Several remarks concerning the local theory of L_{p} spaces

William B. Johnson^{∗}  Gideon Schechtman^{†}

Abstract

For the symmetric X_{p,w}^{n} subspaces of L_{p}, p > 2, we determine the dimension of their approximately Euclidean subspaces and estimate the smallest dimensions of their containing ℓ_{p}^{m} spaces. We also show that a diagonal of the canonical basis of ℓ_{p}, p > 2, with an unconditional basic sequence in L_{p} whose span is complemented, spans a space which is isomorphic to a complemented subspace of L_{p}.

1 Introduction

One of the main results in [BLM] is that for any n and any ε > 0, every n-dimensional subspace, X, of L_{p}, p > 2, (1 + ε)-embeds into ℓ_{p}^{m} for m ≤ c(p, ε)n^{p/2} log n. A result from [BDGJN] implies that, up to the log factor, this is the best possible result for a general subspace of L_{p}. Actually the “worst” known space is X = ℓ_{2}^{n}. However, unlike the corresponding result for 1 ≤ p < 2 (where the dependence of m on n is linear, up to a logarithmic factor), one may ask for conditions on n-dimensional subspaces of L_{p} which guarantee a substantially smaller estimate of the dimension of the containing ℓ_{p}^{m} space. For a long time the authors speculated that a natural such condition could be that X only contains low dimensional Euclidean subspaces. More precisely, we were hoping that if k is the largest dimension of a subspace of X (with X ⊂ L_{p}, p > 2) which is 2-isomorphic to a Euclidean space, then X well embeds into ℓ_{p}^{m} with m ≤ Ck^{p/2}.

The main purpose of this note is to show that this hopeful conjecture fails. The examples are symmetric X_{p} spaces; see the beginning of section 2 for their definition. We do that by determining, for each 0 < w < 1, the best (up to universal constants) dimension of an (approximately) Euclidean subspace of X_{p,w} (in Proposition 1), and by giving, for each 0 < w < 1, a lower bound on the dimension m of an ℓ_{p}^{m} containing well isomorphic copy of X_{p,w} (in Proposition 2). The lower bound on m turns out to be best possible, up to logarithmic factors, in the range w > n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}. In the complementary range w < n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )} we give, in Proposition 3, an upper estimate on m which does not match the lower bound and which is most probably not the best one. It is however better than what was previously known.

∗Supported in part by NSF DMS-0200690, U.S.-Israel Binational Science Foundation and Texas Advanced Research Program 010366-0033-20013

†Supported in part by Israel Science Foundation and U.S.-Israel Binational Science Foundation; participant, NSF Workshop in Linear Analysis and Probability, Texas A&M University.

Section 3 is devoted to another observation regarding subspaces of L_{p}, p > 2; this time complemented subspaces with unconditional basis. We show that given such a space with an unconditional basis {x_{n}}, any diagonal of the given unconditional basis with the unit vector basis of ℓ_{p}, {x_{n} ⊕_{p} α_{n}e_{n}}, spans a space which is well isomorphic to a well complemented subspace of L_{p} (although it may not be complemented in the natural embedding in L_{p} ⊕_{p} ℓ_{p}).

2 Tight embeddings of Euclidean spaces in symmetric X_{p} spaces and of symmetric X_{p} spaces in ℓ_{p} spaces.

Recall that for each n ∈ ℕ, 0 ≤ w ≤ 1, and 2 < p < ∞, X_{p,w}^{n} is ℝ^{n} with the norm

∥(x_{1}, x_{2}, …, x_{n})∥_{X_{p,w}^{n}} = max {(∑_{i=1}^{n} |x_{i}|^{p})^{1/p}, w(∑_{i=1}^{n} |x_{i}|^{2})^{1/2}}.

For w which is o(1) and such that wn^{\frac{1}{2} − \frac{1}{p}} tends to ∞ as n → ∞, we get spaces whose distances from ℓ_{2}^{n} and ℓ_{p}^{n} tend to ∞ with n. All these spaces are well isomorphic to well complemented subspaces of L_{p} [Ro] and up to “well isomorphism” they are all the spaces with good symmetric basis which are well isomorphic to well complemented subspaces of L_{p} [JMST, Theorem 1.1].

Denote by {e_{i}} the canonical basis of ℝ^{n} and let {g_{i}} be independent standard Gaussian random variables. Since 𝔼∥ ∑_{i=1}^{n} g_{i}e_{i}∥_{X_{p,w}^{n}} ≈ wn^{1/2} and ∥ ∑_{i=1}^{n} a_{i}e_{i}∥_{X_{p,w}^{n}} ≤ (∑_{i=1}^{n} |a_{i}|^{2})^{1/2} for all scalars {a_{i}}, it follows from the general theory of Euclidean sections of convex bodies (see, for example, [MS]) that ℓ_{2}^{k} (1 + ε)-embed into X_{p,w}^{n} as long as k ≤ c(ε)w^{2}n. We shall now show that this estimate on the dimension is best possible.

Recall that the Khintchine constant B_{p}, 2 < p < ∞ is the smallest constant such that

Ave_{±}| ∑ ±a_{i}|^{p} ≤ B_{p}^{p}(∑ a_{i}^{2})^{p/2}  (1)

for all sequences {a_{i}} of real numbers. The exact value of B_{p} is known and in particular B_{p}/\sqrt{p} is bounded away from 0 and ∞. The idea of the proofs of the next two propositions is taken from [BDGJN].

Proposition 1 Assume ℓ_{2}^{k} K-embeds into X_{p,w}^{n} and w ≥ n^{\frac{1}{p} − \frac{1}{2}} . Then k ≤ (B_{p} + 1)^{2}K^{2}w^{2}n.

Proof: If ℓ_{2}^{k} K-embeds into X_{p,w}^{n}, then there are numbers {α_{i,j}}_{i=1,j=1}^{k, n} satisfying

(∑_{i=1}^{k} |a_{i}|^{2})^{p/2} ≤ max { ∑_{j=1}^{n} | ∑_{i=1}^{k} a_{i}α_{i,j}|^{p}, w^{p}( ∑_{j=1}^{n} | ∑_{i=1}^{k} a_{i}α_{i,j}|^{2})^{p/2}} ≤ K^{p}(∑_{i=1}^{k} |a_{i}|^{2})^{p/2}.

(2)

For each fixed 1 ≤ l ≤ n, by setting a_{i} = α_{i,l}, we get from the right side of (2)

K^{p}(∑_{i=1}^{k} α_{i,l}^{2})^{p/2} ≥ max { ∑_{j=1}^{n} | ∑_{i=1}^{k} α_{i,l}α_{i,j}|^{p}, w^{p}( ∑_{j=1}^{n} ( ∑_{i=1}^{k} α_{i,l}α_{i,j}|)^{2})^{p/2}}
≥ max {(∑_{i=1}^{k} α_{i,l}^{2})^{p}, w^{p}(∑_{i=1}^{k} α_{i,l}^{2})^{p}}
= (∑_{i=1}^{k} α_{i,l}^{2})^{p},

so that for all 1 ≤ l ≤ n,

(∑_{i=1}^{k} α_{i,l}^{2})^{p/2} ≤ K^{p}.  (3)

Using the left inequality in (2) for ±1 coefficients, averaging over these coefficients, and using Khintchine’s inequality (1), we deduce that

k^{1/2} ≤ B_{p}( ∑_{j=1}^{n}(∑_{i=1}^{k} α_{i,j}^{2})^{p/2})^{1/p} + w(∑_{j=1}^{n} ∑_{i=1}^{k} α_{i,j}^{2})^{1/2}.  (4)

Using (3) in (4) we get

k^{1/2} ≤ B_{p}Kn^{1/p} + wKn^{1/2}.

Since w ≥ n^{\frac{1}{p} − \frac{1}{2}} , we get that k ≤ (B_{p} + 1)^{2}K^{2}w^{2}n.

Next we deal with embeddings of X_{p,w}^{n} into ℓ_{p}^{m}.

Proposition 2 Assume X_{p,w}^{n} K-embed into ℓ_{p}^{m}. Then m ≥ c min{w^{2p}n^{p−1}, n^{p/2}}, where the positive constant c depends only on p and K. More precisely, m ≥ (2B_{p}^{p}K^{p})^{−1}n^{p/2} when w ≥ (2B_{p}^{p}K^{p})^{\frac{1}{2p}} n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )} and m ≥ (2B_{p}^{p}K^{p})^{−2}w^{2p}n^{p−1} when w ≤ (2B_{p}^{p}K^{p})^{\frac{1}{2p}} n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}.

Proof: If X_{p,w}^{n} K-embeds into ℓ_{p}^{m}, then there are {α_{i,j}}_{i=1,j=1}^{n, m} such that

∑_{j=1}^{m} | ∑_{i=1}^{n} a_{i}α_{i,j}|^{p} ≤ max { ∑_{i=1}^{n} |a_{i}|^{p}, w^{p}(∑_{i=1}^{n} a_{i}^{2})^{p/2}} ≤ K^{p} ∑_{j=1}^{m} | ∑_{i=1}^{n} a_{i}α_{i,j}|^{p}  (5)

for all scalars {a_{i}}.

Fix a subset A of {1, 2, …, n}. It follows from (5) that for every 1 ≤ l ≤ n,

max { ∑_{i∈A} |α_{i,l}|^{p}, w^{p}(∑_{i∈A} α_{i,l}^{2})^{p/2}} ≥ ∑_{j=1}^{m} | ∑_{i∈A} α_{i,l}α_{i,j}|^{p} ≥ (∑_{i∈A} α_{i,l}^{2})^{p}

and it follows that

(∑_{i∈A} α_{i,l}^{2})^{p/2} ≤ max {(∑_{i∈A} |α_{i,l}|^{p})^{1/2}, w^{p}}.  (6)

Assume A is of cardinality k. Letting a_{i} = ±1 for i ∈ A and zero elsewhere and averaging over the signs, we get from (5)

max{k, w^{p}k^{p/2}} ≤ B_{p}^{p}K^{p} ∑_{j=1}^{m}(∑_{i∈A} α_{i,j}^{2})^{p/2}.  (7)

Using (6), we get from (7) that

\begin{aligned}max\{k, w^{p}k^{p/2}\} & ≤ B_{p}^{p}K^{p}( ∑_{j=1}^{m}(∑_{i∈A} |α_{i,j}|^{p})^{1/2} + mw^{p}) \\ & ≤ B_{p}^{p}K^{p}(m^{1/2}(∑_{j=1}^{m} ∑_{i∈A} |α_{i,j}|^{p})^{1/2} + mw^{p}). \end{aligned}

Since, by (5), for all i, ∑_{j=1}^{m} |α_{i,j}|^{p} ≤ 1,

w^{p}k^{p/2} ≤ max{k, w^{p}k^{p/2}} ≤ B_{p}^{p}K^{p}(m^{1/2}k^{1/2} + mw^{p}) ≤ 2B_{p}^{p}K^{p} max{m^{1/2}k^{1/2}, mw^{p}} (8)

and this holds for all 1 ≤ k ≤ n. Setting k := n in the extreme sides of (8) yields

m ≥ min{(2B_{p}^{p}K^{p})^{−2}w^{2p}n^{p−1}, (2B_{p}^{p}K^{p})^{−1}n^{p/2}}.

If w ≥ (2B_{p}^{p}K^{p})^{\frac{1}{2p}} n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, we get m ≥ (2B_{p}^{p}K^{p})^{−1}n^{p/2}. If w ≤ (2B_{p}^{p}K^{p})^{\frac{1}{2p}} n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, we get m ≥ (2B_{p}^{p}K^{p})^{−2}w^{2p}n^{p−1}.

Remark: For some time we thought the following might be true: Given a subspace X of L_{p}, 2 < p < ∞, let k be the largest dimension of a 2-Euclidean subspace of X. Then the smallest m such X 2-embed into ℓ_{p}^{m} is at most a constant depending on p times k^{p/2}. That is, the smallest dimension of a containing ℓ_{p} space of a subspace X of L_{p} depends only on the dimension of the largest Euclidean subspace of X. Proposition 2 shows that this conjecture is wrong: For example, for w = n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, the dimension m of ℓ_{p}^{m} which contains a 2-isomorphic copy of X_{p,w}^{n} is, by Proposition 2, at least of order m = n^{p/2} while, by the discussion preceding Proposition 1, ℓ_{2}^{k} (1 + ε)-embed into X_{p,w}^{n} as long as k ≤ c(ε)n^{\frac{1}{p} + \frac{1}{2}} (and (n^{\frac{1}{p} + \frac{1}{2}} )^{\frac{p}{2}} << n^{\frac{p}{2}} ).

For a fixed K and w ≥ c_{p}n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, the result of Proposition 2 is best up to a possible log n factor; it was proved in [BLM] that any n-dimensional subspace of L_{p} (1+ε)-embeds into ℓ_{p}^{m} for m ≤ C(p, ε)n^{p/2} log n. For w ≤ c_{p}n^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, it is not clear if the result obtained here is best possible. In [Sc2] some estimates on the dimension of the containing ℓ_{p} of an X_{p,w}^{n} space are given, which in some cases are better than the general estimate of [BLM]. Using the methods of [BLM] one can somewhat improve these results to get the following result, which however still leaves a gap with the lower bound on m in Proposition 2.

Proposition 3 There is a constant K_{p}, depending only on p > 2, such that, for all 0 < K < ∞, if w ≤ Kn^{\frac{1}{2} ( \frac{1}{p} − \frac{1}{2} )}, then X_{p,w}^{n} K_{p}-embeds into ℓ_{p}^{m} whenever

m ≥ C(K, p)n^{1+ \frac{(p−2)(p−1)}{p}} (log n)w^{2(p−2)}.

C(K, p) depends only on its two arguments.

Note that, up to logarithmic factors, the estimate on m is better than n^{p/2}.

Sketch of proof: As we said above, the proof is a combination of arguments from [Sc2] and [BLM], neither of which is simple, and it does not seem to give the final answer. We thus only sketch the argument.

We first use a specific embedding of X_{p,w}^{n} in L_{p}(0, 1) as is given in Proposition 11 of [Sc2]: There is (sign and permutation) exchangeable sequence {x_{i}}_{i=1}^{n} in L_{p} such that ∥x_{i}∥_{p} = 1 for all i, (∑ x_{i}^{2})^{1/2} ≡ wn^{1/2} and

K_{p}^{−1}∥ ∑ a_{i}x_{i}∥_{p} ≤ max {(∑_{i=1}^{n} |a_{i}|^{p})^{1/p}, w(∑_{i=1}^{n} |a_{i}|^{2})^{1/2}} ≤ K_{p}∥ ∑ a_{i}x_{i}∥_{p}.

Moreover, the span of the x_{i}-s is K_{p}-complemented in L_{p}. Here and below K_{p} denotes a constant depending only on p, not necessarily the same in each instance. Lemma 12 in [Sc2] asserts, in particular, that

∥x∥_{∞} ≤ K_{p} min{n^{1/2}, n^{1−1/p}w^{2}}∥x∥_{p}

for all x in the span of the x_{i}-s. (To add to the confusion resulting from the different notations here and in [Sc2], there is a misprint in equation (23) in [Sc2]: m^{1/q} there should be m^{1/p}; see the bottom of the same page in [Sc2].) Under our assumption on w, this translates to

∥x∥_{∞} ≤ K^{2}K_{p}n^{1−1/p}w^{2}∥x∥_{p}.

Theorem 13 of [Sc2] and its proof shows that there is some choice of m̄ points of (0, 1) with m̄ ≤ K_{p} min{n^{1+p/2}, n^{p}w^{2p}} ≤ K_{p}n^{p}w^{2p} and, letting x̄_{i} be the restriction of x_{i} to these points, considered as elements of Lm̄ ; p (L_{p} on the measure space {1, …, m̄} with the normalized counting measure), we get

\frac{1}{2}∥ ∑ a_{i}x_{i}∥_{p} ≤ ∥ ∑ a_{i}x̄_{i}∥_{p} ≤ 2∥ ∑ a_{i}x_{i}∥_{p}  (9)

and

\frac{1}{2}(∑ a_{i}^{2})^{1/2} ≤ ∥ ∑ a_{i}w^{−1}x̄_{i}∥_{2} ≤ 2(∑ a_{i}^{2})^{1/2}  (10)

for all {a_{i}}_{i=1}^{n}. (Moreover, the span of the x̄_{i}-s is K_{p}-complemented in Lm̄ ; p.) Of course we want to improve the bound on m̄, but we need to use the result above (or something similar) as we shall see shortly.

Note that it is still true that

(∑ x̄_{i}^{2})^{1/2} ≡ wn^{1/2}  (11)

and that

∥x∥_{∞} ≤ K^{2}K_{p}n^{1−1/p}w^{2}∥x∥_{p}  (12)

for all x in the span of the x̄_{i}-s.

We would like to apply now the results and techniques of [BLM] and in particular Theorem 7.3 and its proof. We would like to take advantage of the improved L_{∞} bound in (12) and this leads to a complication since the entropy bounds in [BLM] are obtained using a “change of density” which, if applied, may destroy the L_{∞} bound we have. We thus would like to see that basically the same entropy bounds used in the proof of Theorem 7.3 in [BLM] apply, in our situation, without any change of density.

Let us denote by X_{r} the span of the x̄_{i}-s in Lm̄ ; r. In the notation of [BLM], for any 2 < q < ∞ and any t > 0,

E(B_{X_{p}}, B_{X_{q}}, t) ≤ E(B_{X_{2}}, B_{X_{q}}, t)

where E(U, V, t) denotes the minimal number of translates of tV needed to cover U. Now, by (10), B_{X_{2}} ⊂ 2{∑ a_{i}w^{−1}x̄_{i}; ∑ a_{i}^{2} ≤ 1} =: 2C so that

E(B_{X_{p}}, B_{X_{q}}, t) ≤ E(C, B_{X_{q}}, t/2).

Using now part of the proof of Proposition 4.6 in [BLM] (the part relaying on Proposition 4.2) we get, for M_{X_{q}} computed relative to the euclidean structure given by C and using (11) above, that

M_{X_{q}} ≤ Aq^{1/2}∥(∑ w^{−2}x̄_{i}^{2})^{1/2}∥_{Lm̄ ; q} ≤ Aq^{1/2}n^{1/2}

for A a universal constant. Using Proposition 4.2 in [BLM], we get as in the proof of Proposition 4.6 there that

log E(B_{X_{p}}, B_{X_{q}}, t) ≤ A′nq/t^{2}

for some other universal constant A′. Taking q = log m̄ ≤ K_{p} log n we get

log E(B_{X_{p}}, B_{∞}, t) ≤ K_{p}n(log n)/t^{2},  (13)

so that we recovered the entropy estimate used in the proof of Theorem 7.3 in [BLM]. We now apply this proof to the space X = X_{p}. We can take ε = 1/2 (there is no much point dealing with small ε since we already have a constant K_{p} in the original embedding of X_{p,w}^{n} in L_{p}). Recall that we have an improved estimate on the L_{∞} bound (12) which also implies we can take l = [log(K^{2}K_{p}n^{1−1/p}w^{2})/ log(3/2)] + 1 in the beginning of the proof of Theorem 7.3 in [BLM]. Using the notation of [BLM], we get from (13) the estimates

log ¯ Ā_{k}, log ¯ B̄_{k} ≤ K_{p}n(log n)(4/9)^{k}.  (14)

Looking now at the end of the proof of Theorem 7.3 in [BLM] and using the parameters above (for ε, l and ¯ B̄_{k}), we get the right estimate on m (denoted N in [BLM]).

3 Complemented subspaces of L_{p} with unconditional bases

In this section we prove

Proposition 4 Let {x_{n}} be a (finite or infinite) C-unconditional basic sequence in L_{p}, p > 2, which is K-complemented. Then for all {α_{n}} ⊂ ℝ, the span of {x_{n} ⊕ α_{n}e_{n}} in L_{p} ⊕ ℓ_{p} is K′-isomorphic to a K′-complemented subspace of L_{p}. K′ depends only on K, C, and p. {e_{n}} denotes here the unit vector basis of ℓ_{p}.

Remark. The case when {x_{n}} is equivalent to the unit vector basis of ℓ_{2} is the fundamental result of Rosenthal’s [Ro] upon which this entire note rests. Proof: Denote by {h_{n,i}}_{n=0,i=1}^{∞, 2^{n}} the mean zero L_{∞} normalized Haar functions. We first treat the case where {x_{n}} has the special form

x_{n} = ∑_{i=1}^{2^{kn}} a_{n,i}h_{k_{n},i},

for some subsequence {k_{n}} ⊂ ℕ, and the projection onto the span of {x_{n}} has the special form

P x = ∑ x_{n}^{∗}(x)x_{n}

where

x_{n}^{∗} = ∑_{i=1}^{2^{kn}} b_{n,i}h_{k_{n},i}.

Assume also, as we may, that

∥x_{n}∥ = 2^{−k_{n}/p}( ∑_{i=1}^{2^{kn}} |a_{n,i}|^{p})^{1/p} = 1.

Since the Haar system in its natural order is a monotone basis for L_{p} (see [LT, p. 3]), we have

∥x_{n}^{∗}∥ = 2^{−k_{n}/q}( ∑_{i=1}^{2^{kn}} |b_{n,i}|^{q})^{1/q} ≤ 2∥P ∥.

Let {e_{n,i}}_{n=1,i=1}^{∞, n} be a rearrangement of the unit vector basis of ℓ_{p} and put

y_{n} = α_{n}2^{−k_{n}/p} ∑_{i=1}^{2^{kn}} a_{n,i}e_{k_{n},i} ∈ ℓ_{p}

and

y_{n}^{∗} = α_{n}^{−1}2^{−k_{n}/q} ∑_{i=1}^{2^{kn}} b_{n,i}e_{k_{n},i} ∈ ℓ_{q}.

Then

y_{n}^{∗}(y_{n}) = 1, ∥y_{n}∥_{p} = α_{n}, ∥y_{n}^{∗}∥_{q} ≤ α_{n}^{−1}2∥P ∥

so that P̄(x) = ∑ y_{n}^{∗}(x)y_{n} defines a projection of norm at most 2∥P ∥ from ℓ_{p} onto the closed span of {y_{n}}. Consider the basic sequence

z_{n,i} = a_{n,i}(h_{k_{n},i} ⊕ α_{n}2^{−k_{n}/p}e_{k_{n},i}), n = 1, 2, …, i = 1, 2, …, 2^{k_{n}}

in L_{p} ⊕ ℓ_{p} and its closed span Z. By [KS] (see [Mü] for an alternative proof), Z is K′-isomorphic to a K′-complemented subspace of L_{p}, where K′ depends only on K.

Put

Q = (P, P̄) : Z → L_{p} ⊕ ℓ_{p}

and notice that

Qz_{n,i} = 2^{−k_{n}}a_{n,i}b_{n,i}x_{n} ⊕ α_{n}^{−1}2^{−k_{n}/q}α_{n}2^{−k_{n}/p}a_{n,i}b_{n,i}y_{n} = 2^{−k_{n}}a_{n,i}b_{n,i}(x_{n} ⊕ y_{n}).

In particular, the range of Q is the closed span of {x_{n} ⊕ y_{n}}. Also,

Q(x_{n} ⊕ y_{n}) = Q ∑_{i=1}^{2^{kn}} z_{n,i} = x_{n} ⊕ y_{n} ;

that is, Q is a projection (of norm at most 2∥P ∥) from Z onto the closed span of {x_{n}⊕y_{n}}, and, since Z is well complemented in L_{p} ⊕ ℓ_{p} and the later is isomorphic to L_{p} (with universal constant), it follows that the closed span of {x_{n} ⊕ y_{n}} is well isomorphic to a well complemented subspace of L_{p}.

The sequence {x_{n} ⊕ y_{n}} is clearly isometrically equivalent to {x_{n} ⊕ α_{n}e_{n}}, so this completes the proof of the special case. In this case the unconditional constant of {x_{n}} is no larger than the unconditional constant of the Haar basis, so that K′ depends only on K and p. The reduction of the general case to the special case just treated follows from (the proof in) [Sc1]. This reduction makes the final constant K′ also dependent on C. Here is a sketch of this reduction.

Let P x = ∑ x_{n}^{∗}(x)x_{n} be the given projection. By a standard perturbation argument we may assume that each of the x_{n} and x_{n}^{∗} is a linear combination of indicator functions of dyadic intervals in [0, 1]. It follows that, for some increasing subsequence {k_{n}} ⊂ ℕ,

x_{n} = ∑_{i=1}^{2^{kn}} a_{n,i}|h_{k_{n},i}| and x_{n}^{∗} = ∑_{i=1}^{2^{kn}} b_{n,i}|h_{k_{n},i}|.

Put

z_{n} = ∑_{i=1}^{2^{kn}} a_{n,i}h_{k_{n},i} and z_{n}^{∗} = ∑_{i=1}^{2^{kn}} b_{n,i}h_{k_{n},i}.

The unconditionality of {x_{n}} (and {x_{n}^{∗}}) implies that {x_{n}} is equivalent to {z_{n}}, {x_{n}^{∗}} is equivalent to {z_{n}^{∗}} and Qx = ∑ z_{n}^{∗}(x)z_{n} is a bounded projection. The constants involved depend only on K, C and p.

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[Sc1] Schechtman, G., A remark on unconditional basic sequences in L_{p} (1 < p < ∞). Israel J. Math. 19 (1974), 220–224.

[Sc2] Schechtman, G., Embedding X_{p}^{m} spaces into l_{r}^{n}. Geometrical aspects of functional analysis (1985/86), 53–74, Lecture Notes in Math., 1267, Springer, Berlin, 1987.

William B. Johnson Department Mathematics Texas A&M University College Station, TX, USA E-mail: johnson@math.tamu.edu

Gideon Schechtman Department of Mathematics Weizmann Institute of Science Rehovot, Israel E-mail: gideon@wisdom.weizmann.ac.il