𝑛≥1∶𝑥_𝑛=𝑥^∘𝑛}. According to Theorem 3.3 it yields that ℕ^∗−𝐸=∪_𝑘𝐼_𝑘, where 𝐼_𝑘 are disjoint finite subsets of positive integers such that _𝑛_∈𝐼𝑘𝑥_𝑛=_{𝑛∈𝐼𝑘}𝑥^∘𝑛.(4.8) Assume first that 𝑠<∞ and let 𝜓=(𝜓_𝑛)_𝑛, where 𝜓_𝑛=(𝑥^∘𝑛)^𝑠−1𝑛^{𝑠/𝑝−1}, 𝑛≥1. As before, we have that ‖𝜓‖^𝑠′_{𝑝′,𝑠,}=^∞_𝑛=1𝜓^𝑠′_𝑛𝑛^{𝑠′/𝑝′−1}=‖𝑥^∘‖_𝑝,𝑠.(4.9) Choose 𝑦=(𝑦_𝑛)_𝑛, where 𝑦_𝑛=𝜓_𝑛/‖𝑥^∘‖^{𝑠−1𝑝,𝑠}. Then, ‖𝑦‖_{𝑝′,𝑠′}=1.
From Theorem 3.3, it follows that, for each 𝑘, 𝑥^∘𝑛=𝜆_𝑘𝑛^−𝛼, 𝑛∈𝐼_𝑘. Thus, 𝜓_𝑛=𝜆^𝑘𝑠−1 and ‖𝑥^∘‖^{𝑠−1𝑝,𝑠}_𝑛_∈𝐼𝑘𝑥_𝑛𝑦_𝑛=𝜆^𝑘𝑠−1_𝑛_∈𝐼𝑘𝑥_𝑛=𝜆^𝑘𝑠−1_𝑛_∈𝐼𝑘𝑥^∘𝑛=_{𝑛∈𝐼𝑘}𝑥^∘𝑛^𝑠𝑛^{𝑠/𝑝−1}.(4.10) Moreover, we have that ‖𝑥^∘‖^{𝑠−1𝑝,𝑠}_𝑛_∈𝐸𝑥_𝑛𝑦_𝑛=_𝑛∈𝐸𝑥^∘𝑛^𝑠𝑛^{𝑠/𝑝−1}.(4.11) Thus, we obtain that ∑^∞𝑛=1𝑥_𝑛𝑦_𝑛=‖𝑥^∘‖_𝑝,𝑠. This implies (4.7).
Let now 𝑠=∞. We consider only the nontrivial case 𝐼₁={1,2,…,𝑚}. Let 𝑦=(𝑦_𝑛)_𝑛, where 𝑦_𝑛=⎧⎪⎨⎪⎩∑^𝑚𝑖=11𝑖^−1/𝑝0,iofth𝑛e≤rw𝑚is,e(4.12) and ‖𝑦‖_𝑝^′_,1=1. Then, ^∞_𝑛=1𝑥_𝑛𝑦_𝑛=1∑^𝑚𝑖=1𝑖^{𝑚−1/𝑝}_𝑛=1𝑥_𝑛=1∑^𝑚𝑖=1𝑖^{𝑚−1/𝑝}_𝑛=1𝑥^∘𝑛=𝜆₁=‖𝑥^∘‖_𝑝,∞.(4.13) This implies (4.7), and the proof is complete.

Remark 4.3. Let 1<𝑝<𝑠≤∞ and let 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠 be a nonnegative and nonincreasing sequence. Then, by Remark 3.8, the equality (see (4.2)) ‖𝑥‖^𝑝,𝑠=‖𝑥‖_𝑝,𝑠(4.14) holds if and only if (𝑥_𝑛𝑛^𝛼)_𝑛 is decreasing. Note also that (4.2) holds in general if 1≤𝑠<𝑝 (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 1<𝑝<∞ and 𝑝<𝑠≤∞. Then, for any sequence 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠, it yields that ‖𝑥‖_𝑝,𝑠≤𝐶_𝑝𝑠‖𝑥‖^𝑝,𝑠,(4.15) where 𝐶_𝑝𝑠 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 𝛼₁≥𝛼₂≥⋯𝛼_𝜈 be positive numbers, and let {𝜂_𝑗𝑘} be a (𝑁×𝜈)-matrix of positive numbers, 1≤𝑗≤𝑁,1≤𝑘≤𝜈. Set 𝛽_𝑘=^𝑁_𝑗=1𝜂_𝑗𝑘,𝑘=1,…,𝜈.(5.1) Assume that 𝛽₁+⋯+𝛽_𝑘≥𝛼₁+⋯+𝛼_𝑘(5.2) for any 𝑘=1,…,𝜈. Let 𝜂=max𝜂_𝑗𝑘. Then, for any 𝑗=1,…,𝑁, there exists a permutation {𝜂̃_𝑗𝑘}^𝜈𝑘=1 of the 𝜈-tuple {𝜂_𝑗𝑘}^𝜈𝑘=1 such that 𝛼_𝑘≤𝛽̃_𝑘+𝜂,𝛽̃_𝑘=^𝑁_𝑗=1𝜂̃_𝑗𝑘,(5.3) for any 𝑘=1,…,𝜈.

Our main result in this section reads the following.

Theorem 5.2. Let 1<𝑝<∞ and 1≤𝑠≤∞. Then, for any sequence 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠, ‖𝑥‖^𝑝,𝑠=‖𝑥‖_(𝑝,𝑠).(5.4)

Proof. If 𝑠≤𝑝, then it is well known that (see, e.g., [7, 15]) ‖𝑥‖^𝑝,𝑠=‖𝑥‖_𝑝,𝑠=‖𝑥‖_(𝑝,𝑠).(5.5) Let 1<𝑝<𝑠≤∞. We prove first that ‖𝑥‖^𝑝,𝑠≤‖𝑥‖_(𝑝,𝑠).(5.6) Let 𝑦=(𝑦_𝑛)_𝑛∈ℓ^{𝑝′,𝑠′}, and let 𝑥=_𝑘𝑥^(𝑘).(5.7) Then, by Hölder’s inequality, we have that ^∞_𝑛=1||𝑥_𝑛𝑦_𝑛||≤_𝑘∞_𝑛=1||𝑥^𝑛(𝑘)𝑦_𝑛||≤_𝑘‖‖𝑥^(𝑘)‖‖_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 𝑛₀ such that 𝑥_𝑖=0, for 𝑖>𝑛₀, 𝑛₀∈ℕ^∗. From Theorem 3.3, we have also that 𝑥^∘𝑖=0 for 𝑖>𝑛₁, 𝑛₁∈ℕ^∗. Denote 𝑘₀=max{𝑛₀,𝑛₁}. According to Theorem 4.2, it yields that ‖𝑥‖^𝑝,𝑠=‖𝑥^∘‖_𝑝,𝑠,(5.10) where 𝑥^∘=(𝑥^∘𝑛)_𝑛 is the level sequence of 𝑥=(𝑥_𝑛)_𝑛 with respect to the sequence 𝜑=(𝜑_𝑛)_𝑛, 𝜑_𝑛=𝑛^−𝛼, 𝛼=1−𝑠^/𝑝^.
Let 𝜀>0. Choose 𝛿=𝛿(𝜀,𝑘₀)>0 such that 𝛿<𝜀(1/‖𝑧‖_𝑝,𝑠), with 𝑧=(𝑧_𝑛)_𝑛, where 𝑧_𝑛=1if𝑛≤𝑘⁰0,otherwise,.(5.11)
Choose 𝑁∈ℕ^∗ such that max_𝑖𝑥^∘𝑖<𝑁⋅𝛿.(5.12)
By using Lemma 5.1 with 𝛼_𝑘∶=𝑥_𝑘, 𝛽_𝑘∶=𝑥^∘𝑘, 𝜂_𝑗𝑘∶=𝑥^∘𝑘/𝑁, 1≤𝑘≤𝑘₀, 1≤𝑗≤𝑁, we have that there exists a permutation of {𝜂_𝑗𝑘}, {𝜂̃_𝑗𝑘} such that 𝑥_𝑘≤^𝑁_𝑗=1𝜂̃_𝑗𝑘+𝛿,∀𝑘≤𝑘₀.(5.13)
We define now 𝑡^(𝑗)=(𝑡^𝑘(𝑗))_𝑘, where 𝑡^𝑘(𝑗)∶=𝜂̃_𝑗𝑘. Since 𝑡^(𝑗) is equimeasurable with 𝑥^∘/𝑁, we have that ‖‖𝑡^(𝑗)‖‖_𝑝,𝑠=‖𝑥^∘‖_𝑝,𝑠𝑁,𝑗≤𝑁,(5.14) which, in its turn, implies that ‖𝑥‖_(𝑝,𝑠)≤^𝑁_𝑗=1‖‖𝑡^(𝑗)‖‖_𝑝,𝑠+𝛿⋅‖𝑧‖_𝑝,𝑠≤‖𝑥^∘‖_𝑝,𝑠+𝜀.(5.15) Since 𝜀 was arbitrary, it follows that (5.9) holds. The proof is complete.

Corollary 5.3. Let 𝑥=(𝑥_𝑛)_𝑛∈ℓ^𝑝,𝑠, 1≤𝑝<∞, 1≤𝑠≤∞. Then, ‖𝑥‖_(𝑝,𝑠)=‖𝑥^∗‖_(𝑝,𝑠).(5.16)

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 1<𝑝<𝑠≤∞, and suppose that 𝑥^(𝑘)=(𝑥^𝑛(𝑘))_𝑛∈ℓ^𝑝,𝑠, 𝑘=1,…,𝑁. Then, one has that ‖‖‖‖^𝑁_𝑘=1𝑥^(𝑘)‖‖‖‖_𝑝,𝑠≤𝐶_𝑁𝑝𝑠_𝑘=1‖‖𝑥^(𝑘)‖‖_𝑝,𝑠,(6.1) 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 Kamińska and Parrish [13]. They solved the problem in a completely different way.

Proof. We note that (6.1) is equivalent to the inequality ‖𝑥‖_𝑝,𝑠≤𝐶_𝑝𝑠‖𝑥‖_(𝑝,𝑠),(6.2) where 𝑥=(𝑥_𝑛)_𝑛 is any sequence from ℓ^𝑝,𝑠. Inequality (6.2) follows directly from Theorems 4.4 and 5.2. Moreover, for 𝑥=(𝑥_𝑛)_𝑛 with 𝑥_𝑛=1if𝑛≤𝑘⁰0,otherwise,,(6.3) for a fixed 𝑘₀∈ℕ^∗, we obtain that ‖𝑥‖_𝑝,𝑠=⎛⎜⎜⎝^𝑘0_𝑛=1𝑛^{𝑠/𝑝−1}⎞⎟⎟⎠^1/𝑠.(6.4) From Theorems 4.2 and 5.2, we have that ‖𝑥‖_(𝑝,𝑠)=𝑘⁰∑^𝑘0𝑛=1𝑛^{𝑠′/𝑝′−1}^1/𝑠′(6.5) 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 ℓ^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.

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.