Factorization theorems for multiplication operators on Banach function spaces

Factorization theorems for multiplication

operators on Banach function spaces

E. A. Sanchez Perez

Abstract. Let X Y and Z be Banach function spaces over a measurespace (⌦, ⌃, µ). Consider the spaces of multiplication operators XY 0

from X into the Kothe dual Y 0 of Y , and the spaces XZ and ZY 0

defined in the same way. In this paper we introduce the notion of fac-torization norm as a norm on the product space XZ · ZY 0 ✓ XY 0 thatis defined from some particular factorization scheme related to Z. Inthis framework, a strong factorization theorem for multiplication oper-ators is an equality between product spaces with di↵erent factorizationnorms. Lozanovskii, Reisner and Maurey-Rosenthal theorems are spe-cial cases of this general setting. In this paper we analyze the class d⇤p,Z

of factorization norms, proving some factorization theorems for themwhen p-convexity/p-concavity type properties of the spaces involved areassumed. Some applications in the setting of the product spaces aregiven.

Mathematics Subject Classification (2010). Primary 46E30; Secondary47B38, 46B42, 46B28.

Keywords. Banach function spaces, Kothe dual, generalized dual spaces,multiplication operator, factorizations, product spaces.

1. Introduction

Factorization of operators through Banach function spaces is a common toolfor solving problems in functional analysis. Often, these factorizations occurthrough a multiplication operator defined between Banach function spaces.Maurey-Rosenthal theorems through L

p spaces (see for instance [6, 8]), Pisierfactorization theorems through Lorentz spaces (see [20, 9]), Nikishin theoremthrough weak L

p spaces ([26, Th. III.H.6]) are some relevant examples ofthese results, but this kind or arguments can be found in a lot of di↵erentsettings, in which a multiplication operator plays a fundamental role (see for

The author was supported by the Ministerio de Economıa y Competitividad (Spain) under

grant #MTM2012-36740-C02-02.

2 E. A. Sanchez Perez

example [5]). Sometimes, the only information that gives the factorizationtheorem regarding the function appearing in it is continuity of the multipli-cation operator. However, a deeper knowledge about it improves the power ofthese results specially if it can be given as a new factorization of the multipli-cation operator, since it produces an enrichment of the factorization schemefor the original operator. This is the motivation of the study that we developin this paper in which we introduce a technique for analyzing the factoriza-tion properties of the multiplication operators in a systematic way, followingthe research project that we started in [10].

From the technical point of view, as the reader will notice soon, we mimicin a sense the procedure for constructing reasonable topologies for tensorproducts in the representation theory of operator ideals by means of tensornorms (see [7]). The main di↵erences are that we change tensor products ofBanach spaces by products of function spaces, and the usual Banach spaceduality by the generalized duality for Banach function spaces (see [4, 17,24]). Recently, a new e↵ort has been made in order to develop the theory ofproducts of Banach function spaces and general multiplication operators, thatare a particular —but central— case of the general factorization norms andspaces that we develop here. The reader can find more information on thatin the papers by Kolwicz, Lesnik and Maligranda (see [13] and the referencestherein), Schep ([24], see also [23]), Sukochev and Tomskova (see [22]) andCalabuig, Delgado and the author (see [4, 10]).

2. Factorization norms and strong factorization theorems

Let (⌦,⌃, µ) be a measure space and let X(µ), Y (µ) and Z(µ) be Banachfunction spaces over µ (we shall write X, Y , Z for short if no explicit referenceto the measure is needed). Let Y

0 be the Kothe dual of Y and X

Z , Z

Y

0and

X

Y

0the corresponding spaces of multiplication operators from X to Z, Z to

Y

0 and X to Y

0, respectively. In this paper we continue with the analysis ofthe topological products of function spaces that we started in [10] (see also[4]); with the aim of establishing factorization theorems for multiplicationoperators, we define and characterize the family d

⇤p,Z

of what we call factor-ization norms for the product spaces X

Z ·ZY

0. We show that the elements of

the completion of the product space X

Z ·ZY

0with a factorization norm d

⇤p,Z

always belong to X

Y

0and can be described in terms of a common factor-

ization scheme. We also prove the main factorization theorems for this classof product spaces. In particular, we show that some classical factorizationtheorems (Lozanoskii, Reisner and Maurey-Rosenthal) can be considered asparticular cases of our construction. The class of norms d

⇤p,Z

that we investi-gate here is modeled for satisfying suitable equalities between product spacesunder p-convexity/p-concavity assumptions for the spaces involved. Such anequality is what we call a strong factorization theorem.

Let us fix some introductory ideas. Take three Banach function spacesX, Y and Z such that the spaces X

Z and Z

Y

0are saturated (see Section 2

Factorizations theorems for multiplication operators 3

for the definition). It is easy to check that Z

Y

0= Y

Z

0(see Lemma 3.7 in

[4], or [17]). The product of the space X

Z · Y

Z

0is defined to be the subset

of the space of (classes of µ-a.e. equal) measurable functions L

0(µ) given bythe finite sums of functions f · g, where f 2 X

Z and g 2 Y

Z

0. Note that

such a function defines a multiplication operator from X to Y

0. We analyzesubspaces of X

Y

0instead of X

Y because for the definition of the norms thatwe define in the paper the generalized duality given by the bilinear formZ⇥Z

0 ! R given by the integral is needed (see [4, 13, 17, 24]). However, thereader can notice that a great part of the results of the paper applies also tothe case X

Y .Two natural “extreme” norms can be defined in the space X

Z · Y

Z

0.

The first one (that we will denote by "

Z

) is given by the restriction of theoperator norm to the completion of such subspace of X

Y

0. This is the weaker

“reasonable” topology that we will consider; we use the symbol X

Z

"

Z

Y

Z

0to

denote the Banach function subspace of X

Y

0generated in this way. For the

second one we define the function norm on L

0(µ) given by the formula

⇡(h) := infX

i�1

kfi

kX

Z · kgi

kY

Z0 , h 2 L

0(µ),

where the infimum is computed over all possible dominations of h as |h| P

i�1 |fi

g

i

|, f

i

2 X

Z , g

i

2 Z

Y

0; if there is no such a decomposition, then

⇡(h) = 1 (see [25] for the general theory of function norms, and [10, 13, 24]for the properties of the ⇡-norm, but note that the definitions are slightlydi↵erent in these papers). This norm gives the strongest topology on theproduct X

Z ·Y Z

0that we consider, and defines a function norm that generates

the Banach function space X

Z

⇡Y

Z

0in L

0(µ); under certain requirements anequivalent norm ⇡ for this space can be computed as the infimum of thesingle product of norms of functions f 2 X

Z and g 2 Y

Z

0that provide a

decomposition of |h| as |h| = hg, h 2 X

Z

⇡Y

Z

0(see [4, 10, 13, 24]). It is

easy to check that for every f 2 X

Z

⇡Y

Z

0, "

Z

(f) ⇡(f). We will deal withnorms ↵ on X

Z · Y

Z

0satisfying "

Z

↵ ⇡. We say that such a norm ↵

is a factorization norm if ↵(h) can be computed as an infimum of productsof the norms of suitable operators belonging to a fixed family that factorizethe multiplication operator given by h; a strong factorization theorem is anequality X

Z

↵Z

Y

0= X

Z

�Z

Y

0for a couple of factorization norms ↵ and �.

Relevant known examples of strong factorization theorems for multi-plication operators are the following. Lozanovskii´s theorem ([16, 21]) canbe written as the isometry (L1)Y

⇡Y

L

1= (L1)Y

⇡Y

0 = L

1 = (L1)L

1. If

1/p = 1/q + 1/r, Reisner´s theorem ([21]) gives the equality (Lq)Y

⇡Y

L

p

=(Lq)Y

⇡Y

L

p

= L

r when the adequate requirements on convexity and con-cavity for Y are assumed. Also, Maurey-Rosenthal theorem (see [6, Cor.5] or [19, Ch.6]) for the case of multiplication operators gives the equalityX

Y

0= X

L

p

⇡(Lp)Y

0whenever X is order continuous and p-convex, and Y

0 isp-concave. Other example of the same methodological point of view for the


case of sequence spaces can be found in the analysis of classical inequalitiesexposed in [2].

After this introductory section and the following one —where some no-tation and basic definitions are given—, we present our results in three parts.Section 4 is devoted to the analysis of the properties of the norm d

⇤p,Z

as afactorization norm. The extreme cases (the norms "

Z

and ⇡) are also consid-ered as d

⇤p,Z

norms. In Section 5 we give some applications and prove somefactorization theorems related with the d

⇤p,Z

norms. For instance, Theorem5.1 establishes that, under the adequate convexity properties, the multipli-cation operators of the space defined by d

⇤p,Z

satisfy a special factorizationtheorem that can be written as the equality X

Z

⇡Y

Z

0= X

Z

d

⇤p,Z

Y

Z

0(Sec-

tion 5.1). More applications involving the order continuity of the ⇡ norm arealso given. Finally, we present in Section 5.2 the main results concerning thecomplementary equality X

Z

d

⇤p,Z

Y

Z

0= X

Y

0.

3. Notation and basic concepts

Let (⌦,⌃, µ) be a measure space. Let L

0(µ) be the space of all (classes ofµ-a.e. equal) real functions on ⌦. A Banach function space over µ is a Banachspace X ⇢ L

0(µ) with a norm k · kX

satisfying that if f 2 L

0(µ), g 2 X and|f | |g| µ–a.e. then f 2 X and kfk

X

kgkX

. A Banach function spaceX is order continuous if increasing sequences that are bounded µ-a.e. areconvergent in norm. We say that a Banach function space X has the Fatouproperty if for every sequence (f

n

) ⇢ X such that 0 f

n

" f µ–a.e. andsup

n

kfn

kX

< 1, f 2 X and kfn

kX

" kfkX

. A weaker property for X isgiven by what we call the order semi-continuity ; a Banach function space X

satisfies this property if for every f, f

n

2 X such that 0 f

n

" f µ–a.e.,kf

n

kX

" kfkX

.Throughout the paper the Banach function spaces are considered to be

over the same measure space. If X is a Banach function space, we denoteby X

0 its Kothe dual or associate space, i.e. the Banach function space offunctions that define multiplication operators from X to L

1 (see for example[15]). In general, we identify a function in X

Y with the multiplication operatorthat it defines. The generalized duality induced on X and X

Z by the bilinearmap X ⇥ X

Z ! Z given by the product has been defined and studied byMaligranda and Persson in [17] and also by Calabuig and the authors in [4](see also [10, 3, 13, 24]).

The space of multiplication operators X

Z is a Banach function space(with the natural operator norm) if X

Z is saturated, i.e. if there is no A 2 ⌃with µ(A) > 0 such that f�

A

= 0 µ–a.e. for all f 2 X

Z . Through the paperthe saturation property is always required for all the spaces involved ; this factwill be explicitly mentioned only if we consider that it is specially relevantin the context. Our references for the definition and properties of Banachfunction spaces are [25, Ch. 15], considering the function norm ⇢ defined as⇢(f) = kfk

X

if f 2 X and ⇢(f) = 1 in other case, and [1, 14, 18].


The following notation for norms of sequences of functions involving thegeneralized duality will be used. Let (x

i

) be a sequence in X. Let 1 p 1.Then we write

⇡

p

((xi

)) :=�

X

i�1

kxi

kp

X

�1/p

,

with the obvious modification if p = 1. If ⇡

p

((xi

)) < 1, we say that (xi

) isstrongly p-summable. If (f

i

) a sequence in X

Z , we define the function

w

⇤p,Z

((fi

)) := supx2BX

�

X

i�1

kfi

xkp

Z

�1/p

.

In the case that w

⇤p,Z

((fi

)) < 1, we say that (fi

) is weakly (p, Z)-summable;the “dual” definition and the corresponding analysis can be found in [10].Also, if (f

i

) is a sequence of functions in X

Y and 1 p q 1, we willconsider the operators defined in the natural way, that may change dependingon for which spaces are defined, using the same symbol (f

i

) to denote thecorresponding operator. The reader will find the following three cases, thatwill be used without further explanations; of course, in each one of them thesequence (f

i

) must satisfy an adequate boundedness condition in order theoperator to be well defined.(1) (f

i

) : X ! `

p(Y ), (fi

)(x) := (fi

x), x 2 X.

(2) (fi

) : `

q(X) ! `

p(Y ), (fi

)(xi

) := (fi

x

i

), (xi

) 2 `

q(X).(3) (f

i

) : `

p(X) ! Y, (fi

)(xi

) :=P

i�1 f

i

x

i

, (xi

) 2 `

p(X).

Throughout the paper, if 1 p 1, p

0 denotes the (extended) realnumber satisfying 1/p + 1/p

0 = 1. A Banach function space X is p-convexwith constant M

(p)(X) if for every sequence (xi

) in X,�

�

�

�

X

i�1

|xi

|p�1/p

�

�

�

X

M

(p)(X)⇣

X

i�1

kxi

kp

X

⌘1/p

.

The space X is p-concave if there is a constant M(p)(X) such that for everysequence (x

i

) in X,⇣

X

i�1

kxi

kp

X

⌘1/p

M(p)(X)�

�

�

�

X

i�1

|xi

|p�1/p

�

�

�

X

.

The constants M

(p)(X) and M(p)(X) are the best ones in the above inequal-ities.

A Banach lattice X is said to have a lower p-estimate if and only if thereis a constant k > 0 such that for every element x 2 X,

⇣

X

i�1

kxi

kp

X

⌘1/p

kkxkX

for all the decompositions of x as a pointwise sum of disjoint elements x

i

2 X

(see for instance Definition 1.f.4. [15]). Notice that p-concavity implies lowerp-estimate.


4. The d⇤p,Z-factorization norms

Consider a space Z such that X

Z and Y

Z

0are saturated. If f 2 X

Z andg 2 Y

Z

0= Z

Y

0, the product fg defines an element of X

Y

0. The function

norm "

Z

: L

0(µ) ! R+ [ {1} is defined by

"

Z

(h) := inf{kX

i�1

|fi

g

i

|kX

Y 0 : |h| X

i�1

|fi

g

i

|, f

i

2 X

Z

, g

i

2 Y

Z

0},

where the infimum is computed over all possible k · kX

Y 0 -convergent seriesthat dominates |h| ("

Z

(h) = 1 if no domination occurs). Now, define thespace

X

Z

"

Z

Y

Z

0:=

n

h 2 L

0(µ) : there exist (fi

) ⇢ X

Z

, (gi

) ⇢ Y

Z

0such that

the seriesX

i�1

|fi

g

i

| converges inX

Y

0and |h|

X

i�1

|fi

g

i

|o

.

Consequently, X

Z

"

Z

Z

Y

0can be identified with the (normed) subspace

of all the functions h 2 L

0(µ) that satisfy that "

Z

(h) is finite. Since the spaceX

Y

0is an ideal, we have in particular that X

Z

"

Z

Z

Y

0 ✓ X

Y

0isometrically.

We also assume that X

Z

"

Z

Z

Y

0is saturated; in other case "

Z

maybe just aseminorm. In the case of the other product spaces defined in this section,saturation is also assumed.

Lemma 4.1. Let h 2 X

Z

"

Z

Y

Z

0. Then there is a µ-a.e. convergent series

P

i�1 |f0i

g

0i

| defined by functions f

0i

2 X

Z and g

0i

2 Y

Z

0such that |h| =

P

i�1 |g0i

f

0i

| (convergence in X

Y

0) and "

Z

(h) = khkX

Y 0 . Consequently, thespace (XZ

"

Z

Y

Z

0, "

Z

), where X

Z

"

Z

Y

Z

0:=

�

h 2 L

0(µ) : "

Z

(h) < 1

is(isometrically) a Banach function subspace of X

Y

0that contains the closure

in X

Y

0of the linear combinations of products of elements of X

Z and Y

Z

0.

Proof. Take � > 0. Since h 2 X

Z

"

Z

Y

Z

0, there is a series of functions

P

i�1 |fi

g

i

| convergent in X

Y

0that dominates |h|, f

i

2 X

Z , g

i

2 Y

Z

0, and

kP

i�1 |fi

g

i

|kX

Y 0 "

Z

(h) + �. Consider the function h0 = |h|/P

i�1 |fi

g

i

|.Clearly, h0 1 µ-a.e. and the functions f

0i

:= f

i

h0, i = 1, 2, ... belong to X

Z ,since X

Z is in particular an ideal in L

0(µ). Define g

0i

:= g

i

for every i. Notethat h and

P

i�1 |f0i

g

0i

| are elements of X

Y

0, since it is an ideal, and

khkX

Y 0 kX

i�1

|f0i

g

0i

|kX

Y 0 kX

i�1

|fi

g

i

|kX

Y 0 "

Z

(h) + �.

The seriesP

i�1 |f0i

g

0i

| converges in X

Y

0(and then also µ-a.e.) to |h|, since

k|h|�n

X

i=1

|f0i

g

0i

|kX

Y 0

=�

�

�

h0

�

X

i�1

|fi

g

i

|�n

X

i=1

|fi

g

i

|�

�

�

�

X

Y 0 k

X

i�1

|fi

g

i

|�n

X

i=1

|fi

g

i

|kX

Y 0k ! 0.


Consequently,khk

X

Y 0 = kX

i�1

|f0i

g

0i

|kX

Y 0 = "

Z

(h).

This gives the result on the representation of |h|. Note also that the Riesz-Fischer property is satisfied. To see this, consider a sequence of functions(h

k

) ⇢ X

Z

"

Z

Y

Z

0such that

P

k�1 "

Z

(hk

) < 1. Since X

Y

0is complete, the

function h :=P

i�1 h

i

belongs to X

Y

0and

P

n

k=1 h

i

converges toP

i�1 h

i

in the norm of this space. Also, it can be found for each k a decompositionof |h

k

| as the k · kX

Y 0 -convergent sequenceP

i�1 |fk

i

g

k

i

|, where f

k

i

2 X

Z

and g

k

i

2 Z

Y

0. Therefore, straightforward calculations show that the se-

riesP

n

i=1

P

n

k=1 |fk

i

g

k

i

| converges in X

Y

0to

P

i�1

P

k�1 |fk

i

g

k

i

|, and obvi-ously |h|

P

i�1

P

k�1 |fk

i

g

k

i

| µ-a.e. This shows that h 2 X

Z

"

Z

Y

Z

0, and

so X

Z

"

Z

Y

Z

0is a Banach space.

⇤

Definition 4.2. Let 1 p 1. Consider a couple of Banach function spaces(X, Y ) and a Banach function space Z satisfying that X

Z and Z

Y

0are sat-

urated. We define the function d

⇤p,Z

: X

Y

0 ! R+ [ {1} by the formula

d

⇤p,Z

(h) := inf{w⇤p

0,Z

((fi

)) · ⇡p

((gi

)) : |h| X

i�1

|fi

g

i

|, f

i

2 X

Z

, g

i

2 Z

Y

0}.

Lemma 4.3. d

⇤p,Z

defines a function norm on X

Y

0that is complete and such

that "

Z

d

⇤p,Z

⇡. In particular, X

Z

d

⇤p,Z

Y

Z

0is a Banach function space.

Proof. Let us prove that d

⇤p,Z

satisfies the triangular inequality. Let � > 0.Take a couple of functions f1 and f2 in X

Z

d

⇤p,Z

Y

Z

0and note that we can

always find two series of functionsP

i�1 |g1i

h

1i

| andP

i�1 |g2i

h

2i

| of elementsof X

Z and Y

Z

0dominating |f1| and |f2|, respectively, and satisfying that

w

⇤p

0,Z

((gj

i

)) (d⇤p,Z

(fj

) + �)1/p

0and ⇡

p

((hj

i

)) (d⇤p,Z

(fj

) + �)1/p, j = 1, 2.The domination of |f1 + f2| given by

P

i�1 |g1i

h

1i

|+P

i�1 |g2i

h

2i

| satisfies that

w

⇤p

0,Z

((g1i

) [ (g2i

))⇡p

((h1i

) [ (h2i

))

(w⇤p

0,Z

((g1i

))p

0+ w

⇤p

0,Z

((g2i

))p

0)(⇡

p

((h1i

))p + ⇡

p

((h2i

))p)

(d⇤p,Z

(f1) + d

⇤p,Z

(f2) + 2�)1/p(d⇤p,Z

(f1) + d

⇤p,Z

(f2) + 2�)1/p

0

= d

⇤p,Z

(f1) + d

⇤p,Z

(f2) + 2�

(here the notation (g1i

)[ (g2i

) means the sequence (g11 , g

21 , g

12 , g

22 , g

13 , ...)). Con-

sequently, d

⇤p,Z

is a (lattice) seminorm for the space of functions f satisfyingthat d

⇤p,Z

(f) < 1. This, together with Lemma 4.1, implies that d

⇤p,Z

is infact a function norm. A straightforward argument using the inequalities abovefor infinite sequences of functions shows that d

⇤p,Z

satisfies the Riesz-Fischerproperty, and so X

Z

d

⇤p,Z

Y

Z

0is a Banach function space. Let us prove now


the inequalities "

Z

(f) d

⇤p,Z

(f) ⇡(f) for every f 2 X

Z

d

⇤p,Z

Z

Y

0. For such

a function f and any seriesP

i�1 |fi

g

i

| dominating |f |

"

Z

(f) = supx2BX

kfxkY

0 supx2BX ,y2BY

X

i�1

Z

xy|fi

g

i

| dµ

supx2BX

�

X

i�1

kfi

xkp

0

Z

�1/p

0�

X

i�1

kgi

kp

Z

Y 0

�1/p

.

Thus "

Z

(f) d

⇤p,Z

(f). Also, taking into account

inf{⇡p

0((fi

))⇡p

((gi

)) : |h| X

i�1

|fi

g

i

|, f

i

2 X

Z

, g

i

2 Z

Y

0} = ⇡(h),

(see the comments after Proposition 3.2 in [10]) we obtain d

⇤p,Z

⇡. ⇤

Theorem 4.4. Let 1 < p < 1. The following assertions are equivalent for amultiplication operator h 2 X

Y

0.

(1) h 2 X

Z

d

⇤p,Z

Y

Z

0.

(2) The function h defines a multiplication operator of X

Y

0that factorizes

as

`

p

0(Z)

X(µ)

(fi

)

?

C

-(gi

)`

1(Y 0)

Y

0(µ)-h

6

where w

⇤p

0,Z

((fi

)) < 1, ⇡

p

((gi

)) < 1 and C is defined by C((y0i

)) :=P

i�1 y

0i

, (y0i

) 2 `

1(Y 0).(3) h can be written as an almost everywhere sum of the product of a weakly

(p0, Z)-summable sequence (fi

) of elements of X

Z and other strongly p-summable sequence (g

i

).Moreover, if (1), (2), (3) hold, then d

⇤p,Z

(h) = inf k(fi

)kk(gi

)k, where the infi-mum is computed over all suitable factorizations in (2).

Proof. (1) ) (3). Since d

⇤p,Z

(h) is finite, for any 1 < p < 1 there is acouple of sequences of functions (f0

i

) and (g0i

) such that |h| P

i�1 |f0i

g

0i

|,w

⇤p

0,Z

((f0i

)) < 1 and ⇡

p

((g0i

)) < 1. Let us show that we can obtain a coupleof sequences (f

i

) and (gi

) such that h =P

i�1 f

i

g

i

µ-a.e., w

⇤p

0,Z

((fi

)) < 1and ⇡

p

((gi

)) < 1. Write h as the sum of the positive and the negativeparts h = h

+ � h

�. Let us define the functions of disjoint support '

+ =h

+/

P

i�1 |f0i

g

0i

| and '

� = h

�/

P

i�1 |f0i

g

0i

|, that satisfy that 0 '

+ 1and 0 '

� 1 µ-a.e. Consequently, h =P

i�1 |f0i

g

0i

|'+ �P

i�1 |f0i

g

0i

|'�.


The µ-a.e. pointwise convergence of this sum is guaranteed by the convergenceofP

i�1 |f0i

g

0i

|, since clearly

|X

i�1

|f0i

g

0i

|'+ � |f0i

g

0i

|'�| X

i�1

|f0i

g

0i

|.

Now consider each of the elements |f0i

g

0i

|'+ and |f0i

g

0i

|'� in this sum. Thereare functions 0 f

1i

|f0i

| and 0 g

1i

g

0i

such that f

1i

g

1i

= |f0i

g

0i

|'+,and functions 0 f

2i

|f0i

| and 0 g

2i

g

0i

such that f

2i

g

2i

= |f0i

g

0i

|'�.Note that f

j

i

and g

k

l

can be chosen to be disjointly supported for every i andl whenever j 6= k. The same is true for f

j

i

and f

k

l

, and g

j

i

and g

k

l

, for j 6= k.Therefore, h =

P

i�1(f1i

� f

2i

)(g1i

+ g

2i

), since for every i,

(f1i

� f

2i

)(g1i

+ g

2i

) = f

1i

g

1i

� f

2i

g

1i

+ f

1i

g

2i

� f

2i

g

2i

= f

1i

g

1i

� f

2i

g

2i

.

Moreover, w

⇤p

0,Z

((f1i

� f

2i

)) w

⇤p

0,Z

((f0i

)) and ⇡

p

((g1i

+ g

2i

)) ⇡

p

((g0i

)), bythe lattice properties of the Banach function spaces Z and X

Z .Clearly, (3) implies (2) and (2) implies (1) as a consequence of the

computation of the norms in the factorization, since

k(fi

)kL(X,`

p0 (Z)) = supx2BX

⇣

X

i�1

kfi

xkp

0

Z

⌘1/p

0

= w

⇤p

0,Z

((fi

))

andk(g

i

)kL(`p0 (Z),`1(Y 0)) =

⇣

X

i�1

kgi

kp

Z

Y 0

⌘1/p

= ⇡

p

((gi

)).

Finally, note that the computation at the end of the proof of (1) ) (3) provesin particular that the infimum of the products k(f

i

)k · k(gi

)k for all suitablefactorizations gives d

⇤p,Z

(h).⇤

Remark 4.5. Let us show how the factorization for the norms "

Z

and ⇡

can also be considered in a sense as the extreme cases of the factorizationtheorems for the d

⇤p,Z

norms.(a) The factorization theorem for the "

Z

norm. As a consequence of theisometry X

Y

0⇡Y

0Y 0= X

Y

0⇡L

1(µ) (see [4, 17, 24]), we obtain the fol-lowing result for the "

Z

norm for the particular case Z = Y

0. For afunction h 2 L

0(µ), the following statements are equivalent.(1) h 2 X

Y

0.

(2) There is a factorization for h as

X

h -Y

0

(fi

)

HHHHHHj`

1(Y 0) ��*

(gi

) ,

where (fi

) satisfies w

⇤1,Y

0((fi

)) < 1, and (gi

) is a sequence in`

1(L1).


Moreover, if (1), (2) hold, then khkX

Y 0 = inf k(fi

)kk(gi

)k, where theinfimum is computed over all suitable factorizations as the one given in(2). (Here the norms for the sequences are understood as their normsas operators)

This is a consequence of the following easy arguments. Take a func-tion h 2 X

Y

0and consider the following factorization through `

1(Y 0) ofthe map h : X ! Y

0 given by the sequence (h, 0, 0, ...) 2 `

1(XY

0) and

the map (gi

) : `

1(Y 0) ! Y

0 given by the sequence (gi

), where g1 = �⌦,g

i

= 0 for each i = 2, .... Clearly, khkX

Y 0 = k(h, 0, 0, ...)kk(gi

)k.On the other hand, take a couple of sequences (f

i

) and (gi

) satis-fying the requirements in (2). Since for every x 2 X,

khxk kX

i�1

|fi

g

i

|xkY

0 supi

kgi

kL

1

X

i�1

kfi

xk,

the map is well defined and continuous. Moreover,

khkX

Y 0 kX

i�1

|fi

g

i

|kX

Y 0 X

i�1

kfi

kX

Y 0kgi

kL

1 k(fi

)kk(gi

)k.

This, together with the particular factorization given at the beginningof the proof, gives the formula

khkX

Y 0 = inf k(fi

)kk(gi

)k,

where the infimum is computed over all suitable factorizations, and fin-ishes the proof.

(b) The factorization theorem for the ⇡ norm. Even in the case when wehave no information about the coincidence of ⇡ and ⇡ on X

Z · Y

Z

0,

it is still possible to get a factorization theorem for the elements ofthe ⇡ product; in the case that the equivalence holds, then we have astrong factorization theorem in the sense that has been explained in theIntroduction.

The following assertions are equivalent for a function h 2 X

Y

0.

(1) h 2 X

Z

⇡ Y

Z

0.

(2) There is a real number 1 < p < 1 such that there is a factorizationof h as

`

1(X)

X(µ)

I

?

C

-(fi

)`

p

0(Z) -

`

1(Y 0)(g

i

)

Y

0(µ)-h

6

where I(x) := (x, x, x, ...), x 2 X, and C((y0i

)) =P

i�1 y

0i

, (y0i

) 2`

1(Y 0).


(3) There is a real number 1 < p < 1 (and then for every such numberp) such that h can be written as an almost everywhere sum of theproduct of a strongly p

0-summable sequence (fi

) of elements of X

Z

and other strongly p-summable sequence (gi

) of elements of Z

Y

0.

Moreover, if (1), (2), (3) hold, then ⇡(h) = inf k(fi

)kk(gi

)k, where theinfimum is computed over all the factorizations as in (2).

The implication (1) ) (3) can be shown as (1) ) (3) in Theorem4.4. For (2) ) (1), let us compute the norm of the factorization. Byhypothesis, there are strongly p

0 and p summable sequences (fi

) and(g

i

), respectively, such that |h| P

i�1 |fi

g

i

|, and define a factorizationas the one in (2). But note that

k(fi

)kL(`1(X),`p0 (Z)) = sup

kxikX1

⇣

X

i�1

kfi

x

i

kp

0

Z

⌘1/p

0

=⇣

X

i�1

kfi

kp

0

X

Z

⌘1/p

0

= ⇡

p

0((fi

)),

and, by duality of the spaces `

p and `

p

0,

k(fi

)kL(`p0 (Z),`1(Y 0)) = sup

(zi)2B

`p0 (Z)

⇣

X

i�1

kgi

z

i

kY

0

⌘

=⇣

X

i�1

kgi

kp

Z

Y 0

⌘1/p

= ⇡

p

((gi

)).

The equivalent formula for the norm ⇡ given in the proof of Lemma 4.3and these computations gives also ⇡(h) = inf k(f

i

)kk(gi

)k. (3) ) (2) isdirect, since (2) is just the factorization expression for (3).

5. Applications: p-convexity, p-concavity and strong

factorization theorems for the d⇤p,Z norms

5.1. Coincidence of the ⇡ norm and the d

⇤p,Z

norm

In this section we use the results of the previous ones to obtain strong factor-ization theorems for multiplication operators in X

Z

d

⇤p,Z

Y

Z

0. The key for our

results are the convexity properties of the spaces involved. As we shall show,under these requirements it is possible to improve the factorization for theelements of this product given by Theorem 4.4 in order to obtain a strongfactorization theorem through Z.

Theorem 5.1. Let 1 < p < 1. Let Z be a p

0-convex space and let Y

Z

0be

a p-convex space. Then X

Z

⇡Y

Z

0= X

Z

d

⇤p,Z

Y

Z

0. Consequently, if a function

h allows a decomposition as the one given in (3) of Theorem 4.4, then itfactorizes as

X

h -Y

0,

f

HHHHHHjZ

��*

g

i.e. it can be written as a single product of a function f 2 X

Z and a function


g 2 Z

Y

0. Moreover, in this case

⇡(h) M

(p0)(Z)M (p)(Y Z

0)d⇤

p,z

(h) M

(p0)(Z)M (p)(Y Z

0)⇡(h).

Proof. The inclusion X

Z

⇡Y

Z

0 ✓ X

Z

d

⇤p,Z

Y

Z

0always holds, as a consequence

of Lemma 4.3. On the other hand, if h 2 X

Z

d

⇤p,Z

Y

Z

0, for every couple

of sequences of functions (fi

) and (gi

) such that |h| P

i�1 |fi

g

i

|, wherew

⇤p

0,Z

((fi

)) < 1 and ⇡

p

((gi

)) < 1, we have

k(X

i�1

|fi

|p0)1/p

0k

X

Z · k(X

i�1

|gi

|p)1/pkY

Z0

= supx2BX

k(X

i�1

|fi

|p0)1/p

0xk

Z

· k(X

i�1

|gi

|p)1/pkY

Z0

M

(p0)(Z)M (p)(Y Z

0) sup

x2BX

�

X

i�1

kfi

xkp

0

Z

�1/p

0�

X

i�1

kgi

kp

Y

Z0

�1/p

.

Now, taking into account that

|h| X

i�1

|fi

g

i

| �

X

i�1

|fi

|p0�1/p

0�

X

i�1

|gi

|p�1/p

µ-a.e., we obtain

⇡(h) ⇡(h) M

(p0)(Z)M (p)(Y Z

0) sup

x2BX

�

X

i�1

kfi

xkp

0

Z

�1/p

0�

X

i�1

kgi

kp

Y

Z0

�1/p

.

This gives the result. ⇤

The canonical example of the situation described in Theorem 5.1 is givenwhen Z = L

p

0(µ); this is a consequence of the fact that (Lp

0(µ))0 = L

p(µ)and Y

Z

0is p-convex whenever Z

0 is p-convex. However, there are more casesfor other spaces for which this also hold, as we show in what follows. Recallthat the saturation requirement is always assumed for X

Z , Z

Y

0and X

Y

0; it

holds in all the examples that we explain.

Example 5.2. Clearly, the most interesting examples are found when theconvexity properties of the space Z

Y

0improve the ones of Y

0. Let us showsome of them.(1) Spaces of integrable functions with respect to a vector measure. Let

1 p < 1 and let m be a (countably additive) Banach space val-ued vector measure. Consider the class of spaces L

p(m) and L

p

w

(m) ofp-integrable and weakly p-integrable functions, respectively. Take anyBanach function space X and consider the case Y

0 = L

1(m); remarkthat this example is rather general, since every order continuous Ba-nach function space with a weak order unit can be written (order iso-metrically) as an L

1(m) of a vector measure m (see for instance [19,Prop. 3.9]). The Banach function spaces L

p(m) and L

p

w

(m) are alwaysp-convex; the reader can find more information on these spaces in [19,Ch.3]. Take p such that 1 < p p

0< 1 and Z = L

p

0

w

(m). Then


Z

Y

0= (Lp

0

w

(m))L

1(m) = L

p(m), that is p-convex (see for instance [4,Ex. 4.3], or [19, Prop. 3.43]). Therefore, Theorem 5.1 gives that everyfunction h of

X

L

p0w (m)

d

⇤p,L

p0w (m)

L

p

0

w

(m)L

1(m)= X

L

p0w (m)

d

⇤p,L

p0w (m)

L

p(m)

factorizes through L

p

0

w

(m) as h = fg, where f 2 X

L

p0w (m) and g 2

L

p

0

w

(m)L

1(m)= L

p(m). Similar examples can be found for p-th powersof Banach function spaces, see for instance [4, Prop. 4.2] or [19, 17].

(2) Orlicz spaces. The space of multiplication operators between Orliczspaces can be sometimes represented as other Orlicz space. For the fol-lowing particular choice of Young functions, this can be found in [4,Th.6.7]. Consider three Young functions �,�0,�1 satisfying:

(i) �(st) 12

�

�0(s) + �1(t)�

, for all s, t � 0, and

(ii) ��1(t) ��10 (t)��1

1 (t), for all t � 0.Then, (L�0)L

�= L

�1 . On the other hand, the following character-ization holds for p-convex Orlicz spaces defined on non-atomic measurespaces: An Orlicz space L

'

is p-convex (with p-convexity constant one)if and only if '(s1/p) is convex ([11, Th. 5.1.]). Using both results wecan construct the following example, that illustrates also Theorem 5.1.

Consider the Young functions �,�0,�1 satisfying the requirementsgiven above. Take a Banach function space X such that X

L

�0 is sat-urated, Z = L

�0 and Y such that Y

0 = L

� (for instance, if (L�)0 hasthe Fatou property, take Y = (L�)0). Then Z

Y

0= L

�1 . Suppose alsothat �1(s1/p) and �0(s1/p

0) are convex functions. Then an application

of Theorem 5.1 gives that every function h belonging to

X

L

�0d

⇤p,L

�0 (L�0L

�

) = X

L

�0d

⇤p,L

�0 L

�1

factorizes through an scheme

X

h -L

�,

f

HHHHHHjL

�0��*

g

where f 2 X

L

�0 and h 2 L

�1 . Note that this example generalizesthe case �1(s) = s

p and �0(s) = s

p

0, that gives the spaces Z = L

�0 =L

p

0, Y = L

1 and Z

Y

0= L

�1 = L

p.(3) Lorentz function spaces. Recall that given 1 p < 1, the Lorentz space

⇤p,w

is the subspace of the classes of functions f of L

0 that satisfy thatthe norm

kfkp,w

:=⇣

Z

I

f

⇤p(s)w(s)ds

⌘1/p


is finite, where I = (0, 1] or I = (0,1), and f

⇤ is the decreasing re-arrangement of f . It is known that these spaces are always p-convex withp-convexity constant 1 (see for instance [12, Th. 3] and the referencestherein). Consider the following case. Fix 1 < p < 1. Let Z = ⇤

p

0,w

for some function w. Take numbers 1 r, t < 1 satisfying that r

0t = p

and t p

0, and take Y

0 = L

t. Then it is known (see [4, Prop. 5.3] andthe comments that follow its proof), that since ⇤

p

0,w

is t-convex (recallt p

0), the equalities

(⇤p

0,w

)L

t

=⇣⇣

(⇤p

0,w

)1/t

⌘0⌘t

=⇣

(⇤p

0/t,w

)0⌘

t

hold, where the symbol X

q represents the q-th power of the space X, see[4, 17, 19] for more information. Assume also that ⇤

p

0/t,w

is r-concave(conditions for this to hold are given in [12, Th. 7]). Then (⇤

p

0/t,w

)0 isr

0-convex and so ((⇤p

0/t,w

)0)t is r

0t-convex, i.e. p-convex. Consequently,

if X is a Banach function space such that X

⇤p0,w is saturated, underall the requirements exposed we obtain by Theorem 5.1 that for everyfunction h belonging to

⇣

X

⇤p0,w

⌘

d

⇤p,⇤p0,w

⇣

(⇤p

0/t,w

)0⌘

t

there are functions f 2 X

⇤p0,w and g 2⇣

(⇤p

0/t,w

)0⌘

t

such that h = fg.A particular choice of parameters for which the example is meaningfulare given by p = p

0 = 2, t =p

2 and r

0 =p

2. For w being the constantfunction 1, this example gives also the case of L

p spaces, as in (2).

Let us finish this section by giving some applications related with theorder continuity of the product spaces. Some interesting results follow when ⇡

is equivalent to ⇡ (i.e. there exists a constant C > 0 such that ⇡(z) C ·⇡(z)for all z 2 X⇡Y ). For instance, this happens when there is a factorization forevery function f 2 X

Y

0as the one given by a strong factorization theorem

(see Introduction and [24]). The following result can be applied in this setting.A general factorization theorem also holds for the ⇡ norm, even if it is notequivalent to ⇡, as we have explained in Remark 4.5(b).

Proposition 5.3. Let X, Y be order continuous saturated Banach functionspaces such that X

Y

0is saturated and the norm ⇡ on X⇡Y is equivalent to

⇡. Then, X⇡Y is order continuous. Moreover, in this case, X⇡Y has theFatou property if and only if (XY

0)0 is order continuous.

Proof. First note that the hypothesis guarantees that X⇡Y is a saturatedBanach function space Given z, z

i

2 X⇡Y such that 0 z

i

" z µ–a.e., wehave to prove that ⇡(z � z

i

) ! 0 as i ! 1. Denote by A the support of z.Since ⇡ and ⇡ are equivalent, ⇡(z) < 1. So, there exist x 2 X and y 2 Y

such that z = |xy|. Taking x

i

:= ziz

|x|�A

and y

i

:= ziz

|y|�A

, we have that0 x

i

" |x|�A

µ–a.e. with x

i

, |x|�A

2 X and 0 y

i

" |y|�A

µ–a.e. withy

i

, |y|�A

2 Y . By the order continuity of X and Y , it follows that x

i

! |x|�A


in X and y

i

! |y|�A

in Y . Moreover, x

i

y

i

= z

2i

z

2 |xy|�A

= z

2iz

�

A

z

i

�

A

z

i

and so

0 z � z

i

|xy|� x

i

y

i

= (|xy|� x

i

|y|) + (xi

|y|� x

i

y

i

)= (|xy�

A

|� x

i

|y|) + (xi

|y|�A

� x

i

y

i

)= (|x|�

A

� x

i

)|y| + x

i

(|y|�A

� y

i

) (|x|�

A

� x

i

)|y| + |x|(|y|�A

� y

i

).

Then,

⇡(z � z

i

) k |x|�A

� x

i

kX

· kykY

+ kxkX

· k |y|�A

� y

i

kY

! 0

as i !1. So, X⇡Y is order continuous.Suppose that X⇡Y has the Fatou property. Then, from this hypothesis

we have that X⇡Y = (X⇡Y )00 = (XY

0)0 isometrically (the last equality is

a consequence of direct calculations taking into account [10, Prop. 3.3]). So,(XY

0)0 is order continuous, since we have seen that X⇡Y is so.Conversely, suppose that (XY

0)0 is order continuous. We always have the

continuous inclusion X⇡Y ✓ (X⇡Y )00 = (XY

0)0. Since X⇡Y is order contin-

uous, in particular, it is order semi–continuous. Hence, the above embeddingis an isometry. We end the proof only noting that for two Banach functionspaces E,F such that E is isometrically embedded in F , if F is order contin-uous and has the Fatou property, then E also has the Fatou property. Indeed,given (x

i

) ⇢ E such that 0 x

i

" x and supn

kxi

kE

< 1, by the Fatou prop-erty of F , it follows that x 2 F . Then, since F is order continuous, x

i

! x

in F and we can take a subsequence x

ni ! x µ–a.e. From the fact that E isisometrically embedded in F , we have that kx

ni�x

njkE

= kxni�x

njkF

! 0as i, j ! 1. Then, there exists y 2 E such that x

ni ! y in E. Taking asubsequence of (x

ni) converging µ–a.e. to y, we have that x = y 2 E. More-over, x

i

! x in E, since the norms of E and F coincide on E. In particular,kx

i

kE

" kxkE

. Hence, E has the Fatou property. ⇤The order continuity of the spaces X and Y is not a necessary condition

for X⇡Y to be order continuous: it is enough to consider the case X⇡L

1(µ) =X.

Corollary 5.4. Let Z be a p

0-convex space and let Y

Z

0be an order continuous

p-convex space such that all the elements of X

Y

0factorize as in Theorem 4.4.

Then X

Z

⇡Y

Z

0= X

Y

0, and if X

Z is order continuous, then X

Y

0is order

continuous.

The proof is a direct consequence of the equality X

Z

d

⇤p,Z

Y

Z

0= X

Y

0

given by Theorem 4.4 that is given by hypothesis, Theorem 5.1 and Propo-sition 5.3.

Remark 5.5. Clearly, by Lemma 4.3, under the hypothesis of each of thestrong factorization theorems given in the introduction (Lozanovskii, Reisnerand Maurey-Rosenthal), all the product spaces X

Z

⇡Y

Z

0, X

Z

d

⇤p,Z

Y

Z

0and

X

Z

"

Z

Y

Z

0coincide with X

Y

0.


5.2. Coincidence of the " norm and the d

⇤p,Z

norm

In this section we analyze the complementary case of strong factorizationtheorems for the d

⇤p,Z

norm. We provide conditions under which the spacesX

Z

d

⇤p,Z

Y

Z

0and X

Y

0coincide. This can happen even if there is no coincidence

with the space X

Z

⇡Z

Y

0. Let us show some examples of this situation.

Example 5.6. Banach function spaces satisfying X

Z

d

⇤p,Z

Y

Z

0= X

Y

0but

X

Z

d

⇤p,Z

Y

Z

06= X

Z

⇡Y

Z

0.

(1) Let us consider first a simple case: take X = `

2, Z = `

3 and Y

0 = `

1.We have that X

Z = (`2)`

3= `

1, Z

Y

0= (`3)`

1= `

3/2 and X

Y

0= (`2)`

1= `

2,and then

X

Z

⇡Z

Y

0= (`2)`

3⇡(`3)`

1= `

1⇡`

3/2 = `

3/2,

which do not coincide with X

Y

0= `

2. However, let us show that X

Z

d

⇤2,Z

Z

Y

0=

X

Y

0. Take an element (�

i

) 2 `

2 = (`2)`

1. Then it can be written as the prod-

uct (�i

) · (�i

) =P1

i=1 �

i

�

i

e

i

, where �

i

= 1 for every i and (�i

) = (�i

) ((ei

) isthe canonical basis). But notice that

w

⇤2,Z

((�i

e

i

)) = sup(⌧i)2B`2

(1X

i=1

k⌧i

e

i

k2`

3)1/2 = k(⌧i

)k`

2 = 1, and

⇡2((�i

e

i

)) = (1X

i=1

k�i

e

i

k2)1/2 = k(�i

)k`

2.

Therefore X

Z

d

⇤2,Z

Z

Y

0= (`2)`

3d

⇤2,`

3(`3)`

1= `

2 = X

Y

0.

(2) Consider a measure space (⌦,⌃, µ) and a measurable partition {Ai

}1i=1,

0 < µ(Ai

) for each i 2 N and call µ

i

to the restriction of µ to the set A

i

.Take three sequences of Banach function spaces {X

i

(µi

)}1i=1, {Zi

(µi

)}1i=1 and

{Y 0i

(µi

)}1i=1, i 2 N. Assume also that X

Zii

⇡Y

Z

0i

i

= X

Y

0i

i

isometrically for everyi.

Consider the Banach function spaces X(µ) = �p

0X

i

(µi

), Z(µ) = �q

Z

i

(µi

)and Y (µ) = �1Y

0i

(µi

), where 1 < p

0< q < 1. Note that for the case p = 2,

q = 3 and X

i

(µi

) = Z

i

(µi

) = Y

0(µi

) = R, that satisfy the requirementsabove, we obtain the example given in (1).

By the definition of the spaces, X

Y

0= (�

p

0X

i

(µi

))(�1Zi(µi)) = �p

X

Zii

.Fix h 2 X

Y

0; for every x 2 X we can consider the decompositions of h and x

as the pointwise sumsP1

i=1 h

i

andP1

i=1 x

i

, where h

i

= h�

Ai and x

i

= x�

Ai .By hypothesis for each 0 < " < 1 there is a couple of sequences of elements(f

i

)1i=1 and (g

i

)1i=1 with f

i

2 X

Zii

and g

i

2 Z

Y

0i

i

such that h

i

= f

i

g

i

, kfi

k = 1and (1� ")kg

i

k khi

k, i 2 N.We have that

w

⇤p

0,Z

((fi

)) = supx2BX

�

1X

i=1

kxi

f

i

kp

0

Zi

�1/p

0

supx2BX

�

1X

i=1

kxi

kp

0

Xi

�1/p

0

1,


and

(1� ")⇡p

((gi

)) =�

1X

i=1

kgi

kp(1� ")�1/p

�

1X

i=1

khi

kp

�1/p = khkX

Y 0 .

This proves the equality

X

Z

d

⇤p,Z

Y

Z

0=⇣

(�p

0X

i

(µi

))�qZi(µi)⌘

d

⇤p,Z

⇣

(�q

Z

i

(µi

))�1Y

0i (µi)

⌘

= �p

X

i

(µi

)Y

0i (µi) = (�

p

0X

i

(µi

))�1Y

0i (µi) = X

Y

0,

and, as we have shown in (1), this space do not coincide in general with the⇡-product.

In a sense, this example gives the rule for giving a criterion under whichthe inequality d

⇤p,Z

"

Z

on X

Z

d

⇤p,Z

Y

Z

0holds. In order to do that, we intro-

duce the following concavity type notion, that is weaker than the inequality⇡ "

Z

. We say that X

Y

0satisfies a lower (p, Z)-estimate if there is a constant

K > 0 such that for every function h in the space

inf⇣

1X

i=1

(kfi

kkgi

k)p

⌘1/p

Kkhk

where the infimum is computed over all decomposition of h as a sum ofdisjoint products f

i

g

i

of functions f

i

2 X

Z and g

i

2 Z

Y

0. Note that if

X

Z

⇡Y

Z

0= X

Y

0, this can be obtained as a consequence of the fact that X

Y

0

satisfies a lower p-estimate in the usual sense (see the definition in Section 2,see also [15, Section 1.f]). We say that d

⇤p,Z

(h) can be computed over disjointsums if it coincides with the infimum of w

⇤p

0,Z

((fi

))·⇡((gi

)) when only disjointsums of products f

i

g

i

satisfying |h| P

i�1 |fi

g

i

| are considered.

Theorem 5.7. Let 1 < p < 1 and let X be a Banach function space thatsatisfies a lower p

0-estimate. If X

Y

0satisfies a lower (p, Z)-estimate, then

X

Z

d

⇤p,Z

Y

Z

0= X

Y

0. Moreover, under the assumption that d

⇤p,Z

can be com-puted over disjoint sums the converse is also true.

Proof. Take " > 0. Consider a disjoint representation of h as disjoint productsof elements of X

Z · Y Z

0, h =

P1i=1 f

i

g

i

such that⇣

1X

i=1

(kfi

kkgi

k)p

⌘1/p

Kkfk(1 + ")

and write it as h =P1

i=1fi

kfikgi

kfi

k. Then, if for every x 2 X we write x

i

forx�

Ai , where A

i

is the support of each f

i

, by the lower p

0-estimate of X weobtain

supx2BX

⇣

1X

i=1

kx f

i

kfi

kkp

0⌘1/p

0

supx2BX

⇣

1X

i=1

kxi

kp

0⌘1/p

0

k,

and⇣

1X

i=1

(kkfi

kgi

k)p

⌘1/p

=⇣

1X

i=1

(kfi

kkgi

k)p

⌘1/p

khkK(1 + ").


Since this can be done for every ", we obtain that d

⇤p,Z

(h) kK"

Z

(h), thatimplies the result. A direct computation using the formula of the d

⇤p,Z

normgives the converse result. ⇤

Corollary 5.8. Let 1 < p < 1, let X satisfy a lower p

0-estimate and letX

Y

0satisfy a lower p-estimate. Assume also that there is a measurable par-

tition {Ai

}1i=1 of ⌦ such that the restriction spaces X

i

, Z

i

and Y

0i

satisfyX

Zii

⇡Z

Y

0i

i

= X

Y

0i

i

isometrically for every i 2 N. Then X

Z

d

⇤p,Z

Y

Z

0= X

Y

0.

Remark 5.9. In this paper only the d

⇤p,Z

factorization norm has been de-fined and studied. In the same way, it is possible to define the norms g

⇤p,Z

bycomputing the infimum over the products ⇡

p

0((fi

))w⇤p,Z

((gi

)) and m

⇤p,Z

, com-puting in this case the infimum over all the products w

⇤p

0,Z

((fi

))w⇤p,Z

((gi

)).Similar results to the ones that have been shown for the norm d

⇤p,Z

regardingequality between product spaces associated to the convexity properties of thespaces (Theorem 5.1) can be expected also in these cases.

Remark 5.10. For general operators on Banach function spaces, the Maurey-Rosenthal factorization theory provides factorization of operators throughmultiplication operators. Consequently, the results that have been obtainedhere gives additional information for the development of this theory and forfinding more applications in the theory of operators between Banach spaces.More information about this research program can be found for instance in[5, 6, 8, 9, 19].

The author wants to thank Prof. Olvido Delgado for her help during thepreparation of this paper.

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E. A. Sanchez PerezInstituto Universitario de Matematica Pura y AplicadaUniversidad Politecnica de ValenciaCamino de Vera s/n, 46022 Valencia. Spain.e-mail: [email protected]

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Factorization theorems for multiplication operators on Banach function spaces