Lorentz-Shimogaki-Arazy-Cwikel Theorem Revisited

We present a new approach to Lorentz-Shimogaki and Arazy-Cwikel Theorems which covers all range of $p,q\in (0,\infty]$ for function spaces and sequence spaces. As a byproduct, we solve a conjecture of Levitina and the last two authors.


Introduction
Descriptions of interpolation spaces for couples of L p -spaces, 1 ≤ p ≤ ∞ were extensively researched at the end of the 70's and in the 80's, providing satisfying answers to most problems which were considered relevant at the time.
However, new questions arising from noncommutative analysis recently highlighted some gaps in our knowledge of this subject, especially for the case of p < 1.In this paper, we revisit some important results of the literature ( [20], [1], [25]), generalising them and thus filling some of the holes that were revealed in the theory.In particular, we answer a question asked in [19] by Levitina and the last two authors and already partially studied in [11] regarding the interpolation theory of sequence spaces (see Theorem 1.2).Besides this new result, this paper introduces a general approach which covers the range of all 0 ≤ p ≤ ∞ and is self-contained.It puts an emphasis on the use of the space L 0 of all finitely supported measurable functions.As far as the authors know this space rarely appears in interpolation theory (however, see [15], [2] and [14]).We provide evidence that L 0 is a suitable "left endpoint" on the interpolation scale of L p -spaces, despite its possessing an atypical structure (it is not even an F -space).
Recall that a function space E is an interpolation space for the couple (L p , L q ) if any linear operator T bounded on L p and L q is also bounded on E (see Definition 1.6).This notion provides a way of transfering inequalities well-known in L p -spaces to more exotic ones.To both understand the range of applicability of this technique and be able to check whether it applies to a given function space E, we are interested in simple descriptions of interpolation spaces for the couple (L p , L q ).This problem has a long history starting with seminal Calderón-Mityagin theorem [21] and [8] on the couple (L 1 , L ∞ ) and followed by Lorentz and Shimogaki's [20] results on the couples (L 1 , L q ) and (L p , L ∞ ) with 1 ≤ p, q ≤ ∞, which are stated in terms of various submajorizations, see also [17,Theorem 7.2].
We consider two orders: head majorization and tail majorization.Head majorization coincides with the usual notion of (sub)majorization and already appears in Calderón's work.It is defined on L 1 + L ∞ by: g ≺≺ hd f ⇔ ∀t > 0, t 0 µ(s, g)ds ≤ t 0 µ(s, f )ds.
If moreover f, g ∈ L 1 and f 1 = g 1 , then we write g ≺ hd f .Above and in the remainder of the text µ(g) : t → µ(t, g) denotes the right-continuous decreasing rearangement of g.Note that µ(g) is well-defined if and only if g belongs to L 0 +L ∞ .
Tail majorization is defined on L 0 + L 1 by: If moreover f, g ∈ L 1 and f 1 = g 1 , then we write g ≺ tl f .Remark that g ≺ tl f if and only if f ≺ hd g.
It is well-known that the order ≺≺ hd is strongly linked to the interpolation theory of the couples (L p , L ∞ ).We show that similarly, the order ≺≺ tl is linked to the couples (L 0 , L q ).Combining these two tools, we recover in a self-contained manner characterisations of interpolation spaces for couples of arbitrary L p -spaces, 0 < p ≤ ∞ earlier obtained in [7].Note that tail majorization coincides with the weak supermajorization of [12].
Let X be the linear space of all measurable functions.If not precised otherwise, the underlying measure space we are working on is (0, ∞) equipped with the Lebesgue measure m.We obtain the following: Theorem 1.1.Let E ⊂ X be a quasi-Banach function space (a priori, not necessarily symmetric).Let p, q ∈ (0, ∞) such that p < q.Then: (a) E is an interpolation space for the couple (L p , L ∞ ) if and only if there exists c p,E > 0 such that for any f ∈ E and g ∈ L p + L ∞ , (b) E is an interpolation space for the couple (L 0 , L q ) if and only if there exists c q,E > 0 such that for any f ∈ E and g ∈ L 0 + L q , |g| q ≺≺ tl |f | q ⇒ g ∈ E and g E ≤ c q,E f E ; (c) E is an interpolation space for the couples (L 0 , L q ) and (L p , L ∞ ) if and only if it is an interpolation space for the couple (L p , L q ).This extends results of Lorentz-Shimogaki and Arazy-Cwikel to the quasi-Banach setting and contributes to the two first questions asked by Arazy in [9, p.232] in the particular case of L p -spaces, 0 < p < ∞.As mentioned before, our approach places L 0 as a left endpoint on the interpolation scale of L p -spaces, in sharp contrast to earlier results which focused mostly on Banach spaces and had L 1 playing this part.An advantage of our approach is that it naturally encompasses every symmetric quasi-Banach space since they are all interpolation spaces for the couple (L 0 , L ∞ ) (see [15], [2]).On the contrary, there exist some symmetric Banach spaces which are not interpolation spaces for the couple (L 1 , L ∞ ) (see [24]).This led to some difficulties which were customarily be circumvented with the help of various technical conditions such as the Fatou property (as appears for example in [3]).
Our strategy in this paper is totally different from the techniques used in [21,8,20,25,10,15,1,9,2,11,17,3,7,4] and is based on partition lemmas, which were originally developed in a deep paper due to Braverman and Mekler [6], which lies outside of the realm of interpolation theory.The approach of Braverman and Mekler was subsequently revised and redeveloped in [26] and precisely this revision consitutes the core of our approach in this paper.
We restate partition lemmas based on [26,Proposition 19] in Section 3.These lemmas allow us to restrict head and tail majorizations to very simple situations and reduce the problem to functions taking at most two values.Then, we deduce interpolation results from those structural lemmas.
Note that this scheme of proof is quite direct and in particular, does not involve at any point duality related arguments which are applicable only to Banach spaces ( [20]) or more generally to L-convex quasi-Banach spaces ( [16], [23]).
In Section 5, we pursue the same type of investigation, but in the setting of sequence spaces.The non-diffuse aspect of the underlying measure generates substantial technical difficulties.In particular, we require a new partition lemma which is not as efficient as those in Section 3 (compare Lemmas 5.2 and 3.8).This deficiency has been first pointed out to the authors by Cwikel.However, we are still able to resolve the conjecture of [19] (in the affirmative) by combining Lemma 5.1 with a Boyd-type argument which we borrow from Montgomery-Smith [22].In particular, we substantially strengthen the results in [11].Here is the precise statement that we obtain: Theorem 1.2.Let E ⊂ ℓ ∞ be a quasi-Banach sequence space and q ≥ 1.The following conditions are equivalent: (a) there exists p < q such that E is an interpolation space for the couple (ℓ p , ℓ q ); (b) there exists c > 0 such that for any We freely use results of Cwikel [10] and the first author [7] to avoid repeating too many similar arguments.

Preliminaries
Interpolation spaces.The reader is referred to [5] for more details on interpolation theory and [18] for an introduction to symmetric spaces.In the remainder of this section, p and q will denote two nonnegative reals such that p ≤ q.
Let (Ω, m) be any measure space (in particular the following definitions apply to N equipped with the counting measure i.e sequence spaces).As previously mentioned, L 0 (Ω) ⊂ X (Ω) denotes the set of functions whose supports have finite measures, it is naturally equipped with the group norm The "norm" of a linear operator T : L 0 (Ω) → L 0 (Ω), is defined as follows: .
Definition 1.3.A linear space E ⊂ X (Ω) becomes a quasi-Banach function space when equipped with a complete quasi-norm .E such that Definition 1.5 (Bounded operator on a couple of quasi-Banach function spaces).Let X and Y be quasi-Banach function spaces.We say that a linear operator T is bounded on (X, Y ) if T is defined from X + Y to X + Y and restricts to a bounded operator from X to X and from Y to Y .Set: Les us now recall the precise abstract definition of an interpolation space (see [5], [18]).Definition 1.6 (Interpolation space between quasi-Banach function spaces).Let X, Y and Z be quasi-Banach function spaces on Ω.We say that Z is an interpolation space for the couple (X, Y ) if X ∩ Y ⊂ Z ⊂ X + Y and any bounded operator on (X, Y ) restricts to a bounded operator on Z. Denote by Int(X, Y ) the set of interpolation spaces for the couple (X, Y ).
Theorem 1.7.Let V be a separable topological linear space and let A, B, C ⊂ V be quasi-Banach spaces.If C is an interpolation space for the couple (A, B), then there exists a constant c(A, B, C) > 0 such that for any bounded operator T on (A, B), Proof.In [18, Lemma I.4.3], the assertion is proved for Banach spaces.The argument for quasi-Banach spaces is identical (because it relies on the closed graph theorem, which holds for F -spaces, hence, for quasi-Banach spaces).
If, in particular, 0 < p < q < ∞ and if E is an interpolation space for the couple (L p (Ω), L q (Ω)), then there exists a constant c > 0 called interpolation constant of E for (L p (Ω), L q (Ω)) such that for any bounded operator T on (L p (Ω), L q (Ω)), Finally, let us recall the definition of the K-functional associated to a couple of quasi-Banach spaces.For any t > 0, and f ∈ X + Y , let In addition to that, we need to define an interpolation space between L 0 and a quasi-Banach space.Definition 1.8 (Bounded operator on a couple (L 0 (Ω), Y ) for a quasi-Banach function space Y ).Let Y be a quasi-Banach function space.We say that a linear operator T is bounded on (L 0 (Ω), Y ) if T is defined from L 0 (Ω) + Y to L 0 (Ω) + Y and restricts to a bounded operator from L 0 (Ω) to L 0 (Ω) and from Y to Y. Definition 1.9 (Interpolation space for a couple (L 0 (Ω), Y ) for a quasi-Banach function space Y ).Let Y and Z be quasi-Banach function spaces on Ω.We say that Z is an interpolation space for the couple Y and any bounded operator on (L 0 (Ω), Y ) restricts to a bounded operator on Z. Denote by Int(L 0 (Ω), Y ) the set of interpolation spaces for the couple (L 0 (Ω), Y ).
Symmetry of interpolation spaces.In this subsection, we show that a quasi-Banach interpolation space for a couple of symmetric spaces can always be remormed into a symmetric space.Note that similar results can be found in the literature, see for example [18,Theorem 2.1].
As usual, we will use the term measure preserving for a map ω between measure spaces (Ω 1 , A 1 , m 1 ) and (Ω 2 , A 2 , m 2 ) verifying, and let ε > 0. Assume that µ(f ) = µ(g).There exists a measure preserving map Define, for any n ∈ Z, By assumption, m(F n ) = m(G n ) for every n ∈ Z.Let ω n : G n → F n be an arbitrary measure preserving bijection.Define measure preserving map ω : supp(g) → supp(f ) by concatenating

By asumption for any
Define the measure preserving map ω : supp(g) → supp(f ) by concatenating the ω n 's.For any n ≥ 0 and any t ∈ G n , be quasi-Banach function spaces.Assume that A and B are symmetric and that E is an interpolation space for the couple (A, B).Then E admits an equivalent symmetric quasi-norm.
Proof.Let f ∈ E and g ∈ L 0 + L ∞ .Assume that µ(g) ≤ µ(f ).By Lemma 1.10, there exists a map ω : supp(g) → supp(f ) such that for any t ∈ supp(g), Define, for any h ∈ X (Ω), Since ω is measure preserving, T is bounded on A and B of norm less than 2. Let c E be the interpolation constant of E for the couple (A, B) (as in Theorem 1.7).
We know that T f = g ∈ E and (1.1) Remark 1.12.It is not difficult to see that if the underlying measure space Ω contains both a continuous part and atoms, Lemma 1.11 is no longer true for A = L p (Ω), B = L q (Ω) and p < 1.However, one can observe that if A and B are fully symmetric (i.e.interpolation spaces between L 1 (Ω) and L ∞ (Ω)), Lemma 1.11 remains valid for any Ω.This is reminiscent of the conditions required in [25, Section 4].
2. Interpolation for the couple (L 0 , L q ) In this section, Ω = (0, ∞) (for brevity, we omit Ω in the notations).We investigate some basic properties of the interpolation couple (L 0 , L q ).First, we provide a statement analogous to Theorem 1.7 and applicable to L 0 .
Since the closed graph theorem does not apply to L 0 (it is not an F -space), our proof uses a concrete constructions that relies on the structure of the underlying measure space.
For any f ∈ X , denote by M f the multiplication operator g → f • g.
Theorem 2.1.Let E be a quasi-Banach function space and q ∈ (0, ∞].Assume that E is an interpolation space for the couple (L 0 , L q ).Then, there exists a constant c such that for any contraction T on (L 0 , L q ), T E→E ≤ c.
Obviously, U n and V n are bounded operators on the couple (L 0 , L q ).By assumption, Let us argue by contradiction.For any n ≥ 1, choose an operator T n which is a contraction on (L 0 , L q ) and such that (2.1) It is immediate that where By quasi-triangle inequality, we have Let k n ∈ {1, 2, 3, 4} be such that We, therefore, have Since the S n 's are in direct sum, S L0→L0 = sup n≥1 S n L0→L0 ≤ 1 and Moreover, E is an interpolation space for the couple (L 0 , L q ), it follows that S : Hence, we may assume without loss of generality that E .This contradicts the boundedness of S.
Remark 2.2.Theorem 2.1 above remains true for other underlying measure spaces: • for sequence spaces.Indeed, in the proof of Theorem 2.1, we only use properties of the underlying measure space in the first sentence, namely when we consider a partition of (0, ∞) into countably many sets, each of them isomorphic to (0, ∞).Since a partition satisfying the same property exists for Z + , Theorem 2.1 remains true for interpolation spaces between ℓ 0 and ℓ q .• for (0, 1).The same general idea applies in this case but some modification have to be made because the maps γ n introduced in the proof cannot be assumed to be measure preserving.The details are left to the reader.
Y is symmetric and that E is an interpolation space for the couple (L 0 , Y ).Then E admits an equivalent symmetric quasi-norm.
Proof.The argument follows that in Lemma 1.11 mutatis mutandi.
The following assertion is a special case Theorem 1.1 and an important ingredient in the proof of the latter theorem.
Theorem 2.4.Let E be a quasi-Banach function space and q ∈ (0, ∞).Assume that L 0 ∩ L q ⊂ E ⊂ L 0 + L q and that for any f ∈ E and g ∈ L 0 + L q , Then E is an interpolation space for the couple (L 0 , L q ).The rest of this section is occupied with the proof of Theorem 2.4.Let C E be the concavity modulus of E, that is, (the minimal) constant such that It allows to write a triangle inequality with infinitely many summands.This inequality should be understood in the usual sense: if scalar-valued series in the right hand side converges, then the series on the left hand side converges in E and the inequality holds.
Lemma 2.5.Let E be a quasi-Banach space.We have Proof.The proof is standard and is, therefore, omitted.
Lemma 2.6.Let E ⊂ L 0 + L q be a symmetric quasi-Banach function space.If It is immediate that h n E ≤ 1 and h n q ≥ n.
Let C E be the concavity modulus of the space E and let m ≥ 4C E be a natural number.Set By Lemma 2.5, we have Since h ∈ E, it follows that h ∈ L 0 + L q .Since h = µ(h) is constant on the interval (0, 1), it follows that h ∈ L q .Thus, It follows that h / ∈ L q .This contradiction completes the proof.
Lemma 2.7.Let E be as in Theorem 2.4.If T is a contraction on (L 0 , L q ), then T : E → E.
Proof.Take f ∈ E. Fix t > 0 and let A = A 1 A 2 , where Obviously, m(A) = t and Since t > 0 is arbitrary, it follows that By assumption, T f ∈ E. Since f ∈ E is arbitrary, it follows that T : E → E.
Our next lemma demonstrates the boundedness of the mapping T : E → E established in Lemma 2.7.
Lemma 2.8.Let E be as in Theorem 2.4.If T is a contraction on (L 0 , L q ), then the mapping T : E → E is bounded.
Proof.Using Lemma 2.3, we may assume without loss of generality that E is symmetric.
Let us show that the graph of T : Passing to a subsequence, if needed, we may assume without loss of generality that Clearly, Since T : L 0 → L 0 is a contraction, it follows that In particular, T (f 1n ) → 0 in measure.Since T : L q → L q is a contraction, it follows that Since f 2n is supported on a set of measure 1, it follows that In particular, T (f 2n ) → 0 in measure.Since T : L q → L q is a contraction, it follows that By Lemma 2.6, we have In particular, T (f 3n ) → 0 in measure.
Combining the last 3 paragraphs, we conclude that T f n → 0 in measure.However, T f n → g in E and, therefore, in measure.It follows that g = 0, which contradicts our choice of the sequence (f n ) n≥1 .This contradiction shows that the graph of T : E → E is closed.By the closed graph theorem, T : E → E is bounded.
Proof of Theorem 2.4.Let T be a bounded operator on (L 0 , L q ).Fix n ∈ N such that Let σ 1 n be a dilation operator defined by the usual formula (σ By Lemma 2.8, S : E → E is a bounded mapping.Since E is a symmetric quasi-Banach function space, it follows that σ n : E → E is bounded; this implies T : E → E.
Theorem 2.4 applies in particular to L p -spaces, p ≤ q.We decided to add more precise statement and to provide a direct proof of the latter.
Corollary 2.9.Let p, q ∈ (0, ∞) such that p < q.Then, L p is an interpolation space for the couple (L 0 , L q ).More precisely, if T is a contraction on (L 0 , L q ), then T is a contraction on L p .
Proof.Let us first consider characteristic functions.Let E be a set with finite measure.Since T is a contraction on L 0 , the measure of the support of T (χ E ) is less than m(E).So by Hölder's inequality, setting r = (p −1 − q −1 ) −1 , we have: First, consider the case p ≤ 1.Let f ∈ L p be a step function, i.e.
where a i ∈ C and the sets E i are disjoint sets with finite measure.By the ptriangular inequality: Since T : L p → L p is bounded by Theorem 2.4 and since step functions are dense in L p , it follows that T : L p → L p is a contraction (for p ≤ 1).Now consider the case p > 1.Since p < q, it follows that q > 1.By the preceding paragraph, T : L 1 → L 1 is a contraction.By complex interpolation, T : L p → L p is also contraction.
3. Construction of contractions on (L 0 ,L q ) and (L p ,L ∞ ) Let p, q ∈ (0, ∞).In this section, we are interested in the following question.Given functions f and g in L 0 +L q (resp.L p +L ∞ ), does there exist a bi-contraction T on (L 0 ,L q ) (resp.(L p ,L ∞ )) such that T (f ) = g?We show that such an operator exists provided that |g| q ≺≺ tl |f | q (resp.|g| p ≺≺ hd |f | p .This directly implies a necessary condition for a symmetric space to be an interpolation space for the couple (L 0 , L q ) (resp.(L p , L ∞ )) which will be exploited in the next section.
Our method of proof is very direct.We construct the bi-contraction T as direct sums of very simple operators.This is made possible by three partition lemmas that enable us to understand the orders ≺≺ tl and ≺≺ hd as direct sums of simple situations.
3.1.Partition lemmas.We state our first lemma without proof since it essentially repeats that of Proposition 19 in [26].Lemma 3.1.Let f, g ∈ L 1 be positive decreasing step functions.Assume that g ≺ hd f.There exists a partition {I k , J k } k≥0 of (0, ∞) such that (i) for every k ≥ 0, I k and J k are disjoint intervals of finite length; Remark 3.2.Note that in [26], Proposition 19 is proved for couples of functions f, g of the form with (a i ) an increasing sequence in (0, ∞) and f i , g i ∈ C for any i ∈ N.However, the proof applies with very little modification to couples of decreasing positive step functions as in Lemma 3.1 above.
Our second partition lemma shows that the order ≺≺ hd can be reduced for our purpose to a direct sum of ≺ hd and ≤.Lemma 3.4.Let f, g ∈ L 1 + L ∞ be such that f = µ(f ), g = µ(g) and g ≺≺ hd f.There exists a collection {∆ k } k≥0 of pairwise disjoint sets such that Let Obviously, H is a monotone function, H(t) ≤ t for all t > 0 and Note that Obviously, This proves the claim.We now claim that This proves the claim.It follows from the claim and Scholium 3.
The second assertion is now obvious.
Finally, the third partition lemma deals with describing the order ≺≺ tl in terms of ≺ tl and ≤.Lemma 3.5.Let f, g ∈ L 0 + L 1 be such that f = µ(f ), g = µ(g) and g ≺≺ tl f.There exists a collection {∆ k } k≥0 of pairwise disjoint sets such that Proof.Consider the set {g > f }.Similarly to the previous proof, connected components of the set {g > f } are intervals (closed or not) not reduced to points.Let us enumerate these intervals as (a k , b k ), k ≥ 0. We have Let Obviously, H is a monotone function, H(t) ≥ t for all t > 0 and Obviously, This proves the claim.We now claim that This proves the claim.It follows from the claim and Scholium 3.
it follows that gχ ∆ k ≺ tl f χ ∆ k , which immediately implies the first assertion.By construction, (a k , b k ) ⊂ ∆ k .Thus, The second assertion is now obvious.

Construction of operators.
We repeat the same structure as in the previous subsection, proving four lemmas, each one dealing with a certain order : ≺ hd , ≺ tl , ≺≺ hd and finally ≺≺ tl .
Step 1: First, let us assume that f and g are step functions Apply Lemma 3.1 to the functions f p and g p and let I k and J k be as in Lemma 3.1.Without loss of generality, the interval I k is located to the left of the interval J k .
For every k ≥ 0, define the mapping S k : X (I k ∪ J k ) → X (I k ∪ J k ) as below.The construction of this mapping will depend on whether Therefore, Let l k be a linear bijection from J k to I k .We set Clearly, S k is a contraction in the uniform norm.Let x ∈ L p .We have Also, we have We define S : X → X by: Remark that for any r ∈ [0, ∞], S r→r = sup k≥0 S k Lr→Lr .Hence, It remains only to note that Sf = g.
Step 2: Now, only assume that f and g are positive and non-increasing.Define for any n ∈ Z, Let A be the σ-algebra generated by the intervals (a n , a n+1 ) and (b n , b n+1 ).Define Note that f 0 and g 0 are step functions such that Apply Step 1 to f 0 and g 0 to obtain an operator S and set Clearly, T f = g and Lemma 3.7.Let f, g ∈ L q (0, ∞) be positive non-increasing functions such that g q ≺ tl f q .Let d > 1.There exists a linear operator T : X (0, ∞) → X (0, ∞) such that g = T (f ) and Proof.Following step 2 of Lemma 3.6, we are reduced to dealing with step functions.Apply Lemma 3.1 to the functions g q and f q and let (I k ) k≥1 and (J k ) k≥1 be as in Lemma 3.1.Without loss of generality, the intervals I k is located to the left of the intervals J k .
Let k ≥ 1. Define the mappings S k : The construction of S k will depend on whether f χ Thus, S k L0→L0 ≤ 2. We have Since S : X (0, ∞) → X (0, ∞) is defined as a direct sum: It remains only to note that Sf = g.
There exists a linear operator T : X → X such that g = T (f ) and Proof.Let (∆ k ) k≥0 be as in Lemma 3.4 and let ∆ ∞ be the complement of k≥0 ∆ k .By Lemma 3.6, there exists Obviously, T f = g on (0, ∞) and Lemma 3.9.Let f, g ∈ (L 0 + L q )(0, ∞) be such that |g| q ≺≺ tl |f | q , f = µ(f ) and g = µ(g).There exists a linear operator T : X → X such that g = T (f ) and Proof.Without loss of generality, g = µ(g) and f = µ(f ).Let (∆ k ) k≥0 be as in Lemma 3.5 and let ∆ ∞ be the complement of k≥0 ∆ k .By Lemma 3.7, there exists Set T ∞ = M g f on X (∆ ∞ ).We now set Obviously, T f = g on (0, ∞) and 4. Interpolation spaces for the couple (L p , L q ) In this section, we obtain characterizations of interpolation spaces for the couple (L p , L q ), in terms of the majorization notions studied earlier.The necessity of the condition we consider is a direct consequence of the constructions explained in the previous section.The fact that it is sufficient is shown by linking majorization to the K-functional.Theorem 4.1.Let 0 ≤ p < q ≤ ∞.Let E be a quasi-Banach function space such that E ∈ Int(L p , L q ).There exist c p,E and c q,E in R >0 such that: Proof.By Lemma 1.11 (for p > 0) or Lemma 2.3 (for p = 0), we may assume without loss of generality that E is a symmetric function space.Assume that p = 0. Let f ∈ E and let g ∈ L p + L ∞ be such that |g| p ≺≺ hd |f | p .Since E is symmetric, we may assume without loss of generality that f = µ(f ) and g = µ(g).By Lemma 3.8, there exists an operator T such that T (f ) = g and Recall that L q is an interpolation space for the couple (L p , L ∞ ) (one can take, for example, real or complex interpolation method).Let c p,q be the interpolation constant of L q for the couple (L p , L ∞ ).We have Let c E be the interpolation constant of E for (L p , L q ) (see Theorem 1.7).Then, This proves the first assertion.The proof of the second one follows mutatis mutandi using Lemma 2.9 instead of complex interpolation and (for p = 0) Theorem 2.1 instead of Theorem 1.7.Lemma 4.2.Assume that 0 < p < q < ∞.Let f, g ∈ L p + L q such that f = µ(f ) and g = µ(g).Suppose that at every t > 0, one of the following inequalities holds Then, there exist g 1 , g 2 ∈ (L p + L q ) + satisfying: g = g 1 + g 2 , g p 1 ≺≺ hd f p and g q 2 ≺≺ tl f q .Proof.Set: Since h 1 = µ(h 1 ), the claim follows.
We claim that For t ∈ B, we have For t / ∈ B, we have In either case, This proves the claim.Setting we complete the proof.
Theorem 4.3.Let 0 ≤ p < q ≤ ∞ with either p = 0 or q = ∞.Let E be a quasi-Banach function space.Assume that there exist c p,E and c q,E in R >0 such that: Then E belongs to Int(L p , L q ).
Proof.Assume that p = 0. Let us show that the first condition implies E ⊂ L p + L ∞ .Indeed, assume the contrary and choose f ∈ E such that µ(f )χ (0,1) / ∈ L p .Let Obviously, f n E ≤ µ(f )χ (0,1) E ≤ f E .On the other hand, f n p p χ (0,1) ≺≺ hd f p n .By the first condition on E, we have f n p χ (0,1) E ≤ c p,E f E .However f n p ↑ µ(f )χ (0,1) p = ∞.This contradiction shows that our initial assumption was incorrect.Thus, E ⊂ L p + L ∞ .
A similar argument shows that the second condition implies E ⊂ L 0 + L q .Thus, a combination of both conditions implies E ⊂ L p + L q .
Let T be a contraction on (L p , L q ) and f ∈ E. To conclude the proof, it suffices to show that T f belongs to E. First, note that Assume that p > 0 and q < ∞.Let α −1 = 1 p − 1 q .By Holmstedt formula for the K-functional (see [13]), there exists a constant c p,q > 0 such that for any t ∈ R >0 : Hence, for any given t > 0, we either have By Lemma 4.2, one can write (4.1) µ(T f ) = g 1 + g 2 , g p 1 ≺≺ hd (c p,q µ(f )) p , g 2 ≺≺ tl (c p,q µ(f )) q .By assumption, we have By triangle inequality, we have Assume now that p > 0 and q = ∞.This case is simpler since by the Holmstedt formula (see [13]), there exists c p ∈ R >0 such that for any t ∈ R >0 : This means that |T f | p ≺≺ hd |c p f | p so by assumption (1), T f belongs to E and The case of p = 0 and q < ∞ is given by Theorem 2.4.Theorem 1.1 claimed in the introduction compiles some results of this section.
Proof of Theorem 1.1.The assertion (a) is obtained by combining Theorem 4.1 and Theorem 4.3 with q = ∞.
The assertion (b) is derived similarly from Theorem 4.1 and Theorem 4.3 by applying them with p = 0.
Remark 4.4.In the spirit of Corollary 2.4, we could have used a non-quantitative condition to deal with the case of q = ∞ in Theorem 4.3.Let E be a quasi-Banach function space and p, q ∈ (0, ∞).This means that the two following conditions are equivalent: (ii) there exists c > 0 such that for any Similarly, the two following conditions are equivalent: (ii) there exists c > 0 such that for any f ∈ E, g ∈ L 0 + L q , |g| q ≺≺ tl |f | q ⇒ g ∈ E and g E ≤ c f E .

Interrpolation spaces for couples of ℓ p -spaces
In this section, we show that our approach to the Lorentz-Shimogaki and Arazy-Cwikel theorems also applies to sequence spaces.We follow a structure similar to the previous sections, proving partition lemmas, then constructing bounded operators on couples (ℓ p , ℓ q ) with suitable properties to finally draw conclusions on the interpolation spaces of the couple (ℓ p , ℓ q ).Additional arguments involving Boyd indices will be required to prove Theorem 1.2.
We identify sequences with bounded functions on (0, ∞) which are almost constant on intervals of the form ( 5.1.An interpolation theorem for the couple (ℓ p , ℓ q ).We start with a partition lemma playing, for sequence spaces, the role of Lemma 3.5.
From the definition of i n , we have From the definition of i n+1 , we have Taking the difference of these inequalities, we infer that This proves the second condition.Note that since b ≺≺ tl a, i n ≥ n for any n Hence, by definition of i n , i n+1 > i n for n ∈ I.
From the partition lemma, we deduce an operator lemma similar to Lemma 3.8.It extends Proposition 2 in [4], which is established there for the special case p = 1 by completely different method.Proof.We can assume that both sequences are non-negative and decreasing.Apply Lemma 5.1 to |a| p and |b| p .For every n ∈ Z + , choose a linear form ϕ n on ℓ p of norm less than 1, supported on ∆ n and such that ϕ n (a) = ϕ n (a1 ∆n ) = b n .Define T : x ∈ ℓ p → (ϕ n (x)) n∈Z + .
Proof.The proof of the "only if" implication is identical to the proof of Theorem 4.1 using Lemmas 5.2 and 5.5 instead of Lemmas 3.7 and 3.6.The "if" implication is given by [7, Theorem 4.7].

Upper Boyd index.
Let us now recall the definition of the upper Boyd index, in the case of sequence spaces.For any n ∈ N define the dilation operator: Let E be a symmetric function space.Define the Boyd index associated to E by: Note that since E is a quasi-Banach space, β E < ∞.
In this next proposition, we relate the upper Boyd index to an interpolation property.We follow [22,Theorem 2]. 1 p E ≤ c p,E u E , 0 ≤ u ∈ E. Proof.Let E p be the p−concavification of E, that is, Obviously, E p is a quasi-Banach space.Apply Aoki-Rolewicz theorem to the space E p and fix q = q p,E > 0 such that For every u ∈ E, we have (V (u p )) Let r ∈ (p, β −1 E ).By the definition of β E , there exists c p,E > 0 such that D n E→E ≤ c p,E n 1 r for any n ∈ N. Therefore, (V (u p )) Lemma 5.9.If x = µ(x), then Cx ≤ 3V x for every x ∈ ℓ ∞ .
Proof.Let k ≥ 0. Since x is decreasing, it follows that On the other hand, we have We are now ready to deliver a complete resolution of the conjecture stated by Levitina and two of the authors in [19].

Proposition 5 . 7 .(Lemma 5 . 8 .
Let E be a quasi-Banach symmetric sequence space.Let p < 1 βE .There exists a constant C such that for any u ∈ E and v ∈ ℓ ∞ , satisfying|v| p ≺≺ hd |u| p , we have v ∈ E and v E ≤ C u E .Define the map V : ℓ ∞ → ℓ ∞ by setting V u = ∞ n=0 2 −n D 2 n u,and the map C : ℓ ∞ → ℓ ∞ by If p < 1 βE , then (V (u p ))