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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 682960, 28 pages
Research Article

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

1Institute of Mathematics, PoznaΕ„ University of Technology, Piotrowo 3a, 60-965 PoznaΕ„, Poland
2Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 October 2011; Accepted 20 January 2012

Academic Editor: LechΒ Maligranda

Copyright Β© 2012 Maciej Ciesielski and Anna KamiΕ„ska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 Γ𝑝,π‘€βˆ«={π‘“βˆΆ(π‘“βˆ—βˆ—)𝑝𝑀<∞} for any 0<𝑝<∞ and a nonnegative locally integrable weight function 𝑀, where π‘“βˆ—βˆ— is a maximal function of the decreasing rearrangement π‘“βˆ— for any measurable function 𝑓 on (0,𝛼), with 0<π›Όβ‰€βˆž. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of π‘“βˆˆπΈ, 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 π‘“πœ– of π‘“βˆˆΞ“π‘,𝑀 or π‘“βˆˆΞ“π‘βˆ’1,𝑀, 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 π‘“βˆˆΞ“π‘βˆ’1,𝑀, 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 Γ𝑝,𝑀 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 𝐡(𝑣,πœ–)βŠ‚β„π‘› to this function 𝑓, that is, for a.a. π‘£βˆˆβ„π‘›, 1ξ€œπœ‡(𝐡(𝑣,πœ–))𝐡(𝑣,πœ–)𝑓(𝑑)π‘‘π‘‘βŸΆπ‘“(𝑣)asπœ–βŸΆ0.(1.1) In fact, Lebesgue’s integral average coincides with a best constant approximant on the space 𝐿2(ℝ𝑛) [1]. The Lebesgue Differentiation Theorem (LDT) can be proved as a consequence of the weak maximal inequality π‘‘ξ€·π‘€π»π‘“ξ€Έβˆ—(𝑑)≀4𝑛‖𝑓‖𝐿1(1.2) for the Hardy-Littlewood maximal function 𝑀𝐻𝑓 where π‘‘βˆˆβ„π‘› and π‘“βˆˆπΏ1(ℝ𝑛) [2]. The interesting exploration of LDT was initiated by Stein in [3], who introduced the maximal functions on 𝐿𝑝(ℝ𝑛), associated with integral average, and applied it to obtain differentiation theorem in the notation of the norm in 𝐿𝑝(ℝ𝑛) 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 𝐿𝑝+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 Λ𝑀,πœ™ to the spaces Λ𝑀,πœ™β€² 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 Γ𝑝,𝑀 to the spaces Ξ“π‘βˆ’1,𝑀. 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 Γ𝑝,𝑀.

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 𝑀𝐸(π‘Ÿ)(𝑓) for π‘“βˆˆπΈ, 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 Γ𝑝,𝑀. In both types of LDT we employ the assumption of upper and lower πœ™-estimates of Γ𝑝,𝑀. 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 𝑀𝐸(π‘Ÿ)𝑓, which corresponds to a quasi-norm average of π‘“βˆˆπΈ. In the same spirit we also provide some conditions when the LDT does not hold in 𝐸 or in Γ𝑝,𝑀. 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 Γ𝑝,𝑀 satisfies a lower (resp., an upper) πœ™-estimate, where πœ™ is the fundamental function of Γ𝑝,𝑀. In view of this characterization we investigate pointwise convergence of the best constant approximants π‘“πœ– to 𝑓 as πœ–β†’0 whenever π‘“βˆˆΞ“π‘,𝑀 and 1≀𝑝<∞, as well as the convergence of the extended best constant approximants π‘“πœ– for any π‘“βˆˆΞ“π‘βˆ’1,𝑀 and 1<𝑝<2. We also present examples showing that this assumption is fulfilled by a large class of the spaces Ξ“π‘ž,𝑝. Finally, we investigate relations between maximal functions and the 𝐾-functional of Banach couple (Ξ“π‘ž,𝑝,𝐿∞) in the spirit of the inequalities stated in [4]. We finish Section 3 with an example showing that 𝑀Γ(π‘Ÿ)𝑝,𝑀(𝑓) is not equivalent to the 𝐾-functional of the pair (Γ𝑝,𝑀,𝐿∞).

It is well known that the extension of the best constant approximant operator from Ξ“1,𝑀 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 π‘“βˆˆΞ“π‘βˆ’1,𝑀 and 1<𝑝<∞. To show the uniqueness we need to consider strict monotonicity of the right-hand GΓ’teaux derivative of the norm in Γ𝑝,𝑀 at (π‘“βˆ’π‘’)πœ’π΄ in the direction πœ’π΄ for any π‘“βˆˆΞ“π‘βˆ’1,𝑀 and π‘’βˆˆβ„.

2. Preliminaries

Let ℝ and β„• be the set of real and natural numbers, respectively. For any π΄βŠ‚[0,𝛼) denote 𝐴𝑐=[0,𝛼)⧡𝐴. Let 0<π›Όβ‰€βˆž and πœ‡ be the Lebesgue measure on ℝ. We denote by 𝐿0 the space of all extended real-valued πœ‡-measurable and finite functions a.e. on [0,𝛼). Denote the outer measure on ℝ by πœ‡βˆ—, the support of π‘“βˆˆπΏ0 by π•Š(𝑓)=supp(𝑓), and the restriction of 𝑓 to the set π΄βŠ‚[0,𝛼) by π‘“βˆ£π΄. By a simple (resp., step) function we mean a measurable function π‘“βˆˆπΏ0 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 𝑑𝑓 of a function π‘“βˆˆπΏ0 is given by 𝑑𝑓(πœ†)=πœ‡(π‘ βˆˆ[0,𝛼)∢|𝑓|(𝑠)>πœ†) for all πœ†β‰₯0. Two functions 𝑓,π‘”βˆˆπΏ0 are called equimeasurable, if 𝑑𝑓(πœ†)=𝑑𝑔(πœ†) for all πœ†β‰₯0. We define the decreasing rearrangement for any π‘“βˆˆπΏ0 by π‘“βˆ—(𝑑)=inf{𝑠>0βˆΆπ‘‘π‘“(𝑠)≀𝑑}, 𝑑β‰₯0. For given π‘“βˆˆπΏ0 we denote the maximal function of π‘“βˆ— by π‘“βˆ—βˆ—βˆ«(𝑑)=(1/𝑑)𝑑0π‘“βˆ—(𝑠)𝑑𝑠. It is well known that π‘“βˆ—β‰€π‘“βˆ—βˆ— and π‘“βˆ—βˆ— is decreasing and subadditive, that is, (𝑓+𝑔)βˆ—βˆ—β‰€π‘“βˆ—βˆ—+π‘”βˆ—βˆ— for any 𝑓,π‘”βˆˆπΏ0. For the properties of 𝑑𝑓, π‘“βˆ—, and π‘“βˆ—βˆ— see [2, 8]. A subspace πΈβŠ‚πΏ0 equipped with a quasinorm ‖⋅‖𝐸 is called a quasinormed function space, if the following conditions are satisfied.(1)If π‘“βˆˆπΏ0, π‘”βˆˆπΈ, and |𝑓|≀|𝑔| a.e., then π‘“βˆˆπΈ and ‖𝑓‖𝐸≀‖𝑔‖𝐸.(2)There exists a strictly positive π‘“βˆˆπΈ.

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 π‘“βˆˆπΏ0 and π‘”βˆˆπΈ with 𝑑𝑓=𝑑𝑔, then π‘“βˆˆπΈ and ‖𝑓‖𝐸=‖𝑔‖𝐸 (see [2]). Throughout the paper we use the notation π΄β‰ˆπ΅, which means that the expressions 𝐴 and 𝐡 are equivalent; that is, 𝐴/𝐡 is bounded from both sides. Let 0<𝑝<∞ and π‘€βˆˆπΏ0 be a nonnegative weight function. Lorentz space Γ𝑝,𝑀 is a subspace of 𝐿0 such that‖𝑓‖=‖𝑓‖Γ𝑝,π‘€ξ‚΅ξ€œβˆΆ=𝛼0π‘“βˆ—βˆ—π‘π‘€ξ‚Ά1/𝑝=ξ‚΅ξ€œπ›Ό0π‘“βˆ—βˆ—π‘ξ‚Ά(𝑑)𝑀(𝑑)𝑑𝑑1/𝑝<∞.(2.1) Given a measurable set π΄βŠ‚[0,𝛼) by Γ𝑝,𝑀(𝐴) we denote the set of π‘“βˆˆπΏ0 restricted to 𝐴 and satisfying the above inequality. Unless we say otherwise, throughout the paper we assume that 𝑀 belongs to the class 𝐷𝑝 (in short π‘€βˆˆπ·π‘), whenever it satisfies the following conditions:ξ€œπ‘Š(𝑠)∢=𝑠0𝑀<∞,π‘Šπ‘(𝑠)∢=π‘ π‘ξ€œπ›Όπ‘ π‘‘βˆ’π‘π‘€(𝑑)𝑑𝑑<∞(2.2) for all 0<𝑠≀𝛼 if 𝛼<∞ and for all 0<𝑠<∞ otherwise. These two conditions guarantee that Γ𝑝,𝑀≠{0}. We also assume thatπ‘Šπ‘(𝑠)>0forξ€œ0<𝑠<𝛼,π‘Š(∞)=∞0𝑀=∞if𝛼=∞.(2.3) Under these assumptions (Γ𝑝,𝑀,β€–β‹…β€–) 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,π‘ž<∞ and 𝑀(𝑑)=𝑑𝑝/π‘žβˆ’1, π‘‘βˆˆ(0,𝛼), the space Γ𝑝,𝑀 will be denoted by Ξ“π‘ž,𝑝.

Unless we say otherwise, throughout this paper we assume that πœ™ is the fundamental function of Γ𝑝,𝑀 defined as πœ™(𝑑)=β€–πœ’(0,𝑑)β€–, π‘‘βˆˆ(0,𝛼), and πœ™(0)=0. It is easy to show that the fundamental function πœ™ is strictly increasing and continuous on [0,𝛼), limπ‘‘β†’βˆžπœ™(𝑑)=∞ and limπ‘‘β†’βˆžπ‘“βˆ—(𝑑)=0 for π‘“βˆˆΞ“π‘,𝑀. For more details about the properties of Γ𝑝,𝑀 see [9].

Recall that for given 0<𝑝<∞, classical Lorentz space Λ𝑝,𝑀 is a subspace of 𝐿0 such that‖𝑓‖Λ𝑝,π‘€ξ‚΅ξ€œβˆΆ=𝛼0π‘“βˆ—π‘π‘€ξ‚Ά1/𝑝=ξ‚΅ξ€œπ›Ό0π‘“βˆ—π‘ξ‚Ά(𝑑)𝑀(𝑑)𝑑𝑑1/𝑝<∞.(2.4) In case when π‘Š satisfies the Ξ”2-condition, that is π‘Š(2𝑠)β‰€πΆπ‘Š(𝑠) for all 𝑠>0 and some 𝐢>0, as well as π‘Š(∞)=∞, the space Λ𝑝,𝑀 is a separable r.i. order continuous quasi-Banach function space [9]. The space Λ𝑝,𝑀 is a r.i. Banach function space, whenever the weight 𝑀 is decreasing and 1≀𝑝<∞ [10]. Since π‘“βˆ—β‰€π‘“βˆ—βˆ—, we have the natural inclusion Γ𝑝,π‘€βŠ‚Ξ›π‘,𝑀. Moreover, Γ𝑝,𝑀=Λ𝑝,𝑀 if and only if 𝑀 satisfies condition 𝐡𝑝, (π‘€βˆˆπ΅π‘ for short) which means that there is 𝐴>0 such that for all 𝑠>0 we have π‘Šπ‘(𝑠)β‰€π΄π‘Š(𝑠) [11–13].

Let (Ξ©1,πœ‡1) and (Ξ©2,πœ‡2) be 𝜎-finite measure spaces. A map 𝛾 from Ξ©1 into Ξ©2 is said to be a measure-preserving transformation, if whenever 𝐸 is a πœ‡2-measurable subset of Ξ©2, the set π›Ύβˆ’1(𝐸)={π‘’βˆˆΞ©1βˆΆπ›Ύ(𝑒)∈𝐸} is a πœ‡1-measurable subset of Ξ©1 and πœ‡1(π›Ύβˆ’1(𝐸))=πœ‡2(𝐸). For given subsets 𝐴,π΅βŠ‚β„+ such that πœ‡(𝐴)=πœ‡(𝐡), there exists a measure-preserving transformation π›ΏβˆΆπ΄β†’π΅ [14, Theorem 17, page 410].

Definition 2.1 (see [15]). Let 𝑓,β„ŽβˆˆπΏ0. Denote 𝜏(𝑓,β„Ž)(𝑑)=𝑑𝑓||𝑓||ξ€Έξ€·||𝑓||||𝑓||ξ€Έξ€·||𝑓||||𝑓||ξ€Έ(𝑑)+πœ‡π‘’βˆΆ(𝑒)=(𝑑),β„Ž(𝑒)sign(𝑓(𝑒))>β„Ž(𝑑)sign(𝑓(𝑑))+πœ‡π‘’βˆΆ(𝑒)=(𝑑),β„Ž(𝑒)sign(𝑓(𝑒))=β„Ž(𝑑)sign(𝑓(𝑑)),𝑒≀𝑑(2.5) for all π‘‘βˆˆ[0,𝛼).

In 1970, Ryff proved in [16] that 𝜏(π‘“βˆ£[0,1],0)∢[0,1]β†’[0,1] is a measure-preserving transformation for any π‘“βˆˆπΏ0 and |𝑓|=π‘“βˆ—βˆ˜πœ(|𝑓|[0,1],0) a.e. on [0,1]. In 1993, Carothers et al. established in [15] that 𝜏(𝑓,β„Ž) is a measure preserving transformation from π•Š(𝑓) onto π•Š(π‘“βˆ—) such that |𝑓|=π‘“βˆ—βˆ˜πœ(𝑓,β„Ž) a.e. on π•Š(𝑓) for any π‘“βˆˆπΏ0 with 𝑑𝑓(πœ†)<∞ for all πœ†>0 and any β„ŽβˆˆπΏ0. Notice that for any π‘“βˆˆπΏ0 with 𝑑𝑓(πœ†)<∞ for any πœ†>0 and β„ŽβˆˆπΏ0, if πœ‡(π‘’βˆΆ|𝑓|(𝑒)=𝑣)=0 for every 𝑣>0 we have that 𝜏(𝑓,β„Ž)(𝑑)=𝑑𝑓(|𝑓|(𝑑)) and it is the unique measure-preserving transformation up to measure zero satisfying |𝑓|=π‘“βˆ—βˆ˜πœ(𝑓,β„Ž) a.e. on π•Š(𝑓).

Definition 2.2. Let 𝑓,π‘”βˆˆΞ“π‘,𝑀 and let 𝜏(𝑓,𝑔), 𝜏(π‘”βˆ£π•Š(𝑔)β§΅π•Š(𝑓),0) be measure-preserving transformations given by Definition 2.1. Denote 𝜌(𝑓,𝑔)(ξ‚»πœπ‘ )=(𝑓,𝑔)(𝑠)ifπ‘‘π‘ βˆˆπ•Š(𝑓),𝑓(0)+𝜏(π‘”βˆ£π•Š(𝑔)β§΅π•Š(𝑓),0)(𝑠)ifπ‘ βˆˆπ•Š(𝑔)β§΅π•Š(𝑓).(2.6)

Definition 2.3. Let π‘“βˆˆΞ“π‘,𝑀 and π΄βŠ‚[0,𝛼) with πœ‡(𝐴)<∞. Denote 𝐾(𝑓,𝐴)(1𝑒,𝑑)=π‘‘ξ€œπ΄ξ€·1βˆ’2πœ’{𝑓<𝑒}ξ€Έπœ’(0,𝑑)ξ€·πœŒ((π‘“βˆ’π‘’)πœ’π΄,πœ’π΄)ξ€Έ,𝑆𝑝(𝑓,𝐴)ξ€œ(𝑒)=𝛼0𝐾(𝑓,𝐴)ξ€·(𝑒,𝑑)(π‘“βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)(𝑑)𝑀(𝑑)𝑑𝑑(2.7) for any π‘’βˆˆβ„ and π‘‘βˆˆ(0,𝛼).

Let (𝑋,‖⋅‖𝑋) be a real normed space. Denote by 𝐡𝑋 (resp., 𝑆𝑋) the closed unit ball (resp., the unit sphere) of 𝑋. Assume that π‘Œ is a subset of 𝑋 and π‘₯ is an element of 𝑋. An element Μƒπ‘₯βˆˆπ‘Œ is called best approximant to π‘₯ from π‘Œ ifβ€–π‘₯βˆ’Μƒπ‘₯‖𝑋=infπ‘¦βˆˆπ‘Œβ€–π‘₯βˆ’π‘¦β€–π‘‹.(2.8) A nonempty subset π‘Œ of 𝑋 is a set of uniqueness if for any element π‘₯βˆˆπ‘‹ there is no more than one element Μƒπ‘₯βˆˆπ‘Œ satisfying (2.8). The set π‘Œ is a set of existence if for every element π‘₯βˆˆπ‘‹ there is at least one element Μƒπ‘₯βˆˆπ‘Œ for which condition (2.8) holds. The set π‘Œ is a Chebyshev set if for every element π‘₯βˆˆπ‘‹ there exists exactly one element Μƒπ‘₯βˆˆπ‘Œ satisfying (2.8), that is, if π‘Œ is both a set of uniqueness and a set of existence (for more details see [17]). Let 2π‘Œ be a collection of all subsets of π‘Œ. A set value map π‘‡π‘ŒβˆΆπ‘‹β†’2π‘Œ is said to be best approximant operator, if it assumes for any π‘₯βˆˆπ‘‹ a set of all best approximant elements to π‘₯ from π‘Œ, that is,π‘‡π‘Œξ‚»(π‘₯)=Μƒπ‘₯βˆˆπ‘ŒβˆΆβ€–π‘₯βˆ’Μƒπ‘₯‖𝑋=infπ‘¦βˆˆπ‘Œβ€–π‘₯βˆ’π‘¦β€–π‘‹ξ‚Ό=ξ€½Μƒπ‘₯βˆˆπ‘ŒβˆΆβ€–π‘₯βˆ’Μƒπ‘₯‖𝑋≀‖π‘₯βˆ’π‘¦β€–π‘‹ξ€Ύ.,βˆ€π‘¦βˆˆπ‘Œ(2.9) In case when 𝑋 is a norm function space and π‘Œ is a family of constant functions, then π‘‡π‘Œ is called best constant approximant operator, and each element Μƒπ‘₯βˆˆπ‘‡π‘Œ(π‘₯) is called best constant approximant to π‘₯βˆˆπ‘‹ from π‘Œ. Let π΄βŠ‚[0,𝛼) with 0<πœ‡(𝐴)<∞ and 𝕂(𝐴)={π‘πœ’π΄βˆΆπ‘βˆˆβ„}. It is well known that the set 𝑇𝕂(𝐴)(𝑓) is convex, compact, and a set of existence for all π‘“βˆˆΞ“π‘,𝑀 [18, 19]. Let’s recall some characterizations of best constant approximants over Lorentz spaces Γ𝑝,𝑀.

Theorem 2.4 (see [20, Theorem 7.5]). Let 1≀𝑝<∞ and let π‘“βˆˆΞ“π‘,𝑀⧡𝕂(𝐴). Then π‘’βˆˆπ•‚(𝐴) is the best constant approximant of 𝑓 if and only if 𝑆𝑝(𝑓,𝐴)(𝑒)β‰₯0,𝑆𝑝(βˆ’π‘“,𝐴)(βˆ’π‘’)β‰₯0.(2.10)

Corollary 2.5 (see [20, Corollary 7.3]). For any 1<𝑝<∞ and 𝑀 positive, we have that 𝑇𝕂(𝐴)(𝑓) is Chebyshev set for any π‘“βˆˆΞ“π‘,𝑀; that is, there is an unique best constant approximant π‘’βˆˆπ•‚(𝐴) to 𝑓.

Recently, in [7] it has been developed an existence of extension of the best constant approximant operators from Lorentz space Ξ“1,𝑀 to 𝐿0, if π‘€βˆˆπ·1, and from Γ𝑝,𝑀 to Ξ“π‘βˆ’1,𝑀, if 1<𝑝<∞ and π‘€βˆˆπ·π‘βˆ’1. Now we recall definition of the extended operator 𝑇(𝑝,𝐴) on 𝐿0, if 𝑝=1, and on Ξ“π‘βˆ’1,𝑀, if 1<𝑝<∞.

Definition 2.6 (see [7]). Let π΄βŠ‚(0,𝛼) with 0<πœ‡(𝐴)<∞ and let π‘€βˆˆπ·π‘βˆ’1 if 1<𝑝<∞, and π‘€βˆˆπ·1 if 𝑝=1. Assume that π‘“βˆˆΞ“π‘βˆ’1,𝑀 if 𝑝>1, and π‘“βˆˆπΏ0 if 𝑝=1. Denote 𝑓(𝑝,𝐴)=minπ‘’βˆΆπ‘†π‘(βˆ’π‘“,𝐴),(βˆ’π‘’)β‰₯0𝑓(𝑝,𝐴)=maxπ‘’βˆΆπ‘†π‘(𝑓,𝐴).(𝑒)β‰₯0(2.11) Then the extended best constant approximant operator is given by 𝑇(𝑝,𝐴)𝑓(𝑓)=(𝑝,𝐴)πœ’π΄,𝑓(𝑝,𝐴)πœ’π΄ξ‚„.(2.12)

In fact, any π‘’βˆˆπ‘‡(𝑝,𝐴)(𝑓) is called an extended best constant approximant of 𝑓. Notice that in view of Theorem 2.4, if π‘“βˆˆΞ“π‘,𝑀 for 1≀𝑝<∞, then any π‘’βˆˆπ‘‡(𝑝,𝐴)(𝑓) is a classical best constant approximant of 𝑓.

Definition 2.7 (see [4]). Let 0<π‘Ÿ<∞, (𝐸,‖⋅‖𝐸) be a r.i. quasi-Banach function space, and let Φ∢[0,∞)β†’[0,∞) be an increasing bijection. 𝐸 is said to satisfy an upper (resp., a lower) Ξ¦-estimate for β€–β‹…β€–π‘ŸπΈ, if there exists 𝐢>0 such that for all π‘›βˆˆβ„• and (𝑓𝑖)𝑛𝑖=1βŠ‚πΈ with pairwise disjoint supports we have ‖‖‖‖𝑛𝑖=1π‘“π‘–β€–β€–β€–β€–π‘ŸπΈξƒ©β‰€πΆΞ¦π‘›ξ“π‘–=1Ξ¦βˆ’1ξ€·β€–β€–π‘“π‘–β€–β€–π‘ŸπΈξ€Έξƒͺ,(2.13) respectively Φ𝑛𝑖=1Ξ¦βˆ’1ξ€·β€–β€–π‘“π‘–β€–β€–π‘ŸπΈξ€Έξƒͺ‖‖‖‖≀𝐢𝑛𝑖=1π‘“π‘–β€–β€–β€–β€–π‘ŸπΈ.(2.14)

In the case when Ξ¦(𝑑)=𝑑1/𝑝 and π‘Ÿ=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)={π‘“βˆΆ|𝑓|π‘βˆˆπ‘‹} the 𝑝-convexification of 𝑋 equipped with the quasinorm ‖⋅‖𝑋(𝑝)=β€–|β‹…|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 Γ𝑝,𝑀.

Definition 2.8. Let 0<π‘Ÿ<∞, (𝐸,‖⋅‖𝐸) be a r.i. quasi-Banach function space. For any π‘“βˆˆπΈ we denote π‘“πœ–(π‘Ÿ)β€–β€–(𝑑)=π‘“πœ’π΅(𝑑,πœ–)β€–β€–π‘ŸπΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸(2.15) for all πœ–>0 and π‘‘βˆˆ(0,𝛼). The maximal function 𝑀𝐸(π‘Ÿ)π‘“βˆΆ(0,𝛼)→ℝ is given by 𝑀𝐸(π‘Ÿ)𝑓𝑓(𝑑)=supπœ–(π‘Ÿ)(𝑑)βˆΆπœ–>0,𝐡(𝑑,πœ–)βŠ‚(0,𝛼)(2.16) for any π‘‘βˆˆ(0,𝛼), where 𝐡(𝑑,πœ–)=(π‘‘βˆ’πœ–,𝑑+πœ–).

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 πœ‘(𝑑)=π‘Š(𝑑) on (0,∞). The proof of the proposition is quite standard and is provided for the sake of completeness.

Proposition 2.9. Let 𝛼=∞, π‘“βˆˆπΏ0, and let πœ‘βˆΆ[0,∞)β†’[0,∞) be an increasing continuous function. If limπ‘‘β†’βˆžπœ‘(𝑑)=∞ or 𝑑𝑓(πœ†)<∞ for all πœ†>0, then sup𝑑>0ξ€½πœ‘(𝑑)π‘“βˆ—ξ€Ύ(𝑑)=sup𝑑>0ξ€½ξ€·π‘‘π‘‘πœ‘π‘“.(𝑑)ξ€Έξ€Ύ(2.17)

Proof. Notice first if limπ‘‘β†’βˆžπœ‘(𝑑)=∞ and there exists 𝑑>0 such that 𝑑𝑓(𝑑)=∞, then limπ‘ β†’βˆžπœ‘(𝑠)π‘“βˆ—(𝑠)=∞, and the conclusion follows.
Now assume that βˆ‘π‘“=𝑛𝑖=1π‘Žπ‘–πœ’πΈπ‘– is a nonnegative simple function, where π‘Žπ‘–>π‘Žπ‘–+1 for any 1β‰€π‘–β‰€π‘›βˆ’1, π‘Žπ‘›+1=0, and πΈπ‘–βˆ©πΈπ‘—=βˆ… whenever 𝑖≠𝑗. Then π‘“βˆ—βˆ‘(𝑑)=𝑛𝑖=1π‘Žπ‘–πœ’[π‘šπ‘–βˆ’1,π‘šπ‘–)(𝑑) for any 𝑑>0, where π‘š0=0 and π‘šπ‘–=βˆ‘π‘–π‘—=1πœ‡(𝐸𝑗) for 1≀𝑖≀𝑛. We claim that sup𝑑>0ξ€½πœ‘(𝑑)π‘“βˆ—ξ€Ύ(𝑑)=max1β‰€π‘–β‰€π‘›ξ€½π‘Žπ‘–πœ‘ξ€·π‘šπ‘–.ξ€Έξ€Ύ(2.18) By monotonicity and continuity of πœ‘ we obtain sup𝑑>0{πœ‘(𝑑)π‘“βˆ—(𝑑)}≀max1≀𝑖≀𝑛{π‘Žπ‘–πœ‘(π‘šπ‘–)}. On the other hand we have sup𝑑>0{πœ‘(𝑑)π‘“βˆ—(𝑑)}β‰₯sup𝑑>0{πœ‘(𝑑)π‘Žπ‘–πœ’[π‘šπ‘–βˆ’1,π‘šπ‘–)(𝑑)}=π‘Žπ‘–πœ‘(π‘šπ‘–). Therefore sup𝑑>0{πœ‘(𝑑)π‘“βˆ—(𝑑)}β‰₯max1≀𝑖≀𝑛{π‘Žπ‘–πœ‘(π‘šπ‘–)}, which implies condition (2.18). Now we will show sup𝑑>0ξ€½ξ€·π‘‘π‘‘πœ‘π‘“(𝑑)ξ€Έξ€Ύ=max1β‰€π‘–β‰€π‘›ξ€½π‘Žπ‘–πœ‘ξ€·π‘šπ‘–.ξ€Έξ€Ύ(2.19) For 𝑑>0 we have π‘‘πœ‘(π‘‘π‘“βˆ‘(𝑑))=𝑑𝑛𝑖=1πœ‘(π‘šπ‘–)πœ’[π‘Žπ‘–+1,π‘Žπ‘–)(𝑑)≀max1≀𝑖≀𝑛{π‘Žπ‘–πœ‘(π‘šπ‘–)}, and so sup𝑑>0{π‘‘πœ‘(𝑑𝑓(𝑑))}≀max1≀𝑖≀𝑛{π‘Žπ‘–πœ‘(π‘šπ‘–)}. On the other hand, sup𝑑>0{π‘‘πœ‘(𝑑𝑓(𝑑))}β‰₯π‘Žπ‘–πœ‘(π‘šπ‘–) for every 1≀𝑖≀𝑛, and consequently, sup𝑑>0{π‘‘πœ‘(𝑑𝑓(𝑑))}β‰₯max1≀𝑖≀𝑛{π‘Žπ‘–πœ‘(π‘šπ‘–)}, 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 𝑑𝑓(𝑑)<∞ for any 𝑑>0. Then, by standard arguments of existing a sequence of nonnegative simple functions (𝑓𝑛) such that 𝑑𝑓𝑛↑𝑑𝑓 and π‘“βˆ—π‘›β†‘π‘“βˆ— as π‘›β†’βˆž we can show that sup𝑑>0ξ€½πœ‘(𝑑)π‘“βˆ—ξ€Ύ(𝑑)=limπ‘›β†’βˆžsup𝑑>0ξ€½πœ‘(𝑑)π‘“βˆ—π‘›ξ€Ύ(𝑑)=limπ‘›β†’βˆžsup𝑑>0ξ€½ξ€·π‘‘π‘‘πœ‘π‘“π‘›(𝑑)ξ€Έξ€Ύ=sup𝑑>0ξ€½ξ€·π‘‘π‘‘πœ‘π‘“(𝑑)ξ€Έξ€Ύ(2.20) and conclude the proof.

In fact, Proposition 2.9 describes the largest family of increasing and continuous functions πœ‘, for which (2.17) is satisfied. Indeed, let 𝛼=∞, 𝑓≑1, and πœ‘(𝑑)=2βˆ’(1/(1+𝑑)) for any π‘‘βˆˆ[0,∞). Then π‘“βˆ—β‰‘π‘“, 𝑑𝑓(𝑑)=0 for all 𝑑β‰₯1 and 𝑑𝑓(𝑑)=∞ for any π‘‘βˆˆ[0,1). Clearly, πœ‘ is increasing and continuous and also limπ‘‘β†’βˆžπœ‘(𝑑)=2. Therefore, sup𝑑>0{πœ‘(𝑑)π‘“βˆ—(𝑑)}=2 and sup𝑑>0{π‘‘πœ‘(𝑑𝑓(𝑑))}=∞, 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 π‘“βˆˆπΈ 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 π‘“βˆˆπΈ and for a.a. π‘‘βˆˆ(0,𝛼) we have limπœ–β†’0β€–β€–(π‘“βˆ’π‘“(𝑑))πœ’π΅(𝑑,πœ–)β€–β€–πΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖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<π‘Ÿβ‰€1 and an equivalent π‘Ÿ-norm β€–|β‹…|‖𝐸 to ‖⋅‖𝐸 such that for any π‘“βˆˆπΈ and for a.a. π‘‘βˆˆ(0,𝛼) we getlimπœ–β†’0β€–β€–||π‘“πœ’π΅(𝑑,πœ–)||‖‖𝐸‖‖||πœ’π΅(𝑑,πœ–)||‖‖𝐸=||||.𝑓(𝑑)(3.2) If (𝐸,‖⋅‖𝐸) is a normed space, then the quasinorm β€–|β‹…|‖𝐸 can be replaced by ‖⋅‖𝐸.

In the next proposition we establish measurability of the maximal function 𝑀𝐸(π‘Ÿ)𝑓.

Proposition 3.2. Let 0<π‘Ÿ<∞ and let 𝐸 be a r.i. order continuous quasi-Banach function space. If π‘“βˆˆπΈ, then the maximal function 𝑀𝐸(π‘Ÿ)𝑓 is measurable on (0,𝛼).

Proof. Let πœ–>0 and π‘‘βˆˆ(0,𝛼). We first observe that π‘“πœ–(π‘Ÿ)(𝑑) is continuous on (0,𝛼). In fact, for any 𝑑𝑛→𝑑, limπ‘›β†’βˆžπœ’π΅(𝑑𝑛,πœ–)=πœ’π΅(𝑑,πœ–), and by order continuity of 𝐸 we obtain that limπ‘›β†’βˆžβ€–π‘“πœ’π΅(𝑑𝑛,πœ–)β€–π‘ŸπΈ=β€–π‘“πœ’π΅(𝑑,πœ–)β€–π‘ŸπΈ and limπ‘›β†’βˆžβ€–πœ’π΅(𝑑𝑛,πœ–)‖𝐸=β€–πœ’π΅(𝑑,πœ–)‖𝐸. Now by Fatou’s property of 𝐸 we have limπ›Ώβ†’πœ–βˆ’β€–β€–π‘“πœ’π΅(𝑑,𝛿)‖‖𝐸=β€–β€–π‘“πœ’π΅(𝑑,πœ–)‖‖𝐸,limπ›Ώβ†’πœ–βˆ’β€–β€–πœ’π΅(𝑑,𝛿)‖‖𝐸=β€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸.(3.3) Hence we have 𝑀𝐸(π‘Ÿ)𝑓𝑓(𝑑)=supπœ–(π‘Ÿ),(𝑑)βˆΆπœ–>0,πœ–βˆˆβ„š,𝐡(𝑑,πœ–)βŠ‚(0,𝛼)(3.4) and thus 𝑀𝐸(π‘Ÿ)𝑓 is measurable.

Remark 3.3. If π‘Š satisfies Ξ”2 condition, then we obtain measurability of the maximal function 𝑀Λ(π‘Ÿ)𝑝,𝑀𝑓 for any π‘“βˆˆΞ›π‘,𝑀, analogously as in case of the maximal function 𝑀Γ(π‘Ÿ)𝑝,𝑀𝑔 for any π‘”βˆˆΞ“π‘,𝑀 when 𝛼=∞ and π‘Š(∞)=∞.

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

Theorem 3.4. Let 0<π‘Ÿ<∞. If a r.i. quasi-Banach function space 𝐸 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘ŸπΈ, then there exists 𝐢>0 such that for all π‘“βˆˆπΈ and πœ†>0 we have ξ‚€π‘‘πœ†πœ™π‘€πΈ(π‘Ÿ)𝑓‖(πœ†)β‰€πΆπ‘“β€–π‘ŸπΈ.(3.5)

Proof. Assume that πœ†>0. Denote Ξ©πœ†=ξ‚†π‘‘βˆˆ(0,𝛼)βˆΆπ‘€πΈ(π‘Ÿ).𝑓(𝑑)>πœ†(3.6) Clearly, by Proposition 3.2 we get that Ξ©πœ† is measurable for all πœ†>0. Letting π‘‘βˆˆΞ©πœ†, there exists πœ–π‘‘>0 such that 𝐡(𝑑,πœ–π‘‘)βŠ‚(0,𝛼) and πœ†<π‘“πœ–(π‘Ÿ)𝑑(𝑑).(3.7) Let 𝑐<πœ‡(Ξ©πœ†) and denote ⋃𝐡=π‘‘βˆˆΞ©πœ†π΅(𝑑,πœ–π‘‘). Since Ξ©πœ†βŠ‚π΅, we get 𝑐<πœ‡(𝐡). Hence, by regularity of the Lebesgue measure πœ‡ there is a compact set πΎβŠ‚π΅ such that 𝑐<πœ‡(𝐾). By the fact that a collection 𝐷={𝐡(𝑑,πœ–π‘‘)βˆΆπ‘‘βˆˆΞ©πœ†} 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 {𝐡(π‘‘π‘˜,πœ–π‘‘π‘˜)∢1β‰€π‘˜β‰€π‘›}βŠ‚π· such that βˆ‘πœ‡(𝐾)≀4π‘›π‘˜=1πœ‡(𝐡(π‘‘π‘˜,πœ–π‘‘π‘˜)). Therefore, by πœ™(4𝑠)≀4πœ™(𝑠) and by (3.7) we get ξƒ©πœ™(𝑐)β‰€πœ™(πœ‡(𝐾))≀4πœ™π‘›ξ“π‘˜=1πœ‡ξ€·π΅ξ€·π‘‘π‘˜,πœ–π‘‘π‘˜ξƒͺ≀4πœ™π‘›ξ“π‘˜=1πœ™βˆ’1ξ‚€πœ†βˆ’1β€–β€–π‘“πœ’π΅(π‘‘π‘˜,πœ–π‘‘π‘˜)β€–β€–π‘ŸπΈξ‚ξƒͺ.(3.8) Hence, by assumption that 𝐸 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘ŸπΈ, there is 𝐢>0 such that for any 1β‰€π‘˜β‰€π‘› we get ξƒ©πœ™(𝑐)<4πœ™π‘›ξ“π‘˜=1πœ™βˆ’1ξ‚€πœ†βˆ’1β€–β€–π‘“πœ’π΅(π‘‘π‘˜,πœ–π‘‘π‘˜)β€–β€–π‘ŸπΈξ‚ξƒͺ≀4πΆπœ†β€–β€–β€–β€–π‘›ξ“π‘˜=1π‘“πœ’π΅(π‘‘π‘˜,πœ–π‘‘π‘˜)β€–β€–β€–β€–π‘ŸπΈβ‰€4πΆπœ†β€–π‘“β€–π‘ŸπΈ.(3.9) Since 𝑐<πœ‡(Ξ©πœ†) is arbitrary, we obtain πœ™(πœ‡(Ξ©πœ†))≀(4𝐢/πœ†)β€–π‘“β€–π‘ŸπΈ, 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 πœ™-estimate for ‖⋅‖𝐸, then 𝐸 has LDP, that is, limπœ–β†’0β€–β€–(π‘“βˆ’π‘“(𝑑))πœ’π΅(𝑑,πœ–)β€–β€–πΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=0(3.10) for all π‘“βˆˆπΈ and for a.a. π‘‘βˆˆ(0,∞). If in addition 𝐸 is normable, then limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–πΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=||||.𝑓(𝑑)(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 πœ‡ on ℝ (cf. [8]).
Define an operator 𝐿∢(0,∞)→ℝ by πΏβ„Ž(𝑑)=limsupπœ–β†’0ξ‚†β„Žπœ–(1)(𝑑)=limsupπœ–β†’0ξƒ―β€–β€–β„Žπœ’π΅(𝑑,πœ–)β€–β€–πΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸(3.12) for any β„ŽβˆˆπΈ and π‘‘βˆˆ(0,∞). Assume that πœ–>0 and 𝐡(𝑑,πœ–)βŠ‚(0,∞) and also π‘“βˆˆπΈ, π‘βˆˆβ„. Let 𝑔=π‘πœ’π΄ be a characteristic function of an open interval 𝐴. Notice that for a.a. π‘‘βˆˆ(0,∞) there exist 𝛿𝑑>0 such that for all 0<πœ–<𝛿𝑑 we have either 𝐡(𝑑,πœ–)βŠ‚π΄ or 𝐡(𝑑,πœ–)βŠ‚π΄π‘ and consequently ((π‘”βˆ’π‘)πœ’π΅(𝑑,πœ–))βˆ—=|𝑔(𝑑)βˆ’π‘|πœ’(0,2πœ–). Therefore, (π‘”βˆ’π‘)πœ–(1)β€–β€–(𝑑)=(π‘”βˆ’π‘)πœ’π΅(𝑑,πœ–)β€–β€–πΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=β€–β€–(𝑔(𝑑)βˆ’π‘)πœ’(0,2πœ–)β€–β€–πΈβ€–β€–πœ’(0,2πœ–)‖‖𝐸=||||𝑔(𝑑)βˆ’π‘(3.13) for a.a. π‘‘βˆˆ(0,∞) and for any 0<πœ–<𝛿𝑑. Hence 𝐿(π‘”βˆ’π‘)(𝑑)=limsupπœ–β†’0(π‘”βˆ’π‘)πœ–(1)=||||(𝑑)𝑔(𝑑)βˆ’π‘(3.14) for a.a. π‘‘βˆˆ(0,∞). Observe that the above equation can be proved analogously for any step function 𝑔 with support of finite measure. Let πœ‰ be a constant in the triangle inequality of the quasinorm ‖⋅‖𝐸. Thus 𝐿𝐿||𝑔||ξ€Έ(π‘“βˆ’π‘)(𝑑)β‰€πœ‰(𝐿(π‘“βˆ’π‘”)(𝑑)+𝐿(π‘”βˆ’π‘)(𝑑))=πœ‰(π‘“βˆ’π‘”)(𝑑)+(𝑑)βˆ’π‘(3.15) for a.a. π‘‘βˆˆ(0,∞). Clearly 𝐿(π‘“βˆ’π‘”)(𝑑)≀𝑀𝐸(1)(π‘“βˆ’π‘”)(𝑑), whence 𝑀𝐿(π‘“βˆ’π‘)(𝑑)β‰€πœ‰πΈ(1)||||(π‘“βˆ’π‘”)(𝑑)+𝑔(𝑑)βˆ’π‘(3.16) for a.a. π‘‘βˆˆ(0,∞). Now replacing 𝑐 by 𝑓(𝑑) we get 𝑀𝐿(π‘“βˆ’π‘“(𝑑))(𝑑)β‰€πœ‰πΈ(1)||||(π‘“βˆ’π‘”)(𝑑)+𝑓(𝑑)βˆ’π‘”(𝑑)(3.17) for a.a. π‘‘βˆˆ(0,∞). Define Ω𝑠={π‘‘βˆˆ(0,∞)∢𝐿(π‘“βˆ’π‘“(𝑑))(𝑑)>𝑠}(3.18) for any 𝑠>0. By Proposition 3.2, we have that 𝑀𝐸(1)(π‘“βˆ’π‘”) is πœ‡-measurable. Recall [2, 8] that 𝑑𝑓1+𝑓2(𝑠)≀𝑑𝑓1(𝑠/2)+𝑑𝑓2(𝑠/2) for any 𝑓1,𝑓2∈𝐿0. Thus in view of (3.17) we obtain for 𝑠>0, πœ‡βˆ—ξ€·Ξ©π‘ ξ€Έξ‚΅β‰€πœ‡π‘‘βˆˆ(0,∞)βˆΆπ‘€πΈ(1)||||𝑠(π‘“βˆ’π‘”)(𝑑)+π‘“βˆ’π‘”(𝑑)>πœ‰ξ‚Άβ‰€π‘‘π‘€πΈ(1)(π‘“βˆ’π‘”)𝑠2πœ‰+π‘‘π‘“βˆ’π‘”ξ‚΅π‘ ξ‚Ά.2πœ‰(3.19) Now since πœ™ satisfies the triangle inequality with constant πœ‰, we get πœ™ξ€·πœ‡βˆ—ξ€·Ξ©π‘ ξ‚΅π‘‘ξ€Έξ€Έβ‰€πœ‰πœ™π‘€πΈ(1)(π‘“βˆ’π‘”)𝑠𝑑2πœ‰ξ‚Άξ‚Ά+πœ‰πœ™π‘“βˆ’π‘”ξ‚΅π‘ 2πœ‰ξ‚Άξ‚Ά(3.20) for every 𝑠>0. Observe that for any β„ŽβˆˆπΈ, π‘‘βˆˆ(0,𝛼), we have β„Žβˆ—(𝑑)πœ™(𝑑)β‰€β€–β„Žβˆ—πœ’(0,𝑑)β€–πΈβ‰€β€–β„Žβ€–πΈ. Thus, by Proposition 2.9 we have πœ™ξ‚΅π‘‘π‘“βˆ’π‘”ξ‚΅π‘ β‰€2πœ‰ξ‚Άξ‚Ά2πœ‰π‘ sup𝑑>0ξ€½πœ™(𝑑)(π‘“βˆ’π‘”)βˆ—ξ€Ύβ‰€(𝑑)2πœ‰π‘ β€–π‘“βˆ’π‘”β€–πΈ.(3.21) Furthermore, by Theorem 3.4 there exists 𝐢>0 such that πœ™ξ‚΅π‘‘π‘€πΈ(1)(π‘“βˆ’π‘”)𝑠≀2πœ‰ξ‚Άξ‚Ά2πœ‰πΆπ‘ β€–π‘“βˆ’π‘”β€–πΈ.(3.22) Therefore, for any step function 𝑔 and for all 𝑠>0, πœ™ξ€·πœ‡βˆ—ξ€·Ξ©π‘ β‰€ξ€Έξ€Έ2πœ‰2(𝐢+1)π‘ β€–π‘“βˆ’π‘”β€–πΈ.(3.23) Hence we have πœ™ξ€·πœ‡βˆ—ξ€·Ξ©π‘ ξ€·πœ‡ξ€·Ξ©ξ€Έξ€Έ=πœ™π‘ ξ€Έξ€Έ=0(3.24) for all 𝑠>0. So 𝐿(π‘“βˆ’π‘“(𝑑))(𝑑)=0 for a.a. π‘‘βˆˆ(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 πœ™-estimate of Ξ“π‘ž,𝑝 on (0,∞), for 0<𝑝,π‘ž<∞. Clearly in this case 𝑀(𝑑)=𝑑𝑝/π‘žβˆ’1 satisfies 𝐡𝑝 condition. Thus Λ𝑝,𝑀=Γ𝑝,𝑀 and ‖⋅‖Λ𝑝,π‘€β‰ˆβ€–β‹…β€–Ξ“π‘,𝑀. Hence by Theorems 3 and  7 in [9] and by HΓΆlder’s inequality we obtain the following corollary.

Corollary 3.6. Let 𝛼=∞, 0<𝑝,π‘Ÿ<∞ and, 1<π‘ž<∞.(i)If π‘β‰€π‘ž, then Ξ“π‘ž,𝑝 satisfies a lower πœ™-estimate for the functional β€–β‹…β€–π‘ŸΞ“π‘ž,𝑝, where 1β‰€π‘Ÿ, and an upper πœ™-estimate for β€–β‹…β€–π‘ŸΞ“π‘ž,𝑝, where π‘Ÿβ‰€π‘/π‘ž.(ii)If 𝑝β‰₯π‘ž, then Ξ“π‘ž,𝑝 satisfies an upper πœ™-estimate for the functional β€–β‹…β€–π‘ŸΞ“π‘ž,𝑝, where π‘Ÿβ‰€1, and a lower πœ™-estimate for the functional β€–β‹…β€–π‘ŸΞ“π‘ž,𝑝, where π‘Ÿβ‰₯𝑝.

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

Proposition 3.7. Let 0<𝑝<∞ and 𝛼=∞. If the Lorentz space Γ𝑝,𝑀 satisfies a lower πœ™-estimate for β€–β‹…β€–, then Γ𝑝,𝑀 has LDP. Consequently, for 0<π‘β‰€π‘ž<∞ and 1<π‘ž, Lorentz spaces Ξ“π‘ž,𝑝 on (0,∞) have LDP.

We will show next which spaces (𝐸,‖⋅‖𝐸) do not have LDP with respect to β€–β‹…β€–π‘ŸπΈ. Notice that the ratio β€–π‘“β€–π‘Ÿ1𝐸/β€–π‘”β€–π‘Ÿ2𝐸 can be reduced easily to the quotient β€–π‘“β€–π‘ŸπΈ/‖𝑔‖𝐸 for any 𝑓,π‘”βˆˆπΈ, 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<π‘Ÿ<∞. If order continuous 𝐸 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘ŸπΈ, then for any π‘“βˆˆπΈ and for a.a. π‘‘βˆˆ(0,𝛼) we have limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–π‘ŸπΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=0.(3.25)(ii) Let 0<π‘Ÿ<1. Then for any π‘“βˆˆπΈ and for a.a. π‘‘βˆˆπ•Š(𝑓) we have limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–π‘ŸπΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=∞.(3.26)

Proof. Let 𝐿∢(0,𝛼)→ℝ be the operator on 𝐸 given by 𝐿𝑓(𝑑)=limsupπœ–β†’0{π‘“πœ–(π‘Ÿ)(𝑑)}, π‘“βˆˆπΈ.
(i) Let 𝑔 be a step function with a finite measure support. Notice that for a.a. π‘‘βˆˆ(0,𝛼) and for small enough πœ–>0 we have π‘”πœ–(π‘Ÿ)β€–β€–(𝑑)=π‘”πœ’π΅(𝑑,πœ–)β€–β€–π‘ŸπΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=|𝑔(𝑑)|π‘Ÿβ€–β€–πœ’π΅(𝑑,πœ–)β€–β€–πΈπ‘Ÿβˆ’1.(3.27) Therefore 𝐿𝑔(𝑑)=limsupπœ–β†’0{π‘”πœ–(π‘Ÿ)(𝑑)}=0 and thus 𝐿𝑓(𝑑)β‰€πœ‰(𝐿(π‘“βˆ’π‘”)(𝑑)+𝐿𝑔(𝑑))=πœ‰πΏ(π‘“βˆ’π‘”)(𝑑)β‰€πœ‰π‘€πΈ(π‘Ÿ)(π‘“βˆ’π‘”)(𝑑)(3.28) for a.a. π‘‘βˆˆ(0,𝛼). Denoting Ω𝑠={π‘‘βˆˆ(0,𝛼)βˆΆπΏπ‘“(𝑑)>𝑠}, 𝑠>0, by Proposition 3.2, 𝑀𝐸(π‘Ÿ)(π‘“βˆ’π‘”) is πœ‡-measurable. Now by Theorem 3.4 there exists 𝐢>0 such that for all 𝑠>0, πœ™ξ€·πœ‡βˆ—ξ€·Ξ©π‘ ξ‚΅π‘‘ξ€Έξ€Έβ‰€πœ™π‘€πΈ(π‘Ÿ)(π‘“βˆ’π‘”)ξ‚΅π‘ πœ‰β‰€ξ‚Άξ‚Άπœ‰πΆπ‘ β€–π‘“βˆ’π‘”β€–π‘ŸπΈ,(3.29) which shows that πœ‡(Ω𝑠)=0 and thus 𝐿𝑓(𝑑)=0 a.e.
(ii) Let 𝑓𝑛↑|𝑓| a.e., where 0≀𝑓𝑛 are step functions. Therefore, for a.a. π‘‘βˆˆπ•Š(𝑓) there is π‘›βˆˆβ„• such that 0<𝑓𝑛(𝑑)≀|𝑓(𝑑)| and for small enough πœ–>0 we have ξ€·π‘“π‘›ξ€Έπœ–(π‘Ÿ)‖‖𝑓(𝑑)=π‘›πœ’π΅(𝑑,πœ–)β€–β€–π‘ŸπΈβ€–β€–πœ’π΅(𝑑,πœ–)‖‖𝐸=||𝑓𝑛||(𝑑)π‘Ÿβ€–β€–πœ’π΅(𝑑,πœ–)β€–β€–πΈπ‘Ÿβˆ’1.(3.30) Thus, by the assumption that 0<π‘Ÿ<1 we get 𝑓𝐿𝑓(𝑑)β‰₯𝐿𝑛(𝑑)=liminfπœ–β†’0ξ‚†ξ€·π‘“π‘›ξ€Έπœ–(π‘Ÿ)(𝑑)=∞(3.31) for a.a. π‘‘βˆˆπ•Š(𝑓), and the proof is finished.

The next corollary follows directly from Theorem 3.8.

Corollary 3.9. (i) Let 0<𝑝<∞, 1<π‘Ÿ<∞. If Lorentz space Γ𝑝,𝑀 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘Ÿ, then for any π‘“βˆˆΞ“π‘,𝑀 and for a.a. π‘‘βˆˆ(0,𝛼) we have limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–π‘Ÿβ€–β€–πœ’π΅(𝑑,πœ–)β€–β€–=0.(3.32)(ii) Let 0<𝑝<∞, 0<π‘Ÿ<1. Then for any π‘“βˆˆΞ“π‘,𝑀 and for a.a. π‘‘βˆˆπ•Š(𝑓) we have limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–π‘Ÿβ€–β€–πœ’π΅(𝑑,πœ–)β€–β€–=∞.(3.33)

The next result needed for further applications states conditions, which guarantee that Γ𝑝,𝑀 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘Ÿ, where 0<π‘Ÿ<∞. We omit the proof of the following proposition.

Proposition 3.10. Assume that 0<𝑝,π‘Ÿ<∞, and Lorentz space Γ𝑝,𝑀 satisfies a lower π‘Ÿ-estimate. If πœ™ is concave, then Γ𝑝,𝑀 satisfies a lower πœ™-estimate for β€–β‹…β€–π‘Ÿ.

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

Example 3.11. Let 𝛼=∞, 𝑝=2, and 𝑀(𝑠)=ln(𝑠+1)+1 for all π‘ βˆˆ[0,∞). Then for any π‘“βˆˆΞ“2,𝑀 and for a.a π‘‘βˆˆ(0,∞) we have limπœ–β†’0β€–β€–π‘“πœ’π΅(𝑑,πœ–)β€–β€–2Ξ“2,π‘€β€–β€–πœ’π΅(𝑑,πœ–)β€–β€–Ξ“2,𝑀=0.(3.34)

Proof. Notice that for any π‘‘βˆˆ(0,∞), π‘Š(𝑑)=(𝑑+1)ln(𝑑+1),π‘Š2(𝑑)=𝑑(𝑑+1)ln(𝑑+1)βˆ’π‘‘2ln(𝑑)+𝑑.(3.35) Consequently, β€–β€–πœ’πœ™(𝑑)=(0,𝑑)β€–β€–Ξ“2,𝑀=ξ€·(𝑑+1)2ln(𝑑+1)βˆ’π‘‘2ξ€Έln(𝑑)+𝑑)1/2(3.36) for any π‘‘βˆˆ(0,∞). Since 𝑀 is increasing, 𝑀 satisfies 𝑅𝐡2 condition. Moreover, we have ξ€œ10𝑀(𝑑)𝑑2ξ€œπ‘‘π‘‘=∞1𝑑𝑀(𝑑)𝑑𝑑=∞,ξ‚΅π‘Šπ‘‘π‘‘2(𝑑)𝑑1=ln1+𝑑>0(3.37) for any π‘‘βˆˆ(0,∞), which concludes that π‘Š2(𝑑)/𝑑 is increasing. Hence, by Theorem 3.3 [24] we obtain that Ξ“2,𝑀 satisfies a lower 2-estimate. Now we claim that πœ™ is concave. By simple calculations we get the second derivative on (0,∞), πœ™ξ…žξ…ž(𝑑)=(π‘‘βˆ’ln(𝑑+1))ln(𝑑)βˆ’(𝑑+2)ln(𝑑+1)𝑑2.ln(1+1/𝑑)+(2𝑑+1)ln(𝑑+1)+𝑑(3.38) We observe that 𝑑2ξ‚€1ln1+𝑑+(2𝑑+1)ln(𝑑+1)+𝑑>0,(π‘‘βˆ’ln(𝑑+1))ln(𝑑)<(𝑑+2)ln(𝑑+1).(3.39) Therefore, πœ™ξ…žξ…ž(𝑑)<0 for all π‘‘βˆˆ(0,∞), which implies concavity of πœ™. Hence, by Proposition 3.10 we obtain that Ξ“2,𝑀 satisfies a lower πœ™-estimate for β€–β‹…β€–2Ξ“2,𝑀. 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 π‘‘βˆˆ(0,∞), πœ–>0 be such that 𝐡(𝑑,πœ–)βŠ‚(0,∞). Assume that π‘“πœ–(𝑑)βˆˆπ‘‡(𝑝,𝐡(𝑑,πœ–))(𝑓) for π‘“βˆˆΞ“π‘,𝑀. By definition of the best constant approximant we get||𝑓(𝑑)βˆ’π‘“πœ–||≀‖‖(𝑑)(𝑓(𝑑)βˆ’π‘“)πœ’π΅(𝑑,πœ–)‖‖Γ𝑝,π‘€β€–β€–πœ’π΅(𝑑,πœ–)‖‖Γ𝑝,𝑀+β€–β€–ξ€·π‘“πœ–ξ€Έπœ’(𝑑)βˆ’π‘“π΅(𝑑,πœ–)‖‖Γ𝑝,π‘€β€–β€–πœ’π΅(𝑑,πœ–)‖‖Γ𝑝,𝑀‖‖≀2(𝑓(𝑑)βˆ’π‘“)πœ’π΅(𝑑,πœ–)‖‖Γ𝑝,π‘€β€–β€–πœ’π΅(𝑑,πœ–)‖‖Γ𝑝,𝑀.(3.40) Now applying Proposition 3.7 we get that π‘“πœ–(𝑑)→𝑓(𝑑) as πœ–β†’0. Therefore we get the following theorem.

Theorem 3.12. Let 𝛼=∞ and 1≀𝑝<∞. If the space Γ𝑝,𝑀 satisfies a lower πœ™-estimate for ‖⋅‖Γ𝑝,𝑀, then for every π‘“βˆˆΞ“π‘,𝑀 we have π‘“πœ–(𝑑)βŸΆπ‘“(𝑑)asπœ–βŸΆ0(3.41) for a.a. π‘‘βˆˆ(0,∞), where π‘“πœ–(𝑑)βˆˆπ‘‡(𝑝,𝐡(𝑑,πœ–))(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 π‘“βˆˆΞ“π‘βˆ’1,𝑀 for 1<𝑝<2.

Lemma 3.13. Let 1<𝑝<2, π΄βŠ‚[0,𝛼), 0<πœ‡(𝐴)<∞, and let π‘“βˆˆΞ“π‘βˆ’1,𝑀, 𝑓β‰₯0. Then for any π‘’πœ’π΄βˆˆπ‘‡(𝑝,𝐴)(𝑓) extended best constant approximant of 𝑓 and for a.a. π‘ βˆˆ(0,𝛼) we have ||||π‘’βˆ’π‘“(𝑠)π‘βˆ’1β€–β€–πœ’π΄β€–β€–π‘Ξ“π‘,𝑀‖‖≀5(π‘“βˆ’π‘“(𝑠))πœ’π΄β€–β€–Ξ“π‘βˆ’1π‘βˆ’1,𝑀.(3.42)

Proof. Let π‘ βˆˆ(0,𝛼) and |𝑓(𝑠)|<∞. Since (πœ’π΄)βˆ—βˆ—β‰€1, by subadditivity of the maximal function and the power function π‘£π‘βˆ’1 for 1<𝑝<2 we get ||||π‘’βˆ’π‘“(𝑠)π‘βˆ’1β€–β€–πœ’π΄β€–β€–π‘Ξ“π‘,𝑀=ξ€œπ›Ό0ξ€·πœ’π΄ξ€Έβˆ—βˆ—ξ€·(𝑑)(π‘’βˆ’π‘“(𝑠))πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)β‰€ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑𝛼0ξ€·πœ’π΄ξ€Έβˆ—βˆ—ξ€·(𝑑)(π‘“βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)+β€–β€–(𝑑)𝑀(𝑑)𝑑𝑑(π‘“βˆ’π‘“(𝑠))πœ’π΄β€–β€–Ξ“π‘βˆ’1π‘βˆ’1,𝑀.(3.43) We will finish the proof under assumption that 𝑓(𝑠)β‰₯𝑒. In the other case the proof is similar. Since π‘’πœ’π΄βˆˆπ‘‡(𝑝,𝐴)(𝑓)=[𝑓(𝑝,𝐴)πœ’π΄,𝑓(𝑝,𝐴)πœ’π΄], by Corollary 2.11 [7] and by Hardy-Littlewood inequality we obtain ξ€œπ›Ό0ξ€·πœ’π΄ξ€Έβˆ—βˆ—ξ€·(𝑑)(π‘“βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑≀2𝛼01π‘‘ξ€œπ΄βˆ©{𝑓≀𝑒}πœ’(0,𝑑)ξ€·πœŒ((π‘’βˆ’π‘“)πœ’π΄,πœ’π΄)ξ€Έξ€·π‘‘πœ‡(π‘“βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑≀2𝛼0ξ€·πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—ξ€·(𝑑)(π‘“βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)β€–β€–(𝑑)𝑀(𝑑)𝑑𝑑≀2(π‘“βˆ’π‘“(𝑠))πœ’π΄β€–β€–Ξ“π‘βˆ’1π‘βˆ’1,π‘€ξ€œ+2𝛼0ξ€·πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—ξ€·(𝑑)(𝑓(𝑠)βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)(𝑑)𝑀(𝑑)𝑑𝑑.(3.44) Moreover, by assumption that 1<𝑝<2 we have ξ€·πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—(𝑑)β‰€πœ’(0,πœ‡(𝐴∩{𝑓≀𝑒}))ξ‚΅(𝑑)+πœ‡(𝐴∩{𝑓≀𝑒})π‘‘ξ‚Άπ‘βˆ’1πœ’[πœ‡(𝐴∩{𝑓≀𝑒}),𝛼)=ξ€·πœ’(𝑑)𝐴∩{𝑓≀𝑒}ξ€Έβˆ—βˆ—(π‘βˆ’1)(𝑑)(3.45) for any π‘‘βˆˆ(0,𝛼). Consequently, by the fact that 𝑓(𝑠)β‰₯𝑒 we obtain ξ€œπ›Ό0ξ€·πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—ξ€·(𝑑)(𝑓(𝑠)βˆ’π‘’)πœ’π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)β‰€ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑𝛼0ξ€·(𝑓(𝑠)βˆ’π‘’)πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—(π‘βˆ’1)ξ€·πœ’(𝑑)π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)β‰€ξ€œ(𝑑)𝑀(𝑑)𝑑𝑑𝛼0ξ€·(𝑓(𝑠)βˆ’π‘“)πœ’π΄βˆ©{𝑓≀𝑒}ξ€Έβˆ—βˆ—(π‘βˆ’1)ξ€·πœ’(𝑑)π΄ξ€Έβˆ—βˆ—(π‘βˆ’1)≀‖‖(𝑑)𝑀(𝑑)𝑑𝑑(π‘“βˆ’π‘“(𝑠))πœ’π΄β€–β€–Ξ“π‘βˆ’1π‘βˆ’1,𝑀.(3.46) Hence, by conditions (3.43) and (3.44) we finish the proof.

Corollary 3.14. Let 1<𝑝<∞ and π‘€βˆˆπ·π‘βˆ’1. Assume that πœ™ and πœ“ are the fundamental functions of Γ𝑝,𝑀 and Ξ“π‘βˆ’1,𝑀, respectively. If one of the following conditions(i)there exists 𝐴>0 such that π‘Šπ‘βˆ’1(𝑑)β‰€π΄π‘Šπ‘(𝑑) for all π‘‘βˆˆ(0,𝛼),(ii)π‘€βˆˆπ΅π‘βˆ’1, holds, then πœ™π‘β‰ˆπœ“π‘βˆ’1.

Proof. Notice that for all π‘‘βˆˆ(0,𝛼), πœ™(𝑑)𝑝=π‘Š(𝑑)+π‘‘π‘ξ€œπ›Όπ‘‘π‘ βˆ’π‘π‘€(𝑠)π‘‘π‘ β‰€π‘Š(𝑑)+π‘‘π‘ξ€œπ›Όπ‘‘π‘ βˆ’π‘+1π‘‘βˆ’1𝑀(𝑠)𝑑𝑠=πœ“(𝑑)π‘βˆ’1.(3.47) Suppose now that there is 𝐴>0 such that π‘Šπ‘βˆ’1(𝑑)β‰€π΄π‘Šπ‘(𝑑) for all π‘‘βˆˆ(0,𝛼). Therefore, πœ“(𝑑)π‘βˆ’1=π‘Š(𝑑)+π‘Šπ‘βˆ’1(𝑑)β‰€π‘Š(𝑑)+π΄π‘Šπ‘(𝑑)β‰€π΄πœ™(𝑑)𝑝(3.48) for any π‘‘βˆˆ(0,𝛼), and so πœ™π‘β‰ˆπœ“π‘βˆ’1.
Now assume that π‘€βˆˆπ΅π‘βˆ’1. Then there exists 𝐡>0 such that for all π‘“βˆˆΞ“π‘βˆ’1,𝑀 we have β€–π‘“β€–Ξ“π‘βˆ’1π‘βˆ’1,π‘€β‰€π΅β€–π‘“β€–Ξ›π‘βˆ’1π‘βˆ’1,𝑀. Consequently, πœ“(𝑑)π‘βˆ’1=β€–β€–πœ’(0,𝑑)β€–β€–Ξ“π‘βˆ’1π‘βˆ’1,π‘€β€–β€–πœ’β‰€π΅(0,𝑑)β€–β€–Ξ›π‘βˆ’1π‘βˆ’1,𝑀=π΅π‘Š(𝑑)β‰€π΅πœ™(𝑑)𝑝(3.49) for any π‘‘βˆˆ(0,𝛼), and the proof is completed.

Theorem 3.15. Let 𝛼=∞, 1<𝑝<2, π‘€βˆˆπ·π‘βˆ’1 and π‘“βˆˆΞ“π‘βˆ’1,𝑀, 𝑓β‰₯0. Assume that πœ™ and πœ“ are the fundamental functions of Γ𝑝,𝑀 and Ξ“π‘βˆ’1,𝑀, respectively. If Lorentz space Ξ“π‘βˆ’1,𝑀 satisfies a lower πœ“-estimate for β€–β‹…β€–Ξ“π‘βˆ’1,𝑀 and πœ™π‘β‰ˆπœ“π‘βˆ’1, then π‘“πœ–(𝑑)βŸΆπ‘“(𝑑)asπœ–βŸΆ0(3.50) for a.a. π‘‘βˆˆ(0,∞), where π‘“πœ–(𝑑)βˆˆπ‘‡(𝑝,𝐡(𝑑,πœ–))(𝑓) is an extended best constant approximant of 𝑓.

Proof. Let πœ–>0 and π‘‘βˆˆ(0,∞), |𝑓(𝑑)|<∞. By Lemma 3.13 for any π‘“πœ–(𝑑)βˆˆπ‘‡(𝑝,𝐡(𝑑,πœ–))(𝑓) we get ||π‘“πœ–(||𝑑)βˆ’π‘“(𝑑)≀51/(π‘βˆ’1)β€–β€–(π‘“βˆ’π‘“(𝑑))πœ’π΅(𝑑,πœ–)β€–β€–Ξ“π‘βˆ’1,π‘€β€–β€–πœ’π΅(𝑑,πœ–)‖‖Γ𝑝/(π‘βˆ’1)𝑝,𝑀.(3.51) Since πœ™π‘β‰ˆπœ“π‘βˆ’1, there is 𝐡>0 such that for all 𝛿>0 we have β€–πœ’π΅(𝑑,𝛿)β€–Ξ“π‘βˆ’1,π‘€β‰€π΅β€–πœ’π΅(𝑑,𝛿)‖Γ𝑝/(π‘βˆ’1)𝑝,𝑀. Consequently, ||π‘“πœ–(||𝑑)βˆ’π‘“(𝑑)≀51/(π‘βˆ’1)𝐡‖‖(π‘“βˆ’π‘“(𝑑))πœ’π΅(𝑑,πœ–)β€–β€–Ξ“π‘βˆ’1,π‘€β€–β€–πœ’π΅(𝑑,πœ–)β€–β€–Ξ“π‘βˆ’1,𝑀(3.52) for any π‘‘βˆˆ(0,∞) and πœ–>0. Hence, by Proposition 3.7 and by assumption that Ξ“π‘βˆ’1,𝑀 satisfies a lower πœ“-estimate for β€–β‹…β€–Ξ“π‘βˆ’1,𝑀 we obtain that π‘“πœ–(𝑑)→𝑓(𝑑) as πœ–β†’0 for a.a. π‘‘βˆˆ(0,∞).

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

Corollary 3.16. Let 𝛼=∞ and 1<π‘ž<∞. If one of the following conditions(i)1β‰€π‘β‰€π‘ž and π‘“βˆˆΞ“π‘ž,𝑝,(ii)1<𝑝<2 and π‘“βˆˆΞ“π‘ž,π‘βˆ’1, 𝑓β‰₯0, is satisfied, then for a.a. π‘‘βˆˆ(0,∞) we have π‘“πœ–(𝑑)βŸΆπ‘“(𝑑)asπœ–βŸΆ0,(3.53) where π‘“πœ–(𝑑)βˆˆπ‘‡(𝑝,𝐡(𝑑,πœ–))(𝑓) is the best constant approximant of π‘“βˆˆΞ“π‘,𝑀 and the extended best constant approximant of π‘“βˆˆΞ“π‘βˆ’1,𝑀.

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 𝑀(𝑑)=𝑑(π‘βˆ’1)/π‘žβˆ’1 for π‘‘βˆˆ(0,∞). Since 1<𝑝<2, we have that π‘Šπ‘ž(𝑑)=π‘‘π‘βˆ’1(π‘βˆ’1)/π‘ž,π‘Šπ‘βˆ’1π‘ž(𝑑)=𝑑(π‘βˆ’1)π‘žβˆ’π‘+1(π‘βˆ’1)/π‘ž(3.54) for all π‘‘βˆˆ(0,∞), which implies that π‘€βˆˆπ·π‘βˆ’1βˆ©π΅π‘βˆ’1. Consequently, by Definition 2.6 there is 𝑇(𝑝,𝐡(𝑑,πœ–)) the extended best constant approximant operator on Ξ“π‘ž,π‘βˆ’1. Moreover, by Corollary 3.14 the fundamental functions πœ™ and πœ“ of the spaces Ξ“π‘ž,𝑝 and Ξ“π‘ž,π‘βˆ’1 respectively, satisfy πœ™π‘β‰ˆπœ“π‘βˆ’1. Hence, by Corollary 3.6 we get that Ξ“π‘ž,π‘βˆ’1 satisfies a lower πœ“-estimate for β€–β‹…β€–Ξ“π‘ž,π‘βˆ’1. Finally, by Theorem 3.15 we obtain that π‘“πœ–(𝑑), an extended best constant approximant of 𝑓, converges to 𝑓(𝑑) as πœ–β†’0 for a.a. π‘‘βˆˆ(0,∞).

Observe that the 𝐾-functional of the couple (Ξ“π‘ž,𝑝,𝐿∞) can be expressed equivalently by β€–π‘“βˆ—πœ’(0,𝑑)β€–Ξ“π‘ž,𝑝/πœ™(𝑑) for any π‘“βˆˆΞ“π‘ž,𝑝 and π‘‘βˆˆ(0,∞) [25]. The next result on inequalities between maximal function 𝑀(1)π‘ž,𝑝𝑓 and the 𝐾-functional of (Ξ“π‘ž,𝑝,𝐿∞