Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γ_𝑝,𝑤

Abstract

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces 𝐸 and in particular on Lorentz spaces Γ_𝑝,𝑤={𝑓∶∫(𝑓^∗∗)^𝑝𝑤<∞} for any 0<𝑝<∞ and a nonnegative locally integrable weight function 𝑤, where 𝑓^∗∗ is a maximal function of the decreasing rearrangement 𝑓^∗ for any measurable function 𝑓 on (0,𝛼), with 0<𝛼≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of 𝑓∈𝐸, where 𝐸 is an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximants 𝑓_𝜖 of 𝑓∈Γ_𝑝,𝑤 or 𝑓∈Γ_{𝑝−1,𝑤}, 1<𝑝<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function 𝑓∈Γ_{𝑝−1,𝑤}, 1<𝑝<∞.

1. Introduction

The present paper is devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for best local approximations in r.i. quasi-Banach spaces and Lorentz spaces Γ_𝑝,𝑤 for 0<𝑝<∞. In 1910, Henry Lebesgue has proved one of the most famous differentiation theorem, which establishes a convergence of an integral average of any locally integrable function 𝑓 on the ball 𝐵(𝑣,𝜖)⊂ℝ^𝑛 to this function 𝑓, that is, for a.a. 𝑣∈ℝ^𝑛, 1𝜇(𝐵(𝑣,𝜖))_{𝐵(𝑣,𝜖)}𝑓(𝑡)𝑑𝑡⟶𝑓(𝑣)as𝜖⟶0.(1.1) In fact, Lebesgue’s integral average coincides with a best constant approximant on the space 𝐿²(ℝ^𝑛) [1]. The Lebesgue Differentiation Theorem (LDT) can be proved as a consequence of the weak maximal inequality 𝑡𝑀_𝐻𝑓^∗(𝑡)≤4^𝑛‖𝑓‖_𝐿¹(1.2) for the Hardy-Littlewood maximal function 𝑀_𝐻𝑓 where 𝑡∈ℝ^𝑛 and 𝑓∈𝐿¹(ℝ^𝑛) [2]. The interesting exploration of LDT was initiated by Stein in [3], who introduced the maximal functions on 𝐿^𝑝(ℝ^𝑛), associated with integral average, and applied it to obtain differentiation theorem in the notation of the norm in 𝐿^𝑝(ℝ^𝑛) for 1≤𝑝<∞. In the spirit of this idea many authors developed new techniques of recovering functions in quasi-Banach function spaces. The first results in this subject were obtained by Bastero et al. [4] in 1999, who have investigated Hardy-Littlewood maximal functions and weak maximal inequalities in rearrangement invariant quasi-Banach function spaces. The next paper was published by Mazzone and Cuenya in 2001 [1], about generalizations of the classical Lebesgue differentiation theorem for the best local approximation by constants over balls in 𝐿^𝑝+𝐿^∞ for 0≤𝑝<∞. They evaluated maximal inequalities for the maximal function related to best constant approximants and proved convergence theorem for best constant approximants. In 2008 [5] Levis et al. extended the best constant approximant operator from Orlicz-Lorentz spaces Λ_𝑤,𝜙 to the spaces Λ_𝑤,𝜙^′ and showed monotonicity of the extended operator. In view of this result, in 2009 [6] Levis established maximal inequalities for the maximal function associated with the best constant approximation and proved Lebesgue’s type differentiation theorem for best constant approximants and for integral averages expressed in terms of the modular corresponding to these spaces. Recently, the authors have characterized properties of an expansion of the best constant approximant operator from Lorentz spaces Γ_𝑝,𝑤 to the spaces Γ_{𝑝−1,𝑤}. The present paper is a continuation of the previous results and devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for local approximation in r.i. quasi-Banach space 𝐸 and in particular in Γ_𝑝,𝑤.

The paper consists of three sections and is organized as follows.

In the preliminaries, Section 2, we establish some basic notations and definitions and also recall some auxiliary results, which will be used later.

Sections 3 and 4 consist of the main results of the paper.

We start Section 3 proving measurability of the maximal function 𝑀^𝐸(𝑟)(𝑓) for 𝑓∈𝐸, that corresponds to the quasi-norm average of 𝑓, in r.i. quasi-Banach function spaces 𝐸. Next we establish two types of generalization of LDT in r.i. quasi-Banach function spaces 𝐸 and in Γ_𝑝,𝑤. In both types of LDT we employ the assumption of upper and lower 𝜙-estimates of Γ_𝑝,𝑤. The first main result, in the spirit of Stein [3], has been proved for any order continuous r.i. quasi-Banach function space. The statement is expressed in terms of quasi-norm averages. In order to show it we first prove the inequality for maximal function 𝑀^𝐸(𝑟)𝑓, which corresponds to a quasi-norm average of 𝑓∈𝐸. In the same spirit we also provide some conditions when the LDT does not hold in 𝐸 or in Γ_𝑝,𝑤. Next we continue our discussion with another type of LDT. In order to complete the second main result in this section we characterize conditions for which Lorentz space Γ_𝑝,𝑤 satisfies a lower (resp., an upper) 𝜙-estimate, where 𝜙 is the fundamental function of Γ_𝑝,𝑤. In view of this characterization we investigate pointwise convergence of the best constant approximants 𝑓_𝜖 to 𝑓 as 𝜖→0 whenever 𝑓∈Γ_𝑝,𝑤 and 1≤𝑝<∞, as well as the convergence of the extended best constant approximants 𝑓_𝜖 for any 𝑓∈Γ_{𝑝−1,𝑤} and 1<𝑝<2. We also present examples showing that this assumption is fulfilled by a large class of the spaces Γ_𝑞,𝑝. Finally, we investigate relations between maximal functions and the 𝐾-functional of Banach couple (Γ_𝑞,𝑝,𝐿^∞) in the spirit of the inequalities stated in [4]. We finish Section 3 with an example showing that 𝑀^Γ(𝑟)_𝑝,𝑤(𝑓) is not equivalent to the 𝐾-functional of the pair (Γ_𝑝,𝑤,𝐿^∞).

It is well known that the extension of the best constant approximant operator from Γ_1,𝑤 to 𝐿⁰, or from 𝐿₁ to 𝐿⁰, is a set valued function [1, 7]. Contrary to this in Theorem 4.5 we prove that the extended best constant approximant operator assumes a unique value for any 𝑓∈Γ_{𝑝−1,𝑤} and 1<𝑝<∞. To show the uniqueness we need to consider strict monotonicity of the right-hand Gâteaux derivative of the norm in Γ_𝑝,𝑤 at (𝑓−𝑢)𝜒_𝐴 in the direction 𝜒_𝐴 for any 𝑓∈Γ_{𝑝−1,𝑤} and 𝑢∈ℝ.

2. Preliminaries

Let ℝ and ℕ be the set of real and natural numbers, respectively. For any 𝐴⊂[0,𝛼) denote 𝐴^𝑐=[0,𝛼)⧵𝐴. Let 0<𝛼≤∞ and 𝜇 be the Lebesgue measure on ℝ. We denote by 𝐿⁰ the space of all extended real-valued 𝜇-measurable and finite functions a.e. on [0,𝛼). Denote the outer measure on ℝ by 𝜇_∗, the support of 𝑓∈𝐿⁰ by 𝕊(𝑓)=supp(𝑓), and the restriction of 𝑓 to the set 𝐴⊂[0,𝛼) by 𝑓∣_𝐴. By a simple (resp., step) function we mean a measurable function 𝑓∈𝐿⁰ with a finite measure support, which attains a finite number of values (resp., a finite number of values on a finite number of disjoint intervals). The distribution function 𝑑_𝑓 of a function 𝑓∈𝐿⁰ is given by 𝑑_𝑓(𝜆)=𝜇(𝑠∈[0,𝛼)∶|𝑓|(𝑠)>𝜆) for all 𝜆≥0. Two functions 𝑓,𝑔∈𝐿⁰ are called equimeasurable, if 𝑑_𝑓(𝜆)=𝑑_𝑔(𝜆) for all 𝜆≥0. We define the decreasing rearrangement for any 𝑓∈𝐿⁰ by 𝑓^∗(𝑡)=inf{𝑠>0∶𝑑_𝑓(𝑠)≤𝑡}, 𝑡≥0. For given 𝑓∈𝐿⁰ we denote the maximal function of 𝑓^∗ by 𝑓^∗∗(𝑡)=(1/𝑡)∫^𝑡0𝑓^∗(𝑠)𝑑𝑠. It is well known that 𝑓^∗≤𝑓^∗∗ and 𝑓^∗∗ is decreasing and subadditive, that is, (𝑓+𝑔)^∗∗≤𝑓^∗∗+𝑔^∗∗ for any 𝑓,𝑔∈𝐿⁰. For the properties of 𝑑_𝑓, 𝑓^∗, and 𝑓^∗∗ see [2, 8]. A subspace 𝐸⊂𝐿⁰ equipped with a quasinorm ‖⋅‖_𝐸 is called a quasinormed function space, if the following conditions are satisfied.(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.

If 𝐸 is complete, then it is said to be a quasi-Banach function space. We say that a quasi-Banach function space 𝐸 is rearrangement invariant (r.i. for short), if whenever 𝑓∈𝐿⁰ and 𝑔∈𝐸 with 𝑑_𝑓=𝑑_𝑔, then 𝑓∈𝐸 and ‖𝑓‖_𝐸=‖𝑔‖_𝐸 (see [2]). Throughout the paper we use the notation 𝐴≈𝐵, which means that the expressions 𝐴 and 𝐵 are equivalent; that is, 𝐴/𝐵 is bounded from both sides. Let 0<𝑝<∞ and 𝑤∈𝐿⁰ be a nonnegative weight function. Lorentz space Γ_𝑝,𝑤 is a subspace of 𝐿⁰ such that‖𝑓‖=‖𝑓‖_Γ𝑝,𝑤∶=^𝛼0𝑓^∗∗𝑝𝑤^1/𝑝=^𝛼0𝑓^∗∗𝑝(𝑡)𝑤(𝑡)𝑑𝑡^1/𝑝<∞.(2.1) Given a measurable set 𝐴⊂[0,𝛼) by Γ_𝑝,𝑤(𝐴) we denote the set of 𝑓∈𝐿⁰ restricted to 𝐴 and satisfying the above inequality. Unless we say otherwise, throughout the paper we assume that 𝑤 belongs to the class 𝐷_𝑝 (in short 𝑤∈𝐷_𝑝), whenever it satisfies the following conditions:𝑊(𝑠)∶=^𝑠0𝑤<∞,𝑊_𝑝(𝑠)∶=𝑠^𝑝^𝛼𝑠𝑡^−𝑝𝑤(𝑡)𝑑𝑡<∞(2.2) for all 0<𝑠≤𝛼 if 𝛼<∞ and for all 0<𝑠<∞ otherwise. These two conditions guarantee that Γ_𝑝,𝑤≠{0}. We also assume that𝑊_𝑝(𝑠)>0for0<𝑠<𝛼,𝑊(∞)=^∞0𝑤=∞if𝛼=∞.(2.3) Under these assumptions (Γ_𝑝,𝑤,‖⋅‖) is a rearrangement invariant (r.i. for short) quasi-Banach function space such that it has the Fatou property and the order continuous norm. Letting 0<𝑝,𝑞<∞ and 𝑤(𝑡)=𝑡^{𝑝/𝑞−1}, 𝑡∈(0,𝛼), the space Γ_𝑝,𝑤 will be denoted by Γ_𝑞,𝑝.

Unless we say otherwise, throughout this paper we assume that 𝜙 is the fundamental function of Γ_𝑝,𝑤 defined as 𝜙(𝑡)=‖𝜒_(0,𝑡)‖, 𝑡∈(0,𝛼), and 𝜙(0)=0. It is easy to show that the fundamental function 𝜙 is strictly increasing and continuous on [0,𝛼), lim_𝑡→∞𝜙(𝑡)=∞ and lim_𝑡→∞𝑓^∗(𝑡)=0 for 𝑓∈Γ_𝑝,𝑤. For more details about the properties of Γ_𝑝,𝑤 see [9].

Recall that for given 0<𝑝<∞, classical Lorentz space Λ_𝑝,𝑤 is a subspace of 𝐿⁰ such that‖𝑓‖_Λ𝑝,𝑤∶=^𝛼0𝑓^∗𝑝𝑤^1/𝑝=^𝛼0𝑓^∗𝑝(𝑡)𝑤(𝑡)𝑑𝑡^1/𝑝<∞.(2.4) In case when 𝑊 satisfies the Δ₂-condition, that is 𝑊(2𝑠)≤𝐶𝑊(𝑠) for all 𝑠>0 and some 𝐶>0, as well as 𝑊(∞)=∞, the space Λ_𝑝,𝑤 is a separable r.i. order continuous quasi-Banach function space [9]. The space Λ_𝑝,𝑤 is a r.i. Banach function space, whenever the weight 𝑤 is decreasing and 1≤𝑝<∞ [10]. Since 𝑓^∗≤𝑓^∗∗, we have the natural inclusion Γ_𝑝,𝑤⊂Λ_𝑝,𝑤. Moreover, Γ_𝑝,𝑤=Λ_𝑝,𝑤 if and only if 𝑤 satisfies condition 𝐵_𝑝, (𝑤∈𝐵_𝑝 for short) which means that there is 𝐴>0 such that for all 𝑠>0 we have 𝑊_𝑝(𝑠)≤𝐴𝑊(𝑠) [11–13].

Let (Ω₁,𝜇₁) and (Ω₂,𝜇₂) be 𝜎-finite measure spaces. A map 𝛾 from Ω₁ into Ω₂ is said to be a measure-preserving transformation, if whenever 𝐸 is a 𝜇₂-measurable subset of Ω₂, the set 𝛾⁻¹(𝐸)={𝑢∈Ω₁∶𝛾(𝑢)∈𝐸} is a 𝜇₁-measurable subset of Ω₁ and 𝜇₁(𝛾⁻¹(𝐸))=𝜇₂(𝐸). For given subsets 𝐴,𝐵⊂ℝ₊ such that 𝜇(𝐴)=𝜇(𝐵), there exists a measure-preserving transformation 𝛿∶𝐴→𝐵 [14, Theorem 17, page 410].

Definition 2.1 (see [15]). Let 𝑓,ℎ∈𝐿⁰. Denote 𝜏_(𝑓,ℎ)(𝑡)=𝑑_𝑓||𝑓||(𝑡)++𝜇𝜇𝑢𝑢∶∶||||𝑓𝑓||||((𝑢𝑢))==||||𝑓𝑓||||((𝑡𝑡)),,ℎℎ((𝑢𝑢))ssiiggnn((𝑓𝑓((𝑢𝑢))))>=ℎℎ((𝑡𝑡))ssigignn(𝑓(𝑓(𝑡)()𝑡)),𝑢≤𝑡(2.5) for all 𝑡∈[0,𝛼).

In 1970, Ryff proved in [16] that 𝜏_{(𝑓∣[0,1],0)}∶[0,1]→[0,1] is a measure-preserving transformation for any 𝑓∈𝐿⁰ and |𝑓|=𝑓^∗∘𝜏_{(|𝑓|[0,1],0)} a.e. on [0,1]. In 1993, Carothers et al. established in [15] that 𝜏_(𝑓,ℎ) is a measure preserving transformation from 𝕊(𝑓) onto 𝕊(𝑓^∗) such that |𝑓|=𝑓^∗∘𝜏_(𝑓,ℎ) a.e. on 𝕊(𝑓) for any 𝑓∈𝐿⁰ with 𝑑_𝑓(𝜆)<∞ for all 𝜆>0 and any ℎ∈𝐿⁰. Notice that for any 𝑓∈𝐿⁰ with 𝑑_𝑓(𝜆)<∞ for any 𝜆>0 and ℎ∈𝐿⁰, if 𝜇(𝑢∶|𝑓|(𝑢)=𝑣)=0 for every 𝑣>0 we have that 𝜏_(𝑓,ℎ)(𝑡)=𝑑_𝑓(|𝑓|(𝑡)) and it is the unique measure-preserving transformation up to measure zero satisfying |𝑓|=𝑓^∗∘𝜏_(𝑓,ℎ) a.e. on 𝕊(𝑓).

Definition 2.2. Let 𝑓,𝑔∈Γ_𝑝,𝑤 and let 𝜏_(𝑓,𝑔), 𝜏_{(𝑔∣𝕊(𝑔)⧵𝕊(𝑓),0)} be measure-preserving transformations given by Definition 2.1. Denote 𝜌_(𝑓,𝑔)(𝑠)=𝜏^(𝑓,𝑔)𝑑(𝑠)if𝑠∈𝕊(𝑓),_𝑓(0)+𝜏_{(𝑔∣𝕊(𝑔)⧵𝕊(𝑓),0)}(𝑠)if𝑠∈𝕊(𝑔)⧵𝕊(𝑓).(2.6)

Definition 2.3. Let 𝑓∈Γ_𝑝,𝑤 and 𝐴⊂[0,𝛼) with 𝜇(𝐴)<∞. Denote 𝐾_(𝑓,𝐴)(𝑢,𝑡)=1𝑆_{𝑝(𝑓,𝐴)}𝑡_𝐴1−2𝜒_{𝑓<𝑢}𝜒_(0,𝑡)𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)},(𝑢)=^𝛼0𝐾_(𝑓,𝐴)(𝑢,𝑡)(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡(2.7) for any 𝑢∈ℝ and 𝑡∈(0,𝛼).

Let (𝑋,‖⋅‖_𝑋) be a real normed space. Denote by 𝐵_𝑋 (resp., 𝑆_𝑋) the closed unit ball (resp., the unit sphere) of 𝑋. Assume that 𝑌 is a subset of 𝑋 and 𝑥 is an element of 𝑋. An element 𝑥̃∈𝑌 is called best approximant to 𝑥 from 𝑌 if‖𝑥−𝑥̃‖_𝑋=inf_𝑦∈𝑌‖𝑥−𝑦‖_𝑋.(2.8) A nonempty subset 𝑌 of 𝑋 is a set of uniqueness if for any element 𝑥∈𝑋 there is no more than one element 𝑥̃∈𝑌 satisfying (2.8). The set 𝑌 is a set of existence if for every element 𝑥∈𝑋 there is at least one element 𝑥̃∈𝑌 for which condition (2.8) holds. The set 𝑌 is a Chebyshev set if for every element 𝑥∈𝑋 there exists exactly one element 𝑥̃∈𝑌 satisfying (2.8), that is, if 𝑌 is both a set of uniqueness and a set of existence (for more details see [17]). Let 2^𝑌 be a collection of all subsets of 𝑌. A set value map 𝑇_𝑌∶𝑋→2^𝑌 is said to be best approximant operator, if it assumes for any 𝑥∈𝑋 a set of all best approximant elements to 𝑥 from 𝑌, that is,𝑇_𝑌(𝑥)=𝑥̃∈𝑌∶‖𝑥−𝑥̃‖_𝑋=inf_𝑦∈𝑌‖𝑥−𝑦‖_𝑋=𝑥̃∈𝑌∶‖𝑥−𝑥̃‖^𝑋≤‖𝑥−𝑦‖_𝑋,∀𝑦∈𝑌.(2.9) In case when 𝑋 is a norm function space and 𝑌 is a family of constant functions, then 𝑇_𝑌 is called best constant approximant operator, and each element 𝑥̃∈𝑇_𝑌(𝑥) is called best constant approximant to 𝑥∈𝑋 from 𝑌. Let 𝐴⊂[0,𝛼) with 0<𝜇(𝐴)<∞ and 𝕂(𝐴)={𝑐𝜒_𝐴∶𝑐∈ℝ}. It is well known that the set 𝑇_𝕂(𝐴)(𝑓) is convex, compact, and a set of existence for all 𝑓∈Γ_𝑝,𝑤 [18, 19]. Let’s recall some characterizations of best constant approximants over Lorentz spaces Γ_𝑝,𝑤.

Theorem 2.4 (see [20, Theorem 7.5]). Let 1≤𝑝<∞ and let 𝑓∈Γ_𝑝,𝑤⧵𝕂(𝐴). Then 𝑢∈𝕂(𝐴) is the best constant approximant of 𝑓 if and only if 𝑆^{𝑝(𝑓,𝐴)}(𝑢)≥0,𝑆^{𝑝(−𝑓,𝐴)}(−𝑢)≥0.(2.10)

Corollary 2.5 (see [20, Corollary 7.3]). For any 1<𝑝<∞ and 𝑤 positive, we have that 𝑇_𝕂(𝐴)(𝑓) is Chebyshev set for any 𝑓∈Γ_𝑝,𝑤; that is, there is an unique best constant approximant 𝑢∈𝕂(𝐴) to 𝑓.

Recently, in [7] it has been developed an existence of extension of the best constant approximant operators from Lorentz space Γ_1,𝑤 to 𝐿⁰, if 𝑤∈𝐷₁, and from Γ_𝑝,𝑤 to Γ_{𝑝−1,𝑤}, if 1<𝑝<∞ and 𝑤∈𝐷_𝑝−1. Now we recall definition of the extended operator 𝑇_(𝑝,𝐴) on 𝐿⁰, if 𝑝=1, and on Γ_{𝑝−1,𝑤}, if 1<𝑝<∞.

Definition 2.6 (see [7]). Let 𝐴⊂(0,𝛼) with 0<𝜇(𝐴)<∞ and let 𝑤∈𝐷_𝑝−1 if 1<𝑝<∞, and 𝑤∈𝐷₁ if 𝑝=1. Assume that 𝑓∈Γ_{𝑝−1,𝑤} if 𝑝>1, and 𝑓∈𝐿⁰ if 𝑝=1. Denote 𝑓_(𝑝,𝐴)=min𝑢∶𝑆^{𝑝(−𝑓,𝐴)}(−𝑢)≥0,𝑓_(𝑝,𝐴)=max𝑢∶𝑆^{𝑝(𝑓,𝐴)}(𝑢)≥0.(2.11) Then the extended best constant approximant operator is given by 𝑇_(𝑝,𝐴)(𝑓)=𝑓_(𝑝,𝐴)𝜒_𝐴,𝑓_(𝑝,𝐴)𝜒_𝐴.(2.12)

In fact, any 𝑢∈𝑇_(𝑝,𝐴)(𝑓) is called an extended best constant approximant of 𝑓. Notice that in view of Theorem 2.4, if 𝑓∈Γ_𝑝,𝑤 for 1≤𝑝<∞, then any 𝑢∈𝑇_(𝑝,𝐴)(𝑓) is a classical best constant approximant of 𝑓.

Definition 2.7 (see [4]). Let 0<𝑟<∞, (𝐸,‖⋅‖_𝐸) be a r.i. quasi-Banach function space, and let Φ∶[0,∞)→[0,∞) be an increasing bijection. 𝐸 is said to satisfy an upper (resp., a lower) Φ-estimate for ‖⋅‖^𝑟𝐸, if there exists 𝐶>0 such that for all 𝑛∈ℕ and (𝑓_𝑖)^𝑛𝑖=1⊂𝐸 with pairwise disjoint supports we have ‖‖‖‖^𝑛_𝑖=1𝑓_𝑖‖‖‖‖^𝑟𝐸≤𝐶Φ^𝑛_𝑖=1Φ⁻¹‖‖𝑓_𝑖‖‖^𝑟𝐸,(2.13) respectively Φ^𝑛_𝑖=1Φ⁻¹‖‖𝑓_𝑖‖‖^𝑟𝐸≤𝐶‖‖‖‖^𝑛_𝑖=1𝑓_𝑖‖‖‖‖^𝑟𝐸.(2.14)

In the case when Φ(𝑡)=𝑡^1/𝑝 and 𝑟=1, this definition covers the notions of the upper (resp., lower) 𝑝-estimate [21].

Let 0<𝑝<∞ and 𝑋 be a quasi-Banach function space. We denote by 𝑋^(𝑝)={𝑓∶|𝑓|^𝑝∈𝑋} the 𝑝-convexification of 𝑋 equipped with the quasinorm ‖⋅‖_𝑋^(𝑝)=‖|⋅|^𝑝‖^𝑋1/𝑝. Now we recall the definition of the maximal function for any r.i. quasi-Banach function space, that plays a crucial role in process of generalization of Lebesgue’s Differentiation Theorem in Γ_𝑝,𝑤.

Definition 2.8. Let 0<𝑟<∞, (𝐸,‖⋅‖_𝐸) be a r.i. quasi-Banach function space. For any 𝑓∈𝐸 we denote 𝑓^𝜖(𝑟)(𝑡)=‖‖𝑓𝜒^{𝐵(𝑡,𝜖)}‖‖^𝑟𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸(2.15) for all 𝜖>0 and 𝑡∈(0,𝛼). The maximal function 𝑀^𝐸(𝑟)𝑓∶(0,𝛼)→ℝ is given by 𝑀^𝐸(𝑟)𝑓(𝑡)=sup𝑓^𝜖(𝑟)(𝑡)∶𝜖>0,𝐵(𝑡,𝜖)⊂(0,𝛼)(2.16) for any 𝑡∈(0,𝛼), where 𝐵(𝑡,𝜖)=(𝑡−𝜖,𝑡+𝜖).

We finish the preliminaries with the following proposition needed further. It is a generalization of the well-known result, which in particular can be found in [22] for special case when 𝜑(𝑡)=𝑊(𝑡) on (0,∞). The proof of the proposition is quite standard and is provided for the sake of completeness.

Proposition 2.9. Let 𝛼=∞, 𝑓∈𝐿⁰, and let 𝜑∶[0,∞)→[0,∞) be an increasing continuous function. If lim_𝑡→∞𝜑(𝑡)=∞ or 𝑑_𝑓(𝜆)<∞ for all 𝜆>0, then sup_𝑡>0𝜑(𝑡)𝑓^∗(𝑡)=sup_𝑡>0𝑡𝜑𝑑_𝑓(𝑡).(2.17)

Proof. Notice first if lim_𝑡→∞𝜑(𝑡)=∞ and there exists 𝑡>0 such that 𝑑_𝑓(𝑡)=∞, then lim_𝑠→∞𝜑(𝑠)𝑓^∗(𝑠)=∞, and the conclusion follows.
Now assume that 𝑓=∑^𝑛𝑖=1𝑎_𝑖𝜒_𝐸𝑖 is a nonnegative simple function, where 𝑎_𝑖>𝑎_𝑖+1 for any 1≤𝑖≤𝑛−1, 𝑎_𝑛+1=0, and 𝐸_𝑖∩𝐸_𝑗=∅ whenever 𝑖≠𝑗. Then 𝑓^∗(𝑡)=∑^𝑛𝑖=1𝑎_𝑖𝜒_{[𝑚𝑖−1,𝑚𝑖)}(𝑡) for any 𝑡>0, where 𝑚₀=0 and 𝑚_𝑖=∑^𝑖𝑗=1𝜇(𝐸_𝑗) for 1≤𝑖≤𝑛. We claim that sup_𝑡>0𝜑(𝑡)𝑓^∗(𝑡)=max_{1≤𝑖≤𝑛}𝑎_𝑖𝜑𝑚_𝑖.(2.18) By monotonicity and continuity of 𝜑 we obtain sup_𝑡>0{𝜑(𝑡)𝑓^∗(𝑡)}≤max_{1≤𝑖≤𝑛}{𝑎_𝑖𝜑(𝑚_𝑖)}. On the other hand we have sup_𝑡>0{𝜑(𝑡)𝑓^∗(𝑡)}≥sup_𝑡>0{𝜑(𝑡)𝑎_𝑖𝜒_{[𝑚𝑖−1,𝑚𝑖)}(𝑡)}=𝑎_𝑖𝜑(𝑚_𝑖). Therefore sup_𝑡>0{𝜑(𝑡)𝑓^∗(𝑡)}≥max_{1≤𝑖≤𝑛}{𝑎_𝑖𝜑(𝑚_𝑖)}, which implies condition (2.18). Now we will show sup_𝑡>0𝑡𝜑𝑑_𝑓(𝑡)=max_{1≤𝑖≤𝑛}𝑎_𝑖𝜑𝑚_𝑖.(2.19) For 𝑡>0 we have 𝑡𝜑(𝑑_𝑓(𝑡))=𝑡∑^𝑛𝑖=1𝜑(𝑚_𝑖)𝜒_{[𝑎𝑖+1,𝑎𝑖)}(𝑡)≤max_{1≤𝑖≤𝑛}{𝑎_𝑖𝜑(𝑚_𝑖)}, and so sup_𝑡>0{𝑡𝜑(𝑑_𝑓(𝑡))}≤max_{1≤𝑖≤𝑛}{𝑎_𝑖𝜑(𝑚_𝑖)}. On the other hand, sup_𝑡>0{𝑡𝜑(𝑑_𝑓(𝑡))}≥𝑎_𝑖𝜑(𝑚_𝑖) for every 1≤𝑖≤𝑛, and consequently, sup_𝑡>0{𝑡𝜑(𝑑_𝑓(𝑡))}≥max_{1≤𝑖≤𝑛}{𝑎_𝑖𝜑(𝑚_𝑖)}, which provides (2.19). Both (2.18) and (2.19) show (2.17) for any nonnegative simple function 𝑓.
Now suppose that 𝑓 is a measurable function and 𝑑_𝑓(𝑡)<∞ for any 𝑡>0. Then, by standard arguments of existing a sequence of nonnegative simple functions (𝑓_𝑛) such that 𝑑_𝑓𝑛↑𝑑_𝑓 and 𝑓^∗𝑛↑𝑓^∗ as 𝑛→∞ we can show that sup_𝑡>0𝜑(𝑡)𝑓^∗(𝑡)=lim_𝑛→∞sup_𝑡>0𝜑(𝑡)𝑓^∗𝑛(𝑡)=lim_𝑛→∞sup_𝑡>0𝑡𝜑𝑑_𝑓𝑛(𝑡)=sup_𝑡>0𝑡𝜑𝑑_𝑓(𝑡)(2.20) and conclude the proof.

In fact, Proposition 2.9 describes the largest family of increasing and continuous functions 𝜑, for which (2.17) is satisfied. Indeed, let 𝛼=∞, 𝑓≡1, and 𝜑(𝑡)=2−(1/(1+𝑡)) for any 𝑡∈[0,∞). Then 𝑓^∗≡𝑓, 𝑑_𝑓(𝑡)=0 for all 𝑡≥1 and 𝑑_𝑓(𝑡)=∞ for any 𝑡∈[0,1). Clearly, 𝜑 is increasing and continuous and also lim_𝑡→∞𝜑(𝑡)=2. Therefore, sup_𝑡>0{𝜑(𝑡)𝑓^∗(𝑡)}=2 and sup_𝑡>0{𝑡𝜑(𝑑_𝑓(𝑡))}=∞, which implies that condition (2.17) does not hold.

3. Lebesgue’s Differentiation Theorems

The intention of this section is to establish generalizations of LDT in r.i. quasi-Banach function spaces 𝐸 in terms of the formulas expressed by quasinorm averages. We also focus on convergence of the best and the extended best constant approximant of 𝑓∈𝐸 to 𝑓, which is another type of LDT. First we introduce the notion of the differentiation property for a quasi-Banach function space 𝐸.

Definition 3.1. Let (𝐸,‖⋅‖_𝐸) be a quasi-Banach function space on [0,𝛼). We say that 𝐸 has the Lebesgue differentiation property (LDP), whenever for any 𝑓∈𝐸 and for a.a. 𝑡∈(0,𝛼) we have lim_𝜖→0‖‖(𝑓−𝑓(𝑡))𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=0.(3.1)

Observe that letting (𝐸,‖⋅‖_𝐸) be a quasi-Banach function space on [0,𝛼) with LDP, by the Aoki-Rolewicz theorem [23] there exist 0<𝑟≤1 and an equivalent 𝑟-norm ‖|⋅|‖_𝐸 to ‖⋅‖_𝐸 such that for any 𝑓∈𝐸 and for a.a. 𝑡∈(0,𝛼) we getlim_𝜖→0‖‖||𝑓𝜒_{𝐵(𝑡,𝜖)}||‖‖_𝐸‖‖||𝜒_{𝐵(𝑡,𝜖)}||‖‖_𝐸=||𝑓(𝑡)||.(3.2) If (𝐸,‖⋅‖_𝐸) is a normed space, then the quasinorm ‖|⋅|‖_𝐸 can be replaced by ‖⋅‖_𝐸.

In the next proposition we establish measurability of the maximal function 𝑀^𝐸(𝑟)𝑓.

Proposition 3.2. Let 0<𝑟<∞ and let 𝐸 be a r.i. order continuous quasi-Banach function space. If 𝑓∈𝐸, then the maximal function 𝑀^𝐸(𝑟)𝑓 is measurable on (0,𝛼).

Proof. Let 𝜖>0 and 𝑡∈(0,𝛼). We first observe that 𝑓^𝜖(𝑟)(𝑡) is continuous on (0,𝛼). In fact, for any 𝑡_𝑛→𝑡, lim_𝑛→∞𝜒_{𝐵(𝑡𝑛,𝜖)}=𝜒_{𝐵(𝑡,𝜖)}, and by order continuity of 𝐸 we obtain that lim_𝑛→∞‖𝑓𝜒_{𝐵(𝑡𝑛,𝜖)}‖^𝑟𝐸=‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖^𝑟𝐸 and lim_𝑛→∞‖𝜒_{𝐵(𝑡𝑛,𝜖)}‖_𝐸=‖𝜒_{𝐵(𝑡,𝜖)}‖_𝐸. Now by Fatou’s property of 𝐸 we have lim_{𝛿→𝜖−}‖‖𝑓𝜒_{𝐵(𝑡,𝛿)}‖‖_𝐸=‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸,lim_{𝛿→𝜖−}‖‖𝜒_{𝐵(𝑡,𝛿)}‖‖_𝐸=‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸.(3.3) Hence we have 𝑀^𝐸(𝑟)𝑓(𝑡)=sup𝑓^𝜖(𝑟)(𝑡)∶𝜖>0,𝜖∈ℚ,𝐵(𝑡,𝜖)⊂(0,𝛼),(3.4) and thus 𝑀^𝐸(𝑟)𝑓 is measurable.

Remark 3.3. If 𝑊 satisfies Δ₂ condition, then we obtain measurability of the maximal function 𝑀^Λ(𝑟)_𝑝,𝑤𝑓 for any 𝑓∈Λ_𝑝,𝑤, analogously as in case of the maximal function 𝑀^Γ(𝑟)_𝑝,𝑤𝑔 for any 𝑔∈Γ_𝑝,𝑤 when 𝛼=∞ and 𝑊(∞)=∞.

In view of Theorem  1 in [4], we investigate the so-called weak inequality for the maximal function 𝑀^𝐸(𝑟) whenever 𝐸 is a r.i. quasi-Banach function space.

Theorem 3.4. Let 0<𝑟<∞. If a r.i. quasi-Banach function space 𝐸 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟𝐸, then there exists 𝐶>0 such that for all 𝑓∈𝐸 and 𝜆>0 we have 𝜆𝜙𝑑_𝑀^𝐸(𝑟)_𝑓(𝜆)≤𝐶‖𝑓‖^𝑟𝐸.(3.5)

Proof. Assume that 𝜆>0. Denote Ω_𝜆=𝑡∈(0,𝛼)∶𝑀^𝐸(𝑟)𝑓(𝑡)>𝜆.(3.6) Clearly, by Proposition 3.2 we get that Ω_𝜆 is measurable for all 𝜆>0. Letting 𝑡∈Ω_𝜆, there exists 𝜖_𝑡>0 such that 𝐵(𝑡,𝜖_𝑡)⊂(0,𝛼) and 𝜆<𝑓^𝜖(𝑟)_𝑡(𝑡).(3.7) Let 𝑐<𝜇(Ω_𝜆) and denote 𝐵=⋃_{𝑡∈Ω𝜆}𝐵(𝑡,𝜖_𝑡). Since Ω_𝜆⊂𝐵, we get 𝑐<𝜇(𝐵). Hence, by regularity of the Lebesgue measure 𝜇 there is a compact set 𝐾⊂𝐵 such that 𝑐<𝜇(𝐾). By the fact that a collection 𝐷={𝐵(𝑡,𝜖_𝑡)∶𝑡∈Ω_𝜆} is an open covering of the set 𝐾 and by the Vitali covering lemma [2, Lemma 3.2], there exists a pairwise disjoint finite collection {𝐵(𝑡_𝑘,𝜖_𝑡𝑘)∶1≤𝑘≤𝑛}⊂𝐷 such that 𝜇(𝐾)≤4∑^𝑛𝑘=1𝜇(𝐵(𝑡_𝑘,𝜖_𝑡𝑘)). Therefore, by 𝜙(4𝑠)≤4𝜙(𝑠) and by (3.7) we get 𝜙(𝑐)≤𝜙(𝜇(𝐾))≤4𝜙^𝑛_𝑘=1𝜇𝐵𝑡_𝑘,𝜖_𝑡𝑘≤4𝜙^𝑛_𝑘=1𝜙⁻¹𝜆⁻¹‖‖𝑓𝜒_{𝐵(𝑡𝑘,𝜖𝑡𝑘)}‖‖^𝑟𝐸.(3.8) Hence, by assumption that 𝐸 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟𝐸, there is 𝐶>0 such that for any 1≤𝑘≤𝑛 we get 𝜙(𝑐)<4𝜙^𝑛_𝑘=1𝜙⁻¹𝜆⁻¹‖‖𝑓𝜒_{𝐵(𝑡𝑘,𝜖𝑡𝑘)}‖‖^𝑟𝐸≤4𝐶𝜆‖‖‖‖^𝑛_𝑘=1𝑓𝜒_{𝐵(𝑡𝑘,𝜖𝑡𝑘)}‖‖‖‖^𝑟𝐸≤4𝐶𝜆‖𝑓‖^𝑟𝐸.(3.9) Since 𝑐<𝜇(Ω_𝜆) is arbitrary, we obtain 𝜙(𝜇(Ω_𝜆))≤(4𝐶/𝜆)‖𝑓‖^𝑟𝐸, which finishes the proof.

In the next theorem we present Lebesgue’s differentiation property in the space 𝐸.

Theorem 3.5. Let 𝛼=∞ and let 𝐸 be a r.i. order continuous quasi-Banach function space 𝐸. If 𝐸 satisfies a lower 𝜙-estimate for ‖⋅‖_𝐸, then 𝐸 has LDP, that is, lim_𝜖→0‖‖(𝑓−𝑓(𝑡))𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=0(3.10) for all 𝑓∈𝐸 and for a.a. 𝑡∈(0,∞). If in addition 𝐸 is normable, then lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=||𝑓(𝑡)||.(3.11)

Proof. Observe first that the set of step functions with supports of finite measure is dense in 𝐸. The proof of this observation is standard, by density of the simple functions, which is equivalent to order continuity of 𝐸 and regularity of the Lebesgue measure 𝜇 on ℝ (cf. [8]).
Define an operator 𝐿∶(0,∞)→ℝ by 𝐿ℎ(𝑡)=limsup_𝜖→0ℎ^𝜖(1)(𝑡)=limsup_𝜖→0‖‖ℎ𝜒^{𝐵(𝑡,𝜖)}‖‖_𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸(3.12) for any ℎ∈𝐸 and 𝑡∈(0,∞). Assume that 𝜖>0 and 𝐵(𝑡,𝜖)⊂(0,∞) and also 𝑓∈𝐸, 𝑐∈ℝ. Let 𝑔=𝑏𝜒_𝐴 be a characteristic function of an open interval 𝐴. Notice that for a.a. 𝑡∈(0,∞) there exist 𝛿_𝑡>0 such that for all 0<𝜖<𝛿_𝑡 we have either 𝐵(𝑡,𝜖)⊂𝐴 or 𝐵(𝑡,𝜖)⊂𝐴^𝑐 and consequently ((𝑔−𝑐)𝜒_{𝐵(𝑡,𝜖)})^∗=|𝑔(𝑡)−𝑐|𝜒_(0,2𝜖). Therefore, (𝑔−𝑐)^𝜖(1)(𝑡)=‖‖(𝑔−𝑐)𝜒^{𝐵(𝑡,𝜖)}‖‖_𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=‖‖(𝑔(𝑡)−𝑐)𝜒^(0,2𝜖)‖‖_𝐸‖‖𝜒_(0,2𝜖)‖‖_𝐸=||𝑔(𝑡)−𝑐||(3.13) for a.a. 𝑡∈(0,∞) and for any 0<𝜖<𝛿_𝑡. Hence 𝐿(𝑔−𝑐)(𝑡)=limsup_𝜖→0(𝑔−𝑐)^𝜖(1)(𝑡)=||𝑔(𝑡)−𝑐||(3.14) for a.a. 𝑡∈(0,∞). Observe that the above equation can be proved analogously for any step function 𝑔 with support of finite measure. Let 𝜉 be a constant in the triangle inequality of the quasinorm ‖⋅‖_𝐸. Thus 𝐿(𝑓−𝑐)(𝑡)≤𝜉(𝐿(𝑓−𝑔)(𝑡)+𝐿(𝑔−𝑐)(𝑡))=𝜉𝐿(𝑓−𝑔)(𝑡)+||𝑔(𝑡)−𝑐||(3.15) for a.a. 𝑡∈(0,∞). Clearly 𝐿(𝑓−𝑔)(𝑡)≤𝑀^𝐸(1)(𝑓−𝑔)(𝑡), whence 𝐿(𝑓−𝑐)(𝑡)≤𝜉𝑀^𝐸(1)(𝑓−𝑔)(𝑡)+||𝑔(𝑡)−𝑐||(3.16) for a.a. 𝑡∈(0,∞). Now replacing 𝑐 by 𝑓(𝑡) we get 𝐿(𝑓−𝑓(𝑡))(𝑡)≤𝜉𝑀^𝐸(1)(𝑓−𝑔)(𝑡)+||𝑓(𝑡)−𝑔(𝑡)||(3.17) for a.a. 𝑡∈(0,∞). Define Ω_𝑠={𝑡∈(0,∞)∶𝐿(𝑓−𝑓(𝑡))(𝑡)>𝑠}(3.18) for any 𝑠>0. By Proposition 3.2, we have that 𝑀^𝐸(1)(𝑓−𝑔) is 𝜇-measurable. Recall [2, 8] that 𝑑_𝑓1+𝑓2(𝑠)≤𝑑_𝑓1(𝑠/2)+𝑑_𝑓2(𝑠/2) for any 𝑓₁,𝑓₂∈𝐿⁰. Thus in view of (3.17) we obtain for 𝑠>0, 𝜇_∗Ω_𝑠≤𝜇𝑡∈(0,∞)∶𝑀^𝐸(1)(𝑓−𝑔)(𝑡)+||𝑓−𝑔||(𝑡)>𝑠≤𝑑_{𝑀𝐸(1)(𝑓−𝑔)}𝜉2𝑠𝜉+𝑑^𝑓−𝑔2𝑠𝜉.(3.19) Now since 𝜙 satisfies the triangle inequality with constant 𝜉, we get 𝜙𝜇_∗Ω_𝑠≤𝜉𝜙𝑑_𝑀^𝐸(1)_{(𝑓−𝑔)}2𝑠𝜉+𝜉𝜙𝑑^𝑓−𝑔2𝑠𝜉(3.20) for every 𝑠>0. Observe that for any ℎ∈𝐸, 𝑡∈(0,𝛼), we have ℎ^∗(𝑡)𝜙(𝑡)≤‖ℎ^∗𝜒_(0,𝑡)‖_𝐸≤‖ℎ‖_𝐸. Thus, by Proposition 2.9 we have 𝜙𝑑_𝑓−𝑔2𝑠𝜉≤2𝑠𝜉sup_𝑡>0𝜙(𝑡)(𝑓−𝑔)^∗(𝑡)≤2𝜉𝑠‖𝑓−𝑔‖^𝐸.(3.21) Furthermore, by Theorem 3.4 there exists 𝐶>0 such that 𝜙𝑑_𝑀^𝐸(1)_{(𝑓−𝑔)}2𝑠𝜉≤2𝜉𝑠𝐶‖𝑓−𝑔‖^𝐸.(3.22) Therefore, for any step function 𝑔 and for all 𝑠>0, 𝜙𝜇_∗Ω_𝑠≤2𝜉²(𝐶+1)𝑠‖𝑓−𝑔‖^𝐸.(3.23) Hence we have 𝜙𝜇_∗Ω_𝑠=𝜙𝜇Ω_𝑠=0(3.24) for all 𝑠>0. So 𝐿(𝑓−𝑓(𝑡))(𝑡)=0 for a.a. 𝑡∈(0,∞), which shows the first formula.
The second formula results from the first one since ‖⋅‖_𝐸 is a norm in 𝐸.

Now we characterize the lower and upper 𝜙-estimate of Γ_𝑞,𝑝 on (0,∞), for 0<𝑝,𝑞<∞. Clearly in this case 𝑤(𝑡)=𝑡^{𝑝/𝑞−1} satisfies 𝐵_𝑝 condition. Thus Λ_𝑝,𝑤=Γ_𝑝,𝑤 and ‖⋅‖_Λ𝑝,𝑤≈‖⋅‖_Γ𝑝,𝑤. Hence by Theorems 3 and  7 in [9] and by Hölder’s inequality we obtain the following corollary.

Corollary 3.6. Let 𝛼=∞, 0<𝑝,𝑟<∞ and, 1<𝑞<∞.(i)If 𝑝≤𝑞, then Γ_𝑞,𝑝 satisfies a lower 𝜙-estimate for the functional ‖⋅‖^𝑟Γ_𝑞,𝑝, where 1≤𝑟, and an upper 𝜙-estimate for ‖⋅‖^𝑟Γ_𝑞,𝑝, where 𝑟≤𝑝/𝑞.(ii)If 𝑝≥𝑞, then Γ_𝑞,𝑝 satisfies an upper 𝜙-estimate for the functional ‖⋅‖^𝑟Γ_𝑞,𝑝, where 𝑟≤1, and a lower 𝜙-estimate for the functional ‖⋅‖^𝑟Γ_𝑞,𝑝, where 𝑟≥𝑝.

The immediate consequence of Theorem 3.5 is the next result, which describes under what conditions Γ_𝑝,𝑤 has LDP. The second part follows from the previous corollary.

Proposition 3.7. Let 0<𝑝<∞ and 𝛼=∞. If the Lorentz space Γ_𝑝,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖, then Γ_𝑝,𝑤 has LDP. Consequently, for 0<𝑝≤𝑞<∞ and 1<𝑞, Lorentz spaces Γ_𝑞,𝑝 on (0,∞) have LDP.

We will show next which spaces (𝐸,‖⋅‖_𝐸) do not have LDP with respect to ‖⋅‖^𝑟𝐸. Notice that the ratio ‖𝑓‖^𝑟1_𝐸/‖𝑔‖^𝑟2_𝐸 can be reduced easily to the quotient ‖𝑓‖^𝑟𝐸/‖𝑔‖_𝐸 for any 𝑓,𝑔∈𝐸, where 𝑔≠0. In view of this fact, we state the following theorem.

Theorem 3.8. Let 𝐸 be a r.i. quasi-Banach function space. (i) Let 1<𝑟<∞. If order continuous 𝐸 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟𝐸, then for any 𝑓∈𝐸 and for a.a. 𝑡∈(0,𝛼) we have lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖^𝑟𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=0.(3.25)(ii) Let 0<𝑟<1. Then for any 𝑓∈𝐸 and for a.a. 𝑡∈𝕊(𝑓) we have lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖^𝑟𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=∞.(3.26)

Proof. Let 𝐿∶(0,𝛼)→ℝ be the operator on 𝐸 given by 𝐿𝑓(𝑡)=limsup_𝜖→0{𝑓^𝜖(𝑟)(𝑡)}, 𝑓∈𝐸.
(i) Let 𝑔 be a step function with a finite measure support. Notice that for a.a. 𝑡∈(0,𝛼) and for small enough 𝜖>0 we have 𝑔^𝜖(𝑟)(𝑡)=‖‖𝑔𝜒^{𝐵(𝑡,𝜖)}‖‖^𝑟𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=|𝑔(𝑡)|^𝑟‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖^𝐸𝑟−1.(3.27) Therefore 𝐿𝑔(𝑡)=limsup_𝜖→0{𝑔^𝜖(𝑟)(𝑡)}=0 and thus 𝐿𝑓(𝑡)≤𝜉(𝐿(𝑓−𝑔)(𝑡)+𝐿𝑔(𝑡))=𝜉𝐿(𝑓−𝑔)(𝑡)≤𝜉𝑀^𝐸(𝑟)(𝑓−𝑔)(𝑡)(3.28) for a.a. 𝑡∈(0,𝛼). Denoting Ω_𝑠={𝑡∈(0,𝛼)∶𝐿𝑓(𝑡)>𝑠}, 𝑠>0, by Proposition 3.2, 𝑀^𝐸(𝑟)(𝑓−𝑔) is 𝜇-measurable. Now by Theorem 3.4 there exists 𝐶>0 such that for all 𝑠>0, 𝜙𝜇_∗Ω_𝑠≤𝜙𝑑_𝑀^𝐸(𝑟)_{(𝑓−𝑔)}𝜉𝑠≤𝜉𝑠𝐶‖𝑓−𝑔‖^𝑟𝐸,(3.29) which shows that 𝜇(Ω_𝑠)=0 and thus 𝐿𝑓(𝑡)=0 a.e.
(ii) Let 𝑓_𝑛↑|𝑓| a.e., where 0≤𝑓_𝑛 are step functions. Therefore, for a.a. 𝑡∈𝕊(𝑓) there is 𝑛∈ℕ such that 0<𝑓_𝑛(𝑡)≤|𝑓(𝑡)| and for small enough 𝜖>0 we have 𝑓_𝑛^𝜖(𝑟)(𝑡)=‖‖𝑓^𝑛𝜒_{𝐵(𝑡,𝜖)}‖‖^𝑟𝐸‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_𝐸=||𝑓_𝑛(𝑡)||^𝑟‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖^𝐸𝑟−1.(3.30) Thus, by the assumption that 0<𝑟<1 we get 𝐿𝑓(𝑡)≥𝐿𝑓_𝑛(𝑡)=liminf_𝜖→0𝑓^𝑛^𝜖(𝑟)(𝑡)=∞(3.31) for a.a. 𝑡∈𝕊(𝑓), and the proof is finished.

The next corollary follows directly from Theorem 3.8.

Corollary 3.9. (i) Let 0<𝑝<∞, 1<𝑟<∞. If Lorentz space Γ_𝑝,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟, then for any 𝑓∈Γ_𝑝,𝑤 and for a.a. 𝑡∈(0,𝛼) we have lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖^𝑟‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖=0.(3.32)(ii) Let 0<𝑝<∞, 0<𝑟<1. Then for any 𝑓∈Γ_𝑝,𝑤 and for a.a. 𝑡∈𝕊(𝑓) we have lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖^𝑟‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖=∞.(3.33)

The next result needed for further applications states conditions, which guarantee that Γ_𝑝,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟, where 0<𝑟<∞. We omit the proof of the following proposition.

Proposition 3.10. Assume that 0<𝑝,𝑟<∞, and Lorentz space Γ_𝑝,𝑤 satisfies a lower 𝑟-estimate. If 𝜙 is concave, then Γ_𝑝,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖^𝑟.

The following example shows that Theorem 3.8 is not an empty statement in case of Γ_𝑝,𝑤.

Example 3.11. Let 𝛼=∞, 𝑝=2, and 𝑤(𝑠)=ln(𝑠+1)+1 for all 𝑠∈[0,∞). Then for any 𝑓∈Γ_2,𝑤 and for a.a 𝑡∈(0,∞) we have lim_𝜖→0‖‖𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖^2Γ_2,𝑤‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ2,𝑤=0.(3.34)

Proof. Notice that for any 𝑡∈(0,∞), 𝑊(𝑡)=(𝑡+1)ln(𝑡+1),𝑊₂(𝑡)=𝑡(𝑡+1)ln(𝑡+1)−𝑡²ln(𝑡)+𝑡.(3.35) Consequently, 𝜙(𝑡)=‖‖𝜒_(0,𝑡)‖‖_Γ2,𝑤=(𝑡+1)²ln(𝑡+1)−𝑡²ln(𝑡)+𝑡)^1/2(3.36) for any 𝑡∈(0,∞). Since 𝑤 is increasing, 𝑤 satisfies 𝑅𝐵₂ condition. Moreover, we have ¹⁰𝑤(𝑡)𝑡²𝑑𝑡=^∞1𝑤(𝑡)𝑑𝑡=∞,𝑑𝑑𝑡𝑊²(𝑡)𝑡=ln1+1𝑡>0(3.37) for any 𝑡∈(0,∞), which concludes that 𝑊₂(𝑡)/𝑡 is increasing. Hence, by Theorem 3.3 [24] we obtain that Γ_2,𝑤 satisfies a lower 2-estimate. Now we claim that 𝜙 is concave. By simple calculations we get the second derivative on (0,∞), 𝜙^(𝑡)=(𝑡−ln(𝑡+1))ln(𝑡)−(𝑡+2)ln(𝑡+1)𝑡²ln(1+1/𝑡)+(2𝑡+1)ln(𝑡+1)+𝑡.(3.38) We observe that 𝑡²ln1+1𝑡+(2𝑡+1)ln(𝑡+1)+𝑡>0,(𝑡−ln(𝑡+1))ln(𝑡)<(𝑡+2)ln(𝑡+1).(3.39) Therefore, 𝜙^(𝑡)<0 for all 𝑡∈(0,∞), which implies concavity of 𝜙. Hence, by Proposition 3.10 we obtain that Γ_2,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖^2Γ_2,𝑤. Finally, in view of Corollary 3.9 we finish the proof.

The last part of this section is devoted to a pointwise convergence of the best and the extended best constant approximant of 𝑓 to 𝑓, that is, another type of LDT.

The first result is a corollary of Proposition 3.7. In fact, let 𝑡∈(0,∞), 𝜖>0 be such that 𝐵(𝑡,𝜖)⊂(0,∞). Assume that 𝑓_𝜖(𝑡)∈𝑇_{(𝑝,𝐵(𝑡,𝜖))}(𝑓) for 𝑓∈Γ_𝑝,𝑤. By definition of the best constant approximant we get||𝑓(𝑡)−𝑓_𝜖(𝑡)||≤‖‖(𝑓(𝑡)−𝑓)𝜒^{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤+‖‖𝑓^𝜖(𝑡)−𝑓𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤≤2‖‖(𝑓(𝑡)−𝑓)𝜒^{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ𝑝,𝑤.(3.40) Now applying Proposition 3.7 we get that 𝑓_𝜖(𝑡)→𝑓(𝑡) as 𝜖→0. Therefore we get the following theorem.

Theorem 3.12. Let 𝛼=∞ and 1≤𝑝<∞. If the space Γ_𝑝,𝑤 satisfies a lower 𝜙-estimate for ‖⋅‖_Γ𝑝,𝑤, then for every 𝑓∈Γ_𝑝,𝑤 we have 𝑓_𝜖(𝑡)⟶𝑓(𝑡)as𝜖⟶0(3.41) for a.a. 𝑡∈(0,∞), where 𝑓_𝜖(𝑡)∈𝑇_{(𝑝,𝐵(𝑡,𝜖))}(𝑓) is a best constant approximant of 𝑓.

In order to prove the next approximation theorem let us first establish an inequality related to extended best constant approximants of any function 𝑓∈Γ_{𝑝−1,𝑤} for 1<𝑝<2.

Lemma 3.13. Let 1<𝑝<2, 𝐴⊂[0,𝛼), 0<𝜇(𝐴)<∞, and let 𝑓∈Γ_{𝑝−1,𝑤}, 𝑓≥0. Then for any 𝑢𝜒_𝐴∈𝑇_(𝑝,𝐴)(𝑓) extended best constant approximant of 𝑓 and for a.a. 𝑠∈(0,𝛼) we have ||𝑢−𝑓(𝑠)||^𝑝−1‖‖𝜒_𝐴‖‖^𝑝Γ_𝑝,𝑤≤5‖‖(𝑓−𝑓(𝑠))𝜒_𝐴‖‖^Γ𝑝−1_{𝑝−1,𝑤}.(3.42)

Proof. Let 𝑠∈(0,𝛼) and |𝑓(𝑠)|<∞. Since (𝜒_𝐴)^∗∗≤1, by subadditivity of the maximal function and the power function 𝑣^𝑝−1 for 1<𝑝<2 we get ||𝑢−𝑓(𝑠)||^𝑝−1‖‖𝜒_𝐴‖‖^𝑝Γ_𝑝,𝑤=≤^𝛼0𝜒_𝐴^∗∗(𝑡)(𝑢−𝑓(𝑠))𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝛼0+‖‖(𝑓𝜒_𝐴−𝑓^∗∗((𝑠𝑡)))𝜒(𝑓_𝐴−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡‖‖^Γ𝑝−1_{𝑝−1,𝑤}.(3.43) We will finish the proof under assumption that 𝑓(𝑠)≥𝑢. In the other case the proof is similar. Since 𝑢𝜒_𝐴∈𝑇_(𝑝,𝐴)(𝑓)=[𝑓_(𝑝,𝐴)𝜒_𝐴,𝑓_(𝑝,𝐴)𝜒_𝐴], by Corollary 2.11 [7] and by Hardy-Littlewood inequality we obtain ^𝛼0𝜒_𝐴^∗∗≤2(𝑡_𝛼0)(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡1𝑡_{𝐴∩{𝑓≤𝑢}}≤2_𝛼0𝜒_(0,𝑡)𝜌_{((𝑢−𝑓)𝜒𝐴,𝜒𝐴)}𝑑𝜇(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡≤2‖‖(𝑓𝜒−𝑓(𝑠))_{𝐴∩{𝑓≤𝑢}}𝜒^∗∗_𝐴(𝑡)(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡‖‖^Γ𝑝−1_{𝑝−1,𝑤}+2^𝛼0𝜒_{𝐴∩{𝑓≤𝑢}}^∗∗(𝑡)(𝑓(𝑠)−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡.(3.44) Moreover, by assumption that 1<𝑝<2 we have 𝜒_{𝐴∩{𝑓≤𝑢}}^∗∗(𝑡)≤=𝜒𝜒_{(0,𝜇(𝐴∩{𝑓≤𝑢}))}(𝑡)+𝜇(𝐴∩{𝑡𝑓≤𝑢})^𝑝−1𝜒_{[𝜇(𝐴∩{𝑓≤𝑢}),𝛼)}(𝑡)_{𝐴∩{𝑓≤𝑢}}^{∗∗(𝑝−1)}(𝑡)(3.45) for any 𝑡∈(0,𝛼). Consequently, by the fact that 𝑓(𝑠)≥𝑢 we obtain ^𝛼0𝜒_{𝐴∩{𝑓≤𝑢}}^∗∗≤(𝑡)(𝑓(𝑠)−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝛼0≤(𝑓(𝑠)−𝑢)𝜒_{𝐴∩{𝑓≤𝑢}}^{∗∗(𝑝−1)}(𝑡)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝛼0≤‖‖(𝑓−(𝑓𝑓((𝑠𝑠)))−𝜒_𝐴𝑓)𝜒_{𝐴∩{𝑓≤𝑢}}^{∗∗(𝑝−1)}(𝑡)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡‖‖^Γ𝑝−1_{𝑝−1,𝑤}.(3.46) Hence, by conditions (3.43) and (3.44) we finish the proof.

Corollary 3.14. Let 1<𝑝<∞ and 𝑤∈𝐷_𝑝−1. Assume that 𝜙 and 𝜓 are the fundamental functions of Γ_𝑝,𝑤 and Γ_{𝑝−1,𝑤}, respectively. If one of the following conditions(i)there exists 𝐴>0 such that 𝑊_𝑝−1(𝑡)≤𝐴𝑊_𝑝(𝑡) for all 𝑡∈(0,𝛼),(ii)𝑤∈𝐵_𝑝−1, holds, then 𝜙^𝑝≈𝜓^𝑝−1.

Proof. Notice that for all 𝑡∈(0,𝛼), 𝜙(𝑡)^𝑝=𝑊(𝑡)+𝑡^𝑝^𝛼𝑡𝑠^−𝑝𝑤(𝑠)𝑑𝑠≤𝑊(𝑡)+𝑡^𝑝^𝛼𝑡𝑠^−𝑝+1𝑡⁻¹𝑤(𝑠)𝑑𝑠=𝜓(𝑡)^𝑝−1.(3.47) Suppose now that there is 𝐴>0 such that 𝑊_𝑝−1(𝑡)≤𝐴𝑊_𝑝(𝑡) for all 𝑡∈(0,𝛼). Therefore, 𝜓(𝑡)^𝑝−1=𝑊(𝑡)+𝑊_𝑝−1(𝑡)≤𝑊(𝑡)+𝐴𝑊_𝑝(𝑡)≤𝐴𝜙(𝑡)^𝑝(3.48) for any 𝑡∈(0,𝛼), and so 𝜙^𝑝≈𝜓^𝑝−1.
Now assume that 𝑤∈𝐵_𝑝−1. Then there exists 𝐵>0 such that for all 𝑓∈Γ_{𝑝−1,𝑤} we have ‖𝑓‖^Γ𝑝−1_{𝑝−1,𝑤}≤𝐵‖𝑓‖^Λ𝑝−1_{𝑝−1,𝑤}. Consequently, 𝜓(𝑡)^𝑝−1=‖‖𝜒_(0,𝑡)‖‖^Γ𝑝−1_{𝑝−1,𝑤}≤𝐵‖‖𝜒_(0,𝑡)‖‖^Λ𝑝−1_{𝑝−1,𝑤}=𝐵𝑊(𝑡)≤𝐵𝜙(𝑡)^𝑝(3.49) for any 𝑡∈(0,𝛼), and the proof is completed.

Theorem 3.15. Let 𝛼=∞, 1<𝑝<2, 𝑤∈𝐷_𝑝−1 and 𝑓∈Γ_{𝑝−1,𝑤}, 𝑓≥0. Assume that 𝜙 and 𝜓 are the fundamental functions of Γ_𝑝,𝑤 and Γ_{𝑝−1,𝑤}, respectively. If Lorentz space Γ_{𝑝−1,𝑤} satisfies a lower 𝜓-estimate for ‖⋅‖_{Γ𝑝−1,𝑤} and 𝜙^𝑝≈𝜓^𝑝−1, then 𝑓_𝜖(𝑡)⟶𝑓(𝑡)as𝜖⟶0(3.50) for a.a. 𝑡∈(0,∞), where 𝑓_𝜖(𝑡)∈𝑇_{(𝑝,𝐵(𝑡,𝜖))}(𝑓) is an extended best constant approximant of 𝑓.

Proof. Let 𝜖>0 and 𝑡∈(0,∞), |𝑓(𝑡)|<∞. By Lemma 3.13 for any 𝑓_𝜖(𝑡)∈𝑇_{(𝑝,𝐵(𝑡,𝜖))}(𝑓) we get ||𝑓_𝜖(𝑡)−𝑓(𝑡)||≤5^1/(𝑝−1)‖‖(𝑓−𝑓(𝑡))𝜒_{𝐵(𝑡,𝜖)}‖‖_{Γ𝑝−1,𝑤}‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖^{Γ𝑝/(𝑝−1)}_𝑝,𝑤.(3.51) Since 𝜙^𝑝≈𝜓^𝑝−1, there is 𝐵>0 such that for all 𝛿>0 we have ‖𝜒_{𝐵(𝑡,𝛿)}‖_{Γ𝑝−1,𝑤}≤𝐵‖𝜒_{𝐵(𝑡,𝛿)}‖^{Γ𝑝/(𝑝−1)}_𝑝,𝑤. Consequently, ||𝑓_𝜖(𝑡)−𝑓(𝑡)||≤5^1/(𝑝−1)𝐵‖‖(𝑓−𝑓(𝑡))𝜒^{𝐵(𝑡,𝜖)}‖‖_{Γ𝑝−1,𝑤}‖‖𝜒_{𝐵(𝑡,𝜖)}‖‖_{Γ𝑝−1,𝑤}(3.52) for any 𝑡∈(0,∞) and 𝜖>0. Hence, by Proposition 3.7 and by assumption that Γ_{𝑝−1,𝑤} satisfies a lower 𝜓-estimate for ‖⋅‖_{Γ𝑝−1,𝑤} we obtain that 𝑓_𝜖(𝑡)→𝑓(𝑡) as 𝜖→0 for a.a. 𝑡∈(0,∞).

Now we present a specific family of Lorentz spaces Γ_𝑝,𝑤 for which Theorems 3.12 and 3.15 are fulfilled.

Corollary 3.16. Let 𝛼=∞ and 1<𝑞<∞. If one of the following conditions(i)1≤𝑝≤𝑞 and 𝑓∈Γ_𝑞,𝑝,(ii)1<𝑝<2 and 𝑓∈Γ_{𝑞,𝑝−1}, 𝑓≥0, is satisfied, then for a.a. 𝑡∈(0,∞) we have 𝑓_𝜖(𝑡)⟶𝑓(𝑡)as𝜖⟶0,(3.53) where 𝑓_𝜖(𝑡)∈𝑇_{(𝑝,𝐵(𝑡,𝜖))}(𝑓) is the best constant approximant of 𝑓∈Γ_𝑝,𝑤 and the extended best constant approximant of 𝑓∈Γ_{𝑝−1,𝑤}.

Proof. Suppose that condition (i) is fulfilled. Immediately, by Theorem 3.12 and Corollary 3.6 we complete the first case. Now assume that the condition (ii) is satisfied. Define 𝑤(𝑡)=𝑡^{(𝑝−1)/𝑞−1} for 𝑡∈(0,∞). Since 1<𝑝<2, we have that 𝑊(𝑡)=𝑞𝑝−1𝑡^{(𝑝−1)/𝑞},𝑊_𝑝−1(𝑡)=𝑞(𝑝−1)𝑞−𝑝+1𝑡^{(𝑝−1)/𝑞}(3.54) for all 𝑡∈(0,∞), which implies that 𝑤∈𝐷_𝑝−1∩𝐵_𝑝−1. Consequently, by Definition 2.6 there is 𝑇_{(𝑝,𝐵(𝑡,𝜖))} the extended best constant approximant operator on Γ_{𝑞,𝑝−1}. Moreover, by Corollary 3.14 the fundamental functions 𝜙 and 𝜓 of the spaces Γ_𝑞,𝑝 and Γ_{𝑞,𝑝−1} respectively, satisfy 𝜙^𝑝≈𝜓^𝑝−1. Hence, by Corollary 3.6 we get that Γ_{𝑞,𝑝−1} satisfies a lower 𝜓-estimate for ‖⋅‖_{Γ𝑞,𝑝−1}. Finally, by Theorem 3.15 we obtain that 𝑓_𝜖(𝑡), an extended best constant approximant of 𝑓, converges to 𝑓(𝑡) as 𝜖→0 for a.a. 𝑡∈(0,∞).

Observe that the 𝐾-functional of the couple (Γ_𝑞,𝑝,𝐿^∞) can be expressed equivalently by ‖𝑓^∗𝜒_(0,𝑡)‖_Γ𝑞,𝑝/𝜙(𝑡) for any 𝑓∈Γ_𝑞,𝑝 and 𝑡∈(0,∞) [25]. The next result on inequalities between maximal function 𝑀^(1)𝑞,𝑝𝑓 and the 𝐾-functional of (Γ_𝑞,𝑝,𝐿^∞) follows immediately as a consequence of Theorems 1 and  2 in [4], Corollary 3.6, and Remark 3.3.

Corollary 3.17. Let 𝛼=∞, 𝑝∈[1,∞), and 𝑞∈(0,∞).(i)If 1<𝑝≤𝑞<∞, then there exists 𝐶>0 such that for all 𝑓∈Γ_𝑞,𝑝 and 𝑡>0 we have𝑀^Γ(1)_𝑞,𝑝𝑓^∗(𝑡)≤𝜙𝐶(𝑡)‖‖𝑓^∗𝜒_(0,𝑡)‖‖_Γ𝑞,𝑝.(3.55)(ii)If 1<𝑞≤𝑝<∞, then there exists 𝐶>0 such that for all 𝑓∈Γ_𝑞,𝑝 and 𝑡>0 we get𝑀^Γ(1)_𝑞,𝑝𝑓^∗(𝑡)≥𝜙𝐶(𝑡)‖‖𝑓^∗𝜒_(0,𝑡)‖‖_Γ𝑞,𝑝.(3.56)(iii)If 𝑝=1<𝑞<∞, then there is 𝐶>0 such that for any 𝑓∈Γ_𝑞,𝑝 and 𝑡>0 we obtain the inequality in the condition (i).

The decreasing rearrangement of the maximal function (𝑀^Γ(1)_𝑞,𝑝𝑓)^∗ and 𝐾-functional of (Γ_𝑞,𝑝,𝐿^∞) are not equivalent; that is the opposite inequalities to the ones in Corollary 3.17 do not hold. It is similar as in spaces Λ_𝑞,𝑝, 1≤𝑝<𝑞<∞, as we see in the next example [4].

Example 3.18. Let 1<𝑟<∞. There exists a function 𝑓∈𝐿⁰ such that for all 𝑡>0,‖‖𝑓^∗𝜒_(0,𝑡)‖‖_Γ𝑟,1=^∞0𝑓^∗𝜒_(0,𝑡)^∗∗(𝑠)𝑠^(1/𝑟)−1𝑑𝑠=∞,𝑀^Γ(1)_𝑟,1𝑓^∗(𝑡)<∞.(3.57)

Proof. Since 𝑤(𝑠)=𝑠^(1/𝑟)−1 satisfies 𝐵₁ condition, there is 𝐵>0 such that ‖‖ℎ𝜒_{𝐵(𝑡,𝜖)}‖‖_Λ𝑟,1≤‖‖ℎ𝜒_{𝐵(𝑡,𝜖)}‖‖_Γ𝑟,1≤𝐵‖‖ℎ𝜒_{𝐵(𝑡,𝜖)}‖‖_Λ𝑟,1(3.58) for all ℎ∈Γ_𝑟,1, 𝑡∈(0,∞) and suitable 𝜖>0. Hence 𝑀^Λ(1)_𝑟,1𝑓≥(1/𝐵)𝑀^Γ(1)_𝑟,1𝑓. Therefore, by Remark 3.3 we conclude that 𝑀^Λ(1)_𝑟,1𝑓^∗(𝑡)≥𝐵1𝑀^Γ(1)𝑟,1𝑓^∗(𝑡)(3.59) for all 𝑡>0. Now, by Theorem  3 in [4] there exists 𝑓∈𝐿⁰ such that ‖𝑓^∗𝜒_(0,𝑡)‖_Λ𝑟,1=∞ and (𝑀^Λ(1)_𝑟,1𝑓)^∗(𝑡)<∞ for all 𝑡>0, which implies that ‖𝑓^∗𝜒_(0,𝑡)‖_Γ𝑟,1=∞ and (𝑀^Γ(1)_𝑟,1𝑓)^∗(𝑡)<∞ for any 𝑡>0.

Remark 3.19. Example 3.18 can be expanded to the case when 1≤𝑝<𝑞<∞. Indeed if 1≤𝑝<𝑞<∞ and 𝑟=𝑞/𝑝 then we have Γ_𝑞,𝑝=Γ^(𝑝)𝑟,1 and ‖⋅‖_Γ𝑞,𝑝≈‖|⋅|^𝑝‖^Γ1/𝑝_𝑞,𝑝. As well as (𝑀^Γ(1)_𝑟,1|⋅|^𝑝)^1/𝑝≈𝑀^Γ(1)_𝑞,𝑝 (·) on {𝑓∈Γ_𝑞,𝑝(𝐶)∶𝐶isconvex,𝜇(𝐶)<∞}. This can be obtained by simple observation that the weight function 𝑤(𝑡)=𝑡^1/𝑟−1 satisfies condition 𝐵_𝑠 for any 1/𝑟<𝑠<∞, and then Γ_𝑟,1=Λ_𝑟,1 and Γ_𝑞,𝑝=Λ_𝑞,𝑝 as sets with equivalent norms.

4. Uniqueness of the Extended Best Constant APProximant on Γ_𝑃,𝑊

In this section we prove uniqueness of the expansion of the best constant approximant for any 𝑓∈Γ_{𝑝−1,𝑤}, if 1<𝑝<∞ and 𝑤>0. Namely, we show that the extended best constant approximant operator 𝑇_(𝑝,𝐴) becomes the point value operator. Throughout this section we assume that 𝐴⊂[0,𝛼) is a set of positive and finite measure. Let the truncation of any function 𝑓∈𝐿⁰ be defined as 𝑓^(𝑛)(𝑠)=𝑓(𝑠) if |𝑓(𝑠)|<𝑛, and 𝑓^(𝑛)(𝑠)=sign𝑓(𝑠)𝑛 if |𝑓(𝑠)|≥𝑛.

Lemma 4.1. Let 1<𝑝<∞ and 𝑓∈Γ_{𝑝−1,𝑤}. Then 𝑓^(𝑛)∈Γ_𝑝,𝑤 for all 𝑛∈ℕ.

Proof. Since 𝑓∈Γ_{𝑝−1,𝑤}, we get that 𝑑_𝑓(𝜆)<∞ for every 𝜆>0. Let 𝜖∈(0,1). Then 𝑐_𝑛∶=𝜇𝑠∶||𝑓(𝑠)||≥𝑛≤𝜇𝑠∶||𝑓(𝑠)||>𝑛−𝜖=𝑑_𝑓(𝑛−𝜖)<∞(4.1) for any 𝑛∈ℕ. Moreover, (𝑓^(𝑛))^∗∗≤𝑓^∗∗ and (𝑓^(𝑛))^∗∗≤𝑛 for any 𝑛∈ℕ. Therefore, ^𝛼0𝑓^(𝑛)^∗∗𝑝(𝑡)𝑤(𝑡)𝑑𝑡≤≤𝑛𝑛^𝑝_𝑝𝑊𝑐_𝑛+^𝛼𝑐_𝑛𝑓^(𝑛)^{∗∗(𝑝−1)}(𝑡)𝑓^(𝑛)^∗∗(𝑡)𝑤(𝑡)𝑑𝑡𝑊𝑐_𝑛+𝑛^𝛼𝑐_𝑛𝑓^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡<∞(4.2) for all 𝑛∈ℕ, which concludes the proof.

We omit the standard proof of the next lemma.

Lemma 4.2. Let 𝑓∈𝐿⁰ and 𝑢∈ℝ. Then for a.a. 𝑠∈𝐴 there exists 𝑁(𝑢,𝑠)∈ℕ such that for all 𝑛≥𝑁(𝑢,𝑠) we have the following.(i){𝑣∈𝐴∶𝑓^(𝑛)(𝑣)+𝑓^(𝑛)(𝑠)<2𝑢}={𝑣∈𝐴∶𝑓(𝑣)+𝑓(𝑠)<2𝑢}.(ii){𝑣∈𝐴∶𝑓^(𝑛)(𝑣)=𝑓^(𝑛)(𝑠),𝑣≤𝑠}={𝑣∈𝐴∶𝑓(𝑣)=𝑓(𝑠),𝑣≤𝑠}.(iii){𝑣∈𝐴∶𝑓^(𝑛)(𝑣)>𝑓^(𝑛)(𝑠)}={𝑣∈𝐴∶𝑓(𝑣)>𝑓(𝑠)}.

Now we discuss a convergence of a sequence of functions 𝑆^𝑝(𝑓_(𝑛),𝐴)(𝑢) to the function 𝑆^{𝑝(𝑓,𝐴)}(𝑢) as 𝑛→∞ for all 𝑢∈ℝ and 1<𝑝<∞, whenever 𝑓∈Γ_{𝑝−1,𝑤}.

Theorem 4.3. Let 1<𝑝<∞ and 𝑓∈Γ_{𝑝−1,𝑤}. Then for all 𝑢∈ℝ, lim_𝑛→∞𝑆^𝑝(𝑓_(𝑛),𝐴)(𝑢)=𝑆^{𝑝(𝑓,𝐴)}(𝑢).(4.3)

Proof. Denote 𝐸=𝐴∩{𝑓≥𝑢}. Notice that there exists 𝑁_𝑢∈ℕ such that 𝑁_𝑢>|𝑢| and for all 𝑛≥𝑁_𝑢 we have 𝐸=𝐴∩{𝑓^(𝑛)≥𝑢}. Since 𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)} is a measure-preserving transformation, by Definition 2.3 we get 𝑆^{𝑝(𝑓,𝐴)}(𝑢)=2^𝛼01𝑡_{𝐴∩{𝑓≥𝑢}}−𝜒_(0,𝑡)𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝛼0𝜒_𝐴^∗∗(𝑡)(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡.(4.4) We claim that for a.a. 𝑠∈𝐸 there exists 𝑁(𝑢,𝑠)∈ℕ such that for all 𝑛≥𝑁(𝑢,𝑠) we have 𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠)=𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠).(4.5) Let 𝑠∈𝐸 and |𝑓(𝑠)|<∞. If 𝑓(𝑠)>𝑢, then by Lemma 4.2 there exists 𝑁_(𝑢,𝑠)∈ℕ such that for any 𝑛≥𝑁_(𝑢,𝑠) we obtain 𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠)==+𝜇𝜇𝜇(𝑣𝑣(𝑣∈∈∈𝐴𝐴𝐴∶∶∶𝑓𝑓𝑓(𝑣_(𝑛)()𝑣>)=𝑓𝑓(𝑠()𝑠))+,𝜇𝑣(≤𝑣𝑠∈)𝐴∶𝑓(𝑣)+𝑓(𝑠)<2𝑢)+𝜇𝑣∈𝐴∶𝑓(_(𝑛)𝑣)>𝑓^(𝑛)(𝑠)+𝜇𝑣∈𝐴∶𝑓^(𝑛)(𝑣)+𝑓^(𝑛)(𝑠)<2𝑢(𝑣)=𝑓^(𝑛)(𝑠),𝑣≤𝑠=𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠).(4.6) If 𝑓(𝑠)=𝑢, then again by Lemma 4.2 for 𝑛≥𝑁_(𝑢,𝑠) we have 𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠)==+𝜇𝜇𝜇(𝑣𝑣(𝑣∈∈∈𝐴𝐴𝐴∶∶∶𝑓𝑓𝑣(𝑣_(𝑛)≤)>𝑠)𝑓−(𝜇𝑠)()𝑣+∈𝜇𝐴(𝑣∶∈𝑓𝐴(𝑣∶)=𝑓(𝑓𝑣)(𝑠+),𝑓𝑣(𝑠≤)<𝑠)2𝑢)+𝜇(𝑣∈𝐴∶𝑣≤(𝑣𝑠))−>𝜇𝑓𝑣^(𝑛)(∈𝑠)𝐴+∶𝜇𝑓𝑣_(𝑛)∈𝐴∶𝑓^(𝑛)(𝑣)+𝑓^(𝑛)(𝑠)<2𝑢(𝑣)=𝑓^(𝑛)(𝑠),𝑣≤𝑠=𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}(𝑠).(4.7) Hence we obtain our claim. It follows lim_𝑛→∞_{𝐴∩{𝑓(𝑛)≥𝑢}}𝜒_(0,𝑡)𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}=_{𝐴∩{𝑓≥𝑢}}𝜒_(0,𝑡)𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(4.8) for any 𝑡∈(0,𝛼). Moreover lim_𝑛→∞((𝑓^(𝑛)−𝑓)𝜒_𝐴)^∗∗(𝑡)=0 for all 𝑡∈(0,𝛼), and consequently by triangle inequality for maximal function we obtain lim_𝑛→∞𝑓^(𝑛)−𝑢𝜒_𝐴^∗∗(𝑡)=(𝑓−𝑢)𝜒_𝐴^∗∗(𝑡)(4.9) for all 𝑡∈(0,𝛼). Since 𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)} is the measure-preserving transformation for all 𝑛∈ℕ, by the fact that the power function 𝑣^𝑝−1 for 𝑝>1 is subadditive with a constant 𝐶>0, we have 1𝑡_𝐴∩{𝑓^(𝑛)_≥𝑢}≤𝜒𝜒_(0,𝑡)𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}𝑓^(𝑛)−𝑢𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)_𝐴^∗∗(𝑡)𝑓^(𝑛)−𝑢𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)≤𝐶𝑓𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)+𝑢𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.10) for every 𝑡∈(0,𝛼) and 𝑛∈ℕ. Combining now (4.8), (4.9), and (4.10) we get lim_𝑛→∞^𝛼01𝑡_𝐴∩{𝑓^(𝑛)_≥𝑢}=𝜒_(0,𝑡)𝜌_((𝑓^(𝑛)_{−𝑢)𝜒𝐴,𝜒𝐴)}𝑓^(𝑛)−𝑢𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝛼01𝑡_{𝐴∩{𝑓≥𝑢}}lim_𝑛→∞𝜒_(0,𝑡)𝜌_{((𝑓−𝑢)𝜒𝐴,𝜒𝐴)}(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡,^𝛼0𝜒_𝐴^∗∗(𝑡)𝑓^(𝑛)−𝑢𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡=^𝛼0𝜒_𝐴^∗∗(𝑡)(𝑓−𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡.(4.11) In view of (4.4) we complete the proof.

The following characterization of the function 𝑆^{𝑝(𝑓,𝐴)}(𝑢) is an essential fact in the proof of the main result.

Proposition 4.4. Let 1<𝑝<∞, 𝑤 be a positive weight function and let 𝑓∈Γ_{𝑝−1,𝑤}. Then the function 𝑆^{𝑝(𝑓,𝐴)}(𝑢) is strictly decreasing with respect to 𝑢.

Proof. First we will show that for any 𝑢,𝑣∈ℝ, 𝑢<𝑣 there exist 𝑎∈(𝑢,𝑣] and 𝐵⊂[0,𝛼), 𝜇(𝐵)>0 such that for all 𝑡∈𝐵, 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)<𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡).(4.12)
Suppose for a contrary that there exist 𝑢<𝑣 such that for all 𝑎∈(𝑢,𝑣] and for a.a. 𝑡∈[0,𝛼) we have 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)≥𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡).(4.13) Since 𝐾_(𝑓,𝐴)(𝑎,𝑡) and ((𝑓+𝑎)𝜒_𝐴)^∗∗(𝑡) are continuous functions with respect to 𝑡∈(0,𝛼) and 𝐾_(𝑓,𝐴)(−𝑎,𝑡) is right-continuous with respect to 𝑎∈ℝ, we get that the above inequality is fulfilled for all 𝑡∈(0,𝛼) and for any 𝑎∈[𝑢,𝑣]. By convexity of the function 𝑎→((𝑓+𝑎)𝜒_𝐴)^∗∗𝑝(𝑡) and by Proposition 4.2 [20] we get that (𝑑⁺/𝑑𝑎)((𝑓+𝑎)𝜒_𝐴)^∗∗𝑝(𝑡)=𝑝𝐾_(𝑓,𝐴)(−𝑎,𝑡)((𝑓+𝑎)𝜒_𝐴)^{∗∗(𝑝−1)}(𝑡) is increasing with respect to 𝑎∈ℝ for any 𝑡∈(0,𝛼), which implies that 𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)≤𝐾_(𝑓,𝐴)(−𝑏,𝑡)(𝑓+𝑏)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.14) for any 𝑎<𝑏 and for all 𝑡∈(0,𝛼). Hence 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)=𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.15) for all 𝑎∈[𝑢,𝑣] and 𝑡∈(0,𝛼). Pick up 𝑏∈(𝑢,𝑣]. Denote 𝛿=(𝑏−𝑢)/2. Notice that 𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)=𝐾_(𝑓,𝐴)(−𝑎−𝛿,𝑡)(𝑓+𝑎+𝛿)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.16) for all 𝑎∈[𝑢,𝑢+𝛿] and 𝑡∈(0,𝛼), which yields that ^{𝑢𝑢+𝛿}𝑑⁺𝑑𝑎(𝑓+𝑎)𝜒^𝐴^∗∗𝑝(𝑡)𝑑𝑎=^{𝑢𝑢+𝛿}𝑑⁺𝑑𝑎(𝑓+𝑎+𝛿)𝜒^𝐴^∗∗𝑝=(𝑡)𝑑𝑎_{𝑏𝑢+𝛿}𝑑⁺𝑑𝑎(𝑓+𝑎)𝜒^𝐴^∗∗𝑝(𝑡)𝑑𝑎(4.17) for all 𝑡∈(0,𝛼). Consequently, 𝑓𝜒^𝐴+𝑢+2𝑏𝜒^𝐴^∗∗𝑝(𝑡)=21(𝑓+𝑢)𝜒^𝐴^∗∗𝑝(𝑡)+12(𝑓+𝑏)𝜒^𝐴^∗∗𝑝(𝑡)(4.18) for any 𝑡∈(0,𝛼). Moreover, by subadditivity of the maximal function and by convexity of the power function 𝑠^𝑝 for 𝑝>1 we get 𝑓𝜒^𝐴+𝑢+2𝑏𝜒^𝐴^∗∗𝑝(𝑡)≤21(𝑓+𝑢)𝜒^𝐴^∗∗(𝑡)+12(𝑓+𝑏)𝜒^𝐴^∗∗(𝑡)^𝑝≤12(𝑓+𝑢)𝜒^𝐴^∗∗𝑝(𝑡)+12(𝑓+𝑏)𝜒^𝐴^∗∗𝑝(𝑡)(4.19) for all 𝑡∈(0,𝛼). Therefore, (𝑓+𝑢)𝜒_𝐴+(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡)=(𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)+(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡),(4.20) which implies that ^𝑡0(𝑓+𝑢)𝜒_𝐴+(𝑓+𝑏)𝜒_𝐴^∗(𝑠)−(𝑓+𝑢)𝜒_𝐴^∗(𝑠)−(𝑓+𝑏)𝜒_𝐴^∗(𝑠)𝑑𝑠=0(4.21) for any 𝑡∈(0,𝛼). Hence (𝑓+𝑢)𝜒_𝐴+(𝑓+𝑏)𝜒_𝐴^∗(𝑠)=(𝑓+𝑢)𝜒_𝐴^∗(𝑠)+(𝑓+𝑏)𝜒_𝐴^∗(𝑠)(4.22) for a.a. 𝑠∈(0,𝛼) and for any 𝑏∈[𝑢,𝑣]. By Corollary 9 [8, page 65] we obtain that the functions (𝑓+𝑢)𝜒_𝐴 and (𝑓+𝑏)𝜒_𝐴 for all 𝑏∈[𝑢,𝑣] are of constant sign almost everywhere, that is, sign(𝑓+𝑢)(𝑠)=sign(𝑓+𝑏)(𝑠)(4.23) for a.a. 𝑠∈𝐴 and have a common system of sets {𝐸_𝑡∶𝑡∈(0,𝛼)} such that 𝜇(𝐸_𝑡)=𝑡 and ^𝑡0(𝑓+𝑏)𝜒_𝐴^∗(𝑠)𝑑𝑠=_{𝐸𝑡∩𝐴}||𝑓+𝑏||(𝑠)𝑑𝑠(4.24) for all 𝑡∈(0,𝛼) and for all 𝑏∈[𝑢,𝑣]. We claim that there exist 𝑏∈(𝑢,𝑣] and 𝐵⊂(0,𝛼), 𝜇(𝐵)>0, such that we have either for all 𝑡∈𝐵, (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡),(4.25) or for all 𝑡∈𝐵, (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)>(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡).(4.26) Suppose that the claim does not hold. Then for any 𝑏∈(𝑢,𝑣] and 𝑡∈(0,𝛼) we get ^𝑡0(𝑓+𝑢)𝜒_𝐴^∗(𝑠)𝑑𝑠=^𝑡0(𝑓+𝑏)𝜒_𝐴^∗(𝑠)𝑑𝑠.(4.27) Thus, by the condition (4.24) we obtain _{𝐸𝑡∩𝐴}||𝑓+𝑢||(𝑠)−||𝑓+𝑏||(𝑠)𝑑𝑠=0(4.28) for every 𝑏∈(𝑢,𝑣] and 𝑡∈(0,𝛼). Since 𝐸_𝑟⊂𝐸_𝑡 for any 𝑟<𝑡 and 𝜇(𝐸_𝑡)=𝑡 for any 𝑡∈(0,𝛼), we get that |𝑓(𝑠)+𝑢|=|𝑓(𝑠)+𝑏| for any 𝑏∈(𝑢,𝑣] and for a.a. 𝑠∈𝐴. Hence, by the fact (4.23) we get a contradiction. Now we will consider two cases.
Case 1. Assume that there exist 𝑏∈(𝑢,𝑣] and 𝐵⊂(0,𝛼), 𝜇(𝐵)>0 such that (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡)(4.29) for all 𝑡∈𝐵. Thus and by conditions (4.23) and (4.24) we obtain 0<^𝑡0(𝑓+𝑏)𝜒_𝐴^∗(𝑠)𝑑𝑠−^𝑡0(𝑓+𝑢)𝜒_𝐴^∗(𝑠)𝑑𝑠==(𝑏−_{𝐸𝑡∩𝐴}𝑢||)𝑓(𝑠)_{𝐸𝑡∩𝐴}+𝑏||−||𝑓(𝑠)+𝑢||𝑑𝑠sign(𝑓(𝑠)+𝑢)𝑑𝑠(4.30) for all 𝑡∈𝐵. Since 𝑏−𝑢>0, we have 0<_{𝐸𝑡∩𝐴}sign(𝑓(𝑠)+𝑢)𝑑𝑠(4.31) for any 𝑡∈𝐵. The function 𝑎→((𝑓+𝑎)𝜒_𝐴)^∗∗(𝑡) is strictly increasing on [𝑢,𝑣] and 𝑡∈𝐵. In fact, let 𝜉,𝜂∈[0,𝑣−𝑢] and 𝜉<𝜂. By conditions (4.23) and (4.24) we conclude that (𝑓+=𝑢1+𝜉)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑢+𝜂)𝜒_𝐴^∗∗(𝑡)𝑡_{𝐸𝑡∩𝐴}sign(𝑓(𝑠)+𝑢)(𝑓(𝑠)+𝑢+𝜉)𝑑𝑠−1𝑡_{𝐸𝑡∩𝐴}=𝜉−𝜂sign(𝑓(𝑠)+𝑢)(𝑓(𝑠)+𝑢+𝜂)𝑑𝑠𝑡_{𝐸𝑡∩𝐴}sign(𝑓(𝑠)+𝑢)𝑑𝑠<0(4.32) for any 𝑡∈𝐵. Let 𝑎∈(𝑢,𝑣) and 𝛿_𝑎=min{𝑣−𝑎,𝑎−𝑢}/2. Then, by condition (4.32) we get (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑢+𝜖)𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗(𝑡)(4.33) for every 0<𝜖≤𝛿_𝑎 and for all 𝑡∈𝐵. Repeating calculations in condition (4.32) with 𝑎 instead of 𝑢, in view of (4.23) and (4.24) we have (𝑓+𝑢+𝜖)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)=(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)>0(4.34) for all 0<𝜖≤𝛿_𝑎 and 𝑡∈𝐵. Since the functions ((𝑓+𝑢+𝛿_𝑎)𝜒_𝐴)^∗∗(𝑡) and ((𝑓+𝑎)𝜒_𝐴)^∗∗(𝑡) are continuous and decreasing with respect to 𝑡, there exists a compact interval [𝑐,𝑑]⊂𝐵 such that 𝑓+𝑢+𝛿_𝑎𝜒_𝐴^∗∗(𝑡)<(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)(4.35) for all 𝑡∈[𝑐,𝑑]. Define 𝐹(𝑡)=(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)−𝑓+𝑢+𝛿_𝑎𝜒_𝐴^∗∗(𝑡)(4.36) for any 𝑡∈[𝑐,𝑑]. Since 𝐹 is positive and continuous on [𝑐,𝑑], we get min_{𝑡∈[𝑐,𝑑]}{𝐹(𝑡)}>0. Consequently, there is 𝛾_𝑎∈(0,1) such that 𝑓+𝑢+𝛿_𝑎𝜒_𝐴^∗∗(𝑡)<1−𝛾_𝑎(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)(4.37) for all 𝑡∈[𝑐,𝑑]. Hence, by condition (4.33) we get (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)<<1(𝑓−+𝛾𝑢_𝑎+𝜖)𝜒_𝐴^∗∗(𝑡)<1−𝛾_𝑎(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗(𝑡)(4.38) for every 𝑡∈[𝑐,𝑑] and 0<𝜖≤𝛿_𝑎. Therefore, by (4.34) we obtain (𝑓+𝑢+𝜖)𝜒_𝐴^∗∗𝑝(𝑡)−(𝑓+𝑢)𝜒_𝐴^∗∗𝑝(𝑡)𝜖≤1−𝛾^𝑎^𝑝−1(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗𝑝(𝑡)−(𝑓+𝑎)𝜒_𝐴^∗∗𝑝(𝑡)𝜖(4.39) for any 𝑡∈[𝑐,𝑑] and 0<𝜖≤𝛿_𝑎. Now according to Proposition 4.2 [20] we conclude 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)<𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.40) for all 𝑡∈[𝑐,𝑑], which contradicts condition (4.15) and finishes the first case.Case 2. Suppose that there exist 𝑏∈(𝑢,𝑣] and 𝐵⊂(0,𝛼) with 𝜇(𝐵)>0 such that (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)>(𝑓+𝑏)𝜒_𝐴^∗∗(𝑡)(4.41) for any 𝑡∈𝐵. Analogously as in the previous case we obtain that _{𝐸𝑡∩𝐴}sign(𝑓(𝑠)+𝑢)𝑑𝑠<0(4.42) for all 𝑡∈𝐵. We claim that the function 𝑎→((𝑓+𝑎)𝜒_𝐴)^∗∗(𝑡) is strictly decreasing for any 𝑎∈[𝑢,𝑣]. Let 𝜉,𝜂∈[0,𝑣−𝑢] and 𝜉<𝜂. The conditions (4.23) and (4.24) imply (𝑓+𝑢+𝜉)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑢+𝜂)𝜒_𝐴^∗∗(𝑡)==𝜉1𝑡−𝜂_{𝐸𝑡∩𝐴}||𝑓(𝑠)+𝑢+𝜉||−||𝑓(𝑠)+𝑢+𝜂||𝑑𝑠𝑡_{𝐸𝑡∩𝐴}sign(𝑓(𝑠)+𝑢)𝑑𝑠>0(4.43) for any 𝑡∈𝐵. Therefore, for any 𝑎∈(𝑢,𝑣), 0<𝜖<𝑚𝑖𝑛{𝑣−𝑎,𝑎−𝑢}/2 and for all 𝑡∈𝐵 we have (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)>(𝑓+𝑢+𝜉)𝜒_𝐴^∗∗(𝑡)>(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)>(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗(𝑡).(4.44) Repeating calculation in condition (4.43) with 𝑎 instead of 𝑢, in view of (4.23) and (4.24) we also get (𝑓+𝑢)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑢+𝜖)𝜒_𝐴^∗∗(𝑡)=(𝑓+𝑎)𝜒_𝐴^∗∗(𝑡)−(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗(𝑡)>0(4.45) for all 𝑡∈𝐵. Similarly as in the previous case there exists 𝛾_𝑎>0 such that (𝑓+𝑢+𝜖)𝜒_𝐴^∗∗𝑝(𝑡)−(𝑓+𝑢)𝜒_𝐴^∗∗𝑝(𝑡)𝜖≤1+𝛾^𝑎^𝑝−1(𝑓+𝑎+𝜖)𝜒_𝐴^∗∗𝑝(𝑡)−(𝑓+𝑎)𝜒_𝐴^∗∗𝑝(𝑡)𝜖(4.46) for any 𝑡∈𝐵 and 0<𝜖<𝑚𝑖𝑛{𝑎−𝑢,𝑣−𝑎}/2. Consequently, by Proposition 4.2 [20] we get 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)<𝐾_(𝑓,𝐴)(−𝑎,𝑡)(𝑓+𝑎)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.47) for all 𝑡∈𝐵, which gives us a contradiction with (4.15) and completes the proof of inequality (4.12).
Let now 𝑎,𝑏∈ℝ and 𝑎<𝑏. Denote 𝑣=−𝑎 and 𝑢=−𝑏. Clearly 𝑢<𝑣. By (4.12) there exist 𝑐∈(𝑢,𝑣] and 𝐵⊂[0,𝛼), 𝜇(𝐵)>0 such that 𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)<𝐾_(𝑓,𝐴)(−𝑐,𝑡)(𝑓+𝑐)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)(4.48) for all 𝑡∈𝐵. Hence, by (4.14) and by Proposition  2.4 in [7] we get 𝑆^{𝑝(𝑓,𝐴)}(𝑏)=<𝑆^{𝑝(𝑓,𝐴)}(−𝑢)=_{𝐵∪𝐵𝑐}𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝐵+𝐾_(𝑓,𝐴)(−𝑐,𝑡)(𝑓+𝑐)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡_𝐵𝑐≤𝑆_{𝑝(𝑓,𝐴)}𝐾_(𝑓,𝐴)(−𝑢,𝑡)(𝑓+𝑢)𝜒_𝐴^{∗∗(𝑝−1)}(𝑡)𝑤(𝑡)𝑑𝑡(−𝑐)≤𝑆^{𝑝(𝑓,𝐴)}(−𝑣)=𝑆^{𝑝(𝑓,𝐴)}(𝑎),(4.49) and the proof is done.

Now we establish the main theorem of this section.

Theorem 4.5. Let 1<𝑝<∞ and 𝑤 be a positive weight function. Then the extended best constant approximant operator 𝑇_(𝑝,𝐴) assumes an unique value, that is, for any 𝑓∈Γ_{𝑝−1,𝑤}, 𝑇_(𝑝,𝐴)(𝑓)=𝑓_(𝑝,𝐴)𝜒_𝐴=𝑓_(𝑝,𝐴)𝜒_𝐴.(4.50)

Proof. Suppose that 𝑓_(𝑝,𝐴)<𝑓_(𝑝,𝐴). Then there exist 𝑢,𝑣∈(𝑓_(𝑝,𝐴),𝑓_(𝑝,𝐴)) such that 𝑢<𝑣. By Theorem 2.9 [7] and by Proposition 4.4 we obtain 0≤0𝑆≤𝑆_{(𝑓,𝐴)(−𝑓,𝐴)}𝑓_(𝑝,𝐴)<𝑆^(𝑓,𝐴)(𝑣)<𝑆_(𝑓,𝐴)(𝑢),−𝑓_(𝑝,𝐴)<𝑆^{(−𝑓,𝐴)}(−𝑢)<𝑆_(𝑓,𝐴)(−𝑣).(4.51) Let (𝑓^(𝑛)) be a sequence of truncations of 𝑓. By Theorem 4.3 there exists 𝑁₀∈ℕ such that for all 𝑛≥𝑁₀ we have 0<𝑆_(𝑓^(𝑛)_,𝐴)(𝑣),0<𝑆_(−𝑓^(𝑛)_,𝐴)(−𝑣)(4.52) as well as 0<𝑆_(−𝑓^(𝑛)_,𝐴)(−𝑢),0<𝑆_(𝑓^(𝑛)_,𝐴)(𝑢).(4.53) Choose 𝑛≥𝑁₀. By Lemma 4.1, 𝑓^(𝑛)∈Γ_(𝑝,𝑤) for all 𝑛∈ℕ. Now by conditions (4.52) and (4.53) and by Theorem 2.4 we get that 𝑢,𝑣∈𝑇_(𝑝,𝐴)(𝑓^(𝑛)) for all 𝑛≥𝑁₀. Finally, by Corollary 2.5 we obtain that 𝑇_(𝑝,𝐴)(𝑓^(𝑛)) is unique for 𝑛≥𝑁₀, which implies a contradiction and finishes the proof.