FACTORING MULTI-SUBLINEAR MAPS

GEOFF DIESTEL

Abstract. Using Rademacher type, maximal estimates are es-tablished for k-sublinear operators with values in the space of mea-surable functions. Maurey-Nikishin factorization implies that suchoperators factor through a weak-type Lebesgue space. This ex-tends known results for sublinear operators and improves some re-sults for bilinear operators. For example, any continuous bilinearoperator from a product of type 2 spaces into the space of measur-able functions factors through a Banach space. Also included areapplications for multilinear translation invariant operators.

1. Introduction

Nikishin [23] showed that factorization through a weak-type Lebesguespace is equivalent to a particular maximal estimate as long as the un-derlying measure space is finite. Furthermore, using Rademacher type,Nikishin [23] showed that such an estimate holds for any sublinearoperator which is continuous into the space of measurable functions.This result may be applied to the case of a σ-finite measure to obtaina factorization via multiplication by a positive measurable functionand a change of density. Maurey [22] established an analogous charac-terization of factorization through strong-type Lebesgue spaces. Later,Pisier [27] proved a result intermediate to the those of Nikishin [23] andMaurey [22]. We will refer to these results as Maurey-Nikishin factor-ization. All of these characterizations equate vector-valued estimateswith strong factorization, i.e. factorization given by a multiplicationoperator and a change of density.

In Section 1.1, we formulate our main result Theorem 1.1. Thereduction to the finite measure case appears in Section 1.2 with someclarifying remarks about choosing an appropriate change of density.

Section 2 contains the proof of Theorem 1.1 by proving the corre-sponding k-sublinear maximal estimate, i.e. by proving Theorem 2.1.The analogous maximal estimate and corresponding factorization for

2010 Mathematics Subject Classification. Primary 46A16. Secondary 42B05.Key words and phrases. factorization, type, cotype, bilinear operators, multilin-

ear operators, Fourier series.1

2 GEOFF DIESTEL

Lebesgue space valued operators appears in Section 2.1. These followfrom similar arguments but use Pisier’s [27] analog of Nikishin’s [23]characterization of strong factorization.

In Section 3, Theorem 1.1 is combined with general results for bi-linear operators appearing in [13] which identify additional geometricproperties of the range of a bilinear operator. In particular, a bilinearoperator from a product of type 2 spaces into the space of measurablefunctions must factor through a Banach space.

Section 4 contains applications to general translation invariant op-erators and convergence of multilinear Fourier series which generalizesome results due to Stein [31]. Moreover, we borrow arguments fromthe proof of Maurey’s [22] characterization of strong factorization togive a topological proof that non-trivial k-linear operators which com-mute with translations cannot map into certain Lebesgue spaces whenthe underlying group has infinite Haar measure. This provides a partialextension of a result due to Hormander [8] for linear operators.

Section 5 contains closing remarks and open problems.

1.1. Preliminaries and Main Result. For each nonnegative integerj, let εj represent the j-th Rademacher function defined for t ∈ [0, 1].For 0 < p ≤ 2, a Banach space X has type p if there is a constantTp(X) <∞ such that

(1.1)(E∥∥∥∑

j

εjxj

∥∥∥2)1/2 ≤ Tp(X)(∑

j

‖xj‖p)1/p

for all n and x1, · · · , xn ∈ X. On the other hand, for 2 ≤ q, a Banachspace X has cotype q if there is a constant Cq(X) <∞ such that

(1.2)(∑

j

‖xj‖q)1/q≤ Cq(X)

(E∥∥∥∑

j

εjxj

∥∥∥2)1/2for all n and x1, · · · , xn ∈ X.

For a topological vector space X, a function ‖ · ‖ : X → [0,∞) is aquasi-norm if

(1) For all x ∈ X, ‖x‖ ≥ 0 and x = 0 if and only if ‖x‖ = 0.(2) For any scaler c and all x ∈ X, ‖cx‖ = |c|‖x‖.(3) There is a constant C ≥ 1 so that ‖x+ y‖ ≤ C(‖x‖+ ‖y‖) for

all x, y ∈ X.

Any space X which is complete with respect to a quasi-norm is a quasi-Banach space. The unit ball of a quasi-Banach space X will be denotedby BX . For a quasi-Banach space X, there is always some 0 < p ≤ 1and an equivalent p-norm, i.e. we may assume ‖x+ y‖p ≤ ‖x‖p + ‖y‖pfor all x, y ∈ X. In other words, any quasi-Banach space is p-normable

FACTORING MULTI-SUBLINEAR MAPS 3

for some p > 0. Thus, the notions of type and cotype naturally ex-tend to quasi-Banach spaces. In particular, every quasi-Banach spacehas type. Thus, Maurey-Nikishin factorization applies to operators de-fined on quasi-Banach spaces. Of course, we are using the conventionthat Banach spaces are 1-normable quasi-Banach spaces. For a generaldevelopment of quasi-Banach spaces, refer to Chapter 1 of [16].

Throughout this article, (Ω,Σ, µ) will be a non-atomic σ-finite mea-sure space and f ∈ L0(dµ) if and only if |f(ω)| < ∞ for almost allω ∈ Ω. Of course, f = g in L0 if f(ω) = g(ω) for almost all ω ∈ Ω.We will write L0 for L0(dµ) and only make note of the measure if thereis a change of density. The topology of L0 is defined by convergencein measure and one may always define a probability measure ν anda positive measurable function φ so that f 7→ φf is an isomorphismbetween L0 and L0(dν). Thus, L0 is an F-space.

For 0 < p <∞, f ∈ Lp if

(1.3) ‖f‖pp = p

∫ ∞0

αp−1µ(|f | > α) dα =

∫|f |p dµ <∞

and f ∈ Lp,∞ if ‖f‖p,∞ = supα>0 α · µ(|f | > α)1/p < ∞. Fur-thermore, for any q ∈ (0, p), there exists a constant C(p, q) < ∞ sothat(1.4)

1C(p,q)

‖f‖p,∞ ≤ sup0<µ(E)<∞

µ(E)1/p−1/q(∫

E

|f |q dµ)1/q≤ C(p, q)‖f‖p,∞.

Estimate (1.4) is outlined as a simple exercise in Chapter 1.4 of [5].Hence, Lp,∞ is q-normable for all q ∈ (0, p). In particular, Lp,∞ is aBanach space only if p > 1.

An operator T : X1 × · · · ×Xk → L0 is k-linear(resp. k-sublinear) iffor each 1 ≤ i ≤ k and every choice of xj ∈ Xj, j 6= i, the coordinatemap xi 7→ T (x1, · · · , xk) is linear(resp. sublinear).

A k-sublinear operator T : X1 × · · · × Xk → L0 is continuous at(x1, · · · , xk) if limn |T (xn,1, · · · , xn,k)−T (x1, · · · , xk)| = 0 µ-a.e. when-ever limn xn,i = xi in Xi for each 1 ≤ i ≤ k. If xi = 0 for every1 ≤ i ≤ k, we say T is continuous at zero.

Definition 1.1. For 0 ≤ q < p, a k-sublinear operator T : X1 × · · · ×Xk → Lq strongly factors through Lp,∞ if there is a nonnegative f ∈ Lqsuch that µ

(f = 0 ∩ |T (x1, · · · , xk)| > 0

)= 0 and

‖f−1T (x1, · · · , xk)‖Lp,∞(fqdµ)≤ ‖x1‖ · · · ‖xk‖

for all xi ∈ Xi, 1 ≤ i ≤ k. Strong factorization through Lp is definedanalogously.

4 GEOFF DIESTEL

Notice that the strong factorization of L0-valued operators comeswithout a change of density. Because of this, and because it is some-what standard to do so, we will distinguish between the cases q = 0and q > 0 by viewing factorization in the following equivalent manner.If q = 0, we will write

‖gT (x1, · · · , xk)‖p,∞ ≤ ‖x1‖ · · · ‖xk‖

where g replaces f−1 in Definition 1.1. If q > 0, we write

‖h−1/qT (x1, · · · , xk)‖Lp,∞(dν)≤ ‖x1‖ · · · ‖xk‖

where dν = hdµ and h−1/q replaces f−1 in Definition 1.1. Moreover, ifq > 0, we will state that T strongly factors via a change of density.

Theorem 1.1. Let 0 < p1, · · · , pk ≤ 2, 1/p = 1/p1 + · · · + 1/pk, andXi have type pi for each 1 ≤ i ≤ k. Suppose T : X1 × · · · ×Xk → L0

is a k-sublinear operator which is continuous at zero. Then T stronglyfactors through Lp,∞.

For k = 1 and µ a finite measure, Theorem 1.1 is due to Nikishin [23].It is often pointed out that Nikishin’s result extends to the σ-finite casewithout loss of generality. However, in the σ-finite case, one appliesNikishin’s result to the map x 7→ φT (x) from X into L0(dν) whereν is a probability measure and φ is a positive measurable function sothat f 7→ φf is an isomorphism between L0 and L0(dν). Thus, thereexists g ∈ L0(dν) so that ‖gφT (x)‖Lp,∞(dν) ≤ ‖x‖ for all x ∈ X. Noticethat such an estimate does not automatically imply that x 7→ gT (x)is continuous into Lp,∞. Thus, one could argue that perhaps somegenerality is lost. In the remainder of this section, it will be shownthat a particular isomorphism f 7→ φf between L0 and L0(dν) willalways allow one to conclude that x 7→ gT (x) is continuous into Lp,∞.

Since we are considering operators and spaces with homogeneity, thefollowing proposition is a simple but useful characterization of conti-nuity at zero.

Proposition 1.1. Suppose T : X1×· · ·×Xk → L0 is a k-sublinear op-erator. Then T is continuous at zero if and only if limn T (xn,1, · · · , xn,k) =0 µ-a.e. whenever limn(‖xn,1‖ · · · ‖xn,k‖) = 0.

Proof. Of course we need only show the necessity. Suppose T is con-tinuous at zero and limn(‖x1,n‖ · · · ‖xk,n‖) = 0. We may assume that‖xn,i‖ 6= 0 for all n and i because T is homogeneous.

If we define x′i,n = (‖x1,n‖ · · · ‖xk,n‖)1/kxi,n/‖xi,n‖ for each n andi, limn ‖x′i,n‖ = 0 for every i. Thus, the homogeneity of T and the

FACTORING MULTI-SUBLINEAR MAPS 5

continuity of T at zero imply

limnT (x1,n, · · · , xk,n) = lim

nT (x′1,n, · · · , x′k,n) = 0

µ-a.e.

1.2. Reduction to the finite measure case. Lemma 1.1 show’s thata particular change of density allows for Nikishin’s [23] factorization tocome without a change of density in the σ-finite case.

Lemma 1.1. Let 0 < q < p <∞, ‖h‖1 = 1, and 0 < h ≤ 1. Define νby dν = hdµ. Then there is a constant C = C(p, q) <∞ such that

‖h1/qf‖p,∞ ≤ C‖f‖Lp,∞(dν)

for all f ∈ Lp,∞(dν).

Proof.

‖h1/qf‖p,∞ ≈p,q supµ(E)<∞

µ(E)1/p−1/q(∫

E

|f |q dν)1/q

, (1.4)

≤ sup0<ν(E)<∞

ν(E)1/p−1/q(∫

E

|f |q dν)1/q

, 0 < h ≤ 1

≈p,q ‖f‖Lp,∞(dν), (1.4)

Now, assume Theorem 1.1 holds in the finite measure case. SupposeT : X1 × · · · ×Xk → L0 satisfies the hypotheses of Theorem 1.1 withµ a σ-finite measure. Let 0 < q < p and define h and ν as in Lemma1.1. Of course (x1, · · · , xk) 7→ h−1/qT (x1, · · · , xk) is continuous at zeroas a k-sublinear operator from X1 × · · · ×Xk into L0(dν).

Since we are assuming Theorem 1.1 holds for finite measures, thereis a positive ν-measurable g so that

(1.5) ‖gh−1/qT (x1, · · · , xk)‖Lp,∞(dν)≤ ‖x1‖ · · · ‖xk‖

for all xi ∈ Xi, 1 ≤ i ≤ k. Since ν and µ are defined on the sameσ-algebra, g is µ-measurable.

By Lemma 1.1, there is a constant C(p, q) <∞ so that

‖gT (x1, · · · , xk)‖p,∞ ≤ C(p, q)‖x1‖ · · · ‖xk‖

for all xi ∈ Xi, 1 ≤ i ≤ k. Of course our choice of q depends on p andwe may absorb the constant into g as g is just a positive measurablefunction.

6 GEOFF DIESTEL

2. The Maximal Estimates and Proof of Theorem 1.1

As in the sublinear case, we use the assumption of a finite measureto gain access to homogeneous distributional estimates which are moreuseful than estimates related to the nonhomogeneous metric of L0.

Proposition 2.1. Let T : X1×· · ·×Xk → L0 be a k-sublinear operatorwith µ a finite measure. Then the following are equivalent.

(1) T is continuous at zero.(2) T (BX1 , · · · , BXk) is a bounded set in L0.(3) There is a function C(α) satisfying limα→∞C(α) = 0 and

µ(|T (x1, · · · , xk)| > α‖x1‖ · · · ‖xk‖) ≤ C(α)

for all xi ∈ Xi.

Proposition 2.1 is a simple k-sublinear extension from the presenta-tion of Nikishin’s [23] results appearing in [32]. Proving (1) ⇒ (2) ⇒(3) follows from the homogeneity of k-sublinear operators, Proposi-tion 1.1, and the fact that bounded sets in L0 can be absorbed byany neighborhood of zero. The final implication (3) ⇒ (1) is a trivialconsequence of Proposition 1.1.

Nikishin [23] showed that conditions (N1) and (N2) are equivalentfor a set of measurable functions F when µ is a finite measure.

(N1) There is a positive g ∈ L0 such that

supf∈F‖gf‖p,∞ <∞.

(N2) There exists a function C(α) satisfying limα→∞C(α) = 0 and

µ(supi|cifi| > α) ≤ C(α)

for all n, f1, · · · , fn ∈ F , and scalars c1, · · · , cn satisfying∑

i |ci|p ≤1.

So, if µ is a finite measure and T : X1 × · · · × Xk → L0 is a k-sublinear operator then T strongly factors through Lp,∞ if and only ifthere exists C(α) satisfying limα→∞C(α) = 0 and

(2.6) µ( sup1≤j≤n

|T (xj,1, · · · , xj,k)| > α) ≤ C(α)

for all n, i, and (xj,i)1≤j≤n chosen from Xi to satisfy

n∑j=1

(‖xj,1‖ · · · ‖xj,k‖)p ≤ 1.

Thus, Theorem 1.1 follows from Theorem 2.1.

FACTORING MULTI-SUBLINEAR MAPS 7

Theorem 2.1. Let 0 < p1, · · · , pk ≤ 2, 1/p = 1/p1+· · ·+1/pk, Xi havetype pi for each 1 ≤ i ≤ k, and µ(Ω) <∞. Suppose T : X1×· · ·×Xk →L0 is a k-sublinear operator which is continuous at zero. Then there isa function Cp1,··· ,pk(α) satisfying limα→∞Cp1,··· ,pk(α) = 0 and

µ(supj|T (xj,1, · · · , xj,k)| > α) ≤ Cp1,··· ,pk(α)

for all finite sequences (xj,i)j from Xi satisfying(∑j

(‖xj,1‖ · · · ‖xj,k‖

)p)1/p ≤ 1.

Lemma 2.1 is an adaptation of Nikishin’s arguments as they are pre-sented in [32]. We present the proof of Lemma 2.1 in order to note thespecific distributional estimate to prove the analogous multi-sublinearestimate in Theorem 2.1 by a boot-strapping and homogeneity argu-ment.

Lemma 2.1. Let 0 < p ≤ 2, X be a quasi-Banach space of type p andassume µ is a probability measure. Let T : X → L0 be a sublinear oper-ator continuous at zero. Then there exists a function Cp(α) satisfyinglimα→∞Cp(α) = 0 and

µ(

supj|T (xj)| > α

)≤ 2Tp(X)2α−1 + 2C(

√α).

for all finite sequences (xj)j from X satisfying∑

j ‖xj‖p ≤ 1.

Proof. Proposition 2.1 provides a function C(α) satisfying limα→∞C(α) =0 and µ(|T (x)| > α‖x‖) ≤ C(α) for all x ∈ X.

Fix a finite sequence (xj)j from X with∑

j ‖xj‖p = 1. For each

t ∈ [0, 1] and each k, set gt =∑

j εj(t)xj and gt,k =∑

j δjεj(t)xj whereδk = 1 and δj = −1 for all j 6= k.

Sublinearity implies 2|T (xk)| = |T (gt + gt,k)| ≤ |T (gt)| + |T (gt,k)|.Since t 7→ |T (gt)(ω)| and t 7→ |T (gt,k)(ω)| have the same distribu-tion function for µ-almost all ω ∈ Ω,

∣∣t ∈ [0, 1] : |T (gt)(ω)| ≥|T (gt,k)(ω)|

∣∣ ≥ 1/2 where |A| denotes the Lebesgue measure of a setA.

Since |T (xk)(ω)| does not depend on t, for µ-almost all ω ∈ Ω∣∣t ∈ [0, 1] : |T (gt)(ω)| ≥ maxk|T (xk)(ω)|∣∣ ≥ 1/2.

For every α > 0, µ(

supk |T (xk)| > α)

is less than or equal to

(2.7) 2

∫supk |T (xk)|>α

∣∣t ∈ [0, 1] : |T (gt)| ≥ supk|T (xk)|

∣∣dµ.

8 GEOFF DIESTEL

Since X has type p and µ(|T (x)| > α‖x‖) ≤ C(α) for all x ∈ Xand all α > 0, we can conclude with the following estimate.

µ(

supk|T (xk)| > α

)≤ 2

∫ ∣∣t ∈ [0, 1] : |T (gt)| ≥ α∣∣dµ, (2.7)

≤ 2∣∣t ∈ [0, 1] : ‖gt‖ ≥

√α∣∣+

+2

∫ 1

0

µ(|T (gt)| ≥

√α‖gt‖

)dt

≤ 2Tp(X)2α−1 + 2C(√α)

Now, we prove Theorem 2.1.

Proof. We will assume X1 = X and X2 = Y and prove the 2-sublinearcase for notational simplicity. Only minor adaptations of the argumentsbelow are needed for the general k-sublinear case.

Proposition 2.1 implies there is a C(α) such that limα→∞C(α) = 0and µ(|T (x, y)| > α‖x‖‖y‖) ≤ C(α) for all x ∈ X and y ∈ Y .

Fix y ∈ Y with ‖y‖ = 1. Then the map x 7→ T (x, y) is a sublinearoperator from X into L0 that is continuous at zero. Since Tp1(X) <∞and the function C(α) is independent of y, Lemma 2.1 implies

µ(

supj|T (xj, y)| > α

(∑j

‖xj‖p1)1/p1)

≤ 2Tp1(X)2α−1 + 2C(√α)

for all finite sequences (xj)j from X. Moreover, this estimate is inde-pendent of y so we can fix (xj)j with

∑j ‖xj‖p1 = 1 and consider the

sublinear map y 7→ supj |T (xj, y)| which is continuous at zero. Againby Lemma 2.1,

µ(

supj,k|T (xj, yk)| > α

(∑k

‖yk‖p2)1/p2)

≤ Cp1,p2(α)

where Cp1,p2(α) = 2Tp2(Y )2α−1 + 2(2Tp1(X)2α−1/2 + 2C(α1/4)).Since Cp1,p2(α) is independent of the choices of xj’s and yk’s,

µ(

supj|T (x′j, y

′j)| > α

(∑j

‖x′j‖p1)1/p1(∑

j

‖y′j‖p2)1/p2)

≤ Cp1,p2(α)

for all finite sequences (x′j)j from X and (y′j)j from Y .Let (xj)j and (yj)j be finite sequences fromX and Y respectively. We

may assume xj 6= 0 if and only if yj 6= 0 because T (x, 0) = T (0, y) = 0.Let

x′j =‖yj‖p/p1‖xj‖1−p/p1

xj and y′j =‖xj‖p/p2‖yj‖1−p/p2

yj.

FACTORING MULTI-SUBLINEAR MAPS 9

Then the homogeneity of T and the condition 1/p1 + 1/p2 = 1/p implythat

supj|T (x′j, y

′j)| = sup

j|T (xj, yj)|

and (∑j

‖x′j‖p1)1/p1(∑

j

‖y′j‖p2)1/p2

=(∑

j

‖xj‖p‖yj‖p)1/p

.

Therefore,

µ(

supj|T (xj, yj)| > α

(∑j

(‖xj‖‖yj‖

)p)1/p) ≤ Cp1,p2(α)

and the proof is complete by the equivalence of conditions (N1) and(N2).

2.1. Operators Mapping into Lq. Suppose 0 < q < p ≤ 2 andsuppose T : X → Lq is a continuous linear operator where Tp(X) <∞.Recall that one may obtain the appropriate maximal estimate by usinglinearity and the Kahane-Khintchine inequalities to prove the strongersquare-function estimate, see [22], [1], [32]. For k-linear operators, thiscan be done in the same way by proving a multi-parameter square-function estimate and by using k independent sequences of Rademacherfunctions. Then since maximal functions are controlled by square-functions, one can use homogeneity and the condition 1/p = 1/p1 +· · ·+ 1/pk to get an estimate of the form

‖ supj|T (x1,j, · · · , xk,j)|‖q ≤ C

(∑j

(‖x1,j‖ · · · ‖xk,j‖

)p)1/p.

Since 0 < q < p, Pisier’s [27] Lq version of the equivalence of (N1)and (N2) implies that T : X → Lq strongly factors through Lp,∞ viaa change of density. If k = 1 and p1 = 2, the square function estimateand Maurey’s [22] theorem implies that T strongly factors through L2

via a change of density.To deal with multi-sublinear operators, the maximal estimate fol-

lows by adapting the arguments from the case of L0-valued operators.Again, the k-sublinear maximal estimate will follow from the the sub-linear case by boot-strapping.

Lemma 2.2. Let 0 < q < p ≤ 2 and X have type p. Suppose T :X → Lq is a continuous sublinear operator. Then there is a constant

10 GEOFF DIESTEL

C = C(p, q,X) <∞ such that∥∥ supj|T (xj)|

∥∥q≤ C

(∑j

‖xj‖p)1/p

for all finite sequences (xj)j from X.

Proof. We may assume that µ is a finite measure since ‖f‖q = ‖h−1/qf‖Lq(hdµ)for any positive h ∈ L1 with ‖h‖1 = 1.

Fix a finite sequence (xj)j from X. Using gt and gt,k as defined inthe proof of Lemma 2.1, the following estimate holds.∥∥ sup

k|T (xk)|

∥∥qq

= q

∫ ∞0

αq−1µ(supk|T (xk)| ≥ α) dα (1.3)

≤ 2q

∫ ∞0

αq−1∫ ∣∣t ∈ [0, 1] : |T (gt)(ω)| ≥ α

∣∣ dµ(ω) dα

= 2q

∫ 1

0

∫ ∫ |(T (gt(ω))|0

αq−1 dα dµ(ω) dt (Fubini’s Thm.)

= 2

∫ 1

0

‖T (gt)‖qq dt

≤ 2‖T‖qTp(X)q(∑

j

‖xj‖p)q/p

, (q < 2).

Notice that Lemma 2.2 holds for q ∈ [p, 2]. However, such values ofq have no implications with regard to strong factorization.

Theorem 2.2 follows from Lemma 2.2 in the same manner that The-orem 2.1 followed from Lemma 2.1, i.e. by boot-strapping and thehomogeneity condition 1/p1 + · · ·+ 1/pk = 1/p.

Theorem 2.2. Let 0 < q, p1, · · · , pk ≤ 2, 1/q > 1/p = 1/p1+· · ·+1/pk,and X1, · · · , Xk be quasi-Banach spaces with Xi having type pi. SupposeT : X1 × · · · × Xk → Lq is a continuous k-sublinear operator. Thenthere is a constant C = C(q, p, p1, · · · , pk) <∞ such that

‖ supj|T (xj,1, · · · , xj,k)|‖q ≤ C

(∑j

(‖xj,1‖ · · · ‖xj,k‖

)p)1/pfor all finite sequences (xj,i)j from Xi, 1 ≤ i ≤ k.

Corollary 2.1 is an immediate consequence of Theorem 2.2 and Pisier’s[27] characterization of strong-factorization.

FACTORING MULTI-SUBLINEAR MAPS 11

Corollary 2.1. Let 0 < q, p1, · · · , pk ≤ 2, 1/q > 1/p = 1/p1 + · · · +1/pk, and X1, · · · , Xk be quasi-Banach spaces with Xi having type pi.Suppose T : X1 × · · · ×Xk → Lq is a continuous k-sublinear operator.Then T strongly factors through Lp,∞ via a change of density.

3. Bilinear Maps and Decoupling

The results in [13] were motivated by the following. If X, Y, and Zare quasi-Banach spaces and X⊗Y is the projective tensor product ofX and Y , then a bilinear operator T : X × Y → Z induces a linearmap T ′ : X⊗Y → Z if and only if the convex hull of T (BX , BY ) isbounded, i.e. there is a constant C <∞ such that∥∥∥∑

j

T (xj, yj)∥∥∥ ≤ C

∑j

‖xj‖‖yj‖.

In other words, there is a Banach space Z0, a bilinear map B : X×Y →Z0 and a linear operator L : Z0 → Z such that T = LB.

As discussed in the previous section, the k-linear case of Corollary 2.1can be proven by adapting the arguments in the linear case. If one con-siders operators with values in a quasi-Banach space, mimicking thesearguments amounts to the decoupling of multiparameter Rademacheraverages, see [24], [25], [26], [20], and [21]. In [13], Kalton uses thesetechniques to prove some general factorization results for bilinear op-erators with values in spaces with the decoupling property. For theresults in [13], the main consequence of decoupling is the following.Let T : X × Y → Z be a continuous bilinear map where Z has thedecoupling property. Then there exists a constant C <∞ so that

(3.8)∥∥∥∑

j

T (xj, yj)∥∥∥ ≤ CE

∥∥∥∑j

∑k

εjεkT (xj, yk)∥∥∥

for all finite sequences (xj)j from X and (yj)j from Y . As explained in[13], every quasi-Banach Lattice with nontrivial cotype has the decou-pling property. This is because such spaces satisfy Pisier’s [28] property(α), which Kalton [13] shows implies decoupling.

First, Kalton [13] proves a Maurey-Nikishin style theorem by adapt-ing arguments in [12]. Theorem 3.1 is a simple generalization of thisresult as only the case where both X and Y have type 2 is given in[13].

Theorem 3.1 (Theorem 6.1 in [13]). Let 0 < p1, p2 ≤ 2, 1/p =1/p1 + 1/p2, and X and Y be quasi-Banach spaces with type p1 andp2 respectively. If Z be a quasi-Banach space with the decoupling prop-erty and T : X × Y → Z is a continuous bilinear operator then there

12 GEOFF DIESTEL

is a constant C <∞ such that

(3.9)∥∥∥∑

j

T (xj, yj)∥∥∥ ≤ C

(∑j

(‖xj‖‖yj‖

)p)1/pfor all finite sequences (xj)j from X and (yj)j from Y .

Proof.∥∥∥∑j

T (xj, yj)∥∥∥ ≤ CE

∥∥∥∑j

∑k

εjεkT (xj, yk)∥∥∥, (3.8)

≤ C‖T‖E∥∥∥∑

j

εjxj

∥∥∥∥∥∥∑k

εkyk

∥∥∥, (bilinearity)

≤ C‖T‖(E∥∥∥∑

j

εjxj

∥∥∥2)1/2(E∥∥∥∑k

εkyk

∥∥∥2)1/2≤ C(p1, p2)‖T‖Tp1(X)Tp2(Y )

(∑j

‖xj‖p1)1/p1(∑

j

‖yj‖p2)1/p2

As we have seen, the homogeneity of T and the condition 1/p = 1/p1 +1/p2 imply the desired estimate.

Generally, Theorem 3.1 states that the operator T factors throughp-normable space, i.e. there exists a p-normable space Z ′, a bilinearoperator B : X × Y → Z ′ and a linear operator L : Z ′ → Z so thatT = LB. In other words, (3.9) states that the range of T (normalized),equipped with the quasi-norm of Z is a p-normable space. This ispointed out in [13] for the case p = 1, i.e. when factorization is througha Banach space. Combining Theorem 3.1 with Theorem 1.1 we havethe following.

Theorem 3.2. Let 0 < p1, p2 ≤ 2, 1/p = 1/p1 + 1/p2, and X andY be quasi-Banach spaces with type p1 and p2 respectively. SupposeT : X × Y → L0 is a bilinear operator which is continuous at zero.Then for each q ∈ (0, p), there exists a positive measurable g and aconstant C = C(p, q) <∞ so that∥∥∥∑

j

gT (xj, yj)∥∥∥q≤ C

(∑j

(‖xj‖‖yj‖

)p)1/p.

for all finite sequences (xj)j from X and (yj)j from Y .

Proof. For any q ∈ (0, p), we choose a density h satisfying the hy-potheses of Lemma 1.1 and apply Theorem 1.1 to the map (x, y) 7→h−1/qT (x, y) which is continuous at zero as a map from X × Y into

FACTORING MULTI-SUBLINEAR MAPS 13

L0(hdµ). Since hdµ defines a probability measure, there is a constantC(p, q) <∞ so that

‖gT (x, y)‖q = ‖gh−1/qT (x, y)‖Lq(hdµ) ≤ C(p, q)‖x‖‖y‖

because ‖gh−1/qT (x, y)‖Lq(hdµ) ≤ C(p, q)‖gh−1/qT (x, y)‖Lp,∞(hdµ). SinceLq has the decoupling property, Theorem 3.1 implies there exists a con-stant C ′(p, q) <∞ so that∥∥∥∑

j

gT (xj, yj)∥∥∥q≤ C ′(p, q)

(∑j

(‖xj‖‖yj‖

)p)1/p.

for all finite sequences (xj)j from X and (yj)j from Y .

Corollary 3.1. Let X and Y have type 2 and suppose T : X×Y → L0

is a continuous bilinear operator. Then T factors through a Banachspace.

Proof. Fix 0 < q < 1. By Theorem 3.2 there exists a positive measur-able function g and a constant C <∞ so that∥∥∥∑

j

gT (xj, yj)∥∥∥q≤ C

∑j

‖xj‖‖yj‖

for all finite sequences (xj)j from X and (yj)j from Y . Therefore,T factors through a Banach space because (x, y) 7→ gT (x, y) factorsthrough a Banach space.

Next, Kalton [13] asks whether an analog of Pisier’s [27] Grothendiecktheorem holds, i.e. can the type 2 assumptions on X and Y be replacedwith the assumption that X and Y are quasi-Banach spaces with thebounded approximation property such that X∗ and Y ∗ have cotype 2.Theorem 6.2 from [13] is a partial result in this direction and states thatif, in addition to these assumptions, Z has cotype 2 and the decouplingproperty then

E∥∥∥∑

j

εjT (xj, yj)∥∥∥ ≤ C

∑j

‖xj‖‖yj‖

for all finite sequences (xj)j from X and (yj)j from Y . If Z = Lq forsome 0 < q < 1, then combining this result with Pisier’s [27] theorem,Theorem 6.3 from [13] establishes that T strongly factors through L1,∞via a change of density. Combining this with Theorem 1.1 we have thefollowing.

Theorem 3.3. Let X, Y be quasi-Banach spaces with the bounded ap-proximation property such that X∗ and Y ∗ have cotype 2. Suppose

14 GEOFF DIESTEL

T : X × Y → L0 is a bilinear operator which is continuous at zero.Then T factors through a weak-L1 space.

Proof. Since X, Y are quasi-Banach spaces there are 0 < p1, p2 ≤ 2such that X has type p1 and Y has type p2. Therefore, just as in theproof of Theorem 3.2, for any q < p1p2/(p1 + p2) there is a positivemeasurable g and a constant C(p1, p2, q) <∞ so that

‖gT (x, y)‖q ≤ C(p1, p2, q)‖x‖‖y‖for all x ∈ X and y ∈ Y . By Theorem 6.3 from [13], there exists anessentially positive density h so that

‖gh−1/qT (x, y)‖L1,∞(hdµ)

≤ C ′(p1, p2, q)‖x‖‖y‖

for all x ∈ X and y ∈ Y . Thus, T factors through a weak-L1 space.

4. Translation invariant operators

Suppose µ is the unique Haar measure for a locally compact abeliangroup G. Moreover, assume (Ω,Σ, µ) is σ-finite. For s ∈ G and f ∈ L0,let τs(f)(t) = f(t− s) define the translation operator τs.

Definition 4.1. A k-sublinear operator T : X1 × · · · × Xk → L0 istranslation invariant if for each 1 ≤ i ≤ k there is a family of uniformlybounded isomorphisms (Is,i)s∈G so that for all s ∈ G and all xi ∈ Xi,

τsT (x1, · · · , xk) = T(Is,1(x1), · · · , Is,k(xk)

).

In the case Xi = Lpi(dµ) for each 1 ≤ i ≤ k, we say T commutes withtranslations if

τsT (f1, · · · , fk) = T (τsf1, · · · , τsfk).

Theorem 4.1. Let 0 < p, p1, · · · , pk ≤ 2, 1/p = 1/p1 + · · ·+ 1/pk, andlet Xi have type pi for each 1 ≤ i ≤ k. Suppose T : X1×· · ·×Xk → L0

is a k-sublinear translation invariant operator which is continuous atzero. Then there is a positive measurable function g and a constantC <∞ such that

sups∈G‖τs(g)T (x1, · · · , xk)‖p,∞ ≤ C‖x1‖ · · · ‖xk‖

If G is compact, T : X1×· · ·×Xk → Lp,∞ is continuous. Furthermore,if p1 = 2 and T is linear then T : X1 → L2 is continuous.

Proof. By Theorem 1.1 there is a positive measurable g such that

‖gT (x1, · · · , xk)‖p,∞ ≤ ‖x1‖ · · · ‖xk‖.

FACTORING MULTI-SUBLINEAR MAPS 15

Since T and µ are translation invariant,

sups∈G‖τs(g)T (x1, · · · , xk)‖p,∞ ≤ C‖x1‖ · · · ‖xk‖

where C = sups∈G sup1≤i≤k ‖I−s,i‖ <∞.If G is compact, assume µ is a probability measure. Pick λ > 0 so

that the set E = g > λ satisfies µ(E) ≥ 1/2. Since µ is a translationinvariant measure and T a translation invariant operator, for all s ∈ Gwe have

µ((E + s) ∩ |T (x1, · · · , xk) > α) ≤ λp/αp

for all xi ∈ Xi satisfying ‖x1‖ · · · ‖xk‖ ≤ 1. Since µ(E) ≥ 1/2 and µ isa probability measure, integrating in s implies that

µ(|T (x1, · · · , xk) > α) ≤ 2λp/αp.

Therefore, T : X1×· · ·×Xk → Lp,∞ is continuous. Using Maurey’s [22]theorem, this argument is easily adapted to the case k = 1, p1 = 2, andT is linear to prove T : X → L2 is continuous when G is compact.

4.1. Almost Everywhere Convergence of Fourier Series. Con-sider the case G = Tn where Tn is the n-torus identified with [0, 1]n andwrite dµ(x) = dx as µ is normalized Lebesgue measure. For f ∈ L1∩L2,

f(j) =

∫Tnf(x)e−2πix·j dx

defines the j-the Fourier Coefficient of f where j = (j1, · · · , jn) ∈ Zn.Consider an open set S ⊂ Rnk that contains the origin. For N > 0 letNS be the dilation of S by N . Then for smooth functions f1, · · · , fk,we have

(4.10)k∏

m=1

fm(x) = limN→∞

∑(j1,··· ,jk)∈NS∩Znk

k∏m=1

fm(jm)e2πijm·x

for all x ∈ Tn because such series are absolutely convergent and NSabsorbs all of Znk as N tends to infinity.

Define the maximal operator T ∗S by

T ∗S(f1, · · · , fk)(x) = supN

∣∣∣ ∑(j1,··· ,jk)∈NS∩Znk

k∏t=1

ft(jt)e2πijt·x

∣∣∣.By Theorem 6 from [6], (4.10) holds almost everywhere for all fi ∈ Lpi ,1 ≤ i ≤ k, if there exists a constant C <∞ such that

(4.11) ‖T ∗S(f1, · · · , fk)‖p,∞ ≤ C

k∏i=1

‖fi‖pi .

16 GEOFF DIESTEL

Theorem 4.2. Let 1 ≤ p1, · · · , pk ≤ 2 and 1/p = 1/p1 + · · · + 1/pk.Suppose S ⊂ Rnk is open and contains the origin. Then the followingare equivalent.

(1) For every fi ∈ Lpi, 1 ≤ i ≤ k, (4.10) holds almost everywhere.(2) T ∗S : Lp1 × · · · × Lpk → Lp,∞ is continuous.(3) T ∗S : Lp1 × · · · × Lpk → L0 is continuous at zero.

Proof. The equivalence of (1) and (2) follows from Theorem 6 in [6].Moreover, the implication (2) ⇒ (3) is trivial. Therefore, it sufficesto prove (3) ⇒ (2). However, since normalized Lebesgue measure is aprobability Haar measure and T ∗S is a k-sublinear translation invariantoperator that is continuous at zero, Theorem 4.1 implies (2).

When k = n = 1 and S is the interval (−1, 1), the theorems of Car-leson [2](p = 2) and Hunt [9](1 < p ≤ 2) imply that all the conditionsfrom Theorem 4.2 hold if f ∈ Lp and 1 < p1 = p ≤ 2. Moreover, theclassical example of Kolmogorov [17] shows that none of the conditionshold if p1 = p = 1. If k = 1 < n, and S is the unit ball of Rn, Feffer-man [4] showed that the conditions from Theorem 4.2 may only holdif p1 = p = 2.

The first multilinear extension of Fefferman’s counter-example ap-pears in [3] where, if k = 2 and S is the unit ball in R2n, the conditionsfrom Theorem 4.2 may hold only if 1 < p ≤ 2 ≤ p1, p2 < ∞ and1/p = 1/p1 + 1/p2. A generalization of the counter-example in [3] canbe found in [7].

4.2. Trivial Translation Invariant Operators. Now we turn to thecase of a σ-finite Haar measure µ satisfying µ(G) = ∞. Theorem4.1 hints at a fundamental difference between the finite and infinitemeasure cases for translation invariant operators.

Since it is not so that Lp ⊂ Lq when p > q, we will show thattranslation invariant operators do not continuously map into Lq when1/q > 1/p1 + · · · + 1/pk and each Xi has type pi. Thus, the naturalrange of such operators is Lp,∞ where 1/p = 1/p1+ · · ·+1/pk. This is abasic belief of Harmonic analysts but a proof only exists in the case oflinear operators where the arguments rely on the fact Euclidean spaceis a complete separable metric and the smooth compactly supportedfunctions are dense in Lp for 0 < p < ∞, see [8]. Thus, if T : Lp1 ×· · · × Lpk → Lq is a continuous k-linear operator that commutes withtranslations, it should be true that T 6= 0 if and only if 1/q ≤ 1/p1 +· · ·+ 1/pk. The following general result gives a partial confirmation ofthis belief.

FACTORING MULTI-SUBLINEAR MAPS 17

Theorem 4.3. Let 0 < q, p, p1, · · · , pk ≤ 2, 1/q > 1/p = 1/p1 + · · · +1/pk and assume µ is a unique Haar measure for a locally compactabelian group G such that µ is σ-finite and µ(G) = ∞. Suppose T :X1×· · ·Xk → Lq is a continuous translation invariant k-linear operatorwhere, for each 1 ≤ i ≤ k, Xi has type pi. Then T = 0.

Proof. By Corollary 2.1, there is an essentially positive density h sothat ‖h‖1 = 1 and a constant C <∞ so that

‖h−1/qT (x1, · · · , xk)‖Lp,∞(hdµ)≤ C‖x1‖ · · · ‖xk‖

for all xi ∈ Xi, 1 ≤ i ≤ k. By Holder’s inequality with respect to theprobability measure ν defined by dν = hdµ

‖h−1/qT (x1, · · · , xk)‖Lr(hdµ) ≤ C(q, p, r)‖x1‖ · · · ‖xk‖for r ∈ (q, p). Therefore, Maurey’s [22] theorem implies there is aconstant C <∞ so that

(4.12)∥∥∥(∑

j

∣∣h−1/qT (xj,1, · · · , xj,k)∣∣r)1/r∥∥∥

Lq(dν)≤ 1

for all finite sequences (xj,i)j from Xi satisfying(∑j

(‖xj,1‖ · · · ‖xj,k‖

)r)1/r ≤ C−1

where C is optimal with respect to (4.12).Consider the set S consisting of those nonnegative f ∈ L1(dν) such

that there exists finite sequences (xj,i)i from Xi and t ∈ G so that

f ≤ h−1(∑

j

∣∣τtT (xj,1, · · · , xj,k)∣∣r)q/r

and∑

j

(‖xj,1‖ · · · ‖xj,k‖

)r ≤ C−r. Since q < r, the homogeneity and

translation invariance of T makes it is easy to verify that S is r/q-convex, i.e. if f1, · · · , fn ∈ S and 0 ≤ θ1, · · · , θn ≤ 1 satisfy

∑i θi ≤ 1

then ∑i

θifi ≤(∑

i

θifr/qi

)q/r∈ S.

This means that S is a relatively weakly compact subset of L1(dν).This compactness argument was adapted from the proof of Maurey’stheorem [22], [1], [32].

Since ν is a probability measure, relative weak compactness is equiv-alent to equi-integrablility(see Chapter 5 in [1]), i.e.

limν(E)→0

supf∈S

∫E

f dν = 0.

18 GEOFF DIESTEL

Now, if we assume T 6= 0, we can pick f0 ∈ S of the form h−1|T (x1, · · · , xk)|qsuch that there exists δ > 0 and some set B of finite measure satisfying∫

B

f dν =

∫B

h−1|T (x1, · · · , xk)|q dν =

∫B

|T (x1, · · · , xk)|q dµ > δ.

Then for any t ∈ G, define ft = h−1(τt(hf0)) and notice that ft ∈ S.Moreover, since µ is translation invariant we get∫B+t

ft dν =

∫B+t

|τtT (x1, · · · , xk)|q dµ =

∫B

|T (x1, · · · , xk)|q dµ > δ > 0

for any choice of t ∈ G. Since µ(G) =∞, we may pick (tn)n from G sothat

limnν(B + tn) = lim

n

∫B+tn

hdµ = 0 and limn

∫B+tn

ftn dν > 0.

Under the assumption that T 6= 0, this contradicts that S is equi-integrable. Therefore, T = 0.

5. Concluding Remarks

For p < 2, strong factorization through Lp,∞ is sharp in the sensethat type p subspaces of L0 need not strongly embed into Lp. Forexample, `p embeds into Lp but there is no embedding of `p into Lpsuch that, within the range of the embedding, convergence in Lp isequivalent to convergence in measure. Another important example isthe Ribe space which is a non-locally convex quasi-Banach space whichhas type 1, see [29], [10], [18], and [16]. The Ribe space embeds intoL0 so it strongly embeds into L1,∞. However, since it is not locallyconvex it cannot simply embed into L1. The following question posedby Kwapien [19] remains open: need every Banach subspace of L0 be asubspace of L1? A partial result in this direction is given in [11] where,if 1 ≤ p < 2, it is shown that a Banach space X embeds into Lp if andonly if `p(X) embeds into L0.

In [13], an example is given to show that Theorem 6.3 from [13]cannot be improved to strong factorization through L1 via a changeof density. Essentially the same example shows that Rademacher typedoes not guarantee strong factorization through a strong type spacevia a change of density when k > 1. The example is as follows. LetT : `2 × `2 → L0 be defined by T (v1, v2) = J(v1 · v2) where v1 · v2 ∈ `1is the coordinate-wise multiplication of the sequences v1 and v2 from`2 and J : `1 → L1,∞(0, 1) is an isometric embedding using 1-stablerandom variables. Then T is a continuous map from a product of type2 spaces into Lq(0, 1) for any 0 < q < 1 but cannot strongly factorthrough L1 via a change of density. To adapt this to the case X and

FACTORING MULTI-SUBLINEAR MAPS 19

Y have the bounded approximation property and X∗ and Y ∗ havecotype 2, one need only observe that there is a quotient map Q fromC[0, 1] onto `2. Therefore, a counter-example follows by consideringT ′(f1, f2) = J(Qf1 ·Qf2).

Other notable references are [30], [15] and [14]. in [30], strong fac-torization for positive multilinear operators are established where thep-convexity of the underlying Banach Lattices are considered in placeof type. In [15], the authors prove a dual theorem to Pisier’s [27]. In[14], intersection bodies in convex geometry are shown to be isomor-phic with finite dimensional subspaces of Lq for q < 1. Moreover, thecorresponding Banach-mazur distances depend on q but are indepen-dent of dimension. This result was achieved by proving a version ofMaurey’s [22] theorem for Lp with p < 0. If Theorem 1.2 from [14] canbe extended to include q = 1, intersection bodies and polar projectionbodies would be isomorphically equivalent and this would present a bigstep towards understanding the duality of sections and projections inconvex geometry.

If one considers measurable functions defined on Euclidean spaceequipped with Lebesgue measure, Theorem 4.3 is one step towardsproving the following conjecture about operators that commute withtranslations.

Conjecture 5.1. Let 1 ≤ p1, · · · , pk <∞ and 1/q > 1/p1 + · · ·+1/pk.Suppose T : Lp1 × · · · ×Lpk → Lq be a continuous k-sublinear operatorthat commutes with translations. Then T = 0.

Since the argument in Theorem 4.3 relies on Rademacher type, itonly proves the case that 0 < q < p1, · · · , pk ≤ 2. The case k = 1 wascompletely proven by Hormander [8]. The proof is a simple supportargument using the density of smooth compactly supported functions inLp. Thus, the only linear operator the commutes with translations andmaps down the Lebesgue scale is the zero operator. Currently, theredoes not seem to be a simple adaptation of Hormander’s [8] argumentfor k-linear operators.

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E-mail address: gdiestel@ct.tamus.edu