Boyd Indices of Orlicz–Lorentz Spaces

STEPHEN J. MONTGOMERY-SMITH^{∗} Department of Mathematics, University of Missouri, Columbia, Missouri 65211

ABSTRACT

Orlicz–Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. In this paper, we investigate their Boyd indices. Bounds on the Boyd indices in terms of the Matuszewska–Orlicz indices of the defining functions are given. Also, we give an example to show that the Boyd indices and Zippin indices of an Orlicz–Lorentz space need not be equal, answering a question of Maligranda. Finally, we show how the Boyd indices are related to whether an Orlicz–Lorentz space is p-convex or q-concave.

1 INTRODUCTION

The Boyd indices of a rearrangement invariant space are of fundamental importance. They were originally introduced by Boyd (1969) for the purpose of showing certain interpolation results. Since then, they have played a major role in the theory of rearrangement invariant spaces (see, for example, Bennett and Sharpley (1988), Lindenstrauss and Tzafriri (1979) or Maligranda (1984)).

Orlicz–Lorentz spaces provide a common generalization of Orlicz spaces (see Orlicz(1932) or Luxembourg (1955)) and Lorentz spaces (see Lorentz (1950) or Hunt (1966)), and have been studied by many authors, including, for example, Maligranda (1984), Mastyło (1986) and Kamińska (1990a, 1990b, 1991). In particular, Maligranda posed a question about the Boyd indices of these spaces.

∗Research supported in part by N.S.F. Grants DMS 9001796 and DMS 9001357.

In this paper, we first give some fairly elementary estimates for the Boyd indices of Orlicz–Lorentz spaces. Then we give an example that show that these estimates cannot be improved, thus answering Maligranda’s question. Finally we show how knowledge of the Boyd indices gives information about the p-convexity or q-concavity of the Orlicz–Lorentz space.

2 DEFINITIONS

In discussing Orlicz–Lorentz spaces, it will be convenient to talk about them in the more general framework of rearrangement invariant spaces. Unfortunately, the definitions in the literature usually require that the spaces be quasi-normed, which is not always the case with the Orlicz–Lorentz spaces. For this reason we introduce the following definition of rearrangement invariant spaces.

DEFINITION If (Ω, F, μ) is a measure space, we denote the measurable functions, modulo functions equal to zero almost everywhere, by L_{0}(μ). We say that a Köthe functional is a function ∥ · ∥: L_{0}(μ) → [0, ∞] satisfying

i) if f ∈ L_{0}(μ), then ∥f∥ = 0 ⇔ f = 0; ii) if f ∈ L_{0}(μ) and α ∈ C, then ∥αf∥ = |α| ∥f∥; iii) if f, g ∈ L_{0}(μ), then |f| ≤ |g| ⇒ ∥f∥ ≤ ∥g∥; iv) if f_{n}, f ∈ L_{0}(μ), then |f_{n}| ↗ |f| ⇒ ∥f_{n}∥ → ∥f∥;

v) if f_{n} ∈ L_{0}(μ), then ∥f_{n}∥ → 0 ⇒ f_{n} → 0 in the measure topology. A Köthe space is a pair (X, ∥ · ∥), where ∥ · ∥ is a Köthe functional, and X = {f ∈ L_{0}(μ) : ∥f∥ < ∞}. Usually, we will denote a space by a single letter, X, and denote its functional by ∥ · ∥_{X}.

DEFINITION If f : Ω → C is a measurable function, we define the non-increasing rearrangement of f to be

f^{∗}(x) = sup{ t : μ(|f| ≥ t) ≥ x }.

A rearrangement invariant space is a Köthe space such that if f, g ∈ L_{0}(μ), and f^{∗} ≤ g^{∗}, then ∥f∥ ≤ ∥g∥.

Now we define the Orlicz–Lorentz spaces. We refer the reader to Montgomery-Smith (1992) for a motivation of the following definitions.

DEFINITION A φ-function is a function F : [0, ∞) → [0, ∞) such that i) F (0) = 0; ii) lim_{t→∞} F (t) = ∞; iii) F is strictly increasing; iv) F is continuous; We will say that a φ-function F is dilatory if for some 1 < c_{1}, c_{2} < ∞ we have F (c_{1}t) ≥ c_{2}F (t) for all 0 ≤ t < ∞. We will say that F satisfies the Δ_{2}-condition if F^{−1} is dilatory.

If F is a φ-function, we will define the function F̃(t) to be 1/F (1/t) if t > 0, and 0 if t = 0.

We say that two φ-functions F and G are equivalent (in symbols F ≍ G) if for some number c < ∞ we have that F (c^{−1}t) ≤ G(t) ≤ F (ct) for all 0 ≤ t < ∞.

We will denote the φ-function F (t) = t^{p} by T^{p}.

DEFINITION (See Orlicz (1932) or Luxembourg (1955).) If (Ω, F, μ) is a measure space, and F is a φ-function, then we define the Luxemburg functional of a measurable function f by

∥f∥_{F} = inf { c : ∫_{Ω} F (|f(ω)| /c) dμ(ω) ≤ 1 } ,

The Orlicz space is the associated Köthe space, and is denoted by L_{F} (Ω, F, μ) (or L_{F} (μ), L_{F} (Ω) or L_{F} for short).

DEFINITION If (Ω, F, μ) is a measure space, and F and G are φ-functions, then we define the Orlicz–Lorentz functional of a measurable function f by

∥f∥_{F,G} = ∥f^{∗} ∘ F̃ ∘ G̃^{−1}∥_{G} .

The Orlicz–Lorentz space is the associated Köthe space, and is denoted by L_{F,G}(Ω, F, μ) (or L_{F,G}(μ), L_{F,G}(Ω) or L_{F,G} for short).

We will write L_{F,p}, L_{p,G} and L_{p,q} for L_{F,T^{p}}, L_{T^{p},G} and L_{T^{p},T^{q}} respectively.

It is an elementary matter to show that the Orlicz and Orlicz–Lorentz spaces are rearrangement invariant spaces. We note that ∥ · ∥_{F,F} = ∥ · ∥_{F}, and that ∥χ_{A}∥_{F,G} = F̃^{−1}(μ(A)).

Now we define the various indices that we use throughout this paper. Obviously, the most important of these are the Boyd indices. These were first introduced in Boyd (1969). We will follow Maligranda (1984) for the names of the other indices, but will modify the definitions so as to be consistent with the notation used in Lindenstrauss and Tzafriri (1979). Thus other references to these indices will often reverse the words ‘upper’ and ‘lower’, and use the reciprocals of the indices used here. The Zippin indices were introduced in Zippin (1971), and the Matuszewska–Orlicz indices in Matuszewska and Orlicz (1960 and 1965). The Zippin indices are sometimes called fundamental indices.

DEFINITION For a rearrangement invariant space X, we let the dilation operators d_{a} : X → X be d_{a}f(x) = f(ax) for 0 < a < ∞. We define the lower Boyd index to be

p(X) = sup { p : for some c < ∞ we have ∥d_{a}∥_{X→X} ≤ ca^{−1/p} for a < 1 } .

We define the upper Boyd index to be

q(X) = inf { q : for some c < ∞ we have ∥d_{a}∥_{X→X} ≤ ca^{−1/q} for a > 1 } .

We define the lower Zippin index to be

p_{z}(X) = sup \{ p : \begin{matrix}for some c < ∞ we have ∥d_{a}χ_{A}∥_{X} ≤ ca^{−1/p} ∥χ_{A}∥_{X} \\ for all a < 1 and measurable A \end{matrix} \} .

We define the upper Zippin index to be

q_{z}(X) = inf \{ q : \begin{matrix}for some c < ∞ we have ∥d_{a}χ_{A}∥_{X} ≤ ca^{−1/q} ∥χ_{A}∥_{X} \\ for all a > 1 and measurable A \end{matrix} \} .

DEFINITION For a φ-function F, we define the lower Matuszewska–Orlicz index to be

p_{m}(F) = sup { p : for some c > 0 we have F (at) ≥ c a^{p}F (t) for 0 ≤ t < ∞ and a > 1 } .

We define the upper Matuszewska–Orlicz index to be

q_{m}(F) = inf { q : for some c < ∞ we have F (at) ≤ c a^{q}F (t) for 0 ≤ t < ∞ and a > 1 } .

Thus, for example,

p(L_{p,q}) = q(L_{p,q}) = p_{z}(L_{p,q}) = q_{z}(L_{p,q}) = p_{m}(T^{p}) = q_{m}(T^{p}) = p.

We also note the following elementary proposition about the Matuszewska–Orlicz indices.

PROPOSITION 2.1 Let F be a φ-function.

i) F is dilatory if and only if p_{m}(F) > 0.

ii) F satisfies the Δ_{2}-condition if and only if q_{m}(F) < ∞.

It was conjectured, at one time, that the Boyd and Zippin indices coincide. This is a natural conjecture in view of the fact that these indices do coincide for almost all ‘natural’ rearrangement spaces, for example, the Orlicz spaces and the Lorentz spaces. However Shimogaki (1970) gave an example of a rearrangement invariant Banach space where these indices differ.

Maligranda (1984) posed a conjecture (Problem 6.1) that would imply that the Boyd indices and Zippin indices coincide for the Orlicz–Lorentz spaces. One of the main purposes of this paper is to show that this is not the case.

In the sequel, we will always suppose that the measure space is [0, ∞) with the Lebsgue measure λ.

3 ESTIMATES FOR THE BOYD INDICES OF THE ORLICZ LORENTZ SPACES

The first results that we present give estimates for the Boyd indices. These estimates are not very sophisticated. However, as we will show in Section 4, they cannot be improved, at least in the form in which they are given. It would be nice to give better estimates at some point in the future, which would make use of more detailed structure information of the defining functions of the Orlicz–Lorentz space.

THEOREM 3.1 Let F and G be φ-functions. Then i) p_{m}(F) ≥ p(L_{F,G}) ≥ p_{m}(F ∘ G^{−1})p_{m}(G) ≥ p_{m}(F)p_{m}(G)/q_{m}(G); ii) q_{m}(F) ≤ q(L_{F,G}) ≤ q_{m}(F ∘ G^{−1})q_{m}(G) ≤ q_{m}(F)q_{m}(G)/p_{m}(G).

This will follow from the following propositions.

PROPOSITION 3.2 Let X be a rearrangement invariant space, and let F and G be φfunctions.

i) p(X) ≤ p_{z}(X) and q(X) ≥ q_{z}(X).

ii) p(L_{F}) = p_{z}(L_{F}) and q(L_{F}) = q_{z}(L_{F}).

iii) p_{z}(L_{F,G}) = p_{m}(F) and q_{z}(L_{F,G}) = q_{m}(F).

Proof: See Maligranda (1984) for part (i), and see Lindenstrauss and Tzafriri (1979) for part (ii). Part (iii) is clear.

PROPOSITION 3.3 Let F_{1}, F_{2} and G be φ-functions.

i) p(L_{F_{1},G}) ≥ p_{m}(F_{1} ∘ F_{2}^{−1})p(L_{F_{2},G}).

ii) q(L_{F_{1},G}) ≤ q_{m}(F_{1} ∘ F_{2}^{−1})q(L_{F_{2},G}).

Proof: We will show (i). The proof of (ii) is similar.

We note that if p_{1} < p_{m}(F_{1} ∘ F_{2}^{−1}), and if p_{2} < p(L_{F_{2},G}), then there is a constant c_{1} < ∞ such that for any t ≥ 0 and 0 < a < 1 we have

aF̃_{1} ∘ F̃_{2}^{−1}(t) ≤ F̃_{1} ∘ F̃_{2}^{−1}(c_{1} a^{1/p_{1}}t),

and there is a constant c_{2} < ∞ such that for any f ∈ L_{0} and 0 < b < 1 we have

∥d_{c_{1}b}f∥_{F_{2},G} ≤ c_{2}b^{−1/p_{2}} ∥f∥_{F_{2},G} .

Therefore,

∥d_{a}f∥_{F_{1},G} = ∥x ↦ f^{∗} (a F̃_{1} ∘ G̃^{−1}(x))∥_{G}
≤ ∥x ↦ f^{∗} ∘ F̃_{1} ∘ F̃_{2}^{−1} (c_{1} a^{1/p_{1}} F̃_{2} ∘ G̃^{−1}(x))∥_{G}
≤ c_{2}a^{−1/p_{1}p_{2}} ∥f∥_{F_{1},G} .

Therefore p(L_{F_{1},G}) ≥ p_{1}p_{2}, and the result follows.

Proof of Theorem 3.1: The first inequality follows from Proposition 3.2. The second inequality follows from Propositions (3.2) and (3.3). The third inequality follows because

p_{m}(F ∘ G^{−1}) ≥ p_{m}(F)p_{m}(G^{−1}) = p_{m}(F)/q_{m}(G).

4 BOYD INDICES CAN DIFFER FROM ZIPPIN INDICES

Now we show that Theorem 3.1 cannot be improved. In so doing, we answer Problem 6.1 posed by Maligranda (1984), by showing that the Boyd indices and Zippin indices do not necessarily coincide for the Orlicz–Lorentz spaces.

THEOREM 4.1 Given 0 < p < q < ∞, there is a φ-function G such that p_{m}(G) = p, q_{m}(G) = q, p(L_{1,G}) = p/q, and q(L_{1,G}) = q/p.

We also have the following interesting example, that shows that an Orlicz–Lorentz space need not be quasi normed just because its defining functions are dilatory.

THEOREM 4.2 There is a dilatory φ-function G such that L_{1,G} is not a quasi-Banach space.

At the heart of these results is the following lemma.

LEMMA 4.3 Suppose that 0 < p, q < ∞, a > 1 and n_{0}, n_{1} ∈ N are such that

(n_{1} − n_{0})a^{−p} (1 − a^{−(p+q)}) + a^{−2p−q} = 1.

Suppose that G is a φ-function such that for some L, M > 0 we have that

G̃(Ma^{2n}t) = La^{(p+q)n}t^{p}
G̃(Ma^{2n+1}t) = La^{(p+q)n+p}t^{q}

for 1 ≤ t ≤ a and n_{0} ≤ n ≤ n_{1} + 1. Then for all 0 ≤ θ ≤ inf {q/p, 1}, there are functions f and g such that we have

∥d_{a^{−θ}}f∥_{1,G} = a^{(q/p)θ} ∥f∥_{1,G} and ∥d_{a^{θ}}g∥_{1,G} = a^{−(q/p)θ} ∥g∥_{1,G} .

Proof: We define the functions f and g by

f(Mx) = \{ \begin{matrix}M^{−1}a^{−2n_{0}−3} & if 0 ≤ x < a^{2n_{0}} \\ M^{−1}a^{−2n−3} & if a^{2n} ≤ x < a^{2n+2} and n_{0} ≤ n ≤ n_{1} \\ 0 & if a^{2n_{1}+2} ≤ x \end{matrix}

g(Mx) = \{ \begin{matrix}M^{−1}a^{−2n_{0}−3−(p/q)θ} & if 0 ≤ x < a^{2n_{0}+θ} \\ M^{−1}a^{−2n−3−(p/q)θ} & if a^{2n+θ} ≤ x < a^{2n+2+θ} and n_{0} ≤ n ≤ n_{1} \\ 0 & if a^{2n_{1}+2+θ} ≤ x \end{matrix}

so that g = a^{−(p/q)θ}d_{a^{−θ}}f. Then it is sufficient to show that ∥f∥_{1,G} = ∥g∥_{1,G} = 1. We will only show that ∥g∥_{1,G} = 1, as setting θ = 0 gives the other equality.

First, we note that if

La^{(p+q)n+pθ} ≤ x < La^{(p+q)(n+1)+pθ}

then

Ma^{2n+θ} ≤ G̃^{−1}(x) < Ma^{2n+2+θ}

and so

g^{∗} ∘ G̃^{−1}(x) = M^{−1}a^{−2n−3−(p/q)θ}

implying that

G ∘ g^{∗} ∘ G̃^{−1}(x) = 1/G̃(Ma^{2n+3+(p/q)θ}) = 1/ (La^{(p+q)n+2p+q+pθ}) = L^{−1}a^{−(p+q)n−2p−q−pθ}.

Similarly, if 0 ≤ x < La^{(p+q)n_{0}+pθ}, then G ∘ g^{∗} ∘ G̃^{−1}(x) = L^{−1}a^{−(p+q)n_{0}−2p−q−pθ}. Hence

∫_{0}^{∞} G ∘ g^{∗} ∘ G̃^{−1}(x) dx
= ∑_{n=n_{0}}^{n_{1}} ∫_{La^{(p+q)n+pθ}}^{La^{(p+q)(n+1)+pθ}} G ∘ g^{∗} ∘ G̃^{−1}(x) dx + ∫_{0}^{La^{(p+q)n_{0}+pθ}} G ∘ g^{∗} ∘ G̃^{−1}(x) dx
= ∑_{n=n_{0}}^{n_{1}} (La^{(p+q)(n+1)+pθ} − La^{(p+q)n+pθ}) L^{−1}a^{−n(p+q)−2p−q−pθ}
+La^{(p+q)n_{0}+pθ}L^{−1}a^{−n_{0}(p+q)−2p−q−pθ}
= (n_{1} − n_{0})a^{−p} (1 − a^{−(p+q)}) + a^{−2p−q}
= 1,

as required.

Proof of Theorem 4.1: Construct sequences of numbers a_{k}, b_{k}, M_{k} and N_{k} (k ≥ 0) such that M_{k} and N_{k} are integers, a_{k}, b_{k} > 0,

M_{k}a_{k}^{−p} (1 − a_{k}^{−(p+q)}) + a_{k}^{−2p−q} = 1,
N_{k}b_{k}^{−q} (1 − b_{k}^{−(p+q)}) + b_{k}^{−p−2q} = 1,

a_{k} → ∞, and b_{k} → ∞. Define sequences A_{k} and B_{k} inductively as follows: A_{0} = B_{0} = 1, B_{k} = A_{k}a_{k}^{2M_{k}+2}, and A_{k+1} = B_{k}b_{k}^{2N_{k}+2} for k ≥ 0. Define G by

G(1) = 1,
G(A_{k}a_{k}^{2n}t) = G(A_{k})a_{k}^{(p+q)n}t^{p}
G(A_{k}a_{k}^{2n+1}t) = G(A_{k})a_{k}^{(p+q)n+p}t^{q}

for 0 ≤ n ≤ M_{k} and 1 ≤ t ≤ a_{k},

G(B_{k}b_{k}^{2n}t) = G(B_{k})b_{k}^{(p+q)n}t^{q}
G(B_{k}b_{k}^{2n+1}t) = G(B_{k})b_{k}^{(p+q)n+q}t^{p}

for 0 ≤ n ≤ N_{k} and 1 ≤ t ≤ b_{k}, and

G(t) = G̃(t)

for t < 1. Clearly p_{m}(G) = p and q_{m}(G) = q. From Lemma 4.3, we have that p(L_{1,G}) = p/q and q(L_{1,G}) = q/p.

Proof of Theorem 4.2: Let q = 1, and construct sequences of numbers p_{k}, a_{k} and N_{k} (k ≥ 0) such that N_{k} is an integer, a_{k} > 0,

N_{k}a_{k}^{−p_{k}} (1 − a_{k}^{−(p_{k}+q)}) + a_{k}^{−2p_{k}−q} = 1,

p_{k} → ∞, and a_{k}^{q/p_{k}} → ∞. Define a sequence A_{k} inductively as follows: A_{0} = 1, and A_{k+1} = A_{k}a_{k}^{2N_{k}+2} for k ≥ 0. Define G by

G(1) = 1,
G(A_{k}a_{k}^{2n}t) = G(A_{k})a_{k}^{(p_{k}+q)n}t_{k}^{p}
G(A_{k}a_{k}^{2n+1}t) = G(A_{k})a_{k}^{(p_{k}+q)n+p_{k}}t^{q}

for 0 ≤ n ≤ M_{k} and 1 ≤ t ≤ a_{k}, and

G(t) = G̃(t)

for t > 1. Then p_{m}(G) = 1. From Lemma 4.3, we have that p(L_{1,G}) = 0, and so by Theorem 5.3(ii) below, L_{1,G} cannot be a quasi-Banach space.

5 CONVEXITY AND CONCAVITY OF ORLICZ–LORENTZ SPACES

An important property that one might like to know about Köthe spaces is whether it is p-convex or q-concave for some prescribed p or q. These questions have already been settled for Orlicz spaces and Lorentz spaces.

For Lorentz spaces, it is almost immediate from their definition (Bennett and Sharpley (1988) or Hunt (1966)) that L_{p,q} is q-convex if p ≥ q, and p-concave if p ≤ q. However, outside of these ranges, it is more difficult. In general, it is only the case that L_{p,q} is q∧(p−ϵ)-convex and p ∨ (q + ϵ)-concave. These results are shown in many places, for example, in Bennett and Sharpley (1988) or Hunt (1966). For Orlicz–Lorentz spaces, the same methods of proof work, and we present these results here.

First we define the notions of p-convexity and q-concavity. These notions may also be found in, for example, Lindenstrauss and Tzafriri (1979).

DEFINITION If X is a Köthe space, we say that X is p-convex, respectively q-concave, if for some C < ∞ we have

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

respectively

∥(∑_{i=1}^{n} |f_{i}|^{q})^{1/q}∥_{X} ≥ C^{−1} (∑_{i=1}^{n} ∥f_{i}∥_{X}^{q})^{1/q} ,

for any f_{1}, f_{2}, …, f_{n} ∈ X.

The most elementary result about p-concavity and q-convexity is the following. This corresponds to the result that L_{p,q} is q-convex if p ≥ q, and p-concave if p ≤ q.

THEOREM 5.1 Let F and G be φ-functions.

i) If G ∘ T^{1/p} is equivalent to a convex function and G̃ ∘ F̃^{−1} is concave, then L_{F,G} is

p-convex.

ii) If G ∘ T^{1/q} is equivalent to a concave function and G̃ ∘ F̃^{−1} is convex, then L_{F,G} is

q-concave.

Proof: We will only prove (i), as the proof of (ii) is similar. We first use the identity

∥f∥_{F ∘T^{p},G∘T^{p}} = ∥|f|^{p}∥_{F,G}^{1/p}

to notice that without loss of generality we may take p = 1.

From Hardy, Littlewood and Pólya (1952), Chapter X, it follows that

∥f∥_{F,G} = sup ∥f ∘ σ ∘ F̃ ∘ G̃^{−1}∥_{G} ,

where the supremum is over all measure preserving maps σ : [0, ∞) → [0, ∞). Since G is convex, it follows from Krasnosel’skiĭ and Rutickiĭ (1961) that ∥ · ∥_{G} is 1-convex. Now the result follows easily.

However, if we take the Boyd indices into account, we can also obtain the following results. These correspond to the result that says that L_{p,q} is q ∧(p−ϵ)-convex and p∨(q +ϵ)-concave.

To state and prove these results, it is first necessary to recall notation and results from Montgomery-Smith (1992).

DEFINITION If F and G are φ-functions, then say that F is equivalently less convex than G (in symbols F ≺ G) if G ∘ F^{−1} is equivalent to a convex function. We say that F is equivalently more convex than G (in symbols F ≻ G) if G is equivalently less convex than F.

A φ-function F is said to be an N-function if it is equivalent to a φ-function F_{0} such that F_{0}(t)/t is strictly increasing, F_{0}(t)/t → ∞ as t → ∞, and F_{0}(t)/t → 0 as t → 0.

A φ-function F is said to be complementary to a φ-function G if for some c < ∞ we have

c^{−1}t ≤ F^{−1}(t) · G^{−1}(t) ≤ ct (0 ≤ t < ∞).

If F is an N-function, we will let F^{∗} denote a function complementary to F.

An N-function H is said to satisfy condition (J) if

∥1/H̃^{∗}^{−1}∥_{H^{∗}} < ∞.

To give some intuitive feeling for N-functions that satisfy condition (J), we point out that these are functions that equivalent to slowly rising convex functions, for example,

F (t) = \{ \begin{matrix}t^{1+1/ log(1+t)} & if t ≥ 1 \\ t^{1−1/ log(1+1/t)} & if t ≤ 1. \end{matrix}

THEOREM 5.2 (Montgomery-Smith, 1992) Let F, G_{1} and G_{2} be φ-functions such that one of G_{1} and G_{2} is dilatory, and one of G_{1} or G_{2} satisfies the Δ_{2}-condition. Then the following are equivalent.

i) For some c < ∞, we have that ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}} for all measurable f.

ii) There is an N-function H satisfying condition (J) such that G_{1} ∘ G_{2}^{−1} ≻ H^{−1}.

Now, we are ready to state the main results of this section.

THEOREM 5.3 Let F and G be φ-functions, and 0 < p < ∞.

i) If the lower Boyd index p(L_{F,G}) > p, and if G ≻ H^{−1} ∘ T^{p} for some N-function satisfying condition (J), then L_{F,G} is p-convex. ii) If L_{F,G} is p-convex, then the lower Boyd index p(L_{F,G}) ≥ p, and G ≻ H^{−1} ∘ T^{p} for some N-function satisfying condition (J).

Note that in part (i), it is not sufficient to take p(L_{F,G}) = 1. This is shown by the example L_{1,q} for 1 < q < ∞, which is known to be not 1-convex (Hunt, 1966).

THEOREM 5.4 Let F and G be φ-functions such that G is dilatory and p(L_{F,G}) > 0, and let 0 < q < ∞.

i) If the lower Boyd index q(L_{F,G}) < q, and if T^{q} ∘ G^{−1} ≻ H^{−1} for some N-function satisfying condition (J), then L_{F,G} is q-concave. ii) If L_{F,G} is q-convex, then the lower Boyd index q(L_{F,G}) ≤ p, and T^{q} ∘ G^{−1} ≻ H^{−1} for some N-function satisfying condition (J).

Proof of Theorem 5.3: As in the beginning of the proof of Theorem 5.1, we may suppose without loss of generality that p = 1.

The proof of (i) uses fairly standard techniques (Bennett and Sharpley, 1988). First, by Theorem 5.2, we may assume that G is equivalent to a convex function. Next, for any measurable function f, we define

f^{∗∗}(x) = \frac{1}{x} ∫_{0}^{x} f^{∗}(ξ) dξ = ∫_{0}^{1} d_{a}f^{∗}(x) da.

Then we have the Hardy inequality holding, that is, for some c < ∞ we have that ∥f∥_{F,G} ≤ ∥f^{∗∗}∥_{F,G} ≤ c ∥f∥_{F,G}. The left hand inequality is obvious. For the right hand inequality, since p(L_{F,G}) > 1, we know that for some p > 1 and some c_{1} < ∞ we have that ∥d_{a}∥_{L_{F,G}→L_{F,G}} ≤ c_{1}a^{−1/p} for all a < 1. Hence

∥f^{∗∗}∥_{F,G} = ∥∫_{0}^{1} d_{a}f^{∗} da∥_{F,G}
= ∥∫_{0}^{1} d_{a}f^{∗} ∘ F̃ ∘ G̃^{−1} da∥_{G}
≤ c_{2} ∫_{0}^{1} ∥d_{a}f^{∗} ∘ F̃ ∘ G̃^{−1}∥_{G} da

(as G is equivalent to a convex function)

= c_{2} ∫_{0}^{1} ∥d_{a}f^{∗}∥_{F,G} da
≤ c_{2} ∫_{0}^{1} c_{1}a^{−1/p} da ∥f^{∗}∥_{F,G}
≤ c_{1}c_{2} \frac{p}{p − 1} ∥f∥_{F,G} .

But, the functional that takes f to ∥f^{∗∗}∥_{F,G} is 1-convex. This is because for any x_{0} > 0, we have that

f^{∗∗}(x_{0}) = sup_{λ(A)=x_{0}} ∫_{A} |f(x)| dx.

(See Hardy, Littlewood and Pólya (1952), Chapter X, or Lindenstrauss and Tzafriri (1979).) Hence, (f + g)^{∗∗} ≤ f^{∗∗} + g^{∗∗}. Also, by Krasnosel’skiĭ and Rutickiĭ (1961), it follows that ∥ · ∥_{G} is 1-convex. Therefore,

∥∑_{i=1}^{n} |f_{i}|∥_{F,G} ≤ ∥(∑_{i=1}^{n} |f_{i}|)^{∗∗}∥_{F,G}

≤ ∥∑_{i=1}^{n} f_{i}^{∗∗}∥_{F,G}

= ∥∑_{i=1}^{n} f_{i}^{∗∗} ∘ F̃ ∘ G̃^{−1}∥_{G}
≤ c_{2} ∑_{i=1}^{n} ∥f_{i}^{∗∗} ∘ F̃ ∘ G̃^{−1}∥_{G}
= c_{2} ∑_{i=1}^{n} ∥f_{i}^{∗∗}∥_{F,G}
≤ c c_{2} ∑_{i=1}^{n} ∥f_{i}∥_{F,G} ,

as desired.

To show (ii), we note that if a is the reciprocal of an integer, then there are functions g_{1}, g_{2}, …, g_{a^{−1}}, with disjoint supports, and each with the same distribution as f, so that g_{1} + g_{2} + … + g_{a^{−1}} has the same distribution as d_{a}f. Hence

∥d_{a}f∥_{F,G} ≤ c (∥g_{1}∥_{F,G} + ∥g_{2}∥_{F,G} + … + ∥g_{a^{−1}}∥_{F,G}) = c a^{−1} ∥f∥_{F,G} .

Hence p(L_{F,G}) ≥ 1.

To show that G ≻ H^{−1} for some N-function satisfying condition (J), we note the following inequalities.

∥f∥_{F,G} = ∥∫_{0}^{∞} χ_{|f|≥t} dt∥_{F,G}
≤ c ∫_{0}^{∞} ∥χ_{|f|≥t}∥_{F,G} dt
= c ∫_{0}^{∞} F̃^{−1}(μ{|f| ≥ t}) dt
= c ∥f∥_{F,1} .

Now the result follows immediately from Theorem 5.2.

Proof of Theorem 5.4: As in the proof of Theorem 5.1, we may assume that q = 1. To prove (i) we first note, by Theorem 5.2, we may assume that G^{−1} is equivalent to a convex function. Since G is dilatory, it follows that G is equivalent to a concave function (see Montgomery-Smith (1992), Lemma 5.5.2).

Next, for any measurable function f, we define

f_{∗∗}(x) = f^{∗}(x) + \frac{1}{x} ∫_{x}^{∞} f^{∗}(ξ) dξ = f^{∗}(x) + ∫_{1}^{∞} d_{a}f^{∗}(x) da.

Then, for some c < ∞ we have that ∥f∥_{F,G} ≤ ∥f_{∗∗}∥_{F,G} ≤ c ∥f∥_{F,G}. The left hand inequality is obvious.

For the right hand inequality, we argue as follows. Since q(L_{F,G}) < 1, we know that for some q < 1 and some c_{1} < ∞ we have that ∥d_{a}∥_{L_{F,G}→L_{F,G}} ≤ c_{1}a^{−1/q} for all a > 1. Since G is dilatory, it is easy to see that there is there some p > 0 such that G ∘ T^{1/p} is equivalent to a convex function. Let q < r < 1. Then there is a constant c_{2} < ∞, depending upon r only, such that

f_{∗∗}(x) ≤ c_{2} ((f^{∗}(x))^{p} + \frac{1}{x^{p/r}} ∫_{x}^{∞} ξ^{p/r−1}(f^{∗}(ξ))^{p} dξ)^{1/p}
= c_{2} ((f^{∗}(x))^{p} + ∫_{1}^{∞} a^{p/r−1}(d_{a}f^{∗}(x))^{p} da)^{1/p} .

For if the right hand side is less than or equal to 1, then it is easily seen that

f^{∗}(ξ) ≤ 1 ∧ (\frac{x}{ξ − x})^{1/r} (ξ > x),

and hence

f_{∗∗}(x) ≤ ∫_{1}^{∞} 1 ∧ (θ − 1)^{−1/r} dθ.

Thus we have the following inequalities.

∥f_{∗∗}∥_{F,G} ≤ c_{2} ∥((f^{∗})^{p} + ∫_{1}^{∞} a^{p/r−1}(d_{a}f^{∗})^{p} da)^{1/p}∥_{F,G}

= c_{2} ∥((f^{∗} ∘ F̃ ∘ G̃^{−1})^{p} + ∫_{1}^{∞} a^{p/r−1}(d_{a}f^{∗} ∘ F̃ ∘ G̃^{−1})^{p} da)^{1/p}∥_{G}
≤ c_{3} (∥f^{∗} ∘ F̃ ∘ G̃^{−1}∥_{G}^{p} + ∫_{1}^{∞} a^{p/r−1} ∥d_{a}f^{∗} ∘ F̃ ∘ G̃^{−1}∥_{G}^{p} da)^{1/p}

(as G ∘ T^{1/p} is equivalent to a convex function)

= c_{3} (∥f^{∗}∥_{F,G}^{p} + ∫_{1}^{∞} a^{p/r−1} ∥d_{a}f^{∗}∥_{F,G}^{p} da)^{1/p}
≤ c_{1}c_{3} (1 + ∫_{1}^{∞} a^{p/r−p/q−1} da)^{1/p} ∥f∥_{F,G} .

But, the functional that takes f to ∥f_{∗∗}∥_{F,G} is 1-concave. This is because for any x_{0} > 0, we have that

f_{∗∗}(x_{0}) = \frac{1}{x_{0}} ∫_{0}^{∞} f(ξ) dξ − f^{∗∗}(x_{0}).

Hence, (f +g)_{∗∗} ≥ f_{∗∗}+g_{∗∗}. Also, by an argument similar to that given in M.A. Krasnosel’skiĭ and Rutickiĭ (1961), it follows that ∥ · ∥_{G} is 1-concave. Therefore,

∑_{i=1}^{n} ∥f_{i}∥_{F,G} ≤ ∑_{i=1}^{n} ∥f_{i∗∗}∥_{F,G}
= ∑_{i=1}^{n} ∥f_{i∗∗} ∘ F̃ ∘ G̃^{−1}∥_{G}
≤ c_{3} ∥∑_{i=1}^{n} f_{i∗∗} ∘ F̃ ∘ G̃^{−1}∥_{G}

= c_{3} ∥∑_{i=1}^{n} f_{i∗∗}∥_{F,G}

≤ c_{3} ∥(∑_{i=1}^{n} |f_{i}|)^{∗∗}∥_{F,G}

≤ c c_{1} ∥∑_{i=1}^{n} |f_{i}|∥_{F,G} ,

as desired.

To show (ii), we note that if a is an integer, then there are functions g_{1}, g_{2}, …, g_{a}, with disjoint supports, and each with the same distribution as d_{a}f, so that g_{1} + g_{2} + … + g_{a} has the same distribution as f. Hence

∥f∥_{F,G} ≥ c^{−1} (∥g_{1}∥_{F,G} + ∥g_{2}∥_{F,G} + … + ∥g_{a}∥_{F,G}) = c^{−1}a ∥d_{a}f∥_{F,G} .

Hence q(L_{F,G}) ≤ 1.

To show that G ≺ H for some N-function satisfying condition (J), we note the following inequalities.

∥f∥_{F,G} = ∥∫_{0}^{∞} χ_{|f|≥t} dt∥_{F,G}
≥ c^{−1} ∫_{0}^{∞} ∥χ_{|f|≥t}∥_{F,G} dt
= c^{−1} ∫_{0}^{∞} F̃^{−1}(μ{|f| ≥ t}) dt
= c^{−1} ∥f∥_{F,1} .

Now the result follows immediately from Theorem 5.2.

6 ADDITIONAL COMMENTS

First, we remark that there is another definition of Orlicz–Lorentz spaces given by Torchinsky (1976) (see also Raynaud (1990)). If F and G are φ-functions, then we define

∥f∥_{F,G}^{T} = inf \{ c : ∫_{0}^{∞} G(F̃^{−1}(x)f^{∗}(x)/c) \frac{dx}{x} ≤ 1\} ,

If F is dilatory and satisfy the Δ_{2}-condition, and if G is dilatory, then it is very easy to calculate the Boyd indices of these spaces — they are precisely the same as their corresponding Matuszewska–Orlicz indices. This follows from the fact that under these conditions, ∥χ_{[0,t]}∥_{F,G}^{T} ≈ F̃^{−1}(t) (See Raynaud (1990) for more details).

We also pose some questions.

i) What is the dual of an Orlicz–Lorentz space (when the space itself is 1-convex)? Is it another Orlicz–Lorentz space?

ii) Is it possible to find more precise estimates for the Boyd indices of Orlicz–Lorentz

spaces? An approach to the last problem (at least for giving necessary and sufficient conditions for p(L_{1,G}) = q(L_{1,G}) = 1 is suggested in Montgomery-Smith (1991).

ACKNOWLEDGEMENTS

This paper is an extension of work that I presented in my Ph.D. thesis (1988). I would like to express my thanks to D.J.H. Garling, my Ph.D. advisor, as well as the Science and Engineering Research Council who financed my studies at that time.

I would also like to express gratitude to A. Kamińska, W. Koslowski and N.J. Kalton for their keen interest and useful conversations.

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Index

Lorentz space, 1 Köthe space, 2 non-increasing rearrangement, 2 rearrangement invariant space, 2 φ-function, 2 Luxembourg functional, 3 Orlicz space, 3 Orlicz–Lorentz space, 3 dilation operator, 3 Boyd index, 3 Zippin index, 3 Matuszewska–Orlicz index, 4 p-convex, 8 q-concave, 8 N-function, 8 complementary function, 9 condition (J), 9