Embeddings of rearrangement invariant spaces that are not strictly singular

S.J.Montgomery-Smith (stephen@math.missouri.edu) ^{∗}

University of Missouri

E.M.Semenov (root@func.vsu.su) ^{†}

Voronezh State University

Abstract. We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L_{1}([0, 1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space L_{Φ} with Φ(x) = exp(x^{2}) − 1.

Keywords: rearrangement invariant space, strictly singular mapping, Rademacher function, Orlicz space

A.M.S. Classification (1991): Primary 46E30, 47B38; Secondary 60G50

In this paper we ask the following question. Given a rearrangement invariant space E on [0, 1], when is the natural embedding E ⊂ L_{1}([0, 1]) strictly singular. (We refer the reader to [4] for the definition and properties of rearrangement invariant spaces.) We define a linear map between two normed spaces to be strictly singular if there does not exist an infinite dimensional subspace of the domain upon which the operator is an isomorphism.

This question is a natural extension of similar work by del Amo, Hernández, Sánchez and Semenov [1], when they considered the problem of which embeddings between rearrangement invariant spaces are not disjointly strictly singular. A positive linear operator between two Banach lattices is disjointly strictly singular if there exists an infinite sequence of non-zero disjoint elements in the domain such that the operator is an isomorphism on the span of this sequence. This work [1] contains a number of very sharp results, giving some very clear criteria.

However the question concerning when such maps are strictly singular seems to be more difficult. For this reason, we will restrict ourselves to considering the case when the range is L_{1}([0, 1]). Even then, we do not have complete answers, and in this paper, we leave as many questions unanswered as we answer.

The “other end” was investigated by Novikov [7], who showed that the natural embedding L_{∞}([0, 1]) ⊂ E is strictly singular unless E is

∗ Research supported in part by grants from the N.S.F. and the Research Board of the University of Missouri.

† Research supported in part by a grant from the N.S.F. and a grant from Russia.

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.

equivalent to L_{∞}([0, 1]). (The case when E = L_{p}([0, 1]) follows from a classical result of Grothendieck, Theorem 5.2 of [9].)

In answering our question, there is one rearrangement invariant space that plays a prominent role. This rearrangement invariant space, denoted by G, is the closure of the simple functions in the Orlicz space corresponding to the Orlicz function e^{x^{2}} − 1. (We define Orlicz spaces below.) The reason why this space plays such a role is as follows. Consider the Rademacher functions on [0, 1] given by r_{n}(t) = sign(sin(2^{n}πt)). The following result of Rodin and Semenov [8] is well known.

THEOREM 1. Let E be a rearrangement invariant space on [0, 1]. Then the following are equivalent.

1. The sequence (r_{n}) in E is equivalent to the unit vector basis of ℓ_{2};

2. G embeds continuously into E;

3. There is a constant c > 0 such that ∥∑_{i=1}^{n} r_{i}∥_{E} ≤ c\sqrt{n}.

It is clear that if E is a rearrangement invariant space on [0, 1] that contains G continuously, then the natural embedding E ⊂ L_{1} is not strictly singular. Here, of course, the subspace on which the norms are equivalent is the span of the Rademacher functions.

For this reason, it is natural to pose the following conjecture.

CONJECTURE 2. Let E be a rearrangement invariant space on [0, 1]. Then the natural embedding E ⊂ L_{1} is not strictly singular if and only if G embeds into E continuously.

It will become apparent below that the following conjecture implies the previous one.

CONJECTURE 3. Given x_{1}, …, x_{n} ∈ L_{1}([0, 1]), and a rearrangement invariant space E on [0, 1], there exists signs ϵ_{1}, …, ϵ_{n} = ±1 such that

∥∑_{i=1}^{n} ϵ_{i}x_{i}∥_{L_{1}} ≥ c^{−1} ∥∑_{i=1}^{n} r_{i} ∥x_{i}∥_{1}∥_{E} .

Unfortunately we are not able to prove either of these conjectures without some additional hypotheses.

First, let us introduce some examples of rearrangement invariant spaces. A function Φ : ℝ → [0, ∞) is called an Orlicz function if it is convex, even, and takes zero to zero. The Orlicz space L_{Φ} is the collection of all equivalence classes of measurable functions (where the equivalence relation is equal almost everywhere) on [0, 1] such that the norm:

∥x∥_{Φ} = inf {λ : ∫_{0}^{1} Φ(x(t)/λ) dt ≤ 1}

is finite.

Another class of examples is the Lorentz spaces. If φ : [0, 1] → [0, 1] is increasing and concave with φ(0) = 0 and φ(1) = 1, then the Lorentz space Λ(φ) consists of all equivalence classes of measurable functions on [0, 1] for which the norm

∥x∥_{Λ(φ)} = ∫_{0}^{1} x^{∗}(t) dφ(t)

is finite. Here, as in the rest of the paper, x^{∗} denotes the non-increasing rearrangement of |x|.

One more class of examples is the Marcinkiewicz spaces. If φ : [0, 1] → [0, 1] is as above, then the Marcinkiewicz space M(φ) = Λ(φ)^{∗} consists of all equivalence classes of measurable functions on [0, 1] for which the norm

∥x∥_{M(φ)} = sup_{0<t<1} \frac{∫_{0}^{t} x^{∗}(s) ds}{φ(t)}

is finite.

By definition, G is the closure of the simple functions in the Orlicz space corresponding to the Orlicz function e^{x^{2}} −1. However, it can also be shown that it has an equivalent Marcinkiewicz norm, that is, there is a constant c > 0 such that c^{−1} ∥x∥_{G} ≤ ∥x∥_{M(φ)} ≤ c ∥x∥_{G}, where

φ(t) = \frac{t}{\sqrt{log(e/t)}}.

Another space that will concern us is the space G_{1} = Λ(φ) where

φ(t) = \frac{2}{\sqrt{log(e^{2}/t)}}. It is a simple matter to show that G_{1} embeds con-

tinuously into G. Furthermore, for characteristic functions, the norms on G and G_{1} coincide.

Let us now consider the concept of D-convex rearrangement invariant spaces. This notion was introduced by Kalton [2], and studied extensively by Montgomery-Smith and Semenov [6], where several equivalent properties were given. Perhaps the easiest definition to work with is the following. We will say that a rearrangement invariant space E is D-convex if there is a family of Orlicz functions Φ_{α} : ℝ → [0, ∞), and a constant c > 0, such that

c^{−1} ∥x∥_{E} ≤ sup_{α} ∥x∥_{Φ_{α}} ≤ ∥x∥_{E} .

Note that the Marcinkiewicz spaces are D-convex, because for each 0 < t < 1 the map x ↦ ∫_{0}^{t} x^{∗}(s) ds is equivalent to the Orlicz norm given by the Orlicz function Φ(s) = (s − t^{−1})^{+}.

The best result we have obtained so far regarding Conjectures 2 and 3 is the following.

THEOREM 4. Conjectures 2 and 3 are true if E is D-convex.

From here we are able to get a weaker version of Conjecture 2.

THEOREM 5. Let E be a rearrangement invariant space on [0, 1]. If the natural embedding E ⊂ L_{1} is not strictly singular, then G_{1} embeds into E continuously.

We proceed with the proofs.

Proof of first part of Theorem 4. It is sufficient to consider the case when E is an Orlicz space L_{Φ}, where Φ is an Orlicz function. Suppose that

sup_{ϵ} ∥∑_{i=1}^{n} ϵ_{i}x_{i}∥_{Φ} ≤ 1,

where we write ϵ = (ϵ_{1}, …, ϵ_{n}). Thus

sup_{ϵ} ∫ Φ (∑_{i=1}^{n} ϵ_{i}x_{i}(s)) ds ≤ 1.

Let

F_{ϵ} = {s ∈ [0, 1] : sign(x_{i}(s)) = ϵ_{i}}.

If ϵ = (ϵ_{1}, …, ϵ_{n}) and η = (η_{1}, …, η_{n}), let us set ηϵ = (η_{1}ϵ_{1}, …, η_{n}ϵ_{n}). Then

Ave_{η} Φ (∑_{i=1}^{n} η_{i} ∥x_{i}∥_{1}) = Ave_{η} Φ (∑_{ϵ} ∫ I_{F_{ηϵ}}(s) ∑_{i} η_{i} |x_{i}(s)| ds)

= Ave_{η} Φ (∑_{ϵ} ∫ I_{F_{ηϵ}}(s) ∑_{i} ϵ_{i}x_{i}(s) ds)

≤ Ave_{η} ∫ Φ (∑_{ϵ} I_{F_{ηϵ}}(s) ∑_{i} ϵ_{i}x_{i}(s)) ds

= Ave_{η} ∫ ∑_{ϵ} I_{F_{ηϵ}}(s)Φ (∑_{i} ϵ_{i}x_{i}(s)) ds

= Ave_{ϵ} ∑_{η} ∫ I_{F_{ηϵ}}(s)Φ (∑_{i} ϵ_{i}x_{i}(s)) ds

≤ sup_{ϵ} ∫ Φ (∑_{i} ϵ_{i}x_{i}(s)) ds
≤ 1.

Therefore

∥∑_{i} r_{i} ∥x_{i}∥_{1}∥_{Φ} ≤ 1.

□

Proof of second part of Theorem 4. Suppose that there is an infinite dimensional subspace F ⊂ E such that the norms of L_{1} and E are equivalent on F. By Dvoretzky’s Theorem (see [5] Chapter 4), for each integer n, there is an n-dimensional subspace H of F such that H is 2-isomorphic to Hilbert space. Pick an orthonormal basis x_{1}, …, x_{n} for H. Then there exist signs ϵ_{1}, …, ϵ_{n} = ±1 such that

∥∑_{i} r_{i}∥_{E} ≤ c ∥∑_{i} ϵ_{i}x_{i}∥_{H} = c\sqrt{n}.

Thus the result follows by Theorem 1.

□

In order to prove Theorem 5, we need the following Lemma. This result may also be found in [3] (Theorem 2.5.7.).

LEMMA 6. Given a rearrangement invariant space E, there exists an increasing function φ : [0, 1] → [0, 1] such that ∥x∥_{E} ≥ ∥x∥_{M(φ)}, but if x takes only values 0 or 1, then ∥x∥_{E} = ∥x∥_{M(φ)}.

Proof. Let

φ(t) = \frac{t}{∥I_{[0,t]}∥_{E}} .

The latter property is obvious. To show the former, recall (see for example [4]) the space E′ to be those functions y on [0, 1] for which the norm

∥y∥_{E′} = sup {∫ x(t)y(t) dt : ∥x∥_{E} ≤ 1}

is finite. Notice that

∥I_{[0,t]}∥_{E} · ∥I_{[0,t]}∥_{E′} = t,

that is, ∥I_{[0,1]}∥_{E′} = φ(t). Then the result follows since

∫_{0}^{t} x^{∗}(s) ds ≤ ∥x∥_{E} ∥I_{[0,1]}∥_{E′} .

□

Proof of Theorem 5. Suppose that the embedding E ⊂ L_{1}([0, 1]) is not strictly singular. Produce φ as in Lemma 6. Then it is clear that the embedding M(φ) ⊂ L_{1}([0, 1]) is also not strictly singular. Hence by Theorem 4, G embeds continuously into M(φ).

In particular, this means that there is a constant c > 0 such that for all t ∈ [0, 1] we have

∥I_{[0,t]}∥_{E} = ∥I_{[0,t]}∥_{M(φ)} ≤ c ∥I_{[0,t]}∥_{G} ≤ cψ(t),

where ψ(t) = \frac{2}{\sqrt{log(e^{4}/t)}}. Now, writing x^{∗} = ∫_{0}^{1} I_{[0,t]} d(−x^{∗}(t)), we

obtain that

∥x∥_{E} ≤ ∫_{0}^{1} ∥I_{[0,t]}∥_{E} d(−x^{∗}(t))

≤ c ∫_{0}^{1} ψ(t)d(−x^{∗}(t)) = c ∫_{0}^{1} x^{∗}(t)dψ(t) = c ∥x∥_{G_{1}} .
□

References

1. A.G. del Amo, F.L. Hernández, V.M. Sánchez and E.M. Semenov, Disjointly strictly-singular inclusions between rearrangement invariant spaces, preprint.

2. N.J. Kalton, Representations of Operators between Function Spaces, Indiana U. Math. J. 33, (1984), 639–665.

3. S.G. Kreĭn, Yu. Ī. Petunīn and E.M. Semenov, Interpolation of linear operators. Translated from the Russian by J. Szűcs. Translations of Mathematical Monographs, 54. American Mathematical Society, Providence, R.I., 1982.

4. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II — Function Spaces, Springer-Verlag, Berlin, New York, 1979.

5. V.D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin-New York, 1986.

6. S.J. Montgomery-Smith and E.M. Semenov, Random rearrangements and operators, 25 Years of Voronezh Winter Mathematical School, Proceedings in honor of 80th birthday of S. Krein, A.M.S., Vol. 184, 1998, 157–183.

7. S.Ya. Novikov, Singularities of the embedding operator for symmetric function spaces on [0, 1], (Russian) Mat. Zametki 62 (1997), no. 4, 549–563; translation in Math. Notes 62 (1997), no. 3-4, 457–468 (1998).

8. V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207–222.

9. W. Rudin, Functional analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.

Address for Offprints: Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. or Department of Mathematics, Voronezh State University, Universitetskaya pl.1, Voronezh 394693, Russia.

strict4.tex; 13/01/2000; 23:14; p.8