Research Article | Open Access
Toni Heikkinen, "Generalized Lebesgue Points for Hajłasz Functions", Journal of Function Spaces, vol. 2018, Article ID 5637042, 12 pages, 2018. https://doi.org/10.1155/2018/5637042
Generalized Lebesgue Points for Hajłasz Functions
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 , 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  and independently in , 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.
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 , 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 .
2.1. 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. .
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
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 , 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 .
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 .
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.
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 .
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
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 .