Journal of Function Spaces
Volume 2018 (2018), Article ID 5637042, 12 pages
https://doi.org/10.1155/2018/5637042

Research Article

Generalized Lebesgue Points for Hajłasz Functions

Toni Heikkinen

Department of Mathematics, Aalto University, P.O. Box 11100, 00076 Aalto, Finland

Correspondence should be addressed to Toni Heikkinen; toheikki@gmail.com

Received 10 September 2018; Accepted 29 October 2018; Published 11 November 2018

Academic Editor: Henryk Hudzik

Copyright © 2018 Toni Heikkinen. 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.

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 that

for every . The case was studied in [2, 3]. Recently, in [4] and independently in [5], it was shown that if

for 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 that

for 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 that

for every . Let . Then, for every quasicontinuous , there exists a set with such that

for 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 that

for 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 that

for 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 is

where 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 that

for every and

for 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

where

is 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). Then

whenever , .

Proof. Let . We may assume that . There are functions with such that

By Lemma 5, and . Hence

and 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 obtain

Since 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 so

By (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 gives

as . 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 and, for , let be a Lipschitz function of bounded support such that in . Then, by Lemma 6, . By Theorem 21 and Lemmas 12 and 22, for every , quasievery point is a generalized Lebesgue point of . Hence, for every , quasievery point in is a generalized Lebesgue point of . Thus, by Lemma 16, quasievery point is a generalized Lebesgue point of .

5. Lebesgue Points

In this section we prove Theorem 2.

Lemma 23. Suppose that satisfies (2). Let , , , and where . Then there exists a constant such that is an -gradient of .

Proof. By the Sobolev–Poincaré inequality ([8, Lemma 2.2]), is locally integrable and there exists a constant such that

for every and .
Fix . By the definition of the discrete convolution and by the properties of the functions ,

By Lemma 6, function

is an -gradient of the function for each . A standard chaining argument and (68) imply that, for almost every ,

Since , for almost every , and since the balls have bounded overlap, it follows that

The claim follows by Lemma 5.

Lemma 24. Let be such that outside and suppose that there exists such that . Then there exists a constant such that .

Proof. Let . Then . Indeed, if , then, by definition,

and if one of the points, say , does not belong to , then

Hence, it suffices to show that For almost every ,

Since , we have that . Clearly,

Hence

Theorem 25. Suppose that the assumptions of Theorem 2 are in force. Let be a ball and assume that the sphere is nonempty. Then there exist constants and such that

for every and .

Proof. Let and let be a Lipschitz function such that in and outside . Then, by Lemma 6, and

If , then . Hence,

If , then . In particular, outside . Hence, by Lemmas 23 and 24, and

where . Since is lower semicontinuous, it follows that

By the fact that , assumptions (9) and (79) imply that

and the claim follows.

The proof of the next lemma, which is very similar to the proof of Lemma 22, will be omitted.

Lemma 26. Suppose that is absolutely continuous and that is ball such that, for every ,

whenever . Then, for every quasicontinuous , quasievery point in is a Lebesgue point of .

Proof of Theorem 2. We may assume that contains at least two points. Then can be covered by balls , , where , such that the spheres are nonempty.
Let be quasicontinuous and, for , let be a Lipschitz function of bounded support such that in . Then, by Lemma 6, . By Theorem 25 and Lemmas 12 and 26, for every , quasievery point in is a Lebesgue point of and hence of . Hence, by Lemma 16, quasievery point in is a Lebesgue point of .

6. Proof of Theorem 3

If , then clearly is bounded on . Hence, the first part of Theorem 3 follows from Theorem 1. The second part follows from Theorem 2 via the following lemma.

Lemma 27. Let . If , then the operator is bounded on .

Proof. By assumption, there are constants and such that

for every and . Denote by the decreasing rearrangement of a function , that is,

Then, for every ball , we have

Hence,

Let be such that . By the Hölder inequality, we obtain

Thus, by (85) and (14),

7. Examples

7.1. Lorentz Spaces

For , , and , denote

when and

Then equipped with is a quasi-Banach function space. If , quasinorm is absolutely continuous.

Lemma 28. There exists a constant such that

for every and .

Proof. It follows easily from the definitions that, for every , , , and , we have

Hence, the claim follows from the well-known weak type inequality for the Hardy–Littlewood maximal operator.

The following lemma is an immediate consequence of Lemma 28.

Lemma 29. Let and . There exists a constant such that

for every and . Consequently, .

Lemma 29 and Theorem 3 imply the following result for Lorentz–Hajłasz functions.

Theorem 30. Let , , and let be quasicontinuous. Then quasievery point is a generalized Lebesgue point of . Moreover, if , then quasievery point is a Lebesgue point of .

7.2. Orlicz Spaces

Let be a continuous increasing function and suppose that there exists a constant such that

for every . Then

equipped with a quasinorm

becomes a quasi-Banach function space.

It is easy to see that (96) is equivalent to the existence of constants and such that

whenever .

A function is doubling, if there exists a constant such that

for every . It is easy to see that if is doubling, then is absolutely continuous.

Lemma 31. Let satisfy (99). Then there exists a constant such that

for every and . Consequently, .

Proof. We may assume . Since and, by Lemma 28, for all , we have that

when .

By combining Theorem 3 and Lemma 31, we obtain the following result for Orlicz–Hajłasz spaces .

Theorem 32. Suppose that satisfies (2) and that is doubling and satisfies (96). Let and let be quasicontinuous. Then quasievery point is a generalized Lebesgue point of . Moreover, if satisfies (99) with , then quasievery point is a Lebesgue point of .

7.3. Variable Exponent Spaces

Let be a measurable function. The space consisting of functions for which

for some equipped with quasinorm

is a quasi-Banach function space.

A measurable function is locally log-Hölder continuous if there exists a constant such that

for all

Denote

We need the following result from [19].

Lemma 33. Let be locally log-Hölder continuous with and . Suppose that there exist constants and such that

Then there exists a constant such that

for every and . Consequently, .

The next lemma follows from [38, Lemma 2.3], [38, Corollary 3.5] and from the fact that the function is bi-Lipschitz from to whenever .

Lemma 34. Suppose that is locally log-Hölder continuous and that and . Then, for any ball , there exists a locally log-Hölder continuous extension of such that , and

for some and .

Theorem 35. Suppose that satisfies (2) and that is locally log-Hölder continuous with . Let and let be quasicontinuous.

(1)If , then quasievery point is a generalized Lebesgue point of .(2)If , then quasievery point is a Lebesgue point of .

Proof. Since , has absolutely continuous quasinorm. By Theorem 1, it suffices to show that, for every ball and , there exists a constant such that

for every . By Lemma 34, can be extended to on such that , , and (109) holds true for some and . By Lemma 33, there exists a constant such that

for every . Thus,

for every .
Denote . By Theorem 2, it suffices to show that, for every ball , there exists a constant such that

for every By Lemma 34, can be extended to on such that , , and (109) holds true for some and . Hence, by Lemma 33, and so, by Lemma 27, operator is bounded on . Thus,

for every .

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

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