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t t g∗ (s) ds ≤ f ∗ (s) ds, t> 0 Z0 Z0 (where h∗ denotes the nonincreasing left-continuous rearrangement of |h|), then g ∈ X and kgkX ≤kfkX . Peetre [48, 49] had proved (cf. also a similar result due independently to Oklander [42], and cf. also [32, pp. 158–159]) that the functional t ∗ 1 ∞ 0 f (s) ds is in fact the K-functional of the function f ∈ L + L for the couple (L1, L∞). So the results of [13, 40] naturally suggested the form that analogous resultsR for couples other than (L1, L∞) might take, expressed in terms of the K- functional for those couples. This led many mathematicians to search for such analogous results. Let us mention at least some of the many results of this kind which were obtained: Lorentz-Shimogaki2 [38] ((Lp, L∞) , 1 (1.1) Int (Lp, Lq)= Int L1, Lq ∩ Int (Lp, L∞) . 1Here we are using Calder´on’s terminology. Mityagin formulates this result somewhat differently. 2This paper also describes the interpolation spaces for the couple (L1,Lp) but not in explicit terms of the K-functional for that couple. 3In fact the arguments used in [1] also yield a shorter and simpler proof of this property, at least for couples of exponents p in the range 1 ≤ p ≤ ∞. ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 3 Later on, Bykov and Ovchinnikov obtained a similar result for families of in- terpolation spaces, corresponding to weighted couples of shift-invariant ideal se- quence spaces [10]. On the negative side, Ovchinnikov and Dmitriev [47] showed that the cou- ple (ℓ1(L1),ℓ1(L∞)) of vector-valued sequences is not a Calder´on-Mityagin cou- ple. Neither is the couple (Lp, W 1,p) when p ∈ (2, ∞). (See [14, p. 218].) Later on, Ovchinnikov [44] (see also [39]) proved the same result for the cou- ple (L1 + L∞, L1 ∩ L∞) on (0, ∞). One can find more examples of couples of rearrangement invariant spaces of this kind in Kalton’s work [29]. Many of these results contain a description of interpolation orbits, which cannot be obtained by the real K-method (see e.g. [43], [45], [46], [21], [22]). As shown by Theorem 2.2 of [17, pp. 36–37], if X0 and X1 are both σ-order continuous Banach lattices of measurable functions with the Fatou property on the same underlying σ-finite measure space Ω and if at least one of these spaces does not coincide to within equivalence of norm with some weighted Lp space on Ω. then there exist weight functions w0 and w1 on Ω for which the couple of weighted lattices (X0,w0 ,X1,w1 ) is not a Calder´on-Mityagin couple. We refer to the article [16] for additional details about Calder´on-Mityagin cou- ples. All of the results listed above were obtained for couples of Banach spaces. But there were also some ventures beyond Banach couples. In [57] Sparr was in fact also able to treat couples of weighted Lp spaces for p ∈ (0, ∞) under suitable hypotheses, and then Cwikel [15] considered the couple (ℓp,ℓ∞) also for p in this extended range. New questions have recently arisen (see, for instance, [35], [18]) that require analogous results for more general situations, say, for quasi-Banach couples or even for couples of quasi-normed Abelian groups. The extension of the basic concepts and constructions of interpolation theory to the latter setting was initiated long ago by Peetre and Sparr in [52]. We recall that ℓ0 is the linear space (sometimes considered merely as an Abelian ∞ 4 group) of all eventually zero sequences x = (xk)k=1, equipped with the “norm” kxkℓ0 := card(supp x), where supp x is the support of x. This space is an analogue of the space or normed Abelian group L0, which consists of all measurable functions on (0, ∞) with supports of finite measure, equipped with the quasi- norm kfkL0 := m{t > 0 : f(t) =6 0} ( m is the Lebesgue measure) and of the space of operators S0 (A, B) introduced on p. 249 and p. 256 respectively of [52]. Comparing some simple calculations with L0 in [52] with some quantities appearing implicitly in [34, 35] and [18] can lead one to understand that ℓ0 can play a useful role in studying interpolation properties of ℓp spaces for p> 0. Note also that independently ℓ0 appeared explicitly in [3], where a description of orbits of elements in the couple (ℓ0,ℓ1) is given. 4 Although k·kℓ0 does not satisfy kλfkℓ0 =|λ|kfkℓ0 for scalars λ, it is a (1, ∞)-norm or ∞-norm on the Abelian group ℓ0 in the terminology of [52, p. 219]. 4 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON The two main aims of this paper are, first of all, to completely determine for which values of p and q in the range 0 ≤ p < q ≤ ∞, the couple (ℓp,ℓq) has the Calder´on-Mityagin property and then, secondly, to extend a property analogous to the Arazy-Cwikel property (1.1) to the couples (ℓp,ℓq), with p and q in the enlarged range 0 ≤ p < q ≤∞ and with the role of L1 in (1.1) now played by ℓ0. There are close connections between the present paper and the paper [12]. Although [12] mainly considers the couples (Lp, Lq) of function spaces on (0, ∞), it also deals with interpolation properties of the analogous sequence space couples (ℓp,ℓq) for the range 0 ≤ p < q ≤ ∞. However, in contrast to our paper, the authors of [12] restrict themselves to studying the Calder´on-Mityagin case, i.e., for values q ≥ 1. It seems that some of the results in [12] for this case could be used to establish some of our results, and vice versa. We shall comment more explicitly about connections with [12] at appropriate places in our text, however we have kept our approach almost self-contained. The couple (ℓ0,ℓ∞) has some advantages over the corresponding Banach couple (ℓ1,ℓ∞). In particular, as remarked in [12], it is well-known that there exist sym- metric Banach sequence spaces, which are not interpolation spaces with respect to the latter couple (see e.g. [37, Example 2.a.11, p. 128]. In contrast to that, every symmetric quasi-Banach sequence space E is an interpolation space with respect to the couple (ℓ0,ℓ∞) (this can be obtained by obvious modifications of reasoning in the papers [26] and [2], where the analogous property is proved for the couple (L0, L∞)on(0, ∞) and rearrangement invariant quasi-Banach function spaces). Some other partial results for the couples (ℓp,ℓq), in the non-Banach case, were obtained more recently in [18, 35, 12, 3, 11]. Moreover, in [12], the above Arazy-Cwikel property has been proved for the couple (L0, L∞) of measurable functions on the semi-axis (0, ∞) with the Lebesgue measure. Observe however that there are differences in the properties of the quasi-Banach spaces ℓp and Lp that are essential in our context; for instance, if p ∈ (0, 1), then (ℓp)∗ = ℓ1 while (Lp)∗ = {0} (see Section 2.2). In general, the above-mentioned Brudnyi-Kruglyak result cannot be extended to the class of quasi-Banach couples. Nevertheless, whenever p and q are such that the couple (ℓp,ℓq) is a Calder´on-Mityagin couple (including in the non-Banach case), then every interpolation space with respect to (ℓp,ℓq) can be described by using the real K-method of interpolation. Moreover, but discussion of this is deferred to a forthcoming paper [5], a similar result holds for a rather wide subclass of quasi-Banach couples (the latter paper will also deal with some other related problems). Let us describe now the main results of the paper in more detail. In Sec- tion 2, we give preliminaries with basic definitions and results. So, we address some versions of the Holmstedt inequality and give descriptions of the K- and ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 5 E-functionals for couples of ℓp-spaces. Section 3 contains some auxiliary (appar- ently well-known) results, in particular, an extrapolation theorem for operators bounded on ℓp, 0 In the concluding Section 7 we prove that the couple (ℓp,ℓq), with 0 ≤ p 2. Preliminaries 2.1. Interpolation of operators and the Calder´on-Mityagin property. Let us recall some basic constructions and definitions related to the interpolation theory of operators. For more detailed information we refer to [7, 9, 6, 33, 45]. In this paper we are mainly concerned with interpolation within the class of quasi-Banach sequence spaces while linear bounded operators are considered as the corresponding morphisms. All linear spaces considered will be over the reals. But it should be possible to readily extend much of the theory that we develop also to the case of complex linear spaces. A pair X~ =(X0,X1) of quasi-Banach spaces is called a quasi-Banach couple if X0 and X1 are both linearly and continuously embedded in some Hausdorff linear topological space. In particular, every pair of arbitrary quasi-Banach sequence lattices E0 and E1 forms a quasi-Banach couple, because convergence in a quasi- Banach sequence lattice implies coordinate-wise convergence. For each quasi-Banach couple (X0,X1) we define the intersection X0 ∩ X1 and the sum X0 + X1 as the quasi-Banach spaces equipped with the quasi-norms kxkX0∩X1 := max {kxkX0 , kxkX1 } and kxkX0+X1 := inf {kx0kX0 + kx1kX1 : x = x0 + x1, xi ∈ Xi, i =0, 1} , respectively. A linear space X is called intermediate with respect to a quasi- Banach couple X~ = (X0,X1) (or is said to be between X0 and X1) if it is a quasi-Banach space and satisfies X0 ∩ X1 ⊂ X ⊂ X0 + X1 where both of these inclusions are continuous. If X~ = (X0,X1) and Y~ = (Y0,Y1) are quasi-Banach couples, then we let L(X,~ Y~ ) denote the space of all linear operators T : X0 + X1 → Y0 + Y1 that are bounded from Xi in Yi, i =0, 1, equipped with the quasi-norm (2.1) kT kL ~ ~ := max kT k . (X,Y ) i=0,1 Xi→Yi In the case when Xi = Yi, i =0, 1, we simply write L(X~ ) or L(X0,X1). ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 7 Let X~ = (X0,X1) be a quasi-Banach couple and let X be an intermediate space between X0 and X1. Then, X is called an interpolation space with respect to the couple X~ (or between X0 and X1) if every operator T ∈L(X~ ) is bounded on X. In this case, we write: X ∈ Int(X0,X1). Recall that, by the Aoki-Rolewicz theorem (see e.g. [7, Lemma 3.10.1] ), every quasi-Banach space is a F -space (i.e., the topology in that space is generated by a complete invariant metric). In particular, this applies to the space L(X~ ) which is obviously a quasi-Banach space with respect to the quasi-norm T 7→ max kT kX0→X0 .kT kX1→X1 (cf. (2.1)), and also with respect to the larger quasi- norm T 7→ max kT k , kT k , kT k whenever the quasi-Banach X0→X 0 X1→X1 X→X ~ space X is an interpolation space with respect to the quasi-Banach couple X = (X0,X1). As is well known (see e.g. [54, Theorem 2.2.15]), the Closed Graph Theorem and the equivalent Bounded Inverse Theorem (see e.g. [54, Corollary 2.2.12]) hold for F -spaces. Therefore, by exactly the same reasoning as required for the Banach case (see Theorem 2.4.2 of [7, p. 28]), if X is an interpolation quasi-Banach space with respect to a quasi-Banach couple X~ = (X0,X1), then L ~ there exists a constant C > 0 such that for every T ∈ (X) we have kT kX→X ≤ CkT kL(X~ ). The least constant C, satisfying the last inequality for all such T , is called the interpolation constant of X with respect to the couple X~ . One of the most important ways of constructing interpolation spaces is based on use of the Peetre K-functional, which is defined for an arbitrary quasi-Banach couple (X0,X1), for every x ∈ X0 + X1 and each t> 0 as follows: (2.2) K(t, x; X0,X1) := inf{||x0||X0 + t||x1||X1 : x = x0 + x1, xi ∈ Xi}. For each fixed x ∈ X0+X1 one can easily show that the function t 7→ K(t, x; X0,X1) is continuous, non-decreasing, concave and non-negative on (0, ∞) [7, Lemma 3.1.1]. On the other hand, for each fixed t> 0, the functional x 7→ K(t, x; X0,X1) is an 1 equivalent quasi-norm on X0 + X1. Moreover, if X0 ⊂ X1, then K(t, x; X0,X1)= tkxkX1 , 0 ≤ t ≤ 1. As already discussed at some length in the introduction, for quite a large class of (quasi-)Banach couples, the K-functional can be used to describe all interpolation (quasi-)Banach spaces with respect to those couples. We first need the following definition: Definition 2.1. Let X be an intermediate space with respect to a quasi-Banach couple X~ =(X0,X1). Then, X is said to be a K-monotone space with respect to this couple if whenever elements x ∈ X and y ∈ X0 + X1 satisfy K (t, y; X0,X1) ≤K (t, x; X0,X1) , for all t> 0, it follows that y ∈ X. If it also follows that kykX ≤ C kxkX , for a constant C which does not depend on x and y, then we say that X is a uniform K-monotone space with respect to the couple X~ . The infimum of all constants C with this 8 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON property is referred as the K -monotonicity constant of X. Clearly, each K- monotone space with respect to the couple X~ is an interpolation space between X0 and X1. Note that every K-monotone Banach space with respect to a couple of Ba- nach lattices is also a uniform K-monotone space with respect to this couple [17, Theorem 6.1]. Definition 2.2. Let X~ =(X0,X1) and Y~ =(Y0,Y1) be two quasi-Banach couples and let x ∈ X0 + X1, x =6 0. The orbit Orb(X0,X1)(x; Y0,Y1) of x with respect to the class of operators L(X,~ Y~ ) is the linear space T x : T ∈ L(X,~ Y~ ) . n o This space may be equipped with the quasi-norm defined by L ~ ~ kykOrb(x) := inf kT kL(X,~ Y~ ) : y = T x, T ∈ (X, Y ) . n o In the case when (X0,X1) = (Y0,Y1) we will use the shortened notation Orb(x; X0,X1). Since any orbit Orb (x; X ,X ) can be regarded as a quotient of the quasi- −→ 0 1 Banach space L(X ), it is a quasi-Banach space. If for every nonzero x ∈ X0 + X1 ∗ ∗ ∗ there exists a linear functional x ∈ (X0 + X1) with hx, x i= 6 0 then X0 ∩ X1 is contained in Orb (x; X0,X1) continuously (see e.g. [45, Section 1.6, p. 368]). It is easy to see that then, moreover, each orbit Orb (x; X0,X1) is an interpolation space between X0 and X1. A similar concept may be defined by using the K-functional. Definition 2.3. Let X~ = (X0,X1) and Y~ = (Y0,Y1) be two quasi-Banach cou- ples. The K − orbit of an element x ∈ X0 + X1, x =6 0, which we denote by K − Orb(X0,X1) (x; Y0,Y1) is the space of all y ∈ Y0 + Y1 such that the following quasi-norm K (t, y; Y0,Y1) kykK−Orb(x) := sup t>0 K (t, x; X0,X1) is finite. If (X0,X1)=(Y0,Y1), then we simplify the above notation to K − Orb(x; X0,X1). One can easily check that each K-orbit of an element x ∈ X0 + X1, x =6 0, is an interpolation quasi-normed space between X0 and X1. It is obvious that for all quasi-Banach couples (X0,X1), (Y0,Y1) and each x ∈ X0 + X1 we have 1 (2.3) Orb(X0,X1)(x; Y0,Y1) ⊂ K − Orb(X0,X1)(x; Y0,Y1). ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 9 Definition 2.4. A quasi-Banach couple X~ =(X0,X1) is said to be a Calder´on- Mityagin couple (or to have the Calder´on-Mityagin property) if for each x ∈ X0 + X1 (2.4) K − Orb(x; X0,X1)⊂Orb(x; X0,X1), i.e., if for every y ∈ K − Orb(x; X0,X1) there exists an operator T ∈ L(X~ ) L ~ such that y = T x. If additionally we can choose T ∈ (X) so that kT kL(X~ ) ≤ ~ C kykK−Orb(x), where C is independent of x and y, then X is called a uniform Calder´on-Mityagin couple (or we say that X~ has the uniform Calder´on-Mityagin property). The name of the last property is justified by the fact that historically the first result in this direction was a theorem which describes all interpolation spaces with respect to the Banach couple (L1, L∞), proved independently by Calder´on [13] and Mityagin [40]. In our terminology, this result is equivalent to the assertion that (L1, L∞) is a uniform Calder´on-Mityagin couple. Remark 2.5. The condition that (X0,X1) is a Calder´on-Mityagin couple, obvi- ously implies that every interpolation space with respect to (X0,X1) is also a K-monotone space. Furthermore, if (X0,X1) is a uniform Calder´on-Mityagin couple, this clearly implies that every interpolation space X with interpolation constant C1 is a uniform K-monotone space with K-monotonicity constant not exceeding CC1, where C is the constant appearing in Definition 2.4. 2.2. Some quasi-Banach sequence spaces and quasi-normed groups. As was said above, we will consider, mainly, quasi-Banach spaces which consist of ∞ sequences x =(xk)k=1 of real numbers with the linear coordinate-wise operations. When 0 (2.5) kxkℓ0 := card(supp x) < ∞, 10 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON 0 where supp x := {k ∈ N : xk =6 0}. Observe that ℓ is a linear space with respect to the usual coordinate-wise operations and hence we can consider linear operators defined on ℓ0. However, in constrast to the case of ℓp for every p> 0, ℓ0 is not a quasi-Banach space, but rather a quasi-normed group as defined in [52]. The functional k·kℓ0 , although it is sub-additive, does not have the homogeneity property required for a quasi-norm of a linear space. Indeed, ℓ0 is an Abelian group of sequences, where the group operation is coordinate-wise addition. Remark 2.6. According to the terminology introduced by Peetre and Sparr in Definitions 1.1 and 2.2 of [52, p. 219 and pp. 224-225], ℓ0, when equipped with the functional (2.5) is an example of a quasi-normed group, and, more specifically. it is a (1, 1)-normed Abelian group, and also a (1, 1 | 0)-normed vector space. Sometimes, in places where there could be risk of ambiguity we will use the term group quasi-norm to refer to a functional defined on an Abelian group which is a quasi-norm only in the sense of [52]. The extension of the basic concepts and constructions of the interpolation theory to the class of quasi-normed Abelian groups was initiated by Peetre and Sparr in the above mentioned paper [52] (see also [7, § 3.11] and [9]). In this case the role of morphisms is played, instead of bounded linear operators, by bounded homomorphisms. Recall that a mapping T : X → X on a group X is called a homomorphism on X if T (x+y)= T x+Ty for all x, y ∈ X. As in [52, Definition 1.2, p. 223], a homomorphism T on X is called bounded if kT xk kT kX→X := sup < ∞. x6=0 kxk Note that ℓ0 is complete and is linearly and continuously embedded into the q quasi-Banach space ℓ for every 0 < q ≤ ∞ (the functional kxkℓ0 generates the discrete topology on ℓ0). We shall adopt the following conventions related to homomorphisms which are bounded on the couple (ℓ0,ℓq). Definition 2.7. (i) : For each q with 0 < q ≤ ∞ we let L (ℓ0,ℓq) denote the set of all bounded linear operators on ℓq whose restrictions to ℓ0 are bounded homomorphisms. (ii) : We let Int (ℓ0,ℓq) denote the class of all quasi-normed Abelian groups E which satisfy the continuous inclusions ℓ0 ⊂ E ⊂ ℓq and which are also quasi- Banach spaces with respect to their given group quasi-norms and for which T : E → E is bounded for each T ∈ L(ℓ0,ℓq). L (ℓ0,ℓq) is obviously a linear space and therefore also an Abelian group. Anal- ogously to the usage for couples of quasi-Banach spaces we define kT kL(ℓ0,ℓq) := max(kT kℓ0→ℓ0 , kT kℓq→ℓq ) 0 q for every T in the set L (ℓ ,ℓ ) . Then T 7→ kT kL(ℓ0,ℓq) is a group quasi-norm on this set. ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 11 As shown in Remark 2.2 in [12], using the proof of Theorem 2.1 of that paper, 0 q if E ∈ Int (ℓ ,ℓ ), then there is a constant C such that kT kE→E ≤ C for every 0 q T ∈ L(ℓ ,ℓ ) with kT kL(ℓ0,ℓq) ≤ 1. We adopt a variant of Definition 2.2 and define the orbit of an element x ∈ ℓq with respect to the couple (ℓ0,ℓq) to be the linear space Orb(x; ℓ0,ℓq) of all y ∈ ℓq, representable in the form y = T x, where T is a bounded linear operator in ℓq and is a bounded homomorphism in ℓ0. We can consider this space as a quasi-normed Abelian group by endowing it with the group quasi-norm kykOrb(x) := inf kT kL(ℓ0,ℓq), where the infimum is taken over all T ∈ L(ℓ0,ℓq) such that y = T x. ∞ Given any q ∈ (0, ∞], suppose that x = (xn)n=1 is an arbitrary non-zero 0 q q ∗ element of ℓ + ℓ = ℓ so that xk =6 0 for at least one k ∈ N. For that k let x be q 0 q ∗ the obviously continuous linear functional on ℓ = ℓ + ℓ defined by hy, x i = yk ∞ q ∗ for each element y = (yn)n−1 ∈ ℓ . Since hx, x i= 6 0, we can reason in the same way as in [45, § 1.6, p. 368] (see also Section 2.1 ), and show that Orb(x; ℓ0,ℓq) is an interpolation quasi-normed group between ℓ0 and ℓq. Note that an inspection of the proofs related to a description of orbits of el- ements in the couples (ℓ0,ℓq), 0 < q ≤ ∞, in the papers [2] and [3] shows that these are completely consistent with the above definitions. This fact will allow us to further apply the results of these papers. The space ℓp, with 0 A solid quasi-Banach sequence space has the Fatou property if from xn ∈ E, n = 1, 2,... , supn=1,2,... kxnkE < ∞ and xn → x coordinate-wise as n → ∞ it follows that x ∈ E and kxkE ≤ lim infn→∞ kxnkE. Recall that for all 0 ≤ p 2.3. The Holmstedt formula and related K-functionals. Further, we re- peatedly use the following well-known result due to Holmstedt [25], which is referred usually as the Holmstedt formula. 12 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON Let 0 tα 1/p ∞ 1/q K (t, f; Lp, Lq) ≤ (f ∗ (s))p ds + t (f ∗ (s))q ds 0 tα Z p q Z (2.6) ≤ Cp,qK (t, f; L , L ) , t> 0, where f ∗ is the nonincreasing left-continuous rearrangement of the function |f| and α is given by the formula 1/α =1/p−1/q. Similarly, in the case when q = ∞ we have tp 1/p p ∞ ∗ p p ∞ (2.7) K (t, f; L , L ) ≤ (f (s)) ds ≤ Cp,∞K (t, f; L , L ) , t> 0. Z0 If the underlying measure space is the set of positive integers equipped with the counting measure, the couple (Lp, Lq) can be naturally identified with the p q ∞ ∞ couple (ℓ ,ℓ ) and so, setting f := n=1 fnχ[n−1,n) for every sequence (fn)n=1, we have P p q e p q (2.8) K (t, (fn); ℓ ,ℓ )= K t, f; L (0, ∞) , L (0, ∞) , t> 0. Therefore, since (f)∗ = f ∗, from (2.6)e and (2.7) it follows that α 1/p t p ∞ q 1/q p eq e ∗ ∗ K (t, (fn); ℓ ,ℓ ) ≤ f (s) ds + t f (s) ds 0 tα Z p q Z (2.9) ≤ Cp,qK (t,e(fn); ℓ ,ℓ ) , t> 0, e and (2.10) p 1/p t p p ∞ ∗ p ∞ K (t, (fn); ℓ ,ℓ ) ≤ f (s) ds ≤ Cp,∞K (t, (fn); ℓ ,ℓ ) , t> 0. 0 Z Let us define now, for everye 0 t 1/p ∗ p Ppf (t) := (f (s)) ds , t> 0, Z0 ∞ 1/q ∗ q Qqf (t) := (f (s)) ds , t> 0. Zt By these notations, inequalities (2.9) and (2.10) can be rewritten as follows: p q α α p q (2.11) K (t, (fn); ℓ ,ℓ ) ≤ Ppf (t )+ tQqf (t ) ≤ Cp,qK (t, (fn); ℓ ,ℓ ) , t> 0, and e e p ∞ p p ∞ (2.12) K (t, (fn); ℓ ,ℓ ) ≤ Ppf (t ) ≤ Cp,∞K (t, (fn); ℓ ,ℓ ) , t> 0. e ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 13 In the sequence case, we define the operators Pp and Qq, setting for every ∞ x =(xk)k=1 n 1/p ∗ p N Ppx = ((Ppx)n), (Ppx)n := (xk) , n ∈ , ! Xk=1 ∞ 1/q ∗ q N Qqx = ((Qqx)n), (Qqx)n := (xk) , n ∈ . ! Xk=n ∞ q N Clearly, for all x =(xk)k=1 ∈ ℓ and n ∈ we have (Ppx)n = Ppx(n) and (Qqx)n = Qqx(n). Consequently, inequalities (2.11) and (2.12) imply (2.13) e e 1/α p q 1/α 1/α p q K n , x; ℓ ,ℓ ≤ (Ppx)n + n (Qqx)n ≤ Cp,qK n , x; ℓ ,ℓ , n ∈ N, and 1/p p ∞ 1/p p ∞ (2.14) K n , x; ℓ ,ℓ ≤ (Ppx)n ≤ Cp,∞K n , x; ℓ ,ℓ , n ∈ N. To treat the situation withp = 0, given compatible pair of quasi-normed groups (X0,X1), we introduce the approximation E -functional by E(t, x; X0,X1) := inf{kx − x0kX1 : x0 ∈ X0, kx0kX0 ≤ t}, x ∈ X0 + X1, t> 0 [7, Chapter 7]. Clearly, the mapping t 7→ E(t, x; X0,X1) is a decreasing function on (0, ∞). There is the following connection between the E− and K−functionals: (2.15) K(t, x; X0,X1) = inf (s + tE(s, x; X0,X1)) , t> 0 s>0 [7, § 7.1]. Moreover, it is known (see e.g. [7, Lemma 7.1.3] ) that for every couple of quasi-normed Abelian groups (X0,X1) and arbitrary x ∈ X0 + X1 we have −1 ∗ sup s (K(s, x; X0,X1) − t)= E (t, x; X0,X1), t> 0, s>0 ∗ where E (t, x; X0,X1) is the greatest convex minorant of E(t, x; X0,X1), and also that for each γ ∈ (0, 1) ∗ −1 ∗ E (t, x; X0,X1) ≤E(t, x; X0,X1) ≤ (1 − γ) E (γt, x; X0,X1), t> 0. Assuming now that y ∈ K − Orb(x; X0,X1), with kykK−Orb = C, and applying the above inequalities for γ =1/2, we get ∗ −1 E(2t, y; X0,X1) ≤ 2E (t, y; X0,X1) = 2 sup s (K(s,y; X0,X1) − t) s>0 −1 ∗ ≤ 2C sup s (K(s, x; X0,X1) − t/C)=2CE (t/C, x; X0,X1) s>0 ≤ 2CE(t/C, x; X0,X1), t> 0. Combining the latter inequality together with formula (2.15), we arrive at the following useful implications: (2.16) E(t, y) ≤E(t, x), t> 0 =⇒K(t, y) ≤K(t, x), t> 0, 14 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON and for every C > 0 (2.17) K(t, y) ≤ CK(t, x), t> 0 =⇒E(t, y) ≤ 2CE(t/(2C), x), t> 0, where E(t, z) := E(t, z; X0,X1) and K(t, z) := K(t, z; X0,X1), z ∈ X0 + X1. Further, we will apply the above implications, in particular, to the couple (L0, L∞) of (equivalence classes of) measurable functions on the semi-axis (0, ∞) with the Lebesgue measure m. Here, L0 = L0(0, ∞) is the group (with respect to the usual addition) of all measurable functions on (0, ∞) with supports of finite measure, equipped by the quasi-norm kfkL0 := m{t> 0 : f(t) =06 }. ∞ q Clearly, for any x =(xk)k=1 ∈ ℓ and all t ≥ 0 we have 1/q (Qx) = ∞ (x∗)q if q < ∞ (2.18) E(t, x; ℓ0,ℓq)= [t]+1 k=[t]+1 k x∗ if q = ∞, ( P[t]+1 while for every f ∈ L0 + L∞ and all t> 0 (2.19) E(t, f; L0, L∞)= f ∗(t). We will use the standard (quasi-)Banach space notation (see e.g. [36] and [37]). In particular, throughout the paper, by en , n ∈ N, we denote the vectors ∞ of the standard basis in sequence spaces, and for every sequences x = (xn)n=1, ∞ y =(yn)n=1 we set ∞ hx, yi := xnyn (if the series converges). n=1 X By [t] we denote the integer part of a number t ∈ R and by χA the characteristic function of a set A ⊂ R. In what follows, C, c etc. denote constants whose value may change from line to line or even within lines. 3. Auxiliary results In this section we provide a self-contained presentation of some simple and apparently well-known facts. 3.1. An extension theorem for operators bounded on ℓp-spaces, 0 We begin with proving an auxiliary result, where the following notation will be used. Let T : ℓ0 → ℓ∞ be a bounded linear map. Then T can be identified ∞ ∞ 0 with an infinite matrix {tj,k}j,k=1, where tj,k = hej, T (ek)i . If x = (xn)n=1 ∈ ℓ ∞ then T x = y, y =(yj)j=1 is defined by the finite sum ∞ yj = tj,kxk. Xk=1 0 ∞ For an arbitrary 0 < q ≤∞ let Ωq denote the space of all linear maps T : ℓ → ℓ such that the quantity 0 Θq (T ) := sup kT xkℓq : x ∈ ℓ , kxkℓq =1 ∞ is finite. Clearly, if T ∈ Ωq with the matrix (tj,k)j,k=1, then for each positive ∞ q integer k the sequence tk := (tj,k)j=1 = T (ek) belongs to ℓ and moreover (3.1) ktkkℓq = kT (ek)kℓq ≤ Θq (T ) , k =1, 2,.... Hence, we see that the condition (3.2) sup ktkkℓq < ∞ k=1,2,... is necessary for T ∈ Ωq. Furthermore, we have 0 ∞ ∞ Lemma 3.2. Let T : ℓ → ℓ be a linear map with the matrix (tj,k)j,k=1 . Let ∞ N tk =(tj,k)j=1 ,k ∈ . Then, if 0 < q ≤ 1 we have (i) T ∈ Ωq ⇐⇒ supk ktkkℓq < ∞ and (3.3) Θq (T ) = sup ktkkℓq ; k=1,2,... q (ii) if T ∈ Ωq then there exists an extension T of T to ℓ with (3.4) kT kℓq→ℓq = sup ktekkℓq . k=1,2,... e Proof. (i). If T ∈ Ωq then the reasoning preceding to the lemma implies condition (3.2). ∞ 0 Conversely, assume that we have (3.2). Then, for each x = (xn)n=1 ∈ ℓ , ∞ denoting T x = y, y =(yj)j=1, and taking into account that 0 < q ≤ 1 , we have ∞ ∞ ∞ 1/q q q |yj| = tj,kxk ≤ |tj,k||xk| ≤ |tj,k| |xk| . ! k=1 k=1 k=1 X X X 16 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON Hence, ∞ ∞ ∞ q q q q kT xkℓq = |yj| ≤ |tj,k| |xk| j=1 j=1 X X Xk=1 ∞ ∞ ∞ q q q q = |tj,k| |xk| = ktkkℓq |xk| ≤ k=1 j=1 ! k=1 X X q q X (3.5) ≤ sup ktkkℓq kxkℓq . k=1,2,... Therefore, T ∈ Ωq and Θq (T ) ≤ supk=1,2,... ktkkℓq . Moreover, combining this with inequality (3.1) and taking into account that ℓ0 is dense in ℓq, we get (3.3). 0 q (ii) . Let T ∈ Ωq. Since ℓ is dense in ℓ , it follows from inequality (3.5) that q q ∞ we can define the linear extension T : ℓ → ℓ of T by T x = y, where x =(xj)j=1, ∞ yj = k=1 tj,kxk. Since kT kℓq→ℓq =Θq (T ), then in view of (3.3), formula (3.4) is also verified. e e P e Proof of Theorem 3.1. Let T be the restriction of the given operator S to the 0 space ℓ . Then T ∈ Ωq and Θq (T ) = kSkℓq→ℓq . It is obvious that the extension q ∞ T of T to ℓ defined in Lemma 3.2 is equal to S. Now, if (tj,k)j,k=1 is the matrix 1 associated with T and t = (t )∞ , the embedding ℓq ⊂ ℓr, for q kRkℓr→ℓr =Θr (T ) ≤ Θq (T )= kSkℓq→ℓq , which completes the proof. Corollary 3.3. If 0 ≤ p 3.2. An interpolation property of symmetric quasi-Banach sequence spaces. It is well known that there are symmetric Banach function (resp. se- quence) spaces which are not interpolation spaces with respect to the couple (L1(0, ∞), L∞(0, ∞)) (resp. (ℓ1,ℓ∞)) (see e.g. [55], [33, Theorem II.5.5], [37, Example 2.a.11]). In contrast to that, all symmetric quasi-Banach function (resp. sequence) spaces are interpolation spaces with respect to the couple ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 17 (L0(0, ∞), L∞(0, ∞)) (resp. (ℓ0,ℓ∞)) [2, 26]. Moreover, as we show in this sub- section, such a sequence space is an interpolation space between ℓp and ℓ∞ for some appropriate p> 0. ∞ ∞ For every n ∈ N we define the dilation operator Dn : ℓ → ℓ by ∞ ∞ (3.6) Dn : (xk)k=1 7→ x[k/n] k=1 . Let E be a symmetric quasi-normed sequence space. By the Aoki-Rolewicz theorem (see e.g. [7, Lemma 3.10.1]), we can define a subadditive functional on σE E, which is equivalent to the functional k·kE for some (the Aoki-Rolewicz index) σE > 0. Therefore, since E is symmetric, by the definition of Dn, one can easily 1/σE deduce that kDnkE→E ≤ Cn , n =1, 2,... . Hence, the upper Boyd index qE of E , defined by log kDnkE→E qE := lim , n→∞ log n does not exceed 1/σE and so finite. Proposition 3.4. For every symmetric quasi-Banach sequence space E there exists p> 0 such that E ∈ Int(ℓp,ℓ∞). In the proof of this result we will use Lemma 3.5. ([41, Theorem 1] or [12, Proposition 5.7]). Let E be a symmetric quasi-Banach sequence space and let p ∈ (0, σE). Then, the operator 1/p ∞ 1 n x =(x )∞ 7→ (x∗)p k k=1 n k k=1 ! X n=1 is bounded in E. Proof of Proposition 3.4. By [15, Theorem 3], it suffices to prove that there is a constant C > such that for any x ∈ E and y ∈ ℓ∞ satisfying n n ∗ p ∗ p (yk) ≤ (xk) , n =1, 2,..., Xk=1 Xk=1 we have y ∈ E and kykE ≤ CkxkE. Since n 1 1/p y∗ ≤ (y∗)p , n =1, 2,..., n n k Xk=1 then by the preceding lemma we have n n 1/p 1/p 1 ∗ p 1 ∗ p kykE ≤ (yk) ≤ (xk) ≤ CkxkE. n ! n ! k=1 n E k=1 n E X X 18 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON 4. Arazy-Cwikel type properties for the scale of ℓp-spaces, 0 ≤ p ≤∞. In [1], Arazy and Cwikel have proved that for all 1 ≤ p < q ≤∞ and for each underlying measure space Int (Lp, Lq)= Int L1, Lq ∩ Int (Lp, L∞) . Recently, a similar result has been obtained in [12] for function quasi-Banach spaces on (0, ∞) with the Lebesgue measure, namely, it was proved there that for all 0 Then, Q′ is a subadditive and positively homogeneous operator bounded on ℓq ′ ′ ′ such that kQ kℓ0→ℓ0 ≤kQkℓ0→ℓ0 , kQ kℓq→ℓq ≤kQkℓ1→ℓ1 and Q x = y. ′ Now, we define the linear operator S on the one-dimensional subspace Hx := q ′ ′ ′ {αx, α ∈ R} in ℓ by S (αx) := αy, α ∈ R. Then, S z ≤ Q z, z ∈ Hx, and S′x = y. Hence, by the Hahn-Banach-Kantorovich theorem (see e.g. [53, p. 120]), we can find a linear extension S of S′ to the whole of ℓq such that Sz ≤ Q′z for all z ∈ ℓq. Since the operator S satisfies all the requirements, the proof is completed. Remark 4.4. Alternatively, instead of Theorem 1 from [3], in the proof of Propo- sition 4.3 we may apply Lemma 5.2 from [12] combined with using the dilation operators (see (3.6)). ∞ q ∞ Theorem 4.5. Suppose 0 n 1/p ∞ 1/q n 1/p ∞ 1/q ∗ p 1/α ∗ q ∗ p 1/α ∗ q N (yk) +n (yk) ≤ (xk) +n (xk) , n ∈ , ! ! ! ! Xk=1 Xk=n Xk=1 Xk=n where 1/α =1/p − 1/q. Then, we can find linear operators T : ℓ∞ → ℓ∞ and S : ℓq → ℓq such that 1/p kT kL(ℓp,ℓ∞) ≤ 8 , kSkL(ℓ0,ℓq) ≤ 3, with y = T x + Sx. ∞ ∞ Proof. As above, we can (and will) assume that x =(xn)n=1 and y =(yn)n=1 are nonincreasing and nonnegative sequences. The proof below will be modelled on the arguments used in [1]. Let us define n p p N N A (n) := (xk − yk) , n ∈ , and A := {n ∈ : A (n) ≥ 0} , Xk=1 ∞ q q N N B (n) := (xk − yk) , n ∈ , and B := {n ∈ : B (n) ≥ 0} . Xk=n Then from the assumption of the theorem it follows that A ∪ B = N. Observe that in the case when A = N it follows n n p p N yk ≤ xk, n ∈ , Xk=1 Xk=1 and hence Proposition 4.2 implies that y = T x for some operator T : ℓ∞ → ℓ∞ bounded in the couple (ℓp,ℓ∞) . Similarly, if B = N then ∞ ∞ q q N yn ≤ xk, n ∈ . Xk=n Xk=n Consequently, by Proposition 4.3, y = Sx for some S : ℓq → ℓq bounded in the couple (ℓ0,ℓq). So, in these cases the desired result follows. Assume now that neither A = N nor B = N. We represent A as the union of successive maximal pairwise disjoint intervals of positive integers, i.e., A = 20 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON ∪i∈I1 Ai, where Ai = [ni, mi], mi +1 < ni+1. Let B = ∪i∈I2 Bi be the corresponding union for B. These collections of intervals may be finite or infinite. Let 1 ∈ A, i.e., A1 = [1, m1]. Then, A (min (l, m1)) ≥ 0 for every l ∈ N, and hence we have l min(l,m1) min(l,m1) p ∗ p p (χA1 x)j = xj = A (min (l, m1)) + yj j=1 j=1 j=1 X X X min(l,m1) l p p ∗ ≥ yj = (χA1 y)j . j=1 j=1 X X Suppose now that Ai = [ni, mi], where ni ≥ 2, be a finite interval. Then, c min (l + ni − 1, mi) ∈ A, ni−1 ∈ A for all l ∈ N. Therefore, A (min (l + ni − 1, mi)) ≥ 0, A (ni − 1) ≤ 0 and l l+n −1 min(l+ni−1,mi) p i ∗ p p (χAi x)j = (χAi x)j = xj j=1 j=n j=n X Xi Xi min(l+ni−1,mi) p = A (min (l + ni − 1, mi)) − A (ni − 1) + yj j=n Xi min(l+ni−1,mi) l p p ∗ ≥ yj = (χAi y)j . j=n j=1 Xi X If finally Ai = [ni, ∞) then, for any l ∈ N, we have ni + l − 1 ∈ Ai and c ni − 1 ∈ A . Hence, as above, l n +l−1 n +l−1 p i i ∗ p p (χAi x)j = xj = A (ni + l − 1) − A (ni − 1) + yj j=1 j=n j=n X Xi Xi n +l−1 l i p p ∗ ≥ yj = (χAi y)j . j=n j=1 Xi X Thus, by the estimates obtained and Proposition 4.2, for each i ∈ I1, we can ∞ ∞ 1/p select a linear operator Ti : ℓ → ℓ , kTikL(ℓp,ℓ∞) ≤ 8 , with Ti(χAi x)= χAi y. Now setting T = i∈I1 χAi TiχAi , we see that the operator T is well-defined on ℓ∞, P 1/p kT kL(ℓp,ℓ∞) ≤ sup kTikL(ℓp,ℓ∞) ≤ 8 i∈I1 and T (χAx)= χAy. ′ ′ N Similarly, let Bi = [ni, mi] be a finite interval, and let l ∈ be such that ′ ′ ′ ′ c l ≤ mi − ni + 1. Then, l + ni − 1 ∈ B and mi + 1 ∈ B . Consequently, ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 21 ′ ′ B(l + ni − 1) ≥ 0 and B(mi + 1) ≤ 0. Hence, ′ ∞ m q i ∗ q ′ ′ (χBi x)j = xj = B(l + ni − 1) − B(mi + 1) j=l ′ X j=lX+ni−1 ′ ′ m m ∞ i i q q q ∗ + yj ≥ yj = (χBi y)j . ′ ′ j=l j=lX+ni−1 j=lX+ni−1 X ′ ′ Observe that the latter inequality holds also for l > mi − ni + 1, because in this case its both sides vanish. ′ N In the case when Bi = [ni, ∞) for every l ∈ we get ∞ ∞ ∞ q ∗ q ′ q (χBi x)j = xj = B (ni + l − 1) + yj j=l ′ j=n +l−1 X j=nXi+l−1 Xi ∞ ∞ q q ∗ ≥ yj = (χBi y)j , j=nXi+l−1 Xj=l ′ N because of ni + l − 1 ∈ Bi for all l ∈ . As a result, by using Proposition 4.3, for every i ∈ I2 we can find an operator q q ′ 0 q Si : ℓ → ℓ , kSikL(ℓ ,ℓ ) ≤ 3, with Si(χBi x) = χBi y. Then, the operator S := ∞ q i=1 χBi SiχBi is well-defined on ℓ , ′ P kS kL(ℓ0,ℓq) ≤ sup kSikL(ℓ0,ℓq) ≤ 3 i∈I2 ′ ′ and S (χBx) = χBy. Denoting S := χB\AS , we see that kSkL(ℓ0,ℓq) ≤ 3, and Sx = χB\Ay. Since y = χAy + χB\Ay = T x + Sx, the operators T : ℓ∞ → ℓ∞ and S : ℓq → ℓq satisfy all the requirements and so the proof of the theorem is completed. It is a classical result of the interpolation theory that the couple (Lp, Lq), 1 ≤ p < q ≤ ∞, has the uniform Calder´on-Mityagin property (note that it is a special case of the well-known Sparr theorem, see [57]). The preceding results of this section combined with the observations made in Subsection 2.2 imply the following extension of the above-mentioned theorem for the Banach situation to the quasi-Banach case in the sequence space setting. Corollary 4.6. Let 0 ≤ p < q ≤ ∞ and q ≥ 1. Then (ℓp,ℓq) is a uniform Calder´on-Mityagin couple. ∞ q ∞ q Proof. Let x = (xk)k=1 ∈ ℓ and y = (yk)k=1 ∈ ℓ be two nonincreasing and nonnegative sequences such that (4.1) K(t, y; ℓp,ℓq) ≤K(t, x; ℓp,ℓq), t> 0. We consider four cases depending on values of the numbers p and q separately. 22 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON Let first 0 yk ≤ 2x[k/2], k =1, 2,..., or yk ≤ 2(D2x)k, k =1, 2,..., ∞ ∞ where D2 is the dilation operator (see Section 3.2). For each u = (uk)k=1 ∈ ℓ we define the multiplication operator T by y ∞ Tu = u · k . k 2(D x) 2 k k=1 Obviously, kT kL(ℓ0,ℓ∞) ≤ 1. Therefore, if S := 2TD2, then kSkL(ℓ0,ℓ∞) ≤ 2 and 0 ∞ Sx = y. Thus, y ∈ Orb(x; ℓ ,ℓ ) and kykOrb(x) ≤ 2. 0 Next, if 0 = p Moreover, the quasi-norms of the above spaces are equivalent with constants independent of x ∈ ℓq. Now, we are able to prove the following additivity property for orbits of ele- ments with respect to the couples (ℓp,ℓq), 0 ≤ p < q ≤∞ (cf. [10]). Proposition 4.8. Let 0 ≤ s Proposition 4.9. Let (X0,X3) and (X1,X2) be two couples of quasi-Banach Abelian groups such that X2 ∈ Int (X0,X3) and X3 ∈ Int (X1,X2). Moreover, assume that X2 ∩ X3 ⊆ (X0 ∩ X3)+(X1 ∩ X2) ⊆ (X0 + X3) ∩ (X1 + X2) ⊆ X2 + X3 and Orb(x; X2,X3) = Orb(x; X0,X3)+Orb(x; X1,X2) for every x ∈ X2 + X3. Then, Int (X2,X3)= Int (X0,X3) ∩ Int (X1,X2) . Proof. Suppose first X ∈ Int (X2,X3). If T ∈ L(X0,X3) it follows by interpola- tion that T : X2 → X2 and hence T ∈ L(X2,X3). This implies that T : X → X, i.e., X ∈ Int (X0,X3) . In the same manner one can check that X ∈ Int (X1,X2). Conversely, let X ∈ Int (X0,X3)∩Int (X1,X2). Then, by the first assumption, X2 ∩ X3 ⊆ (X0 ∩ X3)+(X1 ∩ X2) ⊆ X ⊆ (X0 + X3) ∩ (X1 + X2) ⊆ X2 + X3, 24 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON that is, X is an intermediate space between X2 and X3. Moreover, for each x ∈ X we have Orb (x; X0,X3) ⊂ X and Orb(x; X1,X2) ⊂ X. Hence, by the second assumption, Orb (x; X2,X3) ⊂ X. Thus, X ∈ Int (X2,X3), and the desired result follows. Now, we are ready to prove that the full scale of ℓp-spaces, 0 ≤ p ≤∞, possesses the Arazy-Cwikel property. Proof of Theorem 4.1. It suffices to apply Propositions 4.8 and 4.9. In conclusion of this section we deduce the following result, which contains, in particular, a solution of the conjecture stated by Levitina, Sukochev and Zanin in the paper [35] (its earlier version may be found in the preprint [34]). As above, let σE denote the Aoki-Rolewicz index of the quasi-Banach sequence space E (see Section 2.1). Theorem 4.10. Let q > 0 and let E be a quasi-Banach sequence space. The following assertions are equivalent: (i) E ∈ Int(ℓ0,ℓq); p q (ii) E ∈ Int (ℓ ,ℓ ) for each p ∈ (0, σE); 0 q p ∞ (iii) E ∈ Int (ℓ ,ℓ ) and E ∈ Int (ℓ ,ℓ ) for each p ∈ (0, σE). Proof. Observe that from (i) it follows that E is symmetric (see e.g. [12, Lemma 1.11]). Therefore, the implication (i) =⇒ (iii) is an immediate consequence of Propo- sition 3.4 (see also its proof). In turn, the equivalence (ii) ⇐⇒ (iii) follows from Theorem 4.1. Since the implication (iii) =⇒ (i) is obvious, the proof is completed. By the latter result, we are able to determine the exact assumptions under which the above-mentioned Levitina-Sukochev-Zanin conjecture is resolved in af- firmative. To justify this claim, we consider the following conditions, assuming that q > 0 and E is a quasi-Banach sequence space such that E ⊂ ℓ∞: ∞ ∞ ∞ (a) for any x =(xn)n=1 ∈ E and y =(yn)n=1 ∈ ℓ such that ∞ ∞ ∗ q ∗ q N (4.2) (yn) ≤ (xn) , m ∈ , n=m n=m X X we have y ∈ E and kykE ≤ CkxkE, where C depends only on E and q; (b) there exists p ∈ (0, q) such that E ∈ Int(ℓp,ℓq). In [35], the authors are asking if the conditions (a) and (b) are equivalent, and then, in [12], the affirmative answer to this question is given in the case when q ≥ 1 (see also [18]). To show a connection of the above question with Theorem 4.10 , suppose that elements x ∈ E and y ∈ ℓq satisfy the condition K(t, y; ℓ0,ℓq) ≤K(t, x; ℓ0,ℓq), t> 0. ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 25 Then, by implication (2.16), we have ∞ ∞ ∗ q ∗ q N (yn) ≤ 2 (x[n/2]) , m ∈ , n=m n=m X X or equivalently ∞ ∞ ∗ q ∗ q N (yn) ≤ 2 (D2x )n, m ∈ . n=m n=m X X If condition (a) holds, then clearly the space E is symmetric and hence from the latter inequality it follows that kykE ≤ 2CkD2xkE ≤ 4CkxkE, where C is the constant from (a). Therefore, E is a uniform K-monotone space with respect to the couple (ℓ0,ℓq) and hence E ∈ Int (ℓ0,ℓq). Thus, (a) implies condition (i) from Theorem 4.10, and so equivalence (i) ⇐⇒ (ii) of the latter result shows that implication (a)=⇒ (b) holds, in fact, for each q > 0 (including also the non-Calder´on-Mityagin range 0 5. A characterization of couples (ℓp,ℓq) with the Calderon-Mityagin´ property. Here, we prove one of the main results of this paper, showing that (ℓp,ℓq) fails to be a Calder´on-Mityagin couple whenever 0 ≤ p Definition 5.1. Let X~ =(X0,X1) be a quasi-Banach couple (or more generally, a couple of quasi-Banach Abelian groups). Then, we say that y ∈ X0 + X1 is a Calder´on-Mityagin element (CM-element, in brief) with respect to the couple X~ provided if (5.1) K(t, y; X0,X1) ≤K(t, x; X0,X1), t> 0, 26 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON for x ∈ X0 + X1, then there exists an operator T ∈ L(X0,X1) with T x = y. The set of all CM-elements with respect to the couple X~ we will denote by CM(X~ )= CM(X0,X1). Clearly, CM(X~ ) = X0 + X1 if and only if the couple X~ has the Calder´on- Mityagin property. Let a quasi-Banach couple X~ be such that the sum X0 + X1 is continuously embedded into a Banach space Z. Show that then we have CM(X~ ) ⊃ X0 ∩ X1. Indeed, let inequality (5.1) to be hold for some y ∈ X0 ∩ X1 and x ∈ X0 + X1. Clearly, we may assume that x =6 0. Then, according to Hahn-Banach Theorem, ∗ ∗ ∗ ∗ there is a linear functional x ∈ Z , kx kZ∗ = kxkZ , with x (x) = 1. Now, if the ∗ operator T is defined by Tu := x (u)y, then T is bounded from X0 + X1 into X0 ∩ X1 and T x = y. Moreover, for i =0, 1 ∗ ∗ ∗ kTukXi ≤|x (u)|kykXi ≤kx kZ kukZkykXi ≤ CkxkZkykXi kukXi , where C is the embedding constant of X0 + X1 into Z. In particular, since ℓr ⊂ ℓ1 continuously if r ∈ (0, 1), we have that ℓp ⊆ CM(ℓp,ℓq) for all 0 Orb(X0,X1)(x; Y0,Y1)= K − Orb(X0,X1)(x; Y0,Y1) for every Banach couple Y~ =(Y0,Y1) (equivalently, from the inequality K(t, y; Y0,Y1) ≤K(t, x; X0,X1), t> 0, with y ∈ Y0 + Y1 it follows the existence of an operator T ∈ L(X,~ Y~ ) such that T x = y). The main purpose of [22] is an identification of absolutely Calder´on elements for the couple (L1, L∞). Theorem 5.3. Let 0 ≤ p As we will see a little bit later, Theorem 5.3 is a straightforward consequence of the following result. ∞ q p Theorem 5.5. Assume 0 ≤ p Corollary 5.6. Let 0 ≤ p ′ Indeed, for the contrary, assume that g ∈ Orb f,ℓp,ℓr for some r′ ∈ [q, 1). ′ This means that there exists an operator V ∈ L(ℓp,ℓr ) such that V f = g. Since r′ < 1, by Theorem 3.1, V has a linear extension V : ℓ1 → ℓ1 such that ′ ′ kV kℓ1→ℓ1 ≤kV kℓr →ℓr . Then, p 1 p 1 e p 1 K t, g; ℓ ,ℓ = K(t, V f; ℓ ,ℓ ) ≤kV k p r′ K t, f; ℓ ,ℓ , t> 0, e L(ℓ ,ℓ ) which is a contradiction in view of (5.4) (with r = 1). e Now, setting s = p and r′ = q in (5.5) and (5.6) respectively, we see that every element g ∈ ℓq \ ℓp fails to be a CM-element with respect to the couple (ℓp,ℓq). Therefore (see the discussion preceding the statement of Theorem 5.3), CM (ℓp,ℓq)= ℓp. The second assertion of the theorem that (ℓp,ℓq) is not a Calder´on-Mityagin couple is now almost obvious. Indeed, take for g any element from ℓq \ ℓp (say, −σ ∞ p q q g = (n )n=1, where σ ∈ (1/q, 1/p)). Since g 6∈ CM (ℓ ,ℓ ), there is f ∈ ℓ such that g ∈ K − Orb(f; ℓp,ℓq) \ Orb(f; ℓp,ℓq). Clearly, this yields the desired result. Now, we proceed with the proof of Theorem 5.5. Let a, b be arbitrary positive integers. Let Ta,b be the linear map defined on every function h : [0, ∞) → R as follows: h (x) if 0 ∞ r If t h (x) dx > 0 for all t> 0, then we have ∞ ∞ R r r (5.11) (Ta,bh (x)) dx ≤ h (x) dx Zt Zt whenever a positive integer b satisfies the condition ∞ q h (y)r dy r−q (5.12) b> a . ∞ h (y)r dy Rt We postpone the proof of this lemma,R which is a series of elementary calcula- tions, until the end of this section. Proof of Theorem 5.5. As above, for some technical purposes, it will be conve- ∞ nient to consider, instead of sequences h =(hn)n=1, the step functions h(t) defined for t> 0 by ∞ e h(t) := hnχ[n−1,n)(t). n=1 X We will use further an infinitee composition of suitable operators of the form Ta,b ∞ applied to the function g = n=1 gnχ[n−1,n) to obtain as a result the nonnegative and nonincreasing function f ∈ Φ such that the corresponding sequence f = ∞ P (fn)n=1 will possess thee required properties (5.2), (5.3) and (5.4). ∞ ∞ Let (an)n=1 and (bn)n=1 bee two sequences of positive integers such that the ∞ sequence (an)n=1 is strictly increasing and hence limn→∞ an = ∞. Next, choosing ∞ these sequences in a special way, we consider the sequence of functions (Gn)n=1 defined by G1 = Ta1,b1 , Gn = Tan,bn (Gn−1) for n ≥ 2. Then, by the definition of Ta,b, (5.13) Gn (x)= Gm (x) , for each n ≥ m and for all x ∈ [0, am+1) . ∞ Since m=1 [0, am+1) = [0, ∞), there exists the pointwise limit S f (x) := lim Gn (x) , x> 0. n→∞ From (5.13) it follows thate (5.14) f (x)= Gn (x) , for each n > m if x ∈ [0, am+1) . It is clear also that f ∈ Φ and it is a nonnegative and nonincreasing function. e ∞ Now, the main our task is to show that, if the auxiliary sequences (an)n=1 and ∞ ∞ (bn)n=1 are constructede suitably, the sequence (fn)n=1 of values of f will satisfy the conditions (5.2), (5.3) and (5.4). At this stage we will assume that 0 Consequently, for every fixed positive integer n we have ∞ q lim Gn(x) dx =0, y→∞ Zy and hence the set ∞ 1 m ∈ N : G (x)q dx ≤ n n Zm is non-empty. Let ∞ 1 (5.16) γ := min m ∈ N : G (x)q dx ≤ . n n n Zm Next, by (5.14), for each t ≥ 0 and all n satisfying an+1 ≥ t, we have ∞ an+1 ∞ q q q (5.17) Gn (x) dx = Gn (x) dx + Gn (x) dx Zt Zt Zan+1 an+1 ∞ q q = f (x) dx + Gn (x) dx. Zt Zan+1 Observe that for each integer n ≥ 2 thee function Gn depends only on the given function g and previously chosen ak, bk, with 1 ≤ k ≤ n. Consequently, after completing the first n steps, we may select an+1 so that e an+1 ≥ γn, where γn is the positive integer defined in (5.16). Then, ∞ 1 G (x)q dx ≤ , n n Zan+1 and hence, passing to the limit as n tends to infinity in (5.17) , we get ∞ ∞ q q (5.18) f (x) dx = lim Gn (x) dx for each t ≥ 0. n→∞ Zt Zt In particular, settinge t = 0, by (5.15), we have ∞ ∞ ∞ ∞ q q q q fm = f (x) dx = g (x) dx = gm < ∞. m=1 0 0 m=1 X Z Z X Thus, the first desired condition,e (5.2), is established.e Furthermore, repeated applications of (5.8) imply that ∞ ∞ q q Gn (x) dx ≥ g (x) dx for every n ∈ N and all t> 0. Zt Zt Hence, taking limits and using (5.18) yields e ∞ ∞ f (x)q dx ≥ g (x)q dx, t> 0, Zt Zt which implies that ∞e ∞ e q q N gj ≤ fj for all m ∈ . j=m j=m X X ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 31 ∞ Now, recalling that for each nonnegative, nonincreasing sequence h =(hn)n=1 we have K(t, h; ℓ0,ℓq) = inf m + t ·E(m, h; ℓ0,ℓq) , t> 0, m=0,1,2,... where ∞ 1/q 0 q q E(m, h; ℓ ,ℓ )= hj j=m+1 ! X (see e.g. [7, § 7.1] or formulas (2.15) and (2.18) in Section 2.3), we conclude that the second required condition (5.3) holds as well. Thus, it remains only to prove (imposing suitable additional hypotheses on the ∞ ∞ sequences (an)n=1 and (bn)n=1) the last required condition (5.4). Here, we will use the assumption that g∈ / ℓp, and so (5.19) g∈ / Lp (0, ∞) . ∞ Along with the previously introduced sequence (γn)n=1 we will need a new se- ∞ e quence (δn)n=1. For definiteness, we set a1 = b1 = δ1 = 1 and hence G1 = T1,1g = g. Moreover, as was specified above, γ1 is the least positive integer m satisfying ∞ q the inequality m G1(x) dx ≤ 1. e e Suppose that n ≥ 1 and ak, bk, δk,Gk,γk are determined for all 1 ≤ k ≤ n. To pass to the nextR step, we first set (5.20) an+1 := max (γn, δn, an + 1) . Let now δn+1 be the smallest positive integer with the following properties: (5.21) δn+1 > nan+1 and δn+1 an+1 p p p g (x) dx ≥ 2n Gn (x) dx. Z0 Z0 Observe that such an integere exists because of (5.19). Therefore, by (5.7), we infer (5.22) δn+1 an+1 δn+1 p p 1 p lim Tan+1,bGn (x) dx = Gn (x) dx ≤ g (x) dx. b→∞ 2np Z0 Z0 Z0 Combining the fact that g is nonincreasing with (5.19) , we see thate g (x) > 0 and hence δn+1 e g (x)p dx > 0. e Z0 q Moreover, since the functions g and Ge n belong to the intersection Φ ∩ L (0, ∞), the embedding ℓq ⊂ ℓr ensures that r e r (5.23) g ∈ L (0, ∞) and Gn ∈ L (0, ∞) . e 32 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON Then, using (5.10) with the function Gn and the number an+1 instead of h and a respectively, we conclude ∞ r (5.24) lim Tan+1,bGn (x) dx =0 b→∞ Zan+1 As was noted, the function g is strictly positive at all points of (0, ∞). Hence, Gn (x) > 0 if x> 0. Therefore, in view of (5.23), we can apply the last result of Lemma 5.7, according to that,e for each t> 0, every positive integer b, satisfying the condition ∞ G (y)r dy q/(r−q) an+1 n b> ∞ r , R t Gn (y) dy ! satisfies also the inequality R ∞ ∞ r r Tan+1,bGn (x) dx ≤ Gn (x) dx. Zt Zt Thus, from the condition ∞ r q/(r−q) Gn (y) dy b> an+1 ∞ G (y)r dy R an n ! and the fact that ak < an for k By (5.27), (5.14) and the inequality δn+1 ≤ an+2 (see (5.20)), it follows that δn+1 1 δn+1 (5.29) f (x)p dx ≤ g (x)p dx. np Z0 Z0 Moreover, applying (5.26), fore each integer n and em ≥ n + 1 we have ∞ ∞ r r Gm+1 (x) dx ≤ Gm (x) dx, Zan+1 Zan+1 and hence by iteration ∞ ∞ r r Gm (x) dx ≤ Gn+1 (x) dx for each m ≥ n +1. Zan+1 Zan+1 Consequently, for each R ≥ an+1 and all m ≥ n + 1, in view of (5.28), we infer that R ∞ 1 ∞ G (x)r dx ≤ G (x)r dx ≤ g (x)r dx. m n+1 nr Zan+1 Zan+1 Zδn+1 N Since 0 ≤ Gm (x) ≤ g (0) for all x ≥ 0 and m ∈ , using dominatede convergence on the interval [an+1, R], we have R e R ∞ r r 1 r f (x) dx = lim Gm (x) dx ≤ g (x) dx, m→∞ nr Zan+1 Zan+1 Zδn+1 and then, passinge to the limit as R →∞, we obtain e ∞ 1 ∞ (5.30) f (x)r dx ≤ g (x)r dx. nr Zan+1 Zδn+1 By the Holmstedt formulae (2.6), from (5.21), (5.29)e and (5.30) it follows that 1/α p r K(δn+1, f; L (0, ∞), L (0, ∞)) δn+1 1/p ∞ 1/r p 1/α r ≤ fe(x) dx + δn+1 f (x) dx Z0 Zδn+1 1/p 1 δne+1 1 e ∞ 1/r ≤ g (x)p dx + δ1/α g (x)r dx np n+1 nr Z0 Zδn+1 δn+1 1/p ∞ 1/r 1 e p 1/α e r = g (x) dx + δ g (x) dx n n+1 Z0 Zδn+1 ! C ≤ p,r K(δ1/α , eg; Lp(0, ∞), Lr(0, ∞)), n =1e, 2,... n n+1 Hence, for the corresponding sequences f and g it holds e 1/α p r K δn+1, f; ℓ ,ℓ lim =0. n→∞ 1/α p r K δn+1,g; ℓ ,ℓ 34 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON Since δn+1 > an+1, then (5.20) ensures that δn+1 → 0 as n →∞. Thus, relation (5.4) is established, and thus the proof of the theorem in the case when 0 1/p p ∞ K δn+1, f; ℓ ,ℓ lim =0. n→∞ 1/p p ∞ K δn+1,g; ℓ ,ℓ Thus, (5.4) is proved. Since the proofs of (5.2) and (5.3) do not require any modifications in this case, the result follows. Assume that 0 = p Finally, we consider the case when p = 0 and r = ∞. The first part of our process may be conducted in the same way as earlier. Namely, we can find ∞ ∞ ∞ sequences of positive integers (an)n=1,(bn)n=1 and (γn)n=1 such that the sequence ∞ (an)n=1 is strictly increasing, an+1 ≥ γn and the pointwise limit f of the sequence ∞ of functions (Gn)n=1 defined by G1 = Ta1,b1 g = g, Gn = Tan,bn Gn−1, n ≥ 2, satisfies conditions (5.2) and (5.3). To get the remaining conditione (5.4), we need somewhat to modify (in fact, to simplify) the above procedure. ∞ e e In this case the sequence (δn)n=1 is not needed, and we have only to arrange ∞ the choice of the sequence (bn)n=1, which was arbitrary by now. Since g ∈ ℓq \ℓ0, then card(supp g)= ∞. Therefore, the function g(x) is strictly positive for all x > 0. Suppose that n ≥ 1 and ak, bk,Gk,γk are defined for all 1 ≤ k ≤ n (as above, a1 = b1 = γ1 = 1). To pass to the next step,e we set (n + 1)g(a ) q (5.33) a := max (γ , a + 1) and b := n+1 +1. n+1 n n n+1 g(n(a + 1)) n+1 e Then, by the definition of the operators Ta,b, we have for all n ≥ 2 e 1 G (a + 1) ≤ b−1/qg(a ) ≤ g(n(a + 1)). n n n n n n Combining this together with (5.14), wee get e 1 f(a + 1) ≤ g(n(a + 1)), n ≥ 2, n n n which yields for all n ≥e2 e f(x/n) (5.34) lim inf =0. x→∞ g(x) e Let us show that (5.34) implies (5.4) in the case p = 0 and r = ∞. Assuming the contrary, we have for somee positive integer C that (5.35) K(t, g; ℓ0,ℓ∞) ≤ CK(t, f; ℓ0,ℓ∞), t> 0. This implies that K(t, g; L0(0, ∞), L∞(0, ∞)) ≤ CK(t, f; L0(0, ∞), L∞(0, ∞)), t> 0. Consequently, using (2.17) and (2.19), we get e e g(x) ≤ 2Cf(x/C) for all x> 0. Since this inequality contradicts relatione (5.34), inequality (5.35) fails for any C. Therefore, (5.4) holds, ande so the proof of Theorem 5.5 is completed. Proof of Lemma 5.7. Recall that h is assumed to be a nonnegative, nonincreasing function in Φ (i.e., constant on each interval of the form [n − 1, n), n ∈ N). 36 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON We begin with proving (5.7). Fix a ∈ N and t > a. Then, for each b ∈ N, we have t a t p p p (Ta,bh (x)) dx = (Ta,bh (x)) dx + (Ta,bh (x)) dx 0 0 a Z Z a t Z p p (5.36) = h (x) dx + (Ta,bh (x)) dx. Z0 Za Since t t x − a p 0 ≤ (T h (x))p dx = b−p/q h a + dx a.b b Za Za ≤ b−p/q (t − a) h (0)p , it follows that t p lim (Ta.bh (x)) dx =0. b→∞ Za Combining this together with (5.36), we get (5.7). To obtain (5.8) and (5.9), note first that, for t ≥ a, the change of variables gives us ∞ ∞ q q (5.37) (Ta,bh (x)) dx = h (x) dx. Zt Za+(t−a)/b Since b ≥ 1 we have that t ≥ a +(t − a) /b whenever t ≥ a, which together with (5.37) implies (5.8) in the case t ≥ a. If 0 6. About the Sq-property In [18], in connection with the conjecture stated by Levitina, Sukochev and Zanin (see Theorem 4.10 and the subsequent discussion), Cwikel and Nilsson have introduced the following notion. Definition 6.1. Let q ≥ 1 and let E =6 {0} be a normed sequence space, E ⊆ ℓq. Then, E has the Sq-property provided that there is a constant C if, whenever ∞ ∞ q x = (xn)n=1 ∈ E and y = (yn)n=1 ∈ ℓ are two sequences, which satisfy the conditions: ∞ ∞ q q (6.1) |xn| = |yn| n=1 n=1 X X and m m ∗ q ∗ q N (6.2) (xn) ≤ (yn) for all m ∈ , n=1 n=1 X X then it follows that y ∈ E and kykE ≤ C kxkE. It is clear that this definition may be extended to a more general situation when q > 0 and E is a quasi-Banach sequence space. The following result shows that the Sq-property of a quasi-Banach sequence space E is closely related to the fact that E ∈ Int (ℓ0,ℓq). Theorem 6.2. Let 0 Therefore, from the condition (a) it follows that E ∈ Int (ℓ0,ℓq). In the case when q ≥ 1, the converse holds as well, i.e., E has the Sq-property if and only if E ∈ Int (ℓ0,ℓq). ∞ ∞ Proof. (a) =⇒ (b). Assume that sequences x =(xn)n=1 ∈ E and y =(yn)n=1 ∈ ℓq satisfy the condition K(t, y; ℓ0,ℓq) ≤K(t, x; ℓ0,ℓq), t> 0. Then, in the same way as in the end of Section 4, we have ∞ ∞ ∗ q ∗ q N (yn) ≤ 2 (D2x )n, m ∈ , n=m n=m X X ∗ where D2 is the dilation operator, or, denoting un = 2(D2x )n, n =1, 2,... , ∞ ∞ ∗ q q N (6.3) (yn) ≤ un, for all m ∈ . n=m n=m X X Further, we will use a reasoning from the proof of Theorem 5.3 in [18] . q N ∗ Since E ⊆ ℓ , it follows that limn→∞ yn =0. Select n1 ∈ with |yn1 | = y1. Let ∞ ∞ ∗ q ∞ q z =(zn)n=1 be a sequence such that zn = un, n =6 n1, and n=1 (zn) = n=1 un. Then, by (6.3), for all m ∈ N P P ∞ ∞ ∗ q q (zn) ≤ un, n=m n=m X X and hence we have m ∞ ∞ q q q un = un − un n=1 n=1 n=m+1 X X∞ X∞ ∗ q q = (zn) − un n=1 n=m+1 X∞ X∞ m ∗ q ∗ q ∗ q ≤ (zn) − (zn) = (zn) . n=1 n=m+1 n=1 X X X It is clear that every quasi-Banach sequence space, satisfying the Sq -property, is ∗ symmetric. Therefore, u = (un) ∈ E and kukE = 2kD2x kE ≤ 4kxkE. Conse- quently, since E has the Sq-property, combining this together with the preceding relations, we get that (zn) ∈ E and kzkE ≤ 4C kxkE, where C is the Sq-property constant. Moreover, E is solid and |y|≤|z|. Therefore, kykE ≤kzkE ≤ 4C kxkE. Summarizing all, we conclude that E is a uniform K-monotone space with respect to the couple (ℓ0,ℓq) . (b) =⇒ (a). Let now E be a uniform K -monotone space with respect to the 0 q ∞ ∞ q couple (ℓ ,ℓ ) . Suppose that sequences x = (xn)n=1 ∈ E and y = (yn)n=1 ∈ ℓ satisfy conditions (6.1) and (6.2). Then, the same argument as in the first part ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 39 of the proof yields ∞ ∞ ∗ q ∗ q N (yn) ≤ (xn) , m ∈ . n=m n=m X X This inequality, combined with formula (2.18) and implication (2.16), yields K(t, y; ℓ0,ℓq) ≤K(t, x; ℓ0,ℓq), t> 0. Hence, by the assumption, y ∈ E and kykE ≤ C kxkE, where C is the K - monotonicity constant of E with respect to the couple (ℓ0,ℓq). Since every uniform K-monotone space with respect to a couple of quasi-Banach spaces is also an interpolation space with respect to this couple, from (a) it follows that E ∈ Int (ℓ0,ℓq). Finally, assume that q ≥ 1. Then, by Corollary 4.6, every interpolation space between ℓ0 and ℓq is a uniform K-monotone space. This fact, combined together with already proved implication (b) =⇒ (a), implies the last assertion of the theorem. From Theorems 6.2 and 4.10 we get Corollary 6.3. Let q ≥ 1. A solid quasi-Banach sequence space E has the Sq- property if and only if E ∈ Int (ℓp,ℓq) for some p> 0. ∞ Recall now the following definition from [18]. For every sequence x =(xn)n=1 N (N) ∞ (N) and each N ∈ , let (xn )n=1 be the truncated sequence defined by xn = xn if (N) 1 ≤ n ≤ N and xn = 0 if n > N. We say that a normed sequence space E has the weak Fatou property if there is a constant R such that, for every sequence ∞ (N) ∞ N x = (xn)n=1 of nonnegative numbers with (xn )n=1 ∈ E for all N ∈ and (N) supN∈N k(xn )kE < ∞, we have x ∈ E and (N) kxkE ≤ R sup (xn ) E . N∈N Obviously, each normed sequence space with the Fatou property has the weak Fatou property (see Section 2.2. According to the main result of [18], if q ≥ 1, then every normed sequence space E with the weak Fatou property has the Sq-property if and only E is an interpolation space between ℓ1 and ℓq. Moreover, by interpolation, from the assumption E ∈ Int (ℓ1,ℓq) it follows that E ∈ Int (ℓp,ℓq) for all 0 ≤ p < 1. Therefore, applying Theorem 6.2, we get the following result, which in a sense complements Corollary 3.3. Corollary 6.4. Let q > 1 and let E be a solid Banach sequence space with the weak Fatou property. Then, the following conditions are equivalent: (i) E ∈ Int (ℓp,ℓq) for all p ∈ [0, 1); (ii) E ∈ Int (ℓp,ℓq) for some p ∈ [0, 1); (iii) E ∈ Int (ℓ1,ℓq); (iv) E has the Sq-property. 40 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON 7. (ℓp,ℓq) is not a uniform Caldero´ n-Mityagin couple if 0 ≤ p Let Pp and Qq be the operators introduced in Section 2.2. Taking into account the Holmstedt formula (2.13) if p > 0 and relations (2.17) and (2.18) if p = 0, one can easily realize that Theorem 7.1 is a straightforward consequence of the following proposition. Proposition 7.2. Let 0 ≤ p (7.2) (Qqy)n ≤ (Qqx)n, n =1, 2,..., if p =0, such that for every linear operator S : ℓq → ℓq with Sx = y we have (7.3) kSkℓq→ℓq ≥ C. Proof. Taking for y the element e1 of the unit vector basis, we consider the cases p> 0 and p = 0 separately. (a) p> 0. Since every K-functional K(t, x; X0,X1) is an increasing function in 1/α q t, by (2.13), the sum (Ppz)n + n (Qqz)n, for each z ∈ ℓ , is almost increasing in n , i.e., 1/α 1/α (7.4) (Ppz)n + n (Qqz)n ≤ Cp,q (Ppz)m + n (Qqz)m if n ≤ m, where Cp,q ≥ 1 depends only on p and q . Given any constant C > 0, choose a positive integer N so that −1 1/q−1 (7.5) (2Cp,q) N > C. ∞ −1/q Next, we set x = (xn)n=1, where xn = 2Cp,qN if 1 ≤ n ≤ N, and xn = 0 if n > N. Then, N 1/q q (Qqx)1 = xn =2Cp,q. n=1 X Therefore, by (7.4), the right-hand side of inequality (7.1) is not less than 2 for all n ∈ N. On the other hand, (Ppy)n = 1, n ∈ N, and (Qqy)1 =1, (Qqy)n = 0, n ≥ 2. Hence, the left-hand side of (7.1) does not exceed 2, and so the latter inequality holds. ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 41 Let now S be a linear operator such that S : ℓq → ℓq with Sx = y. Clearly, S is defined by a sequence of bounded linear functionals on ℓq. In particular, setting Λ(z) := hSz,e1i, we have Λ (x)= hSx, e1i = hy, e1i = 1 and |Λ(z)|≤kSkkzkℓq . Consequently, if βn := Λ (en), n ∈ N, we have |βn|≤kSk . Hence, N −1/q 1−1/q 1= |Λ(x)| =2Cp,q βnN ≤ 2Cp,q kSk N . j=1 X According to the choice of N in (7.5), this implies that −1 1/q−1 kSk ≥ (2Cp,q) N > C, and in this case the result follows. (b) p = 0. Given constant C > 0 we take now N ∈ N satisfying the inequality 1/q−1 ∞ −1/q N > C. Let x = (xn)n=1, where xn = N if 1 ≤ n ≤ N, and xn = 0 if n > N. As above, we have (Qqx)1 = 1. Therefore, since (Qqy)1 = 1 and (Qqy)n = 0 for all n ≥ 2, inequality (7.2) holds. If S is a linear operator such that S : ℓq → ℓq with Sx = y, the same reasoning as in the case (a) shows that kSk ≥ N 1/q−1 >C, and so the proof is completed. Remark 7.3. It is a long-standing problem in the interpolation theory if a quasi- Banach couple with the Calder´on-Mityagin property possesses as well its uniform version (see, for instance, [28, p. 1150]). In fact, by now this question is unan- swered when restricted to the narrower classes of Banach couples or even of couples of Banach lattices. . References [1] J. Arazy and M. Cwikel, A new characterization of the interpolation spaces between Lp and Lq. Math. Scand., 55 (1984), 253–270. 1, 3, 4, 4 [2] S. V. Astashkin, Interpolation of operators in quasinormed groups of measurable functions, Siberian Math. J. 35 (1994), no. 6, 1075–1082. 1, 2.2, 3.2 [3] S. V. Astashkin, A version of Calder´on-Mityagin theorem for the class of rearrangement invariant groups. Nonlinear Anal., 200:112063, 2020. 1, 2.2, 4, 4.4 [4] S. V. Astashkin, The Rademacher System in Function Spaces. Birkh¨auser, Switzerland, 2020. 1 [5] S. V. Astashkin and P. G. Nilsson, A version of K-divisibility property for some quasi- Banach couples (preprint). 1 [6] C. Bennett and R. Sharpley, Interpolation of operators , London: Academic Press, 1988. 1, 2.1 [7] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976. 1, 2.1, 2.1, 2.1, 2.2, 2.2, 2.2, 2.3, 2.3, 3.2, 5, 5 [8] Yu. A. Brudnyi and N. Ja. Kruglyak, Real interpolation functors. Dokl. Akad. Nauk SSSR, 256 (1981), 14–17 (in Russian), Soviet Math. Dokl., 23 (1981), 5–8. 1 [9] Yu. A. Brudnyi and N. Ya. Kruglyak, Interpolation Functors and Interpolation Spaces 1, North-Holland, 1991. 1, 2.1, 2.2, 5 42 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON [10] Yu. N. Bykov and V.I. Ovchinnikov, Interpolation properties of scales of Banach spaces, Math. Notes 80 (2006), no. 6, 761–769. 1, 4, 4 [11] L. Cadilhac, Majorization, interpolantion and noncommutative Khinchin inequalities, Stu- dia Math. 258 (2021), 1–26. 1 [12] L. Cadilhac, F. Sukochev, and D. Zanin, Lorentz-Shimogaki-Arazy-Cwikel theorem revis- ited. arXiv.org 2009.02145v1 math.FA (2020). 1, 1, 2.2, 3.5, 4, 4.4, 4, 4 [13] A. P. Calder´on, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273–299. 1, 2.1 [14] M. Cwikel, Monotonicity properties of interpolation spaces. Ark. Math. 14 (1976), 213–236. 1 [15] M. Cwikel, Monotonicity properties of interpolation spaces II. Ark. Mat. 19 (1981), 123– 136. 1, 3.2, 4.2 [16] M. Cwikel, Calder´on couples, Encyclopedia of Mathematics. ISBN 1402006098, The current version is accessible at https://encyclopediaofmath.org/wiki/Calder%C3%B3n couples 1 [17] M. Cwikel and P. G. Nilsson, Interpolation of weighted Banach lattices. Memoirs Amer. Math. Soc. 165 (2003) no. 787, 1–105. 1, 2.1 [18] M. Cwikel and P. G. Nilsson, An alternative characterization of normed interpolation spaces between ℓ1 and ℓq, arXiv.org 1711.0809v1 math.FA (2017). 1, 1, 4, 6, 6, 6 [19] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl., 60 (1981), 1–50. 1 w0 w1 [20] V. I. Dmitriev, Interpolation spaces between (L1 ,L1 ) and (L1,L∞) , Math. Notes 17 (1975), no. 5, 433–438. 1 [21] V. I. Dmitriev, On interpolation of operators in Lp spaces, Dokl. Akad. Nauk SSSR 260 (1981), no. 5, 1051–1054 (in Russian). 1 [22] V. I. Dmitriev, On estimates of interpolation orbits of functions from L1 + L∞, Math. Notes 51 (1992), no. 1, 40–47. 1, 5.2 [23] V. I. Dmitriev, S. G. Krein and V. I. Ovchinnikov, Foundations of Linear Operators Inter- polation Theory - In ”Geometry of Linear Spaces and the Operator Theory”, Yaroslavl’, 1977, pp. 31–75 (in Russian). 1 [24] L. Grafakos, Modern Fourier Analysis, New York, Springer, 2009. 1 [25] T. Holmstedt, Interpolation of quasi-normed spaces. Math. Scand. 26 (1970), 177–199. 2.3 [26] H. Hudzik and L. Maligranda, An interpolation theorem in symmetric function F -spaces, Proc. Amer. Math. Soc. 110 (1990), no. 1, 89–96. 1, 3.2 [27] T. P. Hyt¨onen, J. M. A. M. van Neerven, M. C. Veraar, and L. W. Weis, Analysis in Banach Spaces, Vol. II: Probabilistic Methods and Operator Theory. A Series of Modern Surveys in Mathematics, vol. 67, Springer, 2017. 1 [28] N. Kalton and S. Montgomery-Smith, Interpolation of Banach Spaces, Handbook of the Geometry of Banach Spaces, Part 2, North-Holland Publishing Company, (2003), 1131– 1175. 1, 7.3 [29] N. J. Kalton, Calder´on couples of rearrangement invariant spaces. Studia Math. 106 (1993), 233–277. 1, 1 [30] N. J. Kalton, N. T. Peck, and J. W. Roberts, An F -space Sampler, London Mathematical Society Lecture Notes Series, 1984. 3.1 [31] B. S. Kashin and A. A. Saakyan, Orthogonal series, Amer. Math. Soc., Providence RI, 1989. 1 [32] P. Kr´ee, Interpolation d’espaces qui ne sont ni norm´es ni complets. Applications, Ann. Inst. Fourier (Grenoble) 17 (1967), fasc. 2, 137–174 (1968). 1 [33] S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of linear operators, Amer. Math. Soc., Providence, RI, 1982. 1, 2.1, 3.2 ARAZY-CWIKEL AND CALDERON-MITYAGIN´ TYPE PROPERTIES 43 [34] G. Levitina, F. Sukochev, and D. Zanin, Cwikel estimates revisited, arXiv.org 1703.04254v1 math. FA (2017). 1, 1, 1, 4 [35] G. Levitina, F. Sukochev, and D. Zanin, Cwikel estimates revisited, Proc. Lond. Math. Soc. (3), 120 (2020), no. 2, 265–304. 1, 1, 1, 4, 4 [36] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, Berlin, 1979. 1, 2.3 [37] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer-Verlag, Berlin, 1977. 1, 1, 2.3, 3.2 [38] G. G. Lorentz and T. Shimogaki, Interpolation theorems for the pairs of spaces (Lp,L∞) and (L1,Lq). Trans. Amer. Math. Soc., 159 (1971), 207–221. 1 [39] L. Maligranda and V. I. Ovchinnikov, On interpolation between L1 + L∞ and L1 ∩ L∞, J. Funct. Anal. 107 (1992), 342–351. 1 [40] B. S. Mityagin, An interpolation theorem for modular spaces, Proc. Conf. ”Interpol. Spaces and Allied Topics in Anal.” (Lund, 1983), Lecture Notes in Math. 1070 (1984), 10–23 (translated from Mat. Sb. 66 (1965), 473–482 (in Russian)). 1, 2.1 [41] S. J. Montgomery-Smith, The Hardy operator and Boyd indices. In ”Interaction between functional analysis, harmonic analysis, and probability” (Columbia, MO, 1994), vol. 175 of Lecture Notes in Pure and Appl. Math., pp. 359–364. Dekker, New York, 1996. 3.5 [42] E. T. Oklander, On interpolation of Banach Spaces, Ph.D. thesis, University of Chicago, 1964. 1 [43] V. I. Ovchinnikov, Interpolation theorems resulting from an inequality of Grothendieck, Funct. Anal. Appl. 10 (1976), no. 4, 287–294. 1 [44] V. I. Ovchinnikov, On the estimates of interpolation orbits, Math. USSR-Sb. 43 (1982), no. 3, 573–583. 1 [45] V. I. Ovchinnikov, The Method of Orbits in Interpolation Theory, Math. Reports 1 (1984), 349–516. 1, 2.1, 2.1, 2.2 [46] V. I. Ovchinnikov, Interpolation orbits in couples of Lebesgue spaces, Funct. Anal. Appl. 39 (2005), no. 1, 46–56. 1 [47] V. I. Ovchinnikov and V. I. Dmitriev, The limits of the applicability of the K-method of interpolation, in Collection of Articles on Applications of Functional Analysis, pp. 102–108, Voronezh, 1975 (in Russian). 1 [48] J. Peetre, Nouvelles propri´et´es d’espaces d’interpolation. C. R. Acad. Sci. Paris 256 (1963), 1424–1426 (French). 1 [49] J. Peetre, A theory of interpolation of normed spaces. (Lecture notes, Brasilia, 1963.) Notas de matematica, 39 (1968), 1-86. 1 [50] J. Peetre, Banach couples I. Technical report, Lund, 1971. 1 [51] J. Peetre, Remark on the dual of an interpolation space, Math. Scand. 34 (1974), 124-128 3.1 [52] J. Peetre and G. Sparr, Interpolation of normed Abelian groups, Ann. Mat. Pura Appl. (4). 92 (1972), no. 4, 217–262. 1, 4, 2.2, 2.6, 2.2 [53] A. M. Rubinov, Sublinear operators and their applications, Russian Math. Surveys, 32 (1977), no. 4, 115–175. 4 [54] W. Rudin, Functional analysis. Second edition., MacGraw-Hill, New York, 1991. 2.1 [55] G. I. Russu, Symmetric spaces of functions that do not have the majorization property, Mat. Issled. 4 (1969), 82–93 (in Russian). 3.2 [56] A. A. Sedaev and E. M. Semenov, On the possibility of describing interpolation in terms of the K-method of Peetre, Optimizascja 4 (1971), 98–114 (in Russian). 1 [57] G. Sparr, Interpolation of weighted Lp-spaces, Studia Math. 62 (1978), 229–271. 1, 1, 4 [58] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Publ. Comp., Amsterdam, New-York, Oxford, 1978. 1 [59] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, UK, 1991. 44 SERGEY V. ASTASHKIN,MICHAEL CWIKEL, AND PER G. NILSSON 1 Email address: [email protected] Astashkin:Department of Mathematics, Samara National Research Univer- sity, Moskovskoye shosse 34, 443086, Samara, Russia Email address: [email protected] Cwikel: Department of Mathematics, Technion - Israel Institute of Technol- ogy, Haifa 32000, Israel Email address: [email protected] Nilsson: Roslagsgatan 6, 113 55 Stockholm, Sweden Email address: [email protected]