SOME REITERATION RESULTS FOR INTERPOLATION METHODS DEFINED BY MEANS OF POLYGONS

. We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N -tuples of function spaces, of spaces of bounded linear operators and Banach algebras.


Introduction
This paper deals with interpolation methods for finite families (N -tuples) of Banach spaces defined by means of a convex polygon Π in the plane R 2 and a point (α, β) in the interior of Π.These methods were introduced by Cobos and Peetre in [13], further investigations have been done by Cobos, Kühn and Schonbek [10], Cobos, Fernández-Martínez and Schonbek [9], Cobos, Fernández-Martínez and Martínez [7], Ericsson [15], Cobos, Fernández-Martínez, Martínez and Raynaud [8], Cobos and Martín [11] and Fernández-Cabrera and Martínez [18], among other authors.Thinking of the Banach spaces as sitting on the vertices of Π they introduced K-and J-functionals with two parameters and then they define K-and J-spaces by using an (α, β)-weighted L q -norm (the precise definitions are recalled in Section 2).For the special choice of Π as the simplex, these methods give back (the first nontrivial case of) spaces introduced by Sparr [26], and if Π is the unit square they recover spaces studied by Fernandez [16].Other references on interpolation methods for N -tuples can be found in the monographs by Triebel [27] and Brudnyǐ and Krugljak [4].
It was shown in [7] and [15] that reiteration formulae between methods associated to polygons and the real method are important to describe K-and J-spaces in certain cases.In the present paper we continue their research.First we complement a result of Ericsson on interpolation using the unit square of a 4-tuple formed by spaces of class θ j with respect to a couple {X, Y }.As we show with an example, in this result is essential that (α, β) does not lie in any diagonal of the square.The example refers to a 4-tuple of the kind {X, Y, Y, X} with X ֒→ Y .We also characterize the K-spaces generated by this 4-tuple and we show that they are extrapolation spaces when (α, β) is in the diagonal β = 1−α.Then, assuming a mild condition on the θ j and that q takes only the value 1 or ∞, we establish results that work for general polygons Π and for any (α, β) in its interior, even if (α, β) lies in any diagonal of Π. Applications are given to 4-tuples of Lorentz function spaces, Besov spaces, Lorentz operator spaces and N -tuples of spaces of bounded linear operators.We also establish a result on interpolation of Banach algebras.
The paper is organized as follows.In Section 2 we review some basic notions on K-and J-spaces associated to polygons.In Section 3 we show the reiteration results for the unit square and their applications to function spaces and to Lorentz operator spaces.Finally, in Section 4, we establish the results for general polygons.

Preliminaries
By a Banach N -tuple A = {A 1 , . . ., A N } we mean N -Banach spaces A j , j = 1, . . ., N , which are continuously embedded in a common Hausdorff topological vector space.We put Σ Let Π = P 1 • • • P N be a convex polygon in the affine plane R 2 , with vertices P j = (x j , y j ).Given any N -tuple A we imagine each space A j as sitting on the vertex P j and we define K-and J-functionals by Here t and s stand for positive numbers.Now let (α, β) be an interior point of Π, (α, β) ∈ Int Π, and let 1 ≤ q ≤ ∞.The K-space A (α,β),q;K consists of all those a ∈ Σ(A) for which the norm is finite (the integral should be replaced by the supremum if q = ∞).
Note the analogy of these constructions with the real interpolation space (X, Y ) θ,q for Banach couples {X, Y }.The space (X, Y ) θ,q can be described by a similar scheme, but working with R instead of R 2 , with the segment [0, 1] taking the role of the polygon Π and 0 < θ < 1 being an interior point of the segment [0, 1].The space X should be imagined as sitting on the point 0 and Y on the point 1.The relevant functionals are now (see [2] or [27]).
A Banach space Z is said to be an intermediate space with respect to the and Z is said to be of class C J (θ; X, Y ) if there is a constant C > 0 such that a Z ≤ Ct −θ J(t, a) for all a ∈ X ∩ Y.
Here 0 ≤ θ ≤ 1.If Z is of class C K (θ; X, Y ) and of class C J (θ; X, Y ) then we say that Z is of class C(θ; X, Y ).Clearly X is of class C(0; X, Y ) and Y is of class C(1; X, Y ).It is also well-known that for 0 < θ < 1 the real interpolation spaces (X, Y ) θ,q and the complex interpolation spaces (X, Y ) [θ] are spaces of class C(θ; X, Y ) (see [2] or [27]).
If R is any affine bijection on R 2 then K-and J-spaces defined by means of Π and (α, β) coincide (with equivalence of norms) with those defined by means of R(Π) = RP 1 • • • RP N and R(α, β) (see [10], Remark 4.1).We call this fact the property of invariance under affine bijections.
The geometrical elements play an important role in the theory of K-and J-spaces.Indeed, let P α,β be the set of all triples {i, k, r} such that (α, β) belongs to the triangle with vertices P i , P k , P r (see Fig. 2.1).For each {i, k, r} ∈ P α,β let (c i , c k , c r ) be the (unique) barycentric coordinates of (α, β) with respect to P i , P k , P r .It was shown in [6], Thm.1.3, that there is a constant C > 0 such that for any a ∈ ∆(A) we have We also recall that, for any N non-negative real numbers M 1 , . . ., M N we have (see [9], Thm.1.11).It is possible to relate J-and K-spaces generated by an N -tuple A with those spaces generated by a subtuple Ã of A. Next we discuss the case when the subtuple is a 3-tuple.
Let {i, k, r} ∈ P α,β and suppose that (α, β) belongs to the interior of the triangle P i P k P r .If we put Ã = {A i , A k , A r } and we designate by K, J the K-and J-functionals defined by means of the triangle, then we have This yields the continuous embeddings (2.5) If {i, k, r} ∈ P α,β but (α, β) is not in the interior of the triangle, then (α, β) should be in a diagonal of Π. Say, for example, that (α, β) belongs to the diagonal joining P i and P k (see Fig. 2.2).The barycentric coordinates of (α, β) with respect to the points P i , P k , P r are (1 − θ ik , θ ik , 0) for some 0 < θ ik < 1.Then it turns out that (2.6) (see [7], Thm.1.5).

It was shown in
The following result is a consequence of (2.7), (2.8) and the invariance under affine bijection (see [15], Cor. 4).
Let Π = P 1 P 2 P 3 be a triangle, let (α, β) ∈ Int Π with barycentric coordinates (c 1 , c 2 , c 3 ) with respect to P 1 , P 2 , P 3 and let 1 ≤ q ≤ ∞.If A j is a space of class C(θ j ; X, Y ) with 0 ≤ θ j ≤ 1, j = 1, 2, 3, and the θ j are not all equal then we have with equivalent norms 3. Interpolation over the unit square.
In this section we take Π = P 1 P 2 P 3 P 4 equal to the unit square, that is to say, P 1 = (0, 0), P 2 = (1, 0), P 3 = (0, 1), P 4 = (1, 1).Let (α, β) ∈ Int Π such that (α, β) does not lie on any diagonal of Π and let Q = (1/2, 1/2).The point (α, β) is in only one internal triangle P i P k Q and so it is in the two triangles P i P k P r , P i P k P s formed by vertices of Π. Figure 3.1 illustrate the situation for i = 1, k = 2, r = 3, s = 4. Let (c i , c k , c r ) and (d i , d k , d s ) be the barycentric coordinates of (α, β) with respect to P i , P k , P r and P i , P k , P s , respectively.The following results improves [15], Thm. 6, by removing several restrictions on the class of the spaces A j .
Theorem 3.1.Let {X, Y } be a Banach couple, let A j be a space of class C(θ j ; X, Y ), 0 ≤ θ j ≤ 1, j = 1, 2, 3, 4, and let 1 ≤ q ≤ ∞.We suppose that θ i = θ k where i, k are the indices of the vertices P i , P k of the (unique) internal triangle Then we have, with equivalent norms, Proof.First we assume that (α, β) lies in P 1 P 2 Q.Then θ 1 = θ 2 and (α, β) is in the triangles P 1 P 2 P 3 and P 1 P 2 P 4 .Using (2.5) and (2.9) we get In order to check the converse embeddings we consider the affine bijection Now we distinguish two cases.If We have with δα ′ + ρβ ′ = θ 123 and δ ′ α ′ + ρ ′ β ′ = θ 124 .Therefore, by (2.7), (2.8) and the invariance under affine bijection, we derive Again (3.4) holds.This time δα ′ + ρβ ′ = θ 124 and δ ′ α ′ + ρ ′ β ′ = θ 123 .Hence, (3.5) and (3.6) follows as in the previous case.If (α, β) lies in an internal triangle different from P 1 P 2 Q then we use the symmetry of the unit square to lead the situation to the result that we have just established.Assume, for example, that (α, β) is in P 2 P 4 Q (see Fig. 3.2).The remaining cases can be treated in the same way.Then we know with analogous formulae for J-spaces.Hence , write θ * j for the class of B j with respect to {X, Y } and define θ * ikr as in (3.1) but using the barycentric coordinates of (1 − β, 1 − α) and the θ * j .We have θ * 1 = θ 4 = θ 2 = θ * 2 , hence we can apply the result that we have established in the first part of the proof and derive that This proves the K-formula.The J-formula follows similarly.The proof is complete.
Using Theorem 3.1 we can complement [15], Example 1, by reducing the conditions on the parameters.Let us write down the outcome.Take any σ-finite measure space (Ω, µ) and for 1 < p < ∞ and 1 ≤ q ≤ ∞, let L p,q be the Lorentz function space [2] or [27]).We have (L 1 , L ∞ ) θ,q = L p,q for 1/p = 1 − θ.As a direct consequence of Theorem 3.1 we obtain the following.
Corollary 3.2 refers to the case when (α, β) lies in the internal triangle P 1 P 2 Q.Similar results holds when (α, β) is in any of the other three internal triangles.
In order to give a second application we recall the (Fourier-analytical) definition of Besov spaces.Let S(R n ) and S ′ (R n ) be the Schwartz spaces of all rapidly decreasing complex infinitely differentiable functions on R n and the space of tempered distributions on R n , respectively.For f ∈ S ′ (R n ), the Fourier transform and its inverse are defined in the usual way and denoted by f and f , respectively.Let ϕ be a We put ϕ 0 = ϕ and for j ∈ N we write (with the usual modification if q = ∞).We refer to the monographs by Triebel [28], [29], [30] for details on Besov spaces.It is clear from the definition that B s p,q ֒→ B u p,q if u < s.The following interpolation formula holds (B s 0 p,q 0 , B s 1 p,q 1 ) θ,q = B s p,q .
Similar results hold if (α, β) lies in an internal triangle different from If (α, β) lies in any diagonal of the square then it should be in two internal triangles at least (see Fig. 3.3).In this case, Theorem 3.1 is not valid in general even if we assume that θ i = θ j for any triangle P i P k Q containing (α, β).We show it with an example.
Splitting the double integral of the norm of the K-space in the sets Here ∼ equivalence of norms.For the other diagonal we have We may assume, without loss of generality, that the norm of the embedding X ֒→ Y is 1.Then K(t, a) = t a Y for all 0 < t ≤ 1.This yields that for any δ > −1 , 0 < γ ≤ 1 and 0 < θ < 1 (3.9) Let α < 1/2, so −2(1 − α) < −2α.By (3.7), (3.9) and (3.10), we obtain For the other diagonal, from (3.8) and (3.9) we derive The case q = ∞ can be analogously.
Next we specialize Theorem 3.5 to two concrete cases.Let (Ω, µ) be a finite measure space with µ(Ω) = 1.Recall that the Zygmund spaces L log L is formed by all µ-measurable functions f on Ω for which [1]).As it was pointed out in [20], Example 2.6, the space L log L can be obtained by extrapolation from the couple {L ∞ , L 1 }.Indeed, by [2], Thm.5.2.1, the K-functional for {L 1 , L ∞ } is given by As a direct consequence of Theorem 3.5 we get the following result.Corollary 3.6.Let 0 < α < 1.Then Now take a Hilbert space H and let L(H) be the space of all bounded linear operators acting from H into H.The singular numbers of T ∈ L(H) are We refer to [19] for details on these spaces.In a more general way, given 1 < p < ∞ and 1 ≤ q ≤ ∞ the Lorentz operator space L p,q (H) is defined as the collection of all T ∈ L(H) for which (see [27]).Spaces L p,q (H) are the analogues for operators to the Lorentz function spaces L p,q .From the point of view of interpolation theory, both families of spaces behave in a similar way.Namely, (L 1 (H), L(H)) θ,q = L p,q (H) , Hence, writing down Theorem 3.1 for these spaces we obtain a similar result to Corollary 3.2 but replacing L p,q by L p,q (H).
In order to specialize Theorem 3.5, we recall that for 1 ≤ q < ∞ the space L ∞,q (H) is formed by all T ∈ L(H) for which [22] and [14]).These spaces correspond to the limit case p = ∞ in the Lorentz scale but the general theory of Lorentz operator spaces does not cover the case of L ∞,q -spaces (see [14], p. 325).It is shown in [12], Cor.4.3, that (L 1 (H), L(H)) 1,q = L ∞,q (H).
Consequently, Theorem 3.5 gives the following formulae.

Interpolation over general polygons.
In this section we deal with general polygons Π = P 1 • • • P N .Assuming a mild condition on the θ j and that q takes only the values 1 or ∞, we shall establish results that work even if (α, β) lies in any diagonal.
Next we show some direct applications of Theorem 4.1.Assume first that the Banach couple {X, Y } is formed by Banach algebras such that multiplications in X and Y coincide on X ∩ Y .It was shown by Bishop [3] (see also [21] and [5]) that for 0 < θ < 1 the space (X, Y ) θ,1 is a Banach algebra.Multiplication in (X, Y ) θ,1 being the same as in X and Y on X ∩Y .So Theorem 4.1 yields the following result.Corollary 4.2.Let Π = P 1 • • • P N be a convex polygon and let (α, β) ∈ Int Π.Let {X, Y } be a couple of Banach algebras and let A = {A 1 , . . ., A N } be an N -tuple formed by spaces A j of class C(θ j ; X, Y ) where the θ j satisfy the same assumptions as in Theorem 4.1.Then A (α,β),1;J is a Banach algebra.