Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 27, pp. 1–15.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SECOND ORDER SOBOLEV REGULARITY FOR p-HARMONIC

FUNCTIONS IN SU(3)

CHENGWEI YU

Abstract. Let u be a weak solution to the degenerate subelliptic p-Laplacian

equation

∆H,pu(x) =

6∑
i=1

Xi(|∇Hu|p−2Xiu) = 0,

where H is the orthogonal complement of a Cartan subalgebra in SU(3) and

its orthonormal basis is composed of the vector fields X1, . . . , X6. We prove
that when 1 < p < 7/2, the solution u has the second order horizontal Sobolev

W 2,2
H,loc-regularity.

1. Introduction

We consider the group SU(3), that is, the special unitary group of 3×3 complex
matrices endowed with a horizontal vector field ∇H = {X1, X2, . . . , X6}. Let Ω be
a domain in SU(3) and 1 < p <∞. We call a function u as a p-harmonic function

in Ω if u ∈ W 1,p
H,loc(Ω) is a weak solution to the degenerate subelliptic p-Laplacian

equation

∆H,pu(x) =

6∑
i=1

Xi(|∇Hu|p−2Xiu) = 0 in Ω, (1.1)

that is, ∫
Ω

6∑
i=1

|∇Hu|p−2XiuXiφdx = 0, φ ∈ C∞0 (Ω),

where ∇Hu = (X1u,X2u, . . . ,X6u) is the horizontal gradient of a function u ∈
C1(Ω), W 1,p

H,loc(Ω;R) is the first order p-th integrable horizontal local Sobolev space,

that is, all functions u ∈ Lploc(Ω) with its distributional horizontal gradient ∇Hu ∈
Lploc(Ω), see Section 2 for more details.

When p = 2, the p-harmonic functions in SU(3) are usually called as harmonic
functions, and are always smooth as proved by Hörmander [8]. When p 6= 2, for
p-harmonic functions u in SU(3) satisfying

0 < M−1 ≤ |∇Hu|(x) ≤M a.e. in Ω, (1.2)

2020 Mathematics Subject Classification. 35H20, 35B65.

Key words and phrases. p-Laplacian equation; SU(3); W 2,2
H,loc-regularity; Hessian matrix;

p-harmonic function.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted December 2, 2021. Published April 6, 2022.

1



2 C. YU EJDE-2022/27

Domokos-Manfredi [4] also proved that u ∈ C∞. However without assumption
(1.2), one can not expect that u ∈ C∞. Recently, for general p-harmonic function
in SU(3), Domokos-Manfredi [3] built the C0,1-regularity and, when 2 ≤ p < ∞,
the C1,α-regularity.

This article aims to establish the following second order Sobolev regularity for
p-harmonic functions u in SU(3) as below, that is, u ∈ W 2,2

H,loc(Ω). Here for any

function v we say v ∈W 2,2
H,loc(Ω) if v ∈W 1,2

H,loc(Ω) and its second order distributional

horizontal derivative ∇H∇Hv = (XiXjv)1≤i,j≤6 ∈ L2
loc(Ω). For convenience, for

φ ∈ C∞0 (Ω) we write

Kφ = 1 + ‖∇Hφ‖2L∞(Ω) + ‖φ∇T φ‖L∞(Ω). (1.3)

Theorem 1.1. Let 1 < p < 7/2. If u is a p-harmonic function in a domain

Ω ⊂ SU(3), then u ∈ W 2,2
H,loc(Ω). Moreover, when 1 < p ≤ 2, for any φ ∈ C∞0 (Ω)

with 0 ≤ φ ≤ 1, we have∫
Ω

φ2|∇H∇Hu|2dx ≤ c
∫
spt(φ)

|∇Hu|2−pdx+ cK2
φ

∫
spt(φ)

|∇Hu|p+2dx; (1.4)

when 2 < p < 7/2, for any φ ∈ C∞0 (Ω) with 0 ≤ φ ≤ 1, we have∫
Ω

φ6|∇H∇Hu|2dx ≤ cK3
φ

∫
spt(φ)

|∇Hu|p+2dx+ cKφ

∫
Ω

φ4|∇Hu|p−2dx

+ c

∫
Ω

φ6|∇Hu|4−pdx,
(1.5)

where c = c(p) is a positive constant.

Recall that, for p-harmonic functions in Euclidean spaces, their C1,α-regularity
has been established by [18, 17, 7, 9, 16]. Their Sobolev W 2,2-regularity with 1 <
p < 3+ 2

n−2 was proved in [12] (see also [6]). In particular, for p-harmonic functions

in R6, the range of p to get their Sobolev W 2,2
loc -regularity is also 1 < p < 7/2, but

when 7
2 ≤ p < ∞, it remains open to get their W 2,2

loc -regularity; see [6] for more

details. Moreover, for p-harmonic functions in Heisenberg group Hn, their C0,1 and
C1,α-regularity has been established in [2, 5, 11, 13, 15, 19, 14]. If 1 < p ≤ 4 when

n = 1 and 1 < p < 3 + 1
n−1 when n ≥ 2, their horizontal Sobolev HW 2,2

loc -regularity

was established in [5, 10].
To prove Theorem 1.1, it is standard to consider the regularized equation of

subelliptic p-Laplacian equation as did in [3]. To be precise, let u be a p-harmonic
function in Ω. Given any smooth domain U b Ω and δ ∈ (0, 1], denote by uδ ∈
W 1,p
H (U) the weak solution to the regularized equation

6∑
i=1

Xi[(δ + |∇Hv|2)
p−2
2 Xiv] = 0 in U, v − u ∈W 1,p

H,0(U). (1.6)

As for the existence, uniqueness and C∞-regularity of uδ, we refer the reader to
[4, 3] and references therein. It was proved by Domokos-Manfredi [3] (see Theorem
2.3 below) that ∇Huδ ∈ L∞loc(U) uniformly in δ ∈ (0, 1] and also that uδ → u in
C0(U) as δ → 0.

To show Theorem 1.1, it suffices to prove that {uδ}δ∈(0,1] have the following

W 2,2
H,loc(Ω)-regularity uniformly in δ ∈ (0, 1]. Indeed, sending δ → 0, from which

one can conclude Theorem 1.1 in a standard way.



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 3

Theorem 1.2. Let 1 < p < 7/2. If uδ ∈ W 1,p
H,loc(U) is the weak solution to (1.6),

then uδ ∈ W 2,2
H,loc(U) uniformly in δ ∈ (0, 1]. Moreover, when 1 < p ≤ 2, for any

φ ∈ C∞0 (U) with 0 ≤ φ ≤ 1, we have∫
U

φ2|∇H∇Huδ|2dx ≤ c
∫
spt(φ)

(δ + |∇Huδ|2)
2−p
2 dx

+ cK2
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx;

(1.7)

when 2 < p < 7/2, for any φ ∈ C∞0 (U) with 0 ≤ φ ≤ 1, we have∫
U

φ6|∇H∇Huδ|2dx

≤ cK3
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx+ cKφ

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx

+ c

∫
U

φ6(δ + |∇Huδ|2)
4−p
2 dx,

(1.8)

where Kφ is as in (1.3) and the constant c = c(p) > 0.

Below, we outline the idea for proving Theorem 1.2. Our proof is based on
several a priori estimates for uδ established in [3]; see Lemmas 2.1 and 2.2. We
consider two cases: 1 < p ≤ 2 and 2 < p < 7/2.

When 1 < p ≤ 2, we conclude (1.7) from Lemmas 2.1 and 2.2 in a direct way.
In the case 2 < p ≤ 7/2, to obtain (1.8) we use some ideas from [6, 10] to

decompose the horizontal Hessian matrix and then combine a priori estimates in
[3]. We proceed as below. For simplicity we write the subelliptic 2-Laplacian
as ∆0v = ∆0,2v, and write the symmetrization of horizontal hessian ∇H∇Hv =
(XiXjv)1≤i,j≤6 as

D2
0v :=

(XiXjv +XjXiv

2

)
1≤i,j≤6

.

First, the following lemma gives a pointwise estimate of |D2
0u
δ|2, which is inferred

from a fundamental inequality in [6, Lemma 2.1]. See Section 4 for details.

Lemma 1.3. Let 1 < p < 7/2. If uδ ∈ W 1,p
H,loc(U) is the weak solution to (1.6).

Then

|D2
0u
δ|2 ≤ c[|D2

0u
δ|2 − (∆0u

δ)2] in U, (1.9)

where the constant c = c(p) > 0.

Next, we bound the integral of the right-hand side of (1.9); see Section 3 for
details. We denote by ∇T v := (X7v,X8v) the vertical derivative of v.

Lemma 1.4. For any v ∈ C∞(U) and any φ ∈ C∞0 (U), we have∣∣ ∫
U

[|D2
0v|2 − (∆0v)2]φ6dx

∣∣
≤ c

∫
U

|∇Hv|2φ6dx+ c

∫
U

|∇Hv||∇H∇T v|φ6dx

+ c

∫
U

|∇Hv||∇H∇Hv||φ|5[|∇Hφ|+ |φ|]dx,

(1.10)

where c is a positive constant.



4 C. YU EJDE-2022/27

In regards to the term ∫
U

φ6|∇Huδ||∇H∇T uδ|dx

appearing in the right hand side of (1.10), applying some Caccoippoli type in-
equalities established in [3] (see Lemmas 2.1 and 2.2), we have the following upper
bound.

Lemma 1.5. Let 2 < p ≤ 4. If uδ ∈W 1,p
H,loc(U) is the weak solution to (1.6), then

for any φ ∈ C∞0 (U) with 0 ≤ φ ≤ 1, we have∫
U

φ6|∇Huδ||∇H∇T uδ|dx

≤ cK3
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx+ cKφ

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx

+

∫
U

φ6(δ + |∇Huδ|2)
4−p
2 dx,

(1.11)

where Kφ is as in (1.3) and the constant c = c(p) > 0.

On the other hand, we are going to bound |∇H∇Hv|2 via |D2
0v|2 from above.

Denote by Mv the difference between ∇H∇Hv and D2
0v, that is

Mv := ∇H∇Hv −D2
0v =

(XiXjv −XjXiv

2

)
1≤i,j≤6

=
( [Xi, Xj ]v

2

)
1≤i,j≤6

.

Since M is an anti-symmetric matrix (mi,j = −mj,i), we obtain

|∇H∇Hv|2 = |D2
0v|2 + |Mv|2.

We bound the integration of |Mv|2 as follows.

Lemma 1.6. For any v ∈ C∞(U) and any φ ∈ C∞0 (U), we have∫
U

|Mv|2φ6dx ≤ 6

∫
U

|∇Hv||∇H∇T v|φ6dx+ 36

∫
U

|∇Hv||∇T v||φ5∇Hφ|dx

+

∫
U

|∇Hv|2φ6dx.

(1.12)

Finally, combining Lemmas 1.3, 1.4, 1.5 and 1.6, we conclude (1.8) for 2 < p <
7/2.

2. Preliminaries

We recall the special unitary group of 3× 3 complex matrices

{g ∈ GL(3,C) : g · g∗ = I, det g = 1}

as the group SU(3) and define its Lie algebra by

su(3) := {X ∈ gl(3, C) : X +X∗ = 0, trX = 0}.

From this, we give the inner product on SU(3) by

〈X,Y 〉 := −1

2
tr(XY ).



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 5

On the other hand, we note that the two-dimensional maximal torus on SU(3)
is given by the set

T :=
{eia1 0 0

0 eia2 0
0 0 eia3

 : a1, a2, a3 ∈ R, a1 + a2 + a3 = 0
}
.

Then we choose its Lie algebra as the Cartan subalgebra, that is,

T :=
{ia1 0 0

0 ia2 0
0 0 ia3

 : a1, a2, a3 ∈ R, a1 + a2 + a3 = 0
}
.

According to the definition of SU(3), we can obtain its orthonormal basis composed
of the following Gell Mann matrices G:

X1 =

 0 1 0
−1 0 0
0 0 0

 , X2 =

0 i 0
i 0 0
0 0 0

 , X3 =

0 0 0
0 0 1
0 −1 0

 ,

X4 =

0 0 0
0 0 −i
0 −i 0

 , X5 =

 0 0 1
0 0 0
−1 0 0

 , X6 =

0 0 i
0 0 0
i 0 0

 ,

T1 =

−i 0 0
0 i 0
0 0 0

 , T2 =

−i/√3 0 0

0 −i/
√

3 0

0 0 2i/
√

3

 .

Note that T1 and T2 can be generated by the following two vector fields:

X7 = −[X1, X2] =

−2i 0 0
0 2i 0
0 0 0

 , X8 = −[X3, X4] =

0 0 0
0 2i 0
0 0 −2i

 ,

which form an orthonormal basis of the Cartan subalgebra∇T = {X7, X8}. Table 1
provides all the commutators of the vector fields X1, X2, . . . , X8.

Table 1. Commutators in SU(3)

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 −X7 X5 −X6 −X3 X4 4X2 2X2

X2 X7 0 X6 X5 −X4 −X3 −4X1 −2X1

X3 −X5 −X6 0 −X8 X1 X2 2X4 4X4

X4 X6 −X5 X8 0 X2 −X1 −2X3 −4X3

X5 X3 X4 −X1 −X2 0 X8 −X7 2X6 −2X6

X6 −X4 X3 −X2 X1 X7 −X8 0 −2X5 2X5

X7 −4X2 4X1 −2X4 2X3 −2X6 2X5 0 0
X8 −2X2 2X1 −4X4 4X3 2X6 −2X5 0 0

Consider the orthonormal basis of the horizontal subspace H in SU(3); that is,

∇H = {X1, X2, . . . , X6}.



6 C. YU EJDE-2022/27

Note that the matrices G are left-invariant vector fields. According to Table 1, the
basis ∇H satisfies the Hörmander condition at every point of SU(3) and produces
the horizontal distribution of a sub-Riemannian manifold.

We say that the curve γ : [0, T ] → SU(3) is subunitary associated to ∇H if the
following two conditions are met: the curve γ is an absolutely continuous function;
there are measurable functions {αi ∈ L∞[0, T ]}1≤i≤6 such that

γ′(t) =

6∑
i=1

αi(t)Xi(γ(t)) and

6∑
i=1

α2
i (t) ≤ 1 for a.e. t ∈ [0, T ].

Since at every point of SU(3) the basis ∇H satisfies the Hörmander condition, by
[1], for any two given points x, y ∈ SU(3) there exist subunitary curves γ connecting
them. As a result, we define the Carnot-Carathéodory distance in regard to ∇H by

d(x, y) = inf
{
T ≥ 0 : there exists a subunitary curve γ : [0, T ]→ SU(3)

connecting x and y}.

With respect to this distance d, we define the Carnot-Carathéodory balls centered
at x ∈ SU(3) with radius r > 0 by

Br(x) = {y ∈ SU(3) : d(x, y) < r}.

We denote by dx the bi-invariant Harr-measure, by |E| the Lebesgue measure of a
measurable set E ⊂ SU(3) and by

−
∫
E

f dx =
1

|E|

∫
E

f dx

the average of an integrable function f over set E.
In the rest of this section, we recall several a priori uniform estimates for reg-

ularized equation by Domokos-Manfredi [3]; see [3, Corollary 4.1]. Let u be a
p-harmonic function in a domain Ω ⊂ SU(3), where 1 < p <∞. Given any smooth

domain U b Ω and δ ∈ (0, 1], denote by uδ ∈ W 1,p
H (U) the weak solution to the

regularized equation (1.6). We have the following result.

Lemma 2.1. For any φ ∈ C∞0 (U) with 0 ≤ φ ≤ 1, the followings hold:

(i) If β ≥ 0, then∫
U

φ2(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2β |∇H∇T uδ|2dx

≤ c
∫
U

|∇Hφ|2(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2β+2dx

+ c(β + 1)2

∫
U

φ2(δ + |∇Huδ|2)
p
2 |∇T uδ|2βdx.

(2.1)

(ii) If β ≥ 0, then∫
U

φ2(δ + |∇Huδ|2)
p−2
2 +β |∇H∇Huδ|2dx

≤ c(β + 1)4

∫
U

φ2(δ + |∇Huδ|2)
p−2
2 +β |∇T uδ|2dx

+ c(β + 1)2Kφ

∫
spt(φ)

(δ + |∇Huδ|2)
p
2 +βdx,

(2.2)



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 7

(iii) If β ≥ 1, then∫
U

φ2β+2(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2β |∇H∇Huδ|2dx

≤ cβ(β + 1)4β‖∇Hφ‖2βL∞(U)

∫
U

φ2(δ + |∇Huδ|2)
p−2
2 +β |∇H∇Huδ|2dx.

(2.3)

(iv) If β ≥ 1, then∫
U

φ2(δ + |∇Huδ|2)
p−2
2 +β |∇H∇Huδ|2dx

≤ c(β + 1)12Kφ

∫
spt(φ)

(δ + |∇Huδ|2)
p
2 +βdx.

(2.4)

Above Kφ is as in (1.3) and constants c = c(p) > 0.

Combining (2.3) and (2.4), we obtain the following result.

Lemma 2.2. For any β ≥ 1 and any φ ∈ C∞0 (U) with 0 ≤ φ ≤ 1, we have∫
U

φ2β+2(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2β |∇H∇Huδ|2dx

≤ cβ(β + 1)12+4βKβ+1
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p
2 +βdx,

(2.5)

where Kφ is as in (1.3) and the constant c = c(p) > 0.

Moreover, Domokos-Manfredi [3] further established the following uniform gra-
dient estimate and also convergence. We also write u0 = u.

Theorem 2.3. We have ∇Huδ ∈ L∞loc(U ;R6) uniformly in δ ∈ [0, 1) and, for any
ball B2r ⊂ U ,

‖∇Huδ‖L∞(Br) ≤ c(p)
(
−
∫
B2r

(δ + |∇Huδ|2)
p
2

)1/p

. (2.6)

Moreover, uδ → u in C0(Ū).

3. Proofs of lemmas

In this section, we prove Lemmas 1.3, 1.4, 1.5, and 1.6. To prove Lemma 1.3
we need the following pointwise inequality from [6, Lemma 2.1]. To simplify the
following proofs, we write the subelliptic ∞-Laplacian ∆0,∞v of v ∈ C∞ as

∆0,∞v =

6∑
i,j=1

XivXiXjvXjv = (∇Hv)T∇H∇Hv∇Hv = (∇Hv)TD2
0v∇Hv.

Lemma 3.1. For any v ∈ C∞(U), we have∣∣∣|D2
0v∇Hv|2 −∆0v∆0,∞v −

1

2
[|D2

0v|2 − (∆0v)2]|∇Hv|2
∣∣∣

≤ 2[|D2
0v|2|∇Hv|2 − |D2

0v∇Hv|2] in U.
(3.1)

Proof. For each point x̄ ∈ U , we assume that ∇Hv(x̄) 6= 0 below, otherwise (3.1)
obviously holds. In the case ∇Hv(x̄) 6= 0, we may also assume that |∇Hv(x̄)| = 1
below, otherwise we divide both sides by |∇Hv(x̄)|2.



8 C. YU EJDE-2022/27

At x̄, since D2
0v(x̄) is a symmetric matrix, by the linear algebra theory, we

obtain a set of eigenvalues based on the matrix D2
0v(x̄), that is, {λi}6i=1 ⊂ R.

Then according to the linear algebra theory again, there is an orthogonal matrix
O ∈ O(6) such that

OTD2
0vO = diag{λ1, λ2, . . . , λ6}.

Noting that O−1 = OT , we have

|D2
0v|2 = |OTD2

0vO|2 =

6∑
i=1

(λi)
2, ∆0v =

6∑
i=1

λi.

For simplicity, we write OT∇Hv =
∑6
i=1 aiei =: ~a. Thus

∆0,∞v = (∇Hv)TD2
0v∇Hv = (OT∇Hv)T (OTD2

0vO)(OT∇Hv) =

6∑
i=1

λi(ai)
2,

|D2
0v∇Hv|2 = |(OTD2

0vO)(OT∇Hv)|2 =

6∑
i=1

(λi)
2(ai)

2.

By [6, Lemma 2.2] with ~λ := (λ1, λ2, . . . , λ6) and ~a := OT∇Hv, we have∣∣∣|D2
0v∇Hv|2 −∆0v∆0,∞v −

1

2
[|D2

0v|2 − (∆0v)2]|∇Hv|2
∣∣∣

=
∣∣∣ 6∑
i=1

(λi)
2(ai)

2 −
( 6∑
i=1

λi

)[ 6∑
j=1

λj(aj)
2
]
− 1

2

[ 6∑
i=1

(λi)
2 − (

6∑
i=1

λi)
2
]∣∣∣

≤ 2
[ 6∑
i=1

(λi)
2 −

6∑
i=1

(λi)
2(ai)

2
]

= 2[|D2
0v|2|∇Hv|2 − |D2

0v∇Hv|2],

which implies (3.1). �

Now we apply Lemma 3.1 to prove Lemma 1.3.

Proof of Lemma 1.3. Noting that uδ ∈ C∞(U), dividing both sides of (1.6) by

(δ + |∇Huδ|2)
p−4
2 , we have

(p− 2)∆0,∞u
δ + (δ + |∇Huδ|2)∆0u

δ = 0 in U. (3.2)

For any point x̄ ∈ U , we consider two cases: ∇Huδ(x̄) = 0 and ∇Huδ(x̄) 6= 0. In
the case ∇Huδ(x̄) = 0, since

∆0,∞u
δ(x̄) = (∇Huδ(x̄))T∇H∇Huδ(x̄)∇Huδ(x̄) = 0,

Equality (3.2) implies that ∆0u
δ(x̄) = 0. Thus (1.9) holds.

Now we prove (1.9) in the case ∇Huδ(x̄) 6= 0. Applying Lemma 3.1 with v = uδ

and multiplying both sides by (p− 2)2, at x̄ we have

(p− 2)2|D2
0u
δ∇Huδ|2 − (p− 2)2∆0u

δ∆0,∞u
δ

− (p− 2)2

2
[|D2

0u
δ|2 − (∆0u

δ)2]|∇Huδ|2

≤ 2(p− 2)2[|D2
0u
δ|2|∇Huδ|2 − |D2

0v∇Huδ|2].

(3.3)



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 9

Combining (3.3) and (3.2), at x̄ we have

(p− 2)2|D2
0u
δ∇Huδ|2 + (p− 2)(∆0u

δ)2[|∇Huδ|2 + δ]

− (p− 2)2

2
[|D2

0u
δ|2 − (∆0u

δ)2]|∇Huδ|2

≤ 2(p− 2)2[|D2
0u
δ|2|∇Huδ|2 − |D2

0u
δ∇Huδ|2].

By dividing both sides by |∇Huδ(x̄)|2, at x̄ we have

3(p− 2)2 |D2
0u
δ∇Huδ|2

|∇Huδ|2
+ (p− 2)

(∆0u
δ)2

|∇Huδ|2
[|∇Huδ|2 + δ]

≤ (p− 2)2

2
[|D2

0u
δ|2 − (∆0u

δ)2] + 2(p− 2)2|D2
0u
δ|2.

(3.4)

Recalling that

∆0,∞u
δ = (∇Huδ)TD2

0u
δ∇Huδ,

by Hölder’s inequality and (3.2), at x̄ we have

(p− 2)2 |D2
0u
δ∇Huδ|2

|∇Huδ|2
≥ (p− 2)2 |∆0,∞u

δ|2

|∇Huδ|4
≥ (∆0u

δ)2

|∇Huδ|2
[|∇Huδ|2 + δ]. (3.5)

Here we apply Hölder’s inequality to estimate the first inequality in (3.5), and apply
(3.2) to estimate the second inequality.

Combining (3.4) and (3.5), we have

(p+ 1)
( ∆0u

δ

|∇Huδ|

)2

[|∇Huδ|2 + δ] ≤ (p− 2)2

2
[|D2

0u
δ|2 − (∆0u

δ)2] + 2(p− 2)2|D2
0u
δ|2.

Thus

(p+ 1)(∆0u
δ)2 ≤ (p− 2)2

2
[|D2

0u
δ|2 − (∆0u

δ)2] + 2(p− 2)2|D2
0u
δ|2.

From this, subtracting [(p+1)(∆0u
δ)2− (p+1−2(p−2)2)|D2

0u
δ|2] from both sides,

we have

[p+ 1− 2(p− 2)2]|D2
0u
δ|2 ≤

[
p+ 1 +

(p− 2)2

2

]
[|D2

0u
δ|2 − (∆0u

δ)2].

Noting that 1 < p < 7/2 implies

p+ 1− 2(p− 2)2 = (p− 1)(7− 2p) > 0,

we conclude (1.9). �

Proof of Lemma 1.4. For simplicity we write the right-hand side of (1.10) as

R :=c

∫
U

|∇Hv|2φ6dx+ c

∫
U

|∇Hv||∇H∇T v|φ6dx

+ c

∫
U

|∇Hv||∇H∇Hv||φ|5[|∇Hφ|+ |φ|]dx.
(3.6)

Recall that

D2
0v =

(XiXjv +XjXiv

2

)
1≤i,j≤6

, ∆0v =

6∑
i=1

XiXiv.



10 C. YU EJDE-2022/27

Then

[|D2
0v|2 − (∆0v)2]

=

6∑
i,j=1

(XiXjv +XjXiv

2

)2

−
( 6∑
i=1

XiXiv
)2

=

6∑
i,j=1

[1

4
[(XiXjv)2 + (XjXiv)2 + 2XiXjvXjXiv]−XiXivXjXjv

]

=
1

4

6∑
i,j=1

[(XiXjv)2 −XiXivXjXjv] +
1

4

6∑
i,j=1

[(XjXiv)2 −XiXivXjXjv]

+
1

2

6∑
i,j=1

[XiXjvXjXiv −XiXivXjXjv]

=
1

2

6∑
i,j=1

[(XiXjv)2 −XiXivXjXjv]

+
1

2

6∑
i,j=1

[XiXjvXjXiv −XiXivXjXjv].

(3.7)

By this, to prove (1.10), we only need to prove that, for 1 ≤ i, j ≤ 6,∫
U

[(XiXjv)2 −XiXivXjXjv]φ6dx ≤ R, (3.8)∫
U

[XiXjvXjXiv −XiXivXjXjv]φ6dx ≤ R, (3.9)

where R is as in (3.6).
First, we prove (3.8). Integrating by parts, we have∫

U

(XiXjv)2φ6dx = −
∫
U

XjvXiXiXjvφ
6dx− 6

∫
U

XjvXiXjvφ
5Xiφdx.

Since XiXj = XjXi + [Xi, Xj ], we have∫
U

XjvXiXiXjvφ
6dx =

∫
U

XjvXiXjXivφ
6dx+

∫
U

XjvXi[Xi, Xj ]vφ
6dx.

Combining the above two equalities, since XiXj = XjXi + [Xi, Xj ] again, we have∫
U

(XiXjv)2φ6dx =−
∫
U

XjvXjXiXivφ
6dx− 6

∫
U

XjvXiXjvφ
5Xiφdx

−
∫
U

Xjv[Xi, Xj ]Xivφ
6dx−

∫
U

XjvXi[Xi, Xj ]vφ
6dx.

Integrating by parts again, we have∫
U

(XiXjv)2φ6dx =

∫
U

XjXjvXiXivφ
6dx+ 6

∫
U

XjvXiXivφ
5Xjφdx

− 6

∫
U

XjvXiXjvφ
5Xiφdx−

∫
U

Xjv[Xi, Xj ]Xivφ
6dx

−
∫
U

XjvXi[Xi, Xj ]vφ
6dx.

(3.10)



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 11

Table 1 shows that

[Xi, Xj ] =

8∑
k=1

cki,jXk for any i, j ∈ {1, 2, . . . , 8} (3.11)

and that

[Xi, Xj ]Xi =

8∑
k=1

cki,jXkXi

=

8∑
k=1

cki,j (XiXk + [Xk, Xi])

=

8∑
k=1

cki,j

(
XiXk +

8∑
m=1

cmk,iXm

)
for i, j ∈ {1, 2, . . . , 8},

(3.12)

where cki,j and cmk,i are constants and are completely determined by Table 1. Com-

bining (3.10), (3.11) and (3.12), then subtracting
∫
U
XiXivXjXjvφ

6dx from both
sides, by the fact

|∇T v|2 ≤ 2|∇H∇Hv|2,

we obtain (3.8).
Finally, we prove (3.9) in a similar way. Integrating by parts, we have∫
U

XiXjvXjXivφ
6dx = −

∫
U

XjvXiXjXivφ
6dx− 6

∫
U

XjvXjXivφ
5Xiφdx.

Since XiXj = XjXi + [Xi, Xj ], we have∫
U

XjvXiXjXivφ
6dx =

∫
U

XjvXjXiXivφ
6dx+

∫
U

Xjv[Xi, Xj ]Xivφ
6dx.

Combining the above two equalities, by integration by parts again, we have∫
U

XiXjvXjXivφ
6dx

=

∫
U

XjXjvXiXivφ
6dx−

∫
U

Xjv[Xi, Xj ]Xivφ
6dx

+ 6

∫
U

XjvXiXivφ
5Xjφdx− 6

∫
U

XjvXjXivφ
5Xiφdx.

(3.13)

We combine (3.12) and (3.13). Then subtracting
∫
U
XiXivXjXjvφ

6dx from both
sides, by the fact that

|∇T v|2 ≤ 2|∇H∇Hv|2,

we obtain (3.9). �

Proof of Lemma 1.5. Since 2 < p ≤ 4, by Young’s inequality, we have∫
U

φ6|∇Huδ||∇H∇T uδ|dx

≤
∫
U

φ6(δ + |∇Huδ|2)
p−2
2 |∇H∇T uδ|2dx+

∫
U

φ6(δ + |∇Huδ|2)
4−p
2 dx.

(3.14)



12 C. YU EJDE-2022/27

By (2.1) in Lemma 2.1 with β = 0 and φ→ φ3 therein, we have∫
U

φ6(δ + |∇Huδ|2)
p−2
2 |∇H∇T uδ|2dx

≤ c‖∇Hφ‖2L∞(U)

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2dx

+ c

∫
U

φ6(δ + |∇Huδ|2)
p
2 dx.

(3.15)

By Young’s inequality again, that |∇T uδ|2 ≤ 2|∇H∇Huδ|2 and Lemma 2.2 with
β = 1 therein, we have∫

U

φ4(δ + |∇Huδ|2)
p−2
2 |∇T uδ|2dx

≤
∫
U

φ4(δ + |∇Huδ|2)
p−2
2 |∇T uδ|4dx+

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx

≤ cK2
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx+

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx.

(3.16)

Here we apply Young’s inequality to estimate the first inequality in (3.16), and
apply the fact |∇T uδ|2 ≤ 2|∇H∇Huδ|2 and Lemma 2.2 to estimate the second
inequality.

We combine (3.15) and (3.16). Then by Young’s inequality, we have∫
U

φ6(δ + |∇Huδ|2)
p−2
2 |∇H∇T uδ|2dx

≤ cK3
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx+ cKφ

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx.

(3.17)

Combining (3.17) and (3.14), we conclude (1.11). �

Proof of Lemma 1.6. Recall that

Mv =
( [Xi, Xj ]v

2

)
1≤i,j≤6

.

According to Table 1, we have

|Mv|2 =
1

2
[(X7v)2 + (X8v)2 + (X8v −X7v)2] + |∇Hv|2

= (X7v)2 + (X8v)2 −X7vX8v + |∇Hv|2.
Since

2|X7vX8v| ≤ (X7v)2 + (X8v)2,

it remains to bound the integration of (X7v)2 and the integration of (X8v)2.
First, we bound the integration of (X7v)2. Since X7 = −[X1, X2], integration

by parts yields∫
U

(X7v)2φ6dx =

∫
U

(X2X1v −X1X2v)X7vφ
6dx

=

∫
U

X2vX1X7vφ
6dx−

∫
U

X1vX2X7vφ
6dx

+ 6

∫
U

X2vX7vφ
5X1φdx− 6

∫
U

X1vX7vφ
5X2φdx.



EJDE-2022/27 SECOND ORDER SOBOLEV REGULARITY 13

Thus∫
U

(X7v)2φ6dx ≤ 2

∫
U

|∇Hv||∇H∇T v|φ6dx+ 12

∫
U

|∇Hv||∇T v||φ5∇Hφ|dx.

Finally, we bound the integration of (X8v)2 in the same way. Combining these
together, we conclude (1.12). �

4. Proofs of main results

Proof of Theorem 1.2. We consider two cases: 1 < p ≤ 2 and 2 < p < ∞. When
1 < p ≤ 2, applying (2.2) in Lemma 2.1 with β = (2− p)/2 ≥ 0, we have∫

U

φ2|∇H∇Huδ|2dx ≤ cKφ

∫
spt(φ)

(δ + |∇Huδ|2)dx+ c

∫
U

φ2|∇T uδ|2dx. (4.1)

By Young’s inequality, the fact |∇T uδ| ≤ 2|∇H∇Huδ| and Lemma 2.2 with β = 1,
we have∫

U

φ2|∇T uδ|2dx

=

∫
U

φ2(δ + |∇Huδ|2)
2−p
4 (δ + |∇Huδ|2)

p−2
4 |∇T uδ|2dx

≤
∫
spt(φ)

(δ + |∇Huδ|2)
2−p
2 dx+

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 |∇T uδ|4dx

≤
∫
spt(φ)

(δ + |∇Huδ|2)
2−p
2 dx+ cK2

φ

∫
spt(φ)

(δ + |∇Huδ|2)
p
2 +1dx.

(4.2)

Here we apply Young’s inequality to estimate the first inequality in (4.2), and
apply Lemma 2.2 to estimate the second inequality. Combining (4.1) and (4.2), by
Young’s inequality therein, we obtain (1.7).

Now, we consider the case 2 ≤ p < 7/2. Recalling that

|∇H∇Huδ|2 = |D2
0u
δ|2 + |Muδ|2,

by Lemmas 1.3, 1.4, and 1.6, we have∫
U

|∇H∇Huδ|2φ6dx ≤ c
∫
U

|∇Huδ|2φ6dx+ c

∫
U

|∇Huδ||∇H∇T uδ|φ6dx

+ c

∫
U

|∇Huδ||∇H∇Huδ||φ|5[|∇Hφ|+ |φ|]dx.
(4.3)

To obtain (1.8), it remains to estimate the second term in the right-hand of (4.3).
By Lemma 1.5, (4.3) becomes∫

U

|∇H∇Huδ|2φ6dx

≤ c
∫
U

|∇Huδ|2φ6dx+ c

∫
U

|∇Huδ||∇H∇Huδ||φ|5[|∇Hφ|+ |φ|]dx

+ cK3
φ

∫
spt(φ)

(δ + |∇Huδ|2)
p+2
2 dx+ cKφ

∫
U

φ4(δ + |∇Huδ|2)
p−2
2 dx

+ c

∫
U

φ6(δ + |∇Huδ|2)
4−p
2 dx.

By Young’s inequality, we obtain (1.8). �



14 C. YU EJDE-2022/27

Proof of Theorem 1.1. Let Ω be a domain in SU(3). Consider any p-harmonic

function u ∈ W 1,p
H,loc(Ω). Given any smooth domain U b Ω, for p ∈ (1,∞) and

δ ∈ (0, 1], we let uδ ∈ W 1,p
H (U) be a weak solution to (1.6). By Theorem 1.2, we

have that

∇Huδ ∈W 1,2
H,loc(U) uniformly in δ ∈ (0, 1]. (4.4)

Theorem 2.3 shows that

uδ → u in C0(U) as δ → 0, (4.5)

∇Huδ ∈ L∞(U) uniformly in δ ∈ (0, 1]. (4.6)

Combining (4.4) and (4.5), we have

∇Huδ → ∇Hu weakly in W 1,2
H,loc(U) and in L2

loc(U) as δ → 0. (4.7)

By (4.6) and Hölder’s inequality, (4.7) implies that

∇Huδ → ∇Hu in Lqloc(U) for 0 < q <∞ as δ → 0.

By letting δ → 0 in (1.7) and (1.8), we can obtain (1.4) and (1.5). �

Acknowledgments. The author would like to express his gratitude to Yuan Zhou,
Jiayin Liu, and Fa Peng for their fruitful discussions. This work is supported by
grants from the NSF of China, 12025102 and 11871088.

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Chengwei Yu
School of Mathematical Sciences, Beihang University, Haidian District, Beijing 100191,

China

Email address: chengweiyu@buaa.edu.cn


	1. Introduction
	2. Preliminaries
	3. Proofs of lemmas
	4. Proofs of main results
	Acknowledgments

	References