Set-functions and factorization

N.J. Kalton and S.J. Montgomery-Smith^{1} Department of Mathematics University of Missouri-Columbia Columbia, Mo. 65211

Abstract. If ϕ is a submeasure satisfying an appropriate lower estimate we give a quantitative result on the total mass of a measure μ satisfying 0 ≤ μ ≤ ϕ. We give a dual result for supermeasures and then use these results to investigate convexity on non-locally convex quasi-Banach lattices. We then show how to use these results to extend some factorization theorems due to Pisier to the setting of quasi-Banach spaces. We conclude by showing that if X is a quasi-Banach space of cotype two then any operator T : C(Ω) → X is 2-absolutely summing and factors through a Hilbert space and discussing general factorization theorems for cotype two spaces.

1. Introduction.

Let A be an algebra of subsets of some set Ω. Let us say that a set-function ϕ : A → R is monotone if it satisfies ϕ(∅) = 0 and ϕ(A) ≤ ϕ(B) whenever A ⊂ B. We say ϕ is normalized if ϕ(Ω) = 1. A monotone set-function ϕ is a submeasure if

ϕ(A ∪ B) ≤ ϕ(A) + ϕ(B)

whenever A, B ∈ A are disjoint, and ϕ is a supermeasure if

ϕ(A ∪ B) ≥ ϕ(A) + ϕ(B)

whenever A, B ∈ A are disjoint. If ϕ is both a submeasure and supermeasure it is a (finitely additive) measure.

If ϕ and ψ are two monotone set-functions on A we shall say that ϕ is ψ-continuous if lim_{n→∞} ϕ(A_{n}) = 0 whenever lim_{n→∞} ψ(A_{n}) = 0. If ϕ is ψ−continuous and ψ is

1Both authors were supported by grants from the National Science Foundation ϕ−continuous then ϕ and ψ are equivalent. A monotone set-function ϕ is called exhaustive if lim_{n→∞} ϕ(A_{n}) = 0 whenever (A_{n}) is a disjoint sequence in A. The classical (unsolved) Maharam problem ([1], [5], [6] and [15]) asks whether every exhaustive submeasure is equivalent to a measure. A submeasure ϕ is called pathological if whenever λ is a measure satisfying 0 ≤ λ ≤ ϕ then ϕ = 0. The Maharam problem has a positive answer if and only if there is no normalized exhaustive pathological submeasure.

While the Maharam problem remains unanswered, it is known (see e.g. [1] or [15]) that there are non-trivial pathological submeasures. In the other direction it is shown in [6] that if ϕ is a non-trivial uniformly exhaustive submeasure then ϕ cannot be pathological. ϕ is uniformly exhaustive if given ϵ > 0 there exists N ∈ N such that whenever {A_{1}, …, A_{N}} are disjoint sets in A then

min_{1≤i≤N} ϕ(A_{i}) < ϵ.

Let us say that a monotone set-function ϕ satisfies an upper p-estimate where 0 < p < ∞ if ϕ^{p} is a submeasure, and a lower p-estimate if ϕ^{p} is a supermeasure. If ϕ is a normalized submeasure which satisfies a lower p-estimate for some 1 < p < ∞ then ϕ is uniformly exhaustive and hence by results of [6] there is a non-trivial measure λ with 0 ≤ λ ≤ ϕ. In Section 2 we prove this by a direct argument which yields a quantitative estimate that λ can be chosen so that

λ(Ω) ≥ 2(2^{p} − 1)^{−1/p} − 1.

Notice that the expression on the right tends to one as p → 1 so this result can be regarded for p close to 1 as a perturbation result. The dual result for supermeasures (Theorem 2.2) is that if a normalized supermeasure ϕ satisfies an upper p-estimate where 0 < p < 1 then there is a ϕ−continuous measure λ with λ ≥ ϕ and

λ(Ω) ≤ 2(2^{p} − 1)^{−1/p} − 1.

While we believe these results with their relatively simple proofs have interest in their own right, one of our motivations for considering them was to use them in the study of some questions concerning quasi-Banach lattices, or function spaces.

It is well-known ([7]) that a Banach lattice X with a (crude) upper p-estimate is rconvex for every 0 < r < p (for the definitions, see Section 3). This result does not hold for arbitrary quasi-Banach lattices [2]; a quasi-Banach lattice need not be r-convex for any r < ∞. However, it is shown in [2] that if X has a crude lower q-estimate for some q < ∞ then the result is true. We provide first a simple proof of this fact, only depending on the arguments of Section 2. We then investigate this result further, motivated by the fact that if X satisfies a strict upper p-estimate (i.e. with constant one) and a strict lower p-estimate then X is p-convex (and in fact isometric to an L_{p}(μ)−space.) We thus try to estimate the constant of r-convexity M^{(r)}(X) when 0 < r < p and X has a strict upper p-estimate and a strict lower q-estimate where p, q are close. We find that an estimate of the form

log M^{(r)}(X) ≤ cθ(1 + | log θ|)

where c = c(r, p) and θ = q/p − 1. We show by example that such an estimate is best possible. Let us remark that in the case r = 1 < p < q the constant M^{(1)}(X) measures the distance (in the Banach-Mazur sense) of the space X from a Banach lattice.

Finally in Section 4 we apply these results to give extensions of some factorization theorems of Pisier [13] to the non-locally convex setting. Pisier showed the existence of a constant B = B(p) so that if X is a Banach space and T : C(Ω) → X is bounded satisfying for a suitable constant C and all disjointly supported functions f_{1}, …, f_{n} ∈ C(Ω)

(∑_{k=1}^{n} ∥T f_{k}∥^{p})^{1/p} ≤ C max_{1≤k≤n} ∥f_{k}∥

then there is a proabability measure μ on Ω so that for f ∈ C(Ω)

∥T f∥ ≤ BC∥f∥_{L_{p,1}(μ)}

where L_{p,1}(μ) denotes a the Lorentz space L_{p,1} with respect to μ.

Pisier’s approach in [13] uses duality and so cannot be used in the case when X is a quasi-Banach space. Nevertheless the result can be extended and we prove that if 0 < r < 1 there is a constant B = B(r, p) so that if X is r-normable then there exists a probability measure μ so that for all f ∈ C(Ω),

∥T f∥ ≤ BC∥f∥_{L_{p,r}(μ)}.

We apply these results to show if X is a quasi-Banach space of cotype two then any operator T : C(Ω) → X is 2-absolutely summing and so factorizes through a Hilbert space. We conclude by presenting a dual result and make a general conjecture that if X and Y are quasi-Banach spaces such that X^{∗} and Y have cotype two and T : X → Y is an approximable linear operator then T factorizes through a Hilbert space.

2. Submeasures and supermeasures.

Let us define for 0 < p < ∞,

K_{p} = \frac{2}{(2^{p} − 1)^{1/p}} − 1.

Notice that for p close to 1 we have K_{p} ∼ 1 − 4(p − 1) log 2 while for p large we have K_{p} ∼ p^{−1}2^{−p}.

We now state our main result on submeasures with a lower estimate (see Section 1 for the definitions).

Theorem 2.1. Let A be an algebra of subsets of Ω. Suppose that ϕ is a normalized submeasure on A, which satisfies a lower p-estimate where 1 < p < ∞. Then there is measure λ on A with 0 ≤ λ ≤ ϕ and λ(Ω) ≥ K_{p}.

Proof: By an elementary compactness argument we need only prove the result for the case when Ω is finite and A = 2^{Ω}. We fix such an Ω.

Let γ be the greatest constant such that, whenever ϕ is a normalized submeasure on Ω satisfying a lower p-estimate then there is a measure λ with 0 ≤ λ ≤ ϕ and ϕ(Ω) ≥ γ. It follows from a simple compactness argument that there is a normalized submeasure ϕ satisfying a lower p-estimate for which this constant is attained; that is if λ is a measure with 0 ≤ λ ≤ ϕ then λ(Ω) ≤ γ. We choose this ϕ and then pick an optimal measure λ with 0 ≤ λ ≤ ϕ and λ(Ω) = γ.

Let δ = (2^{p} − 1)^{−1/p}. Let E be a maximal subset of Ω such that λ(E) ≥ δϕ(E) and let F = Ω \ E. Suppose A ⊂ F; then

λ(A) + λ(E) = λ(A ∪ E) ≤ δϕ(A ∪ E) ≤ δ(ϕ(A) + ϕ(E)),

and so

(1)  λ(A) ≤ δϕ(A).

Let q be the conjugate index of p, i.e. p^{−1} + q^{−1} = 1. Let ν be any measure on A so that 0 ≤ ν ≤ ϕ. Suppose c_{1}, c_{2} ≥ 0 are such that c_{1}^{q} + c_{2}^{q} = 1. Consider the measure

μ(A) = c_{1}λ(A ∩ E) + c_{2}ν(A ∩ F).

Then for any A

μ(A) ≤ (c_{1}^{q} + c_{2}^{q})^{1/q}(λ(A ∩ E)^{p} + ν(A ∩ F)^{p})^{1/p}
≤ (ϕ(A ∩ E)^{p} + ϕ(A ∩ F)^{p})^{1/p} ≤ ϕ(A).

Hence μ(Ω) ≤ γ which translates as

c_{1}λ(E) + c_{2}ν(F) ≤ γ.

Taking the supremum over all c_{1}, c_{2}, we have:

(2)  λ(E)^{p} + ν(F)^{p} ≤ γ^{p}.

Now take ν(A) = δ^{−1}λ(A ∩ F). It follows from (1) that ν ≤ ϕ and hence from (2),

λ(E)^{p} + δ^{−p}λ(F)^{p} ≤ γ^{p}.

If we set t = λ(E)/γ then

t^{p} + (2^{p} − 1)(1 − t)^{p} ≤ 1

and it follows by calculus that t ≤ \frac{1}{2}. Hence λ(E) ≤ γ/2.

Now, consider the submeasure ψ(A) = ϕ(A ∩ F). By hypothesis on γ there exists a positive measure ν on Ω such that 0 ≤ ν ≤ ψ and ν(Ω) ≥ γψ(Ω). Thus for all A we have 0 ≤ ν(A) ≤ ϕ(A ∩ F), ν(E) = 0 and ν(F) = ν(Ω) ≥ γϕ(F). Returning to equation (2) we have:

λ(E)^{p} + γ^{p}ϕ(F)^{p} ≤ γ^{p}.

However ϕ(F) ≥ 1 − ϕ(E) ≥ 1 − δ^{−1}λ(E). Thus, recalling that t = λ(E)/γ

t^{p} + (1 − δ^{−1}γt)^{p} ≤ 1,

which simplifies to

γ ≥ δ (\frac{1 − (1 − t^{p})^{1/p}}{t} ) .

Since t ≥ \frac{1}{2} it follows again by calculus arguments that the right-hand side is minimized when t = \frac{1}{2} and then

γ ≥ 2δ (1 − (1 − 2^{−p})^{1/p}) = K_{p}

and this completes the proof.

In almost the same manner, we can prove the dual statement for supermeasures.

Theorem 2.2. Let A be an algebra of subsets of a set Ω. Suppose 0 < p < 1 and that ϕ is a normalized supermeasure on Ω which satisfies an upper p-estimate. Then there is a ϕ-continuous measure λ on A such that λ ≥ ϕ, λ(Ω) ≤ K_{p}.

Proof: We first prove the existence of some measure λ with λ ≥ ϕ and λ(Ω) ≤ K_{p} without requiring continuity. As in the preceding proof it will suffice to consider the case when Ω is finite and A = 2^{Ω}. In this case there is a least constant γ < ∞ with the property that if ϕ is a normalized supermeasure on Ω then there is a measure λ ≥ ϕ with λ(Ω) ≤ γ. We again may choose an extremal ϕ and associated extremal λ for which λ(Ω) = γ.

Define δ = (2^{p} − 1)^{−1/p} > 1 we now let E be a maximal subset so that λ(E) ≤ δϕ(E) and defining F = Ω \ E we obtain in this case that if A ⊂ F then λ(A) ≥ ϕ(A).

In this case let q be defined by \frac{1}{q} = \frac{1}{p} − 1. Let ν be any measure on A such that ν(A) ≥ ϕ(A) whenever A ⊂ F. Suppose c_{1}, c_{2} > 0 satisfy c_{1}^{−q} + c_{2}^{−q} = 1. Consider the measure

μ(A) = c_{1}λ(A ∩ E) + c_{2}ν(A ∩ F).

Then for any A,

ϕ(A) ≤ (ϕ(A ∩ E)^{p} + ϕ(A ∩ F)^{p})^{1/p}
≤ (λ(A ∩ E)^{p} + ν(A ∩ F)^{p})^{1/p}
≤ (c_{1}λ(A ∩ E) + c_{2}ν(A ∩ F))(c_{1}^{−q} + c_{2}^{−q})
= μ(A).

Hence μ(Ω) ≥ γ and so

c_{1}λ(E) + c_{2}ν(F) ≥ γ.

Minimizing over c_{1}, c_{2} yields

(3)  λ(E)^{p} + ν(F)^{p} ≥ γ^{p}.

In particular if we let ν(A) = δ^{−1}λ(A ∩ F) and set t = λ(E)/γ we obtain

t^{p} + (2^{p} − 1)(1 − t)^{p} ≥ 1.

Since in this case p < 1 we are led to the conclusion that t ≥ \frac{1}{2}.

Next we consider the supermeasure ψ(A) = ϕ(A ∩ F) and deduce the existence of a measure ν ≥ ψ with ν(Ω) ≤ γψ(Ω) = γϕ(F). In this case (3) gives that

λ(E)^{p} + γ^{p}ϕ(F)^{p} ≥ γ^{p}.

Now as ϕ(F) ≤ 1 − ϕ(E) ≤ 1 − δ^{−1}λ(E) we have:

t^{p} + (1 − δ^{−1}γt)^{p} ≥ 1

and this again leads by simple calculus to the fact that γ ≤ K_{p}. This then completes the proof if we do not require continuity of λ.

Now suppose ϕ is a normalized supermeasure on A satisfying an upper p-estimate. Let λ be a minimal measure subject to the conditions λ ≥ ϕ and λ(Ω) ≤ K_{p}. (It follows from an argument based on Zorn’s Lemma that such a minimal measure exists). Suppose lim_{n→∞} ϕ(F_{n}) = 0. Consider the measures λ_{n}(A) = λ(A ∩ E_{n}) where E_{n} = Ω \ F_{n}. Let U be any free ultrafilter on the natural numbers and define λ_{U}(A) = lim_{U} λ_{n}(A). Clearly λ_{U} ≤ λ. Now for any A

λ_{n}(A) = λ(A ∩ E_{n}) ≥ ϕ(A ∩ E_{n}) ≥ (ϕ(A)^{p} − ϕ(F_{n}))^{1/p}.

Hence λ_{U} = λ by minimality. Thus lim_{U} λ(F_{n}) = 0 for every such ultrafilter and this means lim_{n→∞} λ(F_{n}) = 0.

The following corollary is proved for more general uniformly exhaustive submeasures in [6].

Corollary 2.3. Let A be an algebra of subsets of Ω and let ϕ be a submeasure on Ω such that for some constant c > 0 and some q < ∞, we have:

ϕ(A_{1} ∪ ⋯ ∪ A_{n}) ≥ c(ϕ^{q}(A_{1}) + ⋯ + ϕ^{q}(A_{n}))^{1/q}

whenever A_{1}, …, A_{n} are disjoint. Then there is a measure μ on A such that μ and ϕ are equivalent.

Proof: Define ψ by

ψ(A) = sup(∑_{k=1}^{n} ϕ^{q}(A_{k}))

where the supremum is computed over all n and all disjoint (A_{1}, …, A_{n}) so that A = ∪_{k=1}^{n}A_{k}. It is not difficult to show that ψ is a supermeasure satisfying 1/q−upper estimate and clearly c^{q}ψ ≤ ϕ^{q} ≤ ψ. By Theorem 2.2 we can pick a measure μ ≥ ψ which is equivalent to ψ and hence to ϕ.

3. Convexity in lattices.

Let Ω be a compact Hausdorff space and suppose B(Ω) denotes the σ−algebra of Borel subsets of Ω. Let B(Ω) denote the space of all real-valued Borel functions on Ω. An admissible extended-value quasinorm on B(Ω) is a map f → ∥f∥_{X}, (B(Ω) → [0, ∞]) such that:

(a) ∥f∥_{X} ≤ ∥g∥_{X} for all f, g ∈ B(Ω) with |f| ≤ |g| pointwise.

(b) ∥αf∥_{X} = |α|∥f∥_{X} for f ∈ B(Ω), α ∈ R

(c) There is a constant C so that if f, g ≥ 0 have disjoint supports then ∥f + g∥_{X} ≤ C(∥f∥_{X} + ∥g∥_{X}).

(d) There exists a strictly positive u with 0 < ∥u∥_{X} < ∞.

(e) If f_{n} ≥ 0 and f_{n} ↑ f pointwise, then ∥f_{n}∥_{X} → ∥f∥_{X}. The space X = {f : ∥f∥_{X} < ∞} is then a quasi-Banach function space on Ω equipped with the quasi-norm ∥f∥_{X} (more precisely one identifies functions f, g such that ∥f − g∥_{X} = 0). We say that X is order-continuous if, in addition, we have:

(f) If f_{n} ↓ 0 pointwise and ∥f_{1}∥_{X} < ∞ then ∥f_{n}∥_{X} ↓ 0.

Conversely if X is a quasi-Banach lattice which contains no copy of c_{0} and has a weak order-unit then standard representation theorems can be applied to represent X as an order-continuous quasi-Banach function space on some compact Hausdorff space Ω in the above sense. More precisely, if u is a weak order-unit then there is a compact Hausdorff space Ω and a lattice embedding L : C(Ω) → X so that L[0, χ_{Ω}] = [0, u]. Since X contains no copy of c_{0} we can use a result of Thomas [16] to represent L is the form

Lf = ∫_{Ω} fdΦ

where Φ is regular X-valued Borel measure on Ω. This formula then extends L to all bounded Borel functions. We now define the quasi-Banach function space Y by

∥f∥_{Y} = sup_{n} ∥L(min(|f|, nχ_{Ω})∥_{X}

and it may be verified by standard techniques that L extends to a lattice isomorphism of Y onto X (which is an isometry if we assume that the quasi-norm on X is continuous).

For an arbitrary quasi-Banach function space X and 0 < p < ∞ we define the p-convexity constant M^{(p)}(X) to be the least constant (possibly infinite) such that for f_{1}, …, f_{n} ∈ X

∥(∑_{i=1}^{n} |f_{i}|^{p})^{1/p}∥_{X} ≤ M^{(p)}(∑_{i=1}^{n} ∥f_{i}∥_{X}^{p})^{1/p}

and we let the p-concavity constant M_{(p)}(X) be the least constant such that

M_{(p)}∥(∑_{i=1}^{n} |f_{i}|^{p})^{1/p}∥_{X} ≥ (∑_{i=1}^{n} ∥f_{i}∥_{X}^{p})^{1/p}.

We also let M^{(0)}(X) be the least constant such that

∥|f_{1} … f_{n}|^{1/n}∥_{X} ≤ M^{(0)}(∏_{i=1}^{n} ∥f_{i}∥_{X})^{1/n}.

X is called p-convex if M^{(p)}(X) < ∞ and p-concave if M_{p}(X) < ∞; we will say that X is geometrically convex if M^{(0)}(X) < ∞. In [2] X is called L-convex if it is p-convex for some p > 0; it is follows from [2] and [4] that X is L-convex if and only if it is geometrically convex.

Let us now turn to upper and lower estimates. We say X satisfies a crude upper p-estimate with constant a if for any disjoint f_{1}, …, f_{n} we have

∥f_{1} + … f_{n}∥_{X} ≤ a(∑_{i=1}^{n} ∥f_{i}∥_{X}^{p})^{1/p}

and we say that X satisfies an upper p-estimate if a = 1. We say that X satisfies a crude lower q-estimate with constant b if for any disjoint f_{1}, …, f_{n} we have

b∥f_{1} + … f_{n}∥_{X} ≥ (∑_{i=1}^{n} ∥f_{i}∥_{X}^{p})^{1/p};

and X satisfies a lower q-estimate if b = 1.

Lemma 3.1. Suppose 0 < p < q < ∞. If X is a quasi-Banach function space satisfying a crude upper p-estimate with constant a and a crude lower q-estimate with constant b then there is an equivalent function space quasinorm ∥ ∥_{Y} satisfying an upper p and a lower q−estimate with

∥f∥_{X} ≤ ∥f∥_{Y} ≤ ab∥f∥_{X}.

proof: First we define

∥f∥_{W} = inf(∑_{i=1}^{n} ∥fχ_{A_{k}}∥_{X}^{p})^{1/p}

where the infimum is taken over all possible Borel partitions {A_{1}, …, A_{n}} of Ω. It is clear that ∥f∥_{W} ≤ ∥f∥_{X} ≤ a∥f∥_{W} and it can be verified that W satisfies an upper p-estimate and a crude lower q-estimate with constant b. Next we define

∥f∥_{V} = sup(∑_{i=1}^{n} ∥fχ_{A_{k}}∥_{W}^{q})^{1/q}

and finally set ∥f∥_{Y} = a∥f∥_{V}. We omit the details.

We now give a simple proof of the result proved in [2] that any quasi-Banach function space which satisfies a lower estimate is L-convex. We recall first that if μ is any Borel measure on Ω then L_{p,∞}(μ) is the space of all Borel functions such that

∥f∥_{L_{p,∞}(μ)} = sup_{t>0} t(μ{|f| > t})^{1/p} < ∞.

Theorem 3.2. Let X be a p-normable quasi-Banach function space which satisfies a crude lower q-estimate. Then:

(i) X is r-convex for 0 < r < p (ii) There is a measure μ on Ω such that ∥f∥_{X} = 0 if and only if f = 0 μ−a.e.

proof: We may assume by Lemma 3.1 that X has an upper p-estimate and a lower qestimate. Now suppose f_{1}, …, f_{n} ∈ X_{+} and (∑_{i=1}^{n} f_{i}^{r})^{1/r} = f. Consider the submeasure ϕ(A) = ∥fχ_{A}∥_{X}^{p} for A ∈ B(Ω). This has a lower q/p−estimate and hence there is a Borel measure μ with μ(Ω) ≥ K_{q/p}ϕ(Ω) and such that μ(A) ≤ ∥fχ_{A}∥^{p} for any Borel set A. Thus for any g ∈ B(Ω) (note that μ is supported on the set where f is finite)

∥gf^{−1}∥_{L_{p,∞}(μ)} ≤ ∥g∥_{X}.

Now the space L_{p,∞}(μ) is r-convex with M^{(r)}(L_{p,∞}(μ)) ≤ C = C(p, r). Thus

∥f∥_{X} = ϕ(Ω)^{1/p} ≤ K_{q/p}^{−1/p}∥χ_{Ω}∥_{L_{p,∞}(μ)}

≤ CK_{q/p}^{−1/p}(∑_{i=1}^{n} ∥f_{i}f^{−1}∥_{L_{p,∞}(μ)}^{r})^{1/r}

≤ C′(∑_{i=1}^{n} ∥f_{i}∥_{X}^{r})^{1/r}

where C′ = C′(p, q, r). For (ii) let u be a strictly positive function with 0 < ∥u∥_{X} < ∞ and define ϕ(A) = ∥uχ_{A}∥^{p}; then by Corollary 2.3 there is a measure μ equivalent to ϕ and the conclusion follows quickly.

Now suppose μ is any (finite) Borel measure on Ω. We define the Lorentz space L_{p,q} for 0 < p, q < ∞ by

∥f∥_{L_{p,q}} = (∫_{0}^{∞} \frac{q}{p}t^{q/p−1}f^{∗}(t)^{q}dt)^{1/q} .

Here f^{∗} is the decreasing rearrangement of |f| i.e. f^{∗}(t) = inf_{μ(E)≤t} sup_{ω∉E} |f(ω)|. It can easily seen by integration by parts that

∥f∥_{L_{p,q}} = (∫_{0}^{∞} qt^{q−1}μ(|f| > t)^{q/p}dt)^{1/q}

. It is then clear that if p ≤ q then L_{p,q} satisfies an upper p and a lower q-estimate. If p > q then L_{p,q} has an upper q and a lower p−estimate.

Suppose p < q. We define the functional

(4)  ∥f∥_{Λ_{p,q}} = sup(∑_{i=1}^{n}( inf_{ω∈A_{i}} |f(ω)|)^{q}μ(A_{i})^{q/p})^{1/q}

where the supremum is taken over all Borel partitions {A_{1}, …, A_{n}} of Ω.

Proposition 3.3. Suppose 0 < p < q. Then:

(i) The Λ(p, q)−quasi-norm is the smallest admissible quasi-norm which satisfies a lower q-estimate and such that ∥χ_{A}∥ ≥ μ(A)^{1/p} for any Borel set. (ii) If f ∈ B(Ω) and f^{∗} is the decreasing rearrangement of |f| on [0, ∞) then

(5)  ∥f∥_{Λ_{p,q}} = sup_{T}(∑_{j=1}^{n} f^{∗}(τ_{j})^{q}(τ_{j} − τ_{j−1})^{q/p})^{1/q}

where T = {τ_{0} = 0 < τ_{1} < ⋯ < τ_{n}} runs through all possible finite subsets of R. (iii) If f ∈ B(Ω) then

∥f∥_{Λ_{p,q}} ≤ ∥f∥_{L_{p,q}} ≤ ((1 + θ)^{2(1+θ)}θ^{−θ})^{1/q}∥f∥_{Λ_{p,q}}

where 1 + θ = \frac{q}{p}.

Proof: (i) is clear from the definition.

(ii): Suppose f ∈ B(Ω) and let {A_{1}, …, A_{n}} be any Borel partition of Ω. Suppose that 1 ≤ j, k ≤ n. Suppose inf_{A_{j}} |f| ≤ inf_{A_{k}} |f|. Then it is easy to verify that if we let A′_{k} = {ω ∈ A_{j} ∪ A_{k} : |f(ω)| ≥ inf_{A_{k}} |f|} and let A′_{j} = (A_{j} ∪ A_{k}) \ A′_{k} then the partition obtained by replacing A_{j}, A_{k} by A′_{j}, A′_{k} increases the right-hand side of (4). In particular it follows that (5) defines the Λ_{p,q} quasinorm when μ is nonatomic. Further if f^{∗} is constant on an interval [α, β] it suffices to consider T where no τ_{j} lies in (α, β) and this yields the conclusion for general μ.

(iii): The first inequality in (iii) is immediate from (i) since L_{p,q} satisfies a lower q-estimate. For the right-hand inequality we observe that if h = 1 + θ = q/p:

∥f∥_{L_{p,q}} = (∫_{0}^{∞} \frac{q}{p}t^{q/p−1}(f^{∗}(t))^{q}dt)^{1/q}

≤ ( ∑_{n=−∞}^{∞} \frac{q}{p}(h^{n+1} − h^{n})h^{(n+1)(q/p−1)}f^{∗}(h^{n})^{q})^{1/q}

= ( ∑_{n=−∞}^{∞} h(h − 1)h^{q/p−1}h^{q(n+1)/p}f^{∗}(h^{n+1})^{q})^{1/q}

≤ h^{\frac{2}{p}} (h − 1)^{( \frac{1}{q} − \frac{1}{p})}∥f∥_{Λ_{p,q}}.

The result then follows.

Under the hypothesis p > q we define Λ_{p,q} by

∥f∥_{Λ_{p,q}} = inf(∑_{i=1}^{n}( sup_{ω∈A_{i}} |f(ω)|^{q})μ(A_{i})^{q/p})^{1/q}

where the infimum is again computed over all Borel partitions of Ω. Proposition 3.4 now has an analogue whose proof is very similar and we omit most of the details.

Proposition 3.4. Suppose 0 < q < p. Then:

(i) The Λ(p, q)-quasinorm is the largest admissible quasi-norm which satisfies an upper q-estimate and such that ∥χ_{A}∥ ≤ μ(A)^{1/p}, for any Borel set A. (ii) If f ∈ B(Ω) then

∥f∥_{Λ_{p,q}} = inf_{T}(∑_{j=1}^{n} f^{∗}(τ_{j−1})^{q}(τ_{j} − τ_{j−1})^{q/p})^{1/q}

where T = {τ_{0} = 0 < τ_{1} < ⋯ < τ_{n} = μ(Ω)} runs through all possible finite subsets of [0, μ(Ω)]. (iii) If f ∈ B(Ω) then

∥f∥_{L_{p,q}} ≥ ∥f∥_{Λ_{p,q}} ≥ ((1 + θ)^{−2−θ}θ^{θ})^{1/p}∥f∥_{L_{p,q}}

where θ = \frac{p}{q} − 1.

Proof: We will only proof the second inequality in (iii). We define h = 1 + θ = p/q.

∥f∥_{L_{p,q}} = (∫_{0}^{∞} \frac{q}{p}t^{q/p−1}f^{∗}(t)^{q}dt)^{1/q}

≥ ( ∑_{n=−∞}^{∞} \frac{q}{p}(h^{n+1} − h^{n})h^{n(q/p−1)}f^{∗}(h^{(n+1)})^{q})^{1/q}

= ( ∑_{n=−∞}^{∞} h^{−1}(h − 1)h^{(n−1)q/p}f^{∗}(h^{n})^{q})^{1/q}

≥ (h − 1)^{( \frac{1}{q} − \frac{1}{p})}h^{−1/p−1/q}∥f∥_{Λ_{p,q}}.

The result then follows.

We now immediately deduce the following:

Proposition 3.5. Let X be quasi-Banach function space on Ω satisfying a crude upper p-estimate with constant a and a crude lower q-estimate with constant b. Then if f ∈ X_{+} with ∥f∥_{X} = 1:

(i) There is a probability measure μ on Ω such that f > 0 μ−a.e. and if g ∈ X,

∥gf^{−1}∥_{Λ_{p,q}(μ)} ≤ abK_{q/p}^{−1/p}∥g∥_{X}.

(ii) There is a probability measure λ on Ω such that f > 0 λ−a.e. and if g ∈ X

∥g∥_{X} ≤ abK_{p/q}^{1/q}∥gf^{−1}∥_{Λ_{q,p}}(λ).

proof: We first introduce an equivalent quasinorm ∥ ∥_{Y} with an exact upper p and lower q-estimate as in Lemma 3.1 so that ∥g∥_{X} ≤ ∥g∥_{Y} ≤ ab∥g∥_{X} for all g.

(i)As in Theorem 3.2 we consider the submeasure ϕ(A) = ∥fχ_{A}∥_{Y}^{p} . There is a probability measure μ such that

0 ≤ μ ≤ K_{q/p}^{−1} \frac{ϕ(A)}{ϕ(Ω)}

for all Borel sets A. Then for g ∈ X, and any Borel partition {A_{1}, …, A_{n}} of Ω,

(∑_{i=1}^{n}(inf_{A_{i}} |gf^{−1}|^{q})μ(A_{i})^{q/p})^{1/q} ≤ K_{q/p}^{−1/p}ϕ(Ω)^{−1/p}(∑_{i=1}^{n}(inf_{A_{i}} |gf^{−1}|^{q})∥fχ_{A}∥_{Y}^{q})^{1/q}
≤ K_{q/p}^{−1/p}∥f∥_{Y}^{−1}∥g∥_{Y}
≤ K_{q/p}^{−1/p}ab∥g∥_{X}.

Thus (i) follows. The proof of (ii) is very similar. In this case we consider the supermeasure ϕ(A) = ∥fχ_{A}∥_{Y}^{q}. There is a probability measure λ on Ω such that

0 ≤ K_{p/q}^{−1} \frac{ϕ(A)}{ϕ(Ω)} ≤ λ

for all Borel sets A. Thus for g ∈ X, and any Borel partition {A_{1}, …, A_{n}} of Ω,

∥g∥_{X} ≤ ∥g∥_{Y}

≤ (∑_{i=1}^{n}(sup_{A_{i}} |gf^{−1}|^{p})∥fχ_{A_{i}}∥_{Y}^{p})^{1/p}

≤ (∑_{i=1}^{n}(sup_{A_{i}} |gf^{−1}|^{p})ϕ(A_{i})^{p/q})^{1/p}

≤ K_{p/q}^{1/q}∥f∥_{Y} (∑_{i=1}^{n}(sup_{A_{i}} |gf^{−1}|^{p})λ(A_{i})^{p/q})^{1/p}

and the result follows.

Theorem 3.6. Suppose 0 < r < p < ∞. Then there is a constant c = c(r, p) such that if X is a quasi-Banach function space satisfying an upper p-estimate and a lower q−estimate where q/p = 1 + θ < 2 then log M^{(r)}(X) ≤ θ(c + \frac{1}{p}| log θ|).

Proof: We use Proposition 3.5. Suppose f_{1}, …, f_{n} are nonnegative functions in X with ∥f∥_{X} = 1 where f = (∑_{i=1}^{n} f_{i}^{r})^{1/r}. Then there is a probability measure μ on Ω with f > 0 a.e. and such that if g ∈ X

∥gf^{−1}∥_{Λ_{p,q}} ≤ K_{q/p}^{−1/p}∥g∥_{X}.

Notice that K_{q/p}^{−1} ≤ e^{c_{1}θ} for some c_{1}. We also have q/p ≤ e^{θ} and (q − r)/(p − r) ≤ c_{2}θ where c_{2} depends only on p, r. Let w_{i} = f_{i}f^{−1}.

We note first that for any w ≥ 0 in B(Ω) we have

∫ w^{r}dμ = ∫_{0}^{1} w^{∗}(t)^{r}dt

≤ (\frac{q}{p})^{−r/q}∥w∥_{L_{p,q}}^{r}(∫_{0}^{1} t^{− \frac{r(q−p)}{p(q−r)}} dt)^{1−r/q}

≤ \frac{p}{q}(\frac{q − r}{p − r})^{1−r/q}∥w∥_{L_{p,q}}^{r}

≤ e^{c_{3}θ}∥w∥_{L_{p,q}}^{r}

where c_{3} = c_{3}(r, p). Thus

∫ w^{r}dμ ≤ θ^{−rθ/q}e^{c_{4}θ}∥w∥_{Λ_{p,q}}^{r}

where c_{4} = c_{4}(r, p). Applying this to the w_{i} and summing we have

1 ≤ θ^{−rθ/p}e^{c_{5}θ} ∑_{i=1}^{n} ∥f_{i}∥_{X}^{r},

with c_{5} depending only on r, p. The result now follows. Example: We show that the estimate in the previous theorem is essentially best possible. For convenience we consider the case p = 1 and q = 1 + θ. The conclusion of the theorem is that, for 0 < r < p, M^{(r)}(X) ≤ exp(F_{r}(θ)) where

lim_{θ→0} \frac{F_{r}(θ)}{θ| log θ|} = 1.

Since M^{(0)}(X) ≤ M^{(r)}(X) for all r > 0 a similar conclusion is attained in the case r = 0. We establish a converse by considering only the case r = 0.

For θ > 0 we let X = X_{θ} = Λ_{1,q}[0, 1] and we let κ(θ) = M^{(0)}(X). We will set ϕ(θ) = exp(−| log θ|^{1/2}) so that lim_{θ→0} ϕ(θ) = 0. We further set ψ(θ) = (2^{1/q} − 1)^{−2}; then ψ(θ) = 1 + 4θ log 2 + O(θ^{2}).

We will consider the function f = f_{θ} ∈ X defined by f(t) = t^{−1}χ_{[1−ϕ,1]}. It follows easily from the definition of M^{(0)}(X) that

exp(−\frac{1}{ϕ} ∫_{1−ϕ}^{1} log t dt)∥χ_{[1−ϕ,1]}∥_{X} ≤ κ∥f∥_{X}.

To obtain this one derives the integral version of geometric convexity and applies it to suitable rotations of f. Thus if β(θ) = 1 + (1 − ϕ)ϕ^{−1} log(1 − ϕ) then

e^{β}ϕ ≤ κ∥f∥_{X}.

Turning to the estimation of ∥f∥_{X} we note that ∥f∥_{X}^{q} is the supremum of expressions of the form

∑_{j=1}^{n}(τ_{j} − τ_{j−1})^{q}τ_{j}^{−q}

where 1 − ϕ = τ_{0} < τ_{1} < ⋯ < τ_{n} = 1. Now if τ_{j} > ψτ_{j−1} it can be checked that this expression is increased by interpolating (τ_{j}τ_{j−1})^{1/2} into the partition. We there fore may suppose that we consider only partitions where τ_{j} ≤ ψτ_{j−1}. In this case we estimate:

(τ_{j} − τ_{j−1})^{q}τ_{j}^{−q} ≤ (τ_{j} − τ_{j−1})(τ_{j} − τ_{j−1})^{θ}τ_{j−1}^{−1+θ}
≤ (ψ − 1)^{θ}(τ_{j} − τ_{j−1})τ_{j−1}^{−1}

and after summing we get the estimate

∥f∥_{X}^{q} ≤ (ψ − 1)^{θ}| log(1 − ϕ)|.

Thus

κ ≥ e^{β}ϕ(ψ − 1)^{−θ/q}| log(1 − ϕ)|^{−1/q}.

Now for small θ we can estimate | log(1 − ϕ)| ≤ (1 + ϕ)ϕ. Thus

κ^{q} ≥ e^{qβ}ϕ^{θ}(ψ − 1)^{−θ}(1 + ϕ)^{−1}

so that

lim inf_{θ→0} \frac{log κ}{θ| log θ|} ≥ lim inf_{θ→0} \frac{(log ϕ − log(ψ − 1))}{| log θ|} ≥ 1.

Thus we conclude from this calculation and the theorem that

log κ(θ) = −θ log θ + o(θ| log θ|)

as θ → 0.

4. The factorization theorems of Pisier.

We next show how the results of Proposition 3 quickly give extensions of some factorization theorems due to Pisier [13]. Our approach is valid for quasi-Banach spaces since it does not depend on any duality.

Theorem 4.1. Suppose 0 < r ≤ p < q < ∞. Suppose Ω is a compact Hausdorff space and that Y is an r-Banach space. Suppose T : C(Ω) → Y is a bounded linear operator.

Then the following conditions on T are equivalent:

(i) There is a constant C_{1} so that for any f_{1}, …, f_{n} in C(Ω) with disjoint support we have

(∑_{i=1}^{n} ∥T f_{i}∥^{q})^{1/q} ≤ C_{1} max_{1≤i≤n} ∥f_{i}∥.

(ii) There is a constant C_{2} so that for any f_{1}, …, f_{n} we have

(∑_{i=1}^{n} ∥T f_{i}∥^{q})^{1/q} ≤ C_{2}∥(∑_{i=1}^{n} |f_{i}|^{p})^{1/p}∥.

(iii) There is a constant C_{3} and a probability measure μ on Ω such that for all f ∈ C(Ω),

∥T f∥ ≤ C_{3}∥f∥^{1−p/q}(∫ |f|^{p}dμ)^{1/q}

(iv) There is a constant C_{4} and a probability measure μ on Ω such that for all f ∈ C(Ω),

∥T f∥ ≤ C_{4}∥f∥_{L_{q,r}(μ)}.

proof: Some implications are essentially trivial. Thus (iv) ⇒ (iii) and (ii) ⇒ (i); (iii) ⇒ (ii) is easy and we omit it. It remains to show (i) ⇒ (iv).

To do this we first notice that if f_{n} is a sequence of disjointly supported functions in C(Ω) then T f_{n} → 0. Thus the theorem of Thomas [16] already cited allows us to find a regular Y -valued Borel measure Φ on Ω so that

T f = ∫ fdΦ

and use this formula to extend T to the bounded Borel functions B_{∞}(Ω). It is easy to verify the condition (i) remains in effect for disjointly supported bounded Borel functions.

Now we introduce a quasi-Banach function space Z by defining

∥f∥_{Z} = sup (∑_{i=1}^{n} ∥T g∥^{q})^{1/q} ,

where the supremum is over disjoint g_{i} ∈ B_{∞}(Ω) with |g_{i}| ≤ |f|. It is immediate that ∥ ∥_{Z} satisfies an upper r-estimate and a lower q-estimate. Also, we see that χ_{Ω} ∈ Z. Thus by Proposition 3.5 we can find a probability measure μ on Ω so that for some C, C_{4}

∥g∥_{Z} ≤ C∥g∥_{Λ_{q,r}(μ)} ≤ C_{4}∥g∥_{L_{q,r}(μ)}.

We now prove the companion factorization result for operators on L_{p}-spaces.

Theorem 4.2. Suppose 0 < r ≤ q < s < ∞. Let (Ω, μ) be a σ-finite measure space. Let Y be an r-Banach space, and let T : L_{s}(μ) → Y be a bounded linear operator. Then the following conditions are equivalent:

(i) There is a constant C_{1} so that for any disjoint f_{1}, …, f_{n} in L_{s}(μ) we have:

(∑_{i=1}^{n} ∥T f_{i}∥^{q})^{1/q} ≤ C_{1}∥ ∑_{i=1}^{n} |f_{i}|∥_{L_{s}(μ)}.

(ii) There is a constant C_{2} and a probability measure λ on Ω so that for any f ∈ L_{s}(μ) and any Borel set E we have

∥T (fχ_{E})∥ ≤ C_{2}∥f∥_{L_{s}(μ)}λ(E)^{1/q−1/s}

(iii) There is a constant C_{3} and a probability measure λ on Ω so that for any f ∈ L_{s}(μ) and g ∈ B(Ω) with |g| ≤ 1,

∥T (fg)∥ ≤ C_{3}∥f∥_{L_{s}(μ)}∥g∥_{L_{t,r}(λ)}

where 1/t = 1/q − 1/s. (iv) There exists w ∈ L_{1}(μ) with w ≥ 0 and ∫ w dμ = 1 and a constant C_{4} such that for all f ∈ L_{s}(μ)

∥T f∥ ≤ C_{4}∥fw^{−1/s}∥_{L_{q,r}(wμ)}.

proof: We omit the simple proof that (ii) ⇒ (i). Also (iii) ⇒ (ii) and (iv) ⇒ (ii) are obvious.

We first consider (i) ⇒ (iii). For this direction we define a quasi-Banach function space Z by

∥g∥_{Z} = sup (∑_{i=1}^{n} ∥T (f_{i}g)∥^{q})^{1/q} ,

where the supremum is over disjoint f_{i} ∈ L_{s}(μ) with ∥ ∑_{i=1}^{n} |f_{i}|∥_{s} ≤ 1 and f_{i}g ∈ L_{s}(μ). It is clear that Z satisfies an upper r-estimate. We show that Z satisfies a lower t-estimate, where 1/t = 1/q − 1/s. Let us suppose that we have disjoint g_{1}, …, g_{m} ∈ Z such that

∑_{k=1}^{m} ∥g_{k}∥_{Z}^{t} > 1,

and let g = g_{1} + … g_{m}. Then there are f_{ik} such that

∑_{k=1}^{m} (∑_{i=1}^{n} ∥T (f_{ik}g_{k})∥^{q})^{t/q} = 1

and

∥ ∑_{i=1}^{n} f_{ik}∥_{s} ≤ 1.

We let

α_{k} = (∑_{i=1}^{n} ∥T (f_{ik}g_{k})∥^{q})^{t/qs}

and f′_{ik} = α_{k}f_{ik}. Then we have that

∑_{k=1}^{m} ∑_{i=1}^{n} ∥T (f′_{ik}g)∥^{q} ≥ 1,

and

∥ ∑_{k=1}^{m} ∑_{i=1}^{n} f′_{ik}∥_{s}^{s} ≤ ∑_{k=1}^{m} α_{k}^{s} ≤ 1,

that is, ∥g∥_{Z} ≥ 1.

We also notice that χ_{Ω} ∈ Z. Thus by Propositions 3.4 and 3.5, there is a probability measure λ so that for some C_{3} and all bounded g

∥g∥_{Z} ≤ C_{3}∥g∥_{L_{t,r}(λ)}.

Now we show that (ii) ⇒ (iv). We notice that λ is μ−continuous, and so by the Radon-Nikodym theorem, we can find w = dλ/dμ. Then for any measurable set A we have:

∥T (χ_{A}w^{1/s}∥ ≤ C_{2}∥χ_{A}w^{1/s}∥_{L_{s}(μ)}λ(A)^{1/q−1/s} = C_{2}λ(A)^{1/q}.

Now define a function space W by

∥f∥_{W} = sup_{|g|≤|f|,g∈B_{∞}} ∥T (gw^{1/s})∥.

Then W has an upper r-estimate and

∥χ_{A}∥_{W} ≤ C_{2}λ(A)^{1/q}.

Hence by Proposition 3.4

∥f∥_{W} ≤ C_{4}∥f∥_{L_{q,r}(λ)}

for some C_{4}.

To conclude the paper let us observe that Theorem 4.1 can be used to extend other factorization results to non-locally convex spaces. Let us first recall that an operator T : X → Y where X is a Banach space and Y is a quasi-Banach space is called 2-absolutely summing if there is a constant C so that for x_{1}, …, x_{n} ∈ X we have

(∑_{i=1}^{n} ∥T x_{i}∥^{2})^{1/2} ≤ C max_{∥x^{∗}∥≤1}(∑_{i=1}^{n} |x^{∗}(x_{i})|^{2})^{1/2}.

A quasi-Banach space X is of cotype p if there is a constant C so that if x_{1}, …, x_{n} ∈ X then

Ave _{ϵ_{i}=±1}∥ ∑_{i=1}^{n} ϵ_{i}x_{i}∥ ≤ C(∑_{i=1}^{n} ∥x_{i}∥^{p})^{1/p}.

For the following theorem for Banach spaces see [12], p. 62.

Theorem 4.3. Suppose Ω is a compact Hausdorff space and Y is a quasi-Banach space with cotype two. Suppose T : C(Ω) → Y is a bounded operator. Then T is 2-absolutely summing and hence there is a probability measure μ on Ω and a constant C so that ∥T f∥ ≤ C∥f∥_{L_{2}(μ)} for f ∈ C(Ω).

Proof: We may assume that X is an r-Banach space where 0 < r < 1. We first note that if f_{1}, …, f_{n} ∈ X then since X has cotype two,

(∑_{i=1}^{n} ∥T f_{i}∥^{2})^{1/2} ≤ C_{1}Ave_{ϵ_{i}=±1}∥ ∑_{i=1}^{n} ϵ_{i}T f_{i}∥

≤ C_{1}∥T ∥Ave_{ϵ_{i}=±1}∥ ∑_{i=1}^{n} ϵ_{i}f_{i}∥

≤ C_{1}∥T ∥∥ ∑_{i=1}^{n} |f_{i}|∥

where C_{1} depends on the cotype two constant of X. Applying Theorem 4.1, we see that there is a probability measure ν on Ω and a constant C_{2} so that ∥T f∥ ≤ C_{2}∥f∥_{L_{2,r}(ν)} for f ∈ C(Ω). In particular it follows that ∥T f∥ ≤ C_{3}∥f∥_{L_{4}(ν)}. From this we conclude that if f_{1}, …, f_{n} ∈ C(Ω), using Khintchine’s inequality,

(∑_{i=1}^{n} ∥T f_{i}∥^{2})^{1/2} ≤ C_{1}Ave_{ϵ_{i}=±1}∥ ∑_{i=1}^{n} ϵ_{i}T f_{i}∥

≤ C_{4}Ave_{ϵ_{i}=±1}∥ ∑_{i=1}^{n} ϵ_{i}f_{i}∥_{L_{4}(ν)}

≤ C_{5}∥(∑_{i=1}^{n} |f_{i}|^{2})^{1/2}∥_{L_{4}(ν)}

≤ C_{5}∥(∑_{i=1}^{n} |f_{i}|^{2})^{1/2}∥_{C(Ω)}

so that T is 2-absolutely summing. It now follows from the Grothendieck-Pietsch Factorization Theorem (which applies to non-locally convex X) that there is a probability measure μ on Ω satisfying the conclusions of the Theorem. Remarks: We conclude with some comments on Theorem 4.3. We remark first that the theorem, taking X = L_{1}, gives a circuitous proof of Grothendieck’s inequality, which is equivalent to the assertion that every operator from C(Ω) to L_{1} is 2-absolutely summing.

We also note that there are, by now, many known non-locally convex spaces of cotype two. The most natural examples are the spaces L_{p} when 0 < p < 1; in this case, Theorem 4.3 is known, and is a consequence of work of Maurey [8] (see [17], p. 271). However, more recently Pisier [14] has shown that the spaces L_{p}/H_{p} have cotype two when p < 1 and it follows from work of Xu [18] that the Schatten ideals S_{p} have cotype two when p < 1, and for such examples Theorem 4.3 is apparently new.

We conclude by noting a dual result and then make a conjecture based on these observations. First let us recall that if X is a quasi-Banach space then its dual X^{∗} defined in the usual way is a Banach space; here X^{∗} need not separate the points of X and may indeed reduce to zero. We define the canonical map (not necessarily injective) j : X → X^{∗∗}. The Banach envelope of X is the closure X̂ of j(X).

We shall say that an operator T : X → Y is strongly approximable if T is in the smallest subspace A of L(X, Y) containing the finite-rank operators and closed under the pointwise convergence of uniformly bounded nets.

The following theorem is essentially known.

Theorem 4.4. Suppose (Ω, μ) is a σ−finite measure space. Let X be a quasi-Banach space such that X^{∗} has cotype q < ∞. Then, for 0 < p < 1, there is a constant C = C(p, X) so that if T : X → L_{p}(μ) is a strongly approximable operator then there exists w ≥ 0 with ∫ w^{s}dμ ≤ C, where 1/s = 1/p − 1, and such that ∥w^{−1}T x∥_{L_{1}(μ)} ≤ C∥T ∥∥x∥ for x ∈ X.

Remark: If X is a Banach space, then this theorem is due to Mezrag [9] [10] with no approximability assumptions on T. If X is not locally convex this result is essentially proved in [3] and we here show how to obtain the actual statement from the equivalent Theorem 2.2 of [3]. Note also that for spaces X with trivial dual, the theorem holds vacuously since the only strongly approximable operators are identically zero. Proof: It is shown in Theorem 2.2 of [3] that given ϵ > 0 there exists C_{0}(ϵ) so that, for any probability measure ν, if T : X → L_{p}(ν) is strongly approximable then there exists a Borel subset E of Ω so that ν(E) > 1 − ϵ and ∫_{E} |T x| dμ ≤ C_{0}∥T ∥∥x∥ for x ∈ X. Let us fix ϵ = 1/2 and let C_{0} = C_{0}(1/2). Suppose x_{1}, …, x_{n} ∈ X and let f = ∑_{i=1}^{n} |T x_{i}|. If ∥f∥_{p} > 0, define ν = ∥f∥_{p}^{−p}|f|^{p}μ. Consider the operator S : X → L_{p}(ν) defined by Sx = |f|^{−1}T x (set Sx(ω) = 0 when f(ω) = 0.) Then ∥S∥ ≤ ∥f∥_{p}^{−1}∥T ∥. Thus there is a Borel subset E of Ω with ν(E) > 1/2 and so that ∫_{E} |Sx| dν ≤ C_{0}∥f∥_{p}^{−1}∥T ∥∥x∥ for x ∈ X. Now

\frac{1}{2} ≤ ν(E) = ∫_{E} ∑_{i=1}^{n} |Sx_{i}| dν

≤ C_{0}∥f∥_{p}^{−1}∥T ∥ ∑_{i=1}^{n} ∥x_{i}∥.

and so we obtain an inequality

(∫ (∑_{i=1}^{n} |T x_{i}|)^{p}dμ)^{1/p} ≤ 2C_{0}∥T ∥ ∑_{i=1}^{n} ∥x_{i}∥,

where C_{1} depends only on p, X. Now by the factorization results of Maurey [8] (see [17] p.

264) we obtain the desired conclusion.

Theorem 4.5. Let X be a quasi-Banach space such that X^{∗} has cotype two. Suppose Ω is a compact Hausdorff space and μ is a σ-finite measure on Ω. Then for 0 < p < 1, there is a constant C = C(p, X) so that if T : X → L_{p}(μ) is a strongly approximable bounded operator, then there exists v ≥ 0, with ∫ v^{t}dμ ≤ C where 1/t = 1/p − 1/2, and such that ∥v^{−1}T x∥_{L_{2}(μ)} ≤ C∥T ∥∥x∥ for x ∈ X.

Proof: By Theorem 4.4 we can find w ≥ 0 with ∫ w^{s}dμ ≤ C_{0} such that ∥w^{−1}T x∥_{L_{1}} ≤ C_{0}∥T ∥∥x∥. Now since L_{1} is a Banach space we have ∥wT x∥_{L_{1}} ≤ C_{0}∥T ∥∥x∥X̂ so that wT = Sj where S : X̂ → L_{1} is bounded with ∥S∥ ≤ C_{0}∥T ∥. Since X̂^{∗} = X^{∗} has cotype two and L_{1} has cotype two we can factor S through a Hilbert space [11] and then apply Maurey’s factorization results to obtain u ≥ 0 with ∫ u^{2}dμ ≤ C_{1} and such that ∥u^{−1}Sx∥_{L_{2}} ≤ C_{1}∥T ∥∥x∥ for x ∈ X. Letting v = uw completes the proof. Remarks: We discuss a question motivated by Theorems 4.3 and 4.5. An operator T : X → Y is called approximable if given any compact set K ⊂ X and any ϵ > 0 there exists a finite-rank operator F : X → F with ∥T x − F x∥ < ϵ for x ∈ K. Pisier has shown that if X, Y are Banach spaces such that X^{∗} and Y have cotype two and T : X → Y is an approximable linear operator then T factorizes through a Hilbert space (see [11], [12]). Does the same result hold if we assume X, Y are quasi-Banach spaces? Theorems 4.3 and 4.5 provide some evidence that this perhaps is true.

References.

1. J.P.R. Christensen and W. Herer, On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann. 213 (1975) 203-210.

2. N.J. Kalton, Convexity conditions on non-locally convex lattices, Glasgow Math. J. 25 (1984) 141-152.

3. N.J. Kalton, Banach envelopes of non-locally convex spaces, Canad. J. Math. 38 (1986) 65-86.

4. N.J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Math. 84 (1987) 297-324.

5. N.J. Kalton, The Maharam problem, Expose 18, Seminaire Initiation a l’Analyse, 1988/9 (G. Choquet, G. Godefroy, M. Rogalski, J. Saint Raymond, editors) Université Paris VI/VII, 1991.

6. N.J. Kalton and J.W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983) 803-816.

7. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, II, Function spaces Springer Verlag, Berlin 1979.

8. B. Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans un espace L^{p}, Astérisque 11, 1974.

9. L. Mezrag, Théorèmes de factorisation, et de prolongement pour les opérateurs à vauleurs dans les espaces L^{p}, pour p < 1, Thesis, Université Paris VI, 1984.

10. L. Mezrag, Théorèmes de factorisation, et de prolongement pour les opérateurs à vauleurs dans les espaces L^{p}, pour p < 1, C. R. Acad. Sci. Paris 300 (1985) 299-302.

11. G. Pisier, Une théorème sur les opérateurs entre espaces de Banach qui se factorisent par un espace de Hilbert, Ann. École Norm. Sup. 13 (1980) 23-43.

12. G. Pisier, Factorization of linear operators and geometry of Banach spaces, NSF-CBMS Regional Conference Series No. 60, Providence, 1986.

13. G. Pisier, Factorization of operators through L_{p,∞} or L_{p,1} and non-commutative generalizations, Math. Ann. 276 (1986) 105-136.

14. G. Pisier, A simple proof of a theorem of Jean Bourgain, to appear.

15. M. Talagrand, A simple example of a pathological submeasure, Math. Ann. 252 (1980) 97-102.

16. G.E.F. Thomas, On Radon maps with values in arbitrary topological vector spaces and their integral extensions, unpublished paper, 1972.

17. P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, 1991.

18. Q. Xu, Applications du théorèmes de factorisation pour des fonctions à valeurs operateurs, Studia Math. 95 (1990) 273-292.