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Factorization theorems for quasi-normed spaces

arXiv:math/9211207v1 [math.FA] 2 Nov 1992Factorization theorems for quasi-normed spacesN.J. Kalton and Sik-Chung Tam1Department of MathematicsUniversity of Missouri-ColumbiaColumbia, Mo.

65211Abstract. We extend Pisier’s abstract version of Grothendieck’s theorem to general non-locally convex quasi-Banach spaces.

We also prove a related result on factoring operatorsthrough a Banach space and apply our results to the study of vector-valued inequalitiesfor Sidon sets. We also develop the local theory of (non-locally convex) spaces with dualsof weak cotype 2.1.

Introduction.In [16] (see also [18]) Pisier showed that if X and Y are Banach spaces so that X∗and Y have cotype 2 then any approximable operator u : X →Y factors through a Hilbertspace. This result (referred to as the abstract version of Grothendieck’s theorem in [18])implies the usual Grothendieck theorem by taking the special case X = C(Ω) and Y = L1as explained in [18].Our main result is that the abstract form of Grothendieck’s theorem is valid for quasi-Banach spaces.

To make this precise let us say that an operator u : X →Y between twoquasi-Banach spaces is strongly approximable if it is in the smallest subspace A(X, Y ) ofthe space L(X, Y ) of all bounded operators which contains the finite-rank operators andis closed under pointwise convergence of bounded nets. We define the dual X∗of a quasi-Banach space as the space under all bounded linear functionals; this is always a Banachspace.

Then suppose X, Y are quasi-Banach spaces so that X∗and Y have cotype 2; weprove that if u : X →Y is strongly approximable then u factors through a Hilbert space.Some approximability assumption is necessary even for Banach spaces (cf. [17]).However, in our situation, such an assumption is transparently required because X∗could1The research of both authors was supported by NSF-grant DMS-92013571

have cotype 2 for the trivial reason that X∗= {0}; then the only strongly approximableoperator on X is identically zero. We remark that there are many known examples ofnonlocally convex spaces X with cotype 2 (e.g.

Lp, Lp/Hp and the Schatten ideals Spwhen p < 1 ([20],[23]). Examples of nonlocally convex spaces whose dual have cotype 2 areless visible in nature, but in [7] there is an example of such a space X with an unconditionalbasis so that X∗∼ℓ1.We also give a similar result for factorization through a Banach space; in this case werequire that X∗embeds into an L1−space and that Y has nontrivial cotype.

These resultsare then applied to the study of Sidon sets. We say a quasi-Banach space X is Sidon-regular if for every compact abelian group G and every Sidon subset E = {γn}∞n=1 of thedual group Γ and for every 0 < p ≤∞we have ∥Pnk=1 xkγk∥Lp(G,X) ∼∥Pnk=1 ǫkxk∥Lp(X)where (ǫk) are the Rademacher functions on [0, 1].

It is a well-known result of Pisier [15]that Banach spaces are Sidon-regular but in [9] it is shown that not every quasi-Banachspace is Sidon-regular. We show as a consequence of the above factorization theorems thatany space with nontrivial cotype is Sidon-regular; this includes such spaces as the Schattenideals Sp and the quotient spaces Lp/Hp when 0 < p < 1.Our final section is motivated by the fact that the main factorization theorem sug-gests that quasi-Banach spaces whose duals have cotype 2 have special properties.

On anintuitive level there is no reason to suspect that properties of the dual space will influ-ence the original space very strongly in the absence of local convexity. However we showthat quasi-Banach spaces whose duals have weak cotype 2 can be characterized internallyby conditions dual to the standard characterizations of weak cotype 2 spaces.

We showthat, for example, that X has a dual of weak cotype 2 if and only if its finite-dimensionalquotients have uniformly bounded outer-volume ratios.We refer to [11] for the essential background on quasi-Banach spaces. We will needthe fact that every quasi-normed space can be equivalently normed with an r-norm where0 < r ≤1 (the Aoki-Rolewicz theorem) i.e.

a quasi-norm satisfying ∥x+y∥r ≤∥x∥r +∥y∥r.Let us recall that the Banach envelope ˆX of a quasi-Banach space X is defined tobe the closure of j(X) where j : X →X∗∗is the canonical map (which is not necessarilyinjective). If j is injective (i.e.

X has a separating dual) we regard ˆX as the completion ofX with respect to the norm induced by the convex hull of the closed unit ball BX. If X islocally convex then d(X, ˆX) = ∥IX∥ˆX→X is equal to the minimal Banach-Mazur distancebetween X and a Banach space.2

2. The main factorization theorems.Let us suppose that X and Y are r-Banach spaces where 0 < r ≤1.

Suppose u : X →Y is a bounded linear operator. We will define γ2(u) to be the infimum of ∥v∥∥w∥over allfactorizations u = vw where w : X →H and v : H →Y for some Hilbert space H. Wedefine δ(u) to be the infimum of ∥v∥∥w∥where u = vw and w : X →B and v : B →Yfor some Banach space B.

In the special case when u = IX is the identity operator thenγ2(IX) = dX is the Euclidean distance of X and δ(IX) = δX = d(X, ˆX) (cf. [5], [14]) isthe distance of X to its Banach envelope.Let DN = {−1, +1}N be equipped with normalized counting measure λ and definethe Rademacher functions ǫi(t) = ti on DN for 1 ≤i ≤N.

We define T (N)2(u) to be theleast constant such that Z∥NXi=1ǫiu(xi)∥2dλ!1/2≤T (N)2(u) NXi=1∥xi∥2!1/2for x1, . .

., xN ∈X. We define C(N)2(u) to be the least constant so that NXi=1∥u(xi)∥2!1/2≤C(N)2(u) Z∥NXi=1ǫixi∥2dλ!1/2for x1, .

. ., xN ∈X.

We let T2(u) = supN T (N)2(u) be the type 2 constant of u and C2(u) =supN C(N)2(u) be the cotype two constant of u. In the case when u = IX we let T2(IX) =T2(X) and C2(IX) = C2(X) the type two and cotype two constants of X.Finally we let K(N)(u) be the least constant so that if f ∈L2(DN, X) then∥NXi=1(Zǫiu ◦fdλ)ǫi∥L2(DN,Y ) ≤K(N)(u)∥f∥L2(DN,X).We then let K(u) = supN K(N)(u).

If u = IX then K(IX) = K(X) is the K-convexityconstant of X.We will need the following estimate.Lemma 1. If 0 < r < 1 then there is a constant C = C(r) so that for any r-normed spaceX, we have K(X) ≤CdφX(1 + log dX), where φ = (1/r −1)/(1/r −2).Proof: It is clear that K(X) ≤δXK( ˆX).

Now we have δX ≤CdφX where C = C(r).Lemma 3 of [5]. We also have, by a result of Pisier ([16], [18]) that K( ˆX) ≤C(1 +log d ˆX).It remains to observe that d ˆX ≤dX since any operator u : X →H where H is a Hilbertspace factorizes through the Banach envelope of X with preservation of norm.3

We next discuss some aspects of Lions-Peetre interpolation (see [1] or [2] ). We willonly need to interpolate between pairs of equivalent quasi-norms on a fixed quasi-Banachspace.

Let us suppose that X is an r-Banach space for quasi-norm ∥∥0 and that ∥∥1 is anequivalent r-norm on X; we write Xj = (X, ∥∥j) for j = 0, 1. LetKs(t, x) = inf{(∥x0∥s0 + ts∥x1∥s1)1/s : x = x0 + x1}where r ≤s < ∞.

Then Ks is an r-norm on X. We define∥x∥θ,2 = (θ(1 −θ))1/2Z ∞0K2(t, x)2t1+2θdt1/2for 0 < θ < 1.

We write Xθ,2 = (X, ∥∥θ,2) = (X0, X1)θ,2.We will need some well-known observations.Lemma 2. There exists a C = C(r) so that if ∥∥0 = ∥∥1 then for any x ∈X we haveC−1∥x∥0 ≤∥x∥θ,2 ≤C∥x∥0.Proof: This follows from the simple observation that Kr(t, x) = min(1, t)∥x∥0 and that21/2−1/rKr ≤K2 ≤Kr.Lemma 3.

There exists C = C(r) so that if N ∈N then (ℓN2 (X0), ℓN2 (X1))θ,2 is isometri-cally isomorphic to ℓN2 (Xθ,2).Proof: This follows from the routine estimateK2(t, (x1, . .

., xN)) = (NXi=1K2(t, xi)2)1/2The following lemma is standard.Lemma 4. Suppose (X0, X1) are as above and that (Y0, Y1) is a similar pair of r-normingsof a quasi-Banach space Y .

Let u : X →Y be a bounded linear operator. Then∥u∥Xθ,2→Yθ,2 ≤∥u∥1−θX0→Y0∥u∥θX1→Y1.We now combine these results to give a criterion for convexity of the interpolatedspace.For convenience we will drop the subscript 2 and write Xθ.

We also recall thedefinition of equal norms type p for p ≤2. We let ˆTp(X) be the least constant so that forany N and any x1, .

. ., xN ∈X we have∥NXi=1ǫixi∥Lp(DN,X) ≤ˆTp(X)N 1/p max1≤i≤N ∥xi∥.4

Lemma 5. Suppose (X0, X1) are as above.

Suppose 1 ≤a < ∞and 0 < θ < r/(2 −r).There is a constant C = C(a, θ, r) so that if T2(X0) ≤a, then δXθ ≤C.Proof: Consider the map u : ℓN2 (X) →L2(ΩN, X) defined by u((xi)Ni=1) = PNi=1 ǫixi.Then ∥u∥X0 ≤a. Since ∥∥1 is an r-norm it follows from Holder’s inequality that ∥u∥X1 ≤N 1/r−1/2.

Hence ∥u∥Xθ ≤a1−θN θ(1/r−1/2). Now θ(1/r −1/2) = 1/2 −φ where φ > 0.Assume x1, .

. ., xN ∈X.

Then Z∥NXi=1ǫixi∥2θdλ!1/2≤a1−θN 1−φ max1≤i≤N ∥xi∥θ.This means that ˆTp(Xθ) ≤a1−θ where p = (1 −φ)−1 > 1. Applying Lemma 2 of [5] we getthe lemma.We are now in position to prove the generalization of Pisier’s abstract Grothendiecktheorem.Theorem 6.

Let X, Y be quasi-Banach spaces so that X∗and Y have cotype 2. Thenthere is a constant C so that if u : X →Y is a strongly approximable operator, thenγ2(u) ≤C∥u∥.Proof: We may suppose that both X and Y are r-normed.

Consider first an operatoru : X →Y such that ∥u∥= 1 and γ2(u) < ∞. Then there is a Hilbert space Z and afactorization u = vw where w : X →Z and v : Z →Y satisfy ∥v∥≤2γ2(u) and ∥w∥≤1.We will let Z = Z0 and define Z1 by the quasi-norm ∥z∥1 = max(∥z∥0, ∥v(z)∥Y ).Then T2(Z0) = 1; so we pick 0 < θ < r/(2 −r) depending on r and deduce an estimateδZθ ≤C = C(r).Now consider the map ˜wN : L2(DN, X) →ℓN2 (Z) defined by˜wN(f) =Zw ◦fǫidλNi=1.Clearly for the Euclidean norm ∥∥0 we have ∥˜wN∥0 ≤1.We now consider ∥∥1.

It is routine to see that C2(Z1) ≤1 + C2(Y ) ≤2C2(Y ). Wealso clearly have the estimate ∥z∥0 ≤∥z∥1 ≤2γ2(u)∥z∥0 so that dZ1 ≤2γ2.

From this andLemma 1 we can obtain an estimate K(Z1) ≤C(γ2(u))φ(1 + log γ2(u)) where C = C(r)and φ = (1 −r)/(2 −r). To simplify our estimate we replace this by K(Z1) ≤C(γ2(u))τwhere τ depends only on r and φ < τ < 1.

These estimates combine to give∥˜wN∥1 ≤C(γ2(u))τC2(Y ).5

Interpolation now yields∥˜wN∥θ ≤C(γ2(u))τθC2(Y )θ.Now consider w∗: Z∗θ →X∗. By taking adjoints of ˜wN and observing that L2(DN, X)∗can be identified with L2(DN, X∗) in the standard way we see that we have an estimateT2(w∗: Z∗θ →X∗) ≤C(γ2(u))τθC2(Y )θ.It follows immediately from Maurey’s extension of Kwapien’s theorem [12], [18] Theorem3.4 thatγ2(w∗: Z∗θ →X∗) ≤C(γ2(u))τθC2(Y )θC2(X∗).By duality this gives the same estimate for γ2(w : ˆX →ˆZθ).

Our previous estimate on δZθgives that the norm of the identity map I : ˆZθ →Zθ is bounded by some C = C(r). SinceIX : X →ˆX has norm one, we have:γ2(w : X →Zθ) ≤C(γ2(u))τθC2(Y )θC2(X∗).Now ∥v∥0 ≤2γ2(u) and ∥v∥1 ≤1 by construction.

By interpolation we have ∥v∥θ ≤Cγ2(u)1−θ. Now by factoring through Zθ we obtainγ2(u) ≤C(γ2(u))1−θ+τθC2(Y )θC2(X∗)and soγ2(u) ≤C(C2(Y ))1/(1−τ)(C2(X∗))1/(1−τ)θ.Thus we conclude that if γ2(u) < ∞then γ2(u) ≤C∥u∥where C is a constantdepending only on X, Y.

The remainder of the argument is standard. Let J be the subspaceof L(X, Y ) of all operators for which γ2(u) < ∞.

Then J contains all finite-rank operators.We show it is closed under pointwise convergence of bounded nets. Let (uα) be a boundednet in J converging pointwise to u.

Then sup γ2(uα) = B < ∞. For each α there is aEulcidean seminorm (i.e.

a seminorm obeying the parallelogram law) ∥∥α, on X satisfying∥uα(x)∥≤∥x∥α ≤B∥x∥for x ∈X. By a straightforward compactness argument there is aEuclidean seminorm ∥∥E on X satisfying ∥u(x)∥≤∥x∥E ≤B∥x∥for x ∈X, i.e.

u ∈J .We will next prove a similar result for factorization through a Banach space. We recallthat a quasi-Banach space has cotype q where q ≥2 if there is a constant C so that forevery N and all x1, .

. ., xN ∈X we have:(NXi=1∥xi∥q)1/q ≤C∥NXi=1ǫixi∥Lq(DN,X).6

Lemma 7. Let X be a Banach space so that X∗is isomorphic to a subspace of anL1−space, and suppose Y is a quasi-Banach space of cotype q < ∞.

Then there is aconstant C = C(X, Y ) so that if u : X →Y is a bounded operator then there is a Banachspace Z with T2(Z) ≤C and a factorization u = vw where w : X →Z and v : Z →Ysatisfy ∥v∥∥w∥≤C∥u∥.Proof: By assumption, there is a compact Hausdorffspace Ω0 and an open mappingq : C(Ω) →X∗∗. It follows that there is a constant C0 so that if E is a finite-dimensionalsubspace of X there is a finite rank operator tE : C(Ω) →X∗∗with ∥tE∥< C0 and ifx ∈E with ∥x∥= 1 there exists f ∈C(Ω) with ∥f∥< 2 and tEf = x.

It follows from thePrinciple of Local Reflexivity (cf. [22] p.76) that we can suppose that tE has range in X.We now form an ultraproduct of X and Y .

Let I be the the collection of all finite-dimensional subspaces E of X and let U be an ultrafilter on I containing all sets of theform {E : E ⊃F} where F is a fixed finite-dimensional subspace. Consider the space XUdefined to be the quotient of ℓ∞(I; X) by the subspace cU,0(I; X) of all (xE) such thatlimU xE = 0; XU is thus the space of (equivalence classes of) (xE) normed by limU ∥xE∥.We regard X as a subspace of XU by identifying x with the constant function xE = x forall E. We similarly introduce YU and note that YU has cotype q with the same constantas Y .

We extend u : X →Y to u1 : XU →YU by setting u1((xE)E∈I) = (u(xE)E∈I).Let us also introduce the operator t : C(Ω) →XU by putting t(f) = (tE(f))E∈I. Consideru1t : C(Ω) →YU.

By Theorem 4.1 of [10] there is a regular probabilty measure µ on Ωso that if p = q + 1 then ∥u1t(f)∥≤C∥u∥(R|f|pdµ)1/p. Here C depends on X, Y but noton u.

This implies that u1t = v1j where j : C(Ω) →Lp(µ) is the canonical injection and∥v1∥≤C∥u∥. Let N = v−11 (0) and form the quotient Z1 = Lp/N; let π be the quotientmap.

For each x ∈X there exists f ∈C(Ω) and t(f) = x; this follows from the choice ofultrafilter. Then w(x) = πj(f) is uniquely determined independent of f. Furthermore fcan be chosen so that ∥f∥≤2∥x∥so that ∥w∥≤2.

Let Z be the closure of the range of w.Then clearly v1(π−1(Z)) ⊂Y so that we can define v : Z →Y with ∥v∥≤C and vw = u.Finally T2(Z) ≤T2(Z1) ≤T2(Lp) is bounded by a constant depending only on q.Theorem 8. Suppose X is a quasi-Banach space such that X∗is isomorphic to a subspaceof an L1−space, and Y is a quasi-Banach space of cotype q < ∞.

There is a constant Cso that if u : X →Y is a strongly approximable operator then δ(u) ≤C∥u∥.Proof: Let us assume that X, Y are both r-normed.Suppose first that u : X →Ysatisfies ∥u∥= 1 and δ(u) < ∞. We show that δ(u) ≤C where C depends only on X, Y .By Lemma 7, u can be factored through a Banach space Z satisfying T2(Z) ≤C whereC = C(X, Y ) so that u = vw where w : X →Z with ∥w∥= 1 and v : Z →Y with∥v∥≤Cδ(u).

Let Z = Z0 and introduce Z1 by setting ∥z∥1 = max(∥z∥, ∥v(z)∥). If we pick7

θ < r/(2 −r) then δZθ ≤C where again C depends only on X, Y. Henceδ(u) ≤C∥v∥Zθ→Y ∥w∥X→Zθ.By interpolation this yieldsδ(u) ≤C(δ(u))1−θ,and hence δ(u) ≤C. The remainder of the proof is similar to that of Theorem 6.3.

Applications to Banach envelopes and Sidon sets.It is proved in [7] that if X is a natural quasi-Banach space (i.e. a space isomorphicto a subspace of a space ℓ∞(I; Lp(µi)) with the strong approximation property and if Y isany subspace of X such that Y ∗has cotype q < ∞then Y is locally convex.

We presentnow two variations on this theme.Let us say that a quasi-Banach space X is (isometrically) subordinate to a quasi-Banach space Y if X is (isometrically) isomorphic to a closed subspace of a space ℓ∞(I; Y )for some index set I. Thus a separable space X is natural if it is subordinate to Lp[0, 1]for some 0 < p < 1.Theorem 9.

Let Z be a quasi-Banach space and let X be subordinate to Z. Assume thateither X or Z has the strong approximation property.

Let Y be any subspace of X. Then(1) If Z has cotype 2 and if Y ∗has cotype 2 then Y is locally convex.

(2) If Z has cotype q < ∞and Y ∗is isomorphic to a subspace of an L1−space, then Y islocally convex.Proof: The proofs are essentially identical. We therefore prove only (2).

Let j : Y →ℓ∞(Z) be the inclusion map. Then since j factors through X it is strongly approximable,under either hypothesis.

Let πi : ℓ∞(Z) →Z be the co-ordinate map. Then by Theorem8, we have δ(πij) ≤C for some constant C depending only on Y.

Thus for y ∈Y, ∥y∥=supi ∥πijy∥≤C∥y∥ˆY .Let us now give an application. Suppose G is a compact abelian group with normalizedHaar measure µG, and suppose Γ is the dual group.

We recall that a subset E ⊂Γ is aSidon set if for every ǫγ = ±1 there exists ν ∈M(G) whose Fourier transform satisfiesˆν(γ) = ǫγ.In [9] the first author introduced the property Cp(X) for a subset E of Γ where0 < p ≤∞. We say that E has Cp(X) if there is a constant M so that for any γ1, .

. ., γn ∈E8

and any x1, . .

., xn ∈X we haveM −1∥nXk=1xkǫk∥Lp(Dn,X) ≤∥nXk=1xkγk∥Lp(G,X) ≤M∥nXk=1xkǫk∥Lp(Dn,X)where ǫ1, . .

., ǫn are the Rademacher functions on Dn. Let us say that a quasi-Banachspace X is Sidon-regular if every Sidon set E has property Cp(X) for every 0 < p ≤∞.It is a well-known result of Pisier [15] that every Banach space is Sidon-regular.Byway of contrast, in [9] an example of a quasi-Banach space which is not Sidon-regular isconstructed.

However, every natural space is Sidon-regular. The above results enable usto extend this to a wider class of spaces.Theorem 10.

Let X be a quasi-Banach space of cotype q < ∞. Then every quasi-Banachspace which is subordinate to X, (and, in particular, X itself) is Sidon-regular.Proof: This is very similar to the proof of Theorem 4 in [9].

Suppose G is a compactAbelian group and E is a Sidon subset of Γ. Let s = min(p, 2).

Let Z = Ls(G, X). ThenZ also has cotype q.

To see this we need first to observe that the Kahane-Khintchineinequality holds in an arbitrary quasi-Banach space (Theorem 2.1 of [6]) so that there is aconstant C depending only on X so that if x1, . .

., xn ∈X then Z∥nXk=1ǫkxk∥qdλ!1/q≤C Z∥nXk=1ǫkxk∥sdλ!1/s.Now if f1, . .

., fn ∈Z then, for constants C1, C2, C3 depending only on X, nXk=1∥fk∥q!1/q≤C1 ZG(nXk=1∥fk(t)∥qX)s/qdµG(t)!1/s≤C2ZG ZDn∥nXi=1ǫkfk(t)∥qXdλ!s/qdµG(t)1/s≤C3 ZDnZG∥nXk=1ǫkfk(t)∥sXdµG(t)dλ!1/s≤C3∥nXk=1ǫkfk∥Lq(Dn,Z).Now let En be any sequence of finite subsets of E. Let PEn(Y ) be the space of Y -valued polynomials Pγ∈En yγγ equipped with the Lp(G, Y ) quasi-norm. Then PEn(Y )is isometrically subordinate to Z.

Next equip the finite-dimensional space ℓ∞(En) of all9

bounded functions h : En →C with the quasi-norm of the operator Th : PEn →PEn givenby Th(P yγγ) = P h(γ)yγγ. Let us denote this space Mn.

Then Mn is isometricallysubordinate to Z. Thus the product c0(Mn) is isometrically subordinate to Z and has thestrong approximation property.

However, as in Theorem 4 of [9] the assumption that Eis a Sidon set shows that we have a constant C depending only the Sidon constant of Eso that the envelope norm on Mn satisfies ∥h∥∞≤∥h∥ˆMn ≤C∥h∥∞. Thus the envelopeof c0(Mn) is isomorphic to c0 and Theorem 9(ii) applies to give that this space is locallyconvex so that for some uniform constant C′ we have for every n, ∥h∥Mn ≤C′∥h∥∞.

Asin [9] Theorem 4 this implies that E has property Cp(Y ).Remarks: The above theorem applies to Lp/Hp when p < 1 and to the Schatten idealsSp when p < 1, since these spaces have cotype 2 by recent results of Pisier [20] and Xu[23]. These spaces are known not to be natural; Sp is A-convex (i.e.

has an equivalentplurisubharmonic quasi-norm) while Lp/Hp is not A-convex (see [8]).Let us also remark that if 0 < p < 1 and E is a symmetric p-convex sequence spacewith the Fatou property then we can define an associated Schatten class SE (see Gohberg-Krein [4] for the Banach space versions). Precisely if H is a separable Hilbert space andA is a compact operator with singular values (sn(A)) we say A ∈SE if (sn(A)) ∈E andwe set ∥A∥E = ∥(sn(A))∥E.

It can then be shown that SE is subordinate to Sp. In factwe define a sequence space F by ∥(tn)∥F = sup{∥(sntn)∥p : ∥(sn)∥E ≤1} and it canthen be shown that ∥A∥E = sup{∥AB∥p : ∥B∥F ≤1}.

This result follows quickly from aninequality of Horn (cf. [4] pp.

48-9) thatkXj=1sj(AB)p ≤kXj=1sj(A)psj(B)pfor every k.4. Quasi-Banach spaces with duals of weak cotype 2.Let X be a finite-dimensional continuously quasi-normed quasi-Banach space with unitball BX.

We recall that the volume-ratio of X is defined by vr(X) = (Vol BX/Vol E)1/nwhere E is an ellipsoid of maximal volume contained in BX and n = dim X. We definethe outer volume-ratio of X by vr∗(X) = (Vol F/Vol BX)1/n where F is an ellipsoid ofminimal volume contaning BX.

The Santalo inequality ([19], [21]) shows that vr∗(X) ≥(Vol BX∗/Vol F 0)1/n = vr(X∗). The reverse Santalo inequality of Bourgain and Milman([3],[19]) shows that, if X is normed, vr∗(X) ≤Cvr(X∗) so that vr∗(X) is then equivalentto vr(X∗).

For general quasi-normed spaces the reverse Santalo inequality is not available.10

We recall that a Banach space X is of weak cotype 2 if there exists C so that wheneverH is a finite-dimensional Hilbert space with orthonormal basis (e1, . .

., en) and u : H →Xis a linear operator then ak(u) ≤Ck−1/2ℓ(u) for 1 ≤k ≤n. Hereℓ(u) = (E(∥nXk=1gku(ek)∥2))1/2(for g1, .

. ., gn a sequence of independent normalized Gaussian random variables) andak(u) = inf{∥u −v∥: v : H →X, rank v < k}.

The least such constant C is de-noted by wC2(X). It is known that X is of weak cotype 2 if and only if there exists Cso that vr(E) ≤C for every finite-dimensional subspace of X.

See Pisier [19] for details.It follows quickly that X∗is of weak cotype 2 if and only if vr∗(E) is bounded for allfinite-dimensional quotients of X. We prove in this section that the same characterizationextends to quasi-Banach spaces.We will require a preparatory lemma:Lemma 11.

Let E be an N-dimensional Euclidean space and suppose B is the unit ballof an r-norm on E. Let S be a subspace of E of dimension k. Suppose 1/r = β ∈N. ThenVol (B ∩S)Vol PS⊥(B)Vol B≤Nβkβwhere PS⊥is the orthogonal projection of E onto S⊥.Proof: We duplicate the argument of Lemma 8.8 of [19] (p. 132).

One finds that Vol B ≥aVol (B ∩S)Vol PS⊥B wherea = (N −k)Z 10(1 −tr)k/rtN−k−1dt= N −krZ 10(1 −s)k/rsN/r−k/r−1ds=Nβkβ.Lemma 12. There is a constant C depending only r so that if E is a finite-dimensionalr-normed space then dE ≤Cd2/r−1ˆE, and δE ≤Cd2/r−2ˆE.Proof: By Lemma 3 of [5] we have dE ≤δEd ˆE ≤CdφEd ˆE where φ = (1/r−1)/(1/r−1/2).This proves the first part and the second part follows on reapplying Lemma 3 of [5].11

Theorem 13. Let X be a quasi-Banach space and suppose 0 < α < 1.

Then X∗has weakcotype 2 if and only if there is a constant C so that whenever F is a finite-dimensionalquotient of X, there exists a quotient E of F with dim E ≥αdim F and dE ≤C.Proof: Suppose X∗has weak cotype 2.Then if F is a finite-dimensional quotient,wC2(F ∗) ≤wC2(X∗) and so has a subspace G with dim G ≥αdim F and dG ≤C(where C depends only on X and α). Let E = F/G⊥.

Then d ˆE = dG and so the precedingLemma gives an estimate dE ≤C′ where C′ = C′(α, X).Conversely if X has the given property then it is easy to see that ˆX must also havethe same property and this leads quickly to the fact that X∗has weak cotype 2 by theresults of [13].Proposition 14. Suppose 0 < r < 1 and a ≥1; then there is a constant C = C(a, r) sothat if E is an N-dimensional r-normed space and wC2(E∗) ≤a then vr∗(E) ≤C.Proof: In the argument which follows we use C for a constant which depends only a, rbut may vary from line to line.

It suffices to establish the result when 1/r = β ∈N. LetE be the ellipsoid of minimal volume containing BE.

Using this ellipsoid to introduce aninner-product we can define ∥x∥E∗= supb∈BE |(x, b)|. Then E is the ellipsoid of minimalvolume containing B ˆE = co BE and the ellipsoid of maximal volume contained in BE∗.Now, by imitating the argument of Theorem 8 of [5] we can construct an increasingsequence of subspaces (Wk)∞k=1 of E with dim Wk = N −σk ≥(1−2−k)N and BE∗∩Wk ⊂C23kE for k ≥1.

We let τk = σk −σk+1.It follows from the Hahn-Banach theorem that E ⊃PWkB ˆE ⊃C−12−3kE ∩Wk. Now,identifying Hk = E/W ⊥k with Wk under the quasinorm with unit ball PWk(BE) this impliesthat d ˆHk ≤C23k.

Now from Lemma 12 δHk ≤C2sk for suitable s > 0 depending on r. Weconclude that E ∩Wk ⊂C2tkPWk(BE), where t depends only on r, and C = C(a, r).Let Zk is the orthogonal complement of Wk in Wk+1. Notice that dim Zk = τk.

Nowby Lemma 11, if we set Ak = PWk+1(BE) ∩Zk,Vol PWk(BE)Vol Ak ≤(N −σk+1)βτkβVol PWk+1(BE).Let l be the first index for which Wl = E and so σl = 0. We first estimatel−1Yk=1(N −σk+1)βτkβ=(Nβ)!

((N −σ1)β)!(τ1β)! .

. .

(τl−1β)!=Nβσ1β l−2Yk=1σkβτkβ≤2β(N+Pσk)≤22βN12

Now E ∩Zk ⊂C2t(k+1)Ak and so log2 Vol PZk(BE) ≥−Ckτk + log2 Vol E ∩Zk. (Ifτk = 0 we interpret the relative volume as one).

Summing we obtain since τk ≤2−kN,l−1Xk=1log2 Vol Ak ≥−CN +l−1Xk=1log2 Vol E ∩Zk.Thuslog2 Vol BE = log2 Vol PWl(BE)≥log2 Vol PW1(BE) −CN +l−1Xk=1Vol E ∩Zk≥−CN + log2 Vol E ∩W1 + log2 Vol E ∩W ⊥1≥−CN + log2 Vol E.This completes the proof of the Proposition.Remark: This Proposition can be interpreted as follows. Suppose X is a finite-dimension-al normed space so that wC2(X∗) ≤a.

Consider the set ∂BX of extreme points of BXand form the r-convex hull Ar = cor∂BX. Then although Ar is smaller than BX it is nottoo much smaller, for Vol BX/Vol Ar ≤Cdim E.Theorem 15.

Let X be a quasi-Banach space. Then X∗has weak cotype 2 if and onlythere is a constant C so that vr∗(E) ≤C for every finite-dimensional quotient E of X.Proof: First suppose vr∗(E) ≤C for every finite-dimensional quotient E of X.

Let F bea finite-dimensional subspace of X∗and consider E = X/F ⊥. It is easy to see that theenvelope norm on E is the quotient norm from the envelope norm on X.

Clearly from thedefinition, vr∗( ˆE) ≤vr∗(E) ≤C. Hence by the Santalo inequality vr(F) ≤C.

This showsthat X∗has weak cotype 2.The converse is immediate from Proposition 14.References.1. C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Orlando1988.2.

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