Full text of DOI:10.1112/S0024610703004198

f
C
J. London Math. Soc. (3) 67 (2003) 739–751
2003 London Mathematical Society
DOI: 10.1112/S0024610703004198
LORENTZ SPACES OF VECTOR-VALUED MEASURES
OSCAR BLASCO and PABLO GREGORI
Abstract
Given a non-atomic, finite and complete measure space (Ω, Σ, µ) and a Banach space X, the modulus of
continuity for a vector measure F is defined as the function ωF (t) = supµ(E)6t |F|(E) and the space Vp,q(X)
ꢀ
of vector measures such that t−1/p ωF (t) ∈ Lq((0, µ(Ω)], dt/t) is introduced. It is shown that Vp,q(X)
contains isometrically Lp,q(X) and that Lp,q(X) = Vp,q(X) if and only if X has the Radon–Nikodym
property. It is also proved that Vp,q(X) coincides with the space of cone absolutely summing operators
ꢀ
ꢀ
ꢀ
ꢀ
from Lp ,q into X and the duality Vp,q(X∗) = (Lp ,q (X))∗ where 1/p + 1/pꢀ = 1/q + 1/qꢀ =∞1. Finally,
V
p,q(X) is identified with the interpolation space obtained by the real method (V1(X), V (X))1/p ,q
ꢀ .
Spaces where the variation of F is replaced by the semivariation are also considered.
1. Introduction
Throughout this paper (Ω, Σ, µ) stands for a non-atomic, finite and complete measure
space, X is a (complex or real) Banach space and X∗ is its topological dual space.
As usual pꢀ denotes the conjugate exponent of p, that is, 1/p + 1/pꢀ = 1.
Given a complex-valued measurable function f we denote by µf the distribution
function of f, µf(λ) = µ(Eλ) for λ > 0 where Eλ = {w ∈ Ω : |f(w)| > λ}, by
f∗ the non-increasing rearrangement of f, f∗(t) = inf{λ : µf(λ) 6 t} and we write
ꢀ
t
f∗∗(t) = (1/t) 0 f∗(s) ds.
Then the Lorentz space Lp,q consists of those measurable functions f such that
ꢁfꢁ∗pq < ∞, where

ꢀ
ꢃ
ꢁ∞
1/q
ꢁ
ꢂ
q
q dt

t1/pf∗(t)
0 < p < ∞, 0 < q < ∞,
0 < p 6 ∞, q = ∞.


p
t
ꢁfꢁ∗pq
=
0


sup t1/pf∗(t)
t>0

Of course Lp,p = Lp and if we put ꢁfꢁpq = ꢁf∗∗ꢁp∗q then we get an equivalent
norm and Lp,q for 1 < p 6 ∞, 1 6 q 6 ∞ are Banach spaces.
Let us recall also that simple functions are dense in Lp,q for q = ∞, and ꢀ also
ꢀ
ꢀ
the duality results, (Lp,1)∗ = Lp ,∞ for 1 6 p < ∞, as well as (Lp,q)∗ = Lp ,q for
1 < p, q < ∞.
The reader is referred to [1, 12, 14, 16] for these results and for basic information
on Lorentz spaces.
All these notions make sense also for vector-valued strongly measurable functions
by replacing the modulus by the norm. This leads to the natural definition
of Lorentz–Bochner spaces Lp,q(X), where the norm is defined by ꢁfꢁL
=
p,q(X)
ꢁꢁf(·)ꢁXꢁpq.
In the vector-valued case we still have the density of simple functions, but the
ꢀ
ꢀ
corresponding duality (Lp,q(X))∗ = Lp ,q (X∗) only holds for Banach spaces X such
Received 17 December 2001; revised 13 August 2002.
2000 Mathematics Subject Classification 46E30, 28A33.
Both authors have been partially supported by Spanish grant PB98-0146.

740
oscar blasco and pablo gregori
that X∗ has the Radon–Nikodym property. The reader is referred to [8] for a proof
in the case p = q or to [10, Theorem 3.2] for a proof in the more general case
of the Kothe–Bochner space E(X) for certain Banach lattices including Lp,q. An
¨
identification of the dual space without assumptions on X can be achieved from
some general results on the dual of E(X) where E is a Banach lattice (see [6] for
a description in terms of weakly measurable functions or [10] for a formulation in
terms of vector measures).
Since Lp(X) coincides with Lp,p(X), let us first mention here that in the particular
case of Lebesgue–Bochner spaces Lp(X) the dual can be represented as the space of
ꢀ
X∗-valued measures of pꢀ-bounded variation, denoted by Vp (X∗) (see [9]).
Our objective is to define a space of vector-valued measures in such a way that it
contains Lp,q(X) isometrically and that it coincides with Vp(X) for p = q.
Following [10], one could define Vp,q(X) as the space of vector measures such that
ꢄ
supπ∈D
ꢁ
A∈π(F(A)/µ(A))χAꢁpq < ∞ where the supremum is taken over the set D
of all finite partitions π of Ω, but we would like to present a notion independent of
the knowledge of Lorentz spaces of functions.
In this paper we present a natural definiton of a modulus of continuity of a
vector measure (see Definition 2.1) which will allow to define the space Vp,q(X)
independently of the notion of Lp,q(X) and which extends the previous definition
for measures dG = fdµ, and also coincides with the definition presented above (see
Corollary 2.14).
In the case q = ∞, Marcinkiewicz spaces are denoted Vp,∞(X) and Vp,∞(X) and
ꢀ
defined by the existence of a constant C > 0 for which ꢁF(A)ꢁ 6 Cµ(A)1/p or
ꢀ
|F|(A) 6 Cµ(A)1/p for all A ∈ Σ.
To deal with the case q < ∞, we define two different moduli of continuity for a
vector measure, namely ωꢅ (t) = supµ(E)6tꢁF(E)ꢁ and ωF (t) = supµ(E)6t|F|(E). Then
F
we define the spaces Vp,q(X) and Vp,q(X) consisting of vector measures such that
ꢀ
t
−1/p ωF (t) ∈ Lq((0, µ(Ω)], dt/t) and x∗F ∈ Vp,q(K) for all x∗ ∈ X∗ respectively. Also
a space where ωF is replaced by ωꢅ is considered.
F
The paper is divided into three sections.
In the first section, it is proved that Vp,q(X) contains isometrically Lp,q(X) and
that Lp,q(X) = Vp,q(X) if and only if X has the Radon–ꢀNikodym property. It is
ꢀ
also shown that Vp,q(X∗) coincides with the dual of Lp ,q (X). The next section
deals with identification of the previous spaces of vector-valued measures as spaces
of operators. In particular, we show thaꢀt Vp,q(X) and Vp,q(X) can be described
ꢀ
as spaces of bounded operators from Lp ,q into X and cone absolutely summing
operators respectively.
In the last section, we describe the space as an interpolation space obtained by
interpolation, using the real method, of two natural spaces of vector measures,
namely Vp,q(X) = (V1(X), V∞(X))1/p ,q where V1(X) corresponds to the space of
ꢀ
µ-continuous measures of bounded variation and V∞(X) the subspace of those
measures such that ꢁF(A)ꢁ 6 Cµ(A) for all A ∈ Σ.
2. Marcinkiewicz and Lorentz spaces of vector measures
Let us recall that the variation of a vector measure F : Σ → X at the set E is
ꢄ
given by |F|(E) = supπ
ꢁF(A)ꢁ (πE stands for a finite partition of E and
A∈πE
E
the supremum is taken over all such partitions) and the semivariation is given by

lorentz spaces of vector-valued measures
741
(1)
ꢁFꢁ(A) = sup||x∗||=1 |x∗F|(A). It is worth mentioning (see [8]) that
ꢁFꢁ(A) ≈ sup ꢁF(B)ꢁ.
B⊂A
Let us first introduce the notion of ‘modulus of continuity of a vector measure’.
Definition 2.1. Let (Ω, Σ, µ) be a non-atomic finite measure space and write
I = (0, µ(Ω)]. Let X be a Banach space and let F be an X-valued measure. We
define, for t ∈ I, the functions
ωꢅ (t) = sup ꢁF(E)ꢁ and ωF (t) = sup |F|(E).
F
µ(E)6t
µ(E)6t
Remark 2.2. (a) Taking into account that µ is non-atomic, and using (1), one
easily sees that for all t ∈ I,
ωF (t) = sup |F|(E) and ωꢅ (t) ≈ sup ꢁFꢁ(E).
F
µ(E)=t
µ(E)=t
(b) F ꢃ µ if and only if limt→0 ωꢅ (t) = 0 and, for vector measures of bounded
F
variation, it is also equivalent to limt→0 ωF (t) = 0.
(c) ωF ≡ +∞ if and only if there exists t ∈ I such that ωF (t) = +∞ or, equival-
ently, F is not of bounded variation.
ꢀ
Proposition 2.3. If f ∈ L1(X) and F(E) = E f dµ then tf∗∗(t) = ωF (t) for all
t ∈ I.
ꢀ
Proof. The result follows easily from the facts that |F|(E) =
ꢁfꢁdµ and
E
ꢀ
ꢀ
t
0 f∗(s) ds = supµ(E)6t ꢁfꢁ dµ.
ꢀ
E
Proposition 2.4. Let F be a vector measure. Then either ωF ≡ +∞ on I or ωF is
non-decreasing, continuous and concave.
Proof. Let us assume that ωF (t) < +∞ for some t ∈ I. Hence F has bounded
variation and clearly ωF is non-decreasing.
Let us see first that
ωF (s + h) − ωF (s) > ωF (t + h) − ωF (t)
(2)
for all 0 < s < t < µ(Ω) and 0 < h < µ(Ω) − t.
Indeed, given ε > 0 and t, t + h, s ∈ I, there exist measurable sets Et, Et+h and Es
for which µ(Et) = t, µ(Et+h) = t + h and µ(Es) = s and
ωF (t) − |F|(Et) < ε, ωF (t + h) − |F|(Et+h) < ε, ωF (s) − |F|(Es) < ε.
Let Ah be a measurable set with µ(Ah) = h and Ah ⊂ Et+h\Es. Then
ωF (s + h) > |F|(Es ∪ Ah) = |F|(Es) + |F|(Ah)
> ωF (s) − ε + |F|(Ah) + |F|(Et+h\Ah) − ωF (t)
> ωF (s) − ε + |F|(Et+h) − ωF (t)
> ωF (s) − ε + ωF (t + h) − ε − ωF (t).
Thus for any ε > 0 we obtain (2).

742
oscar blasco and pablo gregori
Therefore for any s, t ∈ I we have
+
|ωF (s) − ωF (t)| 6 ωF (|s − t| ) − ωF (0+)
where ωF (a+) = limh→0 ωF (a + h).
+
Hence ωF is uniformly continuous in I.
It is easy to see from (2) that, for s, t ∈ I,
ꢆ
This fact, together with the continuity, gives the concavity of ωF .
ꢇ
s + t
ωF (s) + ωF (t)
ωF
>
.
2
2
ꢀ
p
A measure F is said to belong to Vp(X) (respectively V (X)) if
ꢈ
ꢊ
1/p
p
ꢉ
ꢁF(A)ꢁ
ꢁFꢁp = sup
< ∞,
µ(A)p−1
π∈D
A∈π
ꢈ
ꢈ
ꢊ
ꢊ
1/p
∗
p
ꢉ
where the supremum is taken over the set D of all finite partitions π of Ω.
|x F(A)|
respectively ꢁ|F|ꢁp = sup
< ∞ ,
µ(A)p−1
π∈D,||x∗||=1
A∈π
ꢀ
In particular, if F ∈ Vp(X) then ꢁF(A)ꢁ 6 Cµ(A)1/p for all measurable sets A.
Definition 2.5. Let (Ω, Σ, µ) be a measure space, X a Banach space and 1 < p 6
∞. Let us define by Vp,∞(µ, X) and Vp,∞(µ, X) the spaces of X-valued measures for
which there exists a constant C > 0 such that
ꢋ
ꢀ ꢌ
ꢀ
ꢁF(A)ꢁ 6 Cµ(A)1/p for any A ∈ Σ equivalently ωꢅ (t) 6 Ct1/p
F
and
ꢋ
We shall consider in these spaces the norms
ꢀ ꢌ
ꢀ
|F|(A) 6 Cµ(A)1/p for any A ∈ Σ equivalently ωF (t) 6 Ct1/p
respectively.
ꢁF(A)ꢁ
ꢀ
p,∞
ꢁFꢁV
ꢁFꢁV
= sup
A∈Σ
= sup t−1/p ωꢅ (t),
(X)
(X)
F
ꢀ
µ(A)1/p
t∈I
|F|(A)
ꢀ
p,∞
= sup
A∈Σ
= sup t−1/p ωF (t).
ꢀ
µ(A)1/p
t∈I
We shall use the notation ꢁ · ꢁp,∞ when the context is not ambiguous, keeping the
notation ꢁ · ꢁp∞ for vector-valued functions.
Remark 2.6. (a) For p = ∞ we clearly have V∞,∞(X) = V∞,∞(X) and the space
coincides with V∞(X) (see [9]).
(b) Observe that, since 1 < p 6 ∞ and µ(Ω) < ∞, measures in Vp,∞(µ, X) are
µ-continuous, and measures in Vp,∞(µ, X) are of bounded variation.
(c) It is clear that, using (1), F ∈ Vp,∞(µ, X) if and only if
ꢀ
ꢁFꢁ(A) 6 Cµ(A)1/p for any A ∈ Σ
if and only if
x∗F ∈ Vp,∞(K) for all x∗ ∈ X∗.

lorentz spaces of vector-valued measures
743
There are many ways to define spaces of vector measures which extend the Lorentz
spaces. We shall try to find a natural way to define Vp,q(X) in such a way that the
map f → dG = fdµ defines an isometric embedding from Lp,q(X) and that the space
coincides with Vp(X) and Vp,∞ for p = q and q = ∞ respectively.
Definition 2.7. Let X be a Banach space, 1 < p < ∞ and 1 6 q 6 ∞. Let us
define Vp,q(µ, X) and Vp,q(µ, X) as the spaces of vector measures such that
ꢅ
ꢆ
ꢇ
ꢆ
ꢇ
ꢀ
dt
t
ꢀ
dt
t
t
−1/p ωF (t) ∈ Lq I,
and
t
−1/p ωꢅ (t) ∈ Lq I,
F
respectively. We consider then the norms
ꢆ
and
ꢇ
ꢁ
1/q
ꢋ
ꢌ
ꢀ
q dt
p,q
ꢁFꢁV
=
t
−1/p ωF (t)
(X)
t
I
ꢆ
ꢇ
ꢁ
1/q
ꢋ
ꢌ
ꢀ
q dt
t
ꢁFꢁ
ꢅ
=
t
−1/p ωꢅ (t)
.
p,q(X)
F
V
I
We shall use the notation ꢁ · ꢁp,q when the context is not ambiguous, keeping the
notation ꢁ · ꢁpq for vector-valued functions.
Definition 2.8. Let X be a Banach space, 1 < p < ∞ and 1 6 q 6 ∞. Let
us define Vp,q(µ, X) as the space of vector measures F : Σ → X such that x∗F ∈
V
p,q(K), for all x∗ ∈ X∗. We consider then the norm
= sup ꢁx∗FꢁV
.
(K)
p,q
p,q
ꢁFꢁV
(X)
ꢁx∗ꢁ61
Let us now recollect some elementary results about these spaces.
Proposition 2.9. Let 1 < p 6 ∞ and 1 6 q 6 ∞. Then the following hold:
(a) (Vp,q(X), ꢁ · ꢁV (X)) and (Vp,q(X), ꢁ · ꢁV (X)) are Banach spaces.
(b) Vp(X) ⊂ Vp,∞(X) ⊂ Vp,∞(X).
p,q
p,q
(c) Vp,q(µ, X) ⊂ Vp,q(X) ⊂ Vp,q(X).
ꢅ
Proof. The proofs of (a), (b) and (c) are easy and left to the reader.
(d) Vp,q(X) ⊂ Vp,∞(X) and Vp,q(X) ⊂ Vp,∞(X).
ꢀ
To see (d) note that, since ωF is non-decreasing, integrating s−q/p −1 from t to
µ(Ω), we have
ꢆ
ꢇ
ꢁ
1/q
µ(Ω)
ꢋ
ꢌ
ꢀ
ꢀ
q ds
s
t
−1/p ωF (t) 6 C1
s
−1/p ωF (s)
+ C2,
t
for some constants C1, C2 > 0. A similar estimate holds for ωꢅ (t).
ꢀ
F
Example 2.10. Let 1 < p, q < ∞ and let us take X = Lp,q(µ, Ω). Consider the
p,q-valued measure F given by F(E) = χE for E ∈ Σ. Then the following hold:
(i) ωF (t) = +∞ for t ∈ I.
L
(ii) ωꢅ (t) = t1/p for t ∈ I.
(iꢀ) F ∈/ Vr,s(X) for any r, s.
F
ꢀ
ꢀ
ꢀ
(iii) For any x∗ = φ ∈ X∗ = Lp ,q , the measure x∗F(E) = E φ dµ for E ∈ Σ.
Therefore the following hold:

744
oscar blasco and pablo gregori
ꢀ
r,s
ꢀ
ꢀ
ꢅ
(ii ) F ∈ V (X) if and only if 1 6 r 6 p and s = ∞ or 1 6 r < p and 1 6 s 6 ∞.
r,s
(iiiꢀ) F ∈ V (X) if and only if r = pꢀ and s > qꢀ or 1 6 r < pꢀ.
ꢀ
ꢀ
ꢀ
ꢀ
pꢀ,qꢀ
In particular, there exist F1 ∈ V (X)\Vp ,q (X) and F2 ∈ Vp ,∞(X)\Vp ,∞(X).
ꢅ
The importance of the following lemma is evident from the several references to
it in the subsequent results.
Lemma 2.11. Let 1 < p < ∞ and 1 6 q 6 ∞. The following are equivalent:
(a) F ∈ Vp,q(X).
ꢀ
(b) There exists ϕ > 0, ϕ ∈ Lp,q such that |F|(A) = A ϕ dµ for all A ∈ Σ.
ꢀ
p,q
Moreover if |F|(A) = A ϕ dµ for all A ∈ Σ for some ϕ > 0 then ꢁFꢁV
= ꢁϕꢁpq.
(X)
Proof. Assume that (a) holds and that q = ∞. Let us take F ∈ Vp,∞(X). Then
using the Radon–Nikodym theorem we find a non-negative function ϕ such that
|F|(A) = A ϕ dµ for all A ∈ Σ. Now observe that
ꢀ
ꢆ
ꢇ
|F|(A)
|F|(A)
p,∞
ꢁFꢁV
= sup
A∈Σ
= sup sup
µ(A)=t
(X)
ꢀ
ꢀ
µ(A)1/p
t1/p
t>0
ꢆ
ꢇ
ꢁ
ꢁ
t
ꢀ
ꢀ
= sup t−1/p sup
ϕ dµ = sup t−1/p ϕ∗(s) ds
t>0
µ(A)=t
t>0
A
0
= sup t1/pϕ∗∗(t) = ꢁϕꢁp∞
.
t>0
For q < ∞ if F ∈ Vp,q(X) then F ∈ Vp,∞(X). From the previous case we find a
ꢀ
non-negative function ϕ such that |F|(A) = A ϕ dµ for all A ∈ Σ. Now, with a look
at Proposition 2.3, it is plain to see that ꢁFꢁp,q = ꢁϕꢁpq.
The converse also follows from Proposition 2.3.
ꢀ
With the help of Lemma 2.11 we can prove the following.
Theorem 2.12. Let X be a Banach space and 1 < p < ∞, 1 6 q 6 ∞. Then the
following hold:
(a) Lp,q(X) is isometrically embedded into Vp,q(X).
(b) Lp,q(X) = Vp,q(X) if and only if X has the Radon–Nikodym property.
ꢀ
Proof. Let f ∈ Lp,q(X), then the measure F(E) = E f dµ belongs to Vp,q(X).
ꢀ
This follows from the fact that |F|(E) =
ꢁfꢁ dµ and Lemma 2.11 for ϕ = ꢁfꢁ.
E
To prove (b), let us assume that Lp,q(X) = Vp,q(X) and let T : L1 → X be a bounded
operator. We need to show that T is representable (see [8]). Note that F(E) = T(χE)
gives a measure in V∞(X), and then F ∈ Vp,q(X). Now by the assumption there exists
ꢀ
ꢀ
ꢀ
f ∈ Lp,q(X) such that F(E) = E f dµ for all E ∈ Σ. This shows that T(ψ) = ψf dµ
for all ψ ∈ L1.
Conversely, let us assume that X has the Radon–Nikodym property and take
F ∈ Vp,q(X). Because F is µ-continuous and with bounded variation, F(E) = E f dµ
for all E ∈ Σ for some f ∈ L1(X). To show that f ∈ Lp,q(X), apply Lemma 2.11 for
ϕ = ꢁfꢁ again.
ꢀ

lorentz spaces of vector-valued measures
745
Lemma 2.13. Let 1 < p < ∞ and 1 6 q 6 ∞ or p = q = ∞. If F ∈ Vp,q(X) then
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢍ
ꢍ
ꢏ
ꢉ
ꢉ
p,q
ꢁFꢁV
≈ sup
|α | ꢁF(A )ꢁ :
αiχA
6 1
(X)
i
i
i
pꢀqꢀ
i
i
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢏ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
= sup
|α | |F|(A ) :
αiχA
6 1 .
i
i
i
pꢀqꢀ
i
i
Proof. If ϕ is the function in Lemma 2.11, we use [1, Theorem 4.7, p. 220]
on the dual norm in the space Lp,q, to show that ꢁFꢁp,q = ꢁϕꢁpq is equivalent to
ꢀ
ꢀ
ꢀ
sup{ ϕ|f| dµ : ꢁfꢁp q 1}.
Ω
Hence
ꢍ
ꢍ
ꢍ
ꢍ
ꢈ
ꢊ
ꢏ
ꢁ
ꢉ
ꢉ
ꢀ
ꢀ
ꢁFꢁp,q ≈ sup
ϕ
|αi|χA dµ : f =
αiχA , ꢁfꢁp q 6 1
i
i
Ω
i
i
ꢎ
ꢎ
ꢏ
ꢁ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
ꢉ
ꢉ
= sup
|αi| ϕ dµ :
αiχA
6 1
i
Ai
pꢀqꢀ
i
i
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢏ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
= sup
|α ||F|(A ) :
αiχA
6 1
i
i
i
pꢀqꢀ
i
i
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢏ
ꢉ
= sup
|α |ꢁF(A )ꢁ :
αiχA
6 1 .
i
i
i
pꢀqꢀ
i
i
The equality of both suprema in the last step follows easily from the definition of
variation |F|(Ai).
ꢀ
Corollary 2.14. Let 1 < p < ∞ and 1 6 q 6 ∞. Then F ∈ Vp,q(X) if and only if
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
F(A)
sup
π∈D
χA
< ∞
pq
ꢎ
ꢎ
ꢎ
ꢎ
µ(A)
A∈π
where the supremum is taken over the set D of all finite partitions π of Ω.
In particular, Vp,p(X) = Vp(X) for all 1 < p 6 ∞.
Proof. Let π be a partition in D. Then
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
F(A)
µ(A)
ꢁF(A)ꢁ
µ(A)
χA
=
χA
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
Lp,q(X)
Lp,q
A∈π
A∈π
ꢍ
ꢍ
ꢏ
ꢁ
ꢉ
ꢁF(A)ꢁ
ꢀ
ꢀ
= sup
ψ dµ : ꢁψꢁp ,q = 1
µ(A)
A
A∈π
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢏ
ꢉ
ꢉ
= sup
α ꢁF(A)ꢁ :
αAχA
= 1 .
A
pꢀ,qꢀ
A∈π
A∈π
ꢀ
ꢄ
In the last equality we have used the ꢀfact that Eπ(ψ) =
A ψ dµ/µ(A)χA defines
A∈π
ꢀ
a bounded operator of norm 1 in Lp ,q
.
ꢀ

746
oscar blasco and pablo gregori
Now, using the convenient results for spaces of measurable functions, we obtain
the following embeddings.
Corollary 2.15. Let 1 < p < ∞, 1 6 q1 6 q2 6 ∞, 1 < p1 6 p2 < ∞ and 1 6 q,
r 6 ∞. Then the following hold:
p,q2
p,q1
p,q2
p,q1
p,q2
(a) Vp,q (X) ⊂ V (X), V (X) ⊂ V (X) and V (X) ⊂ V (X).
1
ꢅ
ꢅ
(b) Vp ,q(X) ⊂ Vp ,r(X), V (X) ⊂ Vp ,r(X) and Vp ,q(X) ⊂ Vp ,r(X).
p2,q
2
1
ꢅ
ꢅ
1
2
1
Our next step is the description of the dual of the vector-valued Lorentz function
spaces in terms of vector measure spaces.
ꢀ
ꢀ
Theorem 2.16. Let 1 < p < ∞ and 1 6 q < ∞ or p = q = 1. Then Vp ,q (X∗) =
[Lp,q(X)]∗.
ꢀ
ꢀ
Proof. Let F ∈ Vp ,q (X∗) and φF be the functional over Lp,q(X) given as usual
ꢄ
ꢄ
by φF ( xiχA ) =
ꢅxi, F(Ai)ꢆ. We have
i
i
i
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢈ
ꢊ
ꢉ
ꢉ
φ
F
xiχA
6
ꢁxiꢁꢁF(Ai)ꢁ
i
i
i
ꢀ
ꢀ
for every simple function. Then Lemma 2.13 gives ꢁφF ꢁ 6 CꢁFꢁp ,q for some
constant C > 0.
Now let φ ∈ [Lp,q(X)]∗. For each A ∈ Σ, Fφ(A) is the element of X∗ such that
Fφ(A)(x) = φ(xχA). Observe that F is well-defined and a countably additive measure
(by continuity of φ). It is plain that φ = φF . Then, we can conclude, since
φ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢍ
ꢍ
ꢍ
ꢍ
ꢏ
ꢉ
ꢉ
ꢁFφꢁp ,q 6 Cꢀ sup
|α | ꢁF(A )ꢁ:
αiχA
6 1
ꢀ
ꢀ
i
i
i
i
i
p,q
ꢐ ꢎ
ꢐ ꢎ
ꢐ ꢎ
ꢎ
ꢏ
ꢏ
ꢏ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
= Cꢀ sup
|α | |ꢅx , F(A )ꢆ :
αiχA
6 1, ꢁxiꢁ 6 1 ∀ i
ꢐ ꢎ
ꢐ ꢎ
i
i
i
i
i
i
p,q
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢉ
ꢉ
= Cꢀ sup
ꢅx , F(A )ꢆ : f =
xiχA ∈ Lp,q(X), ꢁfꢁp,q 6 1
ꢐ
i
i
i
ꢐ
i
i
ꢐ
ꢐ
ꢐ
ꢐ
ꢐ
ꢈ
ꢊ
ꢉ
ꢉ
= Cꢀ sup
φ
F
xiχA
: f =
xiχA ∈ Lp,q(X), ꢁfꢁpq 6 1
i
i
i
i
= Cꢀꢁφꢁ.
ꢀ
ꢀ
ꢀ
Corollary 2.17. Let 1 < p, q < ∞. Then [Lp,q(X)]∗ = Lp ,q (X∗) if and only if X∗
has the Radon–Nikodym property.
3. Vector measures and operators
We comment that a vector measure F provides us with a linear operator TF acting
on characteristic functions as TF (χA) = F(A), and it can be obviously extended by
linearity to the set of step functions.

lorentz spaces of vector-valued measures
In this way we clearly identify
747
V∞(X) = V∞(X) = L(L1, X).
In this section we analyse the cases Vp,q(X) and Vp,q(X).
To describe the spaces Vp,q(X) in terms of operators, we need the following
lemma.
Lemma 3.1. Let 1 < p < ∞ and 1 6 q 6 ∞. If F ∈ Vp,q(X) then
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ ꢎ
ꢎ ꢎ
ꢎ ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢍ
ꢏ
ꢉ
ꢉ
p,q
ꢁFꢁV
≈ sup
α F(A ) :
αiχA
6 1 .
ꢎ ꢎ
(X)
i
i
i
ꢎ ꢎ
pꢀqꢀ
i
i
Proof. Using Lemma 2.13 for the particular case X = K we have
p,q
p,q
ꢁFꢁV
= sup ꢁx∗FꢁV
(X)
ꢁx∗ꢁ61
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢍ
ꢍ ꢑ
ꢍ
ꢏ
ꢉ
ꢉ
ꢇ sup
= sup
= sup
|α ||x∗F(A )|:
αiχA
6 1, ꢁx∗ꢁ 6 1
i
i
i
pꢀqꢀ
i
i
ꢐ
ꢐ ꢎ
ꢐ ꢎ
ꢐ ꢎ
ꢎ
ꢒ
ꢏ
ꢐ
ꢐ
ꢐ
ꢐ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
ꢉ
αiF(Ai), x∗
:
αiχA
6 1, ꢁx∗ꢁ 6 1
ꢐ ꢎ
i
ꢐ ꢎ
pꢀqꢀ
i
i
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ ꢎ
ꢎ ꢎ
ꢎ ꢎ
ꢎ
ꢎ
ꢎ
ꢏ
ꢉ
α F(A ) :
αiχA
6 1 .
ꢎ ꢎ
ꢎ
ꢎ
i
i
i
ꢎ ꢎ
pꢀqꢀ
i
i
ꢀ
The following results are straightforward corollaries.
Theorem 3.2. Let 1 < p < ∞ and 1 6 q < ∞. Then
ꢀ
ꢀ
V
p ,q (X) = L(Lp,q, X).
Corollary 3.3. Let 1 < p < ∞ and 1 6 q < ∞. Then
ꢀ
ꢀ
∗
p,q ˆ
V
p ,q (X∗) = (L ⊗πX) .
ꢀ
ꢀ
From Theorem 3.2, Vp ,q (X) is a subspace of the space of operators from Lp,q to
X. If we want to understand the corresponding class of operators we need to recall
the following notion.
Definition 3.4 (see [15, p. 244]). Let E be a Banach lattice and B a Banach
space. A linear operator from E to B is said to be cone absolutely summing if there
is a constant C > 0 such that for every k ∈ N and every family e1, e2, . . . , ek ∈ E of
positive elements we have
k
k
ꢉ
ꢉ
ꢁT(ek)ꢁB 6 C sup
|ꢅei, e∗ꢆ|.
ꢁe∗ꢁE∗ 61
i=1
i=1
We denote by Π+1 (E, B) the set of such operators and its norm is given by the
infimum of the constants C satisfying the previous inequality.

748
oscar blasco and pablo gregori
Remark 3.5. It is rather easy to give an equivalent defintion (see [15, p. 244]) by
using that if e1, e2, . . . , ek are positive elements in E then
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
k
k
ꢉ
ꢉ
sup
|ꢅe , e∗ꢆ| =
e .
(3)
ꢎ
i
i
ꢎ
ꢁe∗ꢁE∗ 61
i=1
i=1
Let us mention here that Lp,q are, clearly, Banach lattices and the next theorem
allows us to identify the space Π+1 (Lp,q, X).
ꢀ
ꢀ
Theorem 3.6. Let 1 < p < ∞ and 1 6 q < ∞ or p = q = 1. Then Vp ,q (X) =
Π+1 (Lp,q, X).
ꢀ
ꢀ
Proof. Let F ∈ Vp ,q (X) and TF : Lp,q → X be defined as explained at the
beginning of this section. Let us see that T ∈ Π+1 (Lp,q, X). If N ∈ N and f1, f2, . . . , fN
ꢀ
ꢀ
are non-negative functions in Lp,q, then Lemma 2.11 gives the existence of ϕ ∈ Lp ,q
such that for all f ∈ Lp,q
ꢁ
ꢁTF (f)ꢁ 6 |f|ϕdµ.
Ω
Therefore
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢈ
ꢊ
ꢆ
ꢇ
ꢁ
ꢁ
N
N
N
N
ꢉ
ꢉ
ꢉ
ꢉ
ꢀ
ꢀ
.
ꢁTF (fn)ꢁ 6
fnϕ dµ
=
fn ϕ dµ 6
f
n
ꢁFꢁp ,q
Ω
Ω
n=1
n=1
n=1
n=1
p,q
ꢀ
ꢀ
.
Π+1
Hence, from (3), one obtains ꢁTF ꢁ 6 ꢁFꢁp ,q
Now if T ∈ Π+1 (Lp,q, X) and FT : Σ → X is defined by FT (A) = T(χA), it is
ꢄ
obvious that T = TF and FT is countably additive. Let f =
αiχA ∈ Lp,q with
T
i
ꢁfꢁp,q 6 1, then
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢉ
ꢉ
ꢉ
ꢋ
ꢌ
|αi| ꢁFT (Ai)ꢁ =
ꢁT |αi|χA ꢁ 6 ꢁTꢁΠ1
|αi|χA
6 ꢁTꢁΠ1
.
i
i
+
+
i
i
i
p,q
ꢀ
ꢀ
Therefore ꢁFT ꢁp ,q 6 CꢁTꢁΠ1 , and the proof is complete.
ꢀ
+
4. Lorentz spaces and interpolation
We refer the reader to [1] or [2], where a wide study of interpolation spaces is
developed.
Definition 4.1 (the K-functional). Let (X0, X1) be a compatible couple of
Banach spaces. The K-functional can be defined for every f ∈ X0 + X1 and t > 0 by
ꢓ
ꢔ
K(f, t; X0, X1) = inf ꢁf0ꢁX + tꢁf1ꢁX : f = f0 + f1
where the infimum is taken from all the possible representations f = f0 + f1 of f
,
0
1
with f0 ∈ X0 and f1 ∈ X1.
Theorem 4.2 [1, p. 298]. Let (Ω, Σ, µ) be a totally σ-finite measure space, then
ꢁ
t
K(f, t; L1, L∞) = f∗(s) ds = tf∗∗(t).
0

lorentz spaces of vector-valued measures
749
Definition 4.3 [1, p. 299]. Let (X0, X1) be a compatible couple and suppose that
0 < θ < 1, 1 6 q < ∞ or 0 6 θ 6 1, q = ∞. The space (X0, X1)θ,q consists of all f in
X0 + X1 for which the functional

ꢀ
ꢃ
ꢁ∞
1/q

dt
t


[t−θK(f, t)]q
0 < θ < 1, 1 6 q < ∞,
0 6 θ 6 1, q = ∞
0
ꢁfꢁθ,q
=



sup t−θK(f, t)
t>0
is finite (here K(f, t) := K(f, t; X0, X1)).
Let us denote by V1(X) the space of µ-continuous vector measures of bounded
variation and write ꢁFꢁV1(X) = |F|(Ω).
Of course, V∞(X) ⊂ V1(X) and Vp,q(X) ⊂ V1(X) for all 1 < p < ∞ and 1 6 q 6 ∞.
Theorem 4.4. Let F ∈ V1(X) and t > 0. Then
ωF (t) 6 K(F, t; V1(X), V∞(X)) 6 2ωF (t).
Proof. Assume that F = G + H with G ∈ V1(X) and H ∈ V∞(X). It follows
from |F| 6 |G| + |H| that ωF (t) 6 ωG(t) + ωH (t).
Note that
ωG(t) = sup |G|(E) 6 |G|(Ω) = ꢁGꢁV1(X)
µ(E)6t
and
|H|(E)
∞
ωH (t) = sup |H|(E) 6 t sup
= tꢁHꢁV
.
(X)
µ(E)
µ(E)6t
µ(E)6t
Then we get the first estimate by taking the infimum over all decompositions.
Assume now that ωF (t) is finite. Since |F| is a µ-continuous measure, we can find
ꢀ
a function ϕ > 0 such that |F|(A) = A ϕ dµ for every measurable set A.
We take E = {w ∈ Ω : ϕ(w) > ϕ∗(t)}. Since µϕ and ϕ∗ are mutually right-continuous
inverse functions, then µ(E) 6 t.
ꢀ
Let us define G(A) = F(E ∩ A) for A ∈ Σ. In this case |G|(A) = A ϕG dµ for all A,
where ϕG = ϕχE. It is easy to check that
ꢍ
ϕ∗(s)
0 < s < µ(E),
s > µ(E)
ꢀ
ϕ∗G(s) =
0
and from this we can deduce that ꢁGꢁV1(X) = ꢁϕGꢁL1
=
µ(E) ϕ∗(s) ds.
0
Defining H(A) = F(A) − G(A) = F((Ω\E) ∩ A) for all A ∈ Σ, we have |H|(A) =
ꢀ
A ϕH dµ for every A, where ϕH = ϕχΩ\E. It is clear that
ꢍ
µϕ(λ) − µϕ(ϕ∗(t))
0 < λ 6 ϕ∗(t),
λ > ϕ∗(t).
µϕ (λ) =
H
0
Therefore
ϕ∗H (s) = ϕ∗(s + µϕ(ϕ∗(t))).

750
oscar blasco and pablo gregori
Hence
ꢀ
(X)
0s ϕ∗H (θ) dθ
|H|(A)
µ(A)
sup
A
6 sup
6 lim ϕ∗H (s) = ϕ∗(µϕ(ϕ∗(t))) 6 ϕ∗(t)
s→0
s
s>0
6 tϕ∗(t).
∞
and consequently tꢁHꢁV
Now, using the decomposition of F and the properties of ϕ∗, we obtain
ꢁ
ꢁ
µ(E)
t
∞
ꢁGꢁV1(X) + tꢁHꢁV
6
ϕ∗(s) ds + tϕ∗(t) 6 2 ϕ∗(s) ds = 2ωF (t).
(X)
0
0
ꢀ
Theorem 4.5. If 1 < p < ∞ and 1 6 q 6 ∞, then
p,q(X) = (V1(X), V∞(X))θ,q
V
,
where 1/p = 1 − θ.
Using the reiteration theorem (see [1, p. 311]) and Corollary 2.14 we obtain the
following.
Theorem 4.6. If 1 < p1, p2 < ∞ and 1 6 q 6 ∞, then
Vp,q(X) = (Vp1 (X), Vp2 (X))θ,q
,
where 1/p = (1 − θ)/p1 + θ/p2.
Let us mention something about the interpolation when we also change the spaces
in which the measures take values. For the difference between (Vp1 (X1), Vp2 (X2))θ,q
and Vp,q((X1, X2)θ,q) we recall that Vp(X) = Lp(X) if X has the Radon–Nikodym
property and simply refer the reader to the paper by M. Cwikel [7] where it is shown
that for spaces of vector-valued functions the equality may happen only for q = p
where 1/p = 1 − θ/p1 + θ/p2.
Even in the case q = p the expected interpolation result does not hold for
vector-valued measures. The reader is referred to [5] for the difference between
(Vp1 (X1), Vp2 (X2))θ,p and Vp,q((X1, X2)θ,p). Although there the authors deal with
interpolation between spaces of vector-valued harmonic functions, instead of vector-
valued measures, they can be identified according to the results in [4] or [3].
Acknowledgements. We would like to thank the referee for suggestions for
improving the paper.
References
1. C. Bennet and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129 (Academic
Press, New York, 1988).
¨
¨
2. J. Bergh and J. Lofstrom, Interpolation spaces. An introduction (Springer, Berlin, 1973).
3. O. Blasco, ‘Boundary values of vector valued functions in Orlicz–Hardy classes’, Arch. Math. 49
(1987) 434–439.
4. O. Blasco, ‘Boundary values of functions in vector-valued Hardy spaces and geometry of Banach
spaces’, J. Funct. Anal. 78 (1988) 346–364.
5. O. Blasco and Q. Xu, ‘Interpolation between vector-valued Hardy spaces’, J. Funct. Anal. 102 (1991)
331–359.
6. A. V. Bukhvalov, ‘The analytic representation of operators with abstract norm’, Izv. Vyssh. Uchebn.
Zaved. Mat. 11 (1975) 21–32 (Russian).
7. M. Cwikel, ‘On (Lp0 (A0), Lp1 (A1))θ,q’, Proc. Amer. Math. Soc. 44 (1974) 286–292.

lorentz spaces of vector-valued measures
751
8. J. Diestel and J. J. Uhl, Vector measures, Mathematical Surveys 15 (American Mathematical Society,
Providence, RI, 1977).
9. N. Dinculeanu, Vector measures, International Series of Monographs in Pure and Applied Math-
ematics 95 (Pergamon Press, 1967).
10. N. E. Gretsky and J. J. Uhl, ‘Bounded linear operators on Banach function spaces of vector-valued
functions’, Trans. Amer. Math. Soc. 167 (1972) 263–277.
11. P. R. Halmos, Measure theory (Van Nostrand, New York, 1950).
12. R. A. Hunt, ‘On L(p, q) spaces’, Enseign. Math. 4 (1966) 249–276.
13. J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II: Function spaces (Springer, Berlin,
1979).
14. G. G. Lorentz, ‘Some new functional spaces’, Ann. Math. 51 (1950) 37–55.
15. H. H. Schaefer, Banach lattices and positive operators (Springer, New York, 1974).
16. E. M. Stein and G. Weiss, Introduction to Fourier analysis on euclidean spaces (Princeton University
Press, Princeton, 1971).
Oscar Blasco and Pablo Gregori
Departamento de Ana´lisis Matema´tico
Universidad de Valencia
Doctor Moliner, 50
E-46100 Burjassot (Valencia)
Spain
oscar.blasco@uv.es
pablo.gregori@uv.es