Monotonicity and rotundity of Lorentz spaces Γp,w

Nonlinear Analysis 75 (2012) 2713–2723

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Monotonicity and rotundity of Lorentz spaces Γp,w Maciej Ciesielski a , Anna Kamińska b , Paweł Kolwicz a , Ryszard Płuciennik a,∗ a

Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60-965 Poznań, Poland

b

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, United States

article

info

Article history: Received 13 July 2011 Accepted 16 November 2011 Communicated by Enzo Mitidieri MSC: 46E30 46B20 46B42

abstract Criteria for rotundity, strict monotonicity, and lower local uniform monotonicity of the Lorentz spaces Γp,w of maximal functions are given under arbitrary nonnegative weight function w . Necessary conditions are also established for uniform monotonicity of the spaces Γp,w for 1 ≤ p < ∞. Moreover, the spaces Γ1,w that are uniformly monotone are characterized. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Lorentz spaces Uniform monotonicity Lower uniform monotonicity Strict monotonicity Rotundity

1. Introduction The first results on the rotundity properties of Lorentz spaces were established by Halperin in [1] and Altschuler in [2], who characterized the uniform convexity of Λp,w for 1 ≤ p < ∞ and the weight function w . The next amazing result was given in 1991 [3] by Kamińska, who showed the uniform convexity of the generalized Lorentz spaces Λφ,w under some additional assumptions on the weight function w . That result has been generalized in [4] to the class of Calderón–Lozanovski˘ı spaces. The next interesting result appeared in 1995 [5], and was devoted to strict and uniform monotonicity of the Lorentz spaces Λp,w . Recently, monotonicity properties have been studied in several papers [6,4,7–12]. In view of the previous results, the authors of [13] investigated criteria for strict convexity of the Lorentz space Γp,w whenever the weight function w is positive. The present paper is a continuation of the study of rotundity and monotonicity properties of the Lorentz spaces Γp,w . This paper consists of three sections. This first section, the introduction, contains all necessary notation and definitions which we use later. In Section 2, first we show under which conditions the space Γp,w contains an order-isometric copy of l∞ . Next we study necessary and sufficient conditions for strict monotonicity and lower local uniform monotonicity of Γp,w . The final result in this section investigates full criteria for rotundity of Γp,w without the assumption that w is positive. In Section 3, first we introduce the definition of regularity of the fundamental function of Γp,w and establish some auxiliary results which we need later. Next we establish necessary conditions for which the space Γp,w is uniformly monotone. Finally, we present complete criteria for uniform monotonicity of Γ1,w when α < ∞ or α = ∞.

∗

Corresponding author. Tel.: +48 618766051. E-mail addresses: [email protected] (M. Ciesielski), [email protected] (A. Kamińska), [email protected] (P. Kolwicz), [email protected] (R. Płuciennik). 0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.11.011

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M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

Let R, R+ , Q, Q+ , N be the sets of real numbers, non-negative real numbers, rational numbers, positive rational numbers, and positive integers, respectively. As usual, S (X ) (respectively, B(X )) stands for the unit sphere (respectively, the closed unit ball) of a Banach space (X , ‖·‖X ). Let L0 be a set of all (equivalence classes of) extended real-valued Lebesgue measurable functions on (0, α), where 0 < α ≤ ∞. By m we denote the Lebesgue measure on (0, α). For f ∈ L0 , we denote its distribution function by df (λ) = m {x ∈ [0, α) : |f (x)| > λ} ,

λ ≥ 0,

and its decreasing rearrangement by f ∗ (t ) = inf λ > 0 : df (λ) ≤ t ,





t ≥ 0.

Given f ∈ L0 , we denote the maximal function of f ∗ by f ∗∗ (t ) =

1

t

∫

t

f ∗ (s)ds. 0

It is well known that f ∗ ≤ f ∗∗ , f ∗∗ is nonincreasing and subadditive; i.e.,

(f + g )∗∗ ≤ f ∗∗ + g ∗∗

(1)

for any f , g ∈ L0 . For the properties of df , f ∗ , and f ∗∗ , the reader is referred to [14,15]. 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 f ∈ L0 , g ∈ E and |f | ≤ |g | a.e., then f ∈ E and ‖f ‖E ≤ ‖g ‖E . (2) There exists a strictly positive f ∈ E. The norm ‖ · ‖E or the space (E , ‖ · ‖E ) is called order continuous if, for any f ∈ E and |fn | ≤ |f | with |fn | → 0 a.e., we have ‖fn ‖E → 0. We say that E has the Fatou property if, for any sequence (fn ) such that 0 ≤ fn ∈ E for all n ∈ N, f ∈ L0 , fn ↑ f a.e., with supn∈N ‖fn ‖E < ∞, we have f ∈ E and ‖fn ‖E ↑ ‖f ‖E . We say that a Banach function space (E , ‖ · ‖E ) is rearrangement invariant (r.i., for short) if, whenever f ∈ L0 and g ∈ E with df = dg , then f ∈ E and ‖f ‖E = ‖g ‖E . 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 [14]). A Banach function space E is said to be strictly monotone (E ∈ (SM)) if, for each 0 ≤ g ≤ f with g ̸= f , we have ‖g ‖E < ‖f ‖E . We say that E is uniformly monotone (E ∈ (UM)) provided that, for every δ ∈ (0, 1), there exists η ∈ (0, 1) such that, for all 0 ≤ g ≤ f satisfying ‖f ‖E = 1 and ‖g ‖E ≥ δ , we have ‖f − g ‖E ≤ 1 − η (see [16,17]). The lattice E is called lower locally uniformly monotone (E ∈ (LLUM)) if, for any ϵ > 0 and f ∈ E+ , there exists δ > 0 such that, for any g ∈ L0 with 0 ≤ g ≤ f and ‖g ‖E ≥ ϵ , it holds that ‖f − g ‖E ≤ 1 − δ . Recall that a Banach space is rotund (or strictly convex) if ‖x1 + x2 ‖X < 2 whenever x1 and x2 are different points of S (X ). Rotundity is a basic notion in the investigation of geometry of Banach spaces. It is also applied in approximation theory [18,19] and ergodic theory [20]. Monotonicity properties (strict and uniform monotonicity) play an analogous role in the best dominated approximation problems in Banach lattices as do the respective rotundity properties (strict and uniform rotundity) in the best approximation problems in Banach spaces [12]. Moreover, they are very useful in many problems, as for instance in ergodic theory [21], since they provide a tool for estimating a norm. Recall also that strict and uniform monotonicity coincide with rotundity and uniform rotundity, respectively, on an order interval of the positive cone of E [17]. Given 1 ≤ p < ∞ and a nonnegative weight function w ∈ L0 , the Lorentz space Γp,w is a subspace of L0 such that α

∫ ‖f ‖ = ‖f ‖Γp,w :=

f ∗∗p (t )w(t )dt

1/p

< ∞.

0

In order to have Γp,w ̸= {0}, we need to assume that w is from class Dp ; that is, W (s) :=

s

∫

w(t )dt < ∞ and Wp (s) := s 0

p

α

∫

t −p w(t )dt < ∞

s

for all 0 < s ≤ α if α < ∞ and for all 0 < s < ∞ otherwise. It is well known that Γp,w , ‖ · ‖Γp,w is an r.i. Banach function space with the Fatou property. Notice that





 1/p φΓp,w (s) := φp,w (s) = W (s) + Wp (s) for any 0 < s ≤ α if α < ∞ and for all 0 < s < ∞ if α = ∞. It was proved [22] that when α = ∞ the space Γp,w ∞ has an order-continuous norm if and only if 0 w(t )dt = ∞. It is easy to observe that, if α < ∞, then 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 [13,22].

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

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2. Strict monotonicity and rotundity of Γp,w We start this section with a result on an order-isometric copy of l∞ . Next we investigate strict and lower local uniform monotonicity. Proposition 2.1. The space Γp,w contains an order-isometric copy of l∞ if and only if α = ∞ and

∞ 0

w(t )dt < ∞.

Proof (Necessity). Suppose that Γp,w contains an order-isometric copy of l∞ . Then Γp,w is not order continuous, and by the ∞ above discussion we have that α = ∞ and 0 w(t )dt < ∞. Sufficiency. Let α = ∞ and c =

∞

w(t )dt < ∞. Let An be a sequence of pairwise disjoint subsets of the interval (0, ∞) ∞ −1 such that m (An ) = ∞ for every n ∈ N. For any n ∈ N, define fn = c p χAn . Moreover, denoting A = n=1 An , we have 0

1

− 1p

χA . Then (supn fn )∗ = (supn fn )∗∗ = fn∗ = fn∗∗ = c − p χ(0,∞) and ∫ ∞   1p p −1 c p w(t )dt ‖ sup fn ‖Γp,w = ‖fn ‖Γp,w = =1

supn fn = c

n

0

for any n ∈ N. Applying now Theorem 1 in [23], we conclude the proof.



Theorem 2.2. The following conditions are equivalent: (a) Γp,w ∈ (LLUM); (b) Γp,w ∈ (SM); ∞ (c) either α = ∞ and 0 w(t )dt = ∞, or α < ∞ and m ((β, α) ∩ supp w) > 0 for any β ∈ (0, α). Proof. We apply Theorem 1 in [4], which states that E ∈ (LLUM) if and only if E ∈ (SM) and E ∈ (OC ). Hence the implication (a) ⇒ (b) follows immediately. Now, we consider separately the following two cases:  ∞ α = ∞ and α < ∞. Case 1. Suppose that α = ∞. Since the condition 0 w(t )dt = ∞ is equivalent to the fact that Γp,w ∈ (OC ), by Theorem 1 in [4], we conclude that (a) ⇒ (c) and (b) ∧ (c) ⇒ (a). Therefore, to finish the proof in Case 1, it is enough to show that (b) ⇔ (c). ∞ To prove the implication (b) ⇒ (c), suppose to the contrary that W (∞) = 0 w(t )dt < ∞. By Proposition 2.1, the space Γp,w contains an order-isometric copy of l∞ . Consequently, Γp,w cannot be strictly monotone. ∞ To prove the implication (c) ⇒ (b), assume that 0 w(t )dt = ∞ and that f , g ∈ Γp,w are such that 0 ≤ g ≤ f with ∗ ∗ ∗ g ̸= f . Then, by Lemma 3.2 in [24], g ≤ f and g ̸= f ∗ . Moreover, g ∗∗ ≤ f ∗∗ . Since g ∗ ̸= f ∗ , df (λ) < ∞ for all λ > 0, and thus there is a set C ⊂ (0, ∞) of positive measure such that g ∗ (t )χC (t ) < f ∗ (t )χC (t ) for any t ∈ C . Take t0 ∈ C such that m ((0, t0 ) ∩ C ) > 0. Then t0

∫

1

g ∗∗ (t0 ) =

t0

g ∗ (s)ds <

0

t0

∫

1 t0

f ∗ (s)ds = f ∗∗ (t0 ) .

0

Hence, for any t > t0 , g (t ) = ∗∗

<

1

t0

∫

t

g (s)ds + ∗

0

1 t

0



g (s)ds ∗

t0 t0

∫

t

∫

f (s)ds + ∗

t

∫

f (s)ds ∗



= f ∗∗ (t ) .

t0

Since W (t ) < ∞ for any t ∈ (0, ∞) and W (∞) = ∞, there is a set D ⊂ (t0 , ∞) of positive measure such that w(t ) > 0 for any t ∈ D. Therefore

‖g ‖pΓp,w =

∫

<

∫

t0

g ∗∗p (t )w(t )dt +

∞

∫

0

g ∗∗p (t )w(t )dt

t0 t0

f ∗∗p (t )w(t )dt +

∫

0

∞ p

t0

f ∗∗p (t )w(t )dt = ‖f ‖Γp,w ,

which finishes the proof of the implication (c ) ⇒ (b), and completes the proof of Case 1. Case 2. Assume that α < ∞. Then, by order continuity of Γp,w and Theorem 1 in [4], we conclude that (a) ⇔ (b). (b) ⇒ (c). Assume to the contrary that Γp,w ∈ (SM) and that there is β ∈ (0, α) such that w(t ) = 0 for a.e. t ∈ (β, α). Define f = χ(0,α)

and g = χ(0,β) +

1 2

χ(β,α) .

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M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

Then 0 ≤ g ≤ f , g ̸= f , f ∗ = f , and g ∗ = g on (0, α). Moreover, f ∗∗ (t ) = g ∗∗ (t ) = 1 for any t ∈ (0, β), whence β

∫ ‖g ‖Γp,w =

w(t )dt

 1p

0

= ‖f ‖Γp,w .

A contradiction. (c) ⇒ (b). Suppose that m ((β, α) ∩ supp w) > 0 for any β ∈ (0, α). Let f , g ∈ Γp,w be such that 0 ≤ g ≤ f and g ̸= f . Since df (λ) < ∞ for any λ > 0, g ∗ (t ) < f ∗ (t ) for t ∈ C , where C ⊂ (0, α) has positive measure. Take t0 ∈ C such that m ((0, t0 ) ∩ C ) > 0 and m ((t0 , α) ∩ C ) > 0. Then, repeating the same argument as in Case 1, we get that g ∗∗ (t ) < f ∗∗ (t ) for any t ∈ (t0 , α), and consequently ‖g ‖Γp,w < ‖f ‖Γp,w .  In [13, Theorem 3.1], the authors characterize strict convexity of the space Γp,w under the assumption that w(t ) > 0 a.e. on (0, ∞). Namely, they proved that, if the weight function w is positive, then Γp,w is strictly convex if and only if 1 < p < ∞ and W (∞) = ∞ for α = ∞. We present below a criterion for strict convexity of Γp,w without requiring that w is positive. This condition is essentially different from the one presented in [13]. Theorem 2.3. The space Γp,w is rotund if and only if the following conditions are satisfied: (a) 1 < p < ∞; (b) m ((β, γ ) ∩ supp w) > 0 for any interval (β, γ ) ⊂ (0, α) with β < γ ; (c) W (∞) = ∞ whenever α = ∞. Proof (Necessity). Let Γp,w be strictly convex. The necessity of (a) and (c) follows from Theorem 3.1 in [13], because in the proof of these facts the general assumption that the function w is positive was not used. Then it is enough only to prove the necessity of (b). If (b) is not satisfied, then there is an interval (β, γ ), β < γ , such that w(t ) = 0 for a.e. t ∈ (β, γ ). Define f = 2χ(0,β) + χ(β,γ )

1

and g = 2χ



0, 41 (3β+γ )

+ χ 1 (3β+γ ), 1 (β+γ ) + χ 1 (β+γ ),γ  . 4

2

2

2

Then f ̸= g , f = f ∗ , g = g ∗ and f ∗∗ (t ) = g ∗∗ (t ) for any t ∈ [0, β). Moreover, f ∗∗ (γ ) =

1

γ

∫

γ

f ∗ (t )dt =

0

1

γ

[2β + (γ − β)] =

β +γ γ

and g ∗∗ (γ ) =

=

1

γ 1

γ

∫

g ∗ (t )dt

0

[

1

2·

γ

4

(3β + γ ) +

β +γ



−

2

  ] 1 β +γ β +γ γ− = . (3β + γ ) + 4 2 2 γ 1

Hence, by the fact that f (t ) = g (t ) = 0 for t ≥ γ , we conclude that f ∗∗ (t ) = g ∗∗ (t ) also for any t ≥ γ . Obviously, f ∗∗ (t ) ̸= g ∗∗ (t ) only for t ∈ (β, γ ), but then ∗

γ

∫

f

∗∗p

β

∗

γ

∫

(t )w(t )dt =

β

g ∗∗p (t )w(t )dt = 0.

Consequently, β

∫ ‖f ‖Γp,w =

f

∗∗p

(t )w(t )dt +

∫

(t )w(t )dt +

∫

f

β

=

g

∗∗p

∗∗p

β

0

∫

γ

γ

g

(t )w(t )dt + (t )w(t )dt +

∫

∗∗p

β

0 f +g

Consider the function h = 2 . Obviously, h = h∗ (t ) ̸= g ∗ (t ) for t ∈ [β, γ ]. Moreover, h∗∗ (t ) = h

∗∗

(γ ) = =

1

γ

∫

γ

h (t )dt =

1

0 ∫

2

γ

∗

γ

1

0

1

∫

γ

f ∗ (t ) dt +

f ∗ (t ) + g ∗ (t )

1

∫

2

γ

γ 0

f

∗∗p

γ α

g γ

(t )w(t )dt

∗∗p

 1p

(t )w(t )dt

 1p

= ‖g ‖Γp,w .

h∗ , h∗ (t ) = f ∗ (t ) = g ∗ (t ) for t ∈ (0, α) \ [β, γ ] and h∗ (t ) ̸= f ∗ (t ) and f ∗∗ (t ) = g ∗∗ (t ) for t ∈ [0, β) and

γ 0

α

∫

g ∗ (t )dt

dt

 =

β +γ , γ

whence h∗∗ (t ) = f ∗∗ (t ) = g ∗∗ (t ) also for any t ≥ γ . Therefore h∗∗ (t ) ̸= f ∗∗ (t ) and h∗∗ (t ) ̸= g ∗∗ (t ) only for t ∈ [β, γ ], but then w(t ) = 0 for a.e. t ∈ (β, γ ), and consequently ‖h‖Γp,w = ‖f ‖Γp,w = ‖g ‖Γp,w , which contradicts the strict convexity of Γp,w .

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

2717

  Sufficiency. Suppose that conditions (a)–(c) are satisfied. Let f , g ∈ S Γp,w and f = ̸ g. First, suppose that there is a measurable set B of positive measure such that f ∗∗ (t ) ̸= g ∗∗ (t ) for every t ∈ B. Since the function r (s) = sp for 1 < p < ∞ is strictly convex and increasing on the interval (0, ∞), we have  ∗∗ p  p f +g 1 ∗∗ 1 1  ∗∗ p 1  ∗∗ p (t ) ≤ f (t ) + g ∗∗ (t ) < f (t ) + g (t ) (2) 2

2

2

2

2

for all t ∈ B. Now, assume that f ∗∗ (t ) = g ∗∗ (t ) for all t ∈ (0, α). Then we have t

∫

  (f ∗ (s) − g ∗ (s))ds = t f ∗∗ (t ) − g ∗∗ (t ) = 0 0

for any t ∈ (0, α), whence f ∗ (t ) = g ∗ (t )

a.e. in (0, α).

(3)

We claim that there exists a measurable set B of positive measure such that

(f + g )∗∗ (t ) < f ∗∗ (t ) + g ∗∗ (t )

(4)

for every t ∈ B. Suppose to the contrary that

(f + g )∗∗ (t ) = f ∗∗ (t ) + g ∗∗ (t ) for all t > 0. Hence t

∫

(f + g ) (s)ds = ∗

t

∫

0

f (s)ds + ∗

0

t

∫

g ∗ (s) ds 0

for any t > 0. Consequently, (f + g )∗ (t ) = f ∗ (t ) + g ∗ (t ) a.e. in (0, α). Since W (∞) = ∞, by facts 70 and 90 in [15, pp. 89–91], |f + g | (t ) = |f | (t ) + |g | (t ) for almost all t > 0, and there are sets et (f ), et (g ) such that et (f ) = et (g ) and m (et (f )) = t. Hence, by (3), we get

∫ et (f )

|f | (s)ds =

t

∫

f (s)ds = ∗

t

∫

0

g (s)ds = ∗

∫ et (g )

0

|g | (s)ds.

This implies that |f | = |g |, and so f (t ) = g (t ) a.e. in (0, α), which contradicts the assumption that f ̸= g and shows claim (4). Thus



f +g

∗∗

2

(t )

p

<



p

1 ∗∗ 1 f (t ) + g ∗∗ (t ) 2 2

≤

1  ∗∗ p 1  ∗∗ p f (t ) + g (t ) 2 2

(5)

for all t ∈ B. Now, by inequalities (2) and (5), we get



f +g

∗∗

2

(t )

p

<

1  ∗∗ p 1  ∗∗ p f (t ) + g (t ) 2 2

(6)

for any t ∈ B. Take t0 ∈ B. By the continuity of functions f ∗∗ and g ∗∗ , there exists an interval (β, γ ) such that t0 ∈ (β, γ ), and inequality (6) holds for any t ∈ (β, γ ). Hence, by the assumption that m ((β, γ ) ∩ supp w) > 0, we have

   f + g p    2 

α

∫



=

Γp,w

<

∫

γ

2 β ∫ 1

+

∗∗

2

0

1

f +g

(t )

p

w(t )dt

(f (t )) w(t )dt + ∗∗

p

1

∫

γ

(g ∗∗ (t ))p w(t )dt ∫ 1 (f ∗∗ (t ))p w(t )dt + (g ∗∗ (t ))p w(t )dt 2 β

2 (0,α)\(β,γ ) 1 1 = ‖f ‖pΓp,w + ‖g ‖pΓp,w = 1. 2 2

2 (0,α)\(β,γ )

Therefore 12 ‖f + g ‖Γp,w < 1, which finishes the proof of sufficiency.



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M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

3. Uniform monotonicity For p = 1, we provide different criteria on uniform monotonicity of Γp,w for α < ∞ and α = ∞. Let us first introduce a notion of regularity of the fundamental function φp,w which depends on whether α = ∞ or α < ∞. Definition 3.1. The fundamental function φp,w of Γp,w is called regular if inf

t ∈(0,α)

φp,w (t ) > 1. φp,w (t /2)

The following lemma characterizes an equivalent condition of regularity of the concave fundamental function φp,w of Γp,w . Lemma 3.2. Let φp,w be the concave fundamental function of Γp,w . Then φp,w is regular if and only if for any L > 1 there is KL > 1 such that for any t < α/L we have φp,w (Lt ) ≥ KL φp,w (t ). Proof. First, suppose that for any L > 1 there is KL > 1 such that for any t < α/L we have φp,w (Lt ) ≥ KL φp,w (t ). Letting φ

(t )

L = 2, we get that inft ∈(0,α) φ p,w(t /2) > 1. Now, for reverse conclusion, we assume that φp,w is regular. Since φp,w is concave, p,w t there exists a nonnegative decreasing function v such that φp,w (t ) = 0 v(s)ds. By Lemma 6 of [25], there is B > 0 such that φp,w (t ) ≤ Bt v(t ). Let L > 1 and Lt < α . Then 1≤

φp,w (Lt ) BLt v(Lt ) BLt v(Lt ) v(Lt ) ≤ ≤ = BL , φp,w (t ) φp,w (t ) t v(t ) v(t )

which gives

v(Lt ) 1 ≥ . v(t ) BL Hence, we have

 Lt v(s)ds φp,w (Lt ) (L − 1)t v(Lt ) (L − 1)t v(Lt ) L−1 =1+ t ≥1+ ≥1+ ≥1+ 2 , φp,w (t ) φp,w (t ) φp,w (t ) Bt v(t ) B L which completes the proof.



Lemma 3.3. If α = ∞ and φp,w is regular, then Proof. Suppose that

∞ 0

∞ 0

w(t )dt = ∞.

w(t )dt = M < ∞. Then

lim W (s) = lim W (2s) = M .

s→∞

s→∞

Since 0 ≤ sp

∞

∫

t −p w(t )dt ≤ sp s

∞

∫

s−p w(t )dt = s

∞

∫

w(t )dt → 0 s

as s → ∞, we have lim Wp (s) = lim Wp (2s) = 0.

s→∞

s→∞

Hence

 1/p φp,w (2s) W (2s) + Wp (2s) = lim = 1. s→∞ φp,w (s) s→∞ W (s) + Wp (s) lim

Therefore φp,w is not regular.



The next example shows that the implication in Lemma 3.3 cannot be reversed. Example 3.4. Let w(t ) = min 1,



Wp (s) =

1 p

,

and

1 t



for any t ≥ 0. Obviously,

W (s) = 1 + ln s.

∞ 0

w(t )dt = ∞. Taking s > 1, we have

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

2719

Hence

 lim

s→∞

φp,w (2s) φp,w (s)

1 + ln(2s) + 1/p

p = lim

= 1,

1 + ln(s) + 1/p

s→∞

which implies that φp,w is not regular. The next result gives the necessary condition for UM of Γp,w for any p ≥ 1. Theorem 3.5. If the space Γp,w is uniformly monotone, then the fundamental function φp,w is regular. Proof. First, suppose that 0 < α < ∞. Notice that

χ(∗∗ 0,s) (t ) =



for 0 ≤ t ≤ s for t > s.

1 s/t

Let L > 1. Without loss of generality, we can assume that L ≤ 2. Then, for any s > 0 with Ls < α , there is c > 0 such that Ls

[∫ c ‖χ(0,Ls) ‖Γp,w = c

α

∫

w(t )dt + (Ls)p

w(t ) tp

Ls

0

]1/p dt

= 1.

Setting f = c χ(0,Ls) and g = c χ(s,Ls) , we have that 0 ≤ g ≤ f and ‖f ‖Γp,w = 1. Notice that g ∗ = c χ(0,(L−1)s) , and so (L−1)s

[∫ ‖g ‖Γp,w = c

w(t )dt + (L − 1)p sp

0

We claim that ‖g ‖Γp,w ≥ δ :=

L−1

(L−1)s

1 4c p Lp

t

Ls

]1/p

w(t )

(L−1)s

dt

tp

.

. Consider the following two cases.

41/p L  α w(t ) (a) Let (Ls)p Ls t p dt ≥ 4c1p . Then ∫ α ∫ α w(t ) w(t ) p p s dt ≥ s dt ≥ p p

t

α

∫

,

whence

[

‖g ‖Γp,w ≥ c (L − 1) s

p p

(b) Otherwise, if (Ls)p Ls

∫

w(t )dt = 0

It follows that either

1 cp

w(t )

dt <

tp

− (Ls)p

α

∫

(L−1)s

tp

w(t )dt > w(t )dt

1 , 4c p

w(t )

Ls

0

tp

(L−1)s

w(t )

Ls

 (L−1)s

[∫ ‖g ‖Γp,w ≥ c

α

α

∫

1 2c p

]1/p

]1/p ≥

dt

L−1 41/p L

.

then 1

dt >

cp

−

1 4c p

=

3 4c p

 Ls

or (L−1)s w(t )dt >

>c

0

1 21/p c

≥

L−1 41/p L

.

1 . 4c p

If the first inequality holds, then

.

For the second inequality, we have

(L − 1) s

p p

∫

α (L−1)s

w(t ) tp

dt ≥ (L − 1) s

p p

≥

w(t )

Ls

(L−1)s

≥ (L − 1)p sp (L − 1)p

∫ ∫

(L−1)s

∫

Lp

Ls

tp

dt

w(t ) dt (Ls)p

Ls

(L−1)s

w(t )dt >

(L − 1)p 4Lp c p

,

whence

 ∫ ‖g ‖Γp,w ≥ c (L − 1)p sp

α (L−1)s

w(t ) tp

1/p dt

>

L−1 41/p L

,

which proves our claim. Since 0 ≤ g ≤ f , ‖f ‖Γp,w = 1, and ‖g ‖Γp,w ≥ δ , by the uniform monotonicity of Γp,w , there is η = η (δ) > 0 such that ‖f − g ‖Γp,w ≤ 1 − η. Moreover, f − g = c χ[0,s] . Hence

‖f − g ‖Γp,w = c φp,w (s) ≤ 1 − η = (1 − η) c φp,w (Ls).

2720

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

1 Consequently, φp,w (Ls) ≥ 1−η φp,w (s) for any s < α/L; i.e., φp,w is regular. If α = ∞, then we can repeat the above proof, substituting L = 2. 

Before we show the last main result we need the following lemma. Lemma 3.6. Let α = ∞, p = 1 and φ1,w be a regular fundamental function of Γ1,w . Then, for any n ∈ N ∪ {0} and any numbers a, b such that b ≥ a, the inequality

φ1,w (2n a + b) − φ1,w (2n a) ≥ 2



K −1

n+1

2

φ1,w (b)

holds with K from the definition of the regularity of φ1,w . If b < a and 2n b ≥ a for some n ∈ N ∪ {0}, then

φ1,w (b + a) − φ1,w (a) ≥ 2



K −1

n+1

2

φ1,w (b).

Proof. Let b ≥ a. Then, by the concavity of φ1,w (see Proposition 1.2 in [22]), we have

φ1,w (2n a + b) − φ1,w (2n a) ≥ φ1,w (2n b + b) − φ1,w (2n b). Observe that 2n

t

∫

χ[2n b,2n b+b) (s)ds ≥

t

∫

0

χ[2n b,2n+1 b) (s)ds,

0

for any t ∈ (0, ∞). Therefore, by the regularity of φ1,w , we have 1

(φ1,w (2n+1 b) − φ1,w (2n b))  n+1 K −1 1 n φ1,w (b). ≥ n (K − 1)φ1,w (2 b) ≥ 2

φ1,w (2n b + b) − φ1,w (2n b) ≥

2n 2

2

Now, assume that b < a and 2n b ≥ a. Since φ1,w is concave, we have

φ1,w (a + b) − φ1,w (a) ≥ φ1,w (2n b + b) − φ1,w (2n b) ≥ 2



K −1 2

n+1

φ1,w (b). 

In the proof of the next theorem, we will apply the characterization of uniform monotone Banach function spaces proved in [17]. Namely, a Banach function space is uniformly monotone if and only if, for any ε ∈ (0, 1), there is δ(ε) > 0 such that, for any f ∈ E+ with ‖f ‖E = 1 and any measurable set A such that ‖f χA ‖E ≥ ε , we have ‖f − f χA ‖E ≤ 1 − δ(ε). The main idea of the proof of Theorem 3.7 comes from [5], but for α < ∞ we also apply new techniques. Theorem 3.7. The space Γ1,w is uniformly monotone if and only if the fundamental function φ1,w of Γ1,w is regular. Proof. We present the proof for both cases for the sake of completeness. The necessity of the theorem follows from Theorem 3.5. To prove the sufficiency, assume that φp,w is a regular fundamental function and that α ≤ ∞. Let f be a simple function and A be a measurable set such that ‖f ‖Γ1,w = 1 and ‖f χA ‖Γ1,w ≥ ε > 0. We present the proof in two parts. Part A. First, we consider f as a decreasing function given by f =

n −

αi χ[ai−1 ,ai ) ,

i=1

where α1 > α2 > · · · > αn > 0 = αn+1 and 0 = a0 < a1 < · · · < an . Clearly, f = f ∗ and α

∫ 1 ≥ ‖f ‖Γ1,w =

f ∗∗ (t )w(t )dt =

n −

0

(αi − αi+1 )φ1,w (ai ).

i=1

Denote c0 = d0 = 0,

ci =

i −

m([aj−1 , aj ) ∩ A),

di =

j =1

i −

m([aj−1 , aj ) \ A),

j =1

for 1 ≤ i ≤ n. Since ‖f χA ‖Γ1,w ≥ ε , we get

ε≤

α

∫ 0

(f χA )∗∗ (t )w(t )dt =

n − (αi − αi+1 )φ1,w (ci ). i =1

(7)

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

Notice that ai = di + ci for 0 ≤ i ≤ n and (f − f χA )∗ = α

∫

t

 ∫ 1 t

0



F (s)ds w(t )dt = 0

∑n

i =1

2721

(αi − αi+1 )χ[0,di ) . Define F = f − (f − f χA )∗ . Then we have

n − (αi − αi+1 )(φ1,w (ai ) − φ1,w (di )).

(8)

i =1

By the regularity of φ1,w , there is K > 1 such that the inequality φ1,w (s) ≥ K φ1,w (s/2) is satisfied for any s < α . Obviously, K < 2, because φ1,w is concave. Choose r ∈ N so large that K1r ≤ 4ε . The set {1, 2, . . . , n} is denoted by Nn . Define N0 = {i ∈ Nn : 2r di ≥ ci }

and N1 = {i ∈ Nn : 2r di < ci },

and also N2 = {i ∈ N0 : ci ≥ di }, N3 = {i ∈ N0 : 2r ci < di }, N4 = {i ∈ N0 : ci < di and 2r ci ≥ di }. Observe that the sets Nj for j = 1, 2, 3, 4 are disjoint and that their union is equal to the set Nn . Since ‖f ‖Γ1,w ≤ 1, ai ≥ ci , and w is regular, we obtain n − − (αi − αi+1 )φ1,w (ci ) (αi − αi+1 )φ1,w (ai ) ≥

1 ≥

i∈N1

i =1

− − (αi − αi+1 )φ1,w (di ), (αi − αi+1 )φ1,w (2r di ) ≥ K r

>

i∈N1

i∈N1

which gives

− 1 ε (αi − αi+1 )φ1,w (di ) < r ≤ . K

i∈N1

(9)

4

Since ‖f ‖Γ1,w ≤ 1, ai ≥ di , and w is regular, we get

− − (αi − αi+1 )φ1,w (di /2r ) (αi − αi+1 )φ1,w (ci ) < i∈N3

i∈N3

≤

≤

1 − K r i∈N 3 n 1 −

K r i=1

(αi − αi+1 )φ1,w (di ) (αi − αi+1 )φ1,w (ai ) ≤

ε 4

‖f ‖Γ1,w ≤

ε 4

.

(10)

Now, in view of condition (7), either

− ε (αi − αi+1 )φ1,w (ci ) ≥

2

i∈N0

−

or

(αi − αi+1 )φ1,w (ci ) ≥

i∈N1

ε 2

.

(11)

If the second inequality is satisfied, then, by (8) and (9), we get α

∫

t

 ∫ 1 t

0



F (s)ds w(t )dt ≥

− (αi − αi+1 )(φ1,w (ai ) − φ1,w (di ))

≥

− (αi − αi+1 )(φ1,w (ci ) − φ1,w (di ))

0

i∈N1

i∈N1

≥

ε 2

−

ε 4

=

ε 4

.

Now, suppose that the first inequality in (11) holds. By (10), we have

ε 2

≤

−

(αi − αi+1 )φ1,w (ci )

i∈N0

≤

− i∈N2 ∪N4

<

− i∈N2 ∪N4

(αi − αi+1 )φ1,w (ci ) +

− (αi − αi+1 )φ1,w (ci ) i∈N3

ε (αi − αi+1 )φ1,w (ci ) + , 4

(12)

2722

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

whence

−

ε

(αi − αi+1 )φ1,w (ci ) ≥

4

i∈N2 ∪N4

.

(13)

Since ci ≥ di for i ∈ N2 , by the monotonicity, concavity, and regularity of the function φ1,w , we obtain

− − − (αi − αi+1 )(φ1,w (ai ) − φ1,w (di )) ≥ (αi − αi+1 )(φ1,w (2di ) − φ1,w (di )) ≥ (K − 1) (αi − αi+1 )φ1,w (di ) i∈N2

i∈N2

i∈N2

≥ (K − 1)

−

K −1 − (αi − αi+1 )φ1,w (ci /2r ) ≥ (αi − αi+1 )φ1,w (ci ). r 2

i∈N2

(14)

i∈N2

Now, we have to consider the cases α = ∞ and α < ∞ separately. Case 1. Suppose that α = ∞. Since ci < di and 2r ci ≥ di for i ∈ N4 , by Lemma 3.6, we obtain

− − (αi − αi+1 )(φ1,w (ai ) − φ1,w (di )) = (αi − αi+1 )(φ1,w (ci + di ) − φ1,w (di )) i∈N4

i∈N4

 ≥2

K −1 2

 r +1 − (αi − αi+1 )φ1,w (ci ).

(15)

i∈N4

 r +1 , K2−r 1 = (K −21r) . Case 2. Let α < ∞. Take L = 1 + 1/2r . Then, for any i ∈ N4 , α ≥ ai = di + ci ≥ Ldi . Furthermore, by the regularity of φ1,w , there exists KL > 1 such that, if t < α/L, then φ1,w (Lt ) ≥ KL φ1,w (t ). Consequently, − − (αi − αi+1 )(φ1,w (Ldi ) − φ1,w (di )) (αi − αi+1 )(φ1,w (ai ) − φ1,w (di )) ≥ Define γ∞ = min



(K −1)r +1 2r

i∈N4

i∈N4

≥ (KL − 1)

− − (αi − αi+1 )φ1,w (ci ). (αi − αi+1 )φ1,w (di ) ≥ (KL − 1)



Denote γα = min KL − 1,

(K −1) 2r



(16)

i∈N4

i∈N4

.

Now, coming back to both cases, i.e., α ≤ ∞, by (8), (13) and (14) and either (15) for α = ∞ or (16) for α < ∞, we have ∞

∫ 0

t

 ∫ 1 t



−

F (s)ds w(t )dt ≥ 0

(αi − αi+1 )(φ1,w (ai ) − φ1,w (di ))

i∈N2 ∪N4

ε (αi − αi+1 )φ1,w (ci ) ≥ γα .

−

≥ γα

4

i∈N2 ∪N4

(17)

  Taking δ(ε) = min 4ε , γα 4ε and combining the inequalities (12) and (17), we obtain that ∞

∫ 0

t

 ∫ 1 t



F (s)ds w(t )dt ≥ δ(ε). 0

Thus

‖f − f χA ‖Γ1,w =

∞

∫ 0

t

 ∫ 1 t



f (s) − F (s)ds w(t )dt 0

∞

∫ ≤ ‖f ‖Γ1,w −

t

 ∫ 1 t

0

0



F (s)ds w(t )dt ≤ ‖f ‖Γ1,w − δ(ε),

which finishes the proof of Part A. Part B. Now, let f be any simple function with compact support satisfying the assumptions of our theorem. Then, there is a measure-preserving transformation σ : [0, α) → [0, α) such that f ∗ (t ) = f ◦ σ (t ) a.e. on [0, α). Furthermore, (f − f χA )∗ = (f ∗ − f ∗ χσ −1 (A) )∗ and

  ‖f χA ‖Γ1,w = f ◦ σ χσ −1 (A) Γ

1,w

.

Since f ∗ is a decreasing simple function with

 ∗  f χσ −1 (A) 

Γ1,w

  = f ◦ σ χσ −1 (A) Γ

1,w

≥ ε,

M. Ciesielski et al. / Nonlinear Analysis 75 (2012) 2713–2723

2723

by the already proved Part A, there exists δ(ε) > 0 such that

≤ ‖f ‖Γ1,w − δ(ε).

  ‖f − f χA ‖Γ1,w = f ∗ − f ∗ χσ −1 (A) Γ

1,w

The set of simple functions with compact supports is dense in Γ1,w , because W (∞) = ∞. Hence Γ1,w is uniformly monotone.  Remark 3.8. Let p = 1, f ∈ Γ1,w . Since w ∈ D1 , by integration by parts, we obtain α

∫ ‖f ‖Γ1,w =

1 t

0

α

∫ =



f ∗ (s)ds w(t )dt =

f ∗ (t )

α

∫

  ∫ t  w(t ) f ∗ (s)ds − − dt

0

0

α

∫ t

0

α

t

 ∫

w(s) s

t

0



ds dt = ‖f ‖Λ1,v ,

w(s)

where v(t ) = t s ds. Hence Γ1,w = Λ1,v with equality of norms. On the other hand, if p > 1, then Γp,w is only isomorphic with Λp,w if and only if w satisfies condition Bp [26]. In [5], it was proved that the Lorentz space Λ1,v (0, ∞) is UM if and only if the weight v is regular; that is, inft >0

φ1,w (t ) =

∞

∫ t ∫ 0

s

w(u) u



V (2t ) V (t )

> 1, where V (t ) =

t 0

v(s)ds. In view of the above equation, we have

t

∫

v(s)ds = V (t )

du ds = 0 V (2t )

for any t ∈ (0, ∞). Since inft >0 V (t ) > 1 if and only if the fundamental function φ1,w of Γ1,w is regular in the sense of Definition 3.1, we immediately conclude our Theorem 3.7 for α = ∞ in view of [5]. Let us mention that we also present in Theorem 3.7 the case when α < ∞, which had not previously been discussed at all, and required new techniques. Acknowledgments The third and fourth authors were supported partially by the State Committee for Scientific Research, Poland, Grant No. N N210 362236. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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