# High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

Authors: Pengcheng Niu; Kelei Zhang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

Let{
X
1
,
X
2
,
…
,
X
m
} be the basis of space of horizontal vector fields in a Carnot group G
=
(
R
n
;
∘
)
(
m
<
n
). We prove high order Fefferman-Phong type inequalities in G. As applications, we derive a priori L
p
(
G
) estimates for the nondivergence degenerate elliptic operators L
=
-
∑
i
,
j
=
1
m
a
i
j
(
x
)
X
i
X
j
+
V
(
x
) with V
M
O coefficients and a potential V belonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.


---

## Body

## 1. Introduction and the Main Results

The classicalL
p estimates for nondivergence elliptic operators with potentials of the form
(1)
L
u
≡
-
∑
i
,
j
=
1
n
‍
a
i
j
x
u
x
i
x
j
+
V
u
,
x
∈
R
n
,
have been extensively investigated and many results have been proved; see [1–5] and so forth. In particular, when (
a
i
j
)
n
×
n
=
I, the identity matrix, and V belongs to the reverse Hölder class B
q
(
n
/
2
≤
q
<
∞
), Shen [2] established L
p
(
1
<
p
≤
q
) boundedness for the Schrödinger operator -
Δ
+
V and showed that the range of p is optimal. It is noted that V
∈
B
q
(
q
>
1
) means that V
∈
L
loc
⁡
q
(
R
n
), V
≥
0, and there exists a positive constant c such that the reverse Hölder inequality
(2)
B
-
1
∫
B
‍
V
q
x
d
x
1
/
q
≤
c
B
-
1
∫
B
‍
V
x
d
x
holds for every ball B in R
n. More recently, when V
∈
B
q
(
n
/
2
≤
q
<
∞
), a priori L
p
(
R
n
)
(
1
<
p
≤
q
) estimate for L in (1) with V
M
O coefficients has been deduced by Bramanti et al. [1] by using the representation formula for V
u in terms of L
u, which generalized the result in [2]. The aim of this paper is to establish high order Fefferman-Phong type inequalities in Carnot groups and prove L
p regularity of degenerate elliptic operators plus a potential.LetX
1
,
X
2
,
…
,
X
m be horizontal vector fields in a Carnot group G
=
(
R
n
;
∘
), (
m
<
n
) (see Section 2.1). In this paper we consider the nondivergence degenerate elliptic operator of the kind
(3)
L
u
≡
A
u
+
V
u
≡
-
∑
i
,
j
=
1
m
‍
a
i
j
x
X
i
X
j
u
+
V
u
,
where the leading coefficient a
i
j
(
x
) satisfies a
i
j
(
x
)
=
a
j
i
(
x
)
∈
L
∞
(
G
) for i
,
j
=
1
,
…
,
m, and there exists a constant μ
>
0 such that, for any x
∈
G and ξ
∈
R
m,
(4)
μ
ξ
2
≤
∑
i
,
j
=
1
m
‍
a
i
j
(
x
)
ξ
i
ξ
j
≤
μ
-
1
ξ
2
;
furthermore, we assume
(5)
a
i
j
(
x
)
∈
V
M
O
G
,
which shows that, for i
,
j
=
1
,
…
,
m,
(6)
η
i
j
=
sup
⁡
ρ
≤
r
sup
⁡
x
∈
G
B
ρ
x
-
1
∫
B
ρ
x
‍
a
i
j
y
-
a
i
j
B
d
y
⟶
0
,
η
i
j
r
⟶
0
+
.
Here a
i
j
B
=
B
ρ
(
x
)
-
1
∫
B
ρ
(
x
)
‍
a
i
j
(
y
)
d
y. B
ρ
(
x
) denotes a metric ball of radius r and center x associated with the Carnot-Carathéodory distance d (see Section 2) by
(7)
B
ρ
(
x
)
=
B
(
x
,
ρ
)
=
{
y
∈
G
:
d
(
x
,
y
)
<
ρ
}
.As to the potentialV, inspired by Di Fazio and Zamboni [6, Definition 2.4], we introduce the following Stummel type class S
p.Definition 1 (Stummel type class).
LetV
:
G
→
R, 1
<
p
<
∞, r
>
0. One says that V
∈
S
p
(
G
), if for every r
>
0,
(8)
φ
V
r
≔
sup
⁡
x
∈
G
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
∫
d
(
x
,
y
)
<
r
‍
d
x
,
y
2
B
x
,
d
x
,
y
×
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
∫
d
z
,
x
<
r
‍
V
z
×
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
d
y
p
-
1
is finite and
(9)
lim
⁡
r
→
0
φ
V
(
r
)
=
0
,
where d
(
·
,
·
) is the Carnot-Carathéodory distance; see Section 2.Sometimes we will callφ
V
(
r
) the Stummel modulus of V.Remark 2.
We note thatL
∞
(
G
)
⊂
S
p
(
G
)
(
1
<
p
<
∞
) and S
p
(
G
) is the special case of the function class in [7, page 56] with p
=
2 and G
=
R
n
(
n
≥
5
). Also, note that the function V
(
x
)
=
d
(
0
,
x
)
-
2 on G belongs to the classes S
p
(
G
) for p
>
2, where d is the Carnot-Carathéodory distance (see Section 2).Nondivergence degenerate elliptic operators similar to (3) including the form -
∑
i
=
1
m
‍
X
i
2
+
V have been studied by some authors; see [8–11] and so forth. The local L
p estimate for operator (3) with the vanishing potential V
=
0 on the homogeneous group has been verified by Bramanti and Brandolini [12]. For the study of related operators, we refer to [13, 14] and references therein. We will prove regularity for the operator L in (3) on G if a
i
j satisfy (4)-(5) and V
p
∈
S
p
(
G
); see Theorem 3 below. Our methods are different from the Euclidean case by Bramanti et al. [1], where estimates of integral operators and their commutators were used as a main tool.Since Fefferman [15] proved the well-known imbedding inequality
(10)
∫
R
n
‍
V
u
2
d
x
≤
c
∫
R
n
‍
∇
u
2
d
x
,
u
∈
C
0
∞
R
n
,
with V belonging to the classical Morrey class L
r
,
n
-
2
r, 1
<
r
≤
n
/
2, it has been extended to many more general settings and applied to infer regularity for partial differential operators; see [6, 16–19] and so forth. One of the main jobs of this paper is to establish a high order Fefferman-Phong type inequality in Carnot groups (see Theorem 4), which is motivated by Di Fazio and Zamboni [6, Theorem 3.1]. So far as we know, there is not any result in literature on high order Fefferman-Phong inequalities. Using this inequality and proving several estimates with the potential, a priori L
p
(
G
) estimate for L is obtained.We mention that the homogeneous dimensionQ of G, the horizontal gradient X
u, the second order horizontal gradient X
2
u, the horizontal Sobolev spaces H
W
2
,
p
(
G
) and H
W
V
2
,
p
(
G
), the polynomial P
B
(
x
), and the reverse Hölder class B
q in our setting will be described in Section 2. Now we are in a position to state main results.Theorem 3.
Under the assumptions (4)-(5), if V
p
∈
S
p
(
G
)
,
1
<
p
<
∞, then there exists a positive constant c
=
c
(
p
,
μ
,
η
,
V
,
G
) such that, for any u
∈
C
0
∞
G, it follows that
(11)
u
H
W
2
,
p
(
G
)
+
V
u
L
p
(
G
)
≤
c
L
u
L
p
(
G
)
+
u
L
p
(
G
)
,
where η in c depends only on the V
M
O moduli η
i
j of the coefficients a
i
j. Furthermore, (11) holds for u
∈
H
W
V
2
,
p
(
G
).It is noted that theL
p estimates of the operators similar to (3) with discontinuous leading coefficients and bounded lower terms were obtained by Bramanti and Brandolini [12, 20]. Here the potential V in Theorem 3 may be unbounded on G.The key for the proof of Theorem3 is the following high order Fefferman-Phong type inequality.Theorem 4.
LetB
=
B
r
B
(
x
0
) be any metric ball in G. If V
∈
S
p
(
G
)
(
1
<
p
<
∞
), then there exists a first order polynomial P
B
(
x
) such that, for any u
∈
C
∞
(
B
), one has
(12)
∫
B
‍
u
-
P
B
(
x
)
p
V
d
x
≤
c
φ
V
(
2
r
B
)
∫
B
‍
X
2
u
p
d
x
,
where the positive constant c is independent of u and B. Moreover, for any u
∈
C
0
∞
(
B
), one has
(13)
∫
B
‍
u
p
V
d
x
≤
c
φ
V
(
2
r
B
)
∫
B
‍
X
2
u
p
d
x
,
where c
>
0 is independent of u and B.The aboveX
2
u is a set of X
i
X
j
u for all i
,
j
=
1
,
…
,
m. We will define X
2
u precisely in Section 2.Remark 5.
The main difference between Theorem4 and [6, Theorem 3.1] is clear; that is, the right-hand side term X
u
L
p
(
B
) in [6] is replaced by X
2
u
L
p
(
B
) here. Of course, the class involving V is not the same.We observe an important relation between the Stummel class here and the reverse Hölder class: ifV
∈
B
q
∩
L
1
(
G
), q
>
Q
/
2, then V
p
∈
S
p
(
G
), 1
<
p
≤
2
q
/
Q. From it and Theorem 3, the following result follows.Theorem 6.
Under the same assumptions ona
i
j as in Theorem 3, if V
∈
B
q
∩
L
1
(
G
), q
>
Q
/
2, then for 1
<
p
≤
2
q
/
Q and u
∈
C
0
∞
(
B
), the estimate (11) holds.Remark 7.
Whenq
=
Q
/
2 and V
∈
B
Q
/
2
⋂
‍
L
1
(
G
), by the important property of the B
q class (see [21]), there exists ε
>
0 such that V
∈
B
Q
/
2
+
ε. Therefore, estimate (11) holds for 1
<
p
≤
1
+
2
ε
/
Q and u
∈
C
0
∞
(
B
).The paper is organized as follows. In Section2 we recall some basic facts about Carnot groups and function spaces. In Section 3 we first give the proof of Theorem 4. Then combining with the known result in [12, Theorem 2] and proving an estimate with the potential V, we finish the proof of Theorem 3. The proof of Theorem 6 is given in Section 4. In Section 5, we restate Theorems 3 and 4 for the Euclidean case and elliptic operators without proofs.Dependence of Constants. Throughout this paper, the letter c denotes a positive constant which may vary from line to line.


## 2. Preliminaries

### 2.1. Background on Carnot Groups

We collect some facts about Carnot groups that will be needed in the sequel and refer the readers to [22–25] for further details.Definition 8 (Carnot group).
A Carnot groupG
=
(
R
n
;
∘
) is a simply connected nilpotent Lie group such that its Lie algebra g
~ admits a stratification
(14)
g
~
=
V
1
⊕
V
2
⊕
⋯
⊕
V
r
=
⊕
j
=
1
r
V
j
,
where [
V
1
,
V
j
]
=
V
j
+
1, j
=
1
,
…
,
r
-
1, and [
V
1
,
V
r
]
=
{
0
}. Here r is called the step of G.Fork
=
1
,
…
,
r, let X
1
,
k
,
…
,
X
m
k
,
k be a basis of V
k consisting of commutators of length k, where m
k is the dimension of V
k. The horizontal vector fields are ones in the first layer V
1 and for convenience, we set m
1
=
m and denote X
i
,
1
=
X
i, i
=
1
,
…
,
m. Clearly, vector fields X
1
,
…
,
X
m satisfy Hormander’s condition [26].Let{
δ
λ
}
λ
>
0 be a family of nonisotropic dilations on G defined by
(15)
δ
λ
:
G
⟶
G
,
δ
λ
ξ
=
λ
ξ
1
,
λ
2
ξ
2
,
…
,
λ
r
ξ
r
,
for any λ
>
0 and ξ
=
(
ξ
1
,
…
,
ξ
r
)
=
(
x
1,1
,
…
,
x
m
1
,
1
,
…
,
x
1
,
r
,
…
,
x
m
r
,
r
)
∈
G. The integer Q
=
∑
k
=
1
r
‍
k
m
k is said to be the homogeneous dimension of G. In general, we assume Q
≥
4. We call that a vector field X
i
,
k
∈
g
~ is left invariant if for any smooth function f one has
(16)
X
i
,
k
x
(
f
(
y
∘
x
)
)
=
(
X
i
,
k
f
)
(
y
∘
x
)
,
for
any
x
∈
G
,
and X
i
,
k is sth homogeneous if for any smooth function f, it follows that
(17)
X
i
,
k
(
f
(
δ
λ
(
x
)
)
)
=
λ
s
(
X
i
,
k
f
)
(
δ
λ
(
x
)
)
,
for
any
x
∈
G
.As in [23], the homogeneous norm of ξ
∈
G is defined by
(18)
d
G
(
ξ
)
=
∑
k
=
1
r
‍
∑
i
=
1
m
k
‍
ξ
i
,
k
2
r
!
/
k
1
/
2
r
!
.
It is natural to define the pseudo distance by the homogeneous norm
(19)
d
G
(
ξ
,
η
)
=
d
G
(
η
-
1
∘
ξ
)
,
for
any
ξ
,
η
∈
G
,
where η
-
1 is the inverse of η. A polynomial on G [24] is a function which can be expressed in exponential coordinates (see, e.g., [22, Section 2.1
.
9] and [27]) as
(20)
P
(
x
)
=
∑
I
‍
a
I
x
I
,
a
I
∈
R
,
where I
=
(
i
j
,
k
)
j
=
1
,
…
,
m
k
k
=
1
,
…
,
r are multi-indices and
(21)
x
I
=
∏
k
=
1
,
…
,
r
;
j
=
1
,
…
,
m
k
x
j
,
k
i
j
,
k
.
The homogeneous degree of monomial x
I is the sum d
(
I
)
=
∑
k
=
1
r
‍
∑
j
=
1
m
k
‍
k
i
j
,
k and the homogeneous degree of P
(
x
) is max
⁡
{
d
(
I
)
∣
a
I
≠
0
}.From [28], the left invariant vector fields X
1
,
…
,
X
m can induce the corresponding Carnot-Carathéodory distance d: for any δ
>
0, let A
(
δ
) be the set of absolutely continuous curves γ
:
[
0,1
]
→
G such that for a.e. t
∈
0,1,
(22)
γ
′
t
=
∑
i
=
1
m
‍
a
i
(
t
)
X
i
(
γ
(
t
)
)
with
∑
i
=
1
m
‍
a
i
t
≤
δ
.
By [29], it is known that for δ large enough the set A
(
δ
) is nonempty. We define the Carnot-Carathéodory distance by
(23)
d
(
ξ
,
η
)
=
inf
⁡
δ
>
0
∣
∃
δ
∈
A
δ
with
γ
0
=
ξ
,
γ
1
=
η
.
It is well known that the distance d is equivalent to the pseudo distance d
G (see [28]). In this paper, we will mainly use the Carnot-Carathéodory distance d to study regularity of (3). Associated with the distance, we define the metric ball of center ξ and radius r in G by
(24)
B
r
ξ
=
η
∈
G
∣
d
ξ
,
η
<
r
.
The Lebesgue measure in R
n is the Haar measure on G ([25, page 619]). Due to (15), one has
(25)
B
r
(
ξ
)
=
C
Q
r
Q
,
where B
r
(
ξ
) is the measure of B
r
(
ξ
) and C
Q is a positive constant.


### 2.2. Function Spaces

DenoteX
=
(
X
1
,
…
,
X
m
), X
u
=
(
X
1
u
,
…
,
X
m
u
), X
2
u
=
{
X
i
X
j
u
}
i
,
j
=
1
m, X
u
=
∑
i
=
1
m
‍
X
i
u, and X
2
u
=
∑
i
,
j
=
1
m
‍
X
i
X
j
u.Definition 9 (Horizontal Sobolev space).
For anyp
≥
1 and a domain Ω
⊆
G, one defines the Horizontal Sobolev spaces by
(26)
H
W
2
,
p
(
Ω
)
=
u
∈
L
p
Ω
∣
u
H
W
2
,
p
Ω
<
+
∞
with the norm
(27)
u
H
W
2
,
p
(
Ω
)
=
u
L
p
(
Ω
)
+
X
u
L
p
(
Ω
)
+
X
2
u
L
p
(
Ω
)
,
where X
u
L
p
(
Ω
)
=
∑
i
=
1
m
‍
X
i
u
L
p
(
Ω
), X
2
u
L
p
(
Ω
)
=
∑
i
,
j
=
1
m
‍
X
i
X
j
u
L
p
(
Ω
).Analogously to [1], the space H
W
V
2
,
p
(
Ω
) is the closure of C
0
∞
(
Ω
) in the norm
(28)
u
H
W
V
2
,
p
(
Ω
)
=
u
H
W
2
,
p
(
Ω
)
+
V
u
L
p
(
Ω
)
.Definition 10 (Reverse Hölder class).
(1) A nonnegative locallyL
q integrable function V
(
x
) on G is said to belong to the reverse Hölder class B
q
(
1
<
q
<
∞
), if there exists a positive constant c such that
(29)
B
-
1
∫
B
‍
V
q
x
d
x
1
/
q
≤
c
B
-
1
∫
B
‍
V
x
d
x
,
for any metric ball B in G.
(2) LetV
(
x
)
>
0 a.e. and V
x
∈
L
loc
⁡
∞
G; one says V
x
∈
B
∞ if there exists a positive constant c such that
(30)
sup
⁡
B
V
x
≤
c
B
-
1
∫
B
‍
V
x
d
x
,
for
any
metric
ball
B
in
G
.It is easy to see thatB
∞
⊂
B
q
⊂
B
p, 1
<
p
<
q
<
∞.


## 2.1. Background on Carnot Groups

We collect some facts about Carnot groups that will be needed in the sequel and refer the readers to [22–25] for further details.Definition 8 (Carnot group).
A Carnot groupG
=
(
R
n
;
∘
) is a simply connected nilpotent Lie group such that its Lie algebra g
~ admits a stratification
(14)
g
~
=
V
1
⊕
V
2
⊕
⋯
⊕
V
r
=
⊕
j
=
1
r
V
j
,
where [
V
1
,
V
j
]
=
V
j
+
1, j
=
1
,
…
,
r
-
1, and [
V
1
,
V
r
]
=
{
0
}. Here r is called the step of G.Fork
=
1
,
…
,
r, let X
1
,
k
,
…
,
X
m
k
,
k be a basis of V
k consisting of commutators of length k, where m
k is the dimension of V
k. The horizontal vector fields are ones in the first layer V
1 and for convenience, we set m
1
=
m and denote X
i
,
1
=
X
i, i
=
1
,
…
,
m. Clearly, vector fields X
1
,
…
,
X
m satisfy Hormander’s condition [26].Let{
δ
λ
}
λ
>
0 be a family of nonisotropic dilations on G defined by
(15)
δ
λ
:
G
⟶
G
,
δ
λ
ξ
=
λ
ξ
1
,
λ
2
ξ
2
,
…
,
λ
r
ξ
r
,
for any λ
>
0 and ξ
=
(
ξ
1
,
…
,
ξ
r
)
=
(
x
1,1
,
…
,
x
m
1
,
1
,
…
,
x
1
,
r
,
…
,
x
m
r
,
r
)
∈
G. The integer Q
=
∑
k
=
1
r
‍
k
m
k is said to be the homogeneous dimension of G. In general, we assume Q
≥
4. We call that a vector field X
i
,
k
∈
g
~ is left invariant if for any smooth function f one has
(16)
X
i
,
k
x
(
f
(
y
∘
x
)
)
=
(
X
i
,
k
f
)
(
y
∘
x
)
,
for
any
x
∈
G
,
and X
i
,
k is sth homogeneous if for any smooth function f, it follows that
(17)
X
i
,
k
(
f
(
δ
λ
(
x
)
)
)
=
λ
s
(
X
i
,
k
f
)
(
δ
λ
(
x
)
)
,
for
any
x
∈
G
.As in [23], the homogeneous norm of ξ
∈
G is defined by
(18)
d
G
(
ξ
)
=
∑
k
=
1
r
‍
∑
i
=
1
m
k
‍
ξ
i
,
k
2
r
!
/
k
1
/
2
r
!
.
It is natural to define the pseudo distance by the homogeneous norm
(19)
d
G
(
ξ
,
η
)
=
d
G
(
η
-
1
∘
ξ
)
,
for
any
ξ
,
η
∈
G
,
where η
-
1 is the inverse of η. A polynomial on G [24] is a function which can be expressed in exponential coordinates (see, e.g., [22, Section 2.1
.
9] and [27]) as
(20)
P
(
x
)
=
∑
I
‍
a
I
x
I
,
a
I
∈
R
,
where I
=
(
i
j
,
k
)
j
=
1
,
…
,
m
k
k
=
1
,
…
,
r are multi-indices and
(21)
x
I
=
∏
k
=
1
,
…
,
r
;
j
=
1
,
…
,
m
k
x
j
,
k
i
j
,
k
.
The homogeneous degree of monomial x
I is the sum d
(
I
)
=
∑
k
=
1
r
‍
∑
j
=
1
m
k
‍
k
i
j
,
k and the homogeneous degree of P
(
x
) is max
⁡
{
d
(
I
)
∣
a
I
≠
0
}.From [28], the left invariant vector fields X
1
,
…
,
X
m can induce the corresponding Carnot-Carathéodory distance d: for any δ
>
0, let A
(
δ
) be the set of absolutely continuous curves γ
:
[
0,1
]
→
G such that for a.e. t
∈
0,1,
(22)
γ
′
t
=
∑
i
=
1
m
‍
a
i
(
t
)
X
i
(
γ
(
t
)
)
with
∑
i
=
1
m
‍
a
i
t
≤
δ
.
By [29], it is known that for δ large enough the set A
(
δ
) is nonempty. We define the Carnot-Carathéodory distance by
(23)
d
(
ξ
,
η
)
=
inf
⁡
δ
>
0
∣
∃
δ
∈
A
δ
with
γ
0
=
ξ
,
γ
1
=
η
.
It is well known that the distance d is equivalent to the pseudo distance d
G (see [28]). In this paper, we will mainly use the Carnot-Carathéodory distance d to study regularity of (3). Associated with the distance, we define the metric ball of center ξ and radius r in G by
(24)
B
r
ξ
=
η
∈
G
∣
d
ξ
,
η
<
r
.
The Lebesgue measure in R
n is the Haar measure on G ([25, page 619]). Due to (15), one has
(25)
B
r
(
ξ
)
=
C
Q
r
Q
,
where B
r
(
ξ
) is the measure of B
r
(
ξ
) and C
Q is a positive constant.


## 2.2. Function Spaces

DenoteX
=
(
X
1
,
…
,
X
m
), X
u
=
(
X
1
u
,
…
,
X
m
u
), X
2
u
=
{
X
i
X
j
u
}
i
,
j
=
1
m, X
u
=
∑
i
=
1
m
‍
X
i
u, and X
2
u
=
∑
i
,
j
=
1
m
‍
X
i
X
j
u.Definition 9 (Horizontal Sobolev space).
For anyp
≥
1 and a domain Ω
⊆
G, one defines the Horizontal Sobolev spaces by
(26)
H
W
2
,
p
(
Ω
)
=
u
∈
L
p
Ω
∣
u
H
W
2
,
p
Ω
<
+
∞
with the norm
(27)
u
H
W
2
,
p
(
Ω
)
=
u
L
p
(
Ω
)
+
X
u
L
p
(
Ω
)
+
X
2
u
L
p
(
Ω
)
,
where X
u
L
p
(
Ω
)
=
∑
i
=
1
m
‍
X
i
u
L
p
(
Ω
), X
2
u
L
p
(
Ω
)
=
∑
i
,
j
=
1
m
‍
X
i
X
j
u
L
p
(
Ω
).Analogously to [1], the space H
W
V
2
,
p
(
Ω
) is the closure of C
0
∞
(
Ω
) in the norm
(28)
u
H
W
V
2
,
p
(
Ω
)
=
u
H
W
2
,
p
(
Ω
)
+
V
u
L
p
(
Ω
)
.Definition 10 (Reverse Hölder class).
(1) A nonnegative locallyL
q integrable function V
(
x
) on G is said to belong to the reverse Hölder class B
q
(
1
<
q
<
∞
), if there exists a positive constant c such that
(29)
B
-
1
∫
B
‍
V
q
x
d
x
1
/
q
≤
c
B
-
1
∫
B
‍
V
x
d
x
,
for any metric ball B in G.
(2) LetV
(
x
)
>
0 a.e. and V
x
∈
L
loc
⁡
∞
G; one says V
x
∈
B
∞ if there exists a positive constant c such that
(30)
sup
⁡
B
V
x
≤
c
B
-
1
∫
B
‍
V
x
d
x
,
for
any
metric
ball
B
in
G
.It is easy to see thatB
∞
⊂
B
q
⊂
B
p, 1
<
p
<
q
<
∞.


## 3. Proofs of Theorems3 and 4

We first prove Theorem4 and then prove Theorem 3.


### 3.1. Proof of Theorem4

The following lemma is due to Lu and Wheeden [30, 31]. It will play a key role in our proof.Lemma 11.
LetB
=
B
r
B
(
x
0
) be a metric ball in G. If u
∈
C
∞
(
B
), then there exists a first order polynomial P
B
(
x
) such that, for a.e. x
∈
B,
(31)
u
(
x
)
-
P
B
x
≤
c
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
,
where the positive constant c is independent of u, x, and B. Moreover, if u
∈
C
0
∞
(
B
), then for a.e. x
∈
B,
(32)
u
(
x
)
≤
c
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
,
where c
>
0 is independent of u, x, and B.Proof of Theorem4.
By (31), Fubini’s Theorem, and Hölder’s inequality, we have
(33)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
≤
c
∫
B
‍
V
(
x
)
u
(
x
)
-
P
B
(
x
)
p
-
1
×
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
d
x
=
c
∫
B
‍
X
2
u
y
×
d
x
,
y
2
B
x
,
d
x
,
y
∫
B
‍
V
x
u
x
-
P
B
x
p
-
1
×
d
x
,
y
2
B
x
,
d
x
,
y
d
x
d
y
≤
c
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
·
d
x
,
y
2
B
x
,
d
x
,
y
d
x
p
/
(
p
-
1
)
∫
B
‍
d
x
,
y
2
B
x
,
d
x
,
y
∫
B
‍
V
x
u
x
-
P
B
x
p
-
1
×
d
x
,
y
2
B
x
,
d
x
,
y
d
x
p
/
(
p
-
1
)
d
y
(
p
-
1
)
/
p
≡
c
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
·
I
.
Now a computation yields
(34)
I
p
/
(
p
-
1
)
≤
∫
B
‍
∫
B
‍
V
(
z
)
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
·
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
∫
B
‍
u
x
-
P
B
x
p
V
x
×
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
x
d
y
=
∫
B
‍
u
x
-
P
B
x
p
V
x
∫
B
‍
d
x
,
y
2
B
x
,
d
x
,
y
·
∫
B
‍
V
(
z
)
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
d
y
d
x
≤
∫
B
x
0
,
r
B
‍
u
x
-
P
B
x
p
V
x
×
∫
B
(
x
,
2
r
B
)
‍
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
·
d
z
,
y
2
B
z
,
d
z
,
y
∫
B
x
,
2
r
B
‍
V
z
×
d
z
,
y
2
B
z
,
d
z
,
y
d
z
1
/
(
p
-
1
)
d
y
d
x
≤
φ
V
2
r
B
1
/
(
p
-
1
)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
.
Therefore,
(35)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
≤
c
φ
V
2
r
B
1
/
p
∫
B
‍
u
x
-
P
B
x
p
V
x
d
x
(
p
-
1
)
/
p
×
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
.
It implies (12).
By using (32) and repeating the argument above for (12), we immediately obtain (13).


### 3.2. Proof of Theorem3

Let us recallL
p estimates for the operator A
=
∑
i
,
j
=
1
m
‍
a
i
j
X
i
X
j by Bramanti and Brandolini [12, Theorem 2].Lemma 12.
Under the assumptions (4) and (5), for every p
∈
(
1
,
∞
), there exist positive constants c
=
c
(
p
,
μ
,
G
) and r
=
r
(
p
,
μ
,
η
,
G
), where η denotes the V
M
O moduli of coefficients a
i
j, such that, for any u
∈
C
0
∞
(
G
) and sprt u  ⊂
B
r (B
r any metric ball of radius r),
(36)
X
2
u
L
p
(
B
r
)
≤
c
A
u
L
p
(
B
r
)
.Based on it and Theorem4, we have the following L
p estimates for L in (3).Lemma 13.
Under the assumptions (4) and (5), for every p
∈
(
1
,
∞
) and V
p
∈
S
p
(
G
), there exist positive constants r
0
=
r
0
(
p
,
μ
,
η
,
G
) and c
=
c
(
p
,
r
0
,
μ
,
V
,
G
) such that, for any u
∈
C
0
∞
(
G
) and sprt u  ⊂
B
r
0,
(37)
X
2
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
L
u
L
p
(
B
r
0
)
.Proof.
By Theorem4,
(38)
V
u
L
p
(
B
r
0
)
p
≤
c
φ
V
p
(
2
r
0
)
∫
B
r
0
‍
X
2
u
(
x
)
p
d
x
.
Applying Lemma 12, it follows that
(39)
X
2
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
A
u
L
p
B
r
0
+
V
u
L
p
B
r
0
≤
c
L
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
L
u
L
p
(
B
r
0
)
+
φ
V
p
2
r
0
1
/
p
X
2
u
L
p
(
B
r
0
)
.
Choosing r
0
>
0 such that c
φ
V
p
(
2
r
0
)
1
/
p
≤
1
/
2, we derive (37).Proof of Theorem3.
We consult the way in [1, pages 342-343] and apply our previous results. By the basic theorem on the partition of unity (e.g., see [32, page 66]), there exists a partition of unity of nonnegative functions {
φ
i
}
i
=
1
∞ in G such that φ
i
∈
C
0
∞
(
B
r
0
(
z
i
)
) with r
0 in Lemma 13 and a family of metric balls {
B
r
0
z
i
}
i
=
1
∞ satisfying the finite overlapping property. We have from Lemma 13 that
(40)
X
2
u
L
p
G
+
V
u
L
p
G
=
∑
i
‍
X
2
(
φ
i
u
)
L
p
(
G
)
+
∑
i
‍
V
φ
i
u
L
p
(
G
)
≤
c
∑
i
‍
X
2
(
φ
i
u
)
L
p
(
B
r
0
(
z
i
)
)
+
∑
i
‍
V
φ
i
u
L
p
(
B
r
0
(
z
i
)
)
≤
c
∑
i
‍
L
φ
i
u
L
p
B
r
0
z
i
≤
∑
i
‍
L
u
L
p
B
r
0
z
i
+
D
u
L
p
B
r
0
z
i
+
u
L
p
B
r
0
z
i
≤
c
L
u
L
p
(
G
)
+
X
u
L
p
(
G
)
+
u
L
p
(
G
)
.
Combining with the interpolation inequality (see [12, Proposition 2])
(41)
X
u
L
p
(
G
)
≤
ε
X
2
u
L
p
(
G
)
+
2
ε
u
L
p
(
G
)
,
for
any
ε
>
0
,
we obtain (11).


## 3.1. Proof of Theorem4

The following lemma is due to Lu and Wheeden [30, 31]. It will play a key role in our proof.Lemma 11.
LetB
=
B
r
B
(
x
0
) be a metric ball in G. If u
∈
C
∞
(
B
), then there exists a first order polynomial P
B
(
x
) such that, for a.e. x
∈
B,
(31)
u
(
x
)
-
P
B
x
≤
c
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
,
where the positive constant c is independent of u, x, and B. Moreover, if u
∈
C
0
∞
(
B
), then for a.e. x
∈
B,
(32)
u
(
x
)
≤
c
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
,
where c
>
0 is independent of u, x, and B.Proof of Theorem4.
By (31), Fubini’s Theorem, and Hölder’s inequality, we have
(33)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
≤
c
∫
B
‍
V
(
x
)
u
(
x
)
-
P
B
(
x
)
p
-
1
×
∫
B
‍
X
2
u
(
y
)
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
y
d
x
=
c
∫
B
‍
X
2
u
y
×
d
x
,
y
2
B
x
,
d
x
,
y
∫
B
‍
V
x
u
x
-
P
B
x
p
-
1
×
d
x
,
y
2
B
x
,
d
x
,
y
d
x
d
y
≤
c
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
·
d
x
,
y
2
B
x
,
d
x
,
y
d
x
p
/
(
p
-
1
)
∫
B
‍
d
x
,
y
2
B
x
,
d
x
,
y
∫
B
‍
V
x
u
x
-
P
B
x
p
-
1
×
d
x
,
y
2
B
x
,
d
x
,
y
d
x
p
/
(
p
-
1
)
d
y
(
p
-
1
)
/
p
≡
c
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
·
I
.
Now a computation yields
(34)
I
p
/
(
p
-
1
)
≤
∫
B
‍
∫
B
‍
V
(
z
)
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
·
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
∫
B
‍
u
x
-
P
B
x
p
V
x
×
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
d
x
d
y
=
∫
B
‍
u
x
-
P
B
x
p
V
x
∫
B
‍
d
x
,
y
2
B
x
,
d
x
,
y
·
∫
B
‍
V
(
z
)
d
z
,
y
2
B
(
z
,
d
(
z
,
y
)
)
d
z
1
/
(
p
-
1
)
d
y
d
x
≤
∫
B
x
0
,
r
B
‍
u
x
-
P
B
x
p
V
x
×
∫
B
(
x
,
2
r
B
)
‍
d
x
,
y
2
B
(
x
,
d
(
x
,
y
)
)
·
d
z
,
y
2
B
z
,
d
z
,
y
∫
B
x
,
2
r
B
‍
V
z
×
d
z
,
y
2
B
z
,
d
z
,
y
d
z
1
/
(
p
-
1
)
d
y
d
x
≤
φ
V
2
r
B
1
/
(
p
-
1
)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
.
Therefore,
(35)
∫
B
‍
u
(
x
)
-
P
B
(
x
)
p
V
(
x
)
d
x
≤
c
φ
V
2
r
B
1
/
p
∫
B
‍
u
x
-
P
B
x
p
V
x
d
x
(
p
-
1
)
/
p
×
∫
B
‍
X
2
u
(
y
)
p
d
y
1
/
p
.
It implies (12).
By using (32) and repeating the argument above for (12), we immediately obtain (13).


## 3.2. Proof of Theorem3

Let us recallL
p estimates for the operator A
=
∑
i
,
j
=
1
m
‍
a
i
j
X
i
X
j by Bramanti and Brandolini [12, Theorem 2].Lemma 12.
Under the assumptions (4) and (5), for every p
∈
(
1
,
∞
), there exist positive constants c
=
c
(
p
,
μ
,
G
) and r
=
r
(
p
,
μ
,
η
,
G
), where η denotes the V
M
O moduli of coefficients a
i
j, such that, for any u
∈
C
0
∞
(
G
) and sprt u  ⊂
B
r (B
r any metric ball of radius r),
(36)
X
2
u
L
p
(
B
r
)
≤
c
A
u
L
p
(
B
r
)
.Based on it and Theorem4, we have the following L
p estimates for L in (3).Lemma 13.
Under the assumptions (4) and (5), for every p
∈
(
1
,
∞
) and V
p
∈
S
p
(
G
), there exist positive constants r
0
=
r
0
(
p
,
μ
,
η
,
G
) and c
=
c
(
p
,
r
0
,
μ
,
V
,
G
) such that, for any u
∈
C
0
∞
(
G
) and sprt u  ⊂
B
r
0,
(37)
X
2
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
L
u
L
p
(
B
r
0
)
.Proof.
By Theorem4,
(38)
V
u
L
p
(
B
r
0
)
p
≤
c
φ
V
p
(
2
r
0
)
∫
B
r
0
‍
X
2
u
(
x
)
p
d
x
.
Applying Lemma 12, it follows that
(39)
X
2
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
A
u
L
p
B
r
0
+
V
u
L
p
B
r
0
≤
c
L
u
L
p
(
B
r
0
)
+
V
u
L
p
(
B
r
0
)
≤
c
L
u
L
p
(
B
r
0
)
+
φ
V
p
2
r
0
1
/
p
X
2
u
L
p
(
B
r
0
)
.
Choosing r
0
>
0 such that c
φ
V
p
(
2
r
0
)
1
/
p
≤
1
/
2, we derive (37).Proof of Theorem3.
We consult the way in [1, pages 342-343] and apply our previous results. By the basic theorem on the partition of unity (e.g., see [32, page 66]), there exists a partition of unity of nonnegative functions {
φ
i
}
i
=
1
∞ in G such that φ
i
∈
C
0
∞
(
B
r
0
(
z
i
)
) with r
0 in Lemma 13 and a family of metric balls {
B
r
0
z
i
}
i
=
1
∞ satisfying the finite overlapping property. We have from Lemma 13 that
(40)
X
2
u
L
p
G
+
V
u
L
p
G
=
∑
i
‍
X
2
(
φ
i
u
)
L
p
(
G
)
+
∑
i
‍
V
φ
i
u
L
p
(
G
)
≤
c
∑
i
‍
X
2
(
φ
i
u
)
L
p
(
B
r
0
(
z
i
)
)
+
∑
i
‍
V
φ
i
u
L
p
(
B
r
0
(
z
i
)
)
≤
c
∑
i
‍
L
φ
i
u
L
p
B
r
0
z
i
≤
∑
i
‍
L
u
L
p
B
r
0
z
i
+
D
u
L
p
B
r
0
z
i
+
u
L
p
B
r
0
z
i
≤
c
L
u
L
p
(
G
)
+
X
u
L
p
(
G
)
+
u
L
p
(
G
)
.
Combining with the interpolation inequality (see [12, Proposition 2])
(41)
X
u
L
p
(
G
)
≤
ε
X
2
u
L
p
(
G
)
+
2
ε
u
L
p
(
G
)
,
for
any
ε
>
0
,
we obtain (11).


## 4. Proof of Theorem6

Several preliminary conclusions are necessary.Lemma 14.
IfV
∈
B
q, q
>
1, then there exists a constant c
>
0 such that, for any 1
≤
p
<
q
<
∞ and 0
<
r
<
R
<
∞,
(42)
1
B
r
(
x
)
∫
B
r
(
x
)
‍
V
y
p
d
y
≤
c
R
r
p
Q
/
q
1
B
R
(
x
)
∫
B
R
(
x
)
‍
V
(
y
)
d
y
p
.Proof.
By Hölder’s inequality and (29), it yields
(43)
1
B
r
(
x
)
∫
B
r
(
x
)
‍
V
y
p
d
y
≤
1
B
r
(
x
)
∫
B
r
(
x
)
‍
V
y
q
d
y
p
/
q
≤
c
R
r
Q
1
B
R
x
∫
B
R
x
‍
V
y
q
d
y
p
/
q
≤
c
R
r
p
Q
/
q
1
B
R
(
x
)
∫
B
R
(
x
)
‍
V
(
y
)
d
y
p
.Remark 15.
If we takeG
=
R
n and p
=
1, then Lemma 14 gives the version in [2, Lemma 1.2].Lemma 16.
IfV
∈
B
q
∩
L
1
(
G
), q
>
Q
/
2, then V
p
∈
S
p
(
G
), 1
<
p
≤
2
q
/
Q.Proof.
For anyx
∈
G, it follows that
(44)
I
=
:
∫
d
x
,
y
<
r
‍
d
2
x
,
y
B
x
,
d
x
,
y
×
∫
d
z
,
x
<
r
‍
V
p
z
d
2
z
,
y
B
z
,
d
z
,
y
d
z
1
/
(
p
-
1
)
d
y
≤
∫
d
x
,
y
<
r
‍
d
2
x
,
y
B
x
,
d
x
,
y
×
∫
d
z
,
y
<
2
r
‍
V
p
z
d
2
z
,
y
B
z
,
d
z
,
y
d
z
1
/
(
p
-
1
)
d
y
≡
∫
d
(
x
,
y
)
<
r
‍
d
2
(
x
,
y
)
B
(
x
,
d
(
x
,
y
)
)
I
1
1
/
(
p
-
1
)
d
y
.
By (42), it yields
(45)
I
1
≤
∑
k
=
0
∞
‍
∫
r
/
2
k
≤
d
(
z
,
y
)
<
r
/
2
k
-
1
‍
V
p
(
z
)
d
2
(
z
,
y
)
B
(
z
,
d
(
z
,
y
)
)
d
z
≤
c
∑
k
=
0
∞
‍
r
2
k
2
-
Q
∫
d
(
z
,
y
)
<
r
/
2
k
-
1
‍
V
p
(
z
)
d
z
≤
c
r
2
-
p
Q
/
q
R
p
Q
/
q
-
p
Q
∑
k
=
0
∞
‍
1
2
k
2
-
p
Q
/
q
∫
B
y
,
R
‍
V
z
d
z
p
≤
c
r
2
-
p
Q
/
q
R
p
Q
/
q
-
p
Q
∫
G
‍
V
d
z
p
.
Also, we have
(46)
∫
d
(
x
,
y
)
<
r
‍
d
2
(
x
,
y
)
B
(
x
,
d
(
x
,
y
)
)
d
y
≤
∑
k
=
1
∞
‍
∫
r
/
2
k
≤
d
(
x
,
y
)
<
r
/
2
k
-
1
d
2
(
x
,
y
)
B
(
x
,
d
(
x
,
y
)
)
d
y
≤
c
∑
k
=
1
∞
‍
r
2
k
2
-
Q
∫
d
(
x
,
y
)
<
r
/
2
k
-
1
d
y
≤
c
r
2
.
Therefore combining (45) and (46) gets
(47)
I
≤
c
r
(
1
/
(
p
-
1
)
)
(
2
-
p
Q
/
q
)
R
(
p
Q
/
(
p
-
1
)
)
(
1
/
q
-
1
)
×
∫
G
‍
V
(
z
)
d
z
p
/
(
p
-
1
)
·
c
r
2
≤
c
r
(
1
/
(
p
-
1
)
)
2
p
-
p
Q
/
q
R
(
p
Q
/
(
p
-
1
)
)
1
/
q
-
1
×
∫
G
‍
V
z
d
z
p
/
(
p
-
1
)
⟶
0
,
as
r
⟶
0
.
The result is proved.Proof of Theorem6.
By Lemma16 and Theorem 3, we immediately obtain Theorem 6.Remark 17.
In order to assure the convergence of the series∑
k
=
0
∞
‍
(
2
-
k
)
2
-
p
Q
/
q in the proof of Lemma 16, we require the assumption p
≤
2
q
/
Q, which leads to the range of p in Theorem 6 smaller than [1, Theorem 1].


## 5. Results to the Euclidean Case and Elliptic Operators

Here for convenience of readers, we restate Theorems3 and 4 corresponding to the Euclidean case but omit their proofs because the proofs are analogous to Theorems 3 and 4. It will be assumed for the leading coefficients a
i
j in (1) that(
H
1
)
a
i
j
(
x
)
=
a
j
i
(
x
)
∈
L
∞
(
R
n
) for all i
,
j
=
1
,
…
,
n and there exists a positive constant μ such that, for any x
∈
R
n and ξ
∈
R
n,
(48)
μ
-
1
ξ
2
≤
∑
i
,
j
=
1
n
‍
a
i
j
(
x
)
ξ
i
ξ
j
≤
μ
ξ
2
;(
H
2
)
a
i
j
(
x
)
∈
V
M
O
(
R
n
); that is, for i
,
j
=
1
,
…
,
n,
(49)
η
i
j
=
sup
⁡
ρ
≤
r
sup
⁡
x
∈
R
n
B
ρ
x
-
1
∫
B
ρ
x
a
i
j
y
-
a
i
j
B
d
y
⟶
0
,
as
r
⟶
0
+
,
where a
i
j
B
=
B
ρ
(
x
)
-
1
∫
B
ρ
(
x
)
‍
a
i
j
(
y
)
d
y.A functionV
∈
S
p
(
R
n
) for 1
<
p
<
∞ means that, for each r
>
0,
(50)
φ
V
r
≔
sup
⁡
x
∈
R
n
∫
x
-
y
<
r
1
x
-
y
n
-
2
×
∫
z
-
y
<
r
V
(
z
)
z
-
y
n
-
2
d
z
1
/
(
p
-
1
)
d
y
p
-
1
is finite and
(51)
lim
⁡
r
→
0
φ
V
(
r
)
=
0
.Theorem 18.
Under assumptions(
H
1
) and (
H
2
), if V
p
∈
S
p
(
R
n
), 1
<
p
<
∞, then there exists a positive constant c
=
c
(
n
,
p
,
μ
,
η
,
V
) such that, for any u
∈
C
0
∞
R
n, one has
(52)
u
W
2
,
p
(
R
n
)
+
V
u
L
p
(
R
n
)
≤
c
L
u
L
p
(
R
n
)
+
u
L
p
(
R
n
)
,
where η depends only on the V
M
O moduli of the coefficients a
i
j.Remark 19.
IfV
≡
0
,
x
∈
R
n or V
≤
const
.
,
x
∈
R
n, the L
p theory of (1) with discontinuous leading coefficients was intensively studied and the result was proved in [33–36] and so forth. Bramanti et al. [1] obtained a prior L
p
(
R
n
) estimate for (1) with a
i
j
∈
V
M
O and V
∈
B
q. Here B
q
⊄
S
p
(
R
n
) and S
p
(
R
n
)
⊄
B
q.Theorem 20.
LetB
=
B
r
B
(
x
0
) be any ball in R
n. If  V
∈
S
p
(
R
n
), then there exists a first order polynomial P
B
(
x
) in R
n such that, for any u
∈
C
∞
(
B
), one has
(53)
∫
B
‍
u
-
P
B
(
x
)
p
V
d
x
≤
c
φ
V
(
2
r
B
)
∫
B
‍
X
2
u
p
d
x
,
where the positive constant c is independent of u and B. Moreover, for any u
∈
C
0
∞
(
B
), one has
(54)
∫
B
‍
u
p
V
d
x
≤
c
φ
V
(
2
r
B
)
∫
B
‍
X
2
u
p
d
x
,
where c
>
0 is independent of u and B.



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*Source: 274859-2014-11-10.xml*