Uniform λ-property in L1∩L∞
Adam Bohonos and Ryszard Płuciennik
Summary. Here it is proved that the space L1∩ L∞equipped with the stan- dard interpolation norm ∥⋅∥L1∩L∞= max {∥⋅∥L1, ∥⋅∥L∞} has the uniform λ-property if and only if µ(T ) ⩽ 1. Replacing the standard norm with an equivalent one ∥⋅∥′L1∩L∞= ∥⋅∥L1+ ∥⋅∥L∞, a different result is obtained.:
(L1∩ L∞, ∥⋅∥′L1∩L∞) has the uniform λ-property if and only if µ(T ) < ∞.
Keywords λ-property;
uniform λ-property;
interpolation spaces;
convex series representation property
MSC 2010
46E30; 46B20; 46B22 Received: 2016-02-29, Accepted: 2016-04-29
Dedicated to Professor Henryk Hudzik on his 70th birthday in friendship and high esteem.
1. Introduction
Let (X ,∥⋅∥X) be a real Banach space and let B(X) (S(X)) be the closed unit ball (the unit sphere) of X ,. Denote byN the set of positive integers and by R the set of reals.
Before stating our results, we need to recall some notions. A point x∈ S(X) is said to be an extreme point of B(X) if x cannot be written as the arithmetic mean21(y +z) of two distinct points y, z∈ S(X). The set of all extreme points of B (X) is denoted by ext B (X).
Define a function λ∶ B(X) → [0, 1] by the formula
λ(x) = sup{λ ∈ [0, 1] ∶ x = λe + (1 − λ)y, e ∈ ext B (X) , y ∈ B (X)}
Adam Bohonos, School of Mathematics, West Pomeranian University of Technology, Al. Piastów 48/49, 70-310 Szczecin, Poland (e-mail:abohonos@zut.edu.pl)
Ryszard Płuciennik, Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland (e-mail:ryszard.pluciennik@put.poznan.pl)
DOI 10.14708/cm.v55i2.1122 © 2015Polish Mathematical Society
if ext B(X) /= ∅, and λ(x) = 0 for any x ∈ B(X) if ext B (X) = ∅. The function λ is called the λ-function.
A point x∈ B(X) is said to be a λ-point of B(X) if λ(x) > 0. The space X is said to have the λ-property if each x∈ B(X) is a λ-point. Moreover, if
λX= inf{λ (x) ∶ x ∈ S(X)} > 0, then X is said to have the uniform λ-property.
The λ-property was introduced by Aron and Lohman [1,13]. The λ-property is im- portant because for Banach spaces with the λ-property we have B(X) = co (ext B (X)) . Moreover, Aron, Lohman, and Granero proved in [2] that a Banach space X has the λ-pro- perty if and only if it has the convex series representation property, i.e. for each x∈ B (X), there is a sequence(ek) of extreme points of B (X) and a sequence of non-negative real numbers(λk) such that ∑∞k=1λk = 1 and x = ∑∞k=1λkek. It has also been shown in [2]
that the uniform λ-property of a Banach space X is equivalent to the uniform convex se- ries representation property of X , i.e. convex series representation property in which the sequence(λk) does not depend on x.
It is well known that λ(0) = 21 and λ(x) ⩾ max {1
2(1 − ∥x∥X) , λ ( x
∥x∥X) ∥x∥X}
for any x∈ B(X) whenever ext B (X) /= ∅ (see Proposition 2.12 in [6]). It implies that any interior point of B(X) is a λ-point whenever ext B (X) /= ∅. Hence the definition of the λ-property in X can be restricted to the unit sphere S(X). Namely, a Banach space X has the λ-property if and only if ext B(X) /= ∅ and λ(x) > 0 for any x ∈ S(X).
Let(T, Σ, µ) be a measure space with a σ-finite non-atomic and complete measure µ and let L0= L0(T, Σ, µ) be the set of all µ-equivalence classes of real and Σ-measurable functions defined on T .
By a Banach function space (or a Köthe space)(E, ∥⋅∥E) we mean a subspace of L0 which is an ideal in L0, that is, if x ∈ L0(T, Σ, µ) , y ∈ E, and ∣x (t)∣ ⩽ ∣y (t)∣ µ-a.e. on T, then x∈ E and ∥x∥E⩽ ∥y∥E, and there exists x∈ E such that x (t) > 0 for any t ∈ T.
The set E+ = {x ∈ E ∶ x ⩾ 0} is called the positive cone of E. For any subset A ⊂ E define A+= A ∩ E+.
A point x∈ E+∖ {0} is said to be a point of upper monotonicity (x is a UM point, for short) if for any y ∈ E+such that x ⩽ y and y ≠ x, we have ∥x∥E < ∥y∥E. Recall that if each point of E+∖ {0} is a UM point, then we say that E is strictly monotone (E ∈ (SM)) (see [4,7,11]).
For x∈ L0we denote its distribution function by
dx(η) = µ{s ∈ T ∶ ∣x (s)∣ > η}, η ⩾ 0,
and its decreasing rearrangement by
x∗(t) = inf{η > 0 ∶ dx(η) ⩽ t}, t ∈ [0, µ(T)].
Two functions x , y ∈ L0are called equimeasurable (x ∼ y, for short) if dx = dy. We say that a Banach function space(E, ∥ ⋅ ∥E) is rearrangement invariant or symmetric if whenever x∈ L0and y∈ E with x ∼ y, then x ∈ E and ∥x∥E= ∥y∥E(see [3,8]).
The following simple fact will be useful for our considerations.
1.1. Proposition ([5, Proposition 2.1]). Let E be a Köthe space. Then (i) λ(x) = λ (∣x∣) for any x ∈ S (E);
(ii) λ(x) = λ+(x) for any x ∈ S (E)+, where λ+is defined on S(E)+by the formula λ+(x) = sup{λ ∈ [0, 1] ∶ x = λe + (1 − λ)y, e ∈ ext B (X)+, y∈ B (X)}.
In this note, we consider the space L1∩ L∞ that is constructed from the Lebesgue spaces L1and L∞by taking their intersection. This space plays a special role because it is the smallest of all symmetric spaces. The standard norm in L1∩ L∞is defined by
∥x∥L1∩L∞ = max{∥x∥L1,∥x∥L∞}, where
∥x∥L1= ∫T∣x(t)∣ dµ and ∥x∥L∞= sup ess
t∈T ∣x(t)∣ . Moreover we will consider the following norm equivalent to the standard one:
∥x∥′L1∩L∞ = ∥x∥L1+ ∥x∥L∞.
It easy to see that, in contrast to(L1∩ L∞,∥⋅∥L1∩L∞), the space (L1∩ L∞,∥⋅∥′L1∩L∞) is strictly monotone.
1.2. Remark. The space L1∩ L∞can be considered as the Orlicz space LΦ1 ,∞generated by the Orlicz function Φ1 ,∞defined by the formula
Φ1 ,∞(u) =⎧⎪⎪
⎨⎪⎪⎩
∣u∣ for u∈ [−1, 1]
+∞ otherwise.
Namely,
L1∩ L∞= LΦ1 ,∞ = {x ∈ L0 ∶ IΦ1 ,∞(cx) < ∞ for some c > 0},
where IΦ1 ,∞is the modular generated by Φ1 ,∞. Moreover, the norm∥⋅∥L1∩L∞(resp. ∥⋅∥′L1∩L∞) and the Luxemburg (resp. Orlicz) norm coincide (see [9,10,14]). This gives additional mo- tivation to consider the norm∥⋅∥′L1∩L∞.
2. Results
2.1. Lemma. Let E be a strictly monotone symmetric Banach function space. If x , y, z ∈ S(LΦ) and x = λy + (1 − λ)z for some λ ∈ (0, 1), then
(i) supp x= supp y ∪ supp z;
(ii) sign x(t) = sign y(t) = sign z(t) for a.e. t ∈ T.
Proof. Obviously supp x⊂ supp y∪supp z. Suppose, to the contrary, that supp z /⊂ supp x.
Denote
A= supp z ∖ supp x.
Then µ(A) > 0. By strict monotonicity of E, we conclude that
∥zχsupp z∖A∥E < 1.
Since A∩ supp x = ∅, we have
λ y χA+ (1 − λ)zχA= 0.
Hence A⊂ supp y. Therefore
x= λyχsupp y∖A+ (1 − λ)zχsupp z∖A, whence
1= ∥x∥E ⩽ λ ∥yχsupp y∖A∥E+ (1 − λ) ∥zχsupp z∖A∥E< 1.
A contradiction. Hence supp z⊂ supp x. Repeating the same arguments, it can be shown that supp y ⊂ supp x. Consequently, supp x = supp y ∪ supp z, which finishes the proof of(i).
Since
1= ∥x∥E⩽ ∥(1 − λ) ∣y∣ + λ ∣z∣∥E ⩽ (1 − λ) ∥y∥E+ λ ∥z∥E ⩽ 1, by strict monotonicity of E , we conclude
∣(1 − λ)y(t) + λz(t)∣ = (1 − λ) ∣y(t)∣ + λ ∣z(t)∣
for a.e. t∈ T. Hence sign x(t) = sign y(t) = sign z(t) for a.e. t ∈ T.
2.2. Theorem. The space(L1∩ L∞,∥⋅∥L1∩L∞) has the λ-property.
Proof. If µ(T) ⩽ 1, then L1∩ L∞ = L∞ and∥⋅∥L1∩L∞ = ∥⋅∥L∞. It is easy to prove that λ(x) ⩾ 12 for any x∈ L∞, so(L∞,∥⋅∥L∞) has even the uniform λ-property. Assume now that µ(T) > 1. By Corollary 1 in [10], every extreme point of B(L1∩ L∞) is of the form
e(t) = ϰ(t)χA(t),
where µ(A) = 1 and ϰ is a Σ-measurable function such that ∣ϰ(t)∣ = 1 for µ-a.e. t ∈ A.
Let x∈ S (L1∩ L∞) . By the definition of the norm ∥⋅∥L1∩L∞, we conclude that∣x(t)∣ ⩽ 1 for µ-a.e. t∈ supp x. Proposition1.1implies that without loss of generality we can assume that x(t) > 0 for µ-a.e. t ∈ supp x. Hence 0 < x(t) ⩽ 1 for µ-a.e. t ∈ supp x.
If µ(supp x) ⩽ 1, then taking A ∈ Σ such that µ(A) = 1 and supp x ⊂ A, we have x= 1
2 χA+ 1
2(2x − χA) = 1 2
e+ 1 2
y.
Obviously, e= χA∈ ext B (L1∩ L∞) and
∥y∥L1∩L∞ = ∥y∥L1∩L∞ = max {∥y∥L1,∥y∥L∞} = ∥y∥L∞
= ∥2x − χA∥L∞= sup ess
t∈A ∣2x(t) − 1∣ = 1,
i.e. y∈ B (L1∩ L∞) . It follows, by the definition of the function λ, that λ(x) ⩾21. Assume that µ(supp x) > 1. Define
An= {t ∈ T ∶ x (t) > x∗(1) + 1 n}
for any n ∈ N. Then (An) is an increasing sequence of sets such that µ(An) ⩽ 1 for any n∈ N. Denote
B=⋃∞
n=1
An
and
C= {t ∈ T ∶ x (t) ⩾ x∗(1)} .
Obviously, µ(B) ⩽ 1 and µ(C) ⩾ 1. Let A ∈ Σ be such that B ⊂ A ⊂ C and µ(A) = 1. Notice that under the assumption that µ(supp x) > 1 we have 0 < x∗(1) < 1. Indeed, if x∗(1) = 0, then µ(supp x) ⩽ 1, a contradiction. If x∗(1) = 1, then x = χA+ x χsupp xÓAand
∥x∥L1∩L∞⩾ ∥χA+ x χsupp xÓA∥L1= 1 + ∫supp xÓAx(t)dµ > 1, contradicting our assumption that x∈ S (L1∩ L∞) .
It is easy to see that x χA⩾ x∗(1)χAand x χT∖A⩽ x∗(1)χT∖A.
Now, take e = χA and λ = min {x∗(1), 1 − x∗(1)} . Consider the following convex combination
x= x χA+ x χT∖A= λe + (1 − λ)y.
Hence
(1 − λ)y = (x − λe) χA+ x χT∖A.
It remains to show that y∈ B (L1∩ L∞) . We have
∥y∥L1= 1
1− λ∥(x − λe) χA+ x χT∖A∥L1
= 1
1− λ(∫A(x(t) − λ) dµ + ∫T∖Ax(t)dµ)
= 1
1− λ(∫Tx(t)dµ − λµ(A)) ⩽ 1
1− λ(1 − λ) = 1 and
∥y∥L∞= 1
1− λ∥(x − λe) χA+ x χT∖A∥L∞
⩽ 1
1− λmax{∥xχA∥L∞− min {x∗(1), 1 − x∗(1)} , x∗(1)}
⩽ 1
1− λmax{1 − min {x∗(1), 1 − x∗(1)} , x∗(1)}
= 1
1− λmax{1 − x∗(1), x∗(1)}
= 1
1− λ(1 − min {x∗(1), 1 − x∗(1)}) = 1
1− λ(1 − λ) = 1.
Consequently,
∥y∥L1∩L∞ = max {∥y∥L1, ∥y∥L∞} ⩽ 1, i.e. y∈ B (L1∩ L∞) . Therefore
λ(x) ⩾ min{x∗(1), 1 − x∗(1)} > 0, which completes the proof.
2.3. Theorem. The space (L1∩ L∞,∥⋅∥L1∩L∞) has the uniform λ-property if and only if µ(T) ⩽ 1.
Proof. Sufficiency. Suppose that µ(T) ⩽ 1. It was stated at the beginning of the proof of Theorem2.2that(L1∩ L∞,∥⋅∥L1∩L∞) has the uniform λ-property.
Necessity. Assume that µ(T) > 1. Let n0 be the smallest positive integer such that nn+1 ⩽ µ(T) for any n ⩾ n0, n∈ N. Let (An) be a decreasing sequence of measurable sets such that µ(An) = nn+1 for any n⩾ n0. For any n⩾ n0, define
xn= n n+ 1χAn. If n⩾ n0, then
∥xn∥L1∩L∞ = n
n+ 1max{∥χAn∥L1,∥χAn∥L∞} = 1,
i.e. xn ∈ S (L1∩ L∞) . Fix n ⩾ n0. Combining Proposition1.1with Corollary 1 from [10], we conclude that every extreme point of B(L1∩ L∞)+is of the form
e= χA,
where A is an arbitrary measurable subset of T such that µ(A) = 1. Since n+ 1
n = µ(An) = µ(An∖ A) + µ(An∩ A) ⩽ µ(An∖ A) + 1, it follows that
µ(An∖ A) ⩾ 1 n . Let
xn= λe + (1 − λ)yn for some λ∈ (0, 1) and yn∈ B (L1∩ L∞) . Then
(1 − λ)yn= ( n
n+ 1− λ) χAn∩A+ n
n+ 1χAn∖A− λχA∖An, and consequently
1− λ ⩾ (1 − λ) ∥yn∥L1∩L∞
= max {∣ n
n+ 1− λ∣ µ(An∩ A) + n
n+ 1µ(An∖ A) + λµ(A ∖ An), max { n n+ 1, λ}}
⩾ max { n
n+ 1, λ} ⩾ n n+ 1. Hence
λ⩽ 1 − n n+ 1 = 1
n+ 1. Now, by the definition of the function λ(⋅), we get
0< λ(xn) ⩽ 1 n+ 1 → 0.
Therefore, the space(L1∩ L∞,∥⋅∥L1∩L∞) fails to have the uniform λ-property when µ(T) >
1.
Replacing the norm∥⋅∥L1∩L∞with the equivalent one∥⋅∥′L1∩L∞, we get a different cha- racterization of the uniform λ-property in L1∩ L∞.
2.4. Theorem. The space(L1∩ L∞,∥⋅∥′L1∩L∞) has the uniform λ-property if and only if µ(T) < ∞.
Proof. By Corollary 3.3 in [5], the space(L1∩ L∞,∥⋅∥′L1∩L∞) has the λ-property.
Sufficiency. Assume that µ(T) < ∞. Let x ∈ S (L1∩ L∞) . By Proposition1.1, without loss of generality we can assume that x(t) ⩾ 0 for µ-a.e t ∈ T. Denote
A= {t ∈ T ∶ x(t) ⩾ 1
2∥x∥L∞}.
Obviously, µ(A) > 0 and A ⊂ supp x. Define h ∈ L1∩ L∞ by the formula h= ∥x∥L∞χA. Setting z= 2x − h, we have
z= (2x − h)χA+ 2x χsupp x∖A= (2x − ∥x∥L∞)χA+ 2x χsupp x∖A. Using the definition of the set A, we conclude that
(2x(t) − ∥x∥L∞) χA(t) ⩾ 0
for a.e. t∈ A. Moreover, ∥x∥L1 = 1 − ∥x∥L∞ whenever x∈ S (L1∩ L∞) . Then
∥z∥′L1∩L∞ = ∥z∥L∞+ ∥z∥L1 = ∥x∥L∞+ ∥z∥L1
= ∥x∥L∞+ ∫A(2x(t) − ∥x∥L∞) dt + 2 ∫supp x∖Ax(t)dt
= ∥x∥L∞+ 2 ∥x∥L1− ∥x∥L∞µ(A)
= 2 (1 − ∥x∥L∞) + ∥x∥L∞(1 + µ(A))
= 2 − ∥x∥L∞(1 + µ(A)) . On the other hand,
∥h∥′L1∩L∞= ∥h∥L∞+ ∥h∥L1= ∥x∥L∞(1 + µ(A)) . Hence
1
2∥h∥′L1∩L∞+ 1
2∥z∥′L1∩L∞ = 1
2∥x∥L∞(1 + µ(A)) + 1
2[2 − ∥x∥L∞(1 + µ(A))]
= 1 = ∥x∥′L1∩L∞, so∥z∥′L1∩L∞ = 2 ∥x∥′L1∩L∞− ∥h∥′L1∩L∞. Denote
y= z
∥z∥′L1∩L∞
and e= h
∥h∥′L1∩L∞
.
Then e , z∈ S (L1∩ L∞) and
e= h
∥h∥′L1∩L∞ = ∥x∥L∞
∥x∥L∞(1 + µ(A))χA= 1 1+ µ(A)χA.
An application of Corollary 6 from [9] shows that e∈ ext B (L1∩ L∞) . Therefore x= 1
2 h+ 1
2 z
= 1
2∥h∥′L1∩L∞
h
∥h∥′L1∩L∞ + 1 2
1− 12∥h∥′L1∩L∞
1− 12∥h∥′L1∩L∞
z
= 1
2∥h∥′L1∩L∞⋅ e + (1 − 1
2∥h∥′L1∩L∞) z 2− ∥h∥′L1∩L∞
= 1
2∥h∥′L1∩L∞⋅ e + (1 − 1
2∥h∥′L1∩L∞) y,
since 2− ∥h∥′L1∩L∞ = ∥z∥′L1∩L∞. Using the definition of the function λ(⋅) and observing that
1= ∥x∥′L1∩L∞ ⩽ ∥x∥L∞(1 + µ(supp x)) , we get
λ(x) ⩾ 1
2∥h∥′L1∩L∞= 1
2∥x∥L∞(1 + µ(A)) ⩾ 1
2⋅ 1+ µ(A)
1+ µ(supp x) ⩾ 1 2(1 + µ(T)). Since x was an arbitrary element of the unit sphere of L1∩ L∞, we conclude that
λL1∩L∞= inf {λ (x) ∶ x ∈ S(L1∩ L∞)} ⩾ 1
2(1 + µ(T)) > 0,
which shows that(L1∩ L∞,∥⋅∥′L1∩L∞) has the uniform λ-property, whenever µ(T) < ∞.
Necessity. Suppose µ(T) = ∞. By divergence of the harmonic series, to every n ∈ N one can find h(n) ∈ N such that
n− 1 ⩽
h(n)
∑i=1
1 i ⩽ n.
Define
xn= 1 n
h(n)
∑i=1
1 i
χT(n) i
,
where(Ti(n)) is a sequence such that for any n ∈ N the sets Ti(n)(i = 1, 2, . . . , h(n)) are pairwise disjoint and µ(Ti(n)) = 1. Hence
∥xn∥′L1∩L∞ = ∥xn∥L1+ ∥xn∥L∞ = 1 n
h(n)
∑i=1
1 i+ 1
n for any n∈ N. According to the definition of h(n), we have
1⩽ ∥xn∥′L1∩L∞ ⩽ 1 + 1 n
(1)
for any n∈ N. For each n ∈ N define
̃xn= xn
∥xn∥′L1∩L∞
.
Since the space(L1∩ L∞,∥⋅∥′L1∩L∞) has the λ-property (see [5, Corollary 3.3]), there are λn∈ (0, 1), yn∈ S(L1∩ L∞), and en∈ ext B(L1∩ L∞) such that
̃xn= λnen+ (1 − λn)yn
for any n∈ N. By the definition of the norm ∥⋅∥′L1∩L∞and Lemma2.1 (ii), en(t) ∈ [0, 1) and yn(t) ∈ [0, 1)
for µ-a.e. t ∈ T and for any n ∈ N. Since en ∈ ext B (L1∩ L∞) (n ∈ N), it follows from Corollary 6 in [9] that en(n = 1, 2, . . .) is of the following form
en= 1
1+ µ(supp en)χsupp en, where µ(supp en) < ∞. Hence
̃xn= 1 n∥xn∥′L1∩L∞
h(n)
∑i=1
1 i
χT(n) i
= λnen+ (1 − λn)yn
⩾ λn
1+ µ(supp en)χsupp en
for any n∈ N. Consequently,
λnχsupp en ⩽ 1+ µ(supp en) n∥xn∥′L1∩L∞
h(n)
∑i=1
1 i
χT(n) i
. (2)
By Lemma2.1 (i), supp en⊂ supp xn, whence µ(supp en) ⩽ h(n) and at least ⌈µ(supp en)⌉
sets of the collection{Ti(n)}hi=1(n)has a non-empty intersection with supp en(⌈⋅⌉ denotes the ceiling function). Let j0be the smallest integer such that j0 ⩾ µ(supp en) and µ(T(n)j0 ∩ supp en) /= ∅. Then, by the inequalities (1) and (2), for µ-a.e. t∈ T(n)j0 ∩ supp enwe have
λn⩽1+ µ(supp en)
n j0∥xn∥′L1∩L∞ ⩽1+ j0 n j0
. Using the definition of the function λ(⋅), we obtain
λL1∩L∞ = inf {λ (x) ∶ x ∈ S(L1∩ L∞)}
⩽ λ(xn) ⩽ 1+ j0
n j0 → 0 as n → ∞,
whence λL1∩L∞ = 0. Therefore, the space (L1∩ L∞,∥⋅∥′L1∩L∞) fails the uniform λ-property when µ(T) = ∞.
Acknowledgements. This research was supported by the grant 04/43/DSPB/0086 from the Polish Ministry of Science and Higher Education.
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