Journal of Function Spaces and Applications

VolumeΒ 2012Β (2012), Article IDΒ 682960, 28 pages

http://dx.doi.org/10.1155/2012/682960

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

^{1}Institute of Mathematics, PoznaΕ University of Technology, Piotrowo 3a, 60-965 PoznaΕ, Poland^{2}Department 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 KamiΕska. 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.

#### Abstract

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces and in particular on Lorentz spaces for any and a nonnegative locally integrable weight function , where is a maximal function of the decreasing rearrangement for any measurable function on , with . The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of , where 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 of or , , respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function , .

#### 1. Introduction

The present paper is devoted to investigation of maximal inequalities and Lebesgueβs type differentiation theorems for best local approximations in r.i. quasi-Banach spaces and Lorentz spaces for . 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 on the ball to this function , that is, for a.a. ,
In fact, Lebesgueβs integral average coincides with a best constant approximant on the space [1]. The *Lebesgue Differentiation Theorem* (LDT) can be proved as a consequence of the *weak maximal inequality *
for the Hardy-Littlewood maximal function where and [2]. The interesting exploration of LDT was initiated by Stein in [3], who introduced the maximal functions on , associated with integral average, and applied it to obtain differentiation theorem in the notation of the norm in for . 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 for . 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 to the spaces 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 Lebesgueβs 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 to the spaces . The present paper is a continuation of the previous results and devoted to investigation of maximal inequalities and Lebesgueβs type differentiation theorems for local approximation in r.i. quasi-Banach space and in particular in .

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 for , that corresponds to the quasi-norm average of , in r.i. quasi-Banach function spaces . Next we establish two types of generalization of LDT in r.i. quasi-Banach function spaces and in . In both types of LDT we employ the assumption of upper and lower -estimates of . 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 , which corresponds to a quasi-norm average of . In the same spirit we also provide some conditions when the LDT does not hold in or in . 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 satisfies a lower (resp., an upper) -estimate, where is the fundamental function of . In view of this characterization we investigate pointwise convergence of the best constant approximants to as whenever and , as well as the convergence of the extended best constant approximants for any and . We also present examples showing that this assumption is fulfilled by a large class of the spaces . Finally, we investigate relations between maximal functions and the -functional of Banach couple in the spirit of the inequalities stated in [4]. We finish Section 3 with an example showing that is not equivalent to the -functional of the pair .

It is well known that the extension of the best constant approximant operator from to , or from to , 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 and . To show the uniqueness we need to consider strict monotonicity of the right-hand GΓ’teaux derivative of the norm in at in the direction for any and .

#### 2. Preliminaries

Let and be the set of real and natural numbers, respectively. For any denote . Let and be the Lebesgue measure on . We denote by the space of all extended real-valued -measurable and finite functions a.e. on . Denote the outer measure on by , the support of by , and the restriction of to the set by . By a simple (resp., step) function we mean a measurable function 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 * of a function is given by for all . Two functions are called *equimeasurable*, if for all . We define the *decreasing rearrangement* for any by , . For given we denote the *maximal function* of by . It is well known that and is decreasing and subadditive, that is, for any . For the properties of , , and see [2, 8]. A subspace equipped with a quasinorm is called a *quasinormed function space*, if the following conditions are satisfied.(1)If , , and a.e., then and .(2)There exists a strictly positive .

If is complete, then it is said to be a *quasi-Banach function space*. We say that a quasi-Banach function space is *rearrangement invariant* (r.i. for short), if whenever and with , then and (see [2]). Throughout the paper we use the notation , which means that the expressions and are equivalent; that is, is bounded from both sides. Let and be a nonnegative weight function. Lorentz space is a subspace of such that
Given a measurable set by we denote the set of restricted to and satisfying the above inequality. Unless we say otherwise, throughout the paper we assume that belongs to the class (in short ), whenever it satisfies the following conditions:
for all if and for all otherwise. These two conditions guarantee that . We also assume that
Under these assumptions 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 and , , the space will be denoted by .

Unless we say otherwise, throughout this paper we assume that is the fundamental function of defined as , , and . It is easy to show that the fundamental function is strictly increasing and continuous on , and for . For more details about the properties of see [9].

Recall that for given , classical Lorentz space is a subspace of such that In case when satisfies the -condition, that is for all and some , as well as , the space is a separable r.i. order continuous quasi-Banach function space [9]. The space is a r.i. Banach function space, whenever the weight is decreasing and [10]. Since , we have the natural inclusion . Moreover, if and only if satisfies condition , ( for short) which means that there is such that for all we have [11β13].

Let and be -finite measure spaces. A map from into is said to be a *measure-preserving transformation*, if whenever is a -measurable subset of , the set is a -measurable subset of and . For given subsets such that , there exists a measure-preserving transformation [14, Theorem 17, page 410].

*Definition 2.1 (see [15]). * Let . Denote
for all .

In 1970, Ryff proved in [16] that is a measure-preserving transformation for any and a.e. on . In 1993, Carothers et al. established in [15] that is a measure preserving transformation from onto such that a.e. on for any with for all and any . Notice that for any with for any and , if for every we have that and it is the unique measure-preserving transformation up to measure zero satisfying a.e. on .

*Definition 2.2. *Let and let , be measure-preserving transformations given by Definition 2.1. Denote

*Definition 2.3. *Let and with . Denote
for any and .

Let () be a real normed space. Denote by (resp., ) the *closed unit ball* (resp., the *unit sphere*) of . Assume that is a subset of and is an element of . An element is called *best approximant* to from if
A nonempty subset of is a *set of uniqueness* if for any element there is no more than one element satisfying (2.8). The set is a *set of existence* if for every element there is at least one element for which condition (2.8) holds. The set is a *Chebyshev set* if for every element there exists exactly one element satisfying (2.8), that is, if is both a set of uniqueness and a set of existence (for more details see [17]). Let be a collection of all subsets of . A set value map is said to be *best approximant operator*, if it assumes for any a set of all best approximant elements to from , that is,
In case when is a norm function space and is a family of constant functions, then is called *best constant approximant operator,* and each element is called *best constant approximant* to from . Let with and . It is well known that the set is convex, compact, and a set of existence for all [18, 19]. Letβs recall some characterizations of best constant approximants over Lorentz spaces .

Theorem 2.4 (see [20, Theorem 7.5]). *Let and let . Then is the best constant approximant of if and only if
*

Corollary 2.5 (see [20, Corollary 7.3]). *For any and positive, we have that is Chebyshev set for any ; that is, there is an unique best constant approximant to .*

Recently, in [7] it has been developed an existence of extension of the best constant approximant operators from Lorentz space to , if , and from to , if and . Now we recall definition of the extended operator on , if , and on , if .

*Definition 2.6 (see [7]). *Let with and let if , and if . Assume that if , and if . Denote
Then the extended best constant approximant operator is given by

In fact, any is called an *extended best constant approximant* of . Notice that in view of Theorem 2.4, if for , then any is a classical best constant approximant of .

*Definition 2.7 (see [4]). *Let , be a r.i. quasi-Banach function space, and let be an increasing bijection. is said to satisfy an upper (resp., a lower) -estimate for , if there exists such that for all and with pairwise disjoint supports we have
respectively

In the case when and , this definition covers the notions of the upper (resp., lower) -estimate [21].

Let and be a quasi-Banach function space. We denote by the -*convexification* of equipped with the quasinorm . 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 Lebesgueβs Differentiation Theorem in .

*Definition 2.8. *Let , be a r.i. quasi-Banach function space. For any we denote
for all and . The maximal function is given by
for any , where .

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 on . The proof of the proposition is quite standard and is provided for the sake of completeness.

Proposition 2.9. *Let , , and let be an increasing continuous function. If or for all , then
*

*Proof. *Notice first if and there exists such that , then , and the conclusion follows.

Now assume that is a nonnegative simple function, where for any , , and whenever . Then for any , where and for . We claim that
By monotonicity and continuity of we obtain . On the other hand we have . Therefore , which implies condition (2.18). Now we will show
For we have , and so . On the other hand, for every , and consequently, , which provides (2.19). Both (2.18) and (2.19) show (2.17) for any nonnegative simple function .

Now suppose that is a measurable function and for any . Then, by standard arguments of existing a sequence of nonnegative simple functions such that and as we can show that
and conclude the proof.

In fact, Proposition 2.9 describes the largest family of increasing and continuous functions , for which (2.17) is satisfied. Indeed, let , , and for any . Then , for all and for any . Clearly, is increasing and continuous and also . Therefore, and , which implies that condition (2.17) does not hold.

#### 3. Lebesgueβs Differentiation Theorems

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

*Definition 3.1. *Let be a quasi-Banach function space on . We say that has the Lebesgue differentiation property (LDP), whenever for any and for a.a. we have

Observe that letting be a quasi-Banach function space on with LDP, by the Aoki-Rolewicz theorem [23] there exist and an equivalent -norm to such that for any and for a.a. we get If is a normed space, then the quasinorm can be replaced by .

In the next proposition we establish measurability of the maximal function .

Proposition 3.2. *Let and let be a r.i. order continuous quasi-Banach function space. If , then the maximal function is measurable on .*

*Proof. *Let and . We first observe that is continuous on . In fact, for any , , and by order continuity of we obtain that and . Now by Fatouβs property of we have
Hence we have
and thus is measurable.

*Remark 3.3. *If satisfies condition, then we obtain measurability of the maximal function for any , analogously as in case of the maximal function for any when and .

In view of Theoremββ1 in [4], we investigate the so-called weak inequality for the maximal function whenever is a r.i. quasi-Banach function space.

Theorem 3.4. *Let . If a r.i. quasi-Banach function space satisfies a lower -estimate for , then there exists such that for all and we have
*

*Proof. *Assume that . Denote
Clearly, by Proposition 3.2 we get that is measurable for all . Letting , there exists such that and
Let and denote . Since , we get . Hence, by regularity of the Lebesgue measure there is a compact set such that . By the fact that a collection is an open covering of the set and by the Vitali covering lemma [2, Lemma 3.2], there exists a pairwise disjoint finite collection such that . Therefore, by and by (3.7) we get
Hence, by assumption that satisfies a lower -estimate for , there is such that for any we get
Since is arbitrary, we obtain , which finishes the proof.

In the next theorem we present Lebesgueβs differentiation property in the space .

Theorem 3.5. *Let and let be a r.i. order continuous quasi-Banach function space . If satisfies a lower -estimate for , then has LDP, that is,
**
for all and for a.a. . If in addition is normable, then
*

*Proof. *Observe first that the set of step functions with supports of finite measure is dense in . The proof of this observation is standard, by density of the simple functions, which is equivalent to order continuity of and regularity of the Lebesgue measure on (cf. [8]).

Define an operator by
for any and . Assume that and and also , . Let be a characteristic function of an open interval . Notice that for a.a. there exist such that for all we have either or and consequently . Therefore,
for a.a. and for any . Hence
for a.a. . Observe that the above equation can be proved analogously for any step function with support of finite measure. Let be a constant in the triangle inequality of the quasinorm . Thus
for a.a. . Clearly , whence
for a.a. . Now replacing by we get
for a.a. . Define
for any . By Proposition 3.2, we have that is -measurable. Recall [2, 8] that for any . Thus in view of (3.17) we obtain for ,
Now since satisfies the triangle inequality with constant , we get
for every . Observe that for any , , we have . Thus, by Proposition 2.9 we have
Furthermore, by Theorem 3.4 there exists such that
Therefore, for any step function and for all ,
Hence we have
for all . So for a.a. , which shows the first formula.

The second formula results from the first one since is a norm in .

Now we characterize the lower and upper -estimate of on , for . Clearly in this case satisfies condition. Thus and . Hence by Theorems 3 andββ7 in [9] and by HΓΆlderβs inequality we obtain the following corollary.

Corollary 3.6. *Let , and, .*(i)*If , then satisfies a lower -estimate for the functional , where , and an upper -estimate for , where .*(ii)*If , then satisfies an upper -estimate for the functional , where , and a lower -estimate for the functional , where .*

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

Proposition 3.7. *Let and . If the Lorentz space satisfies a lower -estimate for , then has LDP. Consequently, for and , Lorentz spaces on have LDP.*

We will show next which spaces do not have LDP with respect to . Notice that the ratio can be reduced easily to the quotient for any , where . In view of this fact, we state the following theorem.

Theorem 3.8. *Let be a r.i. quasi-Banach function space. (i) Let . If order continuous satisfies a lower -estimate for , then for any and for a.a. we have
**(ii) Let . Then for any and for a.a. we have
*

*Proof. *Let be the operator on given by , .

(i) Let be a step function with a finite measure support. Notice that for a.a. and for small enough we have
Therefore and thus
for a.a. . Denoting , , by Proposition 3.2, is -measurable. Now by Theorem 3.4 there exists such that for all ,
which shows that and thus a.e.

(ii) Let a.e., where are step functions. Therefore, for a.a. there is such that and for small enough we have
Thus, by the assumption that we get
for a.a. , and the proof is finished.

The next corollary follows directly from Theorem 3.8.

Corollary 3.9. *
(i) Let , . If Lorentz space satisfies a lower -estimate for , then for any and for a.a. we have
**(ii) Let , . Then for any and for a.a. we have
*

The next result needed for further applications states conditions, which guarantee that satisfies a lower -estimate for , where . We omit the proof of the following proposition.

Proposition 3.10. *Assume that , and Lorentz space satisfies a lower -estimate. If is concave, then satisfies a lower -estimate for .*

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

*Example 3.11. *Let , , and for all . Then for any and for a.a we have

*Proof. *Notice that for any ,
Consequently,
for any . Since is increasing, satisfies condition. Moreover, we have
for any , which concludes that is increasing. Hence, by Theorem 3.3 [24] we obtain that satisfies a lower 2-estimate. Now we claim that is concave. By simple calculations we get the second derivative on ,
We observe that
Therefore, for all , which implies concavity of . Hence, by Proposition 3.10 we obtain that satisfies a lower -estimate for . 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 to , that is, another type of LDT.

The first result is a corollary of Proposition 3.7. In fact, let , be such that . Assume that for . By definition of the best constant approximant we get Now applying Proposition 3.7 we get that as . Therefore we get the following theorem.

Theorem 3.12. *Let and . If the space satisfies a lower -estimate for , then for every we have
**
for a.a. , where is a best constant approximant of .*

In order to prove the next approximation theorem let us first establish an inequality related to extended best constant approximants of any function for .

Lemma 3.13. *Let , , , and let , . Then for any extended best constant approximant of and for a.a. we have
*

*Proof. *Let and . Since , by subadditivity of the maximal function and the power function for we get
We will finish the proof under assumption that . In the other case the proof is similar. Since , by Corollary 2.11 [7] and by Hardy-Littlewood inequality we obtain
Moreover, by assumption that we have
for any . Consequently, by the fact that we obtain
Hence, by conditions (3.43) and (3.44) we finish the proof.

Corollary 3.14. *Let and . Assume that and are the fundamental functions of and , respectively. If one of the following conditions*(i)*there exists such that for all ,*(ii)*,** holds, then .*

*Proof. *Notice that for all ,
Suppose now that there is such that for all . Therefore,
for any , and so .

Now assume that . Then there exists such that for all we have . Consequently,
for any , and the proof is completed.

Theorem 3.15. *Let , , and , . Assume that and are the fundamental functions of and , respectively. If Lorentz space satisfies a lower -estimate for and , then
**
for a.a. , where is an extended best constant approximant of .*

*Proof. *Let and , . By Lemma 3.13 for any we get
Since , there is such that for all we have . Consequently,
for any and . Hence, by Proposition 3.7 and by assumption that satisfies a lower -estimate for we obtain that as for a.a. .

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

Corollary 3.16. *Let and . If one of the following conditions*(i)* and ,*(ii)* and , ,** is satisfied, then for a.a. we have
**
where is the best constant approximant of and the extended best constant approximant of .*

*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 for . Since , we have that
for all , which implies that . Consequently, by Definition 2.6 there is the extended best constant approximant operator on . Moreover, by Corollary 3.14 the fundamental functions and of the spaces and respectively, satisfy . Hence, by Corollary 3.6 we get that satisfies a lower -estimate for . Finally, by Theorem 3.15 we obtain that , an extended best constant approximant of , converges to as for a.a. .

Observe that the -functional of the couple can be expressed equivalently by for any and [25]. The next result on inequalities between maximal function and the -functional of