On geometric structure of symmetric spaces

J. Math. Anal. Appl. 430 (2015) 98–125

Contents lists available at ScienceDirect

Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

On geometric structure of symmetric spaces Maciej Ciesielski 1 Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland

a r t i c l e

i n f o

Article history: Received 17 February 2015 Available online 16 April 2015 Submitted by Richard M. Aron Keywords: Lorentz spaces Symmetric spaces Local uniform rotundity Strict K-monotonicity

a b s t r a c t In this article we discuss local approach to strict K-monotonicity and local uniform rotundity in symmetric spaces. We prove several general results on local structure of symmetric spaces E showing relation between strict monotonicity and strict K-monotonicity and the Kadec–Klee property for global convergence in measure. We also present the full criteria for points of upper K-monotonicity in Lorentz spaces Γp,w for degenerated weight function w. Next we characterize local uniform rotundity in symmetric spaces E proving several correspondences between x ∈ E a point of local uniform rotundity and its decreasing rearrangement x∗ and absolute value |x|. Finally, we apply these results to find complete criteria for local uniform rotundity of Lorentz spaces Γp,w . © 2015 Elsevier Inc. All rights reserved.

1. Introduction Recently, the geometric structure of the symmetric space has been investigated expansively by many authors [4–6,10,12,13,19,18]. The studies of the local and global properties in symmetric spaces is a key for many different types of branches of mathematics. Indeed, it has been found many applications of the monotonicity and rotundity properties of symmetric spaces in approximation theory (see [9,10,12,20,27]). It is worth mentioning that the monotonicity properties play a similar significant role in the best dominated approximation problems in Banach lattices as the respective rotundity properties do in the best approximation problems in Banach spaces. Moreover, the structure of symmetric spaces plays a crucial role as interpolation spaces between Banach couple (L1 , L∞ ) in an investigation of measurable operators (for more details see [1,13,26]). The natural question of existence of the monotonicity and rotundity properties is a keyword for many employable issues but not always easily solvable problems in the case of certain degenerated symmetric spaces. In [18], H. Hudzik, A. Kamińska and M. Mastyło presented the full criteria for local uniform rotundity (LUR) and strict K-monotonicity (SKM ) in Orlicz–Lorentz spaces Λφ,w under some additional assumptions on the weight function w and Orlicz function φ. Moreover, in [18], it was proved that

1

E-mail address: [email protected]. This research is supported by the grant 04/43/DSPB/0083 from Polish Ministry of Science and Higher Education.

http://dx.doi.org/10.1016/j.jmaa.2015.04.040 0022-247X/© 2015 Elsevier Inc. All rights reserved.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

99

a symmetric rotund and K-monotone space is strictly K-monotone. In [30], A. Medzhitov and P. Sukochev researched in a symmetric space E a correspondence between LUR property and LUR∗ property, that is local uniform rotundity considered on E ∗ the cone of nonnegative and nonincreasing functions of E. It is worth noticing that LUR property with corresponding DGL property was researched intensively in K-interpolation spaces in [15,16] by P.G. Dodds, T.K. Dodds, A.A. Sedaev and F.A. Sukochev. The next interesting results were published in [4], where authors discussed for a symmetric space E a relation between the global version of some rotundity properties P on E and its restricted version to E ∗ . Let’s recall that in [6, Proposition 2.1] it was proved that a separable symmetric space E with the Kadec–Klee property is strictly K-monotone. Moreover, authors in [6] showed that in separable Lorentz spaces Λφ , strict K-monotonicity is equivalent to the Kadec–Klee property (see Theorem 2.11). Recently, in [11] it has been discussed some sufficient and necessary conditions for strict K-monotonicity of some important concrete symmetric spaces. Furthermore, in [11] authors have presented an example of symmetric spaces which are strictly K-monotone and are not rotund. The recent studies of the monotonicity and rotundity properties in certain symmetric spaces have given a positive answer of the precise descriptions of the wanted properties although it required technical proofs (see [8,17,19,25]). In the spirit of finding an applicable local geometry, in this article it is characterized the local version of strict K-monotonicity and local uniform rotundity of symmetric spaces and organized as follows. In Section 2 we recall the necessary notions. In Section 3 we present several results in symmetric spaces devoted to local approach of strict K-monotonicity, strict monotonicity and the Kadec–Klee property for global convergence in measure. Namely, we investigate in a symmetric space a relation between a point of upper (lower) K-monotonicity and a point of upper (lower) monotonicity, respectively. We also show a characterization of a point of upper local uniform K-monotonicity in terms of points of upper K-monotonicity and the Kadec–Klee property for global convergence in measure. In Section 4 we characterize the complete criteria for a point of upper K-monotonicity in Lorentz spaces Γp,w with the degenerated weight function w. Section 5 is devoted to an investigation of a point of local uniform rotundity in symmetric spaces. We answer the essential question whether a point of local uniform rotundity can be equivalently considered only on the positive cone E + of a Banach function space E. This problem is proved without any additional assumption and shows that x ∈ E is a point of local uniform rotundity if and only if |x| is an LUR point in E. We also deliberate the local version of a relation between local uniform rotundity of a symmetric space E and its positive cone E ∗ , of all decreasing rearrangement x∗ of x ∈ E. Namely, we prove under some additional assumptions that a point x is an LUR point if and only if its decreasing rearrangement x∗ is an LUR point. In the final result of this section we improve the answer for well known essential question whether LUR property can be researched only on E ∗ . This problem was establish previously under some additional conditions. In this significant problem we show that a symmetric space E is LUR if and only if the cone E ∗ is LUR. The intention of the last Section 6 is to find a complete criteria of local uniform rotundity in Lorentz spaces Γp,w . Applying obtained results in the previous sections for local approach of strict K-monotonicity and LUR property in symmetric spaces we investigate the equivalent conditions for local uniform rotundity in particular classes of Lorentz spaces Γp,w . We show the class of Banach spaces constructed by the K-method of interpolation which possess LUR property. Finally, we present the necessary conditions for a point of local uniform rotundity in Lorentz spaces Γp,w with the degenerated weight function w.

100

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

2. Preliminaries Let R and N be the sets of reals and positive integers, respectively. As usual S(X) (resp. B(X)) stands for the unit sphere (resp. the closed unit ball) of a Banach space (X, ·X ). Denote by L0 the set of all (equivalence classes of) extended real-valued Lebesgue measurable functions on [0, α), where α = 1 or α = ∞. A Banach lattice (E,  · E ) is called a Banach function space (or a Köthe space) if it is a sublattice of L0 satisfying the following conditions: (1) If x ∈ L0 , y ∈ E and |x| ≤ |y| a.e., then x ∈ E and xE ≤ yE . (2) There exists a strictly positive x ∈ E. By E + we denote the positive cone of E, i.e. E + = {x ∈ E : x ≥ 0}. We use the notation Ac = [0, α)\A for any measurable set A. By μ denote the Lebesgue measure on [0, α). A point x ∈ E is said to be an order continuous point if for any sequence (xn ) ⊂ E + such that xn ≤ |x| and xn → 0 a.e. we have xn E → 0. A Köthe space E is called order continuous (shortly E ∈ (OC )) if every element x of E is an order continuous point (see [28]). Unless we say otherwise, we assume in the whole paper that E has Fatou property, i.e. if (xn ) ⊂ E + , supn∈N xn E < ∞ and xn ↑ x ∈ L0 , then x ∈ E and xn E ↑ xE . A point x ∈ E + \ {0} is called a point of upper monotonicity shortly a UM point (a point of lower monotonicity shortly an LM point) if for any y ∈ E + such that x ≤ y and y = x (y ≤ x and y = x), we have xE < yE (yE < xE ). A point x ∈ E + is said to be a point of upper local uniform monotonicity (lower local uniform monotonicity) shortly a ULUM point (an LLUM point) if for any sequence (xn ) ⊂ E such that x ≤ xn and xn E → xE (0 ≤ xn ≤ x and xn E → xE ), it follows that xn − xE → 0. Recall that if each x ∈ E + \ {0} is a UM point or equivalently if each x ∈ E + \ {0} is an LM point, then we say that E is strictly monotone (E ∈ (SM )) (see [2,19]). Similarly, if each point of E + \ {0} is a ULUM point (an LLUM point), then we say that E is upper locally uniformly monotone shortly E ∈ (ULUM ) (lower local uniform monotone shortly E ∈ (LLUM )). A point x ∈ E is said to be a point of local uniform rotundity (shortly an LUR point) if for any sequence (xn ) ⊂ E such that xn + xE → 2 xE and xn E → xE we have xn − xE → 0. A point x ∈ E is said to be a point of midpoint local uniform rotundity (shortly an MLUR point) if for every sequence (xn ) ⊂ E if xn + xE → xE and xn − xE → xE , then xn E → 0. A Banach function space E is said to be locally uniformly rotund shortly E ∈ (LUR) (midpoint locally uniformly rotund shortly E ∈ (MLUR)) if each point x ∈ E is an LUR point (an MLUR point) in E, respectively. A point x ∈ E is said to be an Hg point in E if for any (xn ) ⊂ E such that xn → x globally in measure and xn E → xE , we have xn − xE → 0. We say that the space E has the Kadec–Klee property for global convergence in measure if each x ∈ E is an Hg point in E. For any function x ∈ L0 we define its distribution function by dx (λ) = μ {s ∈ [0, α) : |x (s)| > λ} ,

λ ≥ 0,

and its decreasing rearrangement by x∗ (t) = inf {λ > 0 : dx (λ) ≤ t} ,

t ≥ 0.

We say that a function x ∈ L0 satisfies condition (+) if there exists τ1 ≥ 0 such that |x(t)| ≥ τ1 for a.e. t ∈ [0, α) and d|x|−τ1 (τ ) < ∞ for any τ > 0 [21], i.e. μ(t ∈ [0, α) : |x(t)| < x∗ (∞)) = 0. The above notion is stated under the convention x∗ (∞) = limt→∞ x∗ (t) if α = ∞ and x∗ (∞) = 0 if α = 1.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

101

Given x ∈ L0 we denote the maximal function of x∗ by 1 x (t) = t ∗∗

t

x∗ (s)ds.

0

It is well known that x∗ ≤ x∗∗ , x∗∗ is decreasing and subadditive, i.e. (x + y)∗∗ ≤ x∗∗ + y ∗∗ for any x, y ∈ L0 . For more properties of dx , x∗ and x∗∗ see [1,26]. Two functions x, y ∈ L0 are said to be equimeasurable (shortly x ∼ y) if dx = dy . A Banach function space (E,  · E ) is called rearrangement invariant (r.i. for short) or symmetric if whenever x ∈ L0 and y ∈ E with x ∼ y, then x ∈ E and xE = yE . Given an r.i. Banach function space E let φE denote its fundamental function, that is φE (t) = χ(0,t) E for any t ∈ [0, α) (see [1]). The notion ≺ is defined for any x, y in L1 + L∞ by x ≺ y ⇔ x∗∗ (t) ≤ y ∗∗ (t) for all t > 0. Let 0 < p < ∞ and w ∈ L0 be a nonnegative weight function, then the Lorentz space Γp,w is a subspace of L0 such that

xΓp,w

⎛ α ⎞1/p  := ⎝ x∗∗p (t)w(t)dt⎠ < ∞. 0

Additionally, we assume that w is from class Dp , i.e. s

α w(t)dt < ∞

W (s) :=

p

and Wp (s) := s

t−p w(t)dt < ∞

s

0

for all 0 < s ≤ 1 if α = 1 and for all 0 < s < ∞ otherwise. These two conditions guarantee that Lorentz   space Γp,w is nontrivial. It is well known that Γp,w ,  · Γp,w is an r.i. quasi-Banach function space with Fatou property. It was proved in [23] that in the case when α = ∞ the space Γp,w has order continuous ∞ norm if and only if 0 w (t) dt = ∞. The spaces Γp,w were introduced by A.P. Calderón in [3] in a similar way as the classical Lorentz spaces Λp,w that is a subspace of L0 with ⎛ xΛp,w = ⎝

α

⎞1/p x∗p (t)w(t)dt⎠

< ∞,

0

where p ≥ 1 and the weight function w is nonnegative and nonincreasing (see [29]). The space Γp,w is an interpolation space between L1 and L∞ yielded by the Lions–Peetre K-method [1,26]. Clearly, Γp,w ⊂ Λp,w . The opposite inclusion Λp,w ⊂ Γp,w is satisfied if and only if w ∈ Bp (see [23]). It is worth mentioning that the spaces Γp,w and Λp,w are also connected by Sawyer’s result (Theorem 1 in [32]; see also [33]), which ∞ states that the Köthe dual of Λp,w , for 1 < p < ∞ and 0 w(t)dt = ∞, coincides with the space Γp ,w ,  p t where 1/p + 1/p = 1 and w(t)

= t/ 0 w(s)ds w(t).

102

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

It is easy to see that for α = 1 by the Lebesgue Dominated Convergence Theorem, Γp,w is order continuous. For more details about the properties of Γp,w the reader is referred to [23]. A symmetric space E is called K-monotone (shortly E ∈ (KM )) if for any x ∈ L1 + L∞ and y ∈ E such that x ≺ y, we have x ∈ E and xE ≤ yE . Recall that a symmetric space is K-monotone if and only if E is exact interpolation space between L1 and L∞ . It is well known that a symmetric spaces with Fatou property or with an order continuous norm is K-monotone (see [26]). For any global property P and any symmetric space E the notation E ∈ (P )∗ means that E satisfies (P )∗ , i.e. the cone E ∗ = {x∗ : x ∈ E} of all decreasing rearrangements of functions in E satisfies P property. Clearly, if E ∈ (P ), then E ∈ (P )∗ . The natural question of the opposite conclusion was investigated by many authors in [4,10,12,30]. A point x ∈ E is a point of upper K-monotonicity (lower K-monotonicity) shortly a UKM point (an LKM point) of E if and only if for any y ∈ E, x∗ = y ∗ with x∗∗ ≤ y ∗∗ (with y ∗∗ ≤ x∗∗ ), we have xE < yE (yE < xE ), respectively. Recall that a symmetric space E is said to be strictly K-monotone (shortly E ∈ (SKM )) if every point of E is a UKM point or equivalently if every point of E is an LKM point. A point x ∈ E is said to be a point of upper local uniform K-monotonicity of E (shortly a ULUKM point) if for any sequence (xn ) ⊂ E such that x∗∗ ≤ x∗∗ n for all n ∈ N and xn E → xE we have x∗ − x∗n E → 0. A point x ∈ E is called a point of lower local uniform K-monotonicity of E (shortly an ∗∗ LLUKM point) if and only if for any (xn ) ⊂ E with x∗∗ for all n ∈ N and xn E → xE we have n ≤ x ∗ ∗ x − xn E → 0. Recall that a symmetric space E is called upper locally uniformly K-monotone shortly E ∈ (ULUKM ) (lower locally uniformly K-monotone shortly (E ∈ (LLUKM )) whenever each point of E is a ULUKM point (an LLUKM point), respectively. For more details the reader is referred to [6,18,11]. Remark 2.1. It is well known by the definition of LUR and LLUM points that for any symmetric space E if x ∈ E is an LUR point, then |x| is an LLUM point. Hence, by Theorem 2.1 in [10] it follows that x is a point of order continuity. 3. Local K-monotonicity In this section we find a relation between local version of strict monotonicity and strict K-monotonicity and also the Kadec–Klee property for global convergence in measure. We start our discussion with some auxiliary result for a point of upper K-monotonicity which might be of interest independently for its application. Remark 3.1. It is easy to notice that for any symmetric space E an element x ∈ E is an LKM point (a UKM point) of E if and only if its decreasing rearrangement x∗ or |x| is an LKM point (a UKM point) of E, respectively. Analogously, it can be established a characterization of ULUKM points and LLUKM points in a symmetric space E. Moreover, it is clear that if x ∈ E is a ULUKM point (an LLUKM point) of E, then x is also a UKM point (an LKM point) of E, respectively. Theorem 3.2. Let E be a symmetric space and x, xn ∈ E for any n ∈ N. If x is a UKM point and x∗∗ ≤ x∗∗ n ∗∗ ∗ ∗ and xn E → xE , then x∗∗ → x and x → x in measure. n n Proof. Since xn E → xE , there exists M > 0 such that xn E ≤ M for any n ∈ N. Since E is symmetric space we have for any n ∈ N and t > 0 (see [1]) x∗n (t)φ(t) ≤ x∗∗ n (t)φ(t) ≤ xn E ≤ M.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

103

Therefore, by Helly’s selection principle passing to subsequence and relabeling if it is necessary there is z ∈ E such that z = z ∗ and x∗n (t) converges to z ∗ (t) for a.a. t ∈ [0, α). Now we show that z ∗ = x∗ . It is obvious that for all 0 < t < s < α, s

x∗n

s →

t

z∗.

(1)

x∗

(2)

t

We claim that t

x∗n

t →

0

0

for any t > 0. Indeed, if it is not true then there exist τ ∈ (0, α),  > 0 and a subsequence (x∗nk ) ⊂ (x∗n ) such that τ

∗

τ

x +≤ 0

x∗nk .

0

By Corollary 2.2 in [6] there is y = y ∗ ∈ E such that x∗∗ ≤ y ∗∗ ≤ x∗∗ nk and τ

∗

τ

x +≤ 0

y∗ .

0

By the last inequality we get x∗ = y ∗ and by assumption that x is a UKM point it follows that xE < yE ≤ xnk E . Thus, by assumption that xn E → xE we conclude a contradiction which proves condition (2). Now according to (1) we obtain x∗ = z ∗ a.e. on [0, α). Hence, we have x∗n → x∗

for a.a. t ∈ [0, α).

(3)

∗∗ Moreover, by condition (2) and by assumption x∗∗ ≤ x∗∗ n and by monotonicity of the maximal function x ∗∗ ∗∗ ∗ it is easy to show that xn converges to x in measure. Now according to condition (3), xn converges to x∗ locally in measure. Therefore, for δ > 0 and s < α,

μ (t ∈ [0, s] : |x∗n (t) − x∗ (t)| > δ) → 0. Clearly, it is enough to consider that α = ∞. Now we continue the proof in two cases. Case 1. Let x∗ (∞) = 0. By (3) and since x∗n is decreasing, for any  > 0 there is 0 < t < α and N ∈ N such that for all n ≥ N , sup |x∗n (t) − x∗ (t)| ≤ .

t≥t

Consequently, μ (t ∈ [t , α) : |x∗n (t) − x∗ (t)| > ) → 0, whence x∗n → x∗ in measure. Case 2. Let x∗ (∞) > 0. Since x∗n is decreasing, by (3), we claim that x∗n (t) ≥ x∗ (∞) for any t ∈ (0, α) and n ∈ N. Indeed, assume that there exists (nk ) ⊂ N such that x∗nk (tk ) < x∗ (∞) for some tk ∈ (0, α). Then

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

104

for every k ∈ N and t ≥ tk we have x∗nk (t) < x∗ (∞). Therefore, for any k ∈ N there is k ∈ (0, 1) such that for any t ≥ tk , x∗ (t) ≥ x∗ (∞) > x∗nk (tk ) + k ≥ x∗nk (t) + k . Hence, for any s > 0 we have t k +s

t k +s

∗

x (t)dt ≥

t k +s

∗

(x∗nk (t) + k )dt

x (∞)dt ≥

tk

tk

tk

t k +s

x∗nk (t)dt + sk

=

(4)

tk

for every k ∈ N. Clearly, by assumption we get t k +s

x∗nk (t)dt

tk +

tk

x∗nk (t)dt

t k +s

x∗nk (t)dt

≥

tk +

tk

0

x∗ (t)dt.

(5)

0

Observe that for any k ∈ N there is 0 < sk < ∞ such that for each s ≥ sk , tk

x∗nk (t)dt

tk < sk +

0

x∗ (t)dt.

0

Consequently, by assumption and by conditions (4) and (5) for any s ≥ sk it follows that t k +s

tk

∗

x (t)dt ≥ 0

t k +s

∗

x∗nk (t)dt + sk

x (t)dt + tk

0

tk >

x∗nk (t)dt

t k +s

x∗nk (t)dt

+ tk

0 t k +s

x∗nk (t)dt

= 0

t k +s

x∗ (t)dt.

≥ 0

Hence we obtain a contradiction, which provides the claim. Now define zn (t) = x∗n (t) − x∗ (∞)

and z(t) = x∗ (t) − x∗ (∞)

for every n ∈ N. Clearly, z = z ∗ , z(∞) = 0 and zn = zn∗ for any n ∈ N. Moreover, by condition (3) we have zn∗ (t) = x∗n (t) − x∗ (∞) → x∗ (t) − x∗ (∞) = z ∗ (t) for a.a. t ∈ [0, α). Now applying case 1 we obtain zn∗ → z ∗ in measure, so also x∗n → x∗ in measure. Finally, compounding both cases, in view of Lemma 2.3 in [6] we complete the proof. 2 Lemma 3.3. Let E be a symmetric space. If x ∈ E + is a UKM point and satisfies condition (+), then x is a UM point.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

105

Proof. Let x ≤ y and x = y where y ∈ E + . Since x satisfies condition (+), there is τ1 = x∗ (∞) ≥ 0 such that |x| ≥ τ1 and d(x−τ1 ) (τ ) < ∞ for all τ > 0. By Lemma 2.1 in [10] there exists a positive measure set B such that x∗ (t) < y ∗ (t) for any t ∈ B. Clearly, x∗ ≤ y ∗ and consequently x∗∗ ≤ y ∗∗ . Therefore, by assumption we have xE < yE , which implies that x is a UM point. 2 The immediate consequence of the previous theorem is the following corollary. Corollary 3.4. Let E be a symmetric space. If x ∈ E + is a UKM point, then x∗ is a UM point. The immediate consequence of Theorem 2.2 in [10] and Lemma 3.3 is the following result. Corollary 3.5. Let E be a symmetric space and x ∈ E + be a UKM point and an Hg point and satisfy condition (+), then x is a ULUM point. Lemma 3.6. Let E be a symmetric space. If x ∈ E + is an LKM point and μ(t ∈ [0, α) : 0 < x(t) ≤ x∗ (∞)) = 0, then x is an LM point. Proof. Let y ≤ x and x = y where y ∈ E + . So, there exists a positive measure set A ⊂ [0, α) such that y(t) < x(t) for all t ∈ A. It is easy to see that A ⊂ supp(x). If x∗ (∞) = 0, then by Lemma 3.2 in [22] we obtain y ∗ = x∗ , by the inequality y ∗∗ ≤ x∗∗ , and by assumption that x is an LKM point we get yE < xE . Now consider x∗ (∞) > 0. Define B = {t : y(t) ≥ x∗ (∞)} and

z=

yχA∩B + xχsupp(x)\(A∩B) ,

if μ(A ∩ B) > 0,

x∗ (∞)χA + xχsupp(x)\A ,

if μ(A ∩ B) = 0.

Then, z = x and y ≤ z ≤ x and also μ(t : 0 < z(t) < x∗ (∞)) = 0. Hence, by assumption μ(t : 0 < x(t) ≤ x∗ (∞)) = 0 and by Lemma 2.1 in [10] we obtain z ∗ = x∗ . Consequently, by the inequality y ∗∗ ≤ z ∗∗ ≤ x∗∗ and by assumption that x is an LKM point we have yE ≤ zE < xE , which implies that x is an LM point. 2 Example 3.7. Now we show that the opposite conclusion in Lemmas 3.3 and 3.6 are not true in general. Observe that the condition (+) and the condition μ(t : 0 < x(t) ≤ x∗ (∞)) = 0 are necessary for x ∈ E + to be a UM point and an LM point, respectively (for more details see [10]). We claim that Lemmas 3.3 and 3.6 failed in the case when the condition (+) and the condition μ(t : 0 < x(t) ≤ x∗ (∞)) = 0 are t omitted, respectively. Indeed, consider the Lorentz space Γp,w with W (t) = 0 w strictly increasing. Define x = χ[1,∞) . Then x∗ (∞) = 1 and μ(t : x(t) < x∗ (∞)) = 1 as well as μ(t : 0 < x(t) ≤ x∗ (∞)) = ∞. By Lemmas 2.3 and 2.4 in [10] it follows that x is no LM point and no UM point in Γp,w . Moreover, by Theorem 12 in [11], the Lorentz space Γp,w is strictly K-monotone, so x is an LKM point and a UKM point. Theorem 3.8. Let E be a symmetric space and x, xn ∈ E with x∗ (∞) = 0 and let: (i) x∗ be an Hg point and a UKM point. ∗∗ (ii) x be a UKM point and if x∗∗ in measure and xn E → xE , then x∗n − x∗ E → 0. n →x (iii) x be a ULUKM point. Then (i) ⇒ (ii) ⇒ (iii). If x∗ is a point of order continuity then (iii) ⇒ (i).

106

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

∗∗ Proof. (i) ⇒ (ii). Let x, xn ∈ E for n ∈ N, x∗∗ in measure and xn E → xE . Then, by property 11o n →x ∗∗ ∗∗ in [26] we get that (xn ) converges to x at each point of continuity of x∗∗ and so

t

(x∗n − x∗ )(s)ds → 0 as n → ∞

β

for all t > β > 0. Denote c = supn∈N xn E . Since xn E converges to xE , we get for any n ∈ N and for any t > 0, x∗n (t)φ(t) ≤ xn E ≤ c. By Helly’s selection principle, passing to subsequence and relabeling if it is necessary, we may assume x∗n converges to a decreasing function z a.e. on [0, α). It is easy to notice that t

(x∗n − z ∗ )(s)ds → 0 as n → ∞

β

for every 0 < β < t. Consequently, for any t > β > 0 we have t

x∗ − z ∗ = 0,

β

which concludes x∗ = z ∗ a.e. on [0, α). Hence, we obtain that x∗n converges to x∗ a.e. on [0, α). By assumption that x∗ (∞) = 0 we get that x∗n converges to x∗ in measure on [0, α). Therefore, by assumption that x∗ is an Hg point and xn E → xE we have x∗n − x∗ E → 0. ∗∗ ∗∗ (ii) ⇒ (iii). Assume x∗∗ ≤ x∗∗ in n and xn E → xE . Immediately, by Theorem 3.2 we have xn → x measure. Consequently, by condition (ii) we finish the proof. (iii) ⇒ (i). Clearly, by Remark 3.1 we obtain that x∗ is a UKM point. Now we show that x∗ is an Hg point in E. Assume x, xn ∈ E, xn → x∗ in measure and xn E → xE . Since x∗n converges to x∗ a.e. on [0, α) (see [26]) it follows that sup{(x∗ − x∗k ) } ↓ 0. +

k≥n

Hence, by order continuity of x∗ we obtain       ∗ (x − x∗n )+  ≤ sup{(x∗ − x∗k )+ } ↓ 0. k≥n  E

(6)

E

By the inequalities x∗n ≤ max{x∗n , x∗ } ≤ x∗n + sup(x∗ − x∗k )+ k≥n

for all n ∈ N and by condition (6) we conclude max{x∗n , x∗ }E → xE .

(7)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

107

∗∗

Clearly, x∗ ≤ max{x∗n , x∗ } and so x∗∗ ≤ (max{x∗n , x∗ }) . Consequently, by condition (7) and by assumption (iii) we get x∗ − max{x∗n , x∗ }E → 0. Since max{x∗n , x∗ } − x∗ E = (x∗n − x∗ )+ E for any n ∈ N we have  ∗  (xn − x∗ )+  → 0. E Thus, by condition (6) we obtain x∗n − x∗ E → 0. Hence, by assumption that xn converges to x∗ in measure, by order continuity of x∗ and by Theorem 2.4 in [13] we conclude xn − x∗ E → 0, which proves that x∗ is an Hg point and completes the proof. 2 4. Points of upper K-monotonicity in Lorentz space Γp,w Proposition 4.1. Let 1 ≤ p < ∞, w be a weight function such that w = 0 on (a, b) and W is strictly increasing on [0, α) \ (a, b). If an element x ∈ Γp,w is a UKM point, then   μ t ∈ (a, b) : x∗ (t) = x∗ (a− ) or x∗ (t) = x∗ (b) = b − a. Proof. Let x ∈ Γp,w be a UKM point. Suppose for a contrary that   μ t ∈ (a, b) : x∗ (t) = x∗ (a− ) or x∗ (t) = x∗ (b) < b − a. Then there exists t ∈ (a, b) such that x∗ (b) < x∗ (t) < x∗ (a− ). Now we continue the proof in two cases. Case 1. Consider γ, β ∈ (a, b) such that x∗ (b) ≤ x∗ (β) < x∗ (γ) ≤ x∗ (a− ). By the right-continuity of the decreasing rearrangement x∗ there is ξ ∈ (γ, β) such that x∗ (β) < x∗ (ξ) ≤ x∗ (γ). Now we discuss case 1 in two parts. Part 1. Assume that x∗ (β) < x∗ (ξ) < x∗ (γ). Denote ξ

∗

∗

β

(x (γ) − x ),

L= γ

P =

(x∗ − x∗ (β)).

ξ

By right-continuity and monotonicity of x∗ and since ξ ∈ (γ, β) we may assume without loss of generality that L > 0 and P > 0. If P ≥ L then, by monotonicity and continuity of the maximal function x∗∗ , we are able to find η ∈ [ξ, β) such that β L= η

Define

(x∗ − x∗ (β)).

(8)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

108

y = x∗ χ[0,γ]∪[ξ,η)∪[β,α) + x∗ (γ)χ(γ,ξ) + x∗ (β)χ[η,β) . First observe that y = y ∗ and x∗ = y ∗ . We claim that x∗∗ ≤ y ∗∗ and xΓp,w = yΓp,w . Notice that if t ∈ (0, γ], then by the definition of y we get x∗∗ (t) = y ∗∗ (t). If t ∈ (γ, ξ], then by monotonicity of x∗ we obtain 1 y (t) = t ∗∗

t

(x∗ χ[0,γ]∪[ξ,η)∪[β,α) + x∗ (γ)χ(γ,ξ) + x∗ (β)χ[η,β) )

0

1 = t

t

1 (x χ[0,γ] + x (γ)χ(γ,ξ) ) = t ∗

∗

0 ∗∗

t

1 x + t ∗

0

t

(x∗ (γ) − x∗ )χ(γ,ξ)

0

≥ x (t). In the case when t ∈ (ξ, η], x∗ (t) = y ∗ (t). In view of the previous inequality, it follows that y ∗∗ (t) ≥ x∗∗ (t) for any t ∈ (ξ, η]. If t ∈ (η, β], then by the definition of L and by (8) we have 1 y (t) = t ∗∗

t

(x∗ χ[0,γ]∪[ξ,η) + x∗ (γ)χ(γ,ξ) + x∗ (β)χ[η,β] )

0

1 = t

t

1 x + t ∗

0

t

t

1 (x (γ) − x )χ(γ,ξ) + t ∗

∗

0

= x∗∗ (t) +

(x∗ (β) − x∗ )χ[η,β]

0

L 1 − t t

t

(x∗ − x∗ (β)) ≥ x∗∗ (t).

η

Hence, since x∗ = y ∗ on (β, α), it is easy to see that x∗∗ (t) = y ∗∗ (t) for any t ∈ (β, α). Consequently, by assumption w = 0 on (a, b) and since [γ, β] ⊂ (a, b) we conclude a p yΓp,w

=

y 0

∗∗p

α w+

y

∗∗p

a w=

x

∗∗p

α

0

b

x∗∗p w = xΓp,w . p

w+ b

Now considering P < L we may choose η ∈ (γ, ξ) such that ξ P =

(x∗ (η) − x∗ ).

(9)

η

Next assuming y = x∗ χ[0,η]∪[β,α) + x∗ (η)χ(η,ξ) + x∗ (β)χ[ξ,β) , it is easy to show that x∗ = y ∗ and yΓp,w = xΓp,w . Now we prove that y ∗∗ ≥ x∗∗ . By the definition of y we have y ∗∗ (t) = x∗∗ (t) for any t ∈ (0, η]. In the case when t ∈ (η, ξ], then by monotonicity of x∗ it follows that y ∗∗ (t) =

1 t

t 0

(x∗ χ[0,η] + x∗ (η)χ(η,ξ) ) = x∗∗ (t) +

1 t

t 0

(x∗ (η) − x∗ )χ(η,ξ) ≥ x∗∗ (t).

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

109

If t ∈ (ξ, β], then by assumption P > 0 and by (9) we conclude 1 y (t) = t ∗∗

t

(x∗ χ[0,η] + x∗ (η)χ(η,ξ) + x∗ (β)χ[ξ,β] )

0

1 = x (t) + t ∗∗

t

1 (x (η) − x )χ(η,ξ) + t ∗

∗

0

= x∗∗ (t) +

1 P − t t

t

(x∗ (β) − x∗ )χ[ξ,β]

0

t

(x∗ − x∗ (β))χ[ξ,β] ≥ x∗∗ (t).

0

Using the above inequality, since x∗ = y ∗ on (β, α), it is easy to notice that y ∗∗ (t) = x∗∗ (t) for any t ∈ (β, α). Finally, combining the cases when P ≥ L and P < L, by assumption that x is a UKM point, we obtain a contradiction. Part 2. Now we may assume that for any s ∈ (a, b) we have either x∗ (s) = x∗ (γ) or x∗ (s) = x∗ (β), otherwise we proceed as in part 1. Denote η = inf{s ∈ (γ, β) : x∗ (s) = x∗ (β)}. Then x∗ (γ) = x∗ (η − ) > x∗ (η) = x∗ (β). If x∗ (a− ) > x∗ (a) = x∗ (γ), then we define δ = (η+a)/2 and  = min{x∗ (a− )−x∗ (a), x∗ (η − )− x∗ (η)} and also y = x∗ + χ[a,δ) − χ[δ,η) . Otherwise, if x∗ (β) = x∗ (b− ) > x∗ (b), then we take δ = (η + b)/2,  = min{x∗ (b− ) − x∗ (b), x∗ (η − ) − x∗ (η)} and y = x∗ + χ[η,δ) − χ[δ,b) . Clearly, x∗ = y ∗ , y ∗ = y. Since w = 0 on (a, b), by the definition of y it follows that yΓp,w = xΓp,w . Furthermore, it is easy to see that x∗∗ ≤ y ∗∗ . Finally, by assumption that x is a UKM point, we get a contradiction. Case 2. Now assume that for all t ∈ (a, b) we have x∗ (t) = c where x∗ (b) < c < x∗ (a− ). Let  = 1 a+b ∗ − ∗ 2 min{x (a ) − c, c − x (b)} and δ = 2 . Define y = x∗ χ[0,a)∪[b,α) + (c + )χ[a,δ) + (c − )χ[δ,b) . Notice that y = y ∗ = x∗ . We claim that y ∗∗ ≥ x∗∗ . Clearly, taking t ∈ (0, a], by the definition of y, we obtain x∗∗ (t) = y ∗∗ (t). Next, for any t ∈ (a, δ] we get 1 y (t) = t ∗∗

t

1 (x χ[0,a) + (c + )χ[a,δ) ) = x (t) + t ∗

∗∗

0

t

(c +  − x∗ )χ[a,δ)

0

= x∗∗ (t) +

(t − a) ≥ x∗∗ (t). t

Furthermore, if t ∈ (δ, b], then by the definition of δ we have y ∗∗ (t) =

1 t

t 0

(x∗ χ[0,a)∪[b,α) + (c + )χ[a,δ) + (c − )χ[δ,b) )

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

110

1 = x (t) + t ∗∗

t

1 (c +  − x )χ[a,δ) + t ∗

0

= x∗∗ (t) +

t

(c −  − x∗ )χ(δ,b)

0

(b − t) ≥ x∗∗ (t). t

Thus, we can easily show that x∗∗ (t) = y ∗∗ (t) for t ∈ (b, α), which proves the claim. Therefore, by assumption w = 0 a.e. on (a, b) it follows that yΓp,w = xΓp,w . Finally, by assumption that x is a UKM point we conclude a contradiction. 2 Proposition 4.2. Let 1 ≤ p < ∞, w be a weight function such that w = 0 on (a, b) and W is strictly increasing on [0, α) \ (a, b) and let x ∈ Γp,w . If the following condition   μ t ∈ (a, b) : x∗ (t) = x∗ (a− ) or x∗ (t) = x∗ (b) = b − a holds, then x is a UKM point. Proof. Let the assumption be satisfied and let y ∈ Γp,w be such that y ∗ = x∗ and x∗∗ ≤ y ∗∗ . Then, there exist t > 0 and  > 0 such that t +

x∗ ≤

0

t

y∗ .

(10)

0

If t ∈ [0, α) \ [a, b] then by continuity of the maximal function there is δ > 0 such that for any s ∈ (t − δ, t + δ) ⊂ [0, α) \ [a, b] we have  + 2

s

∗

s

x ≤ 0

y∗ .

(11)

0

Clearly, by assumption that W is strictly increasing on [0, α) \ (a, b) it follows that μ((c, d) ∩ supp(w)) > 0 for any (c, d) ⊂ [0, α)\(a, b). Therefore, by condition (11) and by superadditivity of the power function up for p ≥ 1 we get t+δ t+δ  p ∗∗p y (s)w(s)ds ≥ w(s)ds x∗∗ (s) + 2s t−δ

t−δ

t+δ t+δ p  w(s) ∗∗p ≥ x (s)w(s)ds + ds (2s)p t−δ

t−δ

t+δ

x∗∗p (s)w(s)ds.

> t−δ

Hence, since x∗∗ ≤ y ∗∗ we conclude yΓp,w > xΓp,w . Now assume that t = a or t = b. Then, replacing the interval (t − δ, t + δ) by the interval (a − δ, a) for t = a and by (b, b + δ) for t = b we can show analogously yΓp,w > xΓp,w . Now suppose that t ∈ (a, b). Without loss of generality we may assume x∗∗ = y ∗∗

on [0, α) \ (a, b),

which concludes that y ∗ = x∗ a.e. on [0, a]. We present the proof in two cases.

(12)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

111

Case 1. Assume that x∗ is constant on (a, b). Then, since y ∗ is decreasing on (a, b) and x∗∗ (b) = y ∗∗ (b) and by condition (10), there exists t0 ∈ (a, b) such that x∗ (t0 ) > y ∗ (t0 ).

(13)

x∗ (s) = x∗ (b).

(14)

We claim that for any s ∈ (a, b),

Indeed, considering the case when x∗ (s) = x∗ (a− ) for all s ∈ (a, b), since x∗ = y ∗ a.e. on [0, a], by condition (10) we get a contradiction. Now according to conditions (13) and (14) we have for all s ∈ [t0 , b], y ∗ (s) ≤ y ∗ (t0 ) < x∗ (t0 ) = x∗ (b), which implies y ∗ (b) < x∗ (b). Thus, by right-continuity of the decreasing rearrangement there is δ > 0 and 0 > 0 such that for all s ∈ [b, b + δ], x∗ (s) > y ∗ (s) + 0 . Hence, by condition (12) we obtain b+δ b+δ b+δ b+δ  b  b   ∗ ∗ ∗ ∗ ∗ y = x + x > y + (y + 0 ) = 0 δ + y∗ , 0

0

b

0

b

0

which gives a contradiction. Case 2. Now we consider that for any t ∈ (a, b) either x∗ (t) = x∗ (a− ) or x∗ (t) = x∗ (b). Denote c = inf{s ∈ (a, b) : x∗ (s) = x∗ (b)}. Clearly, a < c < b. By right-continuity of the decreasing rearrangement x∗ we have x∗ (c− ) > x∗ (c). Furthermore, since x∗∗ ≤ y ∗∗ on [0, α) and x∗ = y ∗ a.e. on [0, a] we obtain x∗ = y ∗ a.e. on [0, c). Since x∗ = x∗ (b) on [c, b), we may proceed similarly as in case 1 and easily get a contradiction which finishes the proof. 2 The immediate consequence of Propositions 4.1 and 4.2 is the following theorem. Theorem 4.3. Let 1 ≤ p < ∞, w be a weight function such that w = 0 on (a, b) and W is strictly increasing on [0, α) \ (a, b). The point x ∈ Γp,w is a UKM point if and only if   μ t ∈ (a, b) : x∗ (t) = x∗ (a− ) or x∗ (t) = x∗ (b) = b − a. 5. Local uniform rotundity in symmetric spaces Recently, H. Hudzik, A. Kamińska and M. Mastyło in Theorem 3 (see [19]) have proved that a Banach function lattice E is locally uniformly rotund if and only if E + possesses the same property. The proof of this result was established, in view of Theorem 1.2 in [14], applying an equivalent norm that is local uniform rotund in the case of the finite measure. Now we present a local approach of Theorem 3 in [19] for the finite and infinite measure which requires a completely different techniques. Theorem 5.1. Let E be a Banach function space. An element x ∈ E is an LUR point if and only if |x| is an LUR point.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

112

Proof. Necessity. Let (xn ) ⊂ E and xn + |x|E → 2 xE

and

xn E → xE .

(15)

Define yn = sign(x)xn + xn χsupp(x)c for any n ∈ N. Notice that |yn | = |xn | for every n ∈ N. Therefore,   |x| + xn E = sign(x)x + xn E = sign(x)x + sign(x)yn + xn χsupp(x)c E   = sign(x)(x + yn ) + yn χsupp(x)c E for any n ∈ N. Hence, since | sign(x)(x + yn ) + yn χsupp(x)c | = |(x + yn )χsupp(x) + yn χsupp(x)c | it follows that   |x| + xn E = (x + yn )χsupp(x) + yn χsupp(x)c E = x + yn E for each n ∈ N. Consequently, by condition (15) we obtain x + yn E → 2 xE

and

yn E → xE .

Thus, by assumption that x is an LUR point we get x − yn E → 0.

(16)

Moreover, for any n ∈ N,   x − yn E = sign(x)|x| − yn χsupp(x) − yn χsupp(x)c E   = sign(x)|x| − yn χsupp(x) − xn χsupp(x)c E   = sign(x)|x| − sign(x)xn − xn χsupp(x)c E   = sign(x)(|x| − xn ) − xn χsupp(x)c E . Now observe that | sign(x)(|x| − xn ) − xn χsupp(x)c | = |(|x| − xn )χsupp(x) − xn χsupp(x)c | and so   x − yn E = sign(x)(|x| − xn ) − xn χsupp(x)c E   = (|x| − xn )χsupp(x) − xn χsupp(x)c E = |x| − xn E which implies the end of the proof of necessity, in view of condition (16). Sufficiency. Let (xn ) ⊂ E and xn + xE → 2 xE Notice,

and

xn E → xE .

(17)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

113

xn + xE ≤ |xn | + |x|E ≤ xn E + xE . Hence, by condition (17) we get |x| + |xn |E → 2 xE and by assumption that |x| is an LUR point in E we conclude |xn | − |x|E → 0.

(18)

Define for any n ∈ N, (n)

= {t : x(t) ≥ 0 and xn (t) ≥ 0},

A2

(n)

= {t : x(t) ≥ 0 and xn (t) < 0},

A4

A1

A3

(n)

= {t : x(t) < 0 and xn (t) < 0},

(n)

= {t : x(t) < 0 and xn (t) ≥ 0}.

We have          0 ≤ max (x − xn )χA(n) ∪A(n)  , (x + xn )χA(n) ∪A(n)  1 2 3 4 E E     ≤ (x − xn )χA(n) ∪A(n) + (x + xn )χA(n) ∪A(n)  1

2

3

E

4

= |x| − |xn |E . Hence, by condition (18) we obtain     (x − xn )χA(n) ∪A(n)  → 0, 1

and

E

2

    (x + xn )χA(n) ∪A(n)  → 0. 3

4

E

(19)

Furthermore,         x + xn E ≤ (x + xn )χA(n) ∪A(n)  + (x + xn )χA(n) ∪A(n)  1 2 3 4 E E     ≤ xE + xn E + (x + xn )χA(n) ∪A(n)  . 3

E

4

Thus, by conditions (17) and (19) it follows that     (x + xn )χA(n) ∪A(n)  → 2 xE . 1

(20)

E

2

Moreover, by (19) and by the triangle inequality for a norm we have         xn χA(n) ∪A(n)  − xχA(n) ∪A(n)  → 0. 1

2

E

1

E

2

Since,             (x + xn )χA(n) ∪A(n)  ≤ xχA(n) ∪A(n)  + xn χA(n) ∪A(n)  ≤ xE + xn E , 1

E

2

1

2

E

1

2

E

according to conditions (17) and (20) we get         xχA(n) ∪A(n)  + xn χA(n) ∪A(n)  → 2 xE . 1

2

E

1

2

E

In consequence, by condition (21) it follows that     xχA(n) ∪A(n)  → xE 1

2

E

and

    xn χA(n) ∪A(n)  → xE . 1

2

E

(21)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

114

Moreover, observe that             2 xχA(n) ∪A(n)  ≤ |x| + |x|χA(n) ∪A(n)  ≤ xE + xχA(n) ∪A(n)  ≤ 2 xE . 1

E

2

1

E

2

1

E

2

Hence,     |x| + |x|χA(n) ∪A(n)  → 2 xE 1

and

E

2

    xχA(n) ∪A(n)  → xE . 1

E

2

Since |x| is an LUR point in E we obtain         xχA(n) ∪A(n)  = |x| − |x|χA(n) ∪A(n)  → 0. 3

E

4

1

E

2

Furthermore, by the triangle inequality we have               xχA(n) ∪A(n)  − xn χA(n) ∪A(n)   ≤ (x + xn )χA(n) ∪A(n)  . 3

4

E

3

4

E

3

E

4

Now applying condition (19) we get     xχA(n) ∪A(n)  → 0 3

4

and

E

    xn χA(n) ∪A(n)  → 0. 3

4

E

Finally, by condition (19) we finish the proof in view of the following inequality,             x − xn E ≤ (x − xn )χA(n) ∪A(n)  + xχA(n) ∪A(n)  + xn χA(n) ∪A(n)  . 1

2

E

3

E

4

3

4

E

2

Theorem 5.2. Let E be a symmetric space and let x ∈ E + be an LUR point, then x∗ is an LUR point. Proof. Let (xn ) ⊂ E and xn + x∗ E → 2 x∗ E

and

xn E → x∗ E .

Since x is an LUR point, by Remark 2.1 it follows that x is a point of order continuity. Hence, by Lemma 2.6 in [10] (see also Proposition 2.3 in [13]) we obtain that x∗ is a point of order continuity and so by Lemma 2.5 in [10] we have x∗ (∞) = 0. Moreover, by Ryff’s theorem [1] there exists a measure preserving transformation σ : supp(x) → supp(x∗ ) such that x∗ ◦ σ = x a.e. on supp(x). Now we consider two cases. Case 1. Assume that μ(supp(x∗ )) < ∞. Let β be a measure preserving transformation from supp(x)c to [μ(supp(x)), ∞) (for more details of construction see [31]). Define  γ=

σ

on supp(x),

β

c

on supp(x) ,

 yn = xn ◦ γ =

xn ◦ σ

on supp(x),

xn ◦ β

on supp(x)c .

Clearly, yn and xn are equimeasurable. Therefore, yn E = xn E → x∗ E = xE and yn + xE = xn ◦ γ + x∗ ◦ γE = xn + x∗ E → 2 xE . Since x is an LUR point we obtain that

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

xn − x∗ E = yn − xE → 0,

115

(22)

which implies that x∗ is an LUR point. Case 2. Assume that μ(supp(x∗ )) = ∞. Now we proceed analogously as for case 1 replacing γ by σ. Indeed, taking yn = xn ◦ σ we get yn + xE = xn ◦ σ + x∗ ◦ σE = xn + x∗ E → 2 xE and yn E = xn ◦ σE = xn E → xE . Hence, by assumption that x is an LUR point it follows that x∗ is an LUR point. 2 A. Medzhitov and P. Sukochev proved in Theorem 1 of [30] that E ∈ (LUR∗ ) implies E ∈ (LUR) for a symmetric space E under some additional assumptions. We will show that the above conclusion holds without any additional conditions. First we present the result which is a localization of Theorem 1 in [30]. Theorem 5.3. Let E be a symmetric space such that y ∗ (∞) = 0 for any y ∈ E and let x ∈ E + . If x∗ is an LUR point, then x is an LUR point. Proof. Let (xn ) ⊂ E, xn E → xE and xn + xE → 2 xE . It is easy to notice that x∗n E → x∗ E .

(23)

Moreover, for any n ∈ N, xn + x ≺ x∗n + x∗ , whence for every n ∈ N, xn + xE ≤ x∗n + x∗ E ≤ x∗n E + x∗ E . Therefore, x∗n + x∗ E → 2 x∗ E .

(24)

Hence, by assumption that x∗ is an LUR point we obtain x∗n − x∗ E → 0.

(25)

Moreover, by Remark 2.1 it follows that x∗ is a point of order continuity and by Lemma 2.6 in [10] (or Proposition 2.3 in [13]) we conclude that x is a point of order continuity. Now, we claim that xn converges to x in measure. First notice that xn + x ≺

x∗ + x∗ (xn + x)∗ + n ≺ x∗n + x∗ 2 2

for any n ∈ N. Thus, by condition (24) we have    (xn + x)∗ x∗n + x∗   → 2 x∗  .  + E   2 2 E

(26)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

116

Furthermore, by the triangle inequality we get    (xn + x)∗ x∗n + x∗   ≤ 1 (xn + x)∗ + 2x∗  + 1 x∗n − x∗   + E E   2 2 2 2 E ≤

1 1 xn + xE + x∗n − x∗ E + x∗ E 2 2

for any n ∈ N. Consequently, by the assumption and by conditions (25) and (26) we obtain    (xn + x)∗  ∗ ∗  + x   → 2 x E . 2 E Hence, by assumption

1 2

xn + xE → xE and by the fact that x∗ is an LUR point we have    (xn + x)∗  ∗  − x   → 0. 2 E

Therefore, by Lemma 2.2 in [13] and by condition (25) we prove the claim. Finally, by Proposition 2.4 in [13] and by condition (25) it follows that xn − xE → 0 and completes the proof. 2 Theorem 5.4. Let E be a symmetric space. Then E is LUR if and only if E is LUR∗ . Proof. Immediately, by Theorems 5.1 and 5.2 we complete the proof of the necessity. Now assume that E is LUR∗ . Then x∗ is an LUR point in E for every x ∈ E and by Lemma 2.5 in [10] we obtain x∗ (∞) = 0. Consequently, Theorems 5.1 and 5.3 imply E is LUR. 2 Now we investigate under which condition a point of local uniform rotundity of a symmetric space E might be considered equivalently on E ∗ , i.e. on the cone of all decreasing rearrangements of elements in a symmetric space E. Let us notice that at the beginning of the proof of Theorem 5.5 we use some similar techniques as in the proof of Theorem 2.1 in [7]. Theorem 5.5. Let E be a symmetric space and let (xn ) ⊂ E, x ∈ E. Then the following conditions: (i) xn E → xE and xn + x∗ E → 2 xE ⇒ xn − x∗ E → 0, (ii) xn E → xE and x∗n + x∗ E → 2 xE ⇒ x∗n − x∗ E → 0, satisfy the implication (i) ⇒ (ii). If E is order continuous then (ii) ⇒ (i). Proof. It is obvious that the conclusion (i) ⇒ (ii) is fulfilled immediately. Now we prove (ii) ⇒ (i). Let xn , x ∈ E for all n ∈ N and xn E → xE

and

xn + x∗ E → 2 xE .

Notice that for every n ∈ N we have xn + x∗ ≺ x∗n + x∗ which concludes xn + x∗ E ≤ x∗n + x∗ E ≤ xn E + xE

(27)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

117

for any n ∈ N. Consequently, by condition (27) we have x∗n + x∗ E → 2 xE

and

xn E → xE .

Hence, by (ii) we obtain x∗n − x∗ E → 0.

(28)

Moreover, observe that x n + x∗ ≺

x∗ + x ∗ (xn + x∗ )∗ + n 2 2

for each n ∈ N. Thus, for all n ∈ N we get    (xn + x∗ )∗ x∗n + x∗   ≤ xn  + x  + xn + x E ≤  E E  2 2 E ∗

which implies    (xn + x∗ )∗ x∗n + x∗   → 2 x .  + E   2 2 E So we have    (xn + x∗ )∗  x∗n − x∗ ∗  + + x   → 2 xE . 2 2 E

(29)

Moreover, by the triangle inequality we get      (xn + x∗ )∗  (xn + x∗ )∗   x∗n − x∗ 1 ∗ ∗   + + x + x ≤ + x∗n − x∗ E     2 2 2 2 E E ≤

1 3 1 xn E + xE + x∗n − x∗ E 2 2 2

for all n ∈ N. Consequently, by (27), (28) and (29) we have    (xn + x∗ )∗  ∗  + x  → 2 xE .  2 E

Clearly, by (27) we obtain    (xn + x∗ )∗   → x ,  E   2 E

whence by condition (ii) we conclude    (xn + x∗ )∗  ∗  − x   → 0. 2 E

(30)

Since E is order continuous, by Theorem 4.8 in [16] (see also Corollary 4.10 in [16] in the case when (M, τ ) is L∞ (0, ∞) with the trace given by the Lebesgue measure) it follows that E has an equivalent symmetric locally uniformly rotund norm | · |E . Thus, by (30) and by the triangle inequality we get

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

118

    ∗ ∗ 1     |(xn + x∗ )∗ |E − |x∗ |E  ≤  (xn + x ) − x∗  → 0, 2    2 E which implies |xn + x∗ |E → 2|x|E .

(31)

Furthermore, by condition (28) and by the inequality ||xn |E − |x|E | ≤ |x∗n − x∗ |E it follows that |xn |E → |x|E . Hence, by (31) and by local uniform convexity of the norm | · |E we obtain |xn − x∗ |E → 0. Finally, by an equivalence of norms ·E and | · |E we complete the proof. 2 Lemma 5.6. Let E be a symmetric space. If x ∈ E is an LUR point in E then x is a lower (an upper) locally uniformly K-monotone point. Proof. Let x ∈ E be an LUR point. Assume xn , yn ∈ E with yn∗ ≺ x∗ ≺ xn ∗ for all n ∈ N and xn E → xE

and

yn E → xE .

(32)

Observe that x∗ ≺

x∗ + x∗n ≺ x∗n 2

and yn∗ ≺

x∗ + yn∗ ≺ x∗ 2

for any n ∈ N. Therefore, since E is a symmetric space we obtain x∗ + x∗n E ≤ x∗n E 2

and

yn∗ E ≤

x∗n + x∗ E → 2 xE

and

yn∗ + x∗ E → 2 xE .

x∗ E ≤

x∗ + yn∗ E ≤ x∗ E 2

for every n ∈ N. Hence,

Thus, in view of Theorem 5.2, by condition (32) and by assumption that x is an LUR point we conclude x∗n − x∗ E → 0 and

yn∗ − x∗ E → 0.

2

6. Local uniform rotundity of Lorentz spaces Γp,w Theorem 6.1. The following conditions are equivalent. (i) 1 < p < ∞ and μ((a, b) ∩ supp(w)) > 0 for any interval (a, b) ⊂ (0, α) with a < b, W (∞) = ∞ whenever α = ∞.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

119

(ii) Γp,w is locally uniformly rotund. (iii) Γp,w is midpoint locally uniformly rotund. (iv) Γp,w is rotund. Proof. Immediately, by the definition of rotundity properties we get (ii) ⇒ (iii) ⇒ (iv). Moreover, by Theorem 2.3 in [8] we have (iv) ⇒ (i). Now we prove (i) ⇒ (ii). Let (i) be satisfied and let x ∈ SΓp,w and (xn ) ⊂ BΓp,w satisfy xn Γp,w → 1 and xn + xΓp,w → 2. By assumption W (∞) = ∞ and by Proposition 1.4 in [23] it follows that Γp,w is order continuous, whence and by Theorems 5.1, 5.3 and 5.5 we may assume that x = x∗ and xn = x∗n for any n ∈ N. Let  > 0, δ > 0. Clearly, for all n ∈ N, x∗∗ n (t)φΓ (t) ≤ xn Γp,w ≤ 1, where φΓ is the fundamental function of Γp,w (see [1]). By the above inequality there is t1 > 0 such that for any t ≥ t1 and n ∈ N we have ∗∗ max{x∗∗ n (t), x (t)} ≤ .

Let A = [0, a) ⊂ [0, α) with t1 ≤ a < ∞. Define for any n ∈ N,

An = {t ∈ A :

|x∗∗ n (t)

∗∗

− x (t)| ≥ δ} ,

Fn =

t∈A:

x∗∗ n (t)

 >

2 W (/4)

1/p 

and

G=



∗∗

t ∈ A : x (t) >

2 W (/4)

1/p  .

Denote Bn = Fn ∪ G for every n ∈ N. Then, we have μ(F  n)

2≥

p xn Γp,w

+

p xΓp,w

x∗∗p n (t)w(t)dt +

≥ 0

μ(F  n)

≥ 0

=

2 w(t)dt + W (/4)

μ(G) 

μ(G) 

x∗∗p (t)w(t)dt

0

2 w(t)dt W (/4)

0

2W (μ(Fn )) + 2W (μ(G)) W (/4)

for any n ∈ N. Hence, W (/4) ≥ W (μ(Fn )) + W (μ(G)) which concludes μ(Bn ) ≤

 , 2

(33)

for every n ∈ N. Denote Ck = {t ∈ A : w(t) > 1/k}. Clearly, Ck ⊂ Ck+1 and A = μ(A \ Ck ) → 0 as k → ∞ and so μ((An \ Bn ) \ Ck ) <

 4

∞ k=1

Ck . Consequently,

(34)

for some k ∈ N which does not depend on n. By uniform convexity of a power function up for p > 1 there exists ξ ∈ (0, 1) such that for any u, v ∈ [0, ∞), |u − v| ≥ δ(W ( 4 )/2)1/p max{u, v} we have

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

120



u+v 2

p ≤

1−ξ p (u + v p ) . 2

≤

1 − ξ ∗∗p (xn (t) + x∗∗p (t)) 2

Therefore, 

∗∗ x∗∗ n (t) + x (t) 2

p

for any t ∈ An \ Bn and n ∈ N. Moreover, for every t > 0 and n ∈ N we get 

∗∗ x∗∗ n (t) + x (t) 2

p ≤

∗∗p x∗∗p (t) n (t) + x . 2

Hence, 

∗∗ x∗∗ n (t) + x (t) 2

p ≤

∗∗p (t) ξ ∗∗p x∗∗p n (t) + x − (xn (t) + x∗∗p (t)) χAn \Bn (t) 2 2

for any t ∈ (0, α) and n ∈ N. Consequently,    xn + x p    2 

α  ≤

Γp,w

∗∗ x∗∗ n (t) + x (t) 2

p w(t)dt

0

α ≤

∗∗p ξ x∗∗p (t) n (t) + x w(t)dt − 2 2

0



 ≤1−ξ An \Bn

 p δ ≤1−ξ 2

An \Bn

p

w(t)dt

for any k, n ∈ N. Therefore, by assumption

 w(t)dt

An \Bn

 w(t)dt ≤ 1 −

(An \Bn )∩Ck

μ((An \ Bn ) ∩ Ck ) ≤

∗∗p (x∗∗p (t)) w(t)dt n (t) + x

  ∗∗  p  xn (t) − x∗∗ (t) p  w(t)dt ≤ 1 − ξ δ    2 2



≤1−ξ An \Bn

∗∗ x∗∗ n (t) + x (t) 2



δp ξ μ ((An \ Bn ) ∩ Ck ) 2p k

1 2

xn + xΓp,w → 1 we obtain

2p k δp ξ

  1 p 1 − p xn + xΓp,w → 0 2

as n → ∞,

which implies μ((An \ Bn ) ∩ Ck ) <

 4

for large enough n ∈ N and for all k ∈ N. Hence, by conditions (33) and (34) it follows that μ(An ) = μ(An \ Bn ) + μ(An ∩ Bn ) ≤ μ(Bn ) + μ((An \ Bn ) ∩ Ck ) + μ((An \ Bn ) \ Ck ) < ,

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

121

∗∗ for large enough n ∈ N, which concludes that (x∗∗ in measure as n → ∞. Now, according n ) converges to x to assumption (i) and by Proposition 2.5 in [11] we get strict K-monotonicity of Γp,w , which means that x ∗∗ is a UKM point. Finally, since x∗∗ in measure and xn Γp,w → xΓp,w , by Theorem 4.1 in [12] and n → x by Theorem 3.8 we complete the proof. 2

Remark 6.2. Now we present a class of symmetric spaces obtained by the K-method of interpolation which are locally uniformly rotund. Let φ0 and φ1 be increasing concave functions on [0, ∞) with φ0 (0+ ) = φ1 (0+ ) = 0 whose quotient λ = φ0 /φ1 is increasing and vanishes continuously at zero and also lim min{φ0 (s), φ1 (s)} = lim λ(s) = ∞.

s→∞

s→∞

By condition 6 in [24], for simplicity we may use the following notations Λφ0 and Λφ1 for Lorentz spaces Λ1,φ0 and Λ1,φ1 , respectively. Considering (Λφ0 , Λφ1 ), a Banach couple of two Lorentz spaces, by Lemma 3.2 in [16] we obtain the formula for the K-functional K(t, x, Λφ0 , Λφ1 ) = inf{x0 Λφ + t x1 Λφ : x = x0 + x1 , x0 ∈ Λφ0 , x1 ∈ Λφ1 } 0

s(t)

1

x∗ (τ )dφ0 (τ ) + t

= 0

∞

x∗ (τ )dφ1 (τ )

s(t)

for any t ∈ (0, ∞) and x ∈ Λφ0 + Λφ1 , where s(t) is uniquely determined by the condition λ(s(t)) = t. Let ρ be a strictly positive weight function on [0, ∞) and 1 < p < ∞ and let w be a weight function such that μ((a, b) ∩ supp(w)) > 0 for any interval (a, b) ⊂ (0, α) with a < b, W (∞) = ∞ whenever α = ∞. Assuming (Λφo , Λφ1 )ρ,Γp,w = {x ∈ Λφo + Λφ1 : ρ(·)K(·, x, Λφ0 , Λφ1 ) ∈ Γp,w } by Corollary 3.7 and Remark (see page 339 in [16]) and by Theorem 6.1 we conclude that the Banach space (Λφo , Λφ1 )ρ,Γp,w equipped with the norm xρ,Γp,w = ρ(·)K(·, x, Λφo , Λφ1 )Γp,w is locally uniformly rotund. t Theorem 6.3. Let 1 < p < ∞, w be a weight function such that W (t) = 0 w is strictly increasing on [0, α) \ (a, b), W (∞) = ∞ and w = 0 on (a, b) and let x ∈ Γp,w \ {0}. If x∗ is an LUR point of Γp,w , then one of the following conditions is satisfied: (i) μ (t ∈ (a, b) : x∗ (t) = x∗ (a− )) = b − a. (ii) μ (t ∈ (a, b) : x∗ (t) = x∗ (b)) = b − a. Proof. Although the following proof is partially similar as the proof of Proposition 4.1 we will present the details of this proof for the sake of convenience. Let x ∈ SΓp,w and x∗ be an LUR point of Γp,w . Suppose for a contrary that (i) and (ii) are not fulfilled. Then either there are γ, β ∈ (a, b) such that x∗ (γ) > x∗ (β) or there exists c ∈ (x∗ (b), x∗ (a− )) such that x∗ (t) = c for all t ∈ (a, b). Now we consider two cases. Case 1. Assume that there exist γ, β ∈ (a, b) such that x∗ (γ) > x∗ (β). Since a < γ < β < b we have x∗ (b) ≤ x∗ (β) < x∗ (γ) ≤ x∗ (a). By the right-continuity of the decreasing rearrangement x∗ there is ξ ∈ (γ, β) such that x∗ (β) < x∗ (ξ) ≤ x∗ (γ). Now we consider case 1 into two parts. Part 1. Assume that x∗ (β) < x∗ (ξ) < x∗ (γ). Denote ξ

∗

∗

(x − x (ξ)),

L= γ

β P = ξ

(x∗ (ξ) − x∗ ).

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

122

By the right-continuity of x∗ we get L > 0 and since β < b we may assume without loss of generality that P > 0. If P ≥ L then, by monotonicity and continuity of the maximal function x∗∗, we are able to find η ∈ (ξ, β] such that η L=

(x∗ (ξ) − x∗ ).

(35)

ξ

Define y = x∗ χ[0,α)\(γ,η) + x∗ (ξ)χ(γ,η) . We claim that x∗ + yΓp,w = 2 and x∗ − yΓp,w > 0. First observe that y is decreasing and x∗ + y = 2x∗ + (x∗ (ξ) − x∗ )χ(γ,η) . Considering t ∈ (0, γ] we have 1 (x + y) (t) = t ∗

∗∗

t



 2x∗ + (x∗ (ξ) − x∗ )χ(γ,η) = 2x∗∗ (t).

0

If t ∈ [η, α), then by condition (35) we obtain 2 (x + y) (t) = t ∗

∗∗

t

1 x + t ∗

ξ

1 (x (ξ) − x ) + t ∗

∗

γ

0

= 2x∗∗ (t) −

η

(x∗ (ξ) − x∗ )

ξ

η

L 1 + t t

(x∗ (ξ) − x∗ ) = 2x∗∗ (t).

ξ

Consequently, since xΓp,w = 1 and by assumption w = 0 on (a, b) and (γ, η) ⊂ (a, b) it follows that ∗

a

x +

p yΓp,w

=

∗

∗∗p

(x + y)

α w+

0

a = 0

(x∗ + y)∗∗p w

b

2p x∗∗p w +

α

2p x∗∗p w = 2p .

b

Moreover, by the definition of y we get yΓp,w = 1. Now we show x∗ − yΓp,w > 0. Observe that 1 (x − y) (t) = t ∗

∗∗

t

(x∗ − x∗ χ[0,α)\(γ,η) − x∗ (ξ)χ(γ,η) )∗

0

=

1 t

t



(x∗ − x∗ (ξ))χ(γ,ξ) + (x∗ − x∗ (ξ))χ(ξ,η)

∗

0

≥

1 t

t 0

(x∗ − x∗ (ξ))χ(γ,ξ) +

1 t

t 0

(x∗ (ξ) − x∗ )χ(ξ,η) = ψ(t)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

123

for any t ∈ (0, α). Clearly, if t ∈ (0, γ], then ψ(t) = 0. Moreover, if t ∈ [η, α), then ψ(t) = assumption W (∞) = ∞ we conclude ∗

α

x −

p yΓp,w

∗

∗∗p

(x − y)

= 0

 =

2L b

α w≥

a p

0

p

Therefore, by

α p

ψ w=

2L t .

ψp w

ψ w+ 0

b

Wp (b) > 0,

which completes the claim. Now considering P < L we may choose η ∈ (γ, ξ) such that ξ P =

(x∗ − x∗ (ξ)).

(36)

η

Next assuming y = x∗ χ[0,α)\(η,β) + x∗ (ξ)χ(η,β) , it is easy to show x∗ + yΓp,w = 2 and yΓp,w = 1. Furthermore,

(x∗ − y)∗∗ (t) ≥

1 t

t

(x∗ − x∗ (ξ))χ(η,ξ) +

1 t

0

t

(x∗ (ξ) − x∗ )χ(ξ,β) ,

0

whence x∗ − yΓp,w ≥ p



2P b

p Wp (b) > 0.

Now combining in part 1 both cases when P ≥ L and P < L we obtain a contradiction with assumption that x∗ is an LUR point. Part 2. Assume that for all t ∈ (a, b) we have either x∗ (γ) = x∗ (t) or x∗ (β) = x∗ (t). Then, we may assume that x∗ (β) = x∗ (β − ) and find ξ ∈ (γ, β) such that x∗ (ξ − ) > x∗ (ξ). Let  = 12 min{β − ξ, ξ − γ} and δ = 12 (x∗ (γ) + x∗ (β)). Define y = x∗ χ[0,α)\[ξ−,ξ+) + δχ[ξ−,ξ+) . Now we prove x∗ + yΓp,w = 2, x∗ − yΓp,w > 0 and yΓp,w = 1. First observe 1 (x + y) (t) = t ∗

∗∗

t



2x∗ + (δ − x∗ )χ(ξ−,ξ+)

∗

0

1 = 2x (t) + t ∗∗

t

1 (δ − x (γ))χ(ξ−,ξ) + t

0

= 2x∗∗ (t) +

1 2t

t 0

∗

t

(δ − x∗ (β))χ(ξ,ξ+)

0

(x∗ (β) − x∗ (γ))χ(ξ−,ξ) +

1 2t

t 0

(x∗ (γ) − x∗ (β))χ(ξ,ξ+)

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

124

for all t > 0. Thus, if t ∈ (0, ξ − ) ∪ [ξ + , α), then (x∗ + y)∗∗ (t) = 2x∗∗ (t). Consequently, since w = 0 on (a, b) and (ξ − , ξ + ) ⊂ (a, b), we get x∗ + yΓp,w = 2. Furthermore, for any t > 0, 1 (x − y) (t) = t ∗

∗∗

t



(x∗ − δ)χ(ξ−,ξ+)

∗

0

1 = 2t

t

 ∗ ∗ (x (γ) − x∗ (β))χ(ξ−,ξ) + (x∗ (β) − x∗ (γ))χ(ξ,ξ+)

0

∗∗ x (γ) − x∗ (β)  χ[0,2) (t), 2 ∗

= whence

x∗ − yΓp,w =

 x∗ (γ) − x∗ (β)  χ[0,2)  > 0. Γp,w 2

Clearly y ∗ = y, and so for all t ∈ (0, α) we have 1 y (t) = t ∗∗

t

(x∗ + (δ − x∗ )χ(ξ−,ξ+) )

0

1 = x (t) + t ∗∗

t

1 (δ − x (γ))χ(ξ−,ξ) + t ∗

0

1 = x (t) + 2t ∗∗

t

(δ − x∗ (β))χ(ξ,ξ+)

0

t

1 (x (β) − x (γ))χ(ξ−,ξ) + 2t ∗

∗

0

t

(x∗ (γ) − x∗ (β))χ(ξ,ξ+) .

0

Thus, if t ∈ [0, ξ − ] ∪ [ξ + , α), then y ∗∗ (t) = x∗∗ (t) and by assumption w = 0 a.e. on (a, b) it follows that yΓp,w = xΓp,w = 1. Finally, we obtain a contradiction in view of assumption that x∗ is an LUR point. Case 2. Assume that x∗ = c on (a, b) for some c ∈ (x∗ (b), x∗ (a− )). Taking δ=

1 min{c − x∗ (b), x∗ (a− ) − c}, 2

and y = x∗ χ[0,α)\(a,b) + (c + δ)χ(a, a+b ) + (c − δ)χ( a+b ,b) 2

2

we may easily prove that yΓp,w = 1, x∗ + yΓp,w = 2 and x∗ − yΓp,w > 0, which contradicts with assumption that x∗ is an LUR point. 2 Acknowledgments We wish to express our gratitude to the reviewer for many valuable suggestions and remarks. References [1] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press Inc., 1988. [2] G. Birkhoff, Lattice Theory, American Mathematical Society, Providence, RI, 1967.

M. Ciesielski / J. Math. Anal. Appl. 430 (2015) 98–125

125

[3] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113–190. [4] J. Cerdá, H. Hudzik, A. Kamińska, M. Mastyło, Geometric properties of symmetric spaces with applications to Orlicz– Lorentz spaces, Positivity 2 (1998) 311–337. [5] J. Cerdá, H. Hudzik, M. Mastyło, On the geometry of some Calderón–Lozanovski˘ı interpolation spaces, Indag. Math. (N.S.) 6 (1) (1995) 35–49. [6] V.I. Chilin, P.G. Dodds, A.A. Sedaev, F.A. Sukochev, Characterizations of Kadec–Klee properties in symmetric spaces of measurable functions, Trans. Amer. Math. Soc. 348 (12) (1996) 4895–4918. [7] V.I. Chilin, A.V. Krygin, F.A. Sukochev, Local uniform and uniform convexity of noncommutative symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 111 (2) (1992) 355–368. [8] M. Ciesielski, A. Kamińska, P. Kolwicz, R. Płuciennik, Monotonicity and rotundity of Lorentz spaces Γp,w , Nonlinear Anal. 75 (2012) 2713–2723. [9] M. Ciesielski, A. Kamińska, R. Płuciennik, Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w , Math. Nachr. 282 (9) (2009) 1242–1264. [10] M. Ciesielski, P. Kolwicz, A. Panfil, Local monotonicity structure of symmetric spaces with applications, J. Math. Anal. Appl. 409 (2014) 649–662. [11] M. Ciesielski, P. Kolwicz, R. Płuciennik, Note on strict K-monotonicity of some symmetric function spaces, Comment. Math. 53 (2) (2013) 311–322. [12] M. Ciesielski, P. Kolwicz, R. Płuciennik, Local approach to Kadec–Klee properties in symmetric function spaces, J. Math. Anal. Appl. 426 (2015) 700–726. [13] M.M. Czerwińska, A. Kamińska, Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators, Studia Math. 201 (3) (2010) 253–285. [14] W.J. Davis, N. Ghoussoub, J. Lindenstrauss, A lattice renorming theorem and applications to vector-valued processes, Trans. Amer. Math. Soc. 263 (2) (1981) 531–540. [15] P.G. Dodds, T.K. Dodds, A.A. Sedaev, F.A. Sukochev, Local uniform convexity and Kadec–Klee type properties in K-interpolation spaces. I. General theory, J. Funct. Spaces Appl. 2 (2) (2004) 125–173. [16] P.G. Dodds, T.K. Dodds, A.A. Sedaev, F.A. Sukochev, Local uniform convexity and Kadec–Klee type properties in K-interpolation spaces. II, J. Funct. Spaces Appl. 2 (3) (2004) 323–356. [17] H. Hudzik, A. Kamińska, Monotonicity properties of Lorentz spaces, Proc. Amer. Math. Soc. 123 (9) (1995) 2715–2721. [18] H. Hudzik, A. Kamińska, M. Mastyło, On geometric properties of Orlicz–Lorentz spaces, Canad. Math. Bull. 40 (3) (1997) 316–329. [19] H. Hudzik, A. Kamińska, M. Mastyło, Monotonicity and rotundity properties in Banach lattices, Rocky Mountain J. Math. 30 (3) (2000) 933–949. [20] H. Hudzik, W. Kurc, Monotonicity properties of Musielak–Orlicz spaces and dominated best approximant in Banach lattices, J. Approx. Theory 95 (3) (1998) 353–368. [21] A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. (Basel) 55 (2) (1990) 173–180. [22] A. Kamińska, Some remarks on Orlicz–Lorentz spaces, Math. Nachr. 147 (1990) 29–38. [23] A. Kamińska, L. Maligranda, On Lorentz spaces Γp,w , Israel J. Math. 140 (2004) 285–318. [24] P. Kolwicz, K. Leśnik, L. Maligranda, Pointwise products of some Banach function spaces and factorization, J. Funct. Anal. 266 (2) (2014) 616–659. [25] P. Kolwicz, R. Płuciennik, Points of upper local uniform monotonicity in Calderón–Lozanowskii spaces, J. Convex Anal. 17 (1) (2010) 111–130. [26] S.G. Krein, Yu.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian). [27] W. Kurc, Strictly and uniformly monotone Musielak–Orlicz spaces and applications to best approximation, J. Approx. Theory 69 (2) (1992) 173–187. [28] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. II. Function Spaces, Springer-Verlag, Berlin–New York, 1979. [29] G.G. Lorentz, On the theory of spaces Λ, Pacific J. Math. 1 (1951) 411–429. [30] A. Medzhitov, P. Sukochev, The property (H) in Orlicz spaces, Bull. Acad. Polon. Sci. Sér. Math. 40 (3–4) (1981) 137–144. [31] H.L. Royden, Real Analysis, third edition, Macmillan Publishing Company, New York, 1988. [32] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (2) (1990) 145–158. [33] V.D. Stepanov, The weighted Hardy’s inequality for nonincreasing functions, Trans. Amer. Math. Soc. 338 (1993) 173–186.