GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 2, 1997, 117-128

IMBEDDINGS BETWEEN WEIGHTED ORLICZ–LORENTZ SPACES

M. KRBEC AND J. LANG

Abstract. We establish necessary and sufficient conditions for imbeddings of weighted Orlicz–Lorentz spaces.

1. Introduction

The purpose of this paper is to present transparent and verifiable necessary and sufficient conditions for imbeddings in a fairly general class of weighted spaces which include important representatives of r. i. spaces as Lorentz and Orlicz spaces.

Necessary and sufficient conditions for imbeddings of weighted spaces L^{p} were found by Avantaggiati [1] and Kabaila [2]; the latter author also considered measures not necessarily absolutely continuous with respect to the Lebesgue measure. The case of Orlicz spaces with certain mild conditions imposed on the growth of Young functions involved was the subject of Krbec and Pick’s paper [3]. Here we shall consider the natural amalgam of Orlicz and Lorentz spaces, permitting one to arrive at integral conditions involving the weights in question. The generality of the concept enables one to give proofs actually simpler than those for Orlicz and/or Lorentz spaces. Observe that abstract conditions in terms of dual spaces can be given in Lorentz spaces (see Pick [4]).

Let us introduce the notation. Throughout the paper, Ω will be a measurable subset of the Euclidean space ℝ^{N}, ϱ and σ will stand for weights in Ω which are measurable, locally integrable, and a.e. positive function in Ω. A Young function F is an even continuous and non-negative function in ℝ^{1}, increasing on (0, ∞), such that lim_{t→0_{+}} F (t) = 0, lim_{t→∞} F (t) = ∞, F (t) = 0

iff t = 0. A Young function F is said to satisfy the global Δ_{2}-condition if there is c > 0 such that F (2t) ≤ cF (t) for all t ∈ ℝ^{1}.

117

1991 Mathematics Subject Classification. 46E35.

Key words and phrases. Orlicz–Lorentz spaces, Orlicz spaces, Lorentz spaces, weights, imbedding theorems.

1072-947X/97/0300-0117$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

The Young functions F_{0} and F_{1} are said to be equivalent (we write F_{0} ∼ F_{1}) if there is a constant c > 0, such that

F_{1}(c^{−1}t) ≤ F_{0}(t) ≤ F_{1}(ct), t > 0.

If F is a Young function, then

comp F (t) = sup{|ts| − F (s); s ∈ ℝ^{1}}

is the complementary function with respect to F. If F is convex, then comp F (t) is equivalent to any Young function M such that

c_{M}^{−1}t ≤ M^{−1}(t)F^{−1}(t) ≤ c_{M}t, t > 0,

where c_{M} is a constant independent of t. In this case, the latter condition is sometimes used as an (equivalent) definition of comp F.

Let F be a Young function and ϱ a weight in Ω. The weighted Orlicz space L_{F,ϱ} = L_{F,ϱ}(Ω) is the linear hull of the weighted Orlicz class

L̃_{F,ϱ} = L̃_{F,ϱ}(Ω) = {f; ∫_{Ω} F (f(x))ϱ(x) dx < ∞}.

The space L_{F,ϱ} is equipped with the Luxemburg functional

∥f∥_{F,ϱ} = inf {λ > 0; ∫_{Ω} F (f(x)/λ) ϱ(x) dx ≤ 1}.

Let f be a measurable function in Ω and m_{ϱ}(f, t) be the weighted distribution function of f, i.e.,

m_{ϱ}(f, λ) = ∫_{{x; |f(x)|>λ}} ϱ(x) dx = ϱ({x; |f(x)| > λ}),

and f_{ϱ}^{∗} be the corresponding weighted nonincreasing rearrangement of f,

f_{ϱ}^{∗}(t) = inf{λ; m_{ϱ}(f, λ) ≤ t}.

Further, let 1 ≤ q, r < ∞. Then

L_{q,r,ϱ} = L_{q,r,ϱ}(Ω) = \{f; ∥f∥_{q,r,ϱ} = (∫_{0}^{∞}[t^{1/q}f_{ϱ}^{∗}(t)]^{r} \frac{dt}{t})^{1/r} < ∞\}

is the weighted Lorentz space. As usual, for r = ∞, we put

L_{q,∞,ϱ} = L_{q,∞,ϱ}(Ω) = {f; ∥f∥_{q,∞,ϱ} = sup_{t>0} t^{1/q}f_{ϱ}^{∗}(t) < ∞}

and let us call the latter space the weak weighted Lorentz (weighted Marcinkiewicz) space.

IMBEDDINGS BETWEEN WEIGHTED ORLICZ–LORENTZ SPACES 119

Let us recall at least some of the basic references concerning the theory of Lorentz and Orlicz spaces such as the well-known monographs of Butzer and Berens [5], Nakano [6], Krasnosel’skii and Rutitskii [7], Musielak [8], and Ren and Rao [9].

Next, we define Orlicz–Lorentz spaces. Because of the nonhomogeneity of Young functions one can do this in several different ways, the L^{p} and L^{p,q} spaces being always included as a special case. We shall follow the definition used in the recent papers, e.g., in Montgomery-Smith [10]: Let F and G be Young functions and ϱ a weight in Ω. For a function h even on ℝ^{1} and positive on (0, ∞) put

h̃(t) = \begin{cases}1/h(1/t), & t > 0, \\ 1/h(−1/t), & t < 0, \\ h(0), & t = 0. \end{cases}

We define the weighted Orlicz–Lorentz space L_{F,G,ϱ} as the set of all measurable f’s on Ω for which the Orlicz–Lorentz functional

∥f∥_{F,G,ϱ} = ∥f_{ϱ}^{∗} ∘ F̃ ∘ G̃^{−1}∥_{G} =
= inf \{λ > 0; ∫_{0}^{∞}G (\frac{f_{ϱ}^{∗}(F̃(G̃^{−1}(t)))}{λ}) dt ≤ 1\}  (1.1)

is finite.

The weak weighted Orlicz (Orlicz–Marcinkiewicz) space L_{F,∞,ϱ} is the set of all measurable f’s on Ω such that their Orlicz–Marcinkiewicz functional

∥f∥_{F,∞,ϱ} = sup_{t>0} F̃^{−1}(t)f_{ϱ}^{∗}(t)  (1.2)

is finite.

We shall write L_{F_{1},G_{1},ϱ} ↪ L_{F_{0},G_{0},σ} if ∥f∥_{F_{0},G_{0},σ} ≤ const ∥f∥_{F_{1},G_{1},ϱ} for all f ∈ L_{F_{1},G_{1},ϱ}.

We shall say that G ≼ G_{1} (on Ω and with respect to ϱ) if L_{F,G_{1},ϱ} ↪ L_{F,G,ϱ} for every Young function F.

Remark 1.1. It is clear that for P and Q equal to power functions we get a weighted Lorentz space. Also, L_{F,F,ϱ} = L_{F,ϱ}.

Observe that for P (t) = t^{p}, Q(t) = t^{q}, and ϱ ≡ 1 the functional in (1.1) becomes

∥f∥_{P,Q,1} = (∫_{0}^{∞}[f^{∗}(t^{p/q})]^{q} dt)^{1/q}

which is equivalent to the usual quasinorm in L_{p,q} and it is actually the expression giving a hint how to reasonably define the L_{F,G,ϱ} spaces.

Remark 1.2. The quantities in (1.1) and (1.2) are not generally norms. Nevertheless, they are quasinorms in the many relevant cases we are interested in.

First, we shall show that if F^{−1} ∈ Δ_{2}, then ∥ . ∥_{F,∞,ϱ} is a quasinorm. Indeed, then F̃^{−1} = F̃^{−1} ∈ Δ_{2}, too, and

∥f + g∥_{F,∞,ϱ} = sup_{t>0} F̃^{−1}(t)(f + g)_{ϱ}^{∗}(t) ≤

≤ c sup_{t>0} F̃^{−1}(t/2) [f_{ϱ}^{∗}(t/2) + g_{ϱ}^{∗}(t/2)] ≤

≤ c (∥f∥_{F,∞,ϱ} + ∥g∥_{F,∞,ϱ}).

Conversely, if ∥ . ∥_{F,∞,ϱ} is a quasinorm, then F^{−1} ∈ Δ_{2} at least near infinity. This can be shown as follows: Let M, N ⊂ Ω be disjoint and such that ϱ(M) = ϱ(N). Then

∥χ_{M} + χ_{N}∥_{F,∞,ϱ} = ∥χ_{M∪N}∥_{F,∞,ϱ} ≤ c [∥χ_{M}∥_{F,∞,ϱ} + ∥χ_{N}∥_{F,∞,ϱ}] .

According to the formula for the Orlicz–Lorentz functional of ∥χ_{A}∥_{F,∞,ϱ} (Lemma 2.1), we have

\frac{1}{F^{−1} (1/2ϱ(M))} ≤ \frac{2c}{F^{−1} (1/ϱ(M))}.

Putting t = 1/2ϱ(M) we get F^{−1}(2t) ≤ 2cF^{−1}(t). If ϱ(Ω) = ∞, this gives directly the Δ_{2}-condition for F^{−1}. If ϱ(Ω) < ∞, then F^{−1} satisfies the Δ_{2}-condition for large t’s.

Now let us consider ∥ . ∥_{F,G,ϱ}. We shall not pursue the sufficient condition in detail as it is not the subject of this paper; nevertheless, let us point out one important case when ∥ . ∥_{F,G,ϱ} is a quasinorm: Suppose that F^{−1} ∈ Δ_{2}. If G ∈ Δ_{2}, then G is c-subadditive, i.e.,

G(t_{1} + t_{2}) ≤ c[G(t_{1}) + G(t_{2})],  t_{1}, t_{2} > 0,  (1.3)

for some c > 0 and there are c_{1} > 1 and c_{2} > 0 such that c_{1}G(t_{1}) ≤ G(c_{2}t_{2}) for all t ∈ ℝ^{1}. In particular, the Orlicz spaces generated by G and αG with α > 0 are the same with equivalent Luxemburg functionals.

Recalling the standard estimate

(f + g)_{ϱ}^{∗}(τ) ≤ f_{ϱ}^{∗}(τ/2) + g_{ϱ}^{∗}(τ/2),

we get

∥f + g∥_{F,G,ϱ} ≤ ∥f_{ϱ}^{∗}(\frac{F̃ ∘ G̃^{−1}(t)}{2}) + g_{ϱ}^{∗}(\frac{F̃ ∘ G̃^{−1}(t)}{2})∥_{G}
= ∥f_{ϱ}^{∗}( ˜2F ∘ G̃^{−1}(t)) + g_{ϱ}^{∗}( ˜2F ∘ G̃^{−1}(t))∥_{G}.  (1.4)

Assuming that F^{−1} ∈ Δ_{2} there is α ∈ (0, 1) such that

˜2F (τ) ≥ F̃(ατ), τ > 0.

Hence

f_{ϱ}^{∗}( ˜2F ∘ G̃^{−1}(t)) ≤ f_{ϱ}^{∗}(F̃(αG̃^{−1}(t))), t > 0.  (1.5)

Further we claim that there is β > 0 such that

G̃(ατ) ≥ βG̃(τ) = { \atop β^{−1}G}(τ).

Indeed, the last inequality is nothing but the Δ_{2}-condition for G in terms of G̃. Therefore, putting τ = G̃^{−1}(t), we get

G̃(αG̃^{−1}(t)) ≥ βt.

Substituting this into (1.5) we have

f_{ϱ}^{∗}( ˜2F ∘ G̃^{−1}(t)) ≤ f_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(βt)).

Returning to (1.4) we see that

∥f + g∥_{F,G,ϱ} ≤ ∥f_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(βt) + g_{ϱ}^{∗}(G̃ ∘ G̃^{−1}(βt)∥_{G},

and, by virtue of (1.3), we get

∥f + g∥_{F,G,ϱ} ≤ ∥f_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(βt))∥_{2cG} + ∥g_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(βt))∥_{2cG} =
= ∥f_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(t))∥_{2cβ^{−1}G} + ∥g_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(t))∥_{2cβ^{−1}G}.

As G ∈ Δ_{2} the functions G and 2cβ^{−1}G generate equivalent Luxemburg functionals.

In the sequel (see the proof of Lemma 2.2) we shall still need a simple condition for the equivalence of the Luxemburg functionals in the Orlicz spaces L_{G} and L_{αG} where α is an arbitrary positive constant. It is clear that the condition

max(α, α^{−1})G(t) ≤ G(βt), t > 0,

for some β ≥ 1 independent of t, is sufficient. Following the above considerations, we see that G ∈ Δ_{2} is enough for this.

Various sufficient conditions for ∥ . ∥_{F,G,ϱ} to be a quasinorm can also be found when more is imposed on the functions F and G.

2. Weighted Imbeddings

In this section we prove the imbedding theorems for weighted strong and weak Lorentz–Orlicz spaces under fairly general conditions on the growth of the Young functions involved. Let us point out a rather surprising fact, namely, that from the point of view of weighted imbeddings there is no essential difference between L_{P,G} and L_{P,∞} spaces (see conditions (ii), (iii), and (iv) of the concluding theorem in this section).

We start with the necessary condition.

Lemma 2.1. Let any of the following condition be satisfied for each measurable f in Ω:

∥f∥_{F_{0},G_{0},σ} ≤ K∥f∥_{F_{1},G_{1},ϱ},  (2.1)
∥f∥_{F_{0},∞,σ} ≤ K∥f∥_{F_{1},∞,ϱ},  (2.2)
∥f∥_{F_{0},∞,σ} ≤ K∥f∥_{F_{1},G_{1},ϱ}.  (2.3)

Then

F̃_{0}^{−1}(σ(A)) ≤ KF̃_{1}^{−1}(ϱ(A))  (2.4)

for every measurable A ⊂ Ω.

Proof. The necessary condition (2.4) follows directly after putting f = χ_{A} in (2.1)–(2.3) and calculating the corresponding norms.

As to ∥f∥_{F_{1},G_{1},ϱ}, we have

∥χ_{A}∥_{F_{1},G_{1},ϱ} = inf {μ > 0; ∫_{0}^{_{G_{1}}_{F_{1}}^{−1}(ϱ(A))}G_{1}(1/μ) dt ≤ 1} =

= inf {μ > 0; G̃_{1}(F̃_{1}^{−1}(ϱ(A)))G_{1}(1/μ) ≤ 1} =

= inf \{μ > 0; \frac{1}{G_{1}(1/(F̃_{1}^{−1}(ϱ(A))))} ≤ \frac{1}{G_{1}(1/μ)}\} =

= F̃_{1}^{−1}(ϱ(A)).

Further,

∥χ_{A}∥_{F_{1},∞,ϱ} = sup_{t>0} F̃_{1}^{−1}(t)(χ_{A})_{ϱ}^{∗}(t) =

= sup_{t>0} F̃_{1}^{−1}(t) inf {λ > 0; ϱ({χ_{A}(x) > λ}) ≤ t} =

= F̃_{1}^{−1}(ϱ(A))

and we are done.

Lemma 2.2. Let K > 1 be such that

F̃_{0}^{−1}(σ(A)) ≤ KF̃_{1}^{−1}(ϱ(A))  for every measurable A ⊂ Ω. (2.5)

Assume that G satisfies the Δ_{2}-condition. Then there is K_{1} > 0 such that

∥f∥_{F_{0},G,σ} ≤ K_{1}∥f∥_{F_{1},G,ϱ} for every f ∈ L_{F_{1},G,ϱ}.

Proof. According to the definition of the Lorentz–Orlicz functional and (2.5) we have

∥f∥_{F_{1},G,ϱ} =inf \{μ>0; ∫_{0}^{∞}G (\frac{1}{μ} [inf\{λ>0; m_{ϱ}(f, λ)≤F̃ ∘ G̃^{−1}(t)\}]) dt≤1\}

and

f_{σ}^{∗}(F̃_{1} ∘ G̃^{−1}(t)) ≥ inf {λ > 0; K^{−1}F̃_{0}^{−1}(m_{σ}(f, λ)) ≤ G̃^{−1}(t)}.

As G ∈ Δ_{2} there is K_{0} > 1 such that

KG̃^{−1}(t) ≤ G̃^{−1}(K_{0}t)

(cf. Remark 1.3) and therefore, after a simple change of variables, we get

∥f∥_{F_{1},G,ϱ} ≥

≥ inf \{μ > 0; ∫_{0}^{∞}G (\frac{1}{μ} [inf\{λ > 0; m_{σ}(f, λ) ≤ F̃_{0}(KG̃^{−1}(t))\}]) dt ≤ 1\} ≥

≥ inf\{μ > 0; ∫_{0}^{∞}G (\frac{1}{μ} [inf \{λ > 0; m_{σ}(f, λ) ≤ F̃_{0}(G̃^{−1}(t))\}]) \frac{dt}{K_{0}} ≤ 1\}.

It is K_{0}^{−1}G ∼ G so that these functions generate the same Orlicz spaces. Hence

K_{1}∥f∥_{F_{1},G,ϱ} ≥ ∥f∥_{F_{0},G,σ}

with a suitable K_{1} > 0.

Next we shall consider weak Orlicz spaces.

Lemma 2.3. Let

F̃_{0}^{−1}(σ(A)) ≤ KF̃_{1}^{−1}(ϱ(A))  (2.6)

for some K > 0 and every measurable A ⊂ Ω. Then

∥f∥_{F_{0},∞,σ} ≤ K∥f∥_{F_{1},∞,ϱ}.

124 M. KRBEC AND J. LANG

Proof. By virtue of (2.6) we get

sup_{t>0} F̃_{0}^{−1}(t)f_{σ}^{∗}(t) = sup_{t>0} F̃_{0}^{−1}(t) inf{λ > 0; m_{σ}(f, λ) ≤ t} ≤

≤ sup_{t>0} F̃_{0}^{−1}(t) inf{λ > 0; F̃_{0}(KF̃_{1}^{−1}(m_{ϱ}(f, λ))) ≤ t} =

= sup_{t>0} t inf{λ > 0; KF̃_{1}^{−1}(m_{ϱ}(f, λ)) ≤ t} =

= K sup_{t>0} F̃_{1}^{−1}(t) inf{λ > 0; m_{ϱ}(f, λ) ≤ t} =

= K∥f∥_{F_{1},∞,ϱ}.

The following lemma links Orlicz–Lorentz and weak Orlicz spaces. Lemma 2.4. Let F and G be arbitrary Young functions. Then

L_{F,G,ϱ} ↪ L_{F,∞,ϱ}.  (2.7)

Proof. By the definition of the Orlicz–Lorentz functional,

∥f∥_{F,∞,ϱ} = sup_{t>0} F̃^{−1}(t) inf{λ > 0; m_{ϱ}(f, λ) ≤ t} =

= sup_{t>0} t inf{λ > 0; m_{ϱ}(f, λ) ≤ F̃(t)} =

= sup_{t>0} G̃^{−1}(t) inf{λ > 0; m_{ϱ}(f, λ) ≤ F̃ ∘ G̃^{−1}(t)}.

On the other hand, for every K > 0,

∥f∥_{F,G,ϱ} = inf \{λ > 0; ∫_{0}^{K} G (\frac{1}{λ}f_{ϱ}^{∗}(F̃ ∘ G̃^{−1}(t))) dt ≤ 1\} ≥

≥ inf \{λ > 0; ∫_{0}^{K} G (\frac{1}{λ}f_{ϱ}^{∗}(F̃(G̃^{−1}(K)))) dt ≤ 1\} ≥

≥ inf \{λ > 0; KG (\frac{1}{λ}f_{ϱ}^{∗}(F̃(G̃^{−1}(K)))) dt ≤ 1\} =

= inf \{λ > 0; \frac{1}{λ}f_{ϱ}^{∗}(F̃(G̃^{−1}(K))) ≤ G^{−1}(1/K)\} =

= inf \{λ > 0; \frac{1}{G^{−1}(1/K)}f_{ϱ}^{∗}(F̃(G̃^{−1}(K))) ≤ λ\} =

= inf {λ > 0; G̃^{−1}(K)f_{ϱ}^{∗}(F̃(G̃^{−1}(K))) ≤ λ} = = G̃^{−1}(K)f_{ϱ}^{∗}(F̃(G̃^{−1}(K))),

which gives (2.7).

Now we are ready to formulate

Theorem 2.5. Let G_{1} ∈ Δ_{2}. Then the following statements are equivalent:

(i) L_{F_{1},G_{1},ϱ} ↪ L_{F_{0},G_{1},σ}, (ii) L_{F_{1},G_{1},ϱ} ↪ L_{F_{0},G_{0},σ} provided G_{1} ≼ G_{0}, (iii) L_{F_{1},G_{1},ϱ} ↪ L_{F_{0},∞,σ}, (iv) L_{F_{1},∞,ϱ} ↪ L_{F_{0},∞,σ},

(v) F̃_{0}^{−1}(σ(A)) ≤ KF̃_{1}^{−1}(ϱ(A)) \begin{matrix}for some K > 0 \\ and every measurable A ⊂ Ω. \end{matrix}

Proof. The necessity of condition (v) follows from Lemma 2.1. The implication (v)⇒(i) was proved in Lemma 2.2. The definition of the ordering G_{1} ≼ G_{0} gives directly (i)⇒(ii) and Lemma 2.4 implies (ii)⇒(iii). Further, Lemma 2.3 gives (v)⇒(iv) and another application of Lemma 2.4 completes the proof by showing that (iv)⇒(iii).

3. More about Weighted Imbeddings

The necessary and sufficient condition (v) for imbeddings (i)–(iv) from Theorem 2.5 is of quite another sort than those previously known for imbeddings of weighted Lebesgue and/or Orlicz spaces. It was proved in [3] that, under some additional assumptions, L_{P,ϱ} ↪ L_{Q,σ} iff σϱ^{−1} ∈ L_{N,ϱ}, where N is the complementary function to QP^{−1}. A natural question is whether the case studied here permits an analogous condition. Let us observe that Theorem 2.5 solves the “nondiagonal” case; therefore one cannot expect a characterization in terms of Lebesgue and Orlicz spaces as in [1] and [3], respectively.

We shall show, however, that a nice condition equivalent to (v) of Theorem 2.5 can be found in important cases. First of all observe that (v) is equivalent to

F̃_{0}^{−1}(σ(A)) ≤ F̃_{2}^{−1}(ϱ(A))  (3.1)

where F_{2}(t) = F_{1}(Kt), and, consequently, equivalent to

\frac{1}{F̃_{0}(F̃_{2}^{−1}(ϱ(A)))} ∫_{A} σ(x) dx ≤ 1 for every measurable A ⊂ Ω. (3.2)

Put H = F_{2} ∘ F_{0}^{−1}. Then H̃^{−1} = F̃_{0} ∘ F̃_{2}^{−1} and we can rewrite (3.2) as

sup_{A⊂Ω} \frac{1}{H̃^{−1}(ϱ(A))} ∫_{A} σ(x) dx < ∞.  (3.3)

We shall show that if F_{2} ∘ F_{0}^{−1} is a convex Young function satisfying the Δ_{2}-condition, then (3.3) is nothing but a characterization of a certain weak Orlicz space. Indeed, following two lemmas hold.

Lemma 3.1. Let H be a Young function and let f ∈ L_{loc}^{1} be such that

sup \frac{1}{H̃^{−1}(ϱ(A))} ∫_{A} |f(x)|ϱ(x) dx < ∞

where the sup is taken over all measurable A ⊂ Ω. Let J be a Young function satisfying

H^{−1}(t)J^{−1}(t) ≥ c_{0}^{−1}t  (3.4)

for some c_{0} > 0 and all t ≥ 0. Then

sup_{t>0} J̃^{−1}(t)f_{ϱ}^{∗}(t) < ∞.

Proof. We have

sup_{t>0} J̃^{−1}(t)f_{ϱ}^{∗}(t) = sup_{t>0} \frac{1}{J^{−1}(1/t)}f_{ϱ}^{∗}(t) ≤ sup_{t>0} \frac{c_{0}H^{−1}(1/t)}{1/t}f_{ϱ}^{∗}(t) =

= sup_{t>0} c_{0}tH^{−1}(1/t)f_{ϱ}^{∗}(t) ≤

≤ c_{0} sup_{t>0} sup_{\begin{matrix}B⊂Ω \\ ϱ(B)=t \end{matrix}} \frac{ϱ(B)}{H̃^{−1}(ϱ(B))}f_{ϱ}^{∗}(ϱ(B)) ≤

≤ c_{0} sup_{B⊂Ω} \frac{ϱ(B)}{H̃^{−1}(ϱ(B))}f_{ϱ}^{∗}(ϱ(B)).  (3.5)

Now we claim that for every B ⊂ Ω there is A ⊂ Ω such that ϱ(A) = ϱ(B) and |f(x)| ≥ f_{ϱ}^{∗}(ϱ(A)) for all x ∈ A. Indeed, it suffices to choose

A = {x ∈ Ω; |f(x)| > λ} ∪ ({x ∈ Ω; |f(x)| = λ} ∩ Ω_{R})

where λ = f_{ϱ}^{∗}(ϱ(B)) and Ω_{R} is a suitable ball centered at the origin. Then (3.5) implies

sup_{t>0} J̃^{−1}(t)f_{ϱ}^{∗}(t) ≤ c_{0} sup_{A⊂Ω} \frac{1}{H̃^{−1}(ϱ(A))} ∫_{A} |f(x)|ϱ(x) dx.

Lemma 3.2. Let H be a convex Young function and let J be complementary to H. Assume that

sup_{t>0} \frac{H′(t)t}{H(t)} = c_{1} < ∞,

and sup_{t>0} J̃^{−1}(t)f_{ϱ}^{∗}(t) = c_{2} < ∞. Then

sup_{A⊂Ω} \frac{1}{H̃^{−1}(ϱ(A))} ∫_{A} |f(x)|ϱ(x) dx < ∞.

Proof. Let A ⊂ Ω be measurable and let f_{|A} be the restriction of f to A. Then

∫_{A} |f(x)|ϱ(x) dx = ∫_{0}^{ϱ(A)}(f_{|A})_{ϱ}^{∗} (λ) dλ ≤ ∫_{0}^{ϱ(A)}f_{ϱ}^{∗}(λ) dλ ≤

≤ c_{2} ∫_{0}^{ϱ(A)}\frac{dλ}{J̃^{−1}(λ)} dλ = c_{2} ∫_{0}^{ϱ(A)}J^{−1}(1/λ) dλ =

= c_{1}c_{2} ∫_{0}^{ϱ(A)}J^{−1}(1/λ) \frac{dλ}{c_{1}} ≤ c_{J}c_{1}c_{2} ∫_{0}^{ϱ(A)}\frac{dλ}{c_{1}λH^{−1}(1/λ)} ≤

≤c_{J}c_{1}c_{2} ∫_{0}^{ϱ(A)}\frac{1}{λH^{−1}(1/λ)} · \frac{1}{H′(H^{−1}(1/λ))λH^{−1}(1/λ)} dλ =

= c_{J}c_{1}c_{2} \frac{1}{H^{−1}(1/ϱ(A))} = c_{J}c_{0}c_{1}H̃^{−1}(ϱ(A)),

where the last step follows by taking the derivative of 1/H(1/t).

Now we are in a position to reformulate Theorem 2.5.

Theorem 3.3. Let F_{0}, F_{1}, G_{0}, and G_{1} be Young functions, G_{1} ∈ Δ_{2}, and let F_{1} ∘ F_{0}^{−1} be a convex Young function, F_{1} ∘ F_{0}^{−1} ∈ Δ_{2}. Then conditions (i)–(iv) from Theorem 2.5 are equivalent to σ/ϱ ∈ L_{J,∞,ϱ} where J is complementary to F_{2} ∘ F_{0}^{−1} where F_{2}(t) = F_{1}(Kt).

It is also worthwhile pointing out the Lorentz space version of the preceding theorem.

Corollary 3.4. Let 1 ≤ p < q < ∞, 1 ≤ r ≤ s ≤ ∞. Then the following statements are equivalent:

(i) L_{q,r,ϱ} ↪ L_{p,r,σ}, (ii) L_{q,r,ϱ} ↪ L_{p,s,σ}, (iii) L_{q,r,ϱ} ↪ L_{p,∞,σ}, (iv) L_{q,∞,ϱ} ↪ L_{p,∞,σ},

(v) σ/ϱ ∈ L_{q/(q−p),∞,ϱ}.

The proof follows immediately by calculation of the complementary function to t ↦ |t|^{q/p}, ∈ ℝ^{1}.

Observe that for r = s imbedding (ii) from Corollary 3.4 was shown to be equivalent to σ(A)^{1/p} ≤ const. ϱ(A)^{1/q} for every measurable A ⊂ Ω by Carro and Soria [11]. They consider more general two-parameter Lorentz spaces which naturally lead to the question about an analogous concept using Orlicz norms instead.

Acknowledgement

The first author was in part supported by Grant No. 201/94/1066 of the Grant Agency of the Czech Republic.

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(Received 14.10.1995)

Authors’ address:

Institute of Mathematics, Czech Academy of Sciences

Žitná 25, 115 67 Prague 1, Czech Republic