Math. Nachr. 279, No. 7, 743 – 755 (2006) / DOI 10.1002/mana.200310390

The Gustavsson–Peetre method for several Banach spaces Pedro Fern´andez–Mart´ınez∗1 , Luis Gonz´alez∗∗1,2 , and Raul ´ Romero∗∗∗3 1

2

3

Departamento de Matem´atica Aplicada, Facultad de Inform´atica, Universidad de Murcia, Campus de Espinardo, 30071 Espinardo (Murcia), Spain Departamento de Matem´aticas, Campus Universitario de Tacira, Universidad de Las Palmas de Gran Canarias, 35017 Las Palmas, Spain Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad Complutense de Madrid, 28040 Madrid, Spain Received 12 December 2003, revised 2 November 2004, accepted 19 December 2004 Published online 4 April 2006 Key words Interpolation methods, Orlicz spaces MSC (2000) 46B70 We extend the Gustavsson–Peetre method to the context of N -tuples of Banach spaces. We give estimates for the norm of the interpolated operator. The method is applied to tuples of weighted Lp -spaces and to tuples of Orlicz spaces identifying the outcoming spaces in both cases. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim !

0

Introduction

The Gustavsson–Peetre method in interpolation was introduced by both authors in 1977, see [12]. This method, also called the ± method, was particularly useful to study interpolation between two Orlicz spaces. Later, in 1982, Gustavsson showed how to use it in interpolating couples of weighted Lp -spaces, see [11]. Besides this, the study of interpolation methods involving more than two spaces was carried out by different authors. Favini gave generalizations of the complex method, [9], while Sparr, [17], and D. L. Fernandez, see [10], studied different extensions of the real method. In 1991 Cobos and Peetre introduced interpolation methods associated to polygons, [7]. These methods agree with Sparr spaces, when the polygon is the simplex and with Fernandez spaces when we work with the unit square. They can be considered as a link between Sparr and Fernandez methods, and offer a common point of view for both of them. The polygon methods were studied during the nineties, they have been a topic of interest for many different authors. Among others, some of the contributions can be found in [3]–[5], [7], and [8]. In the present paper we continue the study of methods that deal with several Banach spaces. Our first goal was the problem of interpolating several, more than two, Orlicz spaces: Given Lϕ , Lϕ1 , . . . , LϕN , Orlicz spaces, we want to know under which conditions Lϕ is an interpolation space with respect to the tuple L = {Lϕ1 , . . . , LϕN }. In the context of couples of spaces, a particular solution for this problem was first given by Calder´on in his famous paper on the complex method, see [2]. Years later the problem was studied by Gustavsson and Peetre and they give a much more general answer in [12] for couples of spaces. In the process of solving the problem they introduce the ± interpolation method. This method appears to be an adequate tool to interpolate Orlicz spaces. More information about interpolation of Orlicz spaces can be found in [14]. Inspired with the ideas of the polygon interpolation methods, and those of the Gustavsson–Peetre method, we give a generalization of the ± method to interpolate tuples of spaces. When we work with tuples of N spaces, we will be aided by a convex polygon of N vertices, Π. We may think of each space of the tuple as sitting on a ∗

∗∗

∗∗∗

Corresponding author: e-mail: [email protected] e-mail: [email protected] e-mail: raul [email protected]

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vertex of the polygon. This approach suggests many of the geometric properties of the polygons methods will be inherit by this new method. In fact, this will be the case. A natural way of extending the ± method to this multidimensional framework is using a minimal description in Aronszjan–Gagliardo sense, see [1] and [13]. Once this is done, a new definition in terms of sequences arises. The possibility of describing the process through sequences helped us to find estimates for the norm of the interpolated operator. Here the influence of the geometry of the underlying polygon plays an important role. This ends the description of the method and will close Section 1. Section 2 is devoted to interpolation of Orlicz spaces. We will work in the more general framework of weighted Orlicz spaces. This will allow some applications to the particular case of interpolation of weighted Lp -spaces, which in the case of couples was studied by Gustavsson in [11]. The result obtained is the following ! Lϕi,j,k (wi,j,k ) (0.1) !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) = P(α,β)

(see page 747 for the definition of P(α,β) and page 748 for those of ϕi,j,k and wi,j,k ). It must be remarked the important role played by the function D(α,β) (find definition in page 747) in the estimate of the norm of the interpolated operator and in the proof of Lemma 2.3. The value of this function was computed by Cobos, Schonbek and one of the present authors in [5]. Previously, the function D(α,β) had appeared in different contexts, see [15] for example.

1

The interpolation method

Recall the Gustavsson–Peetre method can be described as a minimal interpolation method, in the sense of Aronszajn–Gagliardo, see [1], as follows: Given a Banach couple A = (A0 , A1 ) and a real parameter 0 < θ < 1, we define the space % &' "# $ 1 !A0 , A1 "θ = G c0 , c0 (2−m ) ; #∞ θm ( A ) 2 ( * ) ) Tn an ; s.t. #Tn # #an # # $ < ∞ . = n∈N

n∈N

L(c0 ,A )

%∞

1 2θn

$ # Next, assume that A = A1 , . . . , AN is a Banach N -tuple.+Let Π = P1 , . . . , PN be a convex polygon, , with −mx1 −ny1 −mxN −nyN ), . . . , c0 (2 ) . Given vertices Pj = (xj , yj ), and consider also the N -tuple c0 = c0 (2 (α, β) ∈ Int Π we define the space " ' ! A "(α,β) = G c0 , #∞ (2−αm−βn ) ( A ) * ( ) ) Tn an ; s.t. #Tn #L(c0 ,A ) #an #%∞ (2−αm−βn ) < ∞ , = a= n∈N

where #T #L(c0,A ) by the expression

n∈N

+ , = max #T #c0 (2−xj m−yj n ),Aj , 1 ≤ j ≤ N . The norm of this interpolation space is given

#a#(α,β) = inf

(

)

n∈N

#Tn #L(c0 ,A ) #an #%∞ (2−αm−βn )

*

,

where the infimum is taken over all representations of a as above. Next we describe this interpolation space as the orbit of the sequence (2αm+βn )m,n∈Z , see [16]. More precisely we have the following statement where Orb stands for orbit: Theorem 1.1 # $ ! A "(α,β) = Orb c0 (2αm+βn )Z2 , A

with equality of norms.

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# $ P r o o f. We show first the continuous embedding Orb c0 (2αm+βn )Z2 , A &→ ! A "(α,β) . Let T ∈ L(c0 , A ). Since (2αm+βn )Z2 ∈ #∞ (2−αm−βn ), the definition of ! A "(α,β) as a minimal functor shows that a = $ $ # # T (2αm+βn )Z2 ∈ ! A "(α,β) . Moreover, for all T ∈ L(c0 , A ) such that a = T (2αm+βn )Z2 we have #a#%A&(α,β) ≤ #T #L(c0,A ) #(2αm+βn )#%∞ (2−αm−βn ) = #T #L(c0,A )

and taking the infimum over all such operators we get: # $ 1 Orb c0 (2αm+βn )Z2 , A &→ ! A "(α,β) .

$ # Now we prove the reverse inclusion, ! A "(α,β) &→ Orb c0 (2αm+βn )#Z2 , A . Let (xm,n )Z2$ ∈ #∞ (2−αm−βn ), and consider the operator T : Σ(c0 ) → Σ(c0 ) mapping (λm,n ) ! 2−αm−βn λm,n xm,n Z2 . We check the restrictions of this operator are bounded and so T ∈ L(c0 , c0 ). - −αm−βn λm,n xm,n )Z2 - −mx −ny = sup |2−αm−βn xm,n | |λm,n | 2−mxj −nyj -(2 j j) c0 (2 m,n∈Z ≤ (xm,n )m,n∈Z -%∞ (2−αm−βn ) -(λm,n )Z2 -c0 (2−mxj −nyj ) . Hence, for 1 ≤ j ≤ N ,

#T #c0(2−mxj −nyj ),c0 (2−mxj −nyj ) ≤ #(xm,n )Z2 #%∞ (2−αm−βn ) showing that #T #L(c0,c0 ) ≤ #(xm,n )Z2 #%∞ (2−αm−βn ) . Once we have proved that T ∈ L(c0 , c0 ), it is easy to conclude that # # $ $ T (2αm+βn ) = (xm,n )Z2 ∈ Orb c0 (2αm+βn )Z2 , c0 . Finally, we use the fact that ! · "(α,β) is a minimal method to establish the inclusion # $ 1 ! A "(α,β) &→ Orbc0 (2αm+βn )Z2 , A .

The following description of this interpolation method will be more suitable for our purposes. Proposition 1.2 The space ! A "(α,β) consists of all a ∈ Σ( A ) for which there exists a sequence (um,n )Z2 ⊂ ∆( A ) satisfying that ) a = um,n in Σ( A ) , (1.1) m,n∈Z

and that for every F , finite set in Z2 , and any (ξm,n ) ∈ #∞ with #(ξm,n )#%∞ ≤ 1, - ) 2xi m+yi n um,n ξm,n ≤ C, 2αm+βn (m,n)∈F

(1.2)

Ai

for 1 ≤ i ≤ N and some constant C independent of F and (ξm,n ). The norm satisfies the equality #a#% A &(α,β) = inf{C} where the infimum extends over all representations (um,n ) of a as above. % mxi +nyi & 2 um,n Remark 1.3 Condition (1.2) can be expressed by saying that the sequence is weakly αm+βn 2 Z2 unconditionally Cauchy in Ai , for 1 ≤ i ≤ N . This is equivalent to saying that there exists a constant C > 0 verifying that for every finite set, F ⊂ Z2 , and any combination of signs ± - ) 2xi m+yi n um,n ± - ≤ C 2αm+βn (m,n)∈F

Ai

for 1 ≤ i ≤ N .

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P r o o f. Given an element a ∈ Σ( A ) verifying conditions (1.1) and (1.2) put |||a|||(α,β) = inf{C} where C is the constant in condition (1.2) and the infimum extends over all representations as in the # $ statement. We first show that the described elements are in the space Orb c0 (2αm+βn )Z2 , A . Let a ∈ Σ( A ) be described by the statement. Choose (um,n )Z2 ⊂ ∆( A ), verifying conditions (1.1) and (1.2). Define the operator T : Σ(c0 ) → Σ( A ) mapping ) 2−αm−βn λm,n um,n . (1.3) (λm,n ) ! m,n∈Z

Let us check that T belongs to L(c0 , A ). Choose F any finite set in Z2 and (λm,n )Z2 ∈ c0 (2−xj m−yj n ), - ) - ) um,n −αm−βn −xj m−yj n −αm−βn 2 λm,n um,n = 2 2 λm,n −mxj −nyj 2 (m,n)∈F (m,n)∈F Aj Aj + −xj m−yj n , ≤ C sup 2 |λm,n | F

- where C is the constant of the condition (1.2). This shows that -T -c0 (2−xj m−yj n ),Aj ≤ C (1 ≤ j ≤ N ), that is # $ to say, T ∈ L(c0 , A ) and #T #L(c0,A ) ≤ C. Further, we have that for any (λm,n ) ∈ c0 2−mxj −nyj the series T (λm,n ) converges unconditionally in Aj , and so in Σ( A ). Thus for any (λm,n ) ∈ Σ(c0 ) the series in (1.3), T (λm,n ), converges unconditionally in Σ( A ). Using condition (1.1) a =

)

# $ $ # um,n = T (2αm+βn )Z2 ∈ Orb c0 (2αm+βn )Z2 , A ,

m,n∈Z

and

#a#Orb c

αm+βn ) Z2 ,A ) 0 ((2

≤ |||a|||(α,β) = inf{C} .

# $ We prove the reverse inclusion now. Namely, the elements in Orb c0 (2αm+βn )Z2 , A verify conditions (1.1) # αm+βn $ # αm+βn $ and (1.2). Let a ∈ Orb c0 (2 )Z2 , A , say a = T (2 )Z2 for some T ∈ L(c0 , A ). Put (em,n ) the . canonical basis in c0 (Z2 ). Since (2αm+βn )Z2 ∈ Σ(c0 ), the series Z2 2αm+βn em,n converges unconditionally in Σ(c0 ). So ) # $ a = T (2αm+βn )Z2 = 2αm+βn T em,n in Σ( A ) . Z2

Define um,n = 2αm+βn T em,n. Clearly the sequence (um,n )Z2 verifies condition (1.1). On the other hand, for any (ξm,n ) ∈ #∞ and any finite set F in Z2 - ) 2xj m+yj n ξm,n αm+βn um,n 2 (m,n)∈F Aj - ) = ξm,n 2xj m+yj n T em,n (m,n)∈F Aj - ) ≤ #T #c0(2−xj m−yj n ),Aj ξm,n 2xj m+yj n em,n - −xj m−yj n (m,n)∈F

c0 (2

)

≤ #T #L(c0,A ) #(ξm,n )#%∞ .

So (um,n ) verifies condition (1.2) and we have the following inequality of norms |||a|||% A &(α,β) ≤ #a#Orb c c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim !

αm+βn ) Z2 ,A ) 0 ((2

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In what follows we study the norm of the interpolated operator. The estimate we get for this norm is similar to those presented by Cobos, Schonbek and one of the present authors for the polygons methods in [5]. We will use the function D(α,β) : + , + ci cj ck , max Mi Mj Mk (1.4) D(α,β) (M1 , . . . , MN ) = inf max t(xj −α) s(yj −β) Mj = t,s>0 1≤j≤N

{i,j,k}∈P(α,β)

where P(α,β) stands for the set of all triples {i, j, k} such that (α, β) belongs to the triangle Pi , Pj , Pk , (ci , cj , ck ) are the barycentric coordinates of (α, β) with respect to the vertices Pi , Pj , Pk , and Mj ≥ 0 (j = 1, . . . , N ). See [5] for the last equality. Proposition 1.4 Let A and B be two Banach N -tuples, T : A → B be a bounded linear operator, then / 0 - c -T #T #cAii ,Bi #T #Ajj ,Bj #T #cAkk ,Bk . ≤ C max %A& ,% B & (α,β)

(α,β)

{i,j,k}∈P(α,β)

P r o o f. Let a ∈ ! A "(α,β) and choose a sequence, (um,n )Z2 ⊂ ∆( A ), verifying conditions.(1.1) and (1.2). Define a second sequence vm,n = um+ν,n+η where ν and η are fixed integers. Clearly a = m,n∈Z vm,n in . Σ( A ). So T a = Z2 T vm,n , and (T vm,n )m,n∈Z verifies condition (1.1) for T a. Moreover, let F be a finite set in Z2 and (λm,n )m,n∈Z ∈ #∞ (Z2 ), - ) 2xj m+yj n λm,n αm+βn T vm,n 2 Bj (m,n)∈F - ) - 2xj m+yj n ≤ T Aj ,Bj λm,n αm+βn um+ν,n+η 2 (m,n)∈F Aj - ) αν+βη xj (m+ν)+yj (n+η) - 2 2 = T Aj ,Bj λm,n xj ν+yj η α(m+ν)+β(n+η) um+ν,n+η 2 2 Aj (m,n)∈F / 0 ≤ #T #Aj ,Bj C max |λm,n | 2(α−xj )ν+(β−yj )η (m,n)∈F / 0 , + (α−xj )ν+(β−yj )η ≤ C max 2 #T #Aj ,Bj max |λm,n | . 1≤j≤N

(m,n)∈F

This proves that

#T a#% B &(α,β) ≤ max

1≤j≤N

/ 0 2(α−xj )ν+(β−yj )η #T #Aj ,Bj #a#% A &(α,β) .

Take infimum over all possible ν and η to obtain #T #% A &(α,β) ,% B &(α,β) ≤

inf

ν,η∈Z 1≤j≤N

≤ C where C = max

1≤j≤N

+

max

max

/

0 2(α−xj )ν+(β−yj )η #T #Aj ,Bj / 0 c #T #cAii,Bi #T #Ajj ,Bj #T #cAkk ,Bk ,

{i,j,k}∈P(α,β)

+ , + ,, max 1, 2(xj −α) , max 1, 2(yj −β) .

The next section is devoted to the interpolation of N -tuples of Orlicz spaces, which are in the category of tuples of quasi-Banach spaces. In this framework, quasi-Banach tuples, Theorem 1.1 does not hold, so we take as definition the statement of Proposition 1.2. Proposition 1.4 remains true in this case.

2

Interpolation of Orlicz spaces

Let (Ω, µ) be a σ-finite measure space. We will interpolate N -tuples of Orlicz spaces using the ± method. Let ϕ : [0, ∞) → [0, ∞) be an Orlicz function. In our setting this is a continuous strictly increasing function www.mn-journal.com

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satisfying ϕ(0) = 0 and ϕ(t) → ∞ as t → ∞. Besides this, we will require that ϕ verifies an inequality of the type: (2.1)

ϕ(λx) ≤ C(λ)ϕ(x)

for all x ∈ (0, ∞) and some constant C(λ) depending on λ. These conditions are fully satisfied by convex functions and all power functions. The Orlicz space Lϕ consists of all the (equivalent classes of ) measurable functions a on Ω verifying that for some λ > 0 1 2 3 |a| ϕ dµ < ∞ . Ω

λ

Lϕ is equipped with the quasi-norm #a#Lϕ

4 5 1 2 3 |a| = inf λ > 0 s.t. ϕ dµ ≤ 1 . Ω

λ

Recall that for a measurable positive submultiplicative function, s, there exist p0 , p1 in R ∪ {±∞} and a constant C > 0 such that s(x) ≤ C max{xp0 , xp1 } . If ϕ : (0, ∞) → (0, ∞) is a functions that satisfies (2.1), we may consider the submultiplicative function 4 5 ϕ(λx) sϕ (λ) = sup . ϕ(x) x∈(0,∞) The previous remarks assure the existence of an inequality of the form ϕ(λx) ≤ C max{λp0 , λp1 }ϕ(x) with C independent of λ, and p0 ≤ p1 . In this case we say the function ϕ is of lower type p0 and upper type p1 . In order to obtain a wider variety of applications we will work with weighted Orlicz spaces. If w : Ω → R+ is a measurable function and ϕ is an Orlicz function, we define the weighted Orlicz space Lϕ (ω) as the space of all (classes of ) measurable functions f such that f · w ∈ Lϕ . We will also admit the function Φ : [0, ∞) → [0, ∞], defined by Φ(x) = ∞χ(1,∞) (x), so that we can generate L∞ as an Orlicz space. Some notation will be needed to establish the interpolation results: Let ϕ1 , . . . , ϕN be N Orlicz functions and w1 , . . . , wN be N weights functions. We will consider the N -tuple of weighted Orlicz spaces L = {Lϕ1 (w1 ), . . . , Lϕ1 (wN )} . We will also use a convex polygon Π = P1 , . . . , PN , with Pj = (xj , yj ). The pair (α, β) will stand for an interior point of Π. Now, for each (i, j, k) ∈ P(α,β) we define the weight c

wi,j,k = wici wj j wkck and the Orlicz function ϕi,j,k by # −1 $ci # −1 $cj # −1 $ck ϕj ϕk . ϕ−1 i,j,k = ϕi

Let us establish the convention that Φ−1 = 1, allowing us to define ϕi,j,k even when Φ is involved. We start with an auxiliary result. Diag(Π) represents the set of the points of R2 lying on the diagonals of the polygon Π. c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim !

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Lemma 2.1 Let f be a measurable function on Ω. If (α, β) +∈ Diag(Π) there exists a measurable mapping on Ω, x ! (tx , sx ) ∈ R2+ , such that % + # $,& max ϕi,j,k |f (x)| wi,j,k (x) txxl −α syxl −β |f (x)| wl (x) ≤ ϕ−1 l P(α,β)

for 1 ≤ l ≤ N .

P r o o f. Let (i0 , j0 , k0 ) ∈ P(α,β) and consider the measurable sets / # + # $ $,0 Ωi0 ,j0 ,k0 = x ∈ Ω s.t. ϕi0 ,j0 ,k0 |f (x)| wi0 ,j0 ,k0 (x) = max ϕi,j,k |f (x)| wi,j,k (x) . (2.2) P(α,β)

+ , Make the family Ωi,j,k P disjoint so that it forms a partition of Ω. Choose x ∈ Ωi,j,k , for some (i, j, k) ∈ (α,β) P(α,β) , and consider the system of equations in the unknowns t, s, z > 0 # $ ϕi,j,k (|f (x)| wi,j,k (x)) , txl −α syl −β zwl (x) = ϕ−1 l

l = i, j, k.

(2.3)

By taking logarithms we obtain the equivalent system of linear equations in the unknowns log t, log s, log z: " # # $$' ϕ |f (x)| w (xl − α) log t + (yl − β) log s + log z + log wl (x) = log ϕ−1 (x) i,j,k i,j,k l

for l = i, j, k. The determinant of the system is not null since the vertices Pi , Pj , Pk are affinely independent. Thus Cramer’s rule provides a unique solution for the system, say log tx , log sx , log zx . Obviously tx , sx , zx are solution for the initial system and it is an straightforward computation to show that zx = |f (x)|. Further, since tx and sx are sums and products of the measurable functions |f |, w1 , . . . , wN , ϕ1 , . . . , ϕN , −1 2 ϕ−1 1 , . . . , ϕN , the function hi,j,k : Ωi,j,k → R+ define by the rule x ! (tx , sx ) is measurable. We claim that for 1 ≤ l ≤ N # $ ϕi,j,k (|f (x)| wi,j,k (x)) . txl −α syl −β |f (x)| wl (x) ≤ ϕ−1 l Proceed by contradiction: Assume that for some 1 ≤ l ≤ N and some x ∈ Ωi,j,k # $ ϕi,j,k (|f (x)| wi,j,k (x)) . txl −α syl −β |f (x)| wl (x) > ϕ−1 l

It is easy to check that since (α, β)# +∈ Diag(Π), $ we can choose two vertices from the initial triangle {Pi , Pj , Pk }, say Pi , Pk , such that (α, β) ∈ Int Pl , Pi , Pk . For this three vertices we have the equations 6 7 ϕi,j,k (|f (x)| wi,j,k (x)) , txl −α syl −β |f (x)| wl (x) > ϕ−1 l 6 7 ϕi,j,k (|f (x)| wi,j,k (x)) , txi −α syi −β |f (x)| wi (x) = ϕ−1 i 6 7 ϕi,j,k (|f (x)| wi,j,k (x)) . txk −α syk −β |f (x)| wk (x) = ϕ−1 k

Let (cl , ci , ck ) be the barycentric coordinates of (α, β) with respect to the vertices (Pl , Pi , Pk ). Clearly cl , ci , ck > 0. Raising the three equations to its corresponding power and multiplying the three of them we obtain: 6 # $7 |f (x)| wlcl (x)wici (x)wkck (x) > ϕ−1 . l,i,k ϕi,j,k |f (x)| wi,j,k (x) Since ϕl,i,k is an increasing function we have that 6 # 7 $ ϕl,i,k |f (x)| wlcl (x)wici (x)wkck (x) > ϕi,j,k |f (x)| wi,j,k (x) ,

contradicting the choice of x in Ωi,j,k .. Now the measurable function h = P(α,β) hi,j,k · χΩi,j,k proves the theorem.

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Direct computations easily show the lemma holds even when we allow ϕl = Φ for some, or all, 1 ≤ l ≤ N . We are now in condition to establish one of the inclusions announced in (0.1). Theorem 2.2 Assume ϕ1 , . . . , ϕN have positive lower type. If (α, β) +∈ Diag(Π) then !

P(α,β)

Lϕi,j,k (wi,j,k ) &→ !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) .

P r o o f. Let f be an element in the unit ball of 1

Ω

ϕi,j,k (|f | wi,j,k ) dµ ≤ 1 .

8

P(α,β)

Lϕi,j,k (wi,j,k ), then for (i, j, k) ∈ P(α,β)

Let h : Ω → R2+ be the function provided by Lemma 2.1 and the function f . We consider the following partition of Ω: #6 $ 6 $$ Ωm,n = h−1 2m , 2m+1 × 2n , 2n+1 .

# $−1 Put C∗ = min{1, 2xj −α } min{1, 2yj −β } and um,n = C∗−1 |f | χΩm,n . We show the sequence of functions −1 (um,n )Z2 satisfies the conditions that makes C∗ |f | an element of !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) . Let F be a finite set of Z2 and (ξm,n ) ∈ U%∞ . Then for 1 ≤ j ≤ N we have: : ; 9: 1 :) : : : (xj −α)m+(yj −β)n ϕj : ξm,n 2 um,n : wj dµ : : Ω F 1 % & ) ϕj |ξm,n | 2(xj −α)m+(yj −β)n um,n wj dµ ≤ (2.4) ≤

F

Ωm,n

F

Ωm,n

)1

# $ ϕj 2(xj −α)m+(yj −β)n C∗−1 |f | wj dµ .

Since on Ωm,n the inequality 1

Ωm,n

2(xj −α)m+(yj −β)n x −α yj −β txj sx

≤ C∗ holds, then

# $ ϕj 2(xj −α)m+(yj −β)n C∗−1 |f | wj dµ ≤

≤

≤

1

Ωm,n

1

Ωm,n

# $ ϕj C∗ txxj −α syxj −β C∗−1 |f | wj dµ + # $, max ϕi,j,k |f | wi,j,k dµ

P(α,β)

) 1

P(α,β)

Ωm,n

(2.5)

# $ ϕi,j,k |f | wi,j,k dµ .

Equations (2.4) and (2.5) establish that for all 1 ≤ j ≤ N : ; 9: 1 : :) ) ) 1 # $ : : (xj −α)m+(yj −β)n ϕj : ξm,n 2 um,n : wj dµ ≤ ϕi,j,k |f | wi,j,k dµ : : Ω F Z2 P(α,β) Ωm,n 2 3 N . ≤ 3 Now use the fact that ϕj has positive lower type p0 to conclude that < 2 3=1/p0 -) N (xj −α)m+(yj −β)n ξm,n 2 um,n ≤ C , 3 F

c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim !

(2.6)

Lϕj (wj )

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where the constant C, as well as the8type p0 , can be chosen uniformly for all ϕ’s. Restricting F to a singleton in (2.6) we deduce that (um,n )Z2 ⊂ Lϕj (wj ) and (1.2) is verified. Furthermore, from (2.6) the series ) 2(xj −α)m+(yj −β)n um,n Z2

. is weakly unconditionally Cauchy in Lϕj (wj ), and so, see the proof of Prop. 1.2, the series um,n is uncondi. . tionally convergent in ΣLϕj (wj ). Since C∗−1 |f | = um,n point-wise, we conclude that C∗−1 |f | = um,n in ΣLϕj (wj ). We have just proved that |f | ∈ !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) , and so does f . Further < 2 3=1/p0 - - N -f = |f | (α,β) ≤ C∗ C (α,β) 3

establishing the inclusion ! Lϕi,j,k (wi,j,k ) &→ !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) . P(α,β)

We now focus in the proof of the reverse inclusion. We will use three auxiliary results. The following lemma is a generalization of the Carlson type inequality proved in [12] by Gustavsson and Peetre. Lemma 2.3 If (α, β) ∈ Int Pi , Pj , Pk , then there exist C > 0 such that for all sequence (um,n )Z2 ⊂ C and for all finite set F ⊂ Z2 : : 9 ;cl /2 :) : > )# $2 : : (xl −α)m+(yl −β)n um,n : wi,j,k ≤ C |um,n | wl . 2 : : : F

l=i,j,k

F

P r o o f. The inequality is equivalent to : : 9 ;cl /2 :) : > )# $ 2 : : um,n : ≤ C . 2(xl −α)m+(yl −β)n |um,n | : : : F

l=i,j,k

F

Choose arbitrary integers r, s, and sets Λl , l = i, j, k (that we make disjoint in order to form a partition of Z2 ) defined by / + ,0 Λl = (m, n) ∈ Z2 s.t. 2−(xl −α)(m+r)−(yl −β)(n+s) = min 2−(xl −α)(m+r)−(yl−β)(n+s) . l=i,j,k

Then

˛ ˛ ˛X ˛ X ˛ ˛ um,n ˛ ≤ |um,n | ˛ ˛ ˛ F F X X −(x −α)m−(y −β)n (x −α)m+(y −β)n l l = 2 l 2 l |um,n | l=i,j,k Λl ∩F

≤ =

X

l=i,j,k

X

l=i,j,k

≤

X

l=i,j,k

= C

X

l=i,j,k

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X` Λl

−(xl −α)m−(yl −β)n ´2

2

!1/2

Λl ∩F

X` −(x −α)(m+r)−(y −β)(n+s) ´2 l 2 l Λl

X Z2

˘

¯ −(xl −α)m−(yl −β)n 2

min 2 i,j,k

(xl −α)r+(yl −β)s

2

X ` (x −α)m+(y −β)n ´2 l |um,n | 2 l

!1/2

!1/2

Λl ∩F

(xl −α)r+(yl −β)s

X` (x −α)m+(y −β)n ´2 l |um,n | 2 l F

X ` (x −α)m+(y −β)n ´2 l 2 l |um,n |

2(xl −α)r+(yl −β)s

2

!1/2

X`

!1/2

F

(xl −α)m+(yl −β)n

2

´2 |um,n |

!1/2

!1/2

,

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Fern´andez–Mart´ınez, Gonz´alez, and Romero: The ± method

%. + −(x −α)m−(y −β)n ,2 &1/2 l l where C = is finite since (α, β) ∈ Int Pi , Pj , Pk . Z2 mini,j,k 2 Now since r and s are arbitrary integers we can take infimum and conclude that

: : ( 9 ;1/2 * : :) )# $2 : : (xl −α)r+(yl −β)s (xl −α)m+(yl −β)n 2 um,n : ≤ 3C inf max 2 |um,n | : r,s l=i,j,k : : F F 9 ;cl /2 > )# $2 (xl −α)m+(yl −β)n 2 ≤ 3C1 |um,n | , l=i,j,k

F

see [5] for the last inequality. Lemma 2.4 Let (i, j, k) ∈ P(α,β) . If f , fi , fj and fk are measurable functions such that |f | ≤ C |fi |ci |fj |cj |fk |ck # $ ? and Ω ϕl |fl | dµ ≤ K for l = i, j, k and some K > 0, where the ϕ’s have finite upper type p1 , then 1 # $ ϕi,j,k |f | dµ ≤ M Ω

for some M independent of f , fi , fj and fk . # $ P r o o f. Let bl = ϕl |fl | , for l = i, j, k. Using that the ϕ’s are strictly increasing functions we have the inequalities |f | ≤ C |fi |ci |fj |cj |fk |ck # $ci # −1 $cj # −1 $c ϕj (bj ) ϕk (bk ) k = C ϕ−1 i (bi ) > # $c max{bi , bj , bk } l ≤ C ϕ−1 l l=i,j,k

# $ ≤ Cϕ−1 i,j,k max{bi , bj , bk } .

# $ # # $$ So ϕi,j,k |f | ≤ ϕi,j,k Cϕ−1 i,j,k max{bi , bj , bk } . Then since the ϕ’s have finite upper type p1 # $ ϕi,j,k |f | ≤ C1 max{C p0 , C p1 } max{bi , bj , bk } # $ ≤ C1 max{C p0 , C p1 } ϕi (|fi |) + ϕj (|fj |) + ϕk (|fk |) .

(2.7)

Taking integrals in both sides of the inequality 1 ϕi,j,k (|f |) dµ ≤ C1 max{C p0 , C p1 }3K = M . Ω

The result remains true if we admit ϕj = Φ for some j. Let us fix some notation that will be needed in the following lemma. Associated to every finite set F ⊂ Z2 we may consider the Rademacher functions, {rs (t)}0≤s 0 such that for any (um,n )Z2 ⊂ C and any finite set F ⊂ Z2 the following inequalities hold 9 9 :; ;1/2  ;1/2  1 1 9 ::) : ) ) : :  ≤ . C −1 ϕ  |um,n |2 ϕ : rm,n (t)um,n : dt ≤ Cϕ  |um,n |2 : : 0 F

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F

F

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753

This result is proved in [12, Prop. 3.4] for (um,n ) ⊂ R. The usual manipulations establish the complex case (um,n ) ⊂ C. Theorem 2.6 Let ϕ1 , . . . , ϕN be Orlicz functions of positive lower type p0 and finite upper type p1 . If (α, β) +∈ Diag(Π) then ! Lϕi,j,k (wi,j,k ) . !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) &→ P(α,β)

P r o o f. Let f ∈ !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) with #f #(α,β) < 1. There exists a sequence (um,n ) ⊂ . Lϕj (wj ) such that f = Z2 um,n in ΣLϕj (wj ) and verifies that for any finite set F ⊂ Z2 , and (ξm,n ) ∈ #∞ (Z2 ) with #(ξm,n )#%∞ ≤ 1, the inequalities : ; 9: 1 : :) : : (xj −α)m+(yj −β)n ϕj : ξm,n 2 um,n : wj dµ ≤ 1 , 1 ≤ j ≤ N , : : Ω 8

F

hold. In particular, if (rm,n (t))F is the selection of the Rademacher functions described above, we can write : ; 9: 1 11 : :) : : (xj −α)m+(yj −β)n 1 ≥ ϕj : rm,n (t)2 um,n : wj dµ dt : : 0 Ω F : ; 1 1 1 9 ::) : : : (xj −α)m+(yj −β)n ϕj : rm,n (t)2 um,n : wj dt dµ = : : Ω 0 F 9 ; 1/2 1 )# $2 ≥ C −1 ϕj dµ . 2(xj −α)m+(yj −β)n |um,n | Ω

F

Fix (i, j, k) ∈ P(α,β) . Since (α, β) +∈ Diag (Π), Lemma 2.3 assures that : : 9 ;cl /2 : :) > )# ) $2 : : (xl −α)m+(yl −β)n 2 um,n : wi,j,k ≤ |um,n | wi,j,k ≤ C1 |um,n | wl : : : F

F

l=i,j,k

F

for some C1 independent of F . Now Lemma 2.4 establishes : 9: ; 1 : :) : : ϕi,j,k : um,n : wi,j,k dµ ≤ M : : Ω F

where M is independent of F . Apply Fatou’s Lemma to obtain : 9: ; 1 1 :) : # $ : : ϕi,j,k |f | wi,j,k dµ = ϕi,j,k : um,n : wi,j,k dµ ≤ M . : 2 : Ω Ω Z

The fact that ϕi,j,k has positive lower type p0 leads us to the inequality 0 / #f #Lϕi,j,k (wi,j,k ) ≤ max 1, (CM )1/p0

establishing the continuous inclusion

!Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) &→ Lϕi,j,k (wi,j,k ) for all (i, j, k) ∈ P(α,β) . It is easy to check, following the ideas of the proof, that the inclusion remains true if any, or all, of the ϕ’s equals Φ = ∞χ(1,∞) . Moreover, for every (α, β) ∈ Int (Π) and (i, j, k) ∈ P(α,β) the inclusion !LΦ (w1 ), . . . , LΦ (wN )"(α,β) &→ LΦ (wi,j,k )

holds. That is to say, we can include diagonals since there is no need to use Lemma 2.3. Theorems 2.2 and 2.6 give the following straightforward corollary: www.mn-journal.com

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Fern´andez–Mart´ınez, Gonz´alez, and Romero: The ± method

Corollary 2.7 Assume ϕ1 , . . . ϕN have positive lower type and finite upper type. If (α, β) +∈ Diag(Π) then ! !Lϕ1 (w1 ), . . . , LϕN (wN )"(α,β) = Lϕi,j,k (wi,j,k ) . P(α,β)

Next we apply the above result to the interpolation of weighted Lp -spaces. In the case of couples this was studied by Gustavsson with a different approach, see [11]. The techniques used by Gustavsson for couples give only partial results when are applied to the context of N -tuples of weighted Lp -spaces. The result obtained using the interpolation of weighted Orlicz spaces is the following: Corollary 2.8 If (α, β) +∈ Diag(Π) and 0 < p1 , . . . , pN ≤ ∞, then ! !Lp1 (w1 ), . . . , LpN (wN )"(α,β) = Lpi,j,k (wi,j,k ) P(α,β)

where

1 pi,j,k

=

ci pi

+

cj pj

+

ck pk .

P r o o f. Recall that for 0 < p < ∞ Lp = Lϕ (w), where ϕ(t) = tp , and L∞ (w) = LΦ (w). Now apply Corollary 2.7 to complete the proof. Results in the line of Corollary 2.8, this time working with the Cobos–Peetre methods, were also obtained by Cobos and Mart´ın in [6]. They prove that $ # D(α,β) ) , Lp (w1 ), . . . , Lp (wN ) (α,β),p;J = Lp (w where w(x) D = D(α,β) (w1 (x), . . . , wN (x)). This is the same space produced interpolating by the ± method.

3

Other results

As it was mentioned in the Introduction, there are several interpolation methods dealing with finite families of Banach spaces. Favini [9], Sparr [17] and D. L. Fernandez [10], among other authors, developed different methods involving more than two spaces. Restricting ourselves to the generalizations of the real method, some authors find Sparr method for (N + 1)-tuples a more natural tool to interpolate than the Cobos–Peetre method. This suggests the idea of developing a generalized ± method following the Sparr approach rather than the Cobos– Peetre ideas of using polygons. We will not do that in detail here, but for the sake of completeness we include a definition of the method obtained using Sparr’s ideas as well as the interpolation result obtained with this version of the generalized ± method. Definition 3.1 Let A = {A0 , A1 , . . . , AN } be an (N +1)-tuple of Banach spaces. Given θ = (θ0 , θ1 , . . . , θN ) ∈ +1 RN such that 1 = θ0 + θ1 + . . . + θN we define the space ! A "θ as the set of all those elements a in Σ( A ) for + which there exists a sequence (um )ZN in ∆( A ) verifying that: ) (a) a = um in Σ( A ) , ZN

(b)

-) 2 mj ξm PN um ≤ C θ m l l 2 1 F A j

for some C independent of ξ = (ξm ) ∈ U%∞ (ZN ) , any finite set F and 0 ≤ j ≤ N . The space ! A "θ becomes a Banach space when endowed with the norm #a#θ = inf{C} where C runs in the set of all possible constants in (b). If we use the above described interpolation method to interpolate (N + 1)-tuples of Orlicz spaces we obtain the following result: c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim !

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Theorem 3.2 Let L = {Lϕ0 (w0 ), Lϕ1 (w1 ), . . . , LϕN (wN )} be an (N + 1)-tuple of weighted Orlicz spaces. +1 Let θ = (θ0 , θ1 , . . . , θN ) ∈ RN be such that 1 = θ0 + . . . + θN . Define the Orlicz function ϕθ through its + inverse $θ0 # −1 $θ1 $θN # # ϕ−1 ϕ1 = ϕ−1 . . . ϕ−1 . 0 N θ Then, if the functions ϕ0 , ϕ1 , . . . , ϕN have positive lower type and finite upper type, we have the equality !Lϕ0 (w0 ), Lϕ1 (w1 ), . . . , LϕN (wN )"θ = Lϕθ (wθ ) θN where wθ = w0θ0 w1θ1 . . . wN .

Acknowledgements We would like to thank one of the referees of this paper for the kind advices that led us to include §3 “Other results”. Finally, we want to show our most sincere gratitude to F. Cobos for suggesting the study of this method. The first named author has been partially supported by Fundaci´on Seneca, PB/21/FS/02. All authors have been partially supported by Ministerio de Ciencia y Tecnolog´ıa (BFM 2001-1424).

References [1] N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation methods, Ann. Mat. Pura Appl. (4) 68, 51–118 (1965). [2] A. P. Calder´on, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (2), 113–190 (1964). [3] F. Cobos, J. M. Cordeiro, and A. Mart´ınez, On interpolation of bilinear operators by methods associated to polygons, Boll. Un. Mat. Ital. 2 (2), 319–330 (1999). [4] F. Cobos, P. Fernandez–Mart´ınez, and A. Mart´ınez, Measure of non-compactness and interpolation methods associated to polygons, Glasg. Math. J. 41 (1), 65–79 (1999). [5] F. Cobos, P. Fernandez–Mart´ınez, and T. Schonbek, Norm estimates for interpolation methods defined by means of polygons, J. Approx. Theory 80, 321–351 (1995). [6] F. Cobos and J. Mart´ın, On interpolation of function spaces by methods defined by means of polygons, Preprint (2003). [7] F. Cobos and J. Peetre, Interpolation of compact operators: The multidimensional case, Proc. London Math. Soc. (3) 63, 371–400 (1991). [8] S. Ericsson, Certain reiteration and equivalence results for the Cobos–Peetre polygon interpolation method, Math. Scand. 85 (2), 301–319 (1999). [9] A. Favini, Su una estensione del metodo d’interpolazione complesso, Rend. Sem. Mat. Univ. Padova 47, 243–289 (1972). [10] D. L. Fernandez, Interpolation of 2n Banach spaces, Studia Math. 45, 175–201 (1979). [11] J. Gustavsson, On interpolation of weighted Lp -spaces and Ovchinnikov’s theorem, Studia Math. 72 (3), 237–251 (1982). [12] J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1), 33–59 (1977). [13] S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1), 50–73 (1981). [14] L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Mathematics Vol. 5 (Univ. Estadual de Campinas, Campinas SP, Brazil, 1989). [15] L. I. Nikolova, Some estimates of measure of noncompactness for operators acting in interpolation spaces—the multidimensional case, C. R. Acad. Bulgare Sci. 12, 5–8 (1991). [16] V. I. Ovchinnikov, The Method of Orbits in Interpolation Theory, Volume 1, Mathematical Reports (Ed. Acad. Romˆane, Bucharest, 1984). [17] G. Sparr, Interpolation of several Banach spaces, Ann. Math. Pura Appl. (4) 99, 247–316 (1974).

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