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Generalized Kothe p-dual spaces.

1 Introduction and preliminaries

Generalized duality was investigated in 1989 in [15] as spaces of multipliers (see also [2]). A complete bibliography on this topic can be found in the recent papers [10, 11]. The point of view of these papers and the references therein is in general oriented to the study of structural aspects and representation of these spaces of multipliers. The purpose of the present paper is to investigate some geometric aspects of these spaces. To be precise, we study p-concavity, type and cotype of multipliers from Banach function spaces to the space of p-integrable real-valued functions with respect to a positive and finite scalar measure. The advantage of using these spaces lies in the fact that they inherit some geometry from the [L.sup.p]-space, for example the p-convexity (see e.g. [2, Lem. 5.1]). We will also define the natural operator associated with an operator T: X [right arrow] E and the Kothe p-dual [X.sup.p] so as to define the Kothe p-adjoint of T. As applications we will use the properties of the so-called p-th power factorable operators and easy properties of spaces of multipliers in order to factorize the Koo the p-adjoint through an [L.sup.p]-space. We will see that a p-th power factorable operator may factorize through [L.sup.p]-spaces and [L.sup.p,[infinity]]-spaces. Also, we will show that the Kothe r-dual space of a p-convex Banach function space is ij-concave for every q [less than or equal to] 1 such that 1/r = 1/p + 1/q. Computing type and cotype for some Kothe p-dual of an AM-space and applying Kwapienn's Theorem we will obtain a characterization of Hilbert spaces as Kothe 2-dual space and some examples of operators can be written as Koo the p-adjoint operators. We provide an example by using some results of [15] for Orlicz spaces, and results of [9] for the Riesz transform.

Let ([OMEGA], [summation], [mu]) be a complete [sigma]-finite measure space and [L.sup.0] the space of [mu] measurable real-valued functions defined on n that are equal [mu]-a.e.. A (quasi-) Banach function space X is a linear subspace of [L.sup.0], with complete (quasi-)norm [[parallel] * [parallel].sub.x], so that for each g [member of] X and h [member of] [L.sup.0] with [absolute value of (h)] [less than or equal to] [absolute value of (g)] [mu]-a.e., implies that h G X and [[parallel] h [parallel].sub.X] [less than or equal to] [[parallel] g [parallel].sub.X]. Other authors use this space, with slight differences, see e.g. [2, 4, 14, 16]. An element h G X is a weak unit if it is such that h > 0 [mu]-a.e., and it is a weak order unit if it is such that g [conjunction] nh [up arrow] g for every g [member of] [X.sup.+]. For instance if [[chi].sub.[OMEGA]] [member of] X, then it is a weak order unit and [L.sup.[infinity]] [subset or equal to] X (see [16, Prop. 2.2(iv)]). Clearly, a weak order unit is a weak unit. Given 1 [less than or equal to] p [less than or equal to] [infinity], we denote the conjugate of p by p' := p/p - 1 We write Bx for the unit ball of X.

Let 1 [less than or equal to] p [less than or equal to] [infinity]. Recall that a Banach function space X is p-convex if there is a constant [K.sub.p] such that for every finite set [f.sub.1], ..., [f.sub.n] [member of] X, the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds. Let 1 [less than or equal to] q < [infinity]. It is said that X is q-concave if there is a constant [K.sup.q] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

holds. The following definitions can be found in [14, Defn. 1.e.12] and also in [5, Sec. 7.7]. The Rademacher functions are in [L.sup.2] [0, 1] and are defined by

[r.sub.k](t) := [(-1).sup.j], t [member of] [j/[2.sup.k'], [j + 1/[2.sup.k][, j = 0, ..., [2.sup.k] - 1, k = 1, 2, 3, ....

Let E be a Banach space. We say that E has type p [member of] [1, 2] if for every [x.sub.1], ..., [x.sub.n] [member of] E exists C [greater than or equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

And has cotype q [member of] [2, [infinity]] if for every [x.sub.1], ..., [x.sub.n] [member of] E exists C [greater than or equal to] 0

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Every Banach space has cotype [infinity] and type 1 [14, p. 73]. Let us recall Kwapien's characterization of Hilbert spaces [13]: X is isomorphic to a Hilbert space if and only if has type 2 and cotype 2. See [19, Chap. 3] for more details and applications.

Let 1 [less than or equal to] p < [infinity] and X a Banach function space, we call the p-th power space of X the space

[X.sub.[p]] := {f [member of] [L.sup.0] : [[absolute value of (f)].sup.1/p] [member of] X},

equipped with the quasi-norm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This space is a Banach function space if and only if X is p-convex with constant 1 (see e.g. [16, Prop. 2.23(iii)]).

Recall that given two Banach function spaces X and Y over the same measure, the space of multipliers from X to Y is defined by

[X.sup.Y] := {g [member of] L[infinity] : gX [subset or equal to] Y}.

The seminorm [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], gives a Banach function space structure for [X.sup.Y] when the [mu]-a.e. order is considered and X has weak unit (see [15, Prop. 2]). For more details we refer the reader to [2, Sec. 2]. Notice that the Kothe dual X' of X may be characterized by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will use the following characterizations of order continuity and Fatou properties (see [14, p. 28-30] for more details). A Banach function space X is order continuous if and only if X' = [X.sup.*], where [X.sup.*] denotes the topological dual space, and is Fatou if and only if X" = X.

The Kothe p-dual space was introduced in [8, Defn. 2.1] and is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Observe that the classical Lebesgue space [L.sup.p] is obtained as a particular case when X = [L.sup.[infinity]] (see Lemma 2.1(4)). By definition, when X has weak unit, the norm in [X.sup.p] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For example, let 1 [less than or equal to] r [less than or equal to] p [less than or equal to] [infinity] and q [greater than or equal to] 1 be such that 1/r = 1/p + 1/q. Then [L.sup.q] = [([L.sup.p]).sup.r] where the spaces are based on a [sigma]-finite measure (see [15, Prop. 3]). It is important pointing out that the expression [X.sup.p] may be trivial, for example [([L.sup.r]).sup.s] = {0} whenever r < s and the measure [mu] is non-atomic. A sufficient condition can be found in [20, Thm. 1.8], in which is established for the case of non-atomic measure that [X.sup.Y] = { 0} if

inf{p [greater than or equal to] 1 : X is p-concave} < sup{p [greater than or equal to] 1 : Y is p-convex}. (3)

For instance, if X is a Banach function space such that the inclusion [L.sup.[infinity]] [subset] X is proper, then [X.sup.[infinity]] = {0}, since [L.sup.[infinity]] is [infinity]-convex.

Let us define now a class of operators that will be relevant in the paper [16, Defn. 5.1].

Definition 1.1. Let 1 [less than or equal to] p < [infinity], X an order continuous quasi-Banach function space with weak order unit [[chi].sub.[omega]], and E a Banach space. We say that an operator T: X [right arrow] E is p-th power factorable if there exists an operator [T.sub.[p]] : [X.sub.[p]] [right arrow] E, which equals T over X [subset or equal to] [X.sub.[p]]. In other words, the following diagram commutes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [i.sub.[p]] is the natural continuous inclusion.

The following relations will be useful throughout the paper. The main properties of [X.sub.[p]] were presented in [16, Sec. 2.2] for the case of finite measure, which implies in their definition of quasi-Banach function space, that [[chi].sub.[omega]] is a weak order unit in X (see also [20, Sec. 3]). We have replaced the requirement of finite measure by the requirement of [[chi].sub.[omega]] [member of] X. Let us recall some of them.

Proposition 1.2. Let X be a quasi-Banach function space such that [[chi].sub.[omega]] [member of] X.

1. Let 0 < p, q < [infinity]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Let p [member of] (0, [infinity]). Then, X is order continuous if and only if [X.sub.[p]] is order continuous.

3. If 0 < p [less than or equal to] q < [infinity] we have that [X.sub.[p]] [subset or equal to] [X.sub.[q]], in particular X = [X.sub.[1]] [subset or equal to] [X.sub.[p]] for all 1 [less than or equal to] p < [infinity] and [X.sub.[p]] [subset or equal to] X for all 0 < p [less than or equal to] 1.

4. Let 0 < p < [infinity]. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The main duality identification involving [X.sub.[p]] and [X.sup.p] is the last property (4).

The following represents an example of a geometric property that is inherited by the multiplier space and will be useful in what follows. It was established without proof in [2, Lem. 5.1], see also [20, Prop. 3.1].

Lemma 1.3. Let X, Y be Banach function spaces where X has a weak unit and let 1 [less than or equal to] p [less than or equal to] [infinity]. If Y is p-convex, then [X.sup.Y] is p-convex with the same constant.

2 The Kothe p-dual space

In [15, Sec. 2, Cor. 1], [10, Sec. 2] and references therein we can find lists of simple properties involving general spaces of multipliers. Let us now compile some of these properties for the setting of Kothe p-dual and p-th power spaces of Banach function spaces and using our own notation.

Lemma 2.1. Let X and Y be Banach function spaces with weak unit.

1. If 0 < p [less than or equal to] q < [infinity], then [X.sup.q] [subset or equal to] [X.sup.p].

2. If 0 < p [less than or equal to] [infinity] and X [subset or equal to] Y then [Y.sup.p] [subset or equal to] [X.sup.p].

3. If 0 < p [less than or equal to] q [less than or equal to] [infinity], then [([X.sup.p]).sup.q] [subset or equal to] [([X.sup.q]).sup.p].

4. Let 0 < p [less than or equal to] [infinity], then [L.sup.p] = [([L.sup.[infinity]]).sup.p]. The proof is straightforward.

Lemma 2.2. Let Xbea Banach function space such that [[chi].sub.[OMEGA]] [member of] X.

1. If 1 [less than or equal to] p < [infinity], then [X.sup.p] [subset or equal to] [([X.sup.p]).sub.[p]] [subset or equal to] X'.

2. If 0 [less than or equal to] p [less than or equal to] q < [infinity], then [([X.sub.[p]]).sup.q] = [([X.sup.pq]).sup.[p]] and [([X.sup.q]).sub.[p]] = [([X.sub.[p]]).sup.q/p].

3. If 1 [less than or equal to] p [less than or equal to] q [less than or equal to] [infinity], then [([X.sub.[p]]).sup.q] [subset or equal to] [X.sup.p].

4. Ifs [less than or equal to] r [less than or equal to] t and 1 [less than or equal to] p [less than or equal to] q [less than or equal to] [infinity], then

[([X.sub.[q]]).sup.t] [subset or equal to] [([X.sub.[q]]).sup.r] [subset or equal to] [([X.sub.[p]]).sup.r] [subset or equal to] [([X.sub.[p]]).sup.s].

5. If rq [less than or equal to] tp and 1 [less than or equal to] p [less than or equal to] q [less than or equal to] [infinity], then [([X.sub.[p]]).sup.t] [subset or equal to] [([X.sub.[q]]).sup.r].

6. Let q [less than or equal to] r, p [less than or equal to] [infinity] be such that 1/q =1/p + 1/r, if X [subset or equal to] [L.sup.r] then [L.sup.p] [subset or equal to] [X.sup.q].

7. Let 0 < p, q [less than or equal to] [infinity], then [X.sup.p] = [([([X.sub.[1/q]]).sup.pq]).sub.[q]]. In particular X' = [([([X.sub.[1/q]]).sup.q]).sub.[q]].

8. Let 1 [less than or equal to] p [less than or equal to] [infinity], then [X.sup.p] [subset or equal to] ([X.sub.[p]])' [subset or equal to] X'.

9. Let 1 [less than or equal to] p [less than or equal to] [infinity] and let Y be a Banach function space such that [[chi].sub.[OMEGA]] [member of] Y, then Y [subset or equal to] [X.sup.p] if and only if X [subset or equal to] [Y.sup.p].

10. [X.sup.p] [subset or equal to] [L.sup.p].

11. [X.sup.p] = [L.sup.p] if and only if X = [L.sup.[infinity]].

Proof. From (1) to (5) are easy using Proposition 1.2 and Lemma 2.1. Let us prove the rest.

(6) We use Holder's inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

hence [L.sup.p] [subset or equal to] [X.sup.q], which completes the proof.

(7) We proceed directly using the definition of Kothe dual space. [L.sup.p] = [([L.sup.1]).sub.[1/p]], then apply properties (1) and (4) of Proposition 1.2

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(8) Apply Proposition 1.2(3) and Lemma 2.1(2).

(9) By hypothesis and Proposition 1.2(4) we have that [Y.sub.[p]] [subset or equal to] [([X.sup.p]).sub.[p]] = ([X.sub.[p]]))', hence [X.sub.[p]] [subset or equal to] ([X.sub.[p]])" [subset or equal to] ([Y.sub.[p]])'. Let g [member of] [Y.sup.p]. Then, for some C > 0, it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The converse is analogous.

(10) Let be f [member of] X, since [[chi].sub.[OMEGA]] [member of] X then f = f[[chi].sub.[OMEGA]] [member of] [L.sup.p]. In order to prove the continuity we consider the following inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So [X.sup.p] [subset or equal to] [L.sup.p].

(11) We only need to show that [X.sup.p] = [L.sup.p] implies that X = L[degrees][degrees]. It follows from the statement (9), since [L.sup.p] [subset or equal to] [X.sup.p] implies that X [subset or equal to] [([L.sup.p]).sup.p] = [L.sup.[infinity]]. Then X = [L.sup.[infinity]].

Example 2.3. Let [L.sup.p] (m) ([L.sup.p.sub.w] (m)) denote the space of (weakly) p-integrable real-valued functions with respect to the vector measure m. Standard works on this topic are [7, 16], see [6] for the case p = 1. By Lemma 2.2(2) we have

[L.sup.p.sub.w](m) = [([L.sub.1.sub.w](m)).sub.{1/p]] = [([L.sup.1.sub.w](m)").sub.[1/p]] = [([([L.sup.1.sub.w](m)').sup.[1/p]]).sup.p] = [([([L.sup.p.sub.w](m)).sup.p]).sup.p]

Applying [2, Prop. 5.3] we obtain that LU (m) is Fatou and p-convex with constant one.

Let us now study operators defined on Kothe p-dual spaces. They provide a natural p-th power factorization, for < p < o, as the next diagram shows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The next definition and lemma were introduced in [8].

Definition 2.4. Let 1 [less than or equal to] p < [infinity]. Let X be a Banach function space, E a Banach space and T: E [right arrow] X an operator. Then we define an operator [T.sup.p] : [X.sup.p] [right arrow] [E.sup.*] as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that we will call Kothe p-adjoint operator.

The following lemma gives an easy example of p-th power factorable operators. Now, we provide an expression for the extension map, which will be useful in the sequel.

Lemma 2.5. Let 1 [less than or equal to] p < [infinity] and T: E [right arrow] X be an operator, where X is a quasi-Banach function space that contains [[chi].sub.[OMEGA]] and [X.sup.p] is order continuous. If p [greater than or equal to] 1, then the Kothe p-adjoint operator [T.sup.p] is p-th power factorable, and the extension operator is [([T.sup.p]).sub.[p]] = [([i.sub.[p]] [omicron] T).sup.*] = [T.sup.*] [omicron] [i.sup.*.sub.[p]].

Proof. The first assertion is immediate from (4) (see [8, Prop. 2.2]). For the second assertion, by definition of Kothe ;-adjoint we have [T.sup.p] = [([T.sup.p]).sub.[p]] [omicron] [k.sub.[p]], where [k.sub.[p]:] [X.sup.p] [??] [([X.sup.p]).sup.[p]] is the canonic inclusion and [([T.sup.p]).sub.[p]] is the unique extension. On the other hand [i.sub.[p]] : X [??] [X.sub.[p]] is canonic and so is [([i.sub.[p]]).sup.*] : [([X.sub.[p]]).sup.*] = [([X.sup.p]).sub.[p]] = [([X.sup.p]).sub.[p]] [??] [X.sup.*], since X is order continuous by Proposition 1.2(2), so is [X.sub.[p]]. Then, by uniqueness of the extension, we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 2.6. Let us comment the factorization of [T.sup.2] : [X.sup.2] [right arrow] [E.sup.*]. Under the requirements of Lemma 2.5. On one hand, the previous Lemma 2.5 asserts that [T.sup.2] = [([T.sup.2]).sub.[2]] [omicron] [i.sub.[2]]. On the other hand by [20, Prop. 3.1] [X.sup.2] is 2-convex. If X is 4-convex, hence [X.sub.[2]] is 2-convex, and so [([X.sup.2]).sub.[2]] = ([X.sub.[2]])' is 2-concave (see e.g. [4, Lem. 2]). Therefore [i.sub.[2]]: [X.sup.2] [right arrow] [L.sup.2] [right arrow] [([X.sup.2]).sub.[2]] (see [14, Cor. 1.f.15(iii)]). In consequence [T.sup.2] factors through a Hilbert space. In general, when X is (pp')-convex, and T is a positive operator, i.e Tx [greater than or equal to] 0 for every x [greater than or equal to] 0, by means of [4, Cor. 5] we can conclude that [T.sup.p] factors through an [L.sup.p]-space (see [5, Sect. 18.6]).

Example 2.7. Let (R, [summation], [mu]) be a finite measure space. Let [phi] be an Orlicz function, i.e. convex, continuous, increasing and unbounded, defined on [0, [infinity]), so that [phi](0) = 0. The Orlicz space is defined by

[L.sup.[phi]] := {f [member of] [L.sup.0] : inf{[lambda] > 0 : [[integral].sub.R] [phi]([absolute value of (f ([omega]))]/[lambda]) d[mu] [less than or equal to] 1} < [infinity]}.

The one-dimensional Riesz transform R: [L.sup.[phi]] [right arrow] [L.sup.[phi]], defined by

(Rf)(x) := c [[integral].sub.R] [x - y/[absolute value of (x - y)]] f(y) d[mu](y),

where c := [GAMMA] (1/2)/1/2, is continuous (see [9, Thm. 3.11]). Let 1 [less than or equal to] p < [infinity] and let [L.sup.[phi]] := ([L.sup.[phi]0])[2.sup.[pi].sub.p]. For a suitable [[phi].sub.0], it is an Orlicz space again over a non-atomic, finite and positive measure. To be precise, thanks to [15, Thm. 4], [[phi].sub.0] can be chosen as an Orlicz function such that (A): [(uv).sup.2p] [less than or equal to] [[phi].sub.0] (u) + [phi](v) for every u, v [greater than or equal to] 0, and (B): [u.sup.1/(2p)] [less than or equal to] [[phi].sup.-1][(u) [[phi].sup.-1] (u) for every u > 0. [L.sup.[phi]] is Fatou and 2;-convex, as we have seen. Let us choose p = 1. Then by [2, Prop. 5.3]

[L.sup.[phi]] = [([L.sup.[phi]0]).sup.2] [subset or equal to] ([L.sup.[phi]0])' = [L.sup.[phi]1]

where [[phi].sub.1] satisfies (A) and (B) for [[phi].sub.0] and 2; = 1. The Riesz transform R: [L.sup.[phi]1] [right arrow] [L.sup.[phi]1] has Kothe 2-adjoint [R.sup.2:] [L.sup.[phi]] [right arrow] [L.sup.[phi]0], and [R.sup.2] is again a Riesz transform. In virtue of the remark above, it factors through a Hilbert space.

Remark 2.8. In order to summarize the relations and spaces that we considering in this paper we present the following diagram, for 1 [less than or equal to] q [less than or equal to] p < [infinity] and u, v >. X is a Banach function space over a finite measure and E is a Banach space and T: E [right arrow] X is an operator.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 Some geometrical aspects

The geometric structure of the space of multipliers is strongly connected with the notions of concavity and convexity (see e.g. [4, Sect. .c, .d and .e]). Let us now study the p-convexity, q-concavity, type and cotype of the spaces [X.sup.r]. Let us state a first corollary.

Corollary 3.1. Let 1 [less than or equal to] p < [infinity]. Suppose that E is a p'-convex Banach lattice, X is a quasi-Banach function space that contains [[chi].sub.[OMEGA]] and [X.sup.p] is order continuous. If [T.sup.p] : [X.sup.p] [right arrow] [E.sup.*] is positive, then it factors through the space [L.sup.p].

Proof. On one hand, from [14, Prop. .d.4(iii)], [E.sup.*] is p-concave then by [14, Prop. .d.9], Tp is p-concave too, since [T.sup.p] is positive. On the other hand thanks to Lemma 1.3 we have that [X.sup.p] is p-convex, thus Maurey-Rosenthal's Theorem (see e.g. [4, Cor. 5]) ensure us that [T.sup.p] factors by [L.sup.p].

Hence in case that T' is positive and E is oo-convex, e.g. if E = [L.sup.[infinity]], T' will be p-factorable for every p [greater than or equal to] 1, i.e. it factors through [L.sup.p], whenever T' (X') [subset or equal to] [L.sup.1]. It is clear, since in this case T: [L.sup.[infinity]] [right arrow] X, and so T': X' [right arrow] [L.sup.1].

Let p [member of] [1, [infinit]], it is well-known that [X.sup.p] is p-convex with constant 1. (see e.g. [20, Prop 3.1]). However, this result does not hold for the p-concavity. For instance [([L.sup.p]).sup.p] = [L.sup.[infinity]], which is not p-concave for p < [infinity]. This theorem sheds some new light on the p-concave case. In fact it is a generalization of [14, Prop. 1.d.4(i)], see also [12]. The proof is adapted from [18, Lem. 2.2], which is deduced directly from the definitions. Recall that if X has not weak unit, then we only can define a seminorm for [X.sup.Y] ([15, Prop. 2]).

Theorem 3.2. Let 1 [less than or equal to] r, p, q [less than or equal to] [infinity] be such that 1/r = 1/p + 1/q. Given X and Y, two Banach funct on spaces, such that Y s r-concave and X s p-convex w th weak un t, we have that [X.sup.Y] is q-concave.

Proof. Since X has weak unit, [X.sup.Y] is a Banach function space and the definition of q-concavity can be applied. We assume without loss of generality that the involved concavity and convexity constants are equal to 1. On one hand, let us take n [member of] N, [f.sub.1], ..., [f.sub.n] [member of] [X.sup.Y] and [g.sub.1], ..., [g.sub.n] [member of] [B.sub.X]. Thanks to [15, Prop. 3] it is clear that [l.sup.q] = [([l.sup.p]).sup.r], so for an element ([[tau].sub.i]) [member of] [l.sup.q]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

On the other hand, since X is p-convex with constant 1, if [([[lambda].sub.i]).sub.i] [member of] [B.sub.lp], (note that [[lambda].sub.i][g.sub.i] [member of] X) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

hence [([n.summation over (i = 1)] [[absolute value of ([[lambda].sub.i][g.sub.i])].sup.p]).sup.1/p] [member of] [B.sub.X]. So, applying (5), that Y is r-concave and (6), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Let [epsilon] > 0. Choose functions {[g.sub.1], ..., [g.sub.n]} [member of] [B.sub.X] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each i = 1, ..., n. Then applying (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for every [epsilon] > 0, which yields us that [X.sup.Y] is q-concave.

Remark 3.3. The referee suggested the following alternative proof of theorem above under assumption that Y has Fatou property. Let 1 [less than or equal to] p, q, r [less than or equal to] [infinity] with 1/r = 1/p + 1/q. If X is p-convex and Y is r-concave, both with constant 1, then Y is r'-convex and so by [11, Thm. 3] we get that X [??] Y' is q'-convex because 1/q' = 1/r' + 1/p, where X [??] Y' denotes the pointwise product of Banach function spaces in the sense of the cited paper. Then by [11, Cor. 3] and Fatou property of Y we have with equality of the norms

(X [??] Y')' = [X.sup.Y"] = [X.sup.Y], which gives that XY is q-concave.

This theorem sometimes fails without the assumption of p-convexity for X.

Example 3.4. Let 1 [less than or equal to] r, p, q < [infinity] be such that 1/r = 1/p + 1/q. Let X := [L.sup.s]0 for 1 [less than or equal to] r [less than or equal to] s < p, which is not p-convex, then [([L.sup.s]).sup.r] = [l.sup.t], where 1/t = 1/r - 1/s (Y is [l.sup.r] in the previous theorem). Then 1/t = 1/r - 1/s = 1/q + 1p - 1/s and so 1/t - 1/q = 1/p - 1/s < 0 since s < p. Therefore, q < t, and so [([L.sup.s]).sup.r] = [L.sup.t] cannot be q-concave.

Remark 3.5. Let 1 [less than or equal to] q < p [less than or equal to] [infinity]. For the case of Lebesgue spaces we have [([L.sup.q]).sup.p] = {0}, but this is not in general true. Take r > q [greater than or equal to] 1 and choose s [greater than or equal to] 1 so that 1/s = 1/q - 1/r. Then, for p [member of] (q, s) and X := [L.sup.r], we conclude that [X.sup.q] = [L.sup.s] and so [([X.sup.q]).sup.p] = [L.sup.t] [not equal to] {0}, where 1/t = 1/p - 1/s.

The following corollary provides conditions to obtain [([X.sup.q]).sup.p] = {0} when q < p.

Corollary 3.6. Let 1 [less than or equal to] q < p [less than or equal to] [infinity] and let r > 1 such that 1/p < 1/q - 1/r. Let X be a r-convex Banach function with weak unit and based over a non-atomic measure. Then [([X.sup.q]).sup.p] = {0}.

Proof. By the previous theorem [X.sup.q] is s-concave for some s [greater than or equal to] 1 such that 1/q = 1/r + 1/s. Since 1/p < 1/q - 1/r = 1/s, we obtain that s < p and the measure is non-atomic. Then the requirement (3) is satisfied, since [L.sup.p] is p-convex, and [([X.sup.q]).sup.p] = {0}.

To finish we will use the 2-Kothe dual space in order to find a simple characterization for Hilbert spaces. The proof of the following theorem uses the type and cotype inequalities for Lp spaces, which has type min{2, p} and cotype max{2, p} (see [5, Prop. 8.6] or [14, p. 73]). Recall that a Banach function AM-space such that satisfies [parallel] f \[disjunction] g [parallel] = max{[parallel] f [parallel], [parallel] g [parallel]} for f, g > 0, where (f [disjunction] g)([omega]) := max{f ([omega]), g([omega])}. We show that we can provide a direct proof by using this abstract axiomatic definition, instead of writing the direct result for the case X = [L.sup.[infinity]]. We will see in Corollary 3.8 that in fact, the AM-space involved is [L.sup.[infinity]].

Theorem 3.7. Let 1 [less than or equal to] p [less than or equal to] [infinity]. Let Xbea Banach function space with weak unit, which is an AM-space. Then [X.sup.p] has type min{2, p} and cotype max{2, p}.

Proof. Since X has weak unit, [X.sup.Y] is a Banach function space and the definition of type or cotype can be applied. The proof is divided in 2 parts. Step 1: Type. Recall that the norm in [X.sup.p] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [f.sub.1], ..., [f.sub.n] [member of] [X.sup.p] and [epsilon] > 0 we claim that there exists g [member of] [B.sub.X] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where the election of g [member of] [B.sub.X] does not depends on the election of t [member of] [0, 1]. Let us define [[psi].sub.n]: [0, 1] [right arrow] [X.sup.p] by

[[psi].sub.n](t) := [n.summation over (k = 1)] [r.sub.k](t) [f.sub.k].

By definition of the Rademacher functions, ipn has at most [2.sup.n] values, since [r.sub.k] is defined on [2.sup.k] subdivisions of the same length of [0, 1] for k = 1, ..., n. So, it is not hard to realize that [r.sub.n] define the number of possible values of [[psi].sub.n]. Let us select [t.sub.j] [member of] [j/[2.sup.n], [j + 1/[2.sup.n]] [for j = 0, ..., [2.sup.n] - 1, (e.g. [t.sub.j] := j/[2.sup.n]). Then [[psi].sub.n]{t) = [[psi].sub.n] ([t.sub.j]) for t [member of] [j/[2.sup.n], j + 1/[2.sup.n][. By definition, for each j [member of] {0, ..., [2.sup.n] - 1} there exists [g.sub.i] [member of] [B.sub.X] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every t [member of] [j/[2.sup.n], j + 1/[2.sup.n][. Let us define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since, X is an AM-space we have that

[[parallel] g [parallel].sub.X] = max{[[parallel] [g.sub.1] [parallel].sub.X], ..., [[parallel] g[2.sup.n] - 1[parallel].sup.X]} [less than or equal to] 1.

On the other hand [absolute value of ([g.sub.j])] [less than or equal to] [absolute value of (g)], thus [absolute value of ([g.sub.j] [[psi].sub.n](tj))] [less than or equal to] [absolute value of (g[[psi].sub.n]([t.sub.j]))] for all j = 0, ...,[2.sup.n] - 1.

Then [absolute value of ([g.sub.j] [[psi].sub.n] (t))] [less than or equal to] [absolute value of (g[[psi].sub.n] (t))] and [[parallel] [g.sub.j] [[psi].sub.n] (t) [parallel].sub.p] [less than or equal to] [[parallel] g[[psi].sub.n] (t) [parallel].sub.p] for every j = 0, ..., [2.sup.n] - 1,

t [member of] [j/[2.sup.n], [j + 1/[2.sup.n]][. Let t [member of] [0, 1], then there exists; such that [[psi].sub.i] (t) = [[psi].sub.n]([t.sub.j]). Therefore (8) holds.

Let us now compute taking into account that (1) is satisfied in [L.sup.p].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [epsilon] > 0 is arbitrary, we can assert that [X.sup.p] has the same type as [L.sup.p], i.e. min{2, p}.

Step 2: Cotype. Let [f.sub.1], ..., [f.sub.n] [member of] [X.sup.p]. For each k [member of] {1, ..., n}, choose 0 [greater than or equal to] [g.sub.k] [member of] [B.sub.X] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let us define [g.sub.0] := [g.sub.1] [disjunction] *** [disjunction] [g.sub.n]. Since X is an AM-space, [g.sub.0] [member of] [B.sub.X], [g.sub.i] [less than or equal to] [g.sub.0] and [g.sub.i] [absolute value of ([f.sub.i])] [less than or equal to] [g.sub.0] [absolute value of ([f.sub.i])], hence [[absolute value of ([g.sub.i] [f.sub.i])].sup.p] [less than or equal to] [[absolute value of ([g.sub.0] [f.sub.i])].sub.p] for i = 1, ..., n. Therefore, by (2) for Lp we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Which proves that [X.sup.p] has cotype max{2, p}.

Corollary 3.8. Let H be a Banach function space such that [chi][OMEGA] [member of] H. Then H is isomorphic to a Hilbert space if and only if there exists a Banach function AM-space X, such that H = [X.sup.2].

Proof. Assume that H is isomorphic to a Hilbert space, hence H = [H.sup.*]. On one hand, let us prove that [H.sup.2] is an AM-space. By Lemma 2.2(9) taking X := [H.sup.2] and Y := H, we have trivially that H [subset or equal to] [([H.sup.2]).sup.2]. Thus, by Lemma 2.2(10) we have that

H [subset or equal to] [([H.sup.2]).sup.2] [subset or equal to] [L.sup.2], (9)

hence Lemma 2.1(6) (taking X = H, q = 1 and p = r = 2), implies that

[L.sup.2] [subset or equal to] H' [subset or equal to] [H.sup.*] = H. (10)

Then, H = [L.sup.2], thus [H.sup.2] = ([L.sup.2]) = [L.sup.[infinity]], which is an AM-space. On the other hand, inclusions (9) and (10) state that H = [([H.sup.2]).sup.2]. So choose X := [H.sup.2].

For the converse, by the previous Theorem 3.7, we have that [X.sup.2] has type 2 and cotype 2. Applying Kwapien's Theorem we obtain the result.

The Nikishin's Theorem provides the last application of the paper. We recall that the vector measure associated to an operator between Banach function spaces T: X [right arrow] Y, where X is order continuous, is denoted by [m.sub.T] : [summation] [right arrow] E and defined by [m.sub.T] (A) := T([[chi].sub.A]), for every A [member of] [summation]. We refer the reader to [16, Chap. 4] for more information and main properties about this concept. Recall also that a Banach function AL-space is such that satisfies [parallel] f + g [parallel] = [parallel] f [parallel] + [parallel] g [parallel] for f, g > 0 and f [conjunction] g = min{f, g} = 0.

Corollary 3.9. Let X and Y be two Banach function spaces over a positive measure, such that X is order continuous and [[chi].sub.[OMEGA]] [member of] X. Let T: X [right arrow] Y be a p-th power factorable operator. If [L.sup.1]([m.sub.T]) is Fatou and is either q-concave for some q < [infinity] or an AL-space, then the (range) extension [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] factorize through an [L.sup.min{2,p},[infinity]]-space.

Proof. Assume first that [L.sup.1]([m.sub.T]) is q-concave for some q < [infinity]. If p > 2, T is also r-th power factorable for any r < p, then let us choose 1 [less than or equal to] p [less than or equal to] 2. Thanks to Theorem 5.7 in [16], T is extended to the space [L.sup.p]([m.sub.T]), via the integration map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We have that [L.sup.p]([m.sub.T]) is p-convex and [L.sup.1]([m.sub.T]) is q-concave, then [L.sup.p]([m.sub.T]) is also r-concave for an r < [infinity], and then it has type min{2, p} ([14, Prop. 1.f.3(ii)]). Then we can assert that [L.sup.p]([m.sub.T]) has type min{2, p}. Finally, we apply the Nikishin's Theorem (see e.g. [21, Thm. III.H.6]), and obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] factorize through a Lorentz space [L.sup.max{2,p},[infinity]], hence so is i [omicron] T.

Suppose now that [L.sup.1]([m.sub.T]) is an AL-space. Then [L.sup.1]([m.sub.T])' is an AM-space ([1, Thm. 10.15]). Then, since [L.sup.1]([m.sub.T])' is Fatou by Lemma 2.2(7), we have

[L.sup.p]([m.sub.T]) = [([L.sup.1]([m.sub.T])").sub.[1/p]] = [([([L.sup.1]([m.sub.T])'.sub.[1/p]]).sup.p].

Then, applying Theorem 3.7 above we obtain that [L.sup.p]([m.sub.T]) has type min{2, p}. Again Nikishin's Theorem gives the result.

Notice that compactness of the integration map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies that [L.sup.1]([m.sub.T]) is an [L.sup.1]-space and so an AL-space ([17, Thm. 1 and 4]). Therefore, this gives an example of the corollary above. Other conditions to obtain that [L.sup.1] (m) is an AL-space can be found in [3].

Acknowledgment This paper is part of the Ph.D. thesis which I recently have presented in the Polytechnic University of Valencia, which has been supervised by the Professors F. Mayoral and E.A. Sanchez-Perez, whom I am greatly grateful. I would like to thank the referee for carefully reading this manuscript and for the alternative proof of Theorem 3.2 (see Remark 3.3).

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Instituto Universitario de Matematica Pura y Aplicada, Universidad Politecnica de Valencia Camino de Vera s/n, 46022 Valencia. Spain.

email: orlando.galdames@gmail.com

Received by the editors in February 2013 - In revised form in August 2013.

Communicated by F. Bastin.

2010 Mathematics Subject Classification : Primary 46A20, Secondary 32F10, 46B20.
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