Maurey-Nikishin factorization theorems

Factorization Theory

7.1 Maurey-Nikishin factorization theorems

In this section we shall discuss factorization theory of operators with values in theLp-spaces. Here factorization is related to the notion of change of density.

The first factorization result of this type, essentially discovered by Nikishin [157], establishes a criterion for an operator with values in an L_p(µ)-space to factor through L_q(ν) (q > p), where ν is obtained from µ after a suit- able change of density. Nikishin’s motivation came from harmonic analysis rather than Banach space theory, where versions of this result for translation- invariant operators had been known for some time (e.g., in the work of Stein [210]). However, it was the work of Maurey [144] that combined the ideas of Nikishin with the newly evolving theory of Rademacher type to create a very powerful tool.

The proof given below is based on one presented in [221] but is similar to the proof given by Maurey.

Definition 7.1.1.If (Ω,Σ, µ) is aσ-finite measure space then adensity func- tionhon Ω is a measurable function such thath≥0 a.e. and

h dµ= 1.

Theorem 7.1.2.Let µ be a σ-finite measure on some measurable space (Ω,Σ). Suppose that T is an operator from a Banach space X into L_p(µ) and that 1≤p < q <∞. Suppose0< C <∞. Then the following conditions are equivalent:

(a) There exists a density functionhon Ωsuch that

{h>0}|T x|^qh^1−q/pdµ 1/q

≤Cx, x∈X, (7.1) and

µ{ω: |T x(ω)|>0, h(ω) = 0}= 0, x∈X. (7.2) (b) For every finite sequence(x_k)ⁿ_k=1 inX,

ⁿ

k=1

|T xk|^q1/q

p≤C ⁿ

k=1

xk^q1/q

. (7.3)

Interpretation. Condition (a) is to be interpreted in the sense that each function T x is essentially supported on A ={ω ∈ Ω :h(ω) >0}. Thus the operator Sx := h^−1/pT x maps into Lp(Ω, h dµ). However, (a) asserts that S actually maps boundedly into the smaller space L_q(Ω, h dµ). This is the diagram depicting the situation:

X ^T //

L_p(µ)

Lq(hdµ)  //Lp(hdµ)

j

OO

Here j is an isometric embedding ofL_p(h dµ) onto the subspace L_p(A, µ) of L_p(µ), defined byj(f) =f h^1/p.

7.1 Maurey-Nikishin factorization theorems 167 Of course, at a very small cost we could insist thathis a strictly positive density (i.e., h > 0 a.e.) and drop equation (7.2): simply replace h by (1 + v)⁻¹(h+v) where > 0 and v is any strictly positive density. Then j becomes a genuine isometric isomorphism. In this case, however, the norm of S=h^−1/pT is a little greater thanC. Since the precise value ofS is rarely of interest we will often use the theorem is this form. In fact, in a formal sense we could replace (7.1) and (7.2) by

Ω

|T x|^qh^1−q/pdµ 1/q

≤Cx, x∈X,

with the implicit understanding thatT x= 0 a.e. on the set{ω∈Ω :h(ω) = 0} (i.e., whereh^−q/p= 0). We will use this convention later.

Before continuing let us notice that, although we have stated this for gen- eral σ-finite measures, it is enough to prove the theorem under our usual convention that µis a probability measure. Ifµis not a probability measure we choose some strictly positive densityv and set dµ =v dµ; then we define T :X →L_p(µ) by Tx= v⁻¹T x. A quick inspection will show the reader that the statement of the theorem forT implies exactly the same statements for T. Thus we can and do resume our convention that µ is a probability measure.

Proof. (a)⇒(b) Since (Ω, h dµ) is a probability measure space andp < q, the L_p(hdµ)-norm is smaller than the L_q(hdµ)-norm and thus we have

Ω

ⁿ

k=1

|T x_k|^qp/q

dµ 1/p

=

{h>0}

ⁿ

k=1

|T x_k|^qh^−q/pp/q

h dµ 1/p

≤ {h>0}

n k=1

|T x_k|^qh^−qph dµ 1/q

= ⁿ

k=1

{h>0}|T x_k|^qh^−qphdµ 1/q

≤C ⁿ

k=1

x_k^q1/q

.

(b)⇒(a) Let us assume that C is the best constant so that (7.3) holds.

Then sup

ⁿ

k=1

|T xk|^q1/q

p

: (xk)ⁿ_k=1⊂X, n k=1

xk^q ≤C^−q, n∈N

= 1.

LetW₀ be the set of all nonnegative functions inL₁ that are bounded above by functions of the form (n

k=1|T x_k|^q)^p/q, where n ∈ N and (x_k)ⁿ_k=1 ⊂ X withn

k=1x_k^q ≤C^−q, i.e.,

0≤f ≤ⁿ

k=1

|T x_k|^qp/q

; letW be the norm closure ofW0.

W0andW have the following property:

(*) Let r = q/p > 1. Given f₁, . . . , f_n ∈ W₀ [respectively, W] and c₁, . . . , c_n ≥0 withc₁+· · ·+c_n≤1 then(c₁f₁^r+· · ·+c_nf_n^r)^1/r∈W₀ [respectively,W].

To prove (*) it suffices to consider the case ofW₀.Suppose

0≤fk ≤

⎛

⎝^mk

j=1

|T xjk|^q

⎞

⎠

p/q

, 1≤k≤n,

wherem_k

j=1x_jk^q ≤C^−q for 1≤k≤n.Then we also have 0≤

- _n

k=1

c_kf_k^r .1/r

≤

⎛

⎝ⁿ

k=1 m_k

j=1

|T(c_k^1qx_jk)|^q

⎞

⎠

p/q

,

with n

k=1

c_k

m_k

j=1

x_jk^q≤C^−q, and this establishes (*).

Property (*) immediately yields thatW₀(and hence its norm-closureW) is convex. Indeed, iff₁, . . . , f_n∈W₀andc₁, . . . , c_n≥0 withc₁+· · ·+c_n= 1 then by (*) we obtain

n j=1

c_jf_j≤

⎛

⎝ⁿ

j=1

c_jf_j^r

⎞

⎠

1/r

∈W₀.

Using Mazur’s theorem, W is therefore weakly closed. Note that from the choice of C, we have

sup

f∈W₀

f dµ= sup

f∈W

f dµ= 1,

so in particularW is bounded. We next show thatW is weakly compact. This requires to show that it is equi-integrable.

SupposeW is not equi-integrable. Then there is someδ > 0, a sequence (f_n)^∞_n=1in W, and a sequence of disjoint measurable sets (E_n)^∞_n=1 such that

E_n

f_ndµ > δ >0, n∈N.

7.1 Maurey-Nikishin factorization theorems 169

Thus for anyN we have

δN^1−1r ≤N^−1r

max(f₁, f₂, . . . , f_N)dµ

≤ ⎛⎝1

N N j=1

f_j^r

⎞

⎠

1/r

dµ

≤1,

by using (*). This is a contradiction for large enoughN.

HenceW is weakly compact and, since integration is a weakly continuous functional onL₁(µ), it follows that there existsh∈W with

h dµ= 1. (7.4)

Now supposef ∈W.On the one hand, for anyτ >0 we have (1 +τ)⁻¹r(h^r+τ f^r)¹r ∈W,

therefore, by property (*),

(h^r+τ f^r)^1rdµ≤(1 +τ)^1r. (7.5) On the other hand,

(h^r+τ f^r)^1rdµ≥1 +τ^1r

h=0

f dµ. (7.6)

Since 1/r <1, combining (7.5) and (7.6) yields

{h=0}

f dµ= 0. (7.7)

From (7.6) and (7.4) we have

{h>0}

h(1 +τ f^rh^−r)¹r −1

τ dµ≤ (1 +τ)¹r −1

τ , τ >0.

Lettingτ→0 and using Fatou’s lemma we obtain

{h>0}

f^rh^1−rdµ≤1, f ∈W. (7.8) In particular (7.7) and (7.8) hold forf =C^−px^−p|T x|^pwhen 0=x∈X.

This immediately gives (7.2) and (7.1).

Theorem 7.1.3.Let 1 ≤ p < ∞. Suppose that T is an operator from a Banach space X into L_p(µ). If X has type 2 then there exists a constant C=C(p)such that for every finite sequence(x_k)ⁿ_k=1 inX we have

ⁿ

k=1

|T xk|²1/2

p≤C ⁿ

k=1

xk²1/2

.

Proof. By Theorem 6.2.13, for every 1≤p <∞there is a constantc=c(p) such that for any finite set of vectors (x_k)ⁿ_k=1 inX,

ⁿ

k=1

|T x_k|²¹₂

p≤cE n k=1

ε_kT x_k

p≤cTE n k=1

ε_kx_k.

Using Kahane’s inequality and the type 2 ofX, E

n k=1

ε_kx_k≤ E

n k=1

ε_kx_k²1/2

≤T₂(X)ⁿ

k=1

x_k²¹₂ .

SinceL_r(µ) forr≥2 are type-2 spaces, we immediately obtain:

Corollary 7.1.4.

(a) Every operator from a subspace of Lr(µ) (2 ≤r < ∞) into Lp(µ) (1 ≤ p <2) factors through a Hilbert space.

(b) If a Banach spaceX is isomorphic to a closed subspace of bothLp(µ)for some1≤p <2 andL_r(µ)for some2< r <∞, thenX is isomorphic to a Hilbert space.

Corollary 7.1.4 follows immediately from Theorems 7.1.2 and 7.1.3. Curi- ously, the isometric version of (b) does not hold. That is, ifX is isometric to a subspace ofL_p (1≤p <2) and isometric to a subspace ofL_r(2< r <∞), it is not true thatX must be isometric to a Hilbert space. Finite-dimensional counterexamples were given by Koldobsky [112]; however, the following prob- lem is still open (see [114]):

Problem 7.1.5.If an infinite-dimensional Banach space X is isometricto a closed subspace of bothL_p for some 1≤p <2 andL_r for some 2< r <∞, must X be isometric to a Hilbert space?

To push our results further we need a replacement for Theorem 6.2.13 for exponents other than 2.If 1≤q <2 then it turns out that theq-stable random variables constructed in the previous chapter do very nicely. Indeed, we could have used Gaussians in place of Rademachers in the preceding argument.

Lemma 7.1.6.Let 1 ≤p < q < 2. Suppose that γ = (γ_j)^∞_j=1 is a sequence of independent normalized q-stable random variables. Then for any finite se- quence of functions(f_j)ⁿ_j=1 inL_p(µ),

7.1 Maurey-Nikishin factorization theorems 171 ⁿ

j=1

|f_j|^q1/q

p

=c E

n j=1

γ_jf_j^p

p

1/p

, wherec=c(p, q)>0.

Proof. We recall from Theorem 6.4.18 that there is a constantc =c(p, q) so that

⎛

⎝E n j=1

ajγj^p

⎞

⎠

1/p

=c⁻¹

⎛

⎝ⁿ

j=1

|aj|^q

⎞

⎠

1/q

, (aj)ⁿ_j=1⊂R. Using Fubini’s theorem,

n j=1

|fj|^qp_q

dµ=c^pE ⁿ

j=1

γjfj^pdµ, and the lemma follows.

Theorem 7.1.7.Let1≤p <2. Suppose thatT is an operator from a Banach spaceXintoL_p(µ). IfXhas typerfor somep < r <2, then for eachq∈(p, r) there exists a constant C such that

ⁿ

j=1

|T x_j|^q1/q

p≤C ⁿ

j=1

x_j^q1/q

, for every finite sequence (xj)ⁿ_j=1 inX.

Proof. In this proof we will require three mutually independent sequences of independent identically distributed random variables: a sequence (ε_j)^∞_j=1 of Rademachers, a sequence (γ_j)^∞_j=1 of normalizedq-stable random variables and a sequence (η_j)^∞_j=1 of normalized r-stable random variables.

Let (x_j)ⁿ_j=1be a finite sequence inX. By the previous lemma, for a certain constantc=c(p, q) we have

ⁿ

j=1

|T xj|^q1/q

p

=c Eγ

n j=1

γjT xj^p

p

1/p

≤cT Eγ

n j=1

γjxj^p1/p

. Since the normalizedq-stables are symmetric andX has typer,

Eγ n j=1

γ_jx_j^p1/p

=

EγEε n j=1

ε_jγ_jx_j^p1/p

≤ Eγ

⎛

⎝Eε n j=1

ε_jγ_jx_j^r

⎞

⎠

p/r1/p

≤T_r(X) Eγ

ⁿ

j=1

|γ_j|^rx_j^rp/r1/p

. Now notice that

E n j=1

a_jη_j^p=c₁ ⁿ

j=1

|a_j|^rp/r

,

for a certain constant 0< c1=E|η1|^p which is finite sincep < r.Thus letting c2, c3 be positive constants depending only onp, q, andr,

Eγ

ⁿ

j=1

|γj|^rxj^rp/r

=c⁻₁¹EγEη n j=1

ηjγjxj^p

=c⁻₁¹EηEγ n j=1

ηjγjxj^p

=c₂Eη

ⁿ

j=1

|η_j|^qx_j^qp/q

≤c₂ Eη

n j=1

|η_j|^qx_j^qp/q

=c₃ ⁿ

j=1

x_j^qp/q

.

The next result now follows immediately from Theorem 7.1.2:

Theorem 7.1.8.Let X be a Banach space of type r > 1. Suppose that 1 ≤ p < r and that T :X →L_p(µ)is an operator. ThenT factors through L_q(µ) for any p < q < r. More precisely, for each p < q < r there is a strictly positive density function h on Ω so that Sx = h^−1/pT x defines a bounded operator fromL_p(µ)intoL_q(Ω, h dµ).

Note here that there is a fundamental difference between the case of type r <2 and type 2. In the former we only obtain a factorization throughL_q(µ) when q < r. Can we do better and takeq=r? The answer is no and to see why we must consider subspaces ofL_pfor 1≤p <2. This will be the topic of the next section, but let us mention that an improvement is possible: A later theorem of Nikishin [158] implies that T actually factors through the space

“weakL_r.” See [186] and the Problems.

Remark 7.1.9.An examination of the proofs of the theorems of this section shows that the main theorem (Theorem 7.1.8) will also hold if 0 < p < 1, when L_p is no longer a Banach space; in this case we can take r = 1 and