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

Authors: Maciej Ciesielski; Anna Kamińska
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spacesE and in particular on Lorentz spaces Γp,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 fϵ of f∈Γp,w or f∈Γp-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∈Γp-1,w, 1<p<∞.


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## Body

## 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 spacesΓp,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)⊂ℝn to this function f, that is, for a.a. v∈ℝn, (1.1)1μ(B(v,ϵ))∫B(v,ϵ)f(t)dt⟶f(v)asϵ⟶0.
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(1.2)t(MHf)*(t)≤4n‖f‖L1
for the Hardy-Littlewood maximal function MHf where t∈ℝn and f∈L1(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 Λw,ϕ to the spaces Λw,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 Γp,w to the spaces Γp-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 Γp,w.The paper consists of three sections and is organized as follows.In the preliminaries, Section2, we establish some basic notations and definitions and also recall some auxiliary results, which will be used later.Sections3 and 4 consist of the main results of the paper.We start Section3 proving measurability of the maximal function ME(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 Γp,w. In both types of LDT we employ the assumption of upper and lower ϕ-estimates of Γp,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 ME(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 Γp,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 Γp,w satisfies a lower (resp., an upper) ϕ-estimate, where ϕ is the fundamental function of Γp,w. In view of this characterization we investigate pointwise convergence of the best constant approximants fϵ to f as ϵ→0 whenever f∈Γp,w and 1≤p<∞, as well as the convergence of the extended best constant approximants fϵ for any f∈Γp-1,w and 1<p<2. We also present examples showing that this assumption is fulfilled by a large class of the spaces Γq,p. Finally, we investigate relations between maximal functions and the K-functional of Banach couple (Γq,p,L∞) in the spirit of the inequalities stated in [4]. We finish Section 3 with an example showing that MΓp,w(r)(f) is not equivalent to the K-functional of the pair (Γp,w,L∞).It is well known that the extension of the best constant approximant operator fromΓ1,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∈Γp-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 Γp,w at (f-u)χA in the direction χA for any f∈Γp-1,w and u∈ℝ.


## 2. Preliminaries

Letℝ and ℕ be the set of real and natural numbers, respectively. For any A⊂[0,α) denote Ac=[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 functiondf of a function f∈L0 is given by df(λ)=μ(s∈[0,α):|f|(s)>λ) for all λ≥0. Two functions f,g∈L0 are called equimeasurable, if df(λ)=dg(λ) for all λ≥0. We define the decreasing rearrangement for any f∈L0 by f*(t)=inf{s>0:df(s)≤t}, t≥0. For given f∈L0 we denote the maximal function of f* by f**(t)=(1/t)∫0tf*(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 df, 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)
Iff∈L0, g∈E, and |f|≤|g| a.e., then f∈E and ∥f∥E≤∥g∥E.(2)
There exists a strictly positivef∈E.IfE 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 df=dg, 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 Γp,w is a subspace of L0 such that(2.1)‖f‖=‖f‖Γp,w:=(∫0αf**pw)1/p=(∫0αf**p(t)w(t)dt)1/p<∞.
Given a measurable set A⊂[0,α) by Γp,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 Dp (in short w∈Dp), whenever it satisfies the following conditions:(2.2)W(s):=∫0sw<∞,Wp(s):=sp∫sαt-pw(t)dt<∞
for all 0<s≤α if α<∞ and for all 0<s<∞ otherwise. These two conditions guarantee that Γp,w≠{0}. We also assume that(2.3)Wp(s)>0for0<s<α,W(∞)=∫0∞w=∞ifα=∞.
Under these assumptions (Γp,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)=tp/q-1, t∈(0,α), the space Γp,w will be denoted by Γq,p.Unless we say otherwise, throughout this paper we assume thatϕ is the fundamental function of Γp,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), limt→∞ϕ(t)=∞ and limt→∞f*(t)=0 for f∈Γp,w. For more details about the properties of Γp,w see [9].Recall that for given0<p<∞, classical Lorentz space Λp,w is a subspace of L0 such that(2.4)‖f‖Λp,w:=(∫0αf*pw)1/p=(∫0αf*p(t)w(t)dt)1/p<∞.
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 Λp,w is a separable r.i. order continuous quasi-Banach function space [9]. The space Λp,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 Γp,w⊂Λp,w. Moreover, Γp,w=Λp,w if and only if w satisfies condition Bp, (w∈Bp for short) which means that there is A>0 such that for all s>0 we have Wp(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]).
Letf,h∈L0. Denote
(2.5)τ(f,h)(t)=df(|f|(t))+μ(u:|f|(u)=|f|(t),h(u)sign⁡(f(u))>h(t)sign⁡(f(t)))+μ(u:|f|(u)=|f|(t),h(u)sign⁡(f(u))=h(t)sign⁡(f(t)),u≤t)
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 df(λ)<∞ for all λ>0 and any h∈L0. Notice that for any f∈L0 with df(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)=df(|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.
Letf,g∈Γp,w and let τ(f,g), τ(g∣𝕊(g)∖𝕊(f),0) be measure-preserving transformations given by Definition 2.1. Denote
(2.6)ρ(f,g)(s)={τ(f,g)(s)ifs∈S(f),df(0)+τ(g∣S(g)∖S(f),0)(s)ifs∈S(g)∖S(f).Definition 2.3.
Letf∈Γp,w and A⊂[0,α) with μ(A)<∞. Denote
(2.7)K(f,A)(u,t)=1t∫A(1-2χ{f<u})χ(0,t)(ρ((f-u)χA,χA)),S(f,A)p(u)=∫0αK(f,A)(u,t)((f-u)χA)**(p-1)(t)w(t)dt
for any u∈ℝ and t∈(0,α).Let (X,∥·∥X) be a real normed space. Denote by BX (resp., SX) 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(2.8)‖x-x̃‖X=inf⁡y∈Y‖x-y‖X.
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 2Y be a collection of all subsets of Y. A set value map TY:X→2Y 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,(2.9)TY(x)={x̃∈Y:‖x-x̃‖X=inf⁡y∈Y‖x-y‖X}={x̃∈Y:‖x-x̃‖X≤‖x-y‖X,∀y∈Y}.
In case when X is a norm function space and Y is a family of constant functions, then TY is called best constant approximant operator, and each element x̃∈TY(x) is called best constant approximant to x∈X from Y. Let A⊂[0,α) with 0<μ(A)<∞ and 𝕂(A)={cχA:c∈ℝ}. It is well known that the set T𝕂(A)(f) is convex, compact, and a set of existence for all f∈Γp,w [18, 19]. Let’s recall some characterizations of best constant approximants over Lorentz spaces Γp,w.Theorem 2.4 (see [20, Theorem 7.5]).
Let1≤p<∞ and let f∈Γp,w∖𝕂(A). Then u∈𝕂(A) is the best constant approximant of f if and only if
(2.10)S(f,A)p(u)≥0,S(-f,A)p(-u)≥0.Corollary 2.5 (see [20, Corollary 7.3]).
For any1<p<∞ and w positive, we have that T𝕂(A)(f) is Chebyshev set for any f∈Γp,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 Γ1,w to L0, if w∈D1, and from Γp,w to Γp-1,w, if 1<p<∞ and w∈Dp-1. Now we recall definition of the extended operator T(p,A) on L0, if p=1, and on Γp-1,w, if 1<p<∞.Definition 2.6 (see [7]).
LetA⊂(0,α) with 0<μ(A)<∞ and let w∈Dp-1 if 1<p<∞, and w∈D1 if p=1. Assume that f∈Γp-1,w if p>1, and f∈L0 if p=1. Denote
(2.11)f̲(p,A)=min⁡{u:S(-f,A)p(-u)≥0},f¯(p,A)=max⁡{u:S(f,A)p(u)≥0}.
Then the extended best constant approximant operator is given by
(2.12)T(p,A)(f)=[f̲(p,A)χA,f¯(p,A)χA].In fact, anyu∈T(p,A)(f) is called an extended best constant approximant of f. Notice that in view of Theorem 2.4, if f∈Γp,w for 1≤p<∞, then any u∈T(p,A)(f) is a classical best constant approximant of f.Definition 2.7 (see [4]).
Let0<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 ∥·∥Er, if there exists C>0 such that for all n∈ℕ and (fi)i=1n⊂E with pairwise disjoint supports we have
(2.13)‖∑i=1nfi‖Er≤CΦ(∑i=1nΦ-1(‖fi‖Er)),
respectively
(2.14)Φ(∑i=1nΦ-1(‖fi‖Er))≤C‖∑i=1nfi‖Er.In the case whenΦ(t)=t1/p and r=1, this definition covers the notions of the upper (resp., lower) p-estimate [21].Let0<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 Γp,w.Definition 2.8.
Let0<r<∞, (E,∥·∥E) be a r.i. quasi-Banach function space. For any f∈E we denote
(2.15)fϵ(r)(t)=‖fχB(t,ϵ)‖Er‖χB(t,ϵ)‖E
for all ϵ>0 and t∈(0,α). The maximal function ME(r)f:(0,α)→ℝ is given by
(2.16)ME(r)f(t)=sup⁡{fϵ(r)(t):ϵ>0,B(t,ϵ)⊂(0,α)}
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.
Letα=∞, f∈L0, and let φ:[0,∞)→[0,∞) be an increasing continuous function. If limt→∞φ(t)=∞ or df(λ)<∞ for all λ>0, then
(2.17)sup⁡t>0{φ(t)f*(t)}=sup⁡t>0{tφ(df(t))}.Proof.
Notice first iflimt→∞φ(t)=∞ and there exists t>0 such that df(t)=∞, then lims→∞φ(s)f*(s)=∞, and the conclusion follows.
Now assume thatf=∑i=1naiχEi is a nonnegative simple function, where ai>ai+1 for any 1≤i≤n-1, an+1=0, and Ei∩Ej=∅ whenever i≠j. Then f*(t)=∑i=1naiχ[mi-1,mi)(t) for any t>0, where m0=0 and mi=∑j=1iμ(Ej) for 1≤i≤n. We claim that
(2.18)sup⁡t>0{φ(t)f*(t)}=max⁡1≤i≤n{aiφ(mi)}.
By monotonicity and continuity of φ we obtain supt>0{φ(t)f*(t)}≤max1≤i≤n{aiφ(mi)}. On the other hand we have supt>0{φ(t)f*(t)}≥supt>0{φ(t)aiχ[mi-1,mi)(t)}=aiφ(mi). Therefore supt>0{φ(t)f*(t)}≥max1≤i≤n{aiφ(mi)}, which implies condition (2.18). Now we will show
(2.19)sup⁡t>0{tφ(df(t))}=max⁡1≤i≤n{aiφ(mi)}.
For t>0 we have tφ(df(t))=t∑i=1nφ(mi)χ[ai+1,ai)(t)≤max1≤i≤n{aiφ(mi)}, and so supt>0{tφ(df(t))}≤max1≤i≤n{aiφ(mi)}. On the other hand, supt>0{tφ(df(t))}≥aiφ(mi) for every 1≤i≤n, and consequently, supt>0{tφ(df(t))}≥max1≤i≤n{aiφ(mi)}, which provides (2.19). Both (2.18) and (2.19) show (2.17) for any nonnegative simple function f.
Now suppose thatf is a measurable function and df(t)<∞ for any t>0. Then, by standard arguments of existing a sequence of nonnegative simple functions (fn) such that dfn↑df and fn*↑f* as n→∞ we can show that
(2.20)sup⁡t>0{φ(t)f*(t)}=lim⁡n→∞sup⁡t>0{φ(t)fn*(t)}=lim⁡n→∞sup⁡t>0{tφ(dfn(t))}=sup⁡t>0{tφ(df(t))}
and conclude the proof.In fact, Proposition2.9 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, df(t)=0 for all t≥1 and df(t)=∞ for any t∈[0,1). Clearly, φ is increasing and continuous and also limt→∞φ(t)=2. Therefore, supt>0{φ(t)f*(t)}=2 and supt>0{tφ(df(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 spacesE 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
(3.1)lim⁡ϵ→0‖(f-f(t))χB(t,ϵ)‖E‖χB(t,ϵ)‖E=0.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 get(3.2)lim⁡ϵ→0‖|fχB(t,ϵ)|‖E‖|χB(t,ϵ)|‖E=|f(t)|.
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 functionME(r)f.Proposition 3.2.
Let0<r<∞ and let E be a r.i. order continuous quasi-Banach function space. If f∈E, then the maximal function ME(r)f is measurable on (0,α).Proof.
Letϵ>0 and t∈(0,α). We first observe that fϵ(r)(t) is continuous on (0,α). In fact, for any tn→t, limn→∞χB(tn,ϵ)=χB(t,ϵ), and by order continuity of E we obtain that limn→∞∥fχB(tn,ϵ)∥Er=∥fχB(t,ϵ)∥Er and limn→∞∥χB(tn,ϵ)∥E=∥χB(t,ϵ)∥E. Now by Fatou’s property of E we have
(3.3)lim⁡δ→ϵ-‖fχB(t,δ)‖E=‖fχB(t,ϵ)‖E,lim⁡δ→ϵ-‖χB(t,δ)‖E=‖χB(t,ϵ)‖E.
Hence we have
(3.4)ME(r)f(t)=sup⁡{fϵ(r)(t):ϵ>0,ϵ∈Q,B(t,ϵ)⊂(0,α)},
and thus ME(r)f is measurable.Remark 3.3.
IfW satisfies Δ2 condition, then we obtain measurability of the maximal function MΛp,w(r)f for any f∈Λp,w, analogously as in case of the maximal function MΓp,w(r)g for any g∈Γp,w when α=∞ and W(∞)=∞.In view of Theorem  1 in [4], we investigate the so-called weak inequality for the maximal function ME(r) whenever E is a r.i. quasi-Banach function space.Theorem 3.4.
Let0<r<∞. If a r.i. quasi-Banach function space E satisfies a lower ϕ-estimate for ∥·∥Er, then there exists C>0 such that for all f∈E and λ>0 we have
(3.5)λϕ(dME(r)f)(λ)≤C‖f‖Er.Proof.
Assume thatλ>0. Denote
(3.6)Ωλ={t∈(0,α):ME(r)f(t)>λ}.
Clearly, by Proposition 3.2 we get that Ωλ is measurable for all λ>0. Letting t∈Ωλ, there exists ϵt>0 such that B(t,ϵt)⊂(0,α) and
(3.7)λ<fϵt(r)(t).
Let c<μ(Ωλ) and denote B=⋃t∈ΩλB(t,ϵt). Since Ωλ⊂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,ϵt):t∈Ωλ} 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(tk,ϵtk):1≤k≤n}⊂D such that μ(K)≤4∑k=1nμ(B(tk,ϵtk)). Therefore, by ϕ(4s)≤4ϕ(s) and by (3.7) we get
(3.8)ϕ(c)≤ϕ(μ(K))≤4ϕ(∑k=1nμ(B(tk,ϵtk)))≤4ϕ(∑k=1nϕ-1(λ-1‖fχB(tk,ϵtk)‖Er)).
Hence, by assumption that E satisfies a lower ϕ-estimate for ∥·∥Er, there is C>0 such that for any 1≤k≤n we get
(3.9)ϕ(c)<4ϕ(∑k=1nϕ-1(λ-1‖fχB(tk,ϵtk)‖Er))≤4Cλ‖∑k=1nfχB(tk,ϵtk)‖Er≤4Cλ‖f‖Er.
Since c<μ(Ωλ) is arbitrary, we obtain ϕ(μ(Ωλ))≤(4C/λ)∥f∥Er, which finishes the proof.In the next theorem we present Lebesgue’s differentiation property in the spaceE.Theorem 3.5.
Letα=∞ 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,
(3.10)lim⁡ϵ→0‖(f-f(t))χB(t,ϵ)‖E‖χB(t,ϵ)‖E=0
for all f∈E and for a.a. t∈(0,∞). If in addition E is normable, then
(3.11)lim⁡ϵ→0‖fχB(t,ϵ)‖E‖χB(t,ϵ)‖E=|f(t)|.Proof.
Observe first that the set of step functions with supports of finite measure is dense inE. 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 operatorL:(0,∞)→ℝ by
(3.12)Lh(t)=limsup⁡ϵ→0{hϵ(1)(t)}=lim⁡⁡sup⁡ϵ→0{‖hχB(t,ϵ)‖E‖χB(t,ϵ)‖E}
for any h∈E and t∈(0,∞). Assume that ϵ>0 and B(t,ϵ)⊂(0,∞) and also f∈E, c∈ℝ. Let g=bχA be a characteristic function of an open interval A. Notice that for a.a. t∈(0,∞) there exist δt>0 such that for all 0<ϵ<δt we have either B(t,ϵ)⊂A or B(t,ϵ)⊂Ac and consequently ((g-c)χB(t,ϵ))*=|g(t)-c|χ(0,2ϵ). Therefore,
(3.13)(g-c)ϵ(1)(t)=‖(g-c)χB(t,ϵ)‖E‖χB(t,ϵ)‖E=‖(g(t)-c)χ(0,2ϵ)‖E‖χ(0,2ϵ)‖E=|g(t)-c|
for a.a. t∈(0,∞) and for any 0<ϵ<δt. Hence
(3.14)L(g-c)(t)=lim⁡⁡sup⁡ϵ→0{(g-c)ϵ(1)(t)}=|g(t)-c|
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
(3.15)L(f-c)(t)≤ξ(L(f-g)(t)+L(g-c)(t))=ξ(L(f-g)(t)+|g(t)-c|)
for a.a. t∈(0,∞). Clearly L(f-g)(t)≤ME(1)(f-g)(t), whence
(3.16)L(f-c)(t)≤ξ(ME(1)(f-g)(t)+|g(t)-c|)
for a.a. t∈(0,∞). Now replacing c by f(t) we get
(3.17)L(f-f(t))(t)≤ξ(ME(1)(f-g)(t)+|f(t)-g(t)|)
for a.a. t∈(0,∞). Define
(3.18)Ωs={t∈(0,∞):L(f-f(t))(t)>s}
for any s>0. By Proposition 3.2, we have that ME(1)(f-g) is μ-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,
(3.19)μ*(Ωs)≤μ(t∈(0,∞):ME(1)(f-g)(t)+|f-g|(t)>sξ)≤dME(1)(f-g)(s2ξ)+df-g(s2ξ).
Now since ϕ satisfies the triangle inequality with constant ξ, we get
(3.20)ϕ(μ*(Ωs))≤ξϕ(dME(1)(f-g)(s2ξ))+ξϕ(df-g(s2ξ))
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
(3.21)ϕ(df-g(s2ξ))≤2ξssup⁡t>0{ϕ(t)(f-g)*(t)}≤2ξs‖f-g‖E.
Furthermore, by Theorem 3.4 there exists C>0 such that
(3.22)ϕ(dME(1)(f-g)(s2ξ))≤2ξCs‖f-g‖E.
Therefore, for any step function g and for all s>0,
(3.23)ϕ(μ*(Ωs))≤2ξ2(C+1)s‖f-g‖E.
Hence we have
(3.24)ϕ(μ*(Ωs))=ϕ(μ(Ωs))=0
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 upperϕ-estimate of Γq,p on (0,∞), for 0<p,q<∞. Clearly in this case w(t)=tp/q-1 satisfies Bp condition. Thus Λp,w=Γp,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.
Letα=∞, 0<p,r<∞ and, 1<q<∞.(i)
Ifp≤q, then Γq,p satisfies a lower ϕ-estimate for the functional ∥·∥Γq,pr, where 1≤r, and an upper ϕ-estimate for ∥·∥Γq,pr, where r≤p/q.(ii)
Ifp≥q, then Γq,p satisfies an upper ϕ-estimate for the functional ∥·∥Γq,pr, where r≤1, and a lower ϕ-estimate for the functional ∥·∥Γq,pr, where r≥p.The immediate consequence of Theorem3.5 is the next result, which describes under what conditions Γp,w has LDP. The second part follows from the previous corollary.Proposition 3.7.
Let0<p<∞ and α=∞. If the Lorentz space Γp,w satisfies a lower ϕ-estimate for ∥·∥, then Γp,w has LDP. Consequently, for 0<p≤q<∞ and 1<q, Lorentz spaces Γq,p on (0,∞) have LDP.We will show next which spaces(E,∥·∥E) do not have LDP with respect to ∥·∥Er. Notice that the ratio ∥f∥Er1/∥g∥Er2 can be reduced easily to the quotient ∥f∥Er/∥g∥E for any f,g∈E, where g≠0. In view of this fact, we state the following theorem.Theorem 3.8.
LetE be a r.i. quasi-Banach function space. (i) Let 1<r<∞. If order continuous E satisfies a lower ϕ-estimate for ∥·∥Er, then for any f∈E and for a.a. t∈(0,α) we have
(3.25)lim⁡ϵ→0‖fχB(t,ϵ)‖Er‖χB(t,ϵ)‖E=0.(ii) Let 0<r<1. Then for any f∈E and for a.a. t∈𝕊(f) we have
(3.26)lim⁡ϵ→0‖fχB(t,ϵ)‖Er‖χB(t,ϵ)‖E=∞.Proof.
LetL:(0,α)→ℝ be the operator on E given by Lf(t)=limsupϵ→0{fϵ(r)(t)}, f∈E.
(i) Letg 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
(3.27)gϵ(r)(t)=‖gχB(t,ϵ)‖Er‖χB(t,ϵ)‖E=|g(t)|r‖χB(t,ϵ)‖Er-1.
Therefore Lg(t)=limsupϵ→0{gϵ(r)(t)}=0 and thus
(3.28)Lf(t)≤ξ(L(f-g)(t)+Lg(t))=ξL(f-g)(t)≤ξME(r)(f-g)(t)
for a.a. t∈(0,α). Denoting Ωs={t∈(0,α):Lf(t)>s}, s>0, by Proposition 3.2, ME(r)(f-g) is μ-measurable. Now by Theorem 3.4 there exists C>0 such that for all s>0,
(3.29)ϕ(μ*(Ωs))≤ϕ(dME(r)(f-g)(sξ))≤ξCs‖f-g‖Er,
which shows that μ(Ωs)=0 and thus Lf(t)=0 a.e.
(ii) Letfn↑|f| a.e., where 0≤fn are step functions. Therefore, for a.a. t∈𝕊(f) there is n∈ℕ such that 0<fn(t)≤|f(t)| and for small enough ϵ>0 we have
(3.30)(fn)ϵ(r)(t)=‖fnχB(t,ϵ)‖Er‖χB(t,ϵ)‖E=|fn(t)|r‖χB(t,ϵ)‖Er-1.
Thus, by the assumption that 0<r<1 we get
(3.31)Lf(t)≥L(fn)(t)=lim⁡⁡inf⁡ϵ→0{(fn)ϵ(r)(t)}=∞
for a.a. t∈𝕊(f), and the proof is finished.The next corollary follows directly from Theorem3.8.Corollary 3.9.
(i) Let0<p<∞, 1<r<∞. If Lorentz space Γp,w satisfies a lower ϕ-estimate for ∥·∥r, then for any f∈Γp,w and for a.a. t∈(0,α) we have
(3.32)lim⁡ϵ→0‖fχB(t,ϵ)‖r‖χB(t,ϵ)‖=0.(ii) Let 0<p<∞, 0<r<1. Then for any f∈Γp,w and for a.a. t∈𝕊(f) we have
(3.33)lim⁡ϵ→0‖fχB(t,ϵ)‖r‖χB(t,ϵ)‖=∞.The next result needed for further applications states conditions, which guarantee thatΓp,w satisfies a lower ϕ-estimate for ∥·∥r, where 0<r<∞. We omit the proof of the following proposition.Proposition 3.10.
Assume that0<p,r<∞, and Lorentz space Γp,w satisfies a lower r-estimate. If ϕ is concave, then Γp,w satisfies a lower ϕ-estimate for ∥·∥r.The following example shows that Theorem3.8 is not an empty statement in case of Γp,w.Example 3.11.
Letα=∞, p=2, and w(s)=ln(s+1)+1 for all s∈[0,∞). Then for any f∈Γ2,w and for a.a t∈(0,∞) we have
(3.34)lim⁡ϵ→0‖fχB(t,ϵ)‖Γ2,w2‖χB(t,ϵ)‖Γ2,w=0.Proof.
Notice that for anyt∈(0,∞),
(3.35)W(t)=(t+1)ln⁡(t+1),W2(t)=t(t+1)ln⁡(t+1)-t2ln⁡(t)+t.
Consequently,
(3.36)ϕ(t)=‖χ(0,t)‖Γ2,w=((t+1)2ln⁡(t+1)-t2ln⁡(t)+t))1/2
for any t∈(0,∞). Since w is increasing, w satisfies RB2 condition. Moreover, we have
(3.37)∫01w(t)t2dt=∫1∞w(t)dt=∞,ddt(W2(t)t)=ln⁡(1+1t)>0
for any t∈(0,∞), which concludes that W2(t)/t is increasing. Hence, by Theorem 3.3 [24] we obtain that Γ2,w satisfies a lower 2-estimate. Now we claim that ϕ is concave. By simple calculations we get the second derivative on (0,∞),
(3.38)ϕ′′(t)=(t-ln⁡(t+1))ln⁡(t)-(t+2)ln⁡(t+1)t2ln⁡(1+1/t)+(2t+1)ln⁡(t+1)+t.
We observe that
(3.39)t2ln⁡(1+1t)+(2t+1)ln⁡(t+1)+t>0,(t-ln⁡(t+1))ln⁡(t)<(t+2)ln⁡(t+1).
Therefore, ϕ′′(t)<0 for all t∈(0,∞), which implies concavity of ϕ. Hence, by Proposition 3.10 we obtain that Γ2,w satisfies a lower ϕ-estimate for ∥·∥Γ2,w2. 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 off to f, that is, another type of LDT.The first result is a corollary of Proposition3.7. In fact, let t∈(0,∞), ϵ>0 be such that B(t,ϵ)⊂(0,∞). Assume that fϵ(t)∈T(p,B(t,ϵ))(f) for f∈Γp,w. By definition of the best constant approximant we get(3.40)|f(t)-fϵ(t)|≤‖(f(t)-f)χB(t,ϵ)‖Γp,w‖χB(t,ϵ)‖Γp,w+‖(fϵ(t)-f)χB(t,ϵ)‖Γp,w‖χB(t,ϵ)‖Γp,w≤2‖(f(t)-f)χB(t,ϵ)‖Γp,w‖χB(t,ϵ)‖Γp,w.
Now applying Proposition 3.7 we get that fϵ(t)→f(t) as ϵ→0. Therefore we get the following theorem.Theorem 3.12.
Letα=∞ and 1≤p<∞. If the space Γp,w satisfies a lower ϕ-estimate for ∥·∥Γp,w, then for every f∈Γp,w we have
(3.41)fϵ(t)⟶f(t)asϵ⟶0
for a.a. t∈(0,∞), where fϵ(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 functionf∈Γp-1,w for 1<p<2.Lemma 3.13.
Let1<p<2, A⊂[0,α), 0<μ(A)<∞, and let f∈Γp-1,w, f≥0. Then for any uχA∈T(p,A)(f) extended best constant approximant of f and for a.a. s∈(0,α) we have
(3.42)|u-f(s)|p-1‖χA‖Γp,wp≤5‖(f-f(s))χA‖Γp-1,wp-1.Proof.
Lets∈(0,α) and |f(s)|<∞. Since (χA)**≤1, by subadditivity of the maximal function and the power function vp-1 for 1<p<2 we get
(3.43)|u-f(s)|p-1‖χA‖Γp,wp=∫0α(χA)**(t)((u-f(s))χA)**(p-1)(t)w(t)dt≤∫0α(χA)**(t)((f-u)χA)**(p-1)(t)w(t)dt+‖(f-f(s))χA‖Γp-1,wp-1.
We will finish the proof under assumption that f(s)≥u. In the other case the proof is similar. Since uχA∈T(p,A)(f)=[f̲(p,A)χA,f¯(p,A)χA], by Corollary 2.11 [7] and by Hardy-Littlewood inequality we obtain
(3.44)∫0α(χA)**(t)((f-u)χA)**(p-1)(t)w(t)dt≤2∫0α1t∫A∩{f≤u}χ(0,t)(ρ((u-f)χA,χA))dμ((f-u)χA)**(p-1)(t)w(t)dt≤2∫0α(χA∩{f≤u})**(t)((f-u)χA)**(p-1)(t)w(t)dt≤2‖(f-f(s))χA‖Γp-1,wp-1+2∫0α(χA∩{f≤u})**(t)((f(s)-u)χA)**(p-1)(t)w(t)dt.
Moreover, by assumption that 1<p<2 we have
(3.45)(χA∩{f≤u})**(t)≤χ(0,μ(A∩{f≤u}))(t)+(μ(A∩{f≤u})t)p-1χ[μ(A∩{f≤u}),α)(t)=(χA∩{f≤u})**(p-1)(t)
for any t∈(0,α). Consequently, by the fact that f(s)≥u we obtain
(3.46)∫0α(χA∩{f≤u})**(t)((f(s)-u)χA)**(p-1)(t)w(t)dt≤∫0α((f(s)-u)χA∩{f≤u})**(p-1)(t)(χA)**(p-1)(t)w(t)dt≤∫0α((f(s)-f)χA∩{f≤u})**(p-1)(t)(χA)**(p-1)(t)w(t)dt≤‖(f-f(s))χA‖Γp-1,wp-1.
Hence, by conditions (3.43) and (3.44) we finish the proof.Corollary 3.14.
Let1<p<∞ and w∈Dp-1. Assume that ϕ and ψ are the fundamental functions of Γp,w and Γp-1,w, respectively. If one of the following conditions(i)
there existsA>0 such that Wp-1(t)≤AWp(t) for all t∈(0,α),(ii)
w∈Bp-1, holds, then ϕp≈ψp-1.Proof.
Notice that for allt∈(0,α),
(3.47)ϕ(t)p=W(t)+tp∫tαs-pw(s)ds≤W(t)+tp∫tαs-p+1t-1w(s)ds=ψ(t)p-1.
Suppose now that there is A>0 such that Wp-1(t)≤AWp(t) for all t∈(0,α). Therefore,
(3.48)ψ(t)p-1=W(t)+Wp-1(t)≤W(t)+AWp(t)≤Aϕ(t)p
for any t∈(0,α), and so ϕp≈ψp-1.
Now assume thatw∈Bp-1. Then there exists B>0 such that for all f∈Γp-1,w we have ∥f∥Γp-1,wp-1≤B∥f∥Λp-1,wp-1. Consequently,
(3.49)ψ(t)p-1=‖χ(0,t)‖Γp-1,wp-1≤B‖χ(0,t)‖Λp-1,wp-1=BW(t)≤Bϕ(t)p
for any t∈(0,α), and the proof is completed.Theorem 3.15.
Letα=∞, 1<p<2, w∈Dp-1 and f∈Γp-1,w, f≥0. Assume that ϕ and ψ are the fundamental functions of Γp,w and Γp-1,w, respectively. If Lorentz space Γp-1,w satisfies a lower ψ-estimate for ∥·∥Γp-1,w and ϕp≈ψp-1, then
(3.50)fϵ(t)⟶f(t)asϵ⟶0
for a.a. t∈(0,∞), where fϵ(t)∈T(p,B(t,ϵ))(f) is an extended best constant approximant of f.Proof.
Letϵ>0 and t∈(0,∞), |f(t)|<∞. By Lemma 3.13 for any fϵ(t)∈T(p,B(t,ϵ))(f) we get
(3.51)|fϵ(t)-f(t)|≤51/(p-1)‖(f-f(t))χB(t,ϵ)‖Γp-1,w‖χB(t,ϵ)‖Γp,wp/(p-1).
Since ϕp≈ψp-1, there is B>0 such that for all δ>0 we have ∥χB(t,δ)∥Γp-1,w≤B∥χB(t,δ)∥Γp,wp/(p-1). Consequently,
(3.52)|fϵ(t)-f(t)|≤51/(p-1)B‖(f-f(t))χB(t,ϵ)‖Γp-1,w‖χB(t,ϵ)‖Γp-1,w
for any t∈(0,∞) and ϵ>0. Hence, by Proposition 3.7 and by assumption that Γp-1,w satisfies a lower ψ-estimate for ∥·∥Γp-1,w we obtain that fϵ(t)→f(t) as ϵ→0 for a.a. t∈(0,∞).Now we present a specific family of Lorentz spacesΓp,w for which Theorems 3.12 and 3.15 are fulfilled.Corollary 3.16.
Letα=∞ and 1<q<∞. If one of the following conditions(i)
1≤p≤q and f∈Γq,p,(ii)
1<p<2 and f∈Γq,p-1, f≥0, is satisfied, then for a.a. t∈(0,∞) we have
(3.53)fϵ(t)⟶f(t)asϵ⟶0,
where fϵ(t)∈T(p,B(t,ϵ))(f) is the best constant approximant of f∈Γp,w and the extended best constant approximant of f∈Γp-1,w.Proof.
Suppose that condition (i) is fulfilled. Immediately, by Theorem3.12 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
(3.54)W(t)=qp-1t(p-1)/q,Wp-1(t)=q(p-1)q-p+1t(p-1)/q
for all t∈(0,∞), which implies that w∈Dp-1∩Bp-1. Consequently, by Definition 2.6 there is T(p,B(t,ϵ)) the extended best constant approximant operator on Γq,p-1. Moreover, by Corollary 3.14 the fundamental functions ϕ and ψ of the spaces Γq,p and Γq,p-1 respectively, satisfy ϕp≈ψp-1. Hence, by Corollary 3.6 we get that Γq,p-1 satisfies a lower ψ-estimate for ∥·∥Γq,p-1. Finally, by Theorem 3.15 we obtain that fϵ(t), an extended best constant approximant of f, converges to f(t) as ϵ→0 for a.a. t∈(0,∞).Observe that theK-functional of the couple (Γq,p,L∞) can be expressed equivalently by ∥f*χ(0,t)∥Γq,p/ϕ(t) for any f∈Γq,p and t∈(0,∞) [25]. The next result on inequalities between maximal function Mq,p(1)f and the K-functional of (Γq,p,L∞) follows immediately as a consequence of Theorems 1 and  2 in [4], Corollary 3.6, and Remark 3.3.Corollary 3.17.
Letα=∞, p∈[1,∞), and q∈(0,∞).(i)
If1<p≤q<∞, then there exists C>0 such that for all f∈Γq,p and t>0 we have(3.55)(MΓq,p(1)f)*(t)≤Cϕ(t)‖f*χ(0,t)‖Γq,p.(ii)
If1<q≤p<∞, then there exists C>0 such that for all f∈Γq,p and t>0 we get(3.56)(MΓq,p(1)f)*(t)≥Cϕ(t)‖f*χ(0,t)‖Γq,p.(iii)
Ifp=1<q<∞, then there is C>0 such that for any f∈Γq,p and t>0 we obtain the inequality in the condition (i).The decreasing rearrangement of the maximal function(MΓq,p(1)f)* and K-functional of (Γq,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 Λq,p, 1≤p<q<∞, as we see in the next example [4].Example 3.18.
Let1<r<∞. There exists a function f∈L0 such that for all t>0,(3.57)‖f*χ(0,t)‖Γr,1=∫0∞(f*χ(0,t))**(s)s(1/r)-1ds=∞,(MΓr,1(1)f)*(t)<∞.Proof.
Sincew(s)=s(1/r)-1 satisfies B1 condition, there is B>0 such that
(3.58)‖hχB(t,ϵ)‖Λr,1≤‖hχB(t,ϵ)‖Γr,1≤B‖hχB(t,ϵ)‖Λr,1
for all h∈Γr,1, t∈(0,∞) and suitable ϵ>0. Hence MΛr,1(1)f≥(1/B)MΓr,1(1)f. Therefore, by Remark 3.3 we conclude that
(3.59)(MΛr,1(1)f)*(t)≥1B(MΓr,1(1)f)*(t)
for all t>0. Now, by Theorem  3 in [4] there exists f∈L0 such that ∥f*χ(0,t)∥Λr,1=∞ and (MΛr,1(1)f)*(t)<∞ for all t>0, which implies that ∥f*χ(0,t)∥Γr,1=∞ and (MΓr,1(1)f)*(t)<∞ for any t>0.Remark 3.19.
Example3.18 can be expanded to the case when 1≤p<q<∞. Indeed if 1≤p<q<∞ and r=q/p then we have Γq,p=Γr,1(p) and ∥·∥Γq,p≈∥|·|p∥Γq,p1/p. As well as (MΓr,1(1)|·|p)1/p≈MΓq,p(1) (·) on {f∈Γq,p(C):Cisconvex,μ(C)<∞}. This can be obtained by simple observation that the weight function w(t)=t1/r-1 satisfies condition Bs for any 1/r<s<∞, and then Γr,1=Λr,1 and Γq,p=Λq,p as sets with equivalent norms.


## 4. Uniqueness of the Extended Best Constant APProximant onΓP,W

In this section we prove uniqueness of the expansion of the best constant approximant for anyf∈Γp-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.
Let1<p<∞ and f∈Γp-1,w. Then f(n)∈Γp,w for all n∈ℕ.Proof.
Sincef∈Γp-1,w, we get that df(λ)<∞ for every λ>0. Let ϵ∈(0,1). Then
(4.1)cn:=μ(s:|f(s)|≥n)≤μ(s:|f(s)|>n-ϵ)=df(n-ϵ)<∞
for any n∈ℕ. Moreover, (f(n))**≤f** and (f(n))**≤n for any n∈ℕ. Therefore,
(4.2)∫0α(f(n))**p(t)w(t)dt≤npW(cn)+∫cnα(f(n))**(p-1)(t)(f(n))**(t)w(t)dt≤npW(cn)+n∫cnαf**(p-1)(t)w(t)dt<∞
for all n∈ℕ, which concludes the proof.We omit the standard proof of the next lemma.Lemma 4.2.
Letf∈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 functionsS(f(n),A)p(u) to the function S(f,A)p(u) as n→∞ for all u∈ℝ and 1<p<∞, whenever f∈Γp-1,w.Theorem 4.3.
Let1<p<∞ and f∈Γp-1,w. Then for all u∈ℝ,
(4.3)lim⁡n→∞S(f(n),A)p(u)=S(f,A)p(u).Proof.
DenoteE=A∩{f≥u}. Notice that there exists Nu∈ℕ such that Nu>|u| and for all n≥Nu we have E=A∩{f(n)≥u}. Since ρ((f-u)χA,χA) is a measure-preserving transformation, by Definition 2.3 we get
(4.4)S(f,A)p(u)=2∫0α1t∫A∩{f≥u}χ(0,t)(ρ((f-u)χA,χA))((f-u)χA)**(p-1)(t)w(t)dt-∫0α(χA)**(t)((f-u)χA)**(p-1)(t)w(t)dt.
We claim that for a.a. s∈E there exists N(u,s)∈ℕ such that for all n≥N(u,s) we have
(4.5)ρ((f(n)-u)χA,χA)(s)=ρ((f-u)χA,χA)(s).
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
(4.6)ρ((f-u)χA,χA)(s)=μ(v∈A:f(v)>f(s))+μ(v∈A:f(v)+f(s)<2u)+μ(v∈A:f(v)=f(s),v≤s)=μ(v∈A:f(n)(v)>f(n)(s))+μ(v∈A:f(n)(v)+f(n)(s)<2u)+μ(v∈A:f(n)(v)=f(n)(s),v≤s)=ρ((f(n)-u)χA,χA)(s).
If f(s)=u, then again by Lemma 4.2 for n≥N(u,s) we have
(4.7)ρ((f-u)χA,χA)(s)=μ(v∈A:f(v)>f(s))+μ(v∈A:f(v)+f(s)<2u)+μ(v∈A:v≤s)-μ(v∈A:f(v)=f(s),v≤s)=μ(v∈A:f(n)(v)>f(n)(s))+μ(v∈A:f(n)(v)+f(n)(s)<2u)+μ(v∈A:v≤s)-μ(v∈A:f(n)(v)=f(n)(s),v≤s)=ρ((f(n)-u)χA,χA)(s).
Hence we obtain our claim. It follows
(4.8)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))
for any t∈(0,α). Moreover limn→∞((f(n)-f)χA)**(t)=0 for all t∈(0,α), and consequently by triangle inequality for maximal function we obtain
(4.9)lim⁡n→∞((f(n)-u)χA)**(t)=((f-u)χA)**(t)
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 vp-1 for p>1 is subadditive with a constant C>0, we have
(4.10)1t∫A∩{f(n)≥u}χ(0,t)(ρ((f(n)-u)χA,χA))((f(n)-u)χA)**(p-1)(t)≤(χA)**(t)((f(n)-u)χA)**(p-1)(t)≤C((fχA)**(p-1)(t)+(uχA)**(p-1)(t))
for every t∈(0,α) and n∈ℕ. Combining now (4.8), (4.9), and (4.10) we get
(4.11)lim⁡n→∞∫0α1t∫A∩{f(n)≥u}χ(0,t)(ρ((f(n)-u)χA,χA))((f(n)-u)χA)**(p-1)(t)w(t)dt=∫0α1t∫A∩{f≥u}χ(0,t)(ρ((f-u)χA,χA))((f-u)χA)**(p-1)(t)w(t)dt,lim⁡n→∞∫0α(χA)**(t)((f(n)-u)χA)**(p-1)(t)w(t)dt=∫0α(χA)**(t)((f-u)χA)**(p-1)(t)w(t)dt.
In view of (4.4) we complete the proof.The following characterization of the functionS(f,A)p(u) is an essential fact in the proof of the main result.Proposition 4.4.
Let1<p<∞, w be a positive weight function and let f∈Γp-1,w. Then the function S(f,A)p(u) is strictly decreasing with respect to u.Proof.
First we will show that for anyu,v∈ℝ, u<v there exist a∈(u,v] and B⊂[0,α), μ(B)>0 such that for all t∈B,
(4.12)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)<K(f,A)(-a,t)((f+a)χA)**(p-1)(t).
Suppose for a contrary that there existu<v such that for all a∈(u,v] and for a.a. t∈[0,α) we have
(4.13)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)≥K(f,A)(-a,t)((f+a)χA)**(p-1)(t).
Since K(f,A)(a,t) and ((f+a)χA)**(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)χA)**p(t) and by Proposition 4.2 [20] we get that (d+/da)((f+a)χA)**p(t)=pK(f,A)(-a,t)((f+a)χA)**(p-1)(t) is increasing with respect to a∈ℝ for any t∈(0,α), which implies that
(4.14)K(f,A)(-a,t)((f+a)χA)**(p-1)(t)≤K(f,A)(-b,t)((f+b)χA)**(p-1)(t)
for any a<b and for all t∈(0,α). Hence
(4.15)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)=K(f,A)(-a,t)((f+a)χA)**(p-1)(t)
for all a∈[u,v] and t∈(0,α). Pick up b∈(u,v]. Denote δ=(b-u)/2. Notice that
(4.16)K(f,A)(-a,t)((f+a)χA)**(p-1)(t)=K(f,A)(-a-δ,t)((f+a+δ)χA)**(p-1)(t)
for all a∈[u,u+δ] and t∈(0,α), which yields that
(4.17)∫uu+δd+da((f+a)χA)**p(t)da=∫uu+δd+da((f+a+δ)χA)**p(t)da=∫u+δbd+da((f+a)χA)**p(t)da
for all t∈(0,α). Consequently,
(4.18)(fχA+u+b2χA)**p(t)=12((f+u)χA)**p(t)+12((f+b)χA)**p(t)
for any t∈(0,α). Moreover, by subadditivity of the maximal function and by convexity of the power function sp for p>1 we get
(4.19)(fχA+u+b2χA)**p(t)≤(12((f+u)χA)**(t)+12((f+b)χA)**(t))p≤12((f+u)χA)**p(t)+12((f+b)χA)**p(t)
for all t∈(0,α). Therefore,
(4.20)((f+u)χA+(f+b)χA)**(t)=((f+u)χA)**(t)+((f+b)χA)**(t),
which implies that
(4.21)∫0t((f+u)χA+(f+b)χA)*(s)-((f+u)χA)*(s)-((f+b)χA)*(s)ds=0
for any t∈(0,α). Hence
(4.22)((f+u)χA+(f+b)χA)*(s)=((f+u)χA)*(s)+((f+b)χA)*(s)
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)χA and (f+b)χA for all b∈[u,v] are of constant sign almost everywhere, that is,
(4.23)sign⁡(f+u)(s)=sign⁡(f+b)(s)
for a.a. s∈A and have a common system of sets {Et:t∈(0,α)} such that μ(Et)=t and
(4.24)∫0t((f+b)χA)*(s)ds=∫Et∩A|f+b|(s)ds
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,
(4.25)((f+u)χA)**(t)<((f+b)χA)**(t),
or for all t∈B,
(4.26)((f+u)χA)**(t)>((f+b)χA)**(t).
Suppose that the claim does not hold. Then for any b∈(u,v] and t∈(0,α) we get
(4.27)∫0t((f+u)χA)*(s)ds=∫0t((f+b)χA)*(s)ds.
Thus, by the condition (4.24) we obtain
(4.28)∫Et∩A|f+u|(s)-|f+b|(s)ds=0
for every b∈(u,v] and t∈(0,α). Since Er⊂Et for any r<t and μ(Et)=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 existb∈(u,v] and B⊂(0,α), μ(B)>0 such that
(4.29)((f+u)χA)**(t)<((f+b)χA)**(t)
for all t∈B. Thus and by conditions (4.23) and (4.24) we obtain
(4.30)0<∫0t((f+b)χA)*(s)ds-∫0t((f+u)χA)*(s)ds=∫Et∩A|f(s)+b|-|f(s)+u|ds=(b-u)∫Et∩Asign⁡(f(s)+u)ds
for all t∈B. Since b-u>0, we have
(4.31)0<∫Et∩Asign⁡(f(s)+u)ds
for any t∈B. The function a→((f+a)χA)**(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
(4.32)((f+u+ξ)χA)**(t)-((f+u+η)χA)**(t)=1t∫Et∩Asign⁡(f(s)+u)(f(s)+u+ξ)ds-1t∫Et∩Asign⁡(f(s)+u)(f(s)+u+η)ds=ξ-ηt∫Et∩Asign⁡(f(s)+u)ds<0
for any t∈B. Let a∈(u,v) and δa=min{v-a,a-u}/2. Then, by condition (4.32) we get
(4.33)((f+u)χA)**(t)<((f+u+ϵ)χA)**(t)<((f+a)χA)**(t)<((f+a+ϵ)χA)**(t)
for every 0<ϵ≤δa 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
(4.34)((f+u+ϵ)χA)**(t)-((f+u)χA)**(t)=((f+a+ϵ)χA)**(t)-((f+a)χA)**(t)>0
for all 0<ϵ≤δa and t∈B. Since the functions ((f+u+δa)χA)**(t) and ((f+a)χA)**(t) are continuous and decreasing with respect to t, there exists a compact interval [c,d]⊂B such that
(4.35)((f+u+δa)χA)**(t)<((f+a)χA)**(t)
for all t∈[c,d]. Define
(4.36)F(t)=((f+a)χA)**(t)-((f+u+δa)χA)**(t)
for any t∈[c,d]. Since F is positive and continuous on [c,d], we get mint∈[c,d]{F(t)}>0. Consequently, there is γa∈(0,1) such that
(4.37)((f+u+δa)χA)**(t)<(1-γa)((f+a)χA)**(t)
for all t∈[c,d]. Hence, by condition (4.33) we get
(4.38)((f+u)χA)**(t)<((f+u+ϵ)χA)**(t)<(1-γa)((f+a)χA)**(t)<(1-γa)((f+a+ϵ)χA)**(t)
for every t∈[c,d] and 0<ϵ≤δa. Therefore, by (4.34) we obtain
(4.39)((f+u+ϵ)χA)**p(t)-((f+u)χA)**p(t)ϵ≤(1-γa)p-1((f+a+ϵ)χA)**p(t)-((f+a)χA)**p(t)ϵ
for any t∈[c,d] and 0<ϵ≤δa. Now according to Proposition 4.2 [20] we conclude
(4.40)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)<K(f,A)(-a,t)((f+a)χA)**(p-1)(t)
for all t∈[c,d], which contradicts condition (4.15) and finishes the first case.Case 2.
Suppose that there existb∈(u,v] and B⊂(0,α) with μ(B)>0 such that
(4.41)((f+u)χA)**(t)>((f+b)χA)**(t)
for any t∈B. Analogously as in the previous case we obtain that
(4.42)∫Et∩Asign⁡(f(s)+u)ds<0
for all t∈B. We claim that the function a→((f+a)χA)**(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
(4.43)((f+u+ξ)χA)**(t)-((f+u+η)χA)**(t)=1t∫Et∩A|f(s)+u+ξ|-|f(s)+u+η|ds=ξ-ηt∫Et∩Asign⁡(f(s)+u)ds>0
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
(4.44)((f+u)χA)**(t)>((f+u+ξ)χA)**(t)>((f+a)χA)**(t)>((f+a+ϵ)χA)**(t).
Repeating calculation in condition (4.43) with a instead of u, in view of (4.23) and (4.24) we also get
(4.45)((f+u)χA)**(t)-((f+u+ϵ)χA)**(t)=((f+a)χA)**(t)-((f+a+ϵ)χA)**(t)>0
for all t∈B. Similarly as in the previous case there exists γa>0 such that
(4.46)((f+u+ϵ)χA)**p(t)-((f+u)χA)**p(t)ϵ≤(1+γa)p-1((f+a+ϵ)χA)**p(t)-((f+a)χA)**p(t)ϵ
for any t∈B and 0<ϵ<min{a-u,v-a}/2. Consequently, by Proposition 4.2 [20] we get
(4.47)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)<K(f,A)(-a,t)((f+a)χA)**(p-1)(t)
for all t∈B, which gives us a contradiction with (4.15) and completes the proof of inequality (4.12).
Let nowa,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
(4.48)K(f,A)(-u,t)((f+u)χA)**(p-1)(t)<K(f,A)(-c,t)((f+c)χA)**(p-1)(t)
for all t∈B. Hence, by (4.14) and by Proposition  2.4 in [7] we get
(4.49)S(f,A)p(b)=S(f,A)p(-u)=∫B∪BcK(f,A)(-u,t)((f+u)χA)**(p-1)(t)w(t)dt<∫BK(f,A)(-c,t)((f+c)χA)**(p-1)(t)w(t)dt+∫BcK(f,A)(-u,t)((f+u)χA)**(p-1)(t)w(t)dt≤S(f,A)p(-c)≤S(f,A)p(-v)=S(f,A)p(a),
and the proof is done.Now we establish the main theorem of this section.Theorem 4.5.
Let1<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∈Γp-1,w,
(4.50)T(p,A)(f)=f̲(p,A)χA=f¯(p,A)χA.Proof.
Suppose thatf̲(p,A)<f¯(p,A). Then there exist u,v∈(f̲(p,A),f¯(p,A)) such that u<v. By Theorem 2.9 [7] and by Proposition 4.4 we obtain
(4.51)0≤S(f,A)(f¯(p,A))<S(f,A)(v)<S(f,A)(u),0≤S(-f,A)(-f̲(p,A))<S(-f,A)(-u)<S(f,A)(-v).
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
(4.52)0<S(f(n),A)(v),0<S(-f(n),A)(-v)
as well as
(4.53)0<S(-f(n),A)(-u),0<S(f(n),A)(u).
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.



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*Source: 682960-2012-03-20.xml*