Comparison of Orlicz–Lorentz Spaces

S.J. Montgomery-Smith*

Department of Mathematics, University of Missouri, Columbia, MO 65211.

I dedicate this paper to my Mother and Father, who as well as introducing me to mathematics at an early age, shared the arduous task of bringing me up.

ABSTRACT: Orlicz–Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Mastyło, Maligranda, and Kamińska. In this paper, we consider the problem of comparing the Orlicz–Lorentz norms, and establish necessary and sufficient conditions for them to be equivalent. As a corollary, we give necessary and sufficient conditions for a Lorentz–Sharpley space to be equivalent to an Orlicz space, extending results of Lorentz and Raynaud. We also give an example of a rearrangement invariant space that is not an Orlicz–Lorentz space.

1. Introduction

The most well known examples of Banach spaces are the L_{p} spaces. Their definition is very well known: if (Ω, F, μ) is a measure space, and 1 ≤ p ≤ ∞, then for any measurable function f : Ω → ^{I}C, the L_{p}-norm is defined to be

∥f∥_{p} = (∫_{Ω} |f(ω)|^{p} dμ(ω))^{1/p}

for p < ∞, and

∥f∥_{∞} = ess sup_{ω∈Ω} |f(ω)|

for p = ∞. Then we define the Banach space L_{p}(Ω, F, μ) to be the vector space of all measurable functions f : Ω → ^{I}C for which ∥f∥_{p} is finite.

* Research supported in part by N.S.F. Grant DMS 9001796. A.M.S. (1980 ) subject classification: 46E30.

This paper is in final form.

COMPARISON OF ORLICZ–LORENTZ SPACES

It is natural to search for generalizations of these L_{p} spaces. The first examples are the Orlicz spaces. These were first studied by Orlicz [O] and Luxemburg [L]. We say that F : [0, ∞) → [0, ∞) is an Orlicz function if F is non-decreasing and convex with F (0) = 0. Now we define the Luxemburg norm by

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

whenever f is a measurable function, and define the Orlicz space L_{F} (Ω, F, μ) to be those measurable functions f for which ∥f∥_{F} is finite. The Orlicz space L_{F} is a true generalization of L_{p}, at least for p < ∞: if F (t) = t^{p}, then L_{F} = L_{p} with equality of norms.

The other examples are the Lorentz spaces. These were introduced by Lorentz [Lo1], [Lo2]. If f is a measurable function, we define the non-increasing rearrangement of f to be

f^{∗}(x) = sup{ t : μ(|f| ≥ t) ≥ x }.

If 1 ≤ q < ∞, and if w : (0, ∞) → (0, ∞) is a non-increasing function, we define the Lorentz norm of a measurable function f to be

∥f∥_{w,q} = (∫_{0}^{∞} w(x)f^{∗}(x)^{q} dx)^{1/q} .

We define the Lorentz space Λ_{w,q}(Ω, F, μ) to be the space of those measurable functions f for which ∥f∥_{w,q} is finite. These spaces also represent a generalization of the L_{p} spaces: if w(x) = 1 for all 0 ≤ x < ∞, then Λ_{w,p} = L_{p} with equality of norms.

There is one, rather peculiar, choice of the function w which turns out to be rather useful. If 1 ≤ q ≤ p < ∞, we define the spaces L_{p,q} to be Λ_{w,q} with w(x) = \frac{q}{p}x^{q/p−1}. A good reference for a description of these spaces is Hunt [H]. By a suitable change of variables, the L_{p,q} norm may also be defined in the following fashion:

∥f∥_{p,q} = (∫_{0}^{∞} |f^{∗}(x^{p/q})|^{q} dx)^{1/q} .

Thus L_{p,p} = L_{p} with equality of norms. The reason for this definition is that for any measurable set A ∈ F, we have that ∥χ_{A}∥_{p,q} = ∥χ_{A}∥_{p} = μ(A)^{1/p}. Thus L_{p,q} is a space identical to L_{p} for characteristic functions, but ‘glued’ together in a L_{q} fashion.

MONTGOMERY-SMITH

In all the spaces defined above, if we only desire to study quasi-Banach spaces rather than Banach spaces, we may remove some of the restrictions placed upon the defining parameters. Thus with the L_{p} spaces, we need only have p > 0. With the Orlicz spaces L_{F} and the Lorentz space Λ_{w,q}, we may weaken the restrictions that F be convex and that w be non-increasing (we omit details). The spaces Λ_{w,q} so obtained were studied by Sharpley [S], and so we might call then Lorentz–Sharpley spaces. With the L_{p,q} spaces, we need only have 0 < p < ∞ and 0 < q ≤ ∞, where if q = ∞, we define the Lorentz norm by

∥f∥_{p,∞} = sup_{x≥0} x^{1/p}f^{∗}(x).

Now we come to the object of the paper, the Orlicz–Lorentz spaces. These are a common generalization of the Orlicz spaces and the Lorentz spaces. They have been studied by Mastyło (see part 4 of [My]), Maligranda [Ma], and Kamińska [Ka1], [Ka2], [Ka3]. If G is an Orlicz function, and if w : [0, ∞) → [0, ∞) is a non-increasing function, we define the Orlicz–Lorentz norm of a measurable function f to be

∥f∥_{w,G} = inf { c : ∫_{0}^{∞} w(x)G(f^{∗}(x)/c) dx ≤ 1 } .

We define the Orlicz–Lorentz space Λ_{w,G}(Ω, F, μ) to be the vector space of measurable functions f for which ∥f∥_{w,G} is finite. If we do not require that the space be a Banach space, but only a quasi-Banach space, we may weaken the restrictions placed upon G and w as we did for L_{F} and Λ_{w,p} above.

We shall not work with this definition of the Orlicz–Lorentz space, however, but with a different, equivalent definition that bears more resemblance to the spaces L_{p,q}. This definition we give in the following section.

2. Definitions

First we define φ-functions. These replace the notion of Orlicz functions in our discussions.

Definition: A φ-function is a function F : [0, ∞) → [0, ∞) such that i) F (0) = 0; ii) lim_{n→∞} F (t) = ∞; iii) F is strictly increasing; iv) F is continuous; We will say that a φ-function F is dilatory if for some c_{1}, c_{2} > 1 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.

The definition of a φ-function is slightly more restrictive than that of an Orlicz function in that we insist that F be strictly increasing. The notion of dilatory replaces the notion of convexity. The Orlicz spaces generated by dilatory functions are only quasi-Banach spaces, in contrast to those generated by Orlicz functions, which are Banach spaces. The Δ_{2}-condition appears widely in literature about Orlicz spaces.

Definition: If (Ω, F, μ) is a measure space, and F is a φ-function, then we define the Luxemburg functional by

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

for every measurable function f. We define the Orlicz space, L_{F} (Ω, F, μ) (or L_{F} (μ), L_{F} (Ω) or L_{F} for short), to be the vector space of measurable functions f for which ∥f∥_{F} < ∞, modulo functions that are zero almost everywhere.

Now we define the Orlicz–Lorentz spaces.

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} .

We define the Orlicz–Lorentz space, L_{F,G}(Ω, F, μ) (or L_{F,G}(μ), L_{F,G}(Ω) or L_{F,G} for short), to be the vector space of measurable functions f for which ∥f∥_{F,G} < ∞, modulo functions that are zero almost everywhere.

Definition: If (Ω, F, μ) is a measure space, and F is a φ-function, then we define the (weak-)Orlicz–Lorentz functional by

∥f∥_{F,∞} = sup_{x≥0} F̃^{−1}(x)f^{∗}(x).

We define the Orlicz–Lorentz space, L_{F,∞}(Ω, F, μ) (or L_{F,∞}(μ), L_{F,∞}(Ω) or L_{F,∞} for short), to be the vector space of measurable functions f for which ∥f∥_{F,∞} < ∞, modulo functions that are zero almost everywhere.

We see that L_{F,F} = L_{F} with equality of norms, and that if F (t) = t^{p} and G(t) = t^{q}, then L_{F,G} = L_{p,q}, and L_{F,∞} = L_{p,∞}, also with equality of norms. Thus, if F (t) = t^{p}, we shall write L_{p,G} for L_{F,G}, and L_{G,p} for L_{G,F}. Also, if A is any measurable set, then ∥χ_{A}∥_{F,G} = ∥χ_{A}∥_{F,∞} = ∥χ_{A}∥_{F} = F̃^{−1}(μ(A)).

The Orlicz–Lorentz spaces defined here are equivalent to the definition given in the introduction, as we now describe.

Definition: A weight function is a function w : (0, ∞) → (0, ∞) such that

W (t) = ∫_{0}^{t} w(s) ds

is a φ-function.

Then if w is a weight function, and G is a φ-function, then Λ_{w,G} = L_{W̃^{−1}∘G,G}, where

W (t) = ∫_{0}^{t} w(s) ds.

Now let us provide some examples. We define the modified logarithm and the modified exponential functions by

lm(t) = \{ \begin{matrix}1 + log t & if t ≥ 1 \\ 1/(1 + log(1/t)) & if 0 < t < 1 \\ 0 & if t = 0; \end{matrix}

em(t) = lm^{−1}(t) = \{ \begin{matrix}exp(t − 1) & if t ≥ 1 \\ exp(1 − (1/t)) & if 0 < t < 1 \\ 0 & if t = 0. \end{matrix}

These functions are designed so that for large t they behave like the logarithm and the exponential functions, so that lm 1 = 1 and em 1 = 1, and so that l̃m = lm and ẽm = em. Then the functions t^{p}(lm t)^{α} and em(t^{p}) are φ-functions whenever 0 < p < ∞ and −∞ < α < ∞. If the measure space is a probability space, then the Orlicz spaces created using these functions are also known as Zygmund spaces, and the Orlicz–Lorentz spaces L_{t^{p}(lm t)^{α},q} and L_{em(t^{p}),q} are known as Lorentz–Zygmund spaces (see, for example, [B–S]).

Finally, we define the notions of equivalence.

Definition: 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 < ∞.

Definition: We say that two function spaces X and Y on the same measure space are equivalent if for some number c < ∞ we have that f ∈ X ⇔ f ∈ Y with c^{−1} ∥f∥_{X} ≤ ∥f∥_{Y} ≤ c ∥f∥_{X} for all measurable functions f.

3. Survey of Known Comparison Results

There are at least four obvious questions about Orlicz–Lorentz spaces.

i) For which φ-functions F and G is L_{F,G} equivalent to a normed space (or p-convex, or q-concave)?

ii) What are the Boyd indices of the Orlicz–Lorentz spaces? iii) What are necessary and sufficient conditions for L_{F_{1},G_{1}} and L_{F_{2},G_{2}} to be equivalent? iv) Is every rearrangement space equivalent to some Orlicz–Lorentz space? The first and second questions are intimately related, and will be dealt with in another paper [Mo2]. In general, they are very hard to answer. The third question is the subject of this paper. As a corollary, we will also be able to answer the fourth question.

There have already been many comparison results for Lorentz spaces. Indeed, Lorentz himself provided one of the first in 1961 [Lo3]. He found necessary and sufficient conditions for Λ_{w,1} to be equivalent to an Orlicz space.

Definition: A weight function w is said to be strictly monotone if either

i) w is strictly increasing, w(t) → 0 as t → 0 and w(t) → ∞ as t → ∞, or i′) w is strictly decreasing, w(t) → ∞ as t → 0 and w(t) → 0 as t → ∞.

Definition: A strictly monotone weight function is said to satisfy condition (L) if there is a number c < ∞ such that

∫_{0}^{∞} \frac{dt}{w^{−1}(c w(t))} < ∞.

Theorem 3.1. Let w : (0, ∞) → (0, ∞) be a decreasing, strictly monotone weight function. Then the following are equivalent.

i) Λ_{w,1} is equivalent to an Orlicz space. ii) w satisfies condition (L).

The kinds of weight functions that satisfy condition (L) are slowly increasing or slowly decreasing functions. An example that Lorentz implicitly gave is

w(t) = \{ \begin{matrix}t^{−1/ log(1+log t)} & if t ≥ 1 \\ t^{1/ log(1−log t)} & if 0 < t < 1. \end{matrix}

Recently, Raynaud [R] noticed that the above result is also true if w is strictly increasing. He then went on to show the following result.

Theorem 3.2. Let w be a weight function, and 0 < p < ∞. If there are strictly monotone weight functions w_{0} and w_{1} satisfying condition (L) and a number c < ∞ such that

c^{−1} \frac{W (t)}{t} ≤ w_{0}(t)w_{1}(t) ≤ c \frac{W (t)}{t},

then Λ_{w,p} is equivalent to an Orlicz space.

It may seem that the scope of these results is limited, but this is not really the case. Using Lemmas 5.1.2 and 5.1.3 below, one can use these results to find sufficient conditions for equivalence of two Orlicz–Lorentz spaces that are no stronger than the conditions given in this paper.

There are also results due to Bennett and Rudnick [B–R] (see also [B–S]). They proved the following results for probability spaces, but using their methods, it is not too hard to see that these results are true for all measure spaces.

Theorem 3.3. For every 0 < p < ∞, and for every −∞ < α < ∞, we have that L_{t^{p}(lm t)^{α}} and L_{t^{p}(lm t)^{α},p} are equivalent.

Theorem 3.4. For every β > 0, we have that L_{em(t^{β})} and L_{em(t^{β}),∞} are equivalent.

4. Comparison of Orlicz–Lorentz Spaces

In this section, we state the main results of this paper, and give necessary and sufficient conditions for which, given certain restrictions upon G_{1} and G_{2}, we have that ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}}. Thus we find necessary and sufficient conditions for L_{F_{1},G_{1}} and L_{F_{2},G_{2}} to be equivalent.

We first notice that ∥f∥_{p,q_{1}} ≤ ∥f∥_{p,q_{2}} whenever q_{1} ≥ q_{2} (see [H]). This suggests that we have a result something like: if G_{1}∘G_{2}^{−1} is a convex function, then ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}}. And this is indeed the case. However, more is true. For example, if G(t) = t lm t, then it follows from Theorem 3.3 that L_{G,1} is equivalent to L_{G,G}. Thus, it would seem that we only need to know that G_{1} ∘ G_{2}^{−1} is ‘close,’ in some sense, to a convex function.

In this paper, we establish precisely what this notion of closeness is. But, before stating the conditions, we first give a little bit of motivation. We note that a dilatory φfunction G is determined completely, up to equivalence, by its values G(a^{n}), where a > 1 is any fixed number, and n ranges over all integers. Thus, we note that a φ-function G is equivalent to a convex function if and only if for some a > 1 and N ∈ IIN, and all n ∈ ZZ and m ∈ IIN, we have that G(a^{n+m}) ≥ a^{m−N}G(a^{n}) (see Lemma 5.4.2 below).

In all that follows, we take the natural numbers to be IIN = {1, 2, 3, …}.

Definition: Let G be a φ-function. We say that G is i) almost convex if there are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN,

the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≥ a^{m−N}G(a^{n})

is less than b^{m}; ii) almost concave if there are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN,

the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≤ a^{m+N}G(a^{n})

is less than b^{m}; iii) almost linear if there are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN,

the cardinality of the set of n ∈ ZZ such that we do not have

a^{m−N}G(a^{n}) ≤ G(a^{n+m}) ≤ a^{m+N}G(a^{n})

is less than b^{m}; iv) almost constant if there are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN,

the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≤ a^{N}G(a^{n})

is less than b^{m};

v) almost vertical if G^{−1} is almost constant.

We will also express our results in terms of what we shall call condition (J).

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.

Definition: 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.

Definition: 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.

The notation F^{∗} makes sense if F is an N-function, because then there is always a function G complementary to F, and further, if G_{1} and G_{2} are both complementary to F, then G_{1} and G_{2} are equivalent.

Our definition of a complementary function differs from the usual definition. If F is an N-function that is convex, then the complementary function is usually defined by F^{∗}(t) = sup_{s≥0}(st − F (s)). However, it is known that t ≤ F^{−1}(t) · F^{∗}^{−1}(t) ≤ 2t (see [K–R]). Thus our definition is equivalent.

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

∥1/H̃^{∗}^{−1}∥_{H^{∗}} < ∞.

The kinds of N-functions that satisfy condition (J) are slowly rising functions. These are essentially the kinds of Orlicz functions that Lorentz describes in Theorem 1 of his paper [Lo3].

We also describe our results in a third fashion. The following definitions are motivated by the fact that G_{1} ≻ G_{2} if and only if for some c < ∞ and all s ≥ 1 and t > 0 we have that G_{1}(st)/G_{1}(t) ≥ c^{−1}G_{2}(st)/G_{2}(t) (see Lemma 5.4.2).

Definition: Let G_{1} and G_{2} be φ-functions. We say that

i) G_{1} is almost less convex than G_{2} if there are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that we do not have

\frac{G_{1}(a^{n+m})}{G_{1}(a^{n})} ≤ a^{N} \frac{G_{2}(a^{n+m})}{G_{2}(a^{n})}

is less than b^{m}; ii) G_{1} is almost more convex than G_{2} if G_{2} is almost less convex than G_{1}; iii) G_{1} is almost equivalent to G_{2} if there are numbers a > 1, b > 1 and N ∈ IIN such that

for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that we do not have

a^{−N} \frac{G_{2}(a^{n+m})}{G_{2}(a^{n})} ≤ \frac{G_{1}(a^{n+m})}{G_{1}(a^{n})} ≤ a^{N} \frac{G_{2}(a^{n+m})}{G_{2}(a^{n})}

is less than b^{m}.

Now we collect together the comparison results. For all these results, we will assume that the measure space is [0, ∞) with Lebesgue measure. In fact, any non-atomic infinite measure space will do. There are also similar results for non-atomic probability spaces, and IIN with the counting measure (i.e. sequence spaces). We do not give details for these cases. However the idea is that for non-atomic probability spaces, we need only consider the properties of the relevant φ-functions G(t) for large t, and for sequence spaces, their properties for small t. Obviously, if one is only interested in sufficient conditions for Orlicz– Lorentz spaces to be equivalent, one can use any measure space. (Recall that ≍ means equivalent to, see Section 2.)

Proposition 4.1. Let F_{1}, F_{2}, G_{1} and G_{2} be φ-functions.

i) If F_{1} ∘ G_{1}^{−1} ≍ F_{2} ∘ G_{2}^{−1} and G_{1} ≍ G_{2}, and if one of G_{1} or G_{2} is dilatory, then L_{F_{1},G_{1}} and L_{F_{2},G_{2}} are equivalent.

ii) If F_{1} ≍ F_{2}, then L_{F_{1},∞} and L_{F_{2},∞} are equivalent.

Theorem 4.2. Let F, G_{1} and G_{2} be φ-functions. Consider the following statements.

i) For some c < ∞, we have that ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}} for all measurable f. ii) G_{1} ∘ G_{2}^{−1} is almost convex. iii) There is an N-function H satisfying condition (J) such that G_{1} ∘ G_{2}^{−1} ≻ H^{−1}. iv) G_{1} is almost more convex than G_{2}. Then, if one of G_{1} or G_{2} is dilatory, then we have (ii)⇒(i). If one of G_{1} or G_{2} is dilatory and G_{1} satisfies the Δ_{2}-condition, or if G_{2} satisfies the Δ_{2}-condition, then we have (i)⇒(ii). If G_{2} is dilatory and satisfies the Δ_{2}-condition, then we have (ii)⇔(iv). We always have (ii)⇔(iii).

Theorem 4.3. Let F_{1}, F_{2}, G_{1} and G_{2} be φ-functions such that one of G_{1} or G_{2} is dilatory, and that one of G_{1} or G_{2} satisfies the Δ_{2}-condition. Then the following are equivalent.

i) L_{F_{1},G_{1}} and L_{F_{2},G_{2}} are equivalent. ii) F_{1} ≍ F_{2}, and G_{1} ∘ G_{2}^{−1} is almost linear. iii) F_{1} ≍ F_{2}, and there exist N-functions H and K satisfying condition (J) such that

G_{1} ∘ G_{2}^{−1} = H ∘ K^{−1}. iv) F_{1} ≍ F_{2}, and there exist N-functions H and K satisfying condition (J) such that

G_{1} ∘ G_{2}^{−1} = H^{−1} ∘ K.

v) F_{1} ≍ F_{2}, and G_{1} is almost equivalent to G_{2}. vi) F_{1} ≍ F_{2}, and there exist N-functions H and K satisfying condition (J) and a number

c < ∞ such that c^{−1}G_{1}/G_{2} ≤ H/K ≤ c G_{1}/G_{2}. vii) F_{1} ≍ F_{2}, and there exist strictly monotone weight functions w_{0} and w_{1} satisfying

condition (L) and a number c < ∞ such that c^{−1}G_{1}/G_{2} ≤ w_{0}w_{1} ≤ c G_{1}/G_{2}. viii) F_{1} ≍ F_{2}, and there exists an almost linear φ-function F and a number c < ∞ such

that c^{−1}F (t) ≤ tG_{1}(t)/G_{2}(t) ≤ c F (t) for all t > 0.

The condition that one of the φ-functions G_{1} or G_{2} satisfy the Δ_{2}-condition is required, as is shown by the following example. Let G_{1}(t) = em t and G_{2}(t) = em t^{2}. By Theorem 4.6 below, L_{1,G_{1}} and L_{1,G_{2}} are both equivalent to L_{1,∞}. But, it is clear that G_{1} ∘ G_{2}^{−1} is far from being almost linear. The author doesn’t know whether the condition that one of G_{1} or G_{2} be dilatory is needed.

We are also able to obtain certain results stating that in order to compare L_{F_{1},G_{1}} and L_{F_{2},G_{2}}, we need only compare the norms for a certain class of test functions.

Definition: Let T_{1} be the set of functions f : [0, ∞) → [0, ∞) such that for some 0 = a_{0} < a_{1} < a_{2} < … < a_{n} we have that

f(x) = \{ \begin{matrix}1/a_{i} & if a_{i−1} ≤ x < a_{i} and 1 ≤ i ≤ n \\ 0 & otherwise. \end{matrix}

If F is a φ-function, let T_{F} = { F^{−1} ∘ f : f ∈ T_{1}} = { f ∘ F̃^{−1} : f ∈ T_{1}}.

Theorem 4.4. Let F, G_{1} and G_{2} be φ-functions. Suppose that G_{2} is dilatory, and that 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}} whenever f^{∗} ∈ T_{F}. ii) For some c < ∞ we have that ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}} for all measurable f.

Theorem 4.5. Let F_{1}, F_{2}, G_{1} and G_{2} be φ-functions. Suppose that one of G_{1} or G_{2} is dilatory, and that one of G_{1} or G_{2} satisfies the Δ_{2}-condition. Then the following are equivalent.

i) For some c < ∞ we have that c^{−1} ∥f∥_{F_{1},G_{1}} ≤ ∥f∥_{F_{2},G_{2}} ≤ c ∥f∥_{F_{1},G_{1}} whenever f^{∗} ∈ T_{F_{1}}.

ii) For some c < ∞ we have that c^{−1} ∥f∥_{F_{1},G_{1}} ≤ ∥f∥_{F_{2},G_{2}} ≤ c ∥f∥_{F_{1},G_{1}} whenever f^{∗} ∈

T_{F_{2}}. iii) L_{F_{1},G_{1}} and L_{F_{2},G_{2}} are equivalent.

Finally, we give a result for the weak-Orlicz–Lorentz spaces.

Theorem 4.6. Let F_{1}, F_{2} and G be φ-functions. Then the following are equivalent.

i) L_{F_{1},G} and L_{F_{2},∞} are equivalent. ii) F_{1} ≍ F_{2}, and G is almost vertical. iii) F_{1} ≍ F_{2}, and ∥1/G̃^{−1}∥_{G} < ∞.

iv) F_{1} ≍ F_{2}, and G̃^{−1} satisfies condition (L).

It is clear that all the results given in Section 3 follow from these results. We are also able to answer a question of Raynaud, and prove the converse to Theorem 3.2.

Theorem 4.7. Let w be a weight function, and 0 < p < ∞. If Λ_{w,p} is equivalent to an Orlicz space, then there are strictly monotone weight functions w_{0} and w_{1} satisfying condition (L) and a number c < ∞ such that

c^{−1} \frac{W (t)}{t} ≤ w_{0}(t)w_{1}(t) ≤ c \frac{W (t)}{t}.

Proof: This follows immediately from the implication (i)⇒(vii) in Theorem 4.3, and from the observation that a strictly monotone weight function w satisfies condition (L) if and only if w^{p} satisfies condition (L) for any 0 < p < ∞.

5. The Proof of the Results of Section 4

The proofs of the results of Section 4 are rather long. We will split the proof into many lemmas that are grouped into several subsections according to their nature. Many of the lemmas, if not obvious, are at least ‘believable without proof,’ and the reader may pass over them quickly. The key results are contained in Sections 5.3, 5.5 and 5.6.

These proofs could be shortened considerably if we assumed throughout that all φfunctions were dilatory and satisfied the Δ_{2}-condition, but then our results would be correspondingly weaker. In particular, Theorem 6.1 below would be much less general.

5.1. The Elementary Propositions

The first result is obvious, and requires no proof.

Lemma 5.1.1. Let G be a φ-function.

i) If G is dilatory, then for all c_{1} < ∞ there is a number c_{2} < ∞ such that if

∫_{Ω} G ∘ f(ω) dμ(ω) ≤ c_{1},

then ∥f∥_{G} ≤ c_{2}. ii) If G is dilatory, then for all c_{1} > 0 there is a number c_{2} > 0 such that if

∫_{Ω} G ∘ f(ω) dμ(ω) ≥ c_{1},

then ∥f∥_{G} ≥ c_{2}.

Now we have the first result from Section 4.

Proof of Proposition 4.1: This is a simple consequence of Lemma 5.1.1.

The following results describe the basic ‘algebra’ that the Orlicz–Lorentz spaces satisfy. Essentially, they allow one to reduce comparison of Orlicz–Lorentz spaces to the problem of comparing L_{1,G} to L_{1}. The proofs are straightforward, so we omit them.

Lemma 5.1.2. Suppose that F, G_{1} and G_{2} are φ-functions. Then for any number c < ∞ we have that ∥f∥_{F,G_{1}} ≤ c ∥f∥_{F,G_{2}} for all measurable f (respectively, f ∈ T_{F}) if and only if ∥f∥_{1,G_{1}} ≤ c ∥f∥_{1,G_{2}} for all measurable f (respectively, f ∈ T_{1}).

Lemma 5.1.3. Suppose that G_{1}, G_{2} and H are φ-functions.

i) If H is dilatory, then if for some number c_{1} < ∞ we have that ∥f∥_{1,G_{1}} ≤ c_{1} ∥f∥_{1,G_{2}} for all measurable f (respectively, f ∈ T_{1}), then for some number c_{2} < ∞ we have that ∥f∥_{1,G_{1}∘H} ≤ c_{2} ∥f∥_{1,G_{2}∘H} for all measurable f (respectively, f ∈ T_{1}).

ii) If H satisfies the Δ_{2}-condition, then if for some number c_{1} < ∞ we have that ∥f∥_{1,G_{1}∘H} ≤ c_{1} ∥f∥_{1,G_{2}∘H} for all measurable f (respectively, f ∈ T_{1}), then for some number c_{2} < ∞ we have that ∥f∥_{1,G_{1}} ≤ c_{2} ∥f∥_{1,G_{2}} for all measurable f (respectively, f ∈ T_{1}).

5.2. Conditions for Functions to be Dilatory, etc

Here we collect the results that pertain to when a φ-function is dilatory or satisfies the Δ_{2}-condition. The first result is obvious.

Lemma 5.2.1. Let G be a φ-function.

i) If there are numbers a > 1, c_{1} > 1 and c_{2} > 1 such that

c_{1}G(a^{n}) ≤ G(c_{2}a^{n})

except for finitely many n, then G is dilatory. ii) If there are numbers a > 1, c_{1} > 1 and c_{2} > 1 such that

c_{1}G(a^{n}) ≥ G(c_{2}a^{n})

except for finitely many n, then G satisfies the Δ_{2}-condition.

Now we show how the property of G being dilatory or satisfying the Δ_{2}-condition may be captured by the properties of L_{F,G}.

Lemma 5.2.2. Suppose that G is a φ-function. Then the following are equivalent.

i) G is dilatory. ii) There is a number c < ∞ such that we have ∥f∥_{1,G} ≤ c for all functions f : IR → IR

of the form

f(x) = \{ \begin{matrix}a^{−1} & if 0 ≤ x < a \\ b^{−1} & if a ≤ x < b \\ 0 & otherwise, \end{matrix}

where b > a > 0.

Proof: First we will show that (i)⇒(ii). Given a function of the above form, we note that

∫_{0}^{∞} G ∘ f^{∗} ∘ G̃^{−1}(x) dx ≤ G̃(a)G(a^{−1}) + G̃(b)G(b^{−1}) = 2.

Then the result follows immediately from Lemma 5.1.1.

To show that (ii)⇒(i), we will consider functions of the form

f(x) = \{ \begin{matrix}G^{−1}(3^{m}) & if 0 ≤ x < 1/G^{−1}(3^{m}) \\ G^{−1}(3^{n}) & if 1/G^{−1}(3^{m}) ≤ x < 1/G^{−1}(3^{n}) \\ 0 & otherwise , \end{matrix}

where m > n are integers. Then we know that ∥f∥_{1,G} ≤ c, and so

1 ≥ ∫_{0}^{∞} G(c^{−1}f^{∗} ∘ G̃^{−1}(x)) dx ≥ 3^{−m}G(c^{−1}G^{−1}(3^{m})) + \frac{2}{3} 3^{−n}G(c^{−1}G^{−1}(3^{n})).

Therefore, for all except one n ∈ IIN we have that

\frac{2}{3}3^{−n}G(c^{−1}G^{−1}(3^{n})) ≤ \frac{1}{2},

that is,

c^{−1}G^{−1}(3^{n}) ≤ G^{−1}(\frac{3}{4}3^{n}).

By Lemma 5.2.1, G^{−1} satisfies the Δ_{2}-condition, and hence G is dilatory.

Lemma 5.2.3. If F, G_{1} and G_{2} are φ-functions such that G_{1} is dilatory, and such that for some c < ∞ we have ∥f∥_{F,G_{2}} ≤ c ∥f∥_{F,G_{1}} for all f ∈ T_{F}, then G_{2} is dilatory.

Proof: This follows immediately from Lemmas 5.1.2 and 5.2.2.

Lemma 5.2.4. Suppose that G is a φ-function. Consider the following statements.

i) G satisfies the Δ_{2}-condition. ii) Given c > 1, there are numbers d > 1 and N ∈ IIN such that we have ∥f∥_{1,G} ≥ c for

all functions f : IR → IR of the form

f(x) = \{ \begin{matrix}d^{−k_{i}} & if d^{k_{i−1}} ≤ x < d^{k_{i}} and 1 < i ≤ N \\ 0 & otherwise, \end{matrix}

where k_{1} < k_{2} < … < k_{N} are integers, and k_{0} = −∞.

Then (ii)⇒(i). Furthermore, if G is dilatory, then (i)⇒(ii).

Proof: First we will show that (i)⇒(ii) when G is dilatory. Choose d so that G̃(d^{n}) ≥ 2G̃(d^{n−1}) for all n ∈ ZZ. Then if f is of the above form, we have that

∫_{0}^{∞} G ∘ f^{∗} ∘ G̃^{−1}(x) dx ≥ ∑_{i=1}^{N}(G̃(d^{k_{i}}) − G̃(d^{k_{i}−1}))G(d^{−k_{i}}) ≥ N/2.

Thus if G satisfies the Δ_{2}-condition, then by Lemma 5.1.1, there is some N ∈ IIN such that for all f of the above form, we have that ∥f∥_{1,G} ≥ c.

To show that (ii)⇒(i), let us pick c > 2. Then for any function of the above form, we have that

1 < ∫_{0}^{∞} G(2^{−1}f^{∗} ∘ G̃^{−1}(x)) dx ≤ ∑_{i=1}^{N} G̃(d^{k_{i}})G(2^{−1}d^{−k_{i}}).

Therefore, the cardinality of the set of n ∈ IIN such that

G̃(d^{n})G(2^{−1}d^{−n}) ≤ N^{−1}

is less than N. By Lemma 5.2.1, this shows that G satisfies the Δ_{2}-condition.

Lemma 5.2.5. Suppose that F, G_{1} and G_{2} are φ-functions such that one of G_{1} and G_{2} is dilatory. If G_{1} satisfies the Δ_{2}-condition, and for some c < ∞ we have that ∥f∥_{F,G_{2}} ≥ c^{−1} ∥f∥_{F,G_{1}} for all f ∈ T_{F}, then G_{2} satisfies the Δ_{2}-condition.

Proof: By Lemma 5.2.3, we have that G_{1} is dilatory. Now the result follows immediately from Lemmas 5.1.2 and 5.2.4.

5.3. Comparison Conditions for L_{1,G}

In this subsection, we give the key lemma that demonstrates the relationship between the almost convexity of G, and the comparison between L_{1,G} and L_{1}. As a corollary, we will also obtain results that show that in the definition of the ‘almost properties’ that we can take the value of a to be arbitrarily large.

Lemma 5.3.1. Suppose that G is a φ-function. Then the following are equivalent.

i) For some c < ∞, we have that ∥f∥_{1,G} ≤ c ∥f∥_{1} for all measurable f. ii) For some c < ∞, we have that ∥f∥_{1,G} ≤ c ∥f∥_{1} for all f ∈ T_{1}. iii) For all sufficiently large a, there are numbers b > 1 and N ∈ IIN such that for all

m ∈ IIN, the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≥ a^{m−N}G(a^{n})

is less than b^{m}. iv) G is almost convex.

The proof will require the next lemma.

Lemma 5.3.2. Let G be a φ-function. If G is almost convex, then given a′ > 1, there are numbers a > 1, b > 1, c < ∞ and N ∈ IIN such that a > max{a′, b} and such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≥ a^{m−N}G(a^{n})

is less than cb^{m}.

There are similar results if G is almost concave or almost constant.

Proof: There are numbers a > 1, b > 1 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set

A_{m} = { n ∈ ZZ : G(a^{n+m}) < a^{m−N}G(a^{n})}

is less than b^{m}. Pick c ∈ IIN such that a^{c} > b and a^{c} > a′, and let

A′_{m} = { n ∈ ZZ : G(a^{c(n+m)}) < a^{c(m−N)}G(a^{cn})}.

Then, if n ∈ A′_{m}, then at least one of cn, cn + m, …, or cn + (c − 1)m ∈ A_{m}, and hence |A′_{m}| < cb^{m}.

Proof of Lemma 5.3.1: Clearly, (i)⇒(ii) and (iii)⇒(iv). We will show (ii)⇒(iii). By Lemma 5.2.3, we know that G is dilatory. Thus we suppose that a > 2 and G(at) ≥ 2G(t) for all t ≥ 0. Choose N so that a^{N−1} > c. We will prove the result by showing that there cannot be numbers m ∈ IIN and n_{1} < n_{2} < … < n_{a^{m−N+1}} such that

G(a^{n_{i}}) < a^{m−N}G(a^{n_{i}−m}).

For otherwise, consider the function

f(x) = \{ \begin{matrix}a^{n_{i}} & if a^{−n_{i+1}} ≤ x < a^{−n_{i}} and 1 ≤ i ≤ a^{m−N+1} \\ 0 & otherwise, \end{matrix}

where we take n_{a^{m−N+2}} = ∞. Clearly, ∥f∥_{1} ≤ a^{m−N+1}. But also, we have the following inequalities.

∫_{0}^{∞} G(a^{−m}f^{∗} ∘ G̃^{−1}(x)) dx

≥ ∑_{i=1}^{a^{m−N+1}} ∫_{G̃(a^{−n_{i+1}} )}^{G̃(a^{−n_{i}})} G(a^{−m}f^{∗} ∘ G̃^{−1}(x)) dx

≥ \frac{1}{2} ∑_{i=1}^{a^{m−N+1}} \frac{G(a^{−m+n_{i}})}{G(a^{n_{i}})}

≥ a/2 > 1,

where the penultimate inequality follows because G̃(a^{−n_{i}}) > 2G̃(a^{−n_{i+1}}). Thus ∥f∥_{1,G} ≥ a^{m} ≥ a^{N−1} ∥f∥_{1}, which is a contradiction.

Now we show that (iv)⇒(i). By Lemma 5.3.2, there are numbers a > b > 1, c < ∞ and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set

A_{m} = { n ∈ ZZ : G(a^{n}) < a^{m−N}G(a^{n−m})}

is less than cb^{m}. Let {k_{1}, k_{2}, …} be the (possible finite or empty) set of integers not in ⋃_{m=1}^{∞} A_{m}. Define the sequence of sets B_{m} by setting B_{1} = A_{1}, and

B_{m} = {k_{m−1}} ∪ A_{m} \ ⋃_{m′<m} A_{m′}

for m > 1. Then |B_{m}| ≤ cb^{m}.

Choose c_{1} = (a − 1)/a^{4}. Suppose f is a measurable function such that

∥f∥_{1} = ∫_{0}^{∞} f^{∗}(x) dx ≤ c_{1}.

COMPARISON OF ORLICZ–LORENTZ SPACES

For each n ∈ ZZ, let m_{n} ∈ ZZ ∪ {∞} be such that

a^{−m_{n}} ≤ a^{−n}f^{∗}(a^{−n}) ≤ a^{1−m_{n}}.

Then

c_{1} ≥ ∑_{n=−∞}^{∞} ∫_{a^{−n−1}}^{a^{−n}} f^{∗}(x) dx

≥ ∑_{n=−∞}^{∞}(a^{−n} − a^{−n−1})f^{∗}(a^{n})

≥ \frac{a − 1}{a} ∑_{n=−∞}^{∞} a^{−m_{n}}.

Therefore,

∑_{n=−∞}^{∞} a^{−m_{n}} ≤ \frac{a}{a − 1} c_{1} = a^{−3}.

In particular, we note that m_{n} ≥ 3 for all n ∈ ZZ.

Then ∫_{0}^{∞} G ∘ f^{∗} ∘ G̃^{−1}(x) dx

= ∑_{n=−∞}^{∞} ∫_{G̃(a^{−n−1})}^{G̃(a^{−n})} G ∘ f^{∗} ∘ G̃^{−1}(x) dx

≤ ∑_{n=−∞}^{∞} G̃(a^{−n})G(f^{∗}(a^{−n−1}))

≤ ∑_{n=−∞}^{∞} G(a^{2+n−m_{n+1}})/G(a^{n}).

Now let

V = { n ∈ ZZ : n ∉ B_{m} for all m ≤ m_{n+1} − 2}.

Then ∫_{0}^{∞} G ∘ f^{∗} ∘ G̃^{−1}(x) dx

≤ ∑_{n∈V} G(a^{2+n−m_{n+1}})/G(a^{n}) + ∑_{m=1}^{∞} ∑_{n∈B_{m}\V} G(a^{2+n−m_{n+1}})/G(a^{n}).

If n ∈ V, then either n ∉ A_{m_{n+1}−2}, or m_{n+1} = ∞, and so

G(a^{2+n−m_{n+1}})/G(a^{n}) ≤ a^{N+2−m_{n+1}}.

If n ∈ B_{m} \ V, then m ≤ m_{n+1} − 2, and so

G(a^{2+n−m_{n+1}})/G(a^{n}) ≤ G(a^{n−m})/G(a^{n}).

If we also know that m > 1, then n ∉ A_{m−1}, and so

G(a^{2+n−m_{n+1}})/G(a^{n}) ≤ G(a^{1+n−m})/G(a^{n}) ≤ a^{N+1−m}.

Therefore,

∫_{0}^{∞} G ∘ f^{∗} ∘ G̃^{−1}(x) dx

≤ ∑_{n∈V} a^{N+2−m_{n+1}} + ∑_{n∈B_{1}\V} G(a^{n−1})/G(a^{n}) + ∑_{m=2}^{∞} ∑_{n∈B_{m}\V} a^{N+1−m}

≤ a^{N−1} + ∑_{n∈B_{1}} G(a^{n−1})/G(a^{n}) + ∑_{m=2}^{∞} cb^{m}a^{N+1−m},

which is a finite number whose value does not depend on f. However, by Lemma 5.2.1, G is dilatory, and hence by Lemma 5.1.1, ∥f∥_{1,G} is bounded by some number that does not depend on f.

5.4. Convexity and Concavity Conditions

In this subsection, we give basic results about convexity and concavity, and their ‘almost’ equivalents. First, we give a technical lemma whose proof is obvious.

Lemma 5.4.1. Let G be a φ-function. Define a map f : ZZ → ZZ so that for all n ∈ ZZ we have

a^{f(n)−1} ≤ G(a^{n}) < a^{f(n)}.

i) If G is dilatory, then there is a number L ∈ IIN such that f(n_{1}) ≠ f(n_{2}) if |n_{1} − n_{2}| ≥ L.

ii) If G satisfies the Δ_{2}-condition, then there is a number M ∈ IIN such that

|f(n_{1}) − f(n_{2})| ≤ M |n_{1} − n_{2}|

for all n_{1}, n_{2} ∈ ZZ.

Next we give some results about convexity.

Lemma 5.4.2. Let G_{1} and G_{2} be φ-functions. Consider the following statements.

i) G_{1} is equivalently more convex than G_{2}. ii) There is a number c < ∞ such that G_{1} ∘ G_{2}^{−1}(st) ≥ c^{−1}sG_{1} ∘ G_{2}^{−1}(t) for all s ≥ 1 and

t ≥ 0. iii) There is a number c < ∞ such that G_{1}(uv)/G_{1}(v) ≥ c^{−1}G_{2}(uv)/G_{2}(v) for all u ≥ 1

and v > 0. iv) There are numbers a > 1 and N ∈ IIN such that for all m ∈ IIN and n ∈ ZZ, we have

that

\frac{G_{1}(a^{n+m})}{G_{1}(a^{n})} ≥ a^{−N} \frac{G_{2}(a^{n+m})}{G_{2}(a^{n})}.

Then we have (i)⇔(ii)⇔(iii)⇒(iv). If one of G_{1} or G_{2} satisfies the Δ_{2}-condition, then (iv)⇒(iii).

Proof: The implications (i)⇒(ii) and (iii)⇒(iv) are obvious. The implications (ii)⇔(iii) follow by setting t = G_{2}(v) and st = G_{2}(uv).

To show (ii)⇒(i), we let

H_{0}(t) = inf_{s>1} \frac{G_{1} ∘ G_{2}^{−1}(st)}{s} and H(t) = ∫_{0}^{t} \frac{H_{0}(s)}{s} ds.

Then it is easy to see that H is convex, and that H is equivalent to G_{1} ∘ G_{2}^{−1}.

Now suppose that G_{1} satisfies the Δ_{2}-condition. We show (iv)⇒(iii). Let L ∈ IIN be such that G_{1}(at) ≤ a^{L}G_{1}(t) for all t ≥ 0. Suppose that for some u > 1 and v > 0 we have that

\frac{G_{1}(uv)}{G_{1}(v)} < a^{−N−3L} \frac{G_{2}(uv)}{G_{2}(v)}.

Let m and n be such that a^{m} ≤ u < a^{m+1} and a^{n} ≤ v < a^{n+1}. Then

a^{−3L} \frac{G_{1}(a^{m+n+2})}{G_{1}(a^{n})} ≤ \frac{G_{1}(a^{m+n})}{G_{1}(a^{n+1})} < a^{−N−3L} \frac{G_{2}(a^{m+n+2})}{G_{2}(a^{n})},

which is a contradiction.

The argument for when G_{2} satisfies the Δ_{2}-condition is similar.

Now we start looking at the ‘almost properties.’ First we relate almost convexity to almost concavity.

Lemma 5.4.3. Suppose that G is a φ-function.

i) If G is almost convex and satisfies the Δ_{2}-condition, then G^{−1} is almost concave. ii) If G is almost concave, then G^{−1} is almost convex.

This will follow from the next lemma.

Lemma 5.4.4. Suppose that G is a φ-function and that a > 1. Consider the following statements.

i) There are numbers b > 1 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that we do not have

G(a^{n+m}) ≥ a^{m−N}G(a^{n})

is less than b^{m}. ii) There are numbers b > 1 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the

set of n ∈ ZZ such that we do not have

G^{−1}(a^{n+m}) ≤ a^{m+N}G^{−1}(a^{n})

is less than b^{m}. Then (ii)⇒(i). If , in addition, G satisfies the Δ_{2}-condition, then (i)⇒(ii).

Proof: We will show that (i)⇒(ii) when G satisfies the Δ_{2}-condition. Let f : ZZ → ZZ be defined so that

a^{f(n)−1} ≤ G^{−1}(a^{n}) < a^{f(n)}.

Since G^{−1} is dilatory, by Lemma 5.4.1, we know that there is a number L such that for every n ∈ ZZ we have that |f^{−1}({n})| ≤ L.

Let

A_{m} = { n : G^{−1}(a^{n+m}) > a^{m+1+N}G^{−1}(a^{n})}.

Then it can easily be shown that

f(A_{m}) ⊆ { n : G(a^{n+m+N}) < a^{m}G(a^{n})},

and hence |A_{m}| ≤ Lb^{m+N} ≤ b_{0}^{m}, where b_{0} = Lb^{N+1}.

To show (ii)⇒(i) is similar. Let g : ZZ → ZZ be defined so that

a^{g(n)} ≤ G(a^{n}) < a^{g(n)+1}.

Since G is almost convex, it follows that G is dilatory. Now the proof proceeds as in (i)⇒(ii).

Next, we deal with the composition of the ‘almost’ properties. One of the main problems here is that given two φ-functions, each with an ‘almost’ property, is that the a from the definition of the ‘almost’ property for each φ-function could be different. Fortunately, we have already developed the tools to deal with this. First, for ‘almost convexity,’ the implication (iv)⇒(iii) in Lemma 5.3.1 tells us that the a may be any arbitrarily large number. If the φ-function is dilatory, then Lemma 5.4.4 also allows the a to be any arbitrarily large number for the ‘almost concavity’ property. Finally, for other ‘almost’ properties. Lemma 5.3.2 allows us to choose the a to be larger than any given number.

Thus we have the following result.

Lemma 5.4.5. Let G be a φ-function. If G is almost convex and almost concave, then G is almost linear.

Lemma 5.4.6. Let G_{1} and G_{2} be φ-functions.

i) If G_{1} and G_{2} are almost convex, then G_{1} ∘ G_{2} is almost convex. ii) If G_{1} and G_{2} are almost concave, and if G_{2} is dilatory, then G_{1}∘G_{2} is almost concave.

Proof: First we will prove part (ii). By the explanation given above, we may suppose that for one a > 1, there are numbers b_{1} > 1, N_{1} ∈ IIN, b_{2} > 1 and N_{2} ∈ IIN such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that

G_{1}(a^{n+m}) > a^{m+N_{1}}G_{1}(a^{n})

is less than b_{1}^{m}, and the cardinality of the set of n ∈ ZZ such that

G_{2}(a^{n+m}) > a^{m+N_{2}}G_{2}(a^{n})

is less than b_{2}^{m}.

Define a function f : ZZ → ZZ so that for all n ∈ ZZ we have

a^{f(n)−1} ≤ G_{2}(a^{n}) < a^{f(n)}.

Then by Lemma 5.4.1, there is a number L ∈ IIN such that |f^{−1}({n})| ≤ L for all n ∈ ZZ. Then we see that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that

G_{1} ∘ G_{2}(a^{n+m}) > a^{m+N_{1}+N_{2}+1}G_{1} ∘ G_{2}(a^{n})

is less than b_{1}^{m+N_{1}+N_{2}+1} + Lb_{2}^{m+N_{2}+1}. For, if the above holds, and

G_{2}(a^{n+m}) ≤ a^{m+N_{2}}G_{2}(a^{n}),

then

G_{1}(a^{f(n)−1+m+N_{2}+1}) > a^{m+N_{2}+1+N_{1}}G_{1}(a^{f(n)−1}).

The result follows.

To show part (i), we note that as G_{2} is almost convex, that we already know that G_{2} is dilatory. Now the argument follows as in part (i).

Now, we prove two lemmas that are ‘almost’ analogues of Lemma 5.4.2.

Lemma 5.4.7. Let G_{1} and G_{2} be φ-functions such that G_{2} is dilatory and satisfies the Δ_{2}-condition. Then we have the following.

i) G_{1} ∘ G_{2}^{−1} is almost convex if and only if G_{1} is almost more convex than G_{2}. ii) G_{1} ∘ G_{2}^{−1} is almost concave if and only if G_{1} is almost less concave than G_{2}.

Proof: We will show that if G_{1} ∘ G_{2}^{−1} is almost convex, then G_{1} is almost more convex than G_{2}. All the other assertions follow similarly.

So, there are numbers a > 1, b > 2 and N ∈ IIN such that for all m ∈ IIN, the cardinality of

A_{m} = { n : G_{1} ∘ G_{2}^{−1}(a^{n+m}) < a^{m−N}G_{1} ∘ G_{2}^{−1}(a^{n})}

is less than b^{m}. Let us define a map f : ZZ → ZZ so that for all n ∈ ZZ we have that

a^{f(n)−1} ≤ G_{2}(a^{n}) < a^{f(n)}.

Then, by Lemma 5.4.1, there are numbers L, M ∈ IIN such that f(n + L) > f(n) for all n ∈ ZZ, and such that f(m + n) − f(n) ≤ Mm for all m ∈ IIN and n ∈ ZZ.

Now, for each m ∈ IIN, let us consider the cardinality of the set

B_{m} = \{ n : \frac{G_{1}(a^{L(n+m)})}{G_{1}(a^{Ln})} < a^{−N−2} \frac{G_{2}(a^{L(n+m)})}{G_{2}(a^{Ln})}\} .

If n ∈ B_{m}, let n′ = f(Ln) and m′ = f(L(m + n)) − n′. Then

G_{1} ∘ G_{2}^{−1}(a^{n′+m′−1}) < a^{m′−1−N}G_{1} ∘ G_{2}^{−1}(a^{n′}).

Clearly, this is impossible if m′ ≤ 1, and otherwise, this implies that n′ ∈ A_{m′−1}. Since m′ ≤ Mm, we see that

|B_{m}| < ∑_{m=1}^{Mm} b^{m′} ≤ b_{0}^{m},

where b_{0} = b^{M+1}/(b − 1).

Lemma 5.4.8. Let G_{1} and G_{2} be φ-functions such that one of G_{1} or G_{2} is dilatory and one of G_{1} or G_{2} satisfies the Δ_{2}-condition. Then G_{1} ∘ G_{2}^{−1} is almost linear if and only if G_{1} is almost equivalent to G_{2}.

Proof: We note that in either case that if one of G_{1} or G_{2} is dilatory, then both are, and if one of G_{1} or G_{2} satisfies the Δ_{2}-condition, then both do. Now the proof proceeds as in Lemma 5.4.7.

5.5. Condition (L) and Condition (J)

In this subsection, we describe how the notions of satisfying condition (L) or condition (J) relate to the ‘almost’ properties.

Lemma 5.5.1. Let G be a φ-function. Then the following are equivalent.

i) G is almost constant. ii) ∥1/G̃∥_{G^{−1}} < ∞.

iii) G̃ satisfies condition (L).

Proof: This proof is very similar to the proof of Lemma 5.3.1, and so we will omit many details. First we show that (i)⇒(ii). Following the same argument as the proof of (iv)⇒(i) in Lemma 5.3.1, we construct numbers a > b > 1, c < ∞ and N ∈ IIN and a sequence of sets B_{m} such that |B_{m}| ≤ cb^{m}, and such that if n ∈ B_{m} for m > 1 then

G^{−1}(a^{−N}G(a^{n})) ≤ a^{n−m+1}.

Hence,

∫_{0}^{∞} G^{−1}(a^{−N}/G̃(x)) dx

= ∑_{n=−∞}^{∞} ∫_{a^{−n}}^{a^{1−n}} G^{−1}(a^{−N}/G̃(x)) dx

≤ ∑_{m=1}^{∞} ∑_{n∈B_{m}} a^{1−n}G^{−1}(a^{−N}G(a^{n}))

≤ ∑_{n∈B_{1}} a^{1−n}G^{−1}(a^{−N}G(a^{n})) + ∑_{m=2}^{∞} cb^{m}a^{1−n}a^{n−m+1},

which is a finite number. By Lemma 5.2.1, G^{−1} is dilatory, and so the result follows by Lemma 5.1.1.

That (ii)⇒(iii) is straightforward. To show that (iii)⇒(i), choose a > 2, and note that for some N, M ∈ IIN we have that

∫_{0}^{∞} G^{−1}(a^{−N}/G̃(x)) dx ≤ a^{M}.

Then following a similar line of reasoning to that of the proof of (ii)⇒(iii) in Lemma 5.3.1, it is possible to show that there cannot be numbers m ∈ IIN and n_{1} < n_{2} < … < n_{a^{m+2+M}} such that

G(a^{n_{i}}) > a^{N}G(a^{n_{i}−m}).

Lemma 5.5.2. Let H is an N-function. Then the following are equivalent.

i) H is almost concave. ii) H satisfies condition (J). iii) H̃^{∗}^{−1} satisfies condition (L).

Proof: By Lemma 5.4.3, H is almost concave if and only if H^{−1} is almost convex. Clearly, H^{−1} is almost convex if and only if H^{∗}^{−1} is almost constant. Now the result follows by Lemma 5.5.1.

5.6. Condition (J) and the ‘Almost’ Properties

Now, we are ready to establish the relationship between being almost convex or almost concave, and being more or less convex than some N-function satisfying condition (J).

Lemma 5.6.1. Let G be a φ-function. Then we have the following.

i) G is almost convex if and only if there is an N-function H satisfying condition (J) such that G ≻ H^{−1}.

ii) If G is almost concave, then there is an N-function H satisfying condition (J) such

that G ≺ H. iii) If there is an N-function H satisfying condition (J) such that G ≺ H, then G^{−1} is

almost convex.

Proof: We first note that if there is an N-function H satisfying condition (J) such that either G ≻ H^{−1} or G^{−1} ≺ H, then by Lemma 5.4.2, we have that G is almost convex. We will prove the other implication of part (i).

If G is almost convex, then we know that there are numbers a > 1, b > 2 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that

G(a^{n+m}) < a^{m−N}G(a^{n})

is less than b^{m}. Now we define a function L : { a^{n} : n ∈ ZZ} → (0, ∞) by

L(a^{n}) = \{ \begin{matrix}inf_{0=n_{0}<n_{1}<...<n_{K}=n} ∏_{k=1}^{K} a^{N} min \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} & if n ≥ 0 \\ sup_{0=n_{0}>n_{1}>...>n_{K}=n} ∏_{k=1}^{K} a^{−N} max \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} & if n < 0. \end{matrix}

We may extend the domain of L to [0, ∞) ‘log-linearly,’ that is, by setting L(0) = 0, and

L(a^{n}t) = L(a^{n}) exp (\frac{log t}{log a} log (\frac{L(a^{n+1})}{L(a^{n})})) ,

for n ∈ ZZ and 1 ≤ t < a. We notice that L(a^{n+1}) > L(a^{n}) for all n ∈ ZZ, and hence L is a φ-function.

Now, we note that if m ∈ IIN and n ∈ ZZ, then

\frac{L(a^{n+m})}{L(a^{n})} ≤ inf_{n=n_{0}<n_{1}<...<n_{K}=n+m} ∏_{k=1}^{K} a^{N} min \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} .

Thus, we have that L(a^{n+m}) ≤ a^{m+N}L(a^{n}), and so, by Lemma 5.4.2, L^{−1} is equivalent to a convex function. We also have that

\frac{L(a^{n+m})}{L(a^{n})} ≤ a^{N} \frac{G(a^{n+m})}{G(a^{n})},

and therefore, by Lemma 5.4.2 and since L satisfies the Δ_{2}-condition, G ≻ L. We also notice that, since

min \{a^{n_{2}−n_{0}}, \frac{G(a^{n_{2}})}{G(a^{n_{0}})}\} ≥ min \{a^{n_{2}−n_{1}}, \frac{G(a^{n_{2}})}{G(a^{n_{1}})}\} min \{a^{n_{1}−n_{0}}, \frac{G(a^{n_{1}})}{G(a^{n_{0}})}\}

for n_{0} ≤ n_{1} ≤ n_{2}, we have that

\frac{L(a^{n+m})}{L(a^{n})} ≥ a^{−N} inf_{n=n_{0}<n_{1}<...<n_{K}=n+m} ∏_{k=1}^{K} a^{N} min \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} .

Therefore, if L(a^{n+m}) < a^{m−N}L(a^{n}), then for some n ≤ n′ < n′ + m′ ≤ n + m, we have that G(a^{n′+m′}) < a^{m′−N}G(a^{n′}). Therefore, the cardinality of the set of n ∈ ZZ satisfying L(a^{n+m}) < a^{m−N}L(a^{n}) is less than m(b + b^{2} + … + b^{m}), which is less than b_{0}^{m} for b_{0} = 2b^{2}/(b − 1).

Therefore, L is almost convex. Now, we define the φ-function H(t) = L^{−1}(t lm t). It is clear that L ≻ H^{−1}, and hence G ≻ H^{−1}. Since t lm t is easily seen to be almost concave, it follows by Lemmas 5.4.3 and 5.4.6 that H is almost convex. Clearly H is an N-function, and so by Lemma 5.5.2, we have that H satisfies condition (J).

The proof of part (ii) is similar. We know that there are numbers a > 1, b > 2 and N ∈ IIN such that for all m ∈ IIN, the cardinality of the set of n ∈ ZZ such that

G(a^{n+m}) > a^{m+N}G(a^{n})

is less than b^{m}. Now we define a function L : { a^{n} : n ∈ ZZ} → (0, ∞) by

L(a^{n}) = \{ \begin{matrix}sup_{0=n_{0}<n_{1}<...<n_{K}=n} ∏_{k=1}^{K} a^{−N} max \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} & if n ≥ 0 \\ inf_{0=n_{0}>n_{1}>...>n_{K}=n} ∏_{k=1}^{K} a^{N} min \{a^{n_{k}−n_{k−1}}, \frac{G(a^{n_{k}})}{G(a^{n_{k−1}})}\} & if n < 0, \end{matrix}

and extend L ‘log-linearly.’ By the same methods as in the proof of part (i), we see that L is convex, that G ≺ L, and that L is almost concave. Finally, we set H(t) = L(t) lm L(t) to obtain the result.

Lemma 5.6.2. Let G be a φ-function. Then the following are equivalent.

i) G is almost linear. ii) There are N-functions H and K satisfying condition (J) such that G = H ∘ K^{−1}. iii) There are N-functions H and K satisfying condition (J) such that G = H^{−1} ∘ K. iv) There are N-functions H and K satisfying condition (J) and a number c < ∞ such

that c^{−1}G(t)/t ≤ H(t)/K(t) ≤ c G(t)/t for all t > 0.

v) There are strictly monotone weight functions w_{0} and w_{1} satisfying condition (L) and a number c < ∞ such that c^{−1}G(t)/t ≤ w_{0}(t)w_{1}(t) ≤ c G(t)/t for all t ≥ 0.

Before proving this result, we will require a couple of technical lemmas.

Lemma 5.6.3. If G_{1} and G_{2} are equivalent φ-functions, and if one of G_{1} or G_{2} satisfies the Δ_{2}-condition, then there is a number c < ∞ such that c^{−1}G_{1}(t) ≤ G_{2}(t) ≤ c G_{1}(t) for all t ≥ 0.

Lemma 5.6.4. Suppose that F : [0, ∞) → [0, ∞) is a function such that for some numbers c_{1} > 1 and c_{2} > 1 we have that F (c_{1}t) ≥ c_{2}F (t) for all t ≥ 0. Then there is a number c < ∞ and a dilatory φ-function G such that G(c^{−1}t) ≤ F (t) ≤ G(ct) for all t ≥ 0.

Proof: We have that F (c_{1}^{n}t) ≥ c_{2}^{n}F (t) for all n ∈ IIN and t ≥ 0. Then it is clear that

G(t) = sup_{s≥1} s^{− log c_{2}/ log c_{1}}F (st)

satisfies the conclusion of the lemma.

Proof of Lemma 5.6.2: The implications (ii)⇒(i) and (iii)⇒(i) follow from Lemmas 5.4.5 and 5.4.6. The implications (iv)⇒(i) and (v)⇒(i) are obvious.

To show that (i)⇒(ii), we note that, since G is almost convex, by Lemma 5.6.1(i), there is an N-function K_{0} satisfying condition (J) such that G ≻ K_{0}^{−1}. If we let K(t) = K_{0}(t lm t), then we see that H = G ∘ K is an N-function. Since G is almost concave, it follows by Lemma 5.4.6 that H satisfies condition (J). The implication (i)⇒(iii) is similar, using Lemma 5.6.1(ii).

To show (i)⇒(iv), we note that since G is almost concave, by Lemma 5.6.1(ii), there is an N-function H_{0} satisfying condition (J) such that G ≺ H_{0}. Then, from Lemmas 5.4.2 and 5.6.4, it follows that tH_{0}(t)/G(t) is equivalent to a convex function K_{0}. Since G is almost convex, we have that K_{0} is almost concave. Now we let H(t) = H_{0}(t) lm t and K(t) = K_{0}(t) lm t, and the result follows by Lemma 5.6.3.

To show that (iv)⇒(v), by Lemma 5.4.2, we may assume that H and K are convex. Thus, if we let w_{0}(t) = (lm t)H(t)/t and w_{1}(t) = t/(lm t)K(t), then w̃_{0} and ˜1/w_{1} are both almost constant φ-functions. Then it follows from Lemma 5.5.1 that w_{0} and w_{1} satisfy condition (L).

5.7. The Proof of the Results in Section 4

Now we are ready to piece together all the lemmas we have just proved. Proof of Theorem 4.2: First we will show that (ii)⇒(i). By Lemma 5.2.1, we know that G_{2} ∘ G_{1}^{−1} is dilatory, and hence if G_{1} is dilatory, then so is G_{2}. Therefore, we may assume that G_{2} is dilatory.

By Lemma 5.3.1, there is a number c_{1} < ∞ such that ∥f∥_{1,G_{1}∘G_{2}^{−1}} ≤ c_{1} ∥f∥_{1} for all measurable f. Since G_{2} is dilatory, the result follows by follows by Lemmas 5.1.3 and 5.1.2.

Now we show that (i)⇒(ii). By Lemma 5.2.5, we know that G_{2} satisfies the Δ_{2}- condition. Therefore, by Lemmas 5.1.2 and 5.1.3, it follows that there is a number c_{1} < ∞ such that ∥f∥_{1,G_{1}∘G_{2}^{−1}} ≤ c_{1} ∥f∥_{1} for all measurable f. Now the result follows by Lemma 5.3.1.

The implication (ii)⇔(iv) follows from Lemma 5.4.7(i), and (ii)⇔(iii) follows from Lemma 5.6.1.

Proof of Theorem 4.3: First we show that (ii)⇒(i). If G_{1} ∘ G_{2}^{−1} is almost linear, then by Lemma 5.2.1, it follows that G_{1} ∘ G_{2}^{−1} satisfies the Δ_{2}-condition. Therefore, by Lemma 5.4.3, we see that G_{2} ∘ G_{1}^{−1} is almost convex. Therefore, by Theorem 4.2, we have that L_{F_{1},G_{1}} and L_{F_{1},G_{2}} are equivalent. Clearly, we have that both G_{1} and G_{2} are dilatory, and by Proposition 4.1, L_{F_{1},G_{2}} and L_{F_{2},G_{2}} are equivalent. The result follows.

Next we show that (i)⇒(ii). First notice that, since F̃_{1}^{−1}(t) = ∥χ_{[0,t]}∥_{F_{1},G_{1}} and F̃_{2}^{−1}(t) = ∥χ_{[0,t]}∥_{F_{2},G_{2}}, we have that F_{1} ≍ F_{2}. Now let us suppose without loss of generality that G_{1} is dilatory. Then by Proposition 4.1, we have that L_{F_{1},G_{1}} and L_{F_{2},G_{1}} are equivalent, and hence L_{F_{2},G_{1}} and L_{F_{2},G_{2}} are equivalent. Now, by Lemmas 5.2.3 and 5.2.5, both G_{1} and G_{2} are dilatory, and both G_{1} and G_{2} satisfy the Δ_{2}-condition. Therefore, G_{2} ∘ G_{1}^{−1} and G_{1} ∘ G_{2}^{−1} satisfy the Δ_{2}-condition. By Theorem 4.2, both G_{1} ∘ G_{2}^{−1} and G_{2} ∘ G_{1}^{−1} are almost convex. Now the result follows by Lemmas 5.4.2 and 5.4.5.

The implications (ii)⇔(v) follow from Lemma 5.4.8, the implication (v)⇒(viii) follows from Lemmas 5.6.4 and 5.6.3, and the implication (viii)⇒(v) is obvious. Finally, (viii)⇔(iii)⇔(iv)⇔(vi)⇔(vii) all follow from Lemma 5.6.2.

Proof of Theorem 4.4: The implication (ii)⇒(i) is obvious, so we show that (i)⇒(ii). As in the proof of Theorem 4.2, we may suppose that G_{2} satisfies the Δ_{2}-condition. By Lemma 5.1.2, we may assume without loss of generality that F (t) = t. Now the result follows by Lemmas 5.1.3 and 5.3.1, in the same manner as in the proof of Theorem 4.2.

Proof of Theorem 4.5: The implications (iii)⇒(i) and (iii)⇒(ii) are obvious. We show that (i)⇒(ii). First notice that χ_{[0,t)} ∈ T_{F_{1}}, and so as F̃_{1}^{−1}(t) = ∥χ_{[0,t)}∥_{F_{1},G_{1}} and F̃_{2}^{−1}(t) = ∥χ_{[0,t)}∥_{F_{2},G_{2}}, we have that F_{1} and F_{2} are equivalent. Then it is clear that there is a number c < ∞ such that f^{∗} ∈ T_{F_{1}} if and only if there is a function g^{∗} ∈ T_{F_{2}} such that c^{−1}f^{∗} ≤ g^{∗} ≤ c f^{∗}. Similarly, (ii)⇒(i).

Now we show that (i) and (ii)⇒(iii). Let us suppose, without loss of generality, that G_{1} is dilatory. Then by Proposition 4.1, we have that L_{F_{1},G_{1}} and L_{F_{2},G_{1}} are equivalent, and hence L_{F_{2},G_{1}} and L_{F_{2},G_{2}} are equivalent. Now, by Lemma 5.2.3, it follows that if one of G_{1} or G_{2} is dilatory, then both are. Then the result follows by Theorem 4.4.

Proof of Theorem 4.6: We show that (i)⇔(iii). By Proposition 4.1, we know that if F_{1} and F_{2} are equivalent, then L_{F_{1},∞} and L_{F_{2},∞} are equivalent. Also, if L_{F_{1},G} and L_{F_{2},∞} are equivalent, then since F̃_{1}^{−1}(t) = ∥χ_{[0,t]}∥_{F_{1},G} and F̃_{2}^{−1}(t) = ∥χ_{[0,t]}∥_{F_{2},∞}, we have that F_{1} and F_{2} are equivalent. Thus, without loss of generality, we may assume that F_{1} = F_{2} = F.

Now we note that we always have that ∥f∥_{F,∞} ≤ ∥f∥_{F,G}. This follows because for all x ≥ 0, we have that f^{∗} ≥ f^{∗}(x)χ_{[0,x]}, and hence

∥f∥_{F,G} ≥ ∥f^{∗}(x)χ_{[0,x]}∥_{F,G} ≥ f^{∗}(x)F̃^{−1}(x).

Now we show that L_{F,G} is equivalent to L_{F,∞} if and only if ∥1/F̃^{−1}∥_{F,G} < ∞. That

the first statement implies the second is obvious, because ∥1/F̃^{−1}∥_{F,∞} = 1. To show the

second statement from the first, note that if ∥f∥_{F,∞} ≤ 1, then f^{∗}(x) ≤ 1/F̃^{−1}.

But ∥1/F̃^{−1}∥_{F,G} = ∥1/G̃^{−1}∥_{G}, and the result follows. The other implications follow by Lemma 5.5.2.

6. Is Every R.I. Space Equivalent to an Orlicz–Lorentz Space? We can answer the question in the negative easily, as follows. It is well known that L_{1,∞} is not separable. Then it is not hard to see that L_{1,∞}^{0}, the closure of the simple functions in L_{1,∞}, is not an Orlicz–Lorentz space.

However, the reader may consider this cheating. So to avoid all this ‘infinite dimensional nonsense,’ we might ask the following question. Is there a rearrangement invariant space X such that for all Orlicz–Lorentz spaces L_{F,G}, the norms ∥ · ∥_{X} and ∥ · ∥_{F,G} are inequivalent on X ∩L_{F,G}? We answer this question in the positive by the following example.

Theorem 6.1. There is a rearrangement invariant Banach space X, where the measure space is [0, ∞) with Lebesgue measure, such that for every Orlicz–Lorentz space L_{F,G}, the norms ∥ · ∥_{X} and ∥ · ∥_{F,G} are inequivalent on the vector space of simple functions.

Proof: We define the following norm for measurable functions f:

∥f∥_{X} = sup{ ∥f^{∗}g∥_{1} / ∥g∥_{2} : g ∈ T_{t^{2}}}.

We let X be the vector space of all measurable functions f such that ∥f∥_{X} < ∞, modulo functions that are zero almost everywhere. Then it is an easy matter to see that X is a rearrangement invariant space such that ∥g∥_{X} = ∥g∥_{2} for all g ∈ T_{t^{2}}. Thus, if for some φ-functions F and G we have that

c_{1}^{−1} ∥f∥_{X} ≤ ∥f∥_{F,G} ≤ c_{1} ∥f∥_{X}

for all simple functions f, then by Theorem 4.5, we see that there is a constant c_{2} < ∞ such that

c_{2}^{−1} ∥f∥_{X} ≤ ∥f∥_{2} ≤ c_{2} ∥f∥_{X}

for all simple functions f. We will show that this cannot happen.

Define T′ to be the set of functions h : [0, ∞) → [0, ∞) such that for some integers k_{1} < … < k_{n}, and setting k_{0} = −∞, we have that

h(x) = \{ \begin{matrix}2^{−k_{i}} & if 4^{k_{i−1}} ≤ x < 4^{k_{i}} and 1 ≤ i ≤ n \\ 0 & otherwise. \end{matrix}

Then it is an easy matter to see that if g ∈ T_{t^{2}}, then there is a function h ∈ T′ such that h(4x)/2 ≤ g(x) ≤ 2h(x/4). Therefore, we see that

\frac{1}{4} ∥f∥_{X} ≤ sup\{ ∥f^{∗}h∥_{1} / ∥h∥_{2} : h ∈ T′ \} ≤ 4 ∥f∥_{X} .

Now, for each N ∈ IIN, let f_{N} be the simple function

f_{N}(x) = \{ \begin{matrix}1 & if 0 ≤ t < 4 \\ k^{−1/2}2^{−k} & if 4^{k} ≤ t < 4^{k+1} and 1 ≤ k ≤ N \\ 0 & otherwise. \end{matrix}

Then it is easy to see that ∥f_{N}∥_{2} → ∞ as N → ∞. However, a simple, but laborious, calculation shows that there is a number c < ∞ such that ∥f_{N}^{∗}h∥_{1} / ∥h∥_{2} ≤ c for all h ∈ T′, and hence ∥f_{N}∥_{X} ≤ 4c.

7. The Definition of Torchinsky and Raynaud

The definition of the Orlicz–Lorentz spaces presented here is not the only possible definition. In fact, given any weight function w and any φ-functions H and G, one can form the functional

∥f∥_{w,H,G} = ∥w · (f^{∗} ∘ H)∥_{G} .

We have investigated the case when w(x) = 1. However, Torchinsky [T] gave the following definition for the Orlicz–Lorentz functional. If F and G are φ-functions, then we define

∥f∥_{F,G}^{T} = ∥F̃^{−1}(e^{x})f^{∗}(e^{x})∥_{G}

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

and call the corresponding space L_{F,G}^{T} (my notation).

These spaces were investigated by Raynaud [R]. He showed that if F is dilatory and satisfies the Δ_{2}-condition, and if G is dilatory, then ∥χ_{A}∥_{F,G}^{T} ≈ F̃^{−1}(μ(A)) for all measurable A. Thus, it follows that L_{F,G}^{T} and L_{F,G} are equivalent if G(t) = t^{p}.

The comparison results for these spaces are much more straightforward. Raynaud [R] showed that if F_{1} and F_{2} are dilatory and satisfy the Δ_{2}-condition, and if G_{1} and G_{2} are dilatory, then L_{F_{1},G_{1}}^{T} and L_{F_{2},G_{2}}^{T} are equivalent if F_{1} and F_{2} are equivalent, and the sequence spaces l_{G_{1}} and l_{G_{2}} are equivalent. The converse result is also easy to show.

We also comment that the Boyd indices of these spaces are much easier to compute. This will be dealt with more fully in [Mo2].

Also, unlike the Orlicz–Lorentz spaces we have used here, we do not always have that L_{F,F}^{T} is equivalent to the Orlicz space L_{F}. For example, if F (t) = t lm t, then L_{F} is equivalent to L_{F,1}^{T} by Theorem 4.3, and since l_{F} and l_{1} are not equivalent, this is not equivalent to L_{F,F}^{T}.

We finally add that we may define the spaces L_{F,X}, where X is a rearrangement invariant quasi-Banach space on IR satisfying certain mild restrictions. Corresponding to the definition used in this paper, we may define

∥f∥_{F,X} = ∥f^{∗} ∘ F̃ ∘ ϕ_{X} ∥_{X} ,

where ϕ_{X}(t) = ∥χ_{[0,t]}∥_{X} is the fundamental function of X. Corresponding to the definition used by Torchinsky and Raynaud, we may define

∥f∥_{F,X}^{T} = ∥F̃^{−1}(e^{x})f^{∗}(e^{x})∥_{X} .

Acknowledgements

This paper is an extension of work that I presented in my Ph.D. thesis [Mo1]. 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 much gratitude to A. Kamińska, W. Koslowski and N.J. Kalton for their keen interest and useful conversations. I also extend my thanks to the referee for his diligent reading of the final manuscript (although all errors are, of course, my responsibility). Finally, I would like to mention T.N. Prokraz for his great influence on this work.

References

B–R C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissert. Math. 175 (1980),

1–72.

B–S C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press 1988.
H  R.A. Hunt, On L(p, q) spaces, L’Enseignement Math. (2) 12 (1966), 249–275.
Ka1 A. Kamińska, Some remarks on Orlicz–Lorentz spaces, Math. Nachr., to appear.
Ka2 A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math., to appear.
Ka3 A. Kamińska, Uniform convexity of generalized Lorentz spaces, Arch. Math., to appear.

K–R M.A. Krasnosel’skiĭ and Ya.B. Rutickiĭ, Convex Functions and Orlicz Spaces, P. Noodhoof

Ltd., 1961.

Lo1 G.G. Lorentz, Some new function spaces, Ann. Math. 51 (1950), 37–55.
Lo2 G.G. Lorentz, On the theory of spaces Λ, Pac. J. Math. 1 (1951), 411–429.
Lo3 G.G. Lorentz, Relations between function spaces, Proc. A.M.S. 12 (1961), 127–132.
Lu  W.A.J. Luxemburg, Banach Function Spaces, Thesis, Delft Technical Univ., 1955.
Ma  L. Maligranda, Indices and interpolation, Dissert. Math. 234 (1984), 1–49.

My M. Mastyło, Interpolation of linear operators in Calderon–Lozanovskii spaces, Comment.

Math. 26,2 (1986), 247–256.

Mo1 S.J. Montgomery-Smith, The Cotype of Operators from C(K), Ph.D. thesis, Cambridge,

August 1988.

Mo2 S.J. Montgomery-Smith, Boyd Indices of Orlicz–Lorentz spaces, in preparation.

O W. Orlicz, Über eine gewisse Klasse von Räumen vom Typus B, Bull. Intern. Acad. Pol.

8 (1932), 207–220.

R Y. Raynaud, On Lorentz–Sharpley spaces, Proceedings of the Workshop “Interpolation

Spaces and Related Topics”, Haifa, June 1990.

S  R. Sharpley, Spaces Λ_{α}(X) and Interpolation, J. Funct. Anal. 11 (1972), 479–513.

T A. Torchinsky, Interpolation of operators and Orlicz classes, Studia Math. 59 (1976), 177–207.