AN APPLICATION OF A PISIER FACTORIZATION THEOREM TO THE PETTIS INTEGRAL

S.J. Dilworth and Maria Girardi

Séminaire d’Initiation à l’Analyse 1994-1995 (1996) 2001–2009

There are several generalizations of the space L_{1}(R) of Lebesgue integrable functions taking values in the real numbers R (and defined on the usual Lebesgue measure space (Ω, Σ, μ) on [0, 1] ) to a space of strongly-measurable “integrable” (suitably formulated) functions taking values in a Banach space 𝔛.

The most common generalization is the space L_{1}(𝔛) of Bochner-Lebesgue integrable functions. Using the fact [P1, Theorem 1.1] that a strongly-measurable function is essentially separably-valued, one can easily extend Lebesgue’s Differentiation Theorem from L_{1}(R) to L_{1}(𝔛). Specifically [B; cf. DU, Theorem II.2.9], if f ∈ L_{1}(𝔛), then

lim_{h→0} \frac{1}{h} ∫_{t}^{t+h} ∥f(ω) − f(t)∥ dμ(ω) = 0

and so

lim_{h→0} \frac{1}{h} ∫_{t}^{t+h} f(ω) dμ(ω) = f(t)

for almost all t in Ω.

Another generalization of L_{1}(R) is the space P_{1}(𝔛) of strongly-measurable Pettis integrable functions. A function f : Ω → 𝔛 is Pettis integrable if for each E ∈ Σ there is an element x_{E} ∈ 𝔛 satisfying

x^{∗}(x_{E}) = ∫_{E} x^{∗}fdμ

for each x^{∗} in the dual space 𝔛^{∗} of 𝔛. The element x_{E} is called the Pettis integral of f over E and we write

P − ∫_{E} f dμ = x_{E} .

It is clear that L_{1}(𝔛) ⊂ P_{1}(𝔛), while the reverse inclusion holds if and only if 𝔛 is finite dimensional (see e.g. [DG]).

If f ∈ P_{1}(𝔛), then for each x^{∗} ∈ 𝔛^{∗} the function x^{∗}f ∈ L_{1}(R) and so there exists a set A (which depends on x^{∗}) of full measure such that

lim_{h→0} \frac{1}{h} ∫_{t}^{t+h} x^{∗}f(ω) dμ(ω) = x^{∗}f(t)

Typeset by AMS-TEX

for each t ∈ A. In his paper [P1] introducing the Pettis integral, Pettis phrased this by saying that the Pettis integral of a function in P_{1}(𝔛) is pseudo-differentiable . He closed his paper by asking whether the Pettis integral of a function f in P_{1}(𝔛) enjoys the stronger property of being a.e. weakly differentiable ; that is, does there necessarily exist a set A (independent of x^{∗}) of full measure such that

lim_{h→0} \frac{1}{h} ∫_{t}^{t+h} x^{∗}f(ω) dμ(ω) = x^{∗}f(t)

for each t ∈ A and x^{∗} ∈ 𝔛^{∗}, or such that (which is the same thing of course)

weak − lim_{h→0} \frac{1}{h} P − ∫_{t}^{t+h} f(ω) dμ(ω) = f(t)

for each t ∈ A.

If 𝔛 is finite dimensional, then the Pettis integral of a function in P_{1}(𝔛) is a.e. weakly differentiable. R.S. Phillips [Ph] (for 𝔛 = ℓ_{2}) and M.E. Munroe [M] (for 𝔛 = C[0, 1]) each constructed an example of a function in P_{1}(𝔛) whose Pettis integral is not a.e. weakly differentiable. G.E.F. Thomas [T, p. 131] conjectured that such a function in P_{1}(𝔛) exists for every infinite-dimensional Banach space 𝔛.

At the recent May 1993 Kent State University Functional Analysis Conference, Joe Diestel requested a further investigation into Pettis’s question. Independently, V. Kadets [K] recently constructed, for each infinite-dimensional Banach space 𝔛, a function in P_{1}(𝔛) whose Pettis integral fails to be a.e. weakly differentiable; specifically, it fails to be weakly differentiable on a set of positive, but not full, measure.

The first theorem of this paper constructs, for each infinite-dimensional Banach space 𝔛, a function in P_{1}(𝔛) whose Pettis integral is nowhere weakly differentiable. This theorem also addresses the degree of nondifferentiability of the Pettis integral. Our second theorem shows, for arbitrary Banach spaces, that the functions which we construct are close to being optimal with respect to their degree of nondifferentiability. The proof of this theorem uses a factorization theorem of Pisier [P]. From these two theorems it follows (Corollaries 3 and 4) that the cotype of a space is closely tied to the degree of nondifferentiability of the Pettis integral.

Theorem 2 was shown to us by Nigel Kalton in answer to a question posed in a preliminary version of this paper. We are grateful to him for permission to include this result here.

To state our first result we introduce the collection Ψ of all increasing functions ψ: [0, ∞) → [0, ∞) satisfying the growth condition

∑_{n=1}^{∞} ψ(2^{−p_{n−1}}) \sqrt{2^{p_{n}}} < ∞ ,  (†)

for some increasing sequence {p_{n}}_{n=0}^{∞} of integers. Examples of functions in Ψ are

ψ(s) = s^{\frac{1}{2} +ϵ},
ψ(s) = s^{\frac{1}{2}} [\frac{1}{log (1/s)}]^{1+ϵ}  and  ψ(s) = s^{\frac{1}{2}} [\frac{1}{log (1/s)}] [\frac{1}{log log (1/s)}]^{1+ϵ}

for p_{n} = n and any ϵ > 0.

Theorem 1. Let 𝔛 be an infinite-dimensional Banach space. For each ψ ∈ Ψ, there exists f ∈ P_{1}(𝔛) such that

∥P − ∫_{I} f dμ∥_{𝔛} ⩾ ψ (μ (I))  (‡)

for each interval I contained in [0, 1].

Remark. Taking ψ(t) = t^{\frac{3}{4}} gives a Pettis integrable function f such that for each t ∈ Ω,

lim_{h→0} ∥\frac{1}{h} P − ∫_{t}^{t+h} f(ω) dμ(ω)∥_{𝔛} = ∞ .

If the Pettis integral of this f were weakly differentiable at t, then the above limit would be finite.

Sketch of Proof. Let {I_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}} be the dyadic intervals on [0, 1], i.e.

I_{k}^{n} = [\frac{k − 1}{2^{n}}, \frac{k}{2^{n}} ) .

Define inductively a collection {A_{k}^{n} : n = 0, 1, …; k = 1, …, 2^{n}} of disjoint sets of strictly positive measure such that A_{k}^{n} ⊂ I_{k}^{n} (e.g. appropriately chosen “fat Cantor” sets).

Fix K > 1. By a theorem of Mazur there is a basic sequence {x_{n}} in 𝔛 with basis constant at most K. Take a blocking {F_{n}} of the basis with each subspace F_{n} of large enough dimension to find (using the finite-dimensional version of Dvoretzky’s Theorem [D]) a 2^{n}-dimensional subspace E_{n} of F_{n} such that the Banach-Mazur distance between E_{n} and ℓ_{2}^{2^{n}} is less than 2. Note that {E_{n}} forms a finite-dimensional decomposition. Next find operators T_{n}: ℓ_{2}^{2^{n}} → E_{n} such that ∥T_{n}∥ ⩽ 2 and ∥T_{n}^{−1}∥ = 1. Let {u_{k}^{n}: k = 1, … 2^{n}} be the standard unit vectors of ℓ_{2}^{2^{n}} and let e_{k}^{n} ≡ T_{n}u_{k}^{n}.

By the growth condition (†) on ψ, there is an increasing sequence {p_{n}}_{n=0}^{∞} of integers, with p_{0} = 0, satisfying

∑_{n=1}^{∞} ψ(4 · 2^{−p_{n−1}}) \sqrt{2^{p_{n}}} < ∞ .

Define f: [0, 1] → 𝔛 by

f(ω) = ∑_{n=1}^{∞} ∑_{k=1}^{2^{n}} c_{n} \frac{1_{A_{k}^{n}}(ω)}{μ(A_{k}^{n})} e_{k}^{n} ,

where

c_{m} = 2K [ψ (4 · 2^{−p_{n−1}})] · δ_{m,p_{n}} ,

(here δ_{j,k} is the usual Kronecker delta symbol). Clearly, f is strongly measurable. It is straightforward to verify that the Pettis integral of f is

P − ∫_{E} f dμ = ∑_{n=1}^{∞} ∑_{k=1}^{2^{n}} c_{n} (∫_{E} \frac{1_{A_{k}^{n}}}{μ(A_{k}^{n})} dμ) e_{k}^{n} .  (∗)

Next fix an interval I ∈ Σ. Find a dyadic interval I_{j}^{m} ⊂ I such that 4 μ(I_{j}^{m}) ⩾ μ(I) and then find n such that p_{n−1} ⩽ m < p_{n}. Let P be the natural projection from ∑ ⊕E_{j} onto E_{p_{n}}. It follows that

2K ∥P − ∫_{I} f dμ∥_{𝔛} ⩾ c_{p_{n}} [∑_{k=1}^{2^{pn}} |∫_{I} \frac{1_{A_{k}^{pn}}}{μ(A_{k}^{p_{n}})} dμ|^{2}]^{\frac{1}{2}} ,

and so since A_{k}^{p_{n}} ⊂ I_{k}^{p_{n}} ⊂ I_{j}^{m} ⊂ I for some k,

2K ∥P − ∫_{I} f dμ∥_{𝔛} ⩾ c_{p_{n}} = 2K ψ (4 · 2^{−p_{n−1}}) .

But ψ is increasing and 4 · 2^{−p_{n−1}} ⩾ 4 · 2^{−m} ⩾ μ(I) and so (‡) holds. Thus f satisfies the statement of the theorem.

The functions in Ψ can be viewed as indicators of the degree of nondifferentiability (i.e. the poor “averaging behavior”) of the indefinite Pettis integral. For instance, taking

ψ(s) = s^{\frac{1}{2}} [\frac{1}{log (1/s)}]^{1+ϵ} ,

we deduce from Theorem 1 that there exists f ∈ P_{1}(𝔛) such that, not only do we have

lim_{h→0} ∥\frac{1}{h} P − ∫_{t}^{t+h} f(ω) dμ(ω)∥_{𝔛} = ∞ ,

but even worse,

lim_{h→0} h^{\frac{1}{2}} · [log (\frac{1}{h})]^{1+ϵ} ∥\frac{1}{h} P − ∫_{t}^{t+h} f(ω) dμ(ω)∥_{𝔛} = ∞

for all t ∈ Ω.

The next theorem shows that Theorem 1 comes close to describing the worst type of averaging behavior of the Pettis integral that can occur in an arbitrary infinite-dimensional Banach space. In particular, it shows that, for spaces on which the identity operator is (2, 1)-summing (i.e., spaces with the Orlicz property), Theorem 1 fails to hold for the function ψ(s) = s^{\frac{1}{2}} . Thus, the growth condition (†) on ψ ∈ Ψ can not be replaced by ψ(s) = O(s^{\frac{1}{2}} ) as s → 0. We do not know, however, whether it can be replaced by ψ(s) = o(s^{\frac{1}{2}} ) as s → 0.

Theorem 2. If the identity operator on an infinite-dimensional Banach space 𝔛 is (q, 1)-summing for some 2 ⩽ q < ∞, then, for every f ∈ P_{1}(𝔛),

∥P − ∫_{t}^{t+h} f dμ∥_{𝔛} = o (h^{\frac{1}{q}} )

as h → 0^{+} for μ-a.e. t.

The proof below, which uses a factorization theorem of Pisier [P], was pointed out to us by Nigel Kalton.

Sketch of Proof. Fix f ∈ P_{1}(𝔛) for an infinite-dimensional Banach space 𝔛. Consider the operator K: L_{∞} → 𝔛 given by

K(g) = P − ∫_{Ω} g(ω)f(ω) dμ(ω) .

We need to show that

∥K (1_{[0,t+h]}) − K (1_{[0,t]})∥_{𝔛} = o (h^{\frac{1}{q}} )

as h → 0^{+} for μ-a.e. t. Fix ϵ > 0.

Since K is compact and since the dual of L_{∞} has the approximation property, there is [e.g. DU, Thm. VIII.3.6] a decomposition K = K_{1} + K_{2}, with K_{i} ∈ L (L_{∞}, 𝔛), such that K_{1} has finite rank and K_{2} has norm at most ϵ^{2}. It is enough to show that there is some constant A, which depends only on 𝔛 and q, such that for each i,

lim sup_{h→0^{+}} h^{− \frac{1}{q}} ∥K_{i} (1_{[0,t+h]}) − K_{i} (1_{[0,t]})∥_{𝔛} ⩽ A ϵ ,  (♢)

on a set of μ-measure at least 1 − ϵ^{q}.

Towards this, consider [see e.g. R] the natural surjective isometry τ: L_{∞} → C(Δ) for the appropriate extremally disconnected compact Hausdorff space Δ. Recall that τ takes an indicator function of a Borel set in [0, 1] to an indicator function of a clopen set in Δ, say τ (1_{A}) = 1Â in such a way that if A ⊂ B ⊂ Ω, then Â ⊂ B̂ ⊂ Δ and { \atop B \ A} = B̂ \ Â. Let K̂_{i} be the composite map:

K̂_{i} : C(Δ)  ⟶^{τ^{−1}}  L_{∞}  ⟶^{K_{i}}  𝔛 .

It can be shown that K_{1} satisfies

∥K_{1} (1_{[0,t+h]}) − K_{1} (1_{[0,t]})∥_{𝔛} = O(h) μ–a.e.

and so (♢) holds for any q > 1.

Now we deal with K_{2}. Fix 2 ⩽ q < ∞. If the identity operator on 𝔛 is (q, 1)- summing, then [P, Cor. 2.7] there is a probability measure ν on the Borel sets of Δ such that the operator K̂_{2} admits a factorization of the form

C(Δ)  ⟶ _{K_{2}}  𝔛
↘^{J}  ↗^{T}
L_{q,1}(ν)

where J is the natural inclusion map and T is a bounded linear operator with operator norm at most C∥K̂_{2}∥ ≤ Cϵ^{2}, where C depends only on 𝔛 and q. Here, L_{q,1}(ν) is the usual Lorentz space of all real-valued ν-measurable functions f on Δ for which the norm ∥f∥_{q,1} is finite, where

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

and f^{∗} is the non-increasing rearrangement of |f|. As above

∥K_{2} (1_{[0,t+h]}) − K_{2} (1_{[0,t]})∥_{𝔛} = ∥K_{2} (1_{(t,t+h]})∥_{𝔛}
= ∥K̂_{2} (1{ \atop (t,t+h]})∥_{𝔛}
⩽ Cϵ^{2} ∥J (1{ \atop (t,t+h]})∥_{L_{q,1}(ν)} .

Since the non-increasing rearrangement of J (1{ \atop (t,t+h]}) is just the indicator function

of the set [0, ν \binom{}{(t, t + h]}), we have

∥J (1{ \atop (t,t+h]})∥_{L_{q,1}(ν)} = q [ν \binom{}{(t, t + h]}]^{\frac{1}{q}} ,

and so

h^{− \frac{1}{q}} ∥K_{2} (1_{[0,t+h]}) − K_{2} (1_{[0,t]})∥_{𝔛} ⩽ Cqϵ^{2} [\frac{|β (t + h) − β (t)|}{h}]^{\frac{1}{q}} ,

where β: [0, 1] → ℝ is given by β(t) = ν \binom{}{[0, t]}. The function β is increasing and hence differentiable μ-almost everywhere. Thus

lim sup_{h→0^{+}} h^{− \frac{1}{q}} ∥K_{2} (1_{[0,t+h]}) − K_{2} (1_{[0,t]})∥_{𝔛} ⩽ C q ϵ^{2} [β′(t)]^{\frac{1}{q}}

for μ-a.e. t. From ∫_{0}^{1} β′(t) dt ⩽ β(1) − β(0) ⩽ 1, it follows that μ [β′(t) ⩾ ϵ^{−q}] ⩽ ϵ^{q}. Thus, on a set of measure at least 1 − ϵ^{q},

lim sup_{h→0^{+}} h^{− \frac{1}{q}} ∥K_{2} (1_{[0,t+h]}) − K_{2} (1_{[0,t]})∥_{𝔛} ⩽ C q ϵ ,

which implies (♢) for K_{2}.

Recall that the identity operator on a space with finite cotype q is (q, 1)-summing. Indeed, cotype plays a major rôle in the unfolding drama. To see this, consider a space 𝔛 which contains a finite-dimensional decomposition ∑ ⊕E_{n} where the Banach-Mazur distance between E_{n} and ℓ_{p}^{2^{n}} is less than M for each n for some fixed 1 ⩽ p ⩽ ∞ and M > 1. By modifying Mazur’s construction [see e. g. LT] of a basic sequence and using the fact (a simple compactness argument suffices) that finite representability of ℓ_{p} is inherited by subspaces of finite codimension, it is possible to construct such a finite-dimensional decomposition in 𝔛 whenever ℓ_{p} is finitely representable in 𝔛. By the Maurey-Pisier Theorem [MP], ℓ_{q_{0}} is finitely representable in 𝔛 where 2 ⩽ q_{0} ⩽ ∞ and

q_{0} = inf {q: 𝔛 has cotype q} .

In the same spirit as in the proof of Theorem 1 (and with similar notation), for 1 ⩽ p ⩽ ∞ let Ψ_{p} be the collection of all increasing functions ψ: [0, ∞) → [0, ∞) satisfying the growth condition

∑_{n=1}^{∞} ψ (2^{−p_{n−1}}) [2^{p_{n}}]^{\frac{1}{p}} < ∞  (†_{p})

for some increasing sequence {p_{n}}_{n=0}^{∞} of integers (following the convention that 1/∞ is 0). For 1 ⩽ p < ∞, a typical function in Ψ_{p} is

ψ(s) = s^{\frac{1}{p} +ϵ}

with p_{n} = n and for any ϵ > 0. For p = ∞, (†_{p}) reduces to the condition

lim_{s→0^{+}} ψ(s) = 0 .

Fix ψ ∈ Ψ_{p} and find an increasing sequence {p_{n}}_{n=0}^{∞} of integers, with p_{0} = 0, satisfying

∑_{n=1}^{∞} ψ(4 · 2^{−p_{n−1}}) [2^{p_{n}}]^{\frac{1}{p}} < ∞

(again, 1/∞ is 0). Define f: [0, 1] → 𝔛 by

f(ω) = ∑_{n=1}^{∞} ∑_{k=1}^{2^{n}} c_{n} \frac{1_{A_{k}^{n}}(ω)}{μ(A_{k}^{n})} e_{k}^{n} ,

where

c_{m} = 2K [ψ (4 · 2^{−p_{n−1}})] · δ_{m,p_{n}} ,

where K is the finite-dimensional decomposition constant. Minor variations of the proof of Theorem 1 show that this function f satisfies

∥P − ∫_{I} f dμ∥_{𝔛} ⩾ ψ (μ (I))

for each interval I contained in [0, 1].

Theorems 1 and 2, along with the above observations, give the following corollaries.

Corollary 3. Let 𝔛 be an infinite-dimensional Banach space with finite cotype and let q_{0} = inf{q: 𝔛_{0} has cotype q}. Then the following hold.

(1) If p > q_{0}, then for each f ∈ P_{1}(𝔛), we have

∥P − ∫_{t}^{t+h} f dμ∥_{𝔛} = o (h^{\frac{1}{p}} )

as h → 0^{+} for μ-a.e. t.

(2) If p < q_{0}, then there is an f ∈ P_{1}(𝔛) such that

∥P − ∫_{t}^{t+h} f dμ∥_{𝔛} ⩾ h^{\frac{1}{p}}

for all t ∈ [0, 1].

Corollary 4. For an infinite-dimensional Banach space 𝔛, the following are equivalent.

(1) 𝔛 fails cotype.

(2) For each ψ ∈ Ψ_{∞}, there exists f ∈ P_{1}(𝔛) such that

∥P − ∫_{I} f dμ∥_{𝔛} ⩾ ψ (μ (I))

for each interval I contained in [0, 1].

Remark. Note that Corollary 4 proves the existence of a reflexive Banach space for which the Pettis integral has essentially no kind of differentiability property whatsoever.

For further details and results along these lines, we refer the interested reader to [DG2].

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Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

E-mail address: dilworth@math.scarolina.edu and girardi@math.scarolina.edu