Lebesgue ’ s Differentiation Theorems in R . I . Quasi-Banach Spaces and Lorentz Spaces Γ p , w

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,w {f : ∫ f∗∗ w < ∞} 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 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 <∞.


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, ⊂ R n to this function f, that is, for a.a. 1.1 for the Hardy-Littlewood maximal function M H f where t ∈ R n and f ∈ L 1 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 r E 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 r E 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 to f as → 0 whenever f ∈ Γ p,w and 1 ≤ p < ∞, as well as the convergence of the extended best constant approximants f 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 r Γ 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 L 0 , or from L 1 to L 0 , 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.

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 L 0 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 ∈ L 0 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 ∈ L 0 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 It is well known that f * ≤ f * * and f * * is decreasing and subadditive, that is, f g * * ≤ f * * g * * for any f, g ∈ L 0 .For the properties of d f , f * , and f * * see 2, 8 .A subspace E ⊂ L 0 equipped with a quasinorm • E is called a quasinormed function space, if the following conditions are satisfied.
1 If f ∈ L 0 , 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 ∈ L 0 and g ∈ E with 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 ∈ L 0 be a nonnegative weight function.Lorentz space Γ p,w is a subspace of L 0 such that Given a measurable set A ⊂ 0, α by Γ p,w A we denote the set of f ∈ L 0 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: 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 that 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 L 0 such that In case when W satisfies the Δ 2 -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 .
In 1970, Ryff proved in 16 that τ f| 0,1 ,0 : 0, 1 → 0, 1 is a measure-preserving transformation for any f ∈ L 0 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 ∈ L 0 with d f λ < ∞ for all λ > 0 and any h ∈ L 0 .Notice that for any f ∈ L 0 with d f λ < ∞ for any λ > 0 and h ∈ L 0 , 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 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 2 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 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 .
Corollary 2.5 see 20, Corollary 7.3 .For any 1 < p < ∞ and w positive, we have that Recently, in 7 it has been developed an existence of extension of the best constant approximant operators from Lorentz space Γ 1,w to L 0 , if w ∈ D 1 , 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 L 0 , if p 1, and on Then the extended best constant approximant operator is given by 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 • r E , if there exists C > 0 such that for all n ∈ N and f i n i 1 ⊂ E with pairwise disjoint supports we have

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 for any t ∈ 0, α , where B t, t − , t .
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.

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 n i 1 a i χ E i is a nonnegative simple function, where a i > a i 1 for any 1 ≤ i ≤ n − 1, a n 1 0, and 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.

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 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 In view of Theorem 1 in 4 , we investigate the so-called weak inequality for the maximal function M r E 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 • r E , then there exists C > 0 such that for all f ∈ E and λ > 0 we have Proof.Assume that λ > 0. Denote

3.8
Hence, by assumption that E satisfies a lower φ-estimate for • r E , there is C > 0 such that for any 1 ≤ k ≤ n we get 3.9 Since c < μ Ω λ is arbitrary, we obtain φ μ Ω λ ≤ 4C/λ f r E , which finishes the proof.
In the next theorem we present Lebesgue's differentiation property in the space E.
for a.a.t ∈ 0, ∞ .Now replacing c by f t we get for any s > 0. By Proposition 3.2, we have that Thus in view of 3.17 we obtain for s > 0,

3.19
Now since φ satisfies the triangle inequality with constant ξ, we get for every s > 0. Observe that for any h ∈ E, t ∈ 0, α , we have Thus, by Proposition 2.9 we have

3.21
Furthermore, by Theorem 3.4 there exists C > 0 such that

3.22
Therefore, for any step function g and for all s > 0,

3.23
Hence we have 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 • Λ p,w ≈ • Γ p,w .Hence by Theorems 3 and 7 in 9 and by H ölder's inequality we obtain the following corollary.
i If p ≤ q, then Γ q,p satisfies a lower φ-estimate for the functional • r Γ q,p , where 1 ≤ r, and an upper φ-estimate for • r Γ q,p , where r ≤ p/q.
ii If p ≥ q, then Γ q,p satisfies an upper φ-estimate for the functional • r Γ q,p , where r ≤ 1, and a lower φ-estimate for the functional • r Γ 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 • r E .Notice that the ratio f r 1 E / g r 2 E can be reduced easily to the quotient f r E / g E for any f, g ∈ E, where g / 0. In view of this fact, we state the following theorem.

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 .
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, ∞ , > 0 be such that B t, ⊂ 0, ∞ .Assume that f t ∈ T t, f for f ∈ Γ p,w .By definition of the best constant approximant we get

3.40
Now applying Proposition 3.7 we get that f t → f t as → 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 • Γ p,w , then for every f ∈ Γ p,w we have for a.a.t ∈ 0, ∞ , where f t ∈ T p,B t, 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 , by subadditivity of the maximal function and the power function

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

3.44
Moreover, by assumption that 1 < p < 2 we have

3.45
for any t ∈ 0, α .Consequently, by the fact that f s ≥ u we obtain

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 Proof.Notice that for all t ∈ 0, α ,

3.47
Suppose now that there is A > 0 such that W p−1 t ≤ AW p t for all t ∈ 0, α .Therefore, 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 for any t ∈ 0, α , and the proof is completed.
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 • Γ p−1,w and φ p ≈ ψ p−1 , then

3.51
Since φ p ≈ ψ p−1 , there is B > 0 such that for all δ > 0 we have for any t ∈ 0, ∞ and > 0. Hence, by Proposition 3.7 and by assumption that Γ p−1,w satisfies a lower ψ-estimate for • Γ p−1,w we obtain that f t → f t as → 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 where f t ∈ T p,B t, 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 Consequently, by Definition 2.6 there is T p,B t, 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 • Γ q,p−1 .Finally, by Theorem 3.15 we obtain that f t , an extended best constant approximant of f, converges to f t as → 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 Γ q,p /φ t for any f ∈ Γ q,p and t ∈ 0, ∞ 25 .The next result on inequalities between maximal function M 1 q,p f 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

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

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 1 Γ q,p f * 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 .

3.57
Proof.Since w s s 1/r −1 satisfies B 1 condition, there is B > 0 such that . Therefore, by Remark 3.3 we conclude that for all t > 0. Now, by Theorem 3 in 4 there exists f ∈ L 0 such that f * χ 0,t Λ r,1 ∞ and 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 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.

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 ∈ L 0 be defined as Proof.Since f ∈ Γ p−1,w , we get that d f λ < ∞ for every λ > 0. Let ∈ 0, 1 .Then for any n ∈ N.Moreover, f n * * ≤ f * * and f n * * ≤ n for any n ∈ N. Therefore, for all n ∈ N, which concludes the proof.
We omit the standard proof of the next lemma.

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 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
If f s u, then again by Lemma 4.2 for n ≥ N u,s we have

4.7
Hence we obtain our claim.It follows 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 for all t ∈ 0, α .Since ρ f n −u χ A ,χ A 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

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.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, 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 for all t ∈ 0, α .Therefore,

Lemma 4 . 1 .
f n s f s if |f s | < n, and f n s sign f s n if |f s | ≥ n.Let 1 < p < ∞ and f ∈ Γ p−1,w .Then f n ∈ Γ p,w for all n ∈ N.

Theorem 4 . 3 .
Now we discuss a convergence of a sequence of functions S p f n ,A u to the functionS p f,A u as n → ∞ for all u ∈ R and 1 < p < ∞, whenever f ∈ Γ p−1,w .Let 1 < p < ∞ and f ∈ Γ p−1,w .Then for all u ∈ R, 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 χ A ,χ A is a measure-preserving transformation, by Definition 2.3 we get

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.

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 thatK 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 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 B c K f,A −u, t f u χ A * * p−1 t w t dt 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 .8A nonempty subset Y of X is a set of uniqueness if for any element x ∈ X there is no more than one element x ∈ Y satisfying 2.8 .The set Y is a set of existence if for every element x ∈ X there is at least one element x ∈ 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 x ∈ 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, 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 for all > 0 and t ∈ 0, α .The maximal function M r E By monotonicity and continuity of ϕ we obtain sup t>0 {ϕ t f * t } ≤ max 1≤i≤n {a i ϕ m i }.On the other hand we have sup t>0 {ϕ t f * t } ≥ sup t>0 {ϕ t a i χ m i−1 ,m i t } a i ϕ m i .Therefore sup t>0 {ϕ t f * t } ≥ max 1≤i≤n {a i ϕ m i }, which implies condition 2.18 .Now we will show ,a i t ≤ max 1≤i≤n {a i ϕ m i }, and so sup t>0 {tϕ d f t } ≤ max 1≤i≤n {a i ϕ m i }.On the other hand, sup t>0 {tϕ d f t } ≥ a i ϕ m i for every 1 ≤ i ≤ n, and consequently, sup t>0 {tϕ d f t } ≥ max 1≤i≤n {a i ϕ 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 f n ↑ d f and f * n ↑ f * as n → ∞ we can show that | • | E to • E such that for any f ∈ E and for a.a.t ∈ 0, α we get • 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 r E 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 r E f is measurable on 0, α .Proof.Let > 0 and t ∈ 0, α .We first observe that f r t is continuous on 0, α .In fact, for any t n → t, lim n → ∞ χ B t n , χ B t, , and by order continuity of E we obtain that r E f is measurable.Remark 3.3.If W satisfies Δ 2 condition, then we obtain measurability of the maximal function M r Λ p,w f for any f ∈ Λ p,w , analogously as in case of the maximal function M r Γ p,w g for any g ∈ Γ p,w when α ∞ and W ∞ ∞.
Let c < μ Ω λ and denote B t∈Ω λ B t, 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, 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 , t k : 1 ≤ k ≤ n} ⊂ D such that μ K ≤ 4 n k 1 μ B t k , t k .Therefore, by φ 4s ≤ 4φ s and by 3.7 we get 6Clearly, by Proposition 3.2 we get that Ω λ is measurable for all λ > 0. Letting t ∈ Ω λ , there exists t > 0 such that B t, t ⊂ 0, α and 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, ∈ 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 for all f ∈ E and for a.a.t ∈ 0, ∞ .If in addition E is normable, then 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 for any h ∈ E and t ∈ 0, ∞ .Assume that > 0 and B t, ⊂ 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 < < δ t we have either B t, ⊂ A or B t, ⊂ A c and consequently g − c χ B t, * |g t − c|χ 0,2 .Therefore, Theorem 3.8.Let E be a r.i.quasi-Banach function space.iLet 1 < r < ∞.If order continuous E satisfies a lower φ-estimate for • r Proof.Let L : 0, α → R be the operator on E given by Lf t lim sup → 0 {f 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 > 0 we have Now by Theorem 3.4 there exists C > 0 such that for all s > 0, ↑ |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 > 0 we have E , then for any f ∈ E and for a.a.t ∈ 0, α we have a.t ∈ 0, α .Denoting Ω s {t ∈ 0, α : Lf t > s}, s > 0, by Proposition 3.2, M r E f − g is μ-measurable.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 Let f ∈ L 0 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.
By conditions 4.23 and 4.24 we conclude that < ≤ 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 < ≤ 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 ∈ 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 ∈ c, d and 0 < ≤ δ a .Therefore, by 4.34 we obtain ∈ c, d and 0 < ≤ δ a .Now according to Proposition 4.2 20 we conclude ∈ B. Analogously as in the previous case we obtain that ∈ 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 ∈ B. Therefore, for any a ∈ u, v , 0 < < min{v − a, a − u}/2 and for all t ∈ B we have Repeating calculation in condition 4.43 with a instead of u, in view of 4.23 and 4.24 we also get Similarly as in the previous case there exists γ a > 0 such that ∈ B and 0 < < 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.
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.for any t ∈ B. Let a ∈ u, v and δ a min{v − a, a − u}/2.Then, by condition 4.32 we getK f,A −u, t f u χ A * * p−1 t < K f,A −a, t f a χ A * * p−1 t 4.40for 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