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[19] A function ϕ : R → R+ is said to be an Orlicz function if ϕ is convex, even, ϕ(0) = 0, ϕ(u) > 0 for all u > 0, lim ϕ(u) = 0 and lim ϕ(u) = ∞. Its u→0 u u→∞ u complementary function ψ is defined by ψ(v) = sup{|uv| − ϕ(u)} u>0 for all v ∈ R. If ϕ is an Orlicz function, then its complementary function ψ is an Orlicz function. Recall that an Orlicz function ϕ satisfies the condition ∆2 for all value (ϕ ∈ ∆2(R) for short) if there exists a constant K > 0 such that ϕ(2u) ≤ Kϕ(u) holds for all u ∈ R. Analogously, an Orlicz function ϕ satisfies the condition ∆2 for large values (ϕ ∈ ∆2(∞) for short) if there exist a constant K > 0 and a constant u0 ≥ 0 such that ϕ(2u) ≤ Kϕ(u) holds for all u ≥ u0. Sometimes we say ϕ satisfy ∇2 condition for all values (or ∇2 condition for large values) if ψ satisfy ∆2 condition for all values (or ∆2 condition for large values). For more properties of Orlicz function, we may refer to [3, 19]. For any measurable function x : [0,γ) → R, where γ ≤ ∞, its distribution function and decreasing rearrangement are defined as follows

dx(θ)= µ{s ∈ [0,γ): |x(s)| > θ}, ∗ x (t) = inf{θ> 0 : dx(θ) ≤ t}, t ≥ 0, where µ denotes the Lebesgue . A function ω : [0,γ) → R+ is said to be a weight function if it is non-increasing and locally integrable. Define x α = sup{t ≥ 0 : ω(t) > 0}, W (x)= 0 ω(s) ds. The Orlicz-Lorentz space Λϕ,ω is the set of all Lebesgue measurable functions x on [0,γ) such that R γ ∗ ρϕ,ω(λx)= ϕ(λx (t))ω(t) dt < ∞ Z0 for some λ > 0. It is known that the Orlicz-Lorentz space endowed with the Luxemburg norm x kxk = inf ε> 0 : ρ ≤ 1 ϕ,ω ϕ,ω ε [15] n   o is a . If ϕ(t) = t, then Λϕ,ω is the Lorentz function space L1,ω. The L1,ω-norm of x ∈ L1,ω is defined by γ ∗ kxk1,ω = x (t)ω(t) dt. Z0 Obviously,

ρϕ,ω(x)= kϕ ◦ xk1,ω. Recall a Banach lattice E = (E, ≤, k·k) is said to be strictly monotone[2] if x, y ∈ E, 0 ≤ y ≤ x and y =6 x imply that kyk < kxk. A Banach lattice E =(E, ≤, k·k) is said to be lower locally uniformly monotone[12], whenever for any x ∈ (E)+ (the positive cone of E) with kxk = 1 and any ε ∈ (0, 1) there exists δ(x, ε) ∈ (0, 1) such that the conditions 0 ≤ y ≤ x and kyk ≥ ε imply that kx−yk ≤ 1−δ(x, ε). Suppose A1, A2 are the subsets of R. A mapping σ : A1 → A2 NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 3 is called measure preserving transformation if for any measurable set E ⊂ A2, it −1 −1 holds that the set σ (E) ⊂ A1 is also measurable and µ(σ (E)) = µ(E). From [1] we know that if x is a simple function with compact support in Λϕ,ω[0,γ) then there is a measure preserving transformation σ such that γ γ ϕ(x∗)ω = ϕ(x)ω ◦ σ. Z0 Z0 In 1999, Wu and Ren defined the Orlicz norm on the space Λϕ,ω[0,γ) for γ < ∞[24] as γ ◦ ∗ ∗ kxkϕ,ω = sup x (t)y (t)ω(t) dt. ρψ,ω(y)≤1 Z0 In [24] the authors proved that endowed with the Orlicz norm, Λϕ,ω[0,γ) is a ◦ Banach space (denoted by Λϕ,ω[0,γ)), and obtained the following properties of ◦ Λϕ,ω[0,γ): ◦ ◦ (i) Let x ∈ Λϕ,ω[0,γ). If kxkϕ,ω ≤ 1, then ρϕ,ω(x) ≤kxkϕ,ω. ◦ (ii) For any x ∈ Λϕ,ω[0,γ), kxkϕ,ω ≤kxkϕ,ω ≤ 2kxkϕ,ω. (iii) If there exists k > 1 such that ρψ,ω(p(k|x|)) = 1, then γ 1 kxk◦ = x∗(t)p(kx∗(t))ω(t) dt = (1 + ρ (kx)). ϕ,ω k ϕ,ω Z0 ◦ γ ∗ ∗ 1 (iv) kxkϕ,ω = sup x (t)y (t)ω(t) dt = inf (1 + ρϕ,ω(kx)). 0 k>0 k ρψ,ω(y)≤1 (v) For any x ∈ Λϕ,ω[0R,γ),

◦ 1 kxkϕ,ω = (1 + ρϕ,ω(kxx)) kx ∗ ∗∗ ∗ if and only if kx ∈ K(x) = [k ,k ], where k = inf{h > 0 : ρψ,ω(p(h|x|)) ≥ 1} ∗∗ and k = sup{h > 0 : ρψ,ω(p(h|x|)) ≤ 1}. For the sake of convenience, in this paper we will consider γ = 1 whenever γ < +∞.

2. Some Lemmas In 2012, Wang and Chen extended the definition and properties of [24] to γ ≤∞[23], and get the following two lemmas.

◦ Lemma 2.1. (I) inf{k : k ∈ K(x), kxkϕ,ω =1} > 1 if and only if ϕ ∈ ∆2. ◦ (II) The set Q = {K(x): a ≤ kxkϕ,ω ≤ b} is bounded for each b ≥ a > 0 if and only if ϕ ∈∇2. S Lemma 2.2 ([23]). Let A ⊂ [0, ∞) and µA = t. (I) For t< ∞, 1 kχ k◦ = ψ−1 W (t); A ϕ,ω W (t)   (II) For t = ∞ and W (∞) < ∞, 1 kχ k◦ = ψ−1 W (∞). A ϕ,ω W (∞)   4 B. CHEN and W. GONG

By the same method as the proof of Theorem 2 in [16], we can get that if ψ∈ / ∆2(∞) (or ψ∈ / ∆2(R)), then for any ε ∈ (0, 1), there exists a sequence {en} with ρϕ,ω(en)= ε such that n ◦ 1 ε ≤ aiei ≤ 1+ ε, 8 i=1 ϕ,ω X where a ≥ 0 and n a = 1. Therefore we have i i=1 i ◦ ◦ Lemma 2.3. If ψP∈ / ∆2(∞) (or ψ∈ / ∆2(R)), then Λϕ,ω[0, 1) (or Λϕ,ω[0, ∞)) contains ℓ1.

Lemma 2.4 ([11, 17]). The Lorentz function space L1,ω is strictly monotone if γ and only if ω is positive on [0,γ) and 0 ω(t) dt = ∞ whenever γ = ∞.

Lemma 2.5 ([6]). The Lorentz functionR space L1,ω is lower locally uniformly γ monotone if and only if ω is positive on [0,γ) and 0 ω(t) dt = ∞ whenever γ = ∞. R ◦ 3. Non-squareness of Λϕ,ω ◦ ∞ Theorem 3.1. Orlicz-Lorentz space Λϕ,ω[0, ∞) is non-square if and only if 0 ω(t) dt = ∞. R ∞ Proof. (Necessity) If 0 ω(t) dt < ∞, let A ⊂ [0, ∞) and µA = ∞. By Lemma 2.2, we obtain R 1 kχ k◦ = ψ−1 W (∞). A ϕ,ω W (∞)   Let 1 x = χ , −1 1 A ψ W (∞) W (∞) 1 1 y =  χ − χ , −1 1 A1 −1 1 A2 ψ W (∞) W (∞) ψ W (∞) W (∞) ∗ ∗ where A1 ∪ A2 = A, A1∩ A2 = ∅ and µ(A1)= µ(A2)= ∞. Since x = y , we get x + y ◦ x − y ◦ kxk◦ = kyk◦ = = =1. ϕ,ω ϕ,ω 2 2 ϕ,ω ϕ,ω

Which is a contradiction. ◦ (Sufficiency) Let x, y ∈ S(Λϕ,ω[0, ∞)). Fix k1 ∈ K(x), k2 ∈ K(y). Let k = 2k1k2 k1+k2 . Denote

A1 = {t ∈ [0, ∞): x(t)y(t) > 0},

A2 = {t ∈ [0, ∞): x(t)y(t) < 0},

A3 = {t ∈ [0, ∞): x(t)y(t)=0 and max{|x(t)|, |y(t)|} > 0}. NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 5

If t ∈ A1, we can get k(x − y) k k ϕ (t) = ϕ 2 k x(t) − 1 k y(t) 2 k + k 1 k + k 2    1 2 1 2  k k <ϕ 2 k x(t)+ 1 k y(t) k + k 1 k + k 2  1 2 1 2  k2 k1 ≤ ϕ(k1x(t)) + ϕ(k2y(t)). k1 + k2 k1 + k2

If t ∈ A2, there holds k(x + y) k k ϕ (t) = ϕ 2 k x(t)+ 1 k y(t) 2 k + k 1 k + k 2    1 2 1 2  k k <ϕ 2 k x(t) − 1 k y(t) k + k 1 k + k 2  1 2 1 2  k2 k1 ≤ ϕ(k1x(t)) + ϕ(k2y(t)). k1 + k2 k1 + k2

If t ∈ A3, the inequality ϕ(βx) < βϕ(x) for any 0 <β< 1 follows that k(x + y) k k ϕ (t) = ϕ 2 k x(t)+ 1 k y(t) 2 k + k 1 k + k 2    1 2 1 2  k k < max 2 ϕ(k x(t)), 1 ϕ(k y(t)) k + k 1 k + k 2  1 2 1 2  k2 k1 ≤ ϕ(k1x(t)) + ϕ(k2y(t)). k1 + k2 k1 + k2

By the strict monotonicity of L1,ω[0, ∞) we get, if µ(A1) > 0, k(x − y) k k ϕ < 2 kϕ(k x)k + 1 kϕ(k y)k 2 k + k 1 1,ω k + k 2 1,ω   1,ω 1 2 1 2

k2 ◦ k1 ◦ ≤ (k1kxkϕ,ω − 1) + (k2kykϕ,ω − 1) k1 + k2 k1 + k2 ≤ k − 1. Whence ◦ k(x−y) x − y 1+ ρϕ,ω( ) ≤ 2 < 1. 2 k ϕ,ω

Similarly, if µ(A ∪ A ) > 0, there holds 2 3 x + y ◦ < 1. 2 ϕ,ω

In summary

x + y ◦ x − y ◦ min , < 1. 2 2 ( ϕ,ω ϕ,ω)

6 B. CHEN and W. GONG

◦ Theorem 3.2. Orlicz-Lorentz space Λϕ,ω[0, 1) is non-square if and only if α := 1 sup{t ≥ 0 : ω(t) > 0} ∈ ( 2 , 1].

1 Proof. (Necessity) If α ∈ (0, 2 ], set 1 x = χ , −1 1 [0,2α) ψ W (α) W (2α) 1 1 y =  χ − χ . −1 1 [0,α) −1 1 [α,2α] ψ W (α) W (2α) ψ W (α) W (2α)

Clearly,  

x + y ◦ x − y ◦ kxk◦ = kyk◦ = = =1. ϕ,ω ϕ,ω 2 2 ϕ,ω ϕ,ω

◦ Which shows that Λϕ,ω[0, 1) is not non-square, a contradiction. ◦ 2k1k2 (Sufficiency) Let x, y ∈ S(Λϕ,ω[0, 1)). Fix k1 ∈ K(x), k2 ∈ K(y). Let k = k1+k2 . Case 1. Let α = 1. It is similar to the proof of Theorem 3.1. 1 Case 2. Suppose that 2 <α< 1. Define

Ax,y = {t ∈ [0, 1) : max{|x(t)|, |y(t)|} > 0}.

Case 2.1 If µ(Ax,y) ≤ α. We define

∼ x = xχAx,y ◦ σ, ∼ y = yχAx,y ◦ σ, where σ : [0,µ(Ax,y)) → Ax,y is a measure preserving transformation. Obviously, ∼ ∼ ∼ ∼ ∼ ∼ x+y x−y ϕ(x), ϕ(y), ϕ 2 and ϕ 2 are equimeasurable with ϕ(xχAx,y ), ϕ(yχAx,y ), x+y x−y ϕ 2 χAx,y and ϕ 2 χAx,y . Similarly as Case 1, we see  k(x − y) k(x + y) min ρ , ρ ϕ,ω 2 ϕ,ω 2      k(x − y) k(x + y) = min ρ χ , ρ χ ϕ,ω 2 Ax,y ϕ,ω 2 Ax,y   ∼ ∼  ∼ ∼  k x − y k x + y = min ρϕ,ω , ρϕ,ω ( 2 ! 2 !)

Therefore

x + y ◦ x − y ◦ min , < 1. 2 2 ( ϕ,ω ϕ,ω)

NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 7

Case 2.2 Assume µ(Ax,y) >α. By the convexity of ϕ, we obtain k(x + y) ∗ k(x + y) ∗ ϕ (t) = ϕ (t) 2 2       k k ∗ = ϕ 2 k x + 1 k y (t) k + k 1 k + k 2   1 2 1 2  k k ∗ ≤ 2 ϕ(k x)+ 1 ϕ(k y) (t). k + k 1 k + k 2  1 2 1 2  Similarly, for any t ∈ [0, 1), k(x − y) ∗ k k ∗ ϕ (t) ≤ 2 ϕ(k x)+ 1 ϕ(k y) (t). 2 k + k 1 k + k 2     1 2 1 2  In the following we shall show that for some t ∈ [0,α) there holds k(x + y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t) 2 k + k 1 k + k 2     1 2 1 2  or k(x − y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t). 2 k + k 1 k + k 2     1 2 1 2  Suppose k(x ± y) ∗ k(x ± y) ∗ ϕ (t) = ϕ (t) 2 2       k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (t) (3.1) k + k 1 k + k 2  1 2 1 2  for any t ∈ [0,α). Case 2.2.1 Let k k ∗ k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (0) > 2 ϕ(k x)+ 1 ϕ(k y) (t) k + k 1 k + k 2 k + k 1 k + k 2  1 2 1 2   1 2 1 2  for all t>α. Define k k ∗ t = sup s : 2 ϕ(k x)+ 1 ϕ(k y) (s) 0 k + k 1 k + k 2 (  1 2 1 2  k k ∗ > 2 ϕ(k x)+ 1 ϕ(k y) (t) for each t>α . k + k 1 k + k 2  1 2 1 2  )

Obviously, we have 0 < t0 ≤ α and k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (t ) k + k 1 k + k 2 0  1 2 1 2  k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (α) > 0. k + k 1 k + k 2  1 2 1 2  8 B. CHEN and W. GONG

If t0 = α, then k k ∗ k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (s) > 2 ϕ(k x)+ 1 ϕ(k y) (α) k + k 1 k + k 2 k + k 1 k + k 2  1 2 1 2   1 2 1 2  for any s<α, or k k ∗ k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (α) > 2 ϕ(k x)+ 1 ϕ(k y) (t) k + k 1 k + k 2 k + k 1 k + k 2  1 2 1 2   1 2 1 2  for all t>α. If t0 <α, there exists t>α such that k k ∗ k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (s) > 2 ϕ(k x)+ 1 ϕ(k y) (t ) k + k 1 k + k 2 k + k 1 k + k 2 0  1 2 1 2   1 2 1 2  k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (t) k + k 1 k + k 2  1 2 1 2  for any s < t0. By ([20],Property 7, p.64), we can find the set et0 with µ(et0 )= t0 such that t0 k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (t) dt k + k 1 k + k 2 Z0  1 2 1 2  k k = 2 ϕ(k x(t)) + 1 ϕ(k y(t)) dt. k + k 1 k + k 2 Zet0  1 2 1 2  By the proof of ([20], Property 7), we may infer that ∗ k2 k1 k2 k1 ϕ(k1x(s)) + ϕ(k2y(s)) ≥ lim− ϕ(k1x)+ ϕ(k2y) (t) k + k k + k t t k + k k + k 1 2 1 2 → 0  1 2 1 2  for µ-a.e s ∈ et0 . According to the definition of t0, we obtain

k2 k1 ϕ(k1x(s)) + ϕ(k2y(s)) k1 + k2 k1 + k2 k k ∗ > 2 ϕ(k x)+ 1 ϕ(k y) (t) ≥ 0 (3.2) k + k 1 k + k 2  1 2 1 2  for µ-a.e s ∈ et0 and each t > t0. Again using the definition of t0, we get that for

µ-a.e. s ∈ [0, 1)\et0 , there exists t(s) > t0 such that

k2 k1 ϕ(k1x(s)) + ϕ(k2y(s)) k1 + k2 k1 + k2 k k ∗ ≤ 2 ϕ(k x)+ 1 ϕ(k y) (t(s)). (3.3) k + k 1 k + k 2  1 2 1 2  By equality (3.1), let et0 (+) and et0 (−) be sets such that µ(et0 (+)) = µ(et0 (−)) = t0 and t0 k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (t) dt k + k 1 k + k 2 Z0  1 2 1 2  k(x + y) k(x − y) = ϕ (t) dt = ϕ (t) dt. (3.4) 2 2 Zet0 (+)   Zet0 (−)   NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 9

It is similar to the case of the set et0 , for µ-a.e s ∈ et0 (+) and for each t > t0, we get

k(x + y) k k ∗ ϕ (s) > 2 ϕ(k x)+ 1 ϕ(k y) (t). 2 k + k 1 k + k 2    1 2 1 2 

By inequality (3.2) and (3.3) we see et0 (+) ⊂ et0 . Hence by µ(et0 ) = t0 =

µ(et0 (+)), we get et0 (+) = et0 . Analogously, we have et0 = et0 (−). The equalities (3.1) and (3.4) yield

k(x + y) k(x − y) k k ϕ (t) = ϕ (t) = 2 ϕ(k x(t)) + 1 ϕ(k y(t)) 2 2 k + k 1 k + k 2     1 2 1 2 for µ-a.e. t ∈ et0 . By the convexity of ϕ, we know x(t) = y(t) = 0 for µ-a.e. t ∈ et0 . Which is a contradiction with (3.2). Case 2.2.2 Let

k k ∗ k k ∗ 2 ϕ(k x)+ 1 ϕ(k y) (0) = 2 ϕ(k x)+ 1 ϕ(k y) (α) k + k 1 k + k 2 k + k 1 k + k 2  1 2 1 2   1 2 1 2  k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (t) > 0 k + k 1 k + k 2  1 2 1 2  for some t>α. Define

k2 k1 A = t ∈ [0, 1) : ϕ(k1x(t)) + ϕ(k2y(t)) ( k1 + k2 k1 + k2 k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (0) , k + k 1 k + k 2  1 2 1 2  ) k(x + y) A = t ∈ [0, 1) :ϕ (t) + 2 (   k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (0) , k + k 1 k + k 2  1 2 1 2  ) k(x − y) A = t ∈ [0, 1) :ϕ (t) − 2 (   k k ∗ = 2 ϕ(k x)+ 1 ϕ(k y) (0) . k + k 1 k + k 2  1 2 1 2  )

The equality (3.1) and the convexity of ϕ follow that µ(A) >α, A+ ⊂ A, A− ⊂ A and min{µ(A+),µ(A−)} ≥ α. Let A0 = A+ ∩ A− = {t ∈ A : min{|x(t)|, |y(t)|} = 1 0}. Since α> 2 , it is easy to see that µ(A0) > 0. According to the definitions of 10 B. CHEN and W. GONG

A, A+ and A−, we get k(x + y) k(x − y) ϕ (t)χ (t) = ϕ (t)χ (t) 2 A0 2 A0     k2 k1 = ϕ(k1x(t)χA0 (t)) + ϕ(k2y(t)χA0 (t)). k1 + k2 k1 + k2

Hence x(t)χA0 (t) = y(t)χA0 (t) = 0 by the convexity of ϕ. By virtue of A0 ⊂ A, we get a contradiction. From Case 2.2.1 and Case 2.2.2, we infer that there exists t ∈ [0,α) such that the inequality k(x + y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t) 2 k + k 1 k + k 2     1 2 1 2  or k(x − y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t) 2 k + k 1 k + k 2     1 2 1 2  holds. If there exists t ∈ [0,α) such that k(x + y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t), 2 k + k 1 k + k 2     1 2 1 2  then by the right continuity of the rearrangement, we get k(x + y) α k(x + y) ∗ ρ = ϕ (t) ω(t) dt ϕ,ω 2 2   Z0    α k k ∗ < 2 ϕ(k x)+ 1 ϕ(k y) (t)ω(t) dt k + k 1 k + k 2 Z0  1 2 1 2  k k = 2 ϕ(k x)+ 1 ϕ(k y) k + k 1 k + k 2 1 2 1 2 1,ω

k2 ◦ k1 ◦ ≤ (k1kxkϕ,ω − 1) + ( k2kykϕ,ω − 1) k1 + k2 k1 + k2 = k − 1. Similarly, if there exists t ∈ [0,α) such that k(x − y) ∗ k k ∗ ϕ (t) < 2 ϕ(k x)+ 1 ϕ(k y) (t), 2 k + k 1 k + k 2     1 2 1 2  we obtain k(x − y) ρ

NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 11

◦ 4. Local uniform non-squareness of Λϕ,ω ◦ Theorem 4.1. Orlicz-Lorentz space Λϕ,ω[0, ∞) is locally uniformly non-square if and only if (a) ψ ∈ ∆2(R), ∞ (b) 0 ω(t) dt = ∞. R Proof. (Necessity) By Theorem 3.1 and Lemma 2.3, we can see ψ ∈ ∆2(R) and ∞ 0 ω(t) dt = ∞. (Sufficiency) By Lemma 2.1, we see that there exist ξ1,ξ2 ∈ (1, +∞) with R ◦ ◦ ξ1 <ξ2 such that ku ∈ (ξ1,ξ2) for any u ∈ B(Λϕ,ω[0, ∞)). Fix x ∈ S(Λϕ,ω[0, ∞)). ◦ For any y ∈ B(Λϕ,ω[0, ∞)). Fix k1 ∈ K(x), k2 ∈ K(y). Then

ξ k ξ 1 ≤ 2 ≤ 2 . ξ1 + ξ2 k1 + k2 ξ1 + ξ2

2k1k2 Let k = k1+k2 . By the range of k1 and k2, we find

2k1k2 2 2 k = = 1 1 ≤ 1 1 = ξ2. k1 + k2 k1 + k2 ξ2 + ξ2

Certainly there exist t1, t2 ∈ [0, +∞) such that 0 ≤ t1 < t2 < ∞,

t2 t2 ∗ ∗ ϕ(k1x (t))ω(t) dt ≥ ϕ(x (t))ω(t) dt := ξ ∈ (0, 1] (4.1) Zt1 Zt1 ∗ ∗ ∗ and x (s) > x (t) > x (w) for all s ∈ (0, t1), t ∈ (t1, t2) and w > t2 whenever ∗ ∗ t1 > 0, as well as x (t) > x (w) for all t ∈ (t1, t2) and w > t2 whenever t1 = 0.

By ([20], property 7, page 64), there exist sets et1 and et2 with µ(et1 )= t1 and

µ(et2 )= t2 such that

t1 t2 x∗(t) dt = |x(t)| dt and x∗(t) dt = |x(t)| dt Z0 Zet1 Z0 Zet2

(in the case t1 = 0 we have et1 = ∅). It is easy to see that et1 ( et2 . For ∗ ∗ u ∈ [ξ1x (t2),ξ2x (t1)], we can find η ∈ (0, 1) such that

k k ϕ 2 u ≤ (1 − η) 2 ϕ(u). k + k k + k  1 2  1 2 Define

B1 = {t ∈ et2 \et1 : x(t)y(t) ≥ 0},

B2 = {t ∈ et2 \et1 : x(t)y(t) < 0}. 12 B. CHEN and W. GONG

ξ By (4.1), we obtain ρϕ,ω(k1xχet2 \et1 ) ≥ ξ. Obviously, ρϕ,ω(k1xχB1 ) ≥ 2 or ξ ξ ρϕ,ω(k1xχB2 ) ≥ 2 . Assume ρϕ,ω(k1xχB1 ) ≥ 2 , then we have k(x(t) − y(t)) k k ϕ = ϕ 2 k x(t) − 1 k y(t) 2 k + k 1 k + k 2    1 2 1 2  k k ≤ ϕ max 2 k |x(t)|, 1 k |y(t)| k + k 1 k + k 2   1 2 1 2  k k ≤ ϕ 2 k x(t) + ϕ 1 k y(t) k + k 1 k + k 2  1 2   1 2  k2 k1 ≤ (1 − η) ϕ(k1x(t)) + ϕ(k2y(t)) k1 + k2 k1 + k2 for µ-a.e. t ∈ B1. Consequently, k(x(t) − y(t)) k k ηk ϕ ≤ 2 ϕ(k x(t)) + 1 ϕ(k y(t)) − 2 ϕ(k x(t)χ (t)). 2 k + k 1 k + k 2 k + k 1 B1   1 2 1 2 1 2 ξ ηξ Since ρϕ,ω(k1xχB1 ) ≥ 2 , we get kηϕ(k1xχB1 )k1,ω ≥ 2 . Then by the lower local uniform monotonicity of the Lorentz space L1,ω, we obtain k(x − y) k k ηk ϕ ≤ 2 ϕ(k x)+ 1 ϕ(k y) − 2 ϕ(k xχ ) 2 k + k 1 k + k 2 k + k 1 B1   1,ω 1 2 1 2 1 2 1,ω

k1 k2 ηk2 ≤ ϕ(k2y) + ϕ(k1x) − ϕ(k1xχ B1 ) k + k k + k k + k 1 2 1,ω 1 2 1 2 1,ω

k1 k2 k2 ≤ kϕ(k2y)k 1,ω + kϕ(k1x)k1,ω − δ1, k1 + k2 k1 + k2 k1 + k2 ηξ where δ1 = δ1 ϕ(k1x), 2 is a constant only depends on x. That is to say k(x − y)  k k k ρ ≤ 1 ρ (k y)+ 2 ρ (k x) − 2 δ ϕ,ω 2 k + k ϕ,ω 2 k + k ϕ,ω 1 k + k 1   1 2 1 2 1 2 k1 k2 k2 ≤ (k2 − 1) + (k1 − 1) − δ1 k1 + k2 k1 + k2 k1 + k2 k2 ≤ k − 1 − δ1. k1 + k2 By the definition of Orlicz norm, we get

◦ k(x−y) x − y 1+ ρϕ,ω( ) ≤ 2 2 k ϕ,ω

k2 ≤ 1 − δ1 k(k1 + k2) ≤ 1 − δ,

ξ1 where δ = ξ2(ξ1+ξ2) δ1 only depends on x. ξ x+y ◦ If ρϕ,ω(k1xχB2 ) ≥ 2 , we can prove 2 ϕ,ω ≤ 1 − δ analogously.

NON-SQUARENESS AND LOCAL UNIFORM NON-SQUARENESS 13

Theorem 4.2. Let α = sup{t ≥ 0 : ω(t) > 0} = 1, Then Orlicz-Lorentz space ◦ Λϕ,ω[0, 1) is locally uniformly non-square if and only if ψ ∈ ∆2(∞). Proof. (Necessity) By Lemma 2.3, it has been proved. (Sufficiency) By Lemma 2.1, we see that there exist ξ1,ξ2 ∈ (1, +∞) with ◦ ◦ ξ1 < ξ2 such that ku ∈ (ξ1,ξ2) for any u ∈ B(Λϕ,ω[0, 1)). Fix x ∈ S(Λϕ,ω[0, 1)). ◦ 2k1k2 For any y ∈ B(Λϕ,ω[0, 1)). Fix k1 ∈ K(x), k2 ∈ K(y). Let k = k1+k2 and Ax = {t ∈ [0, 1) : |x(t)| > 0}. Obviously, µ(Ax) > 0. Hence, there exist t1, t2 ∗ ∗ such that 0 ≤ t1 < t2 ≤ t0, 0 < x (t2) ≤ x (t1) < ∞ and

t2 t2 ∗ ∗ ϕ(k1x (t))ω(t) dt ≥ ϕ(x (t))ω(t) dt > 0. Zt1 Zt1 Analogously as in the proof of sufficiency in Theorem 4.1, we can find δ > 0 only depends on x such that x − y ◦ x + y ◦ min , ≤ 1 − δ. 2 2 ( ϕ,ω ϕ,ω)

References [1] Bennett C, Sharpley R. Interpolation of Operators. Academic Press, New York, 1988 [2] Birkhoff G. Lattice Theory. Amer. Math. Soc, Providence, RI, 1967 [3] Chen S. Geometry of Orlicz Spaces. Dissert. Math, 1996, 356 [4] Foralewski P, Hudzik H, Kolwicz P. Non-squareness properties of Orlicz-Lorentz function spaces. J. Inequal. Appl, 2013, 32: 25 pp [5] Foralewski P, Hudzik H, Kolwicz P. Non-squareness properties of Orlicz-Lorentz sequence spaces. J. Funct. Anal, 2013, 264(2): 605-629 [6] Foralewski P, Ko´nczak J. Local uniform non-squareness of Orlicz-Lorentz function spaces. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 2019, 113(4): 3425-3493 [7] Garc´ıa-Falset J, Llorens-Fuster E, Mazcu˜nan-Navarro E. Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings. J. Funct. Anal, 2006, 233(2): 494-514 [8] Hudzik H. Some class of uniformly nonsquare Orlicz-Bochner spaces. Comment. Math. Univ. Carolin, 1985, 26(2): 269-274 1 [9] Hudzik H. Uniformly non-ln Orlicz spaces with Luxemburg norm. Studia Math, 1985, 81(3): 271-284 1 [10] Hudzik H. Locally uniformly non-ln Orlicz spaces. Proceedings of the 13th winter school on abstract analysis. Rend. Circ. Mat. Palermo (2) Suppl. 1985, 10(10): 49-56 [11] Hudzik H, Kami´nska A. Monotonicity properties of Lorentz spaces. Proc. Amer. Math. Soc, 1995, 123(9): 2715-2721 [12] Hudzik H, Kami´nska A, Mastylo M. Monotonicity and Rotundity Properties in Banach Lattices. Rocky Mountain J. Math, 2000, 30(3): 933-950 [13] James R. C. Uniformly non-square Banach spaces. Ann. of Math, 1964, 80: 542-550 [14] James R. C. Super-reflexive spaces with bases. Pacific J. Math, 1972, 41: 409-419 [15] Kami´nska A. Some remarks on Orlicz-Lorentz spaces. Math. Nachr, 1990, 147: 29-38 [16] Kami´nska A, Lin P, Sun H. Y. Uniformly normal structure of Orlicz-Lorentz spaces. In- teraction between , harmonic analysis, and probability (Columbia, MO, 1994), 229-238, Lecture Notes in Pure and Appl. Math. 1996, 175, Dekker, New York. [17] Kolwicz P. Rotundity properties in Calder´on-Lozanovskii spaces. Houston J. Math, 2005, 31(3): 883-912 [18] Kondagunta S. Uniformly non-square Orlicz spaces. Nieuw Arch. Wisk, 1966, 14: 31-39 14 B. CHEN and W. GONG

[19] Krasnoselskii M. A, Rutickii Ya .B. Convex Functions and Orlicz Spaces. Problems of Contemporary Mathematics Gosudarstv. Izdat. Fiz.-Mat. Lit. Moscow 1958: 271 pp. [20] Krein SG, Petunin Jul, Semenov EM. Interpolation of Linear Operators. Transl. Math. Monogr, 1982, 54, Amer. Math. Soc, Providence. [21] Prus S. On uniform nonsquareness and uniform normal structure in Banach lattices. J. Convex Anal, 2014, 21(1): 167-177 [22] Shang S, Cui Y. Uniform nonsquareness and locally uniform nonsquareness in Orlicz- Bochner function spaces and applications. J. Funct. Anal, 2014, 267(7): 2056-2076 [23] Wang J, Chen Y. Rotundity and uniform rotundity of Orlicz-Lorentz spaces with the Orlicz norm. Houston J. Math, 2012, 38(1): 131-151 [24] Wu C, Ren L. Strict convexity of Orlicz-Lorentz spaces with an Orlicz norm. J. Math. (Wuhan), 1999, 19(2): 235-240

Department of Mathematics, Anhui Normal University, Wuhu 241000, China. Email address: [email protected]; [email protected]