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

Authors: Junli Zhang; Pengcheng Niu
Journal: Journal of Function Spaces
 (2020)

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## 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 thep-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.


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## Body

## 1. Introduction

In this paper, we concern the generalized Orlicz functional(1)∫ΩΦx,∇Hudx.on the Heisenberg group, where
(2)Φx,∇Hu~φx,∇Hu∈ΦwΩ(we say that f~g if and only if there exist c0 and c1>0 such that c0f≤g≤c1f), ∇Hu=X1u,X2u,⋯,X2nu, ΦwΩ, is the generalized Orlicz space (see Section 2 for details). Write
(3)φB+τ≔supx∈Bφx,τandφB−τ≔infx∈Bφx,τand abbreviate φ±=φΩ±. Here, we assume that(A1) There existsβ∈0,1, such that φ+β≤1≤φ−1.(A2) There existsβ∈0,1, such that for every ball B⊂Ω,
(4)φB+βτ≤φB−τforτ∈1,φB−−11B.(A2)′ There exists β∈0,1, such that for every ball B⊂Ω,
(5)φB+βτ≤φB−τforτ∈1,1diamB.(A3) There existγ−>1 and λ≥1, such that φx,τ/τγ− is λ-almost increasing (λ-almost increasing means that there exists a constant λ≥1 such that φx,s/sγ−≤λφx,τ/τγ− for all s≤τ) with respect to τ>0.(A4) There existγ+>1 and λ≥1, such that φx,τ/τγ+ is λ-almost decreasing with respect to τ>0.Ifφx,τ/τγ− is increasing, then assumption (A3) can be written by (A3)′. If φx,τ/τγ+ is decreasing, then assumption (A4) can be written by (A4)′. Note that γ+≥γ−, and that all these assumptions are invariant to equivalent generalized N-functions and scaling (see (100) below).It is clear that the Euler-Lagrange equation corresponding to functional (1) is
(6)divHφ′x,∇Hu∇Hu∇Hu=0,whereφ′x,τ denotes the derivative of φx,τ with respect to τ. If for any v∈HW01,φΩ, it holds
(7)∫Ωφ′x,∇Hu∇Hu∇Hu·∇Hvdx=0,then we say that u∈HW1,φΩ is a weak solution of (6). When functional Φx,∇Hu in (1) satisfies Φx,τ~Λ+τ2p−2/2τ, the regularity of weak solutions to the corresponding Euler-Lagrange equation has been studied by many scholars. While 0<Λ<1, for p 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 1<p<4, the second order differentiability of weak solutions was deduced by Domokos in [3], which generalized the results in Marchi [4]. While 0≤Λ<1, for p 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 2≤p<4, Mingione, Zatorska-Goldstein, and Zhong in [6] concluded the C1,α regularity of weak solutions by using a double-bootstrap method, energy estimates, and interpolation inequalities; for 1<p<∞, Zhong in [7] got the C1,α regularity of weak solutions by using the energy estimate, the Moser iteration, and the oscillation estimate; Zhang and Niu in [8] proved the Γα regularity of the gradient of weak solutions as Φx,τ~φτ, where φτ belongs to the Orlicz space including the function φτ=Λ+τ2p−2/2τ. We observe that Φx,τ discussed before depends only on the second variable τ. For the more general case depending on two variables x and τ, there is no relevant result on the Heisenberg group. In the Euclidean space, there are many results about the variable exponential case (i.e., Φx,τ~τpx), the p,q-growth case (i.e., c1τp−1≤Φx,τ≤c2τq+1,q>p), and the double phase case (i.e., Φx,τ=τp+axτq), such as [9–14]. Harjulehto, Hästö, and Klen considered the functional ∫ΩΦx,∇udx 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.
Letφ∈ΦwΩ(the generalized orlicz space, see next section). We say that u∈HWloc1,φΩ is a local quasiminimizer of (1), if there exists a constant K≥1 such that
(8)∫v≠0φx,∇Hudx≤K∫v≠0φx,∇Hu+vdx,for any v∈HWloc1,φΩ with sptv⊂Ω.Now, we state the main results:Theorem 2 (Harnack inequality).
Assume thatφ∈ΦwΩsatisfies (A1), (A2), (A2)′, (A3), and (A4), If u∈HWloc1,φΩ is a nonnegative local quasiminimizer of (1), then for a compact set K⊂⊂Ω, there exists R0 as shown in Lemma 27 below such that
(9)esssupQRu≤cessinfQRu+R,
for allR∈0,R0 and cubes Q6R⊂⊂Ω with centered in K, where c depends only on the parameters of (A1), (A2), (A2)′, (A3) and (A4), ℘ and uL∞Q2R.Theorem 3 (Hölder continuity).
Letφ∈ΦwΩ satisfy (A1), (A2), (A2)′, (A3), and (A4). Ifu∈HWloc1,φΩ is a local quasiminimizer of (1), then u∈ClocαΩfor someα>0.This paper is organized as follows. In Section2, we first give the definitions and related knowledge of the Heisenberg Group, then introduce the generalized N-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
(10)φ1x,τ=τpxlog1+τ,φ2x,τ=τp+axτq,φ3x,τ=τpx,φ4x,τ=1pxτpx.Then,Φx,∇Hu in (1) can have the concrete relations
(11)Φx,∇Hu~φ1x,∇Hu,Φx,∇Hu~φ2x,∇Huthedoublephasecase,Φx,∇Hu~φ3x,∇Huthevariableexponentialcase,Φx,∇Hu~φ4x,∇Hu=1px∇Hupx.In this paper, we always denote a positive constant byc which may vary from line to line, x=x1,⋯,x2n,t=x′,t. We assume that Ω⊂ℍn is a bounded domain, Q is a cube whose side length is R in the x′ direction, R2 in the t direction, and its edge is parallel to the coordinate axis and denote diamQ≔2n2+11/4R>2n1/2R. Let cQ be a concentric cube whose side length is c times Q the in x′ direction and c2 times Q in the t direction. For f∈L1Ω, we denote fΩ=1/Ω∫Ωfxdx.


## 2. Preliminaries

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


### 2.1. Heisenberg Groupℍn

The Euclidean spaceℝ2n+1,n≥1 with the group multiplication
(12)x∘y=x1+y1,x2+y2,⋯,x2n+y2n,t+s+12∑i=1nxiyn+i−xn+iyi,wherex=x1,x2,⋯,x2n,t,y=y1,y2,⋯,y2n,s∈ℝ2n+1 leads to the Heisenberg group ℍn. The scaling on ℍn is defined as
(13)δx=δx1,δx2,⋯,δx2n,δ2t.The left invariant vector fields onℍn are of the form
(14)Xi=∂xi−xn+i2∂t,Xn+i=∂xn+i+xi2∂t,1≤i≤n,and a nontrivial commutator on ℍn is
(15)T=∂t=Xi,Xn+i=XiXn+i−Xn+iXi,1≤i≤n.We call thatX1,X2,⋯,X2n are the horizontal vector fields on ℍn and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍn by
(16)∇Hu=X1u,X2u,⋯,X2nu.The homogeneous dimension ofℍn is ℘=2n+2. The Haar measure in ℍn is equivalent to the Lebesgue measure in ℝ2n+1. We denote the Lebesgue measure of a measurable set E⊂ℍn by E. The Carnot-Carathèodary metric (CC-metric) between two points in ℍn is the shortest length of the horizontal curve joining them, denoted by d. The ball defined by the CC-metric is
(17)BRx=y∈ℍn:dy,x<R.One has(18)BRx=R℘B1x.Forx=x1,x2,⋯,x2n,t, its module is defined by
(19)xℍn=∑i=12nxi22+t214.The CC-metricd is equivalent to the Korànyi metric
(20)dx,y=x−1yℍn.


### 2.2. GeneralizedN-Function and Its Related Properties

Definition 4 (generalizedN-function).
A real valued functionφx,τ defined on Ω×0,+∞ is said to be a generalized N-function, and if φx,τ is a Lebesgue measurable with respect to x, the derivative φ′x,τofφx,τ with respect to τ exists, and φ′x,τ is right continuous, nondecreasing, and satisfies φ′x,0=0 and φ′x,τ>0τ>0.
If for anyτ≥0 and x∈Ω, there exists L≥1 such that
(21)ψx,τL≤φx,τ≤ψx,Lτ,then we say functions φ and ψ are equivalent denoted by φ≃ψ. If for any τ≥0, there exists c1>0 such that
(22)φx,2τ≤c1φx,τ;then, we say that φx,τ satisfies the strong Δ2-condition and denotes the minimum constant c1 by Δ2φ. Since
(23)φx,τ≤φx,2τ,the strong Δ2-condition is equivalent to φx,τ~φx,2τ . Obviously, if φ satisfies the strong Δ2-condition, then ~⇔≃. For a family of generalized N-functions, we define
(24)Δ2φλ≔supλΔ2φλ.
Let(25)φ′−1x,τ≔supυ∈0,+∞:φ′x,υ≤τ,forτ≥0.
Ifφ′=φ′x,τ is strictly increasing with respect to τ, then φ′−1 is the inverse function of φ′. Writing
(26)φ∗x,τ≔∫0τφ′−1x,ϑdϑ,τ≥0,then φ∗ is also a generalized N-function and satisfies
(27)φ∗′x,τ=φ′−1x,τ,τ≥0.
Note thatφ∗ is the complementary function of φ and φ∗∗=φ. For any δ>0, there exists cδ depending only on Δ2φ,φ∗, such that
(28)τυ≤δφx,τ+cδφ∗x,υ
for anyτ,υ≥0, and this inequality is called Young’s inequality ([16]). For some a,b>0 and any τ≥0, we denote
(29)ρx,τ=aφx,bτ,then
(30)ρ∗x,τ=aφ∗x,τab.
Ifφ and ρ are generalized N-functions and satisfy φx,τ≤ρx,τ for τ≥0, then for any τ≥0, it holds
(31)ρ∗x,τ≤φ∗x,τ.


### 2.3. Some Function Spaces and Lemmas

We denote the real valued measurable function space byL0Ω. If the generalized N-function φx,τ satisfies the strong Δ2-condition on ℍn, then
(32)Lφℍn≔f∈L0ℍn:∫ℍnφx,fxdx<∞is a Banach space with (Luxemburg) norm
(33)fLφℍn≔infλ>0:∫ℍnφx,fxλdx≤1.We call that it is a generalized Orlicz space or Musielak-Orlicz space denoting byΦwℍn. For Ω⊂ℍn, the generalized Orlicz-Sobolev space HW1,φΩ is defined as
(34)HW1,φΩ≔u:u,∇Hu∈LφΩ,and the local generalized Orlicz-Sobolev space HWloc1,φΩ as
(35)HWloc1,φΩ≔u∣u∈HW1,φΩ′,foranyΩ′⊂⊂Ω.The spaceHW01,φΩ is the closure of C0∞Ω in HW1,φΩ.IfΩ⊂ℍn is bounded and φ∈ΦwΩ satisfies the assumptions (A1) and (A3), then LφΩ↪Lγ−Ω,HW1,φΩ↪HW1,γ−Ω,HW01,φΩ↪HW01,γ−Ω. 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 (Lemmas5–11) that are similar to ones in [16] with some suitable revisions.Lemma 5.
Ifφ∈ΦwΩ, then there exists a generalized N-function ψ increasing strictly such that ψ~φ, and so, ψ is a bijection.Lemma 6.
Letφ∈ΦwΩ, then
(1)
The strongΔ2-condition is equivalent to (A4).(2)
Ifφx,τ is convex with respect to τ, then the strong Δ2-condition is equivalent to (A4)′.Lemma 7.
The assumption (A1) impliesφ−x,1~1.Lemma 8.
Letφ∈ΦwΩ be a bijection with respect to τ or satisfy the strong Δ2-condition. Then,
(1)
The assumption (A4) is equivalent to thatφ−1x,τ/τ1/γ+ is almost increasing uniformally in Ω(2)
The assumption (A3) is equivalent to thatφ−1x,τ/τ1/γ− is almost decreasing uniformally in ΩIfγ+≤℘, then we can get (A2)′ from (A1), (A2), and (A4).Lemma 9.
Letφ∈ΦwΩ satisfy the assumptions (A1), (A2), and (A4). If γ+≤℘, then φx,τ satisfies (A2)′.The proof of Lemma9 is similar to that of Lemma 12 in [16], and a simple distinct is that we should use the fact BR=R℘B1 on the Heisenberg group.Lemma 10.
Letφ∈ΦwΩ satisfy the assumption (A2) or (A2)′. Then, there exists β∈0,1 such that for any ℘Q⊂Ω, we have
(36)φQ+βτ≤φQ−τfor τ∈1,M, where M≔φQ−−11/℘Q under (A2) and M≔1/℘diamQ under (A2)′.The proof of Lemma10 is similar to that of Lemma 13 in [16], but we should employ the statement in the process that on the Heisenberg Group, if B is the smallest ball containing Q, then Q⊂B⊂℘Q.Lemma 11.
Letφ∈ΦwΩ satisfy the assumption (A2) and be a bijection. Then, there exists β∈0,1 such that for any cube Q with ℘Q⊂Ω and ℘Q≤1, we have
(37)βφ−1y,τ≤φ−1x,τfor any τ∈1,1/℘Q and x,y∈Q.For the following lemma, one can refer to [17].Lemma 12 [17].
Ifφ∈ΦwΩ, then φB− satisfies the Jensen type inequality
(38)φB−12B∫Bfdx≤1B∫BφB−fdx.In the generalized Orlicz space, the Hölder inequality with the constant 2 holds, see ([13], Lemma 9). It is stated that for any f,g∈LφΩ, it follows
(39)∫Ωfgdx≤2fφgφ∗.Because the Heisenberg Group is a special case of Carnot groups, the conclusions on Carnot groups are also true onℍn. We write some conclusions in monograph ([18], p276-280) on ℍn. For 0<α<℘, f:ℍn→ℝ, we formally define the Riesz potential operator Iα as
(40)Iαfx=∫Ωfydx,y℘−αdy,where dx,y denotes dy−1∘x. We also call that Iα is the fractional integral of order α, and I1 is abreviated to I.Lemma 13 (Hardy-Littlewood-Sobolev inequality, [18]).
Let1<α<℘, 1<p<℘/α, q>p, and
(41)1q=1p−α℘.
Then, there exists a positive constantc=cα,p such that for every f∈Lpℍn, we have
(42)Iαfq≤cfp.
For a functionf∈Lpℍn,ℂ,1<p<∞, we define the maximal function as
(43)Mfx≔supr>01Bx,r∫Bx,rfydy,x∈ℍn.
One has the statement (maximal function theorem): if1<p<∞, then there exists a positive constant c=cp such that for every f∈Lpℍn,ℂ, we have (see [18])
(44)Mfp≤cfp.Lemma 14 (Sobolev-Stein embedding, (, p280)).
Let1<p<℘. Then, there exists a constant c=cp such that
(45)uq≤c∇Hupfor every u∈C0∞ℍn,ℝ, where
(46)1q=1p−1℘i.e.q=℘p℘−p.Proof.
Foru∈C0∞ℍn,ℝ, the representation formula (5.16) in [18] yields
(47)ux=−∫ℍnΓx−1∘yLuydy,
whereL=∑j=12nXj2, Xj∗=−Xj. By the integrating by parts, we get
(48)ux=∫ℍn∇HΓx−1∘y∇Huydy.
In addition, out of the origin, one sees(49)∇HΓ=βd∇Hd2−℘=2−℘βdd1−℘∇Hd.
Because∇Hd is smooth in ℍn\0 and ∇Hd<1, we obtain that for a constant c>0,
(50)∇HΓ≤cd1−℘.
Using (48), it yields
(51)ux≤c∫ℍn∇Huydx,y1−℘dy=cI1∇Hux.
Then, by Lemma13, we gain
(52)uq≤cI1∇Huq≤c∇Hup,where
(53)1q=1p−1℘i.e.q=℘p℘−p.
This ends the proof.
Noting thatC0∞Br is dense in HW01,qBr, we have the following result from Lemma 14.Lemma 15 (Sobolev inequality).
Let1≤q<℘=2n+2, Br⊂ℍn. For any u∈HW01,qBr, it follows
(54)1Br∫Bru℘q℘−qdx℘−q℘q≤cr1Br∫Br∇Huqdx1q,where c=c℘,q>0.
SinceC∞Ω is not dense in HW1,φΩ, we need to prove.Lemma 16.
LetΩ⊂ℍn, φ∈ΦwΩ, satisfy (A1), (A2), and (A4). If v∈HW1,φΩ with sptv⊂Ω, then v∈HW01,φΩ.Proof.
Becausesptv⊂Ω and sptv are closed, we can find Ω′⊂Ω which is bounded, quasiconvex, and contains sptv. Values outside Ω′ do not affect the claim, so we verify the claim in Ω′. In order to simplify the notation, we might assume that Ω is bounded and quasiconvex . Owing to
(55)∇Hv∗η=∇Hv∗η,forη∈C0∞Ωsimilarly to the proofs of theorems 6.6 and 6.5 in [15], we know that C∞Ω∩HW1,φΩ is dense in HW1,φΩ; then there is a sequence of νi∈C∞Ω∩HW1,φΩ convergencing to v. We take a cut-off function η∈C0∞Ω with 0≤η≤1 and η≡1 in sptv and see ηνi∈C0∞Ω. Since
(56)v−ηνiHW1,φΩ=ηv−νiHW1,φΩ≤v−νiHW1,φΩ→0,it follows
(57)v∈HW01,φΩ.Lemma 17.
LetQ⊂ℍnwithQbeing finite. Ifφ∈ΦwΩ, then for all v∈HW01,φΩ, we have
(58)∫Qφv℘℘−1dx≤c∫Qφ∇Hvdx℘℘−1,where c depends only on ℘ and Q.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 ℍn.Lemma 18.
Suppose thatφ∈Φw3Q satisfies (A1), (A2), and (A3), then there exists β∈0,1 such that for all f1∈Lφ3Q with
(59)f1Lφ3Q≤1,sptf1⊂Q×0∪1,∞,we have
(60)∫Qφx,βMf1dx≤c∫Qφx,f1dx,where c depends only on ℘ and the parameters of (A1), (A2), and (A3).
The proof is similar to the proof of Lemma23 in [16].


## 2.1. Heisenberg Groupℍn

The Euclidean spaceℝ2n+1,n≥1 with the group multiplication
(12)x∘y=x1+y1,x2+y2,⋯,x2n+y2n,t+s+12∑i=1nxiyn+i−xn+iyi,wherex=x1,x2,⋯,x2n,t,y=y1,y2,⋯,y2n,s∈ℝ2n+1 leads to the Heisenberg group ℍn. The scaling on ℍn is defined as
(13)δx=δx1,δx2,⋯,δx2n,δ2t.The left invariant vector fields onℍn are of the form
(14)Xi=∂xi−xn+i2∂t,Xn+i=∂xn+i+xi2∂t,1≤i≤n,and a nontrivial commutator on ℍn is
(15)T=∂t=Xi,Xn+i=XiXn+i−Xn+iXi,1≤i≤n.We call thatX1,X2,⋯,X2n are the horizontal vector fields on ℍn and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍn by
(16)∇Hu=X1u,X2u,⋯,X2nu.The homogeneous dimension ofℍn is ℘=2n+2. The Haar measure in ℍn is equivalent to the Lebesgue measure in ℝ2n+1. We denote the Lebesgue measure of a measurable set E⊂ℍn by E. The Carnot-Carathèodary metric (CC-metric) between two points in ℍn is the shortest length of the horizontal curve joining them, denoted by d. The ball defined by the CC-metric is
(17)BRx=y∈ℍn:dy,x<R.One has(18)BRx=R℘B1x.Forx=x1,x2,⋯,x2n,t, its module is defined by
(19)xℍn=∑i=12nxi22+t214.The CC-metricd is equivalent to the Korànyi metric
(20)dx,y=x−1yℍn.


## 2.2. GeneralizedN-Function and Its Related Properties

Definition 4 (generalizedN-function).
A real valued functionφx,τ defined on Ω×0,+∞ is said to be a generalized N-function, and if φx,τ is a Lebesgue measurable with respect to x, the derivative φ′x,τofφx,τ with respect to τ exists, and φ′x,τ is right continuous, nondecreasing, and satisfies φ′x,0=0 and φ′x,τ>0τ>0.
If for anyτ≥0 and x∈Ω, there exists L≥1 such that
(21)ψx,τL≤φx,τ≤ψx,Lτ,then we say functions φ and ψ are equivalent denoted by φ≃ψ. If for any τ≥0, there exists c1>0 such that
(22)φx,2τ≤c1φx,τ;then, we say that φx,τ satisfies the strong Δ2-condition and denotes the minimum constant c1 by Δ2φ. Since
(23)φx,τ≤φx,2τ,the strong Δ2-condition is equivalent to φx,τ~φx,2τ . Obviously, if φ satisfies the strong Δ2-condition, then ~⇔≃. For a family of generalized N-functions, we define
(24)Δ2φλ≔supλΔ2φλ.
Let(25)φ′−1x,τ≔supυ∈0,+∞:φ′x,υ≤τ,forτ≥0.
Ifφ′=φ′x,τ is strictly increasing with respect to τ, then φ′−1 is the inverse function of φ′. Writing
(26)φ∗x,τ≔∫0τφ′−1x,ϑdϑ,τ≥0,then φ∗ is also a generalized N-function and satisfies
(27)φ∗′x,τ=φ′−1x,τ,τ≥0.
Note thatφ∗ is the complementary function of φ and φ∗∗=φ. For any δ>0, there exists cδ depending only on Δ2φ,φ∗, such that
(28)τυ≤δφx,τ+cδφ∗x,υ
for anyτ,υ≥0, and this inequality is called Young’s inequality ([16]). For some a,b>0 and any τ≥0, we denote
(29)ρx,τ=aφx,bτ,then
(30)ρ∗x,τ=aφ∗x,τab.
Ifφ and ρ are generalized N-functions and satisfy φx,τ≤ρx,τ for τ≥0, then for any τ≥0, it holds
(31)ρ∗x,τ≤φ∗x,τ.


## 2.3. Some Function Spaces and Lemmas

We denote the real valued measurable function space byL0Ω. If the generalized N-function φx,τ satisfies the strong Δ2-condition on ℍn, then
(32)Lφℍn≔f∈L0ℍn:∫ℍnφx,fxdx<∞is a Banach space with (Luxemburg) norm
(33)fLφℍn≔infλ>0:∫ℍnφx,fxλdx≤1.We call that it is a generalized Orlicz space or Musielak-Orlicz space denoting byΦwℍn. For Ω⊂ℍn, the generalized Orlicz-Sobolev space HW1,φΩ is defined as
(34)HW1,φΩ≔u:u,∇Hu∈LφΩ,and the local generalized Orlicz-Sobolev space HWloc1,φΩ as
(35)HWloc1,φΩ≔u∣u∈HW1,φΩ′,foranyΩ′⊂⊂Ω.The spaceHW01,φΩ is the closure of C0∞Ω in HW1,φΩ.IfΩ⊂ℍn is bounded and φ∈ΦwΩ satisfies the assumptions (A1) and (A3), then LφΩ↪Lγ−Ω,HW1,φΩ↪HW1,γ−Ω,HW01,φΩ↪HW01,γ−Ω. 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 (Lemmas5–11) that are similar to ones in [16] with some suitable revisions.Lemma 5.
Ifφ∈ΦwΩ, then there exists a generalized N-function ψ increasing strictly such that ψ~φ, and so, ψ is a bijection.Lemma 6.
Letφ∈ΦwΩ, then
(1)
The strongΔ2-condition is equivalent to (A4).(2)
Ifφx,τ is convex with respect to τ, then the strong Δ2-condition is equivalent to (A4)′.Lemma 7.
The assumption (A1) impliesφ−x,1~1.Lemma 8.
Letφ∈ΦwΩ be a bijection with respect to τ or satisfy the strong Δ2-condition. Then,
(1)
The assumption (A4) is equivalent to thatφ−1x,τ/τ1/γ+ is almost increasing uniformally in Ω(2)
The assumption (A3) is equivalent to thatφ−1x,τ/τ1/γ− is almost decreasing uniformally in ΩIfγ+≤℘, then we can get (A2)′ from (A1), (A2), and (A4).Lemma 9.
Letφ∈ΦwΩ satisfy the assumptions (A1), (A2), and (A4). If γ+≤℘, then φx,τ satisfies (A2)′.The proof of Lemma9 is similar to that of Lemma 12 in [16], and a simple distinct is that we should use the fact BR=R℘B1 on the Heisenberg group.Lemma 10.
Letφ∈ΦwΩ satisfy the assumption (A2) or (A2)′. Then, there exists β∈0,1 such that for any ℘Q⊂Ω, we have
(36)φQ+βτ≤φQ−τfor τ∈1,M, where M≔φQ−−11/℘Q under (A2) and M≔1/℘diamQ under (A2)′.The proof of Lemma10 is similar to that of Lemma 13 in [16], but we should employ the statement in the process that on the Heisenberg Group, if B is the smallest ball containing Q, then Q⊂B⊂℘Q.Lemma 11.
Letφ∈ΦwΩ satisfy the assumption (A2) and be a bijection. Then, there exists β∈0,1 such that for any cube Q with ℘Q⊂Ω and ℘Q≤1, we have
(37)βφ−1y,τ≤φ−1x,τfor any τ∈1,1/℘Q and x,y∈Q.For the following lemma, one can refer to [17].Lemma 12 [17].
Ifφ∈ΦwΩ, then φB− satisfies the Jensen type inequality
(38)φB−12B∫Bfdx≤1B∫BφB−fdx.In the generalized Orlicz space, the Hölder inequality with the constant 2 holds, see ([13], Lemma 9). It is stated that for any f,g∈LφΩ, it follows
(39)∫Ωfgdx≤2fφgφ∗.Because the Heisenberg Group is a special case of Carnot groups, the conclusions on Carnot groups are also true onℍn. We write some conclusions in monograph ([18], p276-280) on ℍn. For 0<α<℘, f:ℍn→ℝ, we formally define the Riesz potential operator Iα as
(40)Iαfx=∫Ωfydx,y℘−αdy,where dx,y denotes dy−1∘x. We also call that Iα is the fractional integral of order α, and I1 is abreviated to I.Lemma 13 (Hardy-Littlewood-Sobolev inequality, [18]).
Let1<α<℘, 1<p<℘/α, q>p, and
(41)1q=1p−α℘.
Then, there exists a positive constantc=cα,p such that for every f∈Lpℍn, we have
(42)Iαfq≤cfp.
For a functionf∈Lpℍn,ℂ,1<p<∞, we define the maximal function as
(43)Mfx≔supr>01Bx,r∫Bx,rfydy,x∈ℍn.
One has the statement (maximal function theorem): if1<p<∞, then there exists a positive constant c=cp such that for every f∈Lpℍn,ℂ, we have (see [18])
(44)Mfp≤cfp.Lemma 14 (Sobolev-Stein embedding, (, p280)).
Let1<p<℘. Then, there exists a constant c=cp such that
(45)uq≤c∇Hupfor every u∈C0∞ℍn,ℝ, where
(46)1q=1p−1℘i.e.q=℘p℘−p.Proof.
Foru∈C0∞ℍn,ℝ, the representation formula (5.16) in [18] yields
(47)ux=−∫ℍnΓx−1∘yLuydy,
whereL=∑j=12nXj2, Xj∗=−Xj. By the integrating by parts, we get
(48)ux=∫ℍn∇HΓx−1∘y∇Huydy.
In addition, out of the origin, one sees(49)∇HΓ=βd∇Hd2−℘=2−℘βdd1−℘∇Hd.
Because∇Hd is smooth in ℍn\0 and ∇Hd<1, we obtain that for a constant c>0,
(50)∇HΓ≤cd1−℘.
Using (48), it yields
(51)ux≤c∫ℍn∇Huydx,y1−℘dy=cI1∇Hux.
Then, by Lemma13, we gain
(52)uq≤cI1∇Huq≤c∇Hup,where
(53)1q=1p−1℘i.e.q=℘p℘−p.
This ends the proof.
Noting thatC0∞Br is dense in HW01,qBr, we have the following result from Lemma 14.Lemma 15 (Sobolev inequality).
Let1≤q<℘=2n+2, Br⊂ℍn. For any u∈HW01,qBr, it follows
(54)1Br∫Bru℘q℘−qdx℘−q℘q≤cr1Br∫Br∇Huqdx1q,where c=c℘,q>0.
SinceC∞Ω is not dense in HW1,φΩ, we need to prove.Lemma 16.
LetΩ⊂ℍn, φ∈ΦwΩ, satisfy (A1), (A2), and (A4). If v∈HW1,φΩ with sptv⊂Ω, then v∈HW01,φΩ.Proof.
Becausesptv⊂Ω and sptv are closed, we can find Ω′⊂Ω which is bounded, quasiconvex, and contains sptv. Values outside Ω′ do not affect the claim, so we verify the claim in Ω′. In order to simplify the notation, we might assume that Ω is bounded and quasiconvex . Owing to
(55)∇Hv∗η=∇Hv∗η,forη∈C0∞Ωsimilarly to the proofs of theorems 6.6 and 6.5 in [15], we know that C∞Ω∩HW1,φΩ is dense in HW1,φΩ; then there is a sequence of νi∈C∞Ω∩HW1,φΩ convergencing to v. We take a cut-off function η∈C0∞Ω with 0≤η≤1 and η≡1 in sptv and see ηνi∈C0∞Ω. Since
(56)v−ηνiHW1,φΩ=ηv−νiHW1,φΩ≤v−νiHW1,φΩ→0,it follows
(57)v∈HW01,φΩ.Lemma 17.
LetQ⊂ℍnwithQbeing finite. Ifφ∈ΦwΩ, then for all v∈HW01,φΩ, we have
(58)∫Qφv℘℘−1dx≤c∫Qφ∇Hvdx℘℘−1,where c depends only on ℘ and Q.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 ℍn.Lemma 18.
Suppose thatφ∈Φw3Q satisfies (A1), (A2), and (A3), then there exists β∈0,1 such that for all f1∈Lφ3Q with
(59)f1Lφ3Q≤1,sptf1⊂Q×0∪1,∞,we have
(60)∫Qφx,βMf1dx≤c∫Qφx,f1dx,where c depends only on ℘ and the parameters of (A1), (A2), and (A3).
The proof is similar to the proof of Lemma23 in [16].


## 3. Local Boundedness

Unless otherwise specified, we will use the following notations. Suppose that0∈Ω⊂ℍn,0<R<R0≤1/2,
(61)QR≔Q0,Ris a cube whose side length is R in the x′ direction, R2 in the t direction, and its edge is parallel to the coordinate axis with centered 0 and denotes diamQ≔2n2+11/4R>2n1/2R,
(62)Ak,R+≔QR∩u>k,u+≔maxu,0.Lemma 19 [16].
Suppose thatF is a bounded nonnegative function in r,R and W satisfies the strong Δ2-condition in 0,∞, if there exists θ∈0,1 such that for any r≤ι<s≤R,
(63)Fι≤W1s−ι+θFs,then we have
(64)Fr≤cW1R−r,where c depends only on θ and Δ2W.Lemma 20 (Caccioppoli inequality).
Letφ∈ΦwΩ and u∈HWloc1,φΩ be a local quasiminimizer of (1). Then for all k∈ℝ, there holds
(65)∫Ak,r+φx,∇Hu−k+dx≤c∫Ak,R+φx,u−kR−rdx,where c depends on K in definition 1 and R.The proof is similar to Lemma27 in [16].Lemma 21.
Suppose thatφ∈ΦwΩ satisfies (A1), (A2), (A3), and (A4) and define
(66)φ¯x,τ≔φQR−τχ0,1τ+φx,τχ1,∞τ.
Ifu∈HWloc1,φΩ satisfies (65), R/2≤ι<s≤R,Q3R⊂Ω, then there exists R0 such that
(67)∫Qιφ¯x,u−k+dx≤c∫Qsφ¯x,u−h+φ¯x,k−hdxα∫Qsφ¯x,u−h+s−ιdx
asR∈0,R0,k−h≥s−ι, where α≔γ−/℘2γ+−℘γ+−γ−, c, depends only on the parameters of (A1), (A2), (A3), and (A4), Δ2φ and ℘. Here, R0 satisfies ∫3QR0φx,∇Hudx≤1.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.
Leth<k,Q≔Qι+s/2ι<s, η∈C0∞Q, satisfy
(68)0≤η≤1,∇Hη≤4s−ιandη=1inQι.
Denoting(69)v≔u−k+η,A≔v>1∩Q,B≔v≤1∩Q,andD≔u>k∩Q,then we have
(70)∫Qιφ¯x,u−k+dx≤∫Qφ¯x,u−k+ηdx=∫Aφ¯x,vdx+∫Bφ¯x,vdx.
We first estimate the integral∫Aφ¯x,vdx. If τ>1, then from (A3), (A4), and Lemma 7,
(71)φ¯x,τ℘/℘−1≥cτγ−/℘−1φ¯x,τ≥cφ¯x,τφ¯x,1γ−/℘−1γ+φ¯x,τ~φ¯x,τ1+γ−/℘−1γ+≔φ¯x,τ1+ε,so
(72)φ¯x,τ≤cφ¯x,τ℘/℘−11/1+ε,ε=γ−/℘−1γ+.
We use it and the Hölder inequality to gain(73)∫Aφ¯x,vdx≤c∫Aφ¯x,v℘/℘−11/1+εdx≤cAε/1+ε∫Aφ¯x,v℘/℘−1dx1/1+ε.
Denotingf1≔∇Hvχ∇Hv>1, then we know that from (51),
(74)v≤c1I∇Hv≤c1I∇Hvχ∇Hv≤1+c1If1.
IfR0 is small enough, then
(75)I∇Hvχ∇Hv≤1≤IχQ≤cR0≤12c1.
Because ofv>1 in the set A, it yields from (74) and (75) that
(76)v≤12+c1If1≤v2+c1If1.
Therefore, in the setA, it implies
(77)v≤2c1If1.
Suppose thatR0 is small enough such that 3QR0≤1, then from Lemma 13 and f1Lφ3Q≤cMf1, we know
(78)If1℘/℘−1≤cMf1.
Noting(79)f1Lφℍn=f1Lφ3Q≤∇HuLφ3Q≤1,
the conditions of Lemma18 are satisfied. Now, we combine (77), (78), the strong Δ2-condition and Lemma 18 to obtain
(80)∫Aφ¯x,v℘/℘−1dx≤c∫Aφ¯x,If1℘/℘−1dx≤c∫Aφ¯x,Mf1dx≤c∫Aφ¯x,∇Hvdx.
Denoting(81)φ~τ≔φQR−τχ0,1τ+τγ−φQR+1χ1,∞τ,
we see from (A3) that for everyx,
(82)φ~τ≤φQR−τχ0,1τ+φx,τφx,1φQR+1χ1,∞τ≤φQR−τχ0,1τ+φx,τχ1,∞τ=φ¯x,τ.
Let us estimate the measure ofA. When v∈A, we have χAv=1,
(83)φ~v℘/℘−1=vγ−φQR+1℘/℘−1≥φx,v℘/℘−1≥1,and deduce from Lemmas 16, 17, and (82) that
(84)A≤∫Qφ~v℘/℘−1dx≤c∫Qφ~∇Hvdx℘/℘−1≤c∫Qφ¯x,∇Hvdx℘/℘−1.
On the other hand,A≤D, so
(85)Aε/1+ε≤Dε/℘1+εAε℘−1/℘1+ε.
Combining (73), (80), (85), (84), and φ¯≤φ, we know
(86)∫Aφ¯x,vdx≤cDε/℘1+εAε℘−1/℘1+ε∫Aφ¯x,∇Hvdx1/1+ε≤cDα∫Qφx,∇Hvdx,
whereα=ε/℘1+ε.
Next, we estimate the integral∫Bφ¯x,vdx. Observing that φ¯=φ~ and they do not depend on x in the set B, the usual Hölder inequality yields
(87)∫Bφ¯x,vdx≤∫Dφ~vdx≤D1/℘∫Qφ~v℘/℘−1dx℘−1/℘.
Because ofD≤Q≤1,α=ε/℘1+ε<1/℘, one has D1/℘≤Dα. Using Lemmas 16, 17, and cφ~≤φ¯≤φ, it deduces
(88)∫Bφ¯x,vdx≤Dα∫Qφx,∇Hvdx.
In conclusion, the integrals∫Aφ¯x,vdx and ∫Bφ¯x,vdx have same upper bound. At present, we estimate the integral ∫Qφx,∇Hvdx . Using the
(89)∇Hv≤∇Hu−k++∇Hηu−k+,∇Hη≤4s−ι,strong Δ2-condition and (65), we obtain
(90)∫Qφx,∇Hvdx≤c∫Qφx,∇Hu−k+dx+∫Dφx,u−k+s−ιdx≤c∫Qsφx,u−k+s−ιdx.
Note thatu>k in D, so u−h>k−h≥s−ι. Thus,
(91)φx,u−k+s−ι≤φx,u−h+s−ιχD≤φ¯x,u−h+s−ι.
Sinceφ¯ is increasing, it follows
(92)D=∫QsχDdx≤∫Qsφ¯x,u−h+φ¯x,k−h+dx.
Combining (86)–(92), we get
(93)∫Qφ¯x,vdx≤c∫Qsφ¯x,u−h+φ¯x,k−h+dxα∫Qsφ¯x,u−h+s−ιdx.
Returning to (70), it reaches (67).Lemma 22 [16].
Letα>0 and Θi be a sequence of real numbers satisfying
(94)Θi+1≤DℓiΘi1+α
forD>0,ℓ>1. If
(95)Θ0≤D−1αℓ−1α2,then Θi→0 as i→∞.Lemma 23.
Letφ∈ΦwΩ satisfy (A1), (A2), (A3), and (A4). If u∈HWloc1,φΩ satisfies (65) and R0∈0,1 as shown in Lemma 21, then for 0<R≤R0 and any k0∈ℝ, we have
(96)esssupQR/2u≤k0+1+cR−γ+αγ−∫QRφx,u−k0+dx1/γ−,
whereα as shown in Lemma 21, c depends only on the parameters of (A1), (A2), (A3), and (A4), Δ2φ and ℘.
The proof is similar to Theorem 4.11 in [16].We use (96) for u and −u to immediately obtain the following.Theorem 24 (local boundedness).
Letφ∈ΦwΩ 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.
Letφ∈ΦwΩ satisfy (A1) and (A4) and define
(97)φ~x,τ≔τγ+φ−1χ0,1τ+φx,τχ1,∞τ.
Ifu∈HWloc1,φΩ satisfies (65), R/2≤ι<s≤R,QR⊂Ω, then we have
(98)∫Qιφ~QR−u−k+dx≤c∫Qsφ~x,u−h+φ~x,k−hdx1℘∫Qsφ~x,u−h+s−ιdxas k−h≥s−ι, where c depends only on the parameters of (A1) and (A4), Δ2φ and ℘.The proof is similar to the proof of Lemma31 in [16], and a necessary change is to replace the Sobolev inequality in the Euclidean space to the inequality in Lemma 17.Fors>0, define
(99)usx≔usxs=usx1,⋯,sx2n,s2ts,(100)φsx,fx≔φsx,fx.Note that(101)∇Husx=∇Husx,Δ2φs=Δ2φand the constants of (A2) and (A2)′ will be changed under this scaling (100).Lemma 26.
Letφ∈ΦwΩ. Ifu∈HWloc1,φΩ is a local quasiminimizer of (1) in QsR with s∈0,1,QsR⊂⊂Ω, then us is a local quasiminimizer of the functional ∫φsx,∇Hudx in QR.Proof.
Supposevs∈HW1,φQR and sptvs⊂QR, obviously, we have
(102)v≠0=svs≠0⊂QsR.
Using the transformation of variable and the quasiminimality ofu in QsR, it implies
(103)∫vs≠0φsx,∇Husxdx=∫vs≠0φsx,∇Husxdx=s−℘∫v≠0φz,∇Huzdz≤Ks−℘∫v≠0φz,∇Hu+vzdz=K∫vs≠0φsx,∇Hus+vsxdx.Lemma 27.
Letφ∈ΦwΩ satisfy (A1), (A2)′, and (A4). If u∈HWloc1,φΩ is a bounded local quasiminimizer of (1) in QsR, then for 0<R<R0<1/32n℘u∞,s∈0,1,QsR⊂⊂Ω,k0>−2usL∞QR, we have
(104)esssupQR/2us−k0≤c∫QRus−k0+γ+dx1/γ++R,where R0 depends only on ℘ and uL∞QsR, c depends only on the parameters of (A1), (A2)′, and (A4), ℘,R,Δ2φ, and uL∞QsR does not depend on s.Proof.
The proof follows the line proving proposition 28 in [16]. Writing
(105)φ~x,τ≔τγ+φ−1χ0,1τ+φx,τχ1,∞τ.
Assumingτ0∈0,3u∞/s, we claim
(106)φ~sQR+βτ0≤φ~sQR−τ0⇔φ~QsR+βτ0≤φ~QsR−τ0.
In fact, obviously, it holds(107)φ~sQR+τ=supx∈QRφ~sx,τ=supx∈QRτγ+φs−1χ0,1τ+φsx,τχ1,∞τ=τγ+infx∈QRφsx,1χ0,1τ+supx∈QRφsx,τχ1,∞τ=1sτγ+infx∈QsRφx,1χ0,1τ+supx∈QsRφx,τχ1,∞τ=1sφ~QsR+τ.
Similarly, we haveφ~sQR−τ=1/sφ~QsR−τ, so two inequalities in (106) are equivalent.
In the sequel, we prove the latter of (106). Let us consider cases τ0<1/β and τ0≥1/β.
(1)
Ifτ0<1/β, then(108)φ~QsR+βτ0=βτ0γ+φ−1.
On one hand,βτ0γ+φ−1≤τ0γ+φ−1; on the other hand, (A4) and definition 4 yield
(109)βτ0γ+φ−1≤φ−βτ0≤φ−τ0,so
(110)φ~QsR+βτ0≤τ0γ+φ−1χ0,1τ0+φQsR−τ0χ1,∞τ0=φ~QsR−τ0.(2)
Ifτ0≥1/β, then(111)φ~QsR+βτ0=φQsR+βτ0,φ~QsR−τ0=φQsR−τ0.
By Lemma10, for τ0∈1,1/℘diamQsR, we have φQsR+βτ0≤φQsR−τ0; so when 3u∞/s≤1/℘diamQsR<1/2n℘sR (i.e., R0<1/32n℘u∞), it holds
(112)φQsR+βτ0≤φQsR−τ0.
Thus, the latter of (106) is proved.
Denote(113)d0≔max∫QRus−k0+γ+dx1/γ+,R,d≔Md0,where M≥1 is to be determined. Note d0∈0,3u∞/s by using k0>−2usL∞QR. Denoting
(114)ψ≔φ~sQR+,it deduces by the almost increasing of φx,τ, (106), and the strong Δ2-condition that
(115)φ~sx,d≥LMφ~sx,d0≥LMψβd0≥cLMψd0.
Fori∈ℕ, we take
(116)ki≔k0+d1−2−i,σi≔R21+2−i,and see ki+1−ki≥σi+1−σi. Defining
(117)ϕi≔∫Qσiφ~sx,us−ki+dxand applying the strong Δ2-condition and (106), we get
(118)ϕi+1=∫Qσi+1φ~sx,us−ki+1+dx≤c∫Qσi+1φ~sx,βus−ki+1+dx≤c∫Qσi+1φ~sQR+βus−ki+1+dx≤c∫Qσi+1φ~sQR−us−ki+1+dx.
By Lemma26, us is a local quasiminimizer, so we know from Lemma 20 that us satisfies (65). Hence, using (118) and Lemma 25 yields
(119)ϕi+1≤c∫Qσi+1φ~sQR−us−ki+1+dx≤c∫Qσiφ~sx,us−ki+φ~sx,ki+1−kidx1/℘∫Qσiφ~sx,us−ki+σi−σi+1dx=c∫Qσiφ~sx,us−ki+φ~sx,d2−i−1dx1/℘∫Qσiφ~sx,us−ki+R2−i−2dx.
By (A4), definition4, (106), and the strong Δ2-condition, we have
(120)φ~sx,Md02−i−1≥cM2−i+1γ+φ~sx,d0≥cM2−i+1γ+φ~s−d0≥cM2−i+1γ+φ~s+βd0≥cM2−i+1γ+ψd0,φ~sx,us−ki+R2−i−2≤cR2−i−2−γ+φ~sx,us−ki+,so it deduces from (119) that
(121)ϕi+1≤c∫Qσiφ~sx,us−ki+M2−i+1γ+ψd0dx1/℘∫QσiR2−i−2−γ+φ~sx,us−ki+dx=c2−i+1γ+Mψd0−1/℘R2−i−2−γ+ϕi1+1/℘=c2iγ+1+1/℘M−1/℘ψd0−1/℘R−γ+ϕi1+1/℘.
Selectα≔1/℘,D≔cR−γ+M−αψd0−α,ℓ≔21+αγ+ in Lemma 22. If
(122)ϕ0≤c2−1+αα2γ+Rγ+αMψd0,then we have by Lemma 22 that
(123)us≤k∞=k0+Md0almost everywhere in Qσ∞=QR/2.
The remaining job is to prove (122). We will point out that ϕ0/ψd0 has a bound not depending on s and then choose M such that (4.9) holds. Since τ−γ+φ~sx,τ is almost decreasing, it follows that τ−γ+ψτ is also almost decreasing, and τ−1ψ−1τγ+ is almost increasing by Lemma 8. Then, we know from ([15], Lemma 5) that ψ−1γ+ is equivalent to a convex function ς. Because of φ~s≤ψ, it yields by the Jensen inequality that
(124)ψ−1ϕ0=ψ−1∫QRφ~sx,us−k0+dx≤ψ−1∫QRψus−k0+dx~ς∫QRψus−k0+dx1/γ+≤∫QRςψus−k0+dx1/γ+~∫QRus−k0+γ+dx1/γ+≤d0.
Sinceψ satisfies the strong Δ2-condition, we have
(125)ϕ0≤ψcd0≤cψd0,and (122) is proved.Theorem 28.
Letφ∈ΦwΩ satisfy (A1), (A2)′ and (A4), and u∈HWloc1,φΩ be a bounded local quasiminimizer of (1). When 0<r<R0,k∈ℝ, we have
(126)esssupQr/2u−k≤c1Qr∫Qru−k+γ+dx1γ++r,where c depends only on the parameters of (A1), (A2)′, and (A4), Δ2φ, ℘,R0 and uL∞Qr.
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.
Letφ∈ΦwΩ satisfy (A1), (A2)′, and (A4) and u∈HWloc1,φΩ be a bounded local quasiminimizer of (1). When 0<r<R0,k∈ℝ, we have for any ε∈0,1,
(127)esssupQεRu−k≤c11−ε℘QR∫QRu−k+γ+dx1γ++R,where c does not depend on R and ε.
The proof is similar to Corollary 5.8 in [16].Corollary 30.
Letφ∈ΦwΩ satisfy (A1), (A2)′, and (A4) and u∈HWloc1,φΩ be a bounded local quasiminimizer of (1). When 0<r<R0,k∈ℝ, we have for any q∈0,∞,
(128)esssupQR/2u−k≤c1QR∫QRu−k+qdx1q+R,where c does not depend on R and depends only on the parameters of (A1), (A2)′, and (A4), Δ2φ, ℘,R0, and uL∞Qr.
The proof is similar to Corollary 5.9 in [16].


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

Denote(129)Dθ≔u<θ∩QR.Lemma 31.
Ifu≥0, −u satisfies (128) with q=1, c=c1,k=−θ, and for some θ>0,
(130)Dθ≤12c1QR,then
(131)essinfQR2u+c1R≥θ2.
The proof is similar to Lemma 6.1 in [16].Lemma 32.
Letφ∈ΦwΩ satisfy (A1), (A2), (A2)′, (A4), and (A3), and u∈HWloc1,φΩ be a bounded local quasiminimizer of (1), R0 as shown in Lemma 27. If u≥0, and for all r∈0,R0, θ>0, and κ∈0,1, there exists μ>0 such that
(132)Dθ≤κQR;then, we have
(133)essinfQR2u+cR≥μθ.
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 thatQR⊂ℍn,E and G(E⊂G⊂QR) are measurable sets, if there exists 0<δ<1 such that
(134)E≤δQR;(1)
For any cubeQ with Q∩E≥δQ, it holds Q′⊂G, where Q′ is a subset of Q
Then,(135)E≤δG.Proof.
From the Calderón-Zygmund decomposition Theorem ([20], p17), we know that for every f∈L1QR and 0<δ<1, there is a sequence of disjoint cubes Qj such that for almost all x∈QR\∪jQj, we have
(136)fx≤δandδ≤1Qj∫Qjfdx<2℘δ.
If we takef=χE, then
(137)E⊂∪jQj∪N,N=0andδ≤E∩QjQj<2℘δ.
Let us decomposeQj until the cube Q~j satisfies E∩Q~j/Q~j<δ; so,
(138)E\N⊂∪jQ~j⊂G.
Hence, it follows(139)E≤∑jE∩Q~j≤δ∑jQ~j≤δG.Lemma 34.
Supposeu≥0 and for all R∈0,R0 and every κ>0, there exists μ>0 such that if
(140)Dθ≤κQR,then
(141)essinfQR2u+R≥μθ;then, there exists q>0 such that for every R∈0,R0, we have
(142)1QR∫QRuqdx1q≤cessinfQR2u+R.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 0<δ<1,κ≔1−δ and write
(143)Esμi,z,R≔y∈QRz:uy+R>sμi.
Take(144)E=Esi≔y∈QR:uy+R>sμi,G=Esi+1in Lemma 5.3, where μ is the constant in (141) with respect to κ. Suppose that for some r<R and z∈QR, we have
(145)Qrz∩Esi≥δQrz.
Therefore, it follows under (145) that
(146)δQr≤Qrz∩Esi=Qrz∩QR∩uy+R>sμi=QR∩Qrz∩uy+R>sμi=QR∩Esμi,z,r≤Esμi,z,r,Dsμi,z,r=uy<sμi∩Qrz≤uy+r<sμi∩Qrz≤Qr−Esμi,z,r≤1−δQr=κQr,so (140) holds. Thus,
(147)essinfQz,r2u+r≥sμi+1.
From the above discussion, we can infer (147) from the hypothesis (145), so Qr/2z∩QR⊂Esi+1. 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 Esi+1=QR, which implies
(148)u+R≥essinfQR/2u+R≥sμi+1.
If (1) in Lemma33 holds, then Esi+1≥δ−1Esi, which gives
(149)Esj≥δ−1Esj−1≥δ−2Esj−2≥⋯≥δ−jEs0.
IfEs0>0, we choose j to be the smallest integral satisfying j≥1/logδlogEs0/QR, and then
(150)δjQR≤δ1logδlogEs0QRQR=δlogEs0−logQRlogδQR=Es0≤δjEsj,so Esj≥QR, i.e., Esj=QR, Thus, as (1), we have
(151)essinfQR/2u+R≥sμj.
Combining (1) and (2), we obtain(152)essinfQR/2u+R≥sμj+1=csμ1logδlogEs0QR=csμlogEs0QR1logδ=csEs0QRlogμlogδ.
If we writeξ=essinfQR/2u+R,a=logδ/logμ, then
(153)Es0≤cQRξas−a.
Taking0<q<a, it gets
(154)∫QRu+Rqdx=q∫0∞sq−1Es0ds=q∫0ξsq−1Es0ds+q∫ξ∞sq−1Es0ds=q∫0ξsq−1QRds+q∫ξ∞sq−1Es0ds≤QRξq+cQRξa∫ξ∞sq−a−1ds=cQRξq,so
(155)1QR∫QRuqdx1/q≤cessinfQR/2u+R.
Thus, (142) is proved.Lemma 35 (weak Harnack inequality).
Letφ∈ΦwΩ satisfy (A1), (A2), (A2)′, (A4), and (A3). If u∈HWloc1,φΩ is a nonnegative local quasiminimizer of (1), R0 as shown in Lemma 27, then for all R∈0,R0 and k∈ℝ, we have
(156)1QR∫QRuqdx1q≤cessinfQR2+R,where c depends only on the parameters of (A1), (A2), (A2)′, (A4), and (A3), ℘,R0, and uL∞QR.Proof.
According to Lemma32, we see that the conditions of Lemma 34 are satisfied, so (156) is true by using Lemma 34.Proof of Theorem 2.
Combining Corollary30 and Lemma 35, we prove immediately Theorem 2.Proof of Theorem 3.
The Harnack inequality in Theorem2 implies the Hölder continuity.



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