#### Abstract

Let be a quasi-Banach function space over a doubling metric measure space . Denote by the generalized upper Boyd index of . We show that if and has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function . Moreover, if , then quasievery point is a Lebesgue point of . As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

#### 1. Introduction and Main Results

Let be a doubling metric measure space. By the Lebesgue differentiation theorem, almost every point of a locally integrable function is a Lebesgue point. As expected, for smoother functions, the set of non-Lebesgue points is smaller. In [1], Kinnunen and Latvala showed that, for a quasicontinuous Hajłasz–Sobolev function , , there exists a set of -capacity zero such thatfor every . The case was studied in [2, 3]. Recently, in [4] and independently in [5], it was shown that iffor every , and , then for every quasicontinuous with , (1) holds true outside a set of -capacity zero.

If we replace integral averages in (1) by medians, then the result holds true also for small . For , , and , denote If and is quasicontinuous, then by [4, Theorem 1.2], there exists a set of -capacity zero such thatfor every and .

In this paper, we will study the existence of (generalized) Lebesgue points for functions whose Hajłasz gradient belongs to a general quasi-Banach function space . This approach allows us to simultaneously cover, for example, Orlicz–Hajłasz, Lorentz–Hajłasz, and variable exponent Hajłasz functions.

For , , , and , denote Operator and its variants have turned out to be useful in harmonic analysis and in the theory of function spaces; see, for example, [4, 6–24].

Theorem 1. *Let be doubling. Suppose that has absolutely continuous quasinorm and that, for every and for every ball , there exists a constant such thatfor every . Let . Then, for every quasicontinuous , there exists a set with such thatfor every and .*

We say that a point satisfying (7) for every is a* generalized Lebesgue point* of .

The (restricted) Hardy–Littlewood maximal function of a locally integrable function is As usual, we denote .

Theorem 2. *Let and let satisfy (2). Denote . Suppose that has absolutely continuous quasinorm and that, for every ball , there exists a constant such thatfor every . Then, for every quasicontinuous , quasievery point is a Lebesgue point of .*

For any quasi-Banach function space over , define ; The* generalized upper Boyd index* of is The generalized upper Boyd index was introduced in [20], where it was shown that the Hardy–Littlewood maximal operator is bounded on if and only if . As a corollary of Theorems 1 and 2 we have the following result.

Theorem 3. *Suppose that has absolutely continuous quasinorm and that satisfies (2). Let and let be quasicontinuous. *(i)*If , then quasievery point is a generalized Lebesgue point of .*(ii)*If , then quasievery point is a Lebesgue point of .*

#### 2. Preliminaries

##### 2.1. Basic Assumptions

In this paper, is a metric measure space equipped with a metric and a Borel regular, doubling outer measure , for which the measure of every ball is positive and finite. The doubling property means that there exists a constant such that for every ball , where and .

The doubling condition is equivalent to the existence of constants and such that (2) holds for every , , and .

* The integral average* of a locally integrable function over a set of positive and finite measure is

By , we denote the characteristic function of a set and by , the extended real numbers . is the set of all measurable, almost everywhere finite functions . In general, and are positive constants whose values are not necessarily same at each occurrence. When we want to stress that the constant depends on other constants or parameters , we write .

##### 2.2. Quasi-Banach Function Spaces

A quasinorm on a subspace of is a functional such that(i) a.e.;(ii) for every ;(iii)there exists a constant such that .

A quasi-Banach function space over is a subspace of equipped with a complete quasinorm that has the following properties:(i) and a.e. and ;(ii);(iii) a.e. .

By the Aoki–Rolewicz theorem ([25, 26]), there exists a constant such thatfor all .

A quasinorm on is absolutely continuous if whenever and is a decreasing sequence of sets such that .

##### 2.3. Hajłasz Spaces

Let . A measurable function is an *-gradient* of a function if there exists a set with such that, for all ,The collection of all -gradients of is denoted by .

The* homogeneous Hajłasz space * consists of measurable functions for which is finite. The* Hajłasz space * is equipped with the norm

Hajłasz spaces were introduced in [27] for , and in [28] for fractional scales. Recall that, for , (see [27]), whereas for , coincides with the Hardy–Sobolev space by [29, Thm 1].

Next two lemmas for -gradients follow easily from the definition; see [30, Lemmas 2.4] and [1, Lemma 2.6].

Lemma 4. *Let , , and . Then is an -gradient of functions and .*

Lemma 5. *Let and , . Let and . If , then .*

The following lemma is essential [4, Lemma 7.2].

Lemma 6. *Let and let be a measurable set. Let be a measurable function with and let be a bounded -Lipschitz function supported in . Then Consequently, there exists a constant such that for every .*

Lemma 7. *Let and suppose that has absolutely continuous quasinorm. Then s-Hölder continuous functions are dense in .*

*Proof. *Let and let be the exceptional set for (15). Then is -Hölder continuous with constant in the set . By [31], there is an extension of to such that is -Hölder continuous with constant . It is easy to see that ; see [32, Proposition 4.5]. By the absolute continuity of , and as .

##### 2.4. -Median

Let . The -median of a measurable function over a set of finite measure is Note that if and , then is finite.

In the following lemma, we list some basic properties of the -median. Properties (a), (b), (d), (f), and (g) follow from [21, Propositions 1.1 and 1.2] and (h) and (i) from [21, Theorem 2.1]. The remaining properties (c) and (e) follow immediately from the definition.

Lemma 8. *The -median has the following properties: *(a)*If , then .*(b) *If almost everywhere, then .*(c) *If and , then .*(d) *If , then .*(e) *If , then .*(f) *.*(g) *.*(h) *If is continuous, then for every ,*(i) *If , then there exists a set with such that* *for every and .*

##### 2.5. Discrete Maximal Functions

In this subsection, we define “discrete” versions of the Hardy–Littlewood maximal function and the median maximal function

We first recall the definition of a discrete convolution. Discrete convolutions are basic tools in harmonic analysis in homogeneous spaces; see, for example, [33, 34]. The following lemma is well known.

Lemma 9. *For every , there exists a collection of balls , where , and functions , , with the following properties: *(a)*The balls , , cover .*(b)*.*(c)*For every , is -Lipschitz, on and outside .*(d)*.** Here, the constant depends only on the doubling constant .*

For each scale , we choose a collection of balls and a collection of functions satisfying conditions (a)–(d) of Lemma 9.

*Definition 10. *The* discrete convolution* of a locally integrable function at the scale iswhere and are the chosen collections of balls and functions for the scale .

The* discrete maximal function* of is ,

The discrete maximal function, which can be seen as a smooth version of the Hardy–Littlewood maximal function, was introduced in [1].

Similarly, we define median versions of a discrete convolution and a discrete maximal function.

*Definition 11. *Let . The* discrete **-median convolution* of a function at scale is where and are as in Definition 10.

The* discrete *-*median maximal function* of is ,

As a supremum of continuous functions, the discrete maximal functions are lower semicontinuous and hence measurable.

Lemma 12. *Let . There exists a constant such thatfor every andfor every and .*

*Proof. *We will prove (30). The proof of (29) is similar. Let , , and let be as in Definition 11. If , then . By the doubling property, , and so, by Lemma 8(c), . Since , it follows that On the other hand, since the balls , , cover , there is such that . By the doubling property, , and so, by Lemma 8(c), . Since on , we have that The claim (30) follows immediately from these estimates.

#### 3. Capacity

In this section, we define the -capacity and prove some of its basic properties.

*Definition 13. *Let . The -capacity of a set is whereis a set of admissible functions for the capacity. We say that a property holds* quasieverywhere* if it holds outside a set of -capacity zero.

*Remark 14. *Lemma 4 easily implies that where .

*Remark 15. *It is easy to see that the -capacity is an outer capacity, which means that

The -capacity is not generally subadditive, but for most purposes, it suffices that it satisfies inequality (37) below.

Lemma 16. *Let and let be the constant from (14). Thenwhenever , .*

*Proof. *Let . We may assume that . There are functions with such that By Lemma 5, and . Henceand the claim follows by letting .

A function is* quasicontinuous* if for every , there exists a set such that and the restriction of to is continuous. By Remark 15, the set can be chosen to be open.

The following lemma follows from a result of Kilpeläinen [35].

Lemma 17. *Suppose that and are quasicontinuous. If almost everywhere in an open set , then quasieverywhere in .*

Lemma 18. *Suppose that has absolutely continuous quasinorm. Then, for every , there exists quasicontinuous such that almost everywhere.*

*Proof. *Suppose first that . By Lemma 7, there are continuous functions converging to in such that for every . Moreover, by [36, Lemma 3.3], we may assume that pointwise almost everywhere. Denote and . Then in for every . Hence converges pointwise in and the convergence is uniform in . By continuity, and so for every . Hence, by Lemma 16, which implies that . It follows that the function is quasicontinuous. Moreover, almost everywhere.

Suppose then that . Let . For , let be a Lipschitz function of bounded support such that in . Then, by Lemma 6, . By the first part of the proof, there exists quasicontinuous such that almost everywhere. Since almost everywhere in , Lemma 17 implies that there exists with such that in . It follows that the limit exists in , where, by Lemma 16, is of -capacity zero. Define . Then, clearly almost everywhere. Let . For every , there exists with such that is continuous. It follows that is continuous and, by Lemma 16, .

The following lemma gives a useful characterization of the capacity in terms of quasicontinuous functions. The proof of the lemma is essentially same as the proof of [37, Theorem 3.4]. For , denote

Lemma 19. *Suppose that has absolutely continuous quasinorm. Then for every .*

*Proof. *To prove the first inequality, let and let be a quasicontinuous representative of . Since in some open set containing and almost everywhere, it follows that almost everywhere in . Since is quasicontinuous, the equality actually holds quasieverywhere in . Hence quasieverywhere in , which implies that .

For the second inequality, let . By truncation, we may assume that . Fix , and choose an open set with so that on and that is continuous in . By continuity, there is an open set such that Clearly, . Choose such that and that . Define . Then on which is an open neighbourhood of . Hence and so Since and are arbitrary, the desired inequality follows.

#### 4. Generalized Lebesgue Points

In this section, we prove the first main result of the paper, Theorem 1. The main ingredient of the proof of is a capacitary weak type estimate, Theorem 21.

Lemma 20. *Let , , and . Let and . Then there exists a constant such that is an -gradient of .*

*Proof. *Let . By the definition of the discrete -median convolution and by the properties of the functions , By Lemma 6, function is an -gradient of function for each .

Let , where is the exceptional set for (15). Using Lemma 8 and the facts that and , we obtainSince for almost every and since the balls have bounded overlap, it follows that for every . Consequently, by Lemma 5, as desired.

Theorem 21. *Suppose that the assumptions of Theorem 1 are in force. Then, for every ball and for every , there exists a constant such that for every and .*

*Proof. *Since when , it suffices to prove the claim for . Let and let be a Lipschitz function such that in and outside . Then in and soBy (30) and (6), and, by Lemmas 20 and 6 and (6),Hence, . Since is lower semicontinuous, . Thus, and the claim follows.

Lemma 22. *Suppose that has absolutely continuous quasinorm and that is a ball such that, for every and ,whenever . Then, for every quasicontinuous , quasievery point in is a generalized Lebesgue point of .*

*Proof. *By Lemma 7, continuous functions are dense in . Let be quasicontinuous and let , , be continuous such that as . Denote . Fix and . By Lemma 8,Hence, by Lemma 16,By assumption, as . Since is quasicontinuous, Lemma 19 givesas . It follows that for every and . Denote Then and so, by Lemma 16, . Since, by Lemma 8, the claim follows.

*Proof of Theorem 1. *Let be quasicontinuous. Fix