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

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 
 
 p
 
 -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.

It is clear that the Euler-Lagrange equation corresponding to functional (1) is where φ ′ ðx, τÞ denotes the derivative of φðx, τÞ with respect to τ. If for any v ∈ HW then we say that u ∈ HW 1,φ ðΩÞ is a weak solution of (6). When functional Φðx, j∇ H ujÞ in (1) satisfies Φðx, τÞð Λ + τ 2 Þ p−2/2 τ, 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 C 1,α regularity of weak solutions by using a double-bootstrap method, energy estimates, and interpolation inequalities; for 1 < p < ∞, Zhong in [7] got the C 1,α regularity of weak solutions by using the energy estimate, the Moser iteration, and the oscillation estimate; Zhang and Niu in [8] proved the Γ α regularity of the gradient of weak solutions as Φðx, τÞ~φðτÞ, where φðτÞ belongs to the Orlicz space including the function φðτÞ = ðΛ + τ 2 Þ p−2/2 τ.
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.
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 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 Then, Φðx, j∇ H ujÞ in (1) can have the concrete relations 2 Journal of Function Spaces In this paper, we always denote a positive constant by c which may vary from line to line, x = ðx 1 ,⋯,x 2n , tÞ = ðx′, tÞ. We assume that Ω ⊂ ℍ n is a bounded domain, Q is a cube whose side length is R in the x′ direction, R 2 in the t direction, and its edge is parallel to the coordinate axis and denote diamQ ≔ ðð2nÞ 2 + 1Þ 1/4 R > ð2nÞ 1/2 R. Let cQ be a concentric cube whose side length is c times Q the in x′ direction and

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

Heisenberg Group
where x = ðx 1 , x 2 ,⋯,x 2n , tÞ, y = ðy 1 , y 2 ,⋯,y 2n , sÞ ∈ ℝ 2n+1 leads to the Heisenberg group ℍ n . The scaling on ℍ n is defined as The left invariant vector fields on ℍ n are of the form and a nontrivial commutator on ℍ n is We call that X 1 , X 2 , ⋯, X 2n are the horizontal vector fields on ℍ n and T the vertical vector field. Denote the horizontal gradient of a smooth function u on ℍ n by The homogeneous dimension of ℍ n is ℘ = 2n + 2. The Haar measure in ℍ n is equivalent to the Lebesgue measure in ℝ 2n+1 . We denote the Lebesgue measure of a measurable set E ⊂ ℍ n by jEj. The Carnot-Carathèodary metric (CCmetric) between two points in ℍ n is the shortest length of the horizontal curve joining them, denoted by d. The ball defined by the CC-metric is One has For x = ðx 1 , x 2 ,⋯,x 2n , tÞ, its module is defined by The CC-metric d is equivalent to the Korànyi metric
If for any τ ≥ 0 and x ∈ Ω, there exists L ≥ 1 such that then we say functions φ and ψ are equivalent denoted by φ ≃ ψ. If for any τ ≥ 0, there exists c 1 > 0 such that then, we say that φðx, τÞ satisfies the strong Δ 2 -condition and denotes the minimum constant c 1 by Δ 2 ðφÞ. Since the strong Δ 2 -condition is equivalent to φðx, τÞ~φðx, 2τÞ . Obviously, if φ satisfies the strong Δ 2 -condition, then~⇔ ≃ . For a family of generalized N-functions, we define If φ ′ = φ ′ ðx, τÞ is strictly increasing with respect to τ, then φ * is also a generalized N-function and satisfies Note that φ * is the complementary function of φ and ðφ * Þ * = φ. For any δ > 0, there exists c δ depending only on for any τ, υ ≥ 0, and this inequality is called Young's inequality ( [16]). For some a, b > 0 and any τ ≥ 0, we denote then If φ and ρ are generalized N-functions and satisfy φðx, τÞ ≤ ρðx, τÞ for τ ≥ 0, then for any τ ≥ 0, it holds 2.3. Some Function Spaces and Lemmas. We denote the real valued measurable function space by L 0 ðΩÞ. If the generalized N-function φðx, τÞ satisfies the strong Δ 2 -condition on ℍ n , then is a Banach space with (Luxemburg) norm We call that it is a generalized Orlicz space or Musielak-Orlicz space denoting by Φ w ðℍ n Þ. For Ω ⊂ ℍ n , the generalized Orlicz-Sobolev space HW 1,φ ðΩÞ is defined as and the local generalized Orlicz-Sobolev space HW 1,φ loc ðΩÞ as The space HW 0 1,φ ðΩÞ is the closure of C ∞ 0 ðΩÞ in H W 1,φ ðΩÞ.
We now describe their proofs (Lemmas 5-11) that are similar to ones in [16] with some suitable revisions.
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 jB R j = R ℘ jB 1 j on the Heisenberg group. Lemma 10. Let φ ∈ Φ w ðΩÞ satisfy the assumption (A2) or (A2)′. Then, there exists β ∈ ð0, 1Þ such that for any 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 B is the smallest ball containing Q, then Q ⊂ B ⊂ ffiffiffi ℘ p Q.
For the following lemma, one can refer to [17].
Lemma 12 [17]. If φ ∈ Φ w ðΩÞ, then φ − B 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 f , g ∈ L φ ðΩÞ, it follows Because the Heisenberg Group is a special case of Carnot groups, the conclusions on Carnot groups are also true on ℍ n . We write some conclusions in monograph ( [18], p276-280) on ℍ n . For 0 < α < ℘, f : ℍ n → ℝ, we formally define the Riesz potential operator I α as where dðx, yÞ denotes dðy −1 ∘ xÞ. We also call that I α is the fractional integral of order α, and I 1 is abreviated to I.
Lemma 13 (Hardy-Littlewood-Sobolev inequality, [18]). Let 1 < α < ℘, 1 < p < ℘/α, q > p, and Then, there exists a positive constant c = cðα, pÞ such that for every f ∈ L p ðℍ n Þ, we have For a function f ∈ L p ðℍ n , ℂÞ, 1 < p < ∞, we define the maximal function as One has the statement (maximal function theorem): if 1 < p < ∞, then there exists a positive constant c = cðpÞ such that for every f ∈ L p ðℍ n , ℂÞ, we have (see [18]) Lemma 14 (Sobolev-Stein embedding, (, p280)). Let 1 < p < ℘ . Then, there exists a constant c = cðpÞ such that for every u ∈ C ∞ 0 ðℍ n , ℝÞ, where Proof. For u ∈ C ∞ 0 ðℍ n , ℝÞ, the representation formula (5.16) in [18] yields where L = ∑ 2n j=1 X 2 j , X * j = −X j . By the integrating by parts, we get In addition, out of the origin, one sees Because ∇ H d is smooth in ℍ n \ f0g and j∇ H dj < 1, we obtain that for a constant c > 0, Using (48), it yields Then, by Lemma 13, we gain where This ends the proof. Noting that C ∞ 0 ðB r Þ is dense in HW 1,q 0 ðB r Þ, we have the following result from Lemma 14.
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 ℍ n . Lemma 18. Suppose that φ ∈ Φ w ð3QÞ satisfies (A1), (A2), and (A3), then there exists β ∈ ð0, 1Þ such that for all f 1 ∈ L φ ð3QÞ with we have where c depends only on ℘ and the parameters of (A1), (A2), and (A3). The proof is similar to the proof of Lemma 23 in [16].

Local Boundedness
Unless otherwise specified, we will use the following notations. Suppose that 0 ∈ Ω ⊂ ℍ n , 0 < R < R 0 ≤ 1/2, is a cube whose side length is R in the x′ direction, R 2 in the t direction, and its edge is parallel to the coordinate axis with centered 0 and denotes diamQ ≔ ðð2nÞ 2 + 1Þ 1/4 R > ð2nÞ 1/2 R, Lemma 19 [16]. Suppose that F is a bounded nonnegative function in ½r, R and W satisfies the strong Δ 2 -condition in ½0, ∞Þ, if there exists θ ∈ ½0, 1Þ such that for any r ≤ ι < s ≤ R, then we have where c depends only on θ and Δ 2 ðWÞ.
Lemma 20 (Caccioppoli inequality). Let φ ∈ Φ w ðΩÞ and u ∈ HW 1,φ loc ðΩÞ be a local quasiminimizer of (1). Then for all k ∈ ℝ, there holds where c depends on K in definition 1 and R.
The proof is similar to Lemma 27 in [16].

Journal of Function Spaces
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.
The proof is similar to Theorem 4.11 in [16].
We use (96) for u and −u to immediately obtain the following.