Hölder Regularity of Quasiminimizers to Generalized Orlicz Functional on the Heisenberg Group

Zhang, Junli; Niu, Pengcheng

doi:https://doi.org/10.1155/2020/8838654

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Volume 2020 | Article ID 8838654 | https://doi.org/10.1155/2020/8838654

Hölder Regularity of Quasiminimizers to Generalized Orlicz Functional on the Heisenberg Group

Junli Zhang¹and Pengcheng Niu¹

Academic Editor: Calogero Vetro

Received22 Aug 2020

Accepted20 Oct 2020

Published25 Nov 2020

Abstract

In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.

1. Introduction

In this paper, we concern the generalized Orlicz functional on the Heisenberg group, where (we say that if and only if there exist and such that ), , , is the generalized Orlicz space (see Section 2 for details). Write and abbreviate . Here, we assume that

(A1) There exists , such that .

(A2) There exists , such that for every ball ,

(A3) There exist and , such that is -almost increasing (-almost increasing means that there exists a constant such that for all ) with respect to .

(A4) There exist and , such that is -almost decreasing with respect to .

If is increasing, then assumption (A3) can be written by (A3). If is decreasing, then assumption (A4) can be written by (A4). Note that , and that all these assumptions are invariant to equivalent generalized -functions and scaling (see (100) below).

It is clear that the Euler-Lagrange equation corresponding to functional (1) is

where denotes the derivative of with respect to . If for any , it holds then we say that is a weak solution of (6). When functional in (1) satisfies , the regularity of weak solutions to the corresponding Euler-Lagrange equation has been studied by many scholars. While , for not being far from 2, Manfredi and Mingione in [1] got the Hölder continuity of the ordinary gradient of weak solutions and derived smoothness of weak solutions by using the method in [2]; for , the second order differentiability of weak solutions was deduced by Domokos in [3], which generalized the results in Marchi [4]. While , for not being far from 2, Domokos and Manfredi in [5] used the Calderón-Zygmund theory on the Heisenberg group to study regularity of weak solutions; for , Mingione, Zatorska-Goldstein, and Zhong in [6] concluded the regularity of weak solutions by using a double-bootstrap method, energy estimates, and interpolation inequalities; for , Zhong in [7] got the regularity of weak solutions by using the energy estimate, the Moser iteration, and the oscillation estimate; Zhang and Niu in [8] proved the regularity of the gradient of weak solutions as , where belongs to the Orlicz space including the function . We observe that discussed before depends only on the second variable . For the more general case depending on two variables and , there is no relevant result on the Heisenberg group. In the Euclidean space, there are many results about the variable exponential case (i.e., ), the -growth case (i.e., ), and the double phase case (i.e., ), such as [9–14]. Harjulehto, Hästö, and Klen considered the functional including the above three cases and proved the existence of quasiminimizers in [15]. Whereafter, Harjulehto, Hästö, and Toivanen in [16] obtained the Hölder regularity of quasiminimizers by using the De Giorgi-Moser iteration and some tools in harmonic analysis.

In this paper, we consider the Hölder regularity of the quasiminimizers of the functional (1) inspired by [16]. The main difference is that we need to use the Sobolev inequality, the Riesz potential, and the maximal function on the Heisenberg group. In addition, to derive the regularity, we prove a covering Lemma by the Calderón-Zygmund decomposition on the Heisenberg group.

Before stating the main results, we give the following definition.

Definition 1. Let(the generalized orlicz space, see next section). We say that is a local quasiminimizer of (1), if there exists a constant such that for any with .

Now, we state the main results:

Theorem 2 (Harnack inequality). Assume that satisfies (A1), (A2), (A2), (A3), and (A4), If is a nonnegative local quasiminimizer of (1), then for a compact set , there exists as shown in Lemma 27 below such that for all and cubes with centered in , where depends only on the parameters of (A1), (A2), (A2), (A3) and (A4), and .

Theorem 3 (Hölder continuity). Let satisfy (A1), (A2), (A2), (A3), and (A4). is a local quasiminimizer of (1), then .

This paper is organized as follows. In Section 2, we first give the definitions and related knowledge of the Heisenberg Group, then introduce the generalized -function and its related properties. Some definitions of function spaces and some known Lemmas are given. In Section 3, we use the De Giorgi-Moser iteration to obtain the local boundedness of the quasiminimizer. As the result of the third section, when the radius approaches 0, the constant will blow up; so in Section 4, the upper bound of the solution is improved, but the solution is needed to bounded. In Section 5, on the basis of the results obtained in Section 4, we first prove a covering Lemma by the Calderón-Zygmund decomposition and then use it to obtain the Harnack inequality and Hölder continuity.

The common generalized Orlicz functions [see [15]] involves

Then, in (1) can have the concrete relations

In this paper, we always denote a positive constant by which may vary from line to line, . We assume that is a bounded domain, is a cube whose side length is in the direction, in the direction, and its edge is parallel to the coordinate axis and denote . Let be a concentric cube whose side length is times the in direction and times in the direction. For , we denote .

2. Preliminaries

In this section, we first recall the related knowledge of the Heisenberg Group, then introduce the definition of the generalized -function and some properties related to it. Finally, some function spaces and lemmas are given.

2.1. Heisenberg Group

The Euclidean space with the group multiplication

where leads to the Heisenberg group . The scaling on is defined as

The left invariant vector fields on are of the form and a nontrivial commutator on is

We call that are the horizontal vector fields on and the vertical vector field. Denote the horizontal gradient of a smooth function on by

The homogeneous dimension of is . The Haar measure in is equivalent to the Lebesgue measure in . We denote the Lebesgue measure of a measurable set by . The Carnot-Carathèodary metric (CC-metric) between two points in is the shortest length of the horizontal curve joining them, denoted by . The ball defined by the CC-metric is

One has

For its module is defined by

The CC-metric is equivalent to the Korànyi metric

2.2. Generalized -Function and Its Related Properties

Definition 4 (generalized -function). A real valued function defined on is said to be a generalized -function, and if is a Lebesgue measurable with respect to , the derivative of with respect to exists, and is right continuous, nondecreasing, and satisfies and .
If for any and , there exists such that then we say functions and are equivalent denoted by . If for any , there exists such that then, we say that satisfies the strong -condition and denotes the minimum constant by . Since the strong -condition is equivalent to . Obviously, if satisfies the strong -condition, then . For a family of generalized -functions, we define Let If is strictly increasing with respect to , then is the inverse function of . Writing then is also a generalized -function and satisfies Note that is the complementary function of and . For any , there exists depending only on , such that for any , and this inequality is called Young’s inequality ([16]). For some and any , we denote then If and are generalized -functions and satisfy for , then for any , it holds

2.3. Some Function Spaces and Lemmas

We denote the real valued measurable function space by . If the generalized N-function satisfies the strong -condition on , then is a Banach space with (Luxemburg) norm

We call that it is a generalized Orlicz space or Musielak-Orlicz space denoting by . For the generalized Orlicz-Sobolev space is defined as and the local generalized Orlicz-Sobolev space as

The space is the closure of in .

If is bounded and satisfies the assumptions (A1) and (A3), then . For their proofs, one can refer to Lemmas 4.4, 6.2, and 6.9 in [15] with some evident changes.

We now describe their proofs (Lemmas 5–11) that are similar to ones in [16] with some suitable revisions.

Lemma 5. If, then there exists a generalized -function increasing strictly such that , and so, is a bijection.

Lemma 6. Let, then (1)The strong -condition is equivalent to (A4).(2)If is convex with respect to , then the strong -condition is equivalent to (A4).

Lemma 7. The assumption (A1) implies .

Lemma 8. Let be a bijection with respect to or satisfy the strong -condition. Then, (1)The assumption (A4) is equivalent to that is almost increasing uniformally in (2)The assumption (A3) is equivalent to that is almost decreasing uniformally in

If , then we can get (A2) from (A1), (A2), and (A4).

Lemma 9. Let satisfy the assumptions (A1), (A2), and (A4). If , then satisfies (A2).

The proof of Lemma 9 is similar to that of Lemma 12 in [16], and a simple distinct is that we should use the fact on the Heisenberg group.

Lemma 10. Let satisfy the assumption (A2) or (A2). Then, there exists such that for any , we have for , where under (A2) and under (A2).

The proof of Lemma 10 is similar to that of Lemma 13 in [16], but we should employ the statement in the process that on the Heisenberg Group, if is the smallest ball containing , then .

Lemma 11. Let satisfy the assumption (A2) and be a bijection. Then, there exists such that for any cube with and , we have for any and .

For the following lemma, one can refer to [17].

Lemma 12 [17]. If , then satisfies the Jensen type inequality

In the generalized Orlicz space, the Hölder inequality with the constant 2 holds, see ([13], Lemma 9). It is stated that for any , it follows

Because the Heisenberg Group is a special case of Carnot groups, the conclusions on Carnot groups are also true on . We write some conclusions in monograph ([18], p276-280) on . For , , we formally define the Riesz potential operator as where denotes . We also call that is the fractional integral of order , and is abreviated to .

Lemma 13 (Hardy-Littlewood-Sobolev inequality, [18]). Let , , , and Then, there exists a positive constant such that for every , we have For a function , we define the maximal function as One has the statement (maximal function theorem): if , then there exists a positive constant such that for every , we have (see [18])

Lemma 14 (Sobolev-Stein embedding, (, p280)). Let . Then, there exists a constant such that for every , where

Proof. For , the representation formula (5.16) in [18] yields where , . By the integrating by parts, we get In addition, out of the origin, one sees Because is smooth in and , we obtain that for a constant , Using (48), it yields Then, by Lemma 13, we gain where This ends the proof.
Noting that is dense in , we have the following result from Lemma 14.

Lemma 15 (Sobolev inequality). Let , . For any , it follows where
Since is not dense in , we need to prove.

Lemma 16. Let, , satisfy (A1), (A2), and (A4). If with , then .

Proof. Because and are closed, we can find which is bounded, quasiconvex, and contains . Values outside do not affect the claim, so we verify the claim in . In order to simplify the notation, we might assume that is bounded and quasiconvex . Owing to similarly to the proofs of theorems 6.6 and 6.5 in [15], we know that is dense in ; then there is a sequence of convergencing to . We take a cut-off function with and in and see . Since it follows

Lemma 17. Let withbeing finite. If, then for all , we have where depends only on and .

The proof is similar to the proof of Lemma 24 in [16], and it only needs to change the classical Sobolev inequality in the Euclidean space into the Sobolev inequality (54) on .

Lemma 18. Suppose that satisfies (A1), (A2), and (A3), then there exists such that for all with we have where depends only on and the parameters of (A1), (A2), and (A3).
The proof is similar to the proof of Lemma 23 in [16].

3. Local Boundedness

Unless otherwise specified, we will use the following notations. Suppose that , is a cube whose side length is in the direction, in the direction, and its edge is parallel to the coordinate axis with centered 0 and denotes ,

Lemma 19 [16]. Suppose that is a bounded nonnegative function in and satisfies the strong -condition in , if there exists such that for any , then we have where depends only on and .

Lemma 20 (Caccioppoli inequality). Let and be a local quasiminimizer of (1). Then for all , there holds where depends on in definition 1 and .

The proof is similar to Lemma 27 in [16].

Lemma 21. Suppose that satisfies (A1), (A2), (A3), and (A4) and define If satisfies (65), , then there exists such that as , where , , depends only on the parameters of (A1), (A2), (A3), and (A4), and . Here, satisfies .

Though the proof is similar to Lemma 4.6 in [16], we need to replace the results about the Riesz potential and maximal function with our Lemma 13 and (51). For completeness, let us write the detailed proof.

Proof. Let , , satisfy Denoting then we have We first estimate the integral . If , then from (A3), (A4), and Lemma 7, so We use it and the Hölder inequality to gain Denoting , then we know that from (51), If is small enough, then Because of in the set , it yields from (74) and (75) that Therefore, in the set , it implies Suppose that is small enough such that , then from Lemma 13 and , we know Noting the conditions of Lemma 18 are satisfied. Now, we combine (77), (78), the strong -condition and Lemma 18 to obtain Denoting we see from (A3) that for every , Let us estimate the measure of . When , we have , and deduce from Lemmas 16, 17, and (82) that On the other hand, , so Combining (73), (80), (85), (84), and , we know where .
Next, we estimate the integral . Observing that and they do not depend on in the set , the usual Hölder inequality yields Because of , one has . Using Lemmas 16, 17, and , it deduces In conclusion, the integrals and have same upper bound. At present, we estimate the integral . Using the strong -condition and (65), we obtain Note that in , so . Thus, Since is increasing, it follows Combining (86)–(92), we get Returning to (70), it reaches (67).

Lemma 22 [16]. Let and be a sequence of real numbers satisfying for . If then as .

Lemma 23. Let satisfy (A1), (A2), (A3), and (A4). If satisfies (65) and as shown in Lemma 21, then for and any , we have where as shown in Lemma 21, depends only on the parameters of (A1), (A2), (A3), and (A4), and .
The proof is similar to Theorem 4.11 in [16].

We use (96) for and to immediately obtain the following.

Theorem 24 (local boundedness). Let satisfy (A1), (A2), (A3), and (A4), then every local quasiminimizer of (1) is locally bounded.

4. Improvement of the Upper Bound of Bounded Solutions

Lemma 25. Let satisfy (A1) and (A4) and define If satisfies (65), , then we have as , where depends only on the parameters of (A1) and (A4), and .

The proof is similar to the proof of Lemma 31 in [16], and a necessary change is to replace the Sobolev inequality in the Euclidean space to the inequality in Lemma 17.

For , define

Note that and the constants of (A2) and (A2) will be changed under this scaling (100).

Lemma 26. Let . If is a local quasiminimizer of (1) in with , then is a local quasiminimizer of the functional in .

Proof. Suppose and , obviously, we have Using the transformation of variable and the quasiminimality of in , it implies

Lemma 27. Let satisfy (A1), (A2), and (A4). If is a bounded local quasiminimizer of (1) in , then for , we have where depends only on and , depends only on the parameters of (A1), (A2), and (A4), , and does not depend on .

Proof. The proof follows the line proving proposition 28 in [16]. Writing Assuming , we claim In fact, obviously, it holds Similarly, we have , so two inequalities in (106) are equivalent.
In the sequel, we prove the latter of (106). Let us consider cases and . (1)If , thenOn one hand, ; on the other hand, (A4) and definition 4 yield so (2)If , thenBy Lemma 10, for , we have ; so when (i.e., ), it holds Thus, the latter of (106) is proved.
Denote where is to be determined. Note by using . Denoting it deduces by the almost increasing of , (106), and the strong -condition that For , we take and see . Defining and applying the strong -condition and (106), we get By Lemma 26, is a local quasiminimizer, so we know from Lemma 20 that satisfies (65). Hence, using (118) and Lemma 25 yields By (A4), definition 4, (106), and the strong -condition, we have so it deduces from (119) that Select in Lemma 22. If then we have by Lemma 22 that almost everywhere in .
The remaining job is to prove (122). We will point out that has a bound not depending on and then choose such that (4.9) holds. Since is almost decreasing, it follows that is also almost decreasing, and is almost increasing by Lemma 8. Then, we know from ([15], Lemma 5) that is equivalent to a convex function . Because of , it yields by the Jensen inequality that Since satisfies the strong -condition, we have and (122) is proved.

Theorem 28. Let satisfy (A1), (A2) and (A4), and be a bounded local quasiminimizer of (1). When , we have where depends only on the parameters of (A1), (A2), and (A4), , and .
The proof is similar to Theorem 5.7 in [16] by using Lemma 27 here. We omit it and describe two interesting corollaries.

Corollary 29. Let satisfy (A1), (A2), and (A4) and be a bounded local quasiminimizer of (1). When , we have for any , where does not depend on and .
The proof is similar to Corollary 5.8 in [16].

Corollary 30. Let satisfy (A1), (A2), and (A4) and be a bounded local quasiminimizer of (1). When , we have for any , where does not depend on and depends only on the parameters of (A1), (A2), and (A4), , , and .
The proof is similar to Corollary 5.9 in [16].

5. Weak Harnack Inequality and the Proof of the Main Result

Denote

Lemma 31. If, satisfies (128) with , , and for some , then The proof is similar to Lemma 6.1 in [16].

Lemma 32. Let satisfy (A1), (A2), (A2), (A4), and (A3), and be a bounded local quasiminimizer of (1), as shown in Lemma 27. If , and for all , , and , there exists such that then, we have The proof is similar to the proof of Lemma 6.2 in [16]. It only needs to replace theorem 3.16 in [19] with Lemma 15 in this paper.

Lemma 33 (covering Lemma). Suppose that and () are measurable sets, if there exists such that (1)For any cube with , it holds , where is a subset of Then,

Proof. From the Calderón-Zygmund decomposition Theorem ([20], p17), we know that for every and , there is a sequence of disjoint cubes such that for almost all , we have If we take , then Let us decompose until the cube satisfies ; so, Hence, it follows

Lemma 34. Suppose and for all and every , there exists such that if then then, there exists such that for every , we have

Proof. The proof is similar to the proof of Lemma 6.3 in [16], and the key difference is that we replace the covering lemma there with Lemma 5.3. Fix and write Take in Lemma 5.3, where is the constant in (141) with respect to . Suppose that for some and , we have Therefore, it follows under (145) that so (140) holds. Thus, From the above discussion, we can infer (147) from the hypothesis (145), so . It deduces that (2) in Lemma 33 holds. Hence, we know that from Lemma 33 that if (1) in Lemma 33 does not holds, then , which implies If (1) in Lemma 33 holds, then , which gives If , we choose to be the smallest integral satisfying , and then so , i.e., , Thus, as (1), we have Combining (1) and (2), we obtain If we write , then Taking , it gets so Thus, (142) is proved.

Lemma 35 (weak Harnack inequality). Let satisfy (A1), (A2), (A2), (A4), and (A3). If is a nonnegative local quasiminimizer of (1), as shown in Lemma 27, then for all and , we have where depends only on the parameters of (A1), (A2), (A2), (A4), and (A3), , and .

Proof. According to Lemma 32, we see that the conditions of Lemma 34 are satisfied, so (156) is true by using Lemma 34.

Proof of Theorem 2. Combining Corollary 30 and Lemma 35, we prove immediately Theorem 2.

Proof of Theorem 3. The Harnack inequality in Theorem 2 implies the Hölder continuity.

Data Availability

No data used.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This work was Supported by the National Natural Science Foundation of China (Grant No. 11771354) and the Fundamental Research Funds for the Central Universities (Grant No. 310201911cx013).

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Copyright © 2020 Junli Zhang and Pengcheng Niu. 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.

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