1. Introduction
Since the differential form has been proposed by Elie Cartan, its development is obvious. Differential form theory has been applied to many fields, such as partial differential equations, nonlinear analysis, and control theory. Particularly, it is an important part in differential forms to research norm inequalities for an operator or composite operator.
L
p
norm has been studied very thoroughly [1–3], so we try to establish the Lipschitz norm and BMO norm inequalities for the composition of the HardyLittlewood maximal operator and potential operator. It is useful to explore the properties of the weighted norms; see [4–7]. So, we also research the weighted Lipschitz norm and the weighted BMO norm inequalities.
As the traditional notations, we write
Ω
for a bounded convex domain in
ℝ
n
,
n
≥
2
, endowed with the usual Lebesgue measure denoted by

Ω

.
B
and
σ
B
are concentric balls, with
diam
(
σ
B
)
=
σ
diam
(
B
)
.
ω
denotes a weight defined by
ω
∈
L
loc
(
ℝ
n
)
, and
ω
>
0
a.e.. The
l
form
, denoted by
Λ
l
=
Λ
l
(
ℝ
n
)
, is a
l
vector
, spanned by exterior products
e
I
=
e
i
1
∧
e
i
2
∧
⋯
∧
e
i
l
, for all ordered
l
tuples
I
=
(
i
1
,
i
2
,
…
,
i
l
)
,
1
≤
i
1
<
i
2
<
⋯
<
i
l
≤
n
. The
l
form
u
(
x
)
=
Σ
I
u
I
(
x
)
d
x
I
is called a differential
l
form, if
u
I
is differential. We use
D
′
(
Ω
,
Λ
l
)
to denote the differential
l
form and
L
s
(
Ω
,
Λ
l
)
to denote the
l
form
u
(
x
)
on
Ω
satisfying
∫
Ω

u
I

s
<
∞
. In particular, we know that the
0
form is a function. We define the exterior derivative
d
:
D
′
(
Ω
,
Λ
l
)
→
D
′
(
Ω
,
Λ
l
+
1
)
by
(1)
d
(
u
(
x
)
)
=
∑
k
=
1
n
∑
1
≤
i
1
<
i
2
<
⋯
<
i
l
≤
n
∂
u
i
1
i
2
⋯
i
l
(
x
)
∂
x
k
×
d
x
k
∧
d
x
i
1
∧
⋯
∧
d
x
i
l
,
l
=
0,1
,
…
,
n

1
.
We define
⋆
by
(2)
⋆
u
=
sign
(
π
)
u
i
1
i
2
⋯
i
l
(
x
)
d
x
j
1
∧
⋯
∧
d
x
j
n

l
,
where
π
=
(
i
1
,
…
,
i
l
,
j
1
,
…
,
j
n

l
)
is a permutation of
(
1
,
…
,
n
)
and
sign
(
π
)
is the signature of the permutation. Now, we can define the Hodge codifferential
d
⋆
like this
(3)
d
⋆
=
(

1
)
n
l
+
1
⋆
d
⋆
:
D
′
(
Ω
,
Λ
l
+
1
)
⟶
D
′
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n

1
.
The differential form we research here satisfies the nonhomogeneous
A
harmonic equation
(4)
d
⋆
A
(
x
,
d
u
)
=
B
(
x
,
d
u
)
,
where
A
:
Ω
×
Λ
l
(
ℝ
n
)
→
Λ
l
(
ℝ
n
)
and
B
:
Ω
×
Λ
l
(
ℝ
n
)
→
Λ
l

1
(
ℝ
n
)
satisfy the conditions
(5)

A
(
x
,
ξ
)

≤
a

ξ

p

1
,
A
(
x
,
ξ
)
·
ξ
≥

ξ

p
,

B
(
x
,
ξ
)

≤
b

ξ

p

1
for almost every
x
∈
Ω
and all
ξ
∈
∧
l
(
ℝ
n
)
. Here
a
,
b
>
0
are constants and
1
<
p
<
∞
is a fixed exponent associated with (4).
2. Lipschitz and BMO Norm Inequalities
In this section, we introduce some definitions. Then, we give main lemmas used in theorems. Finally, we give the norm comparison estimates for the composite operator.
For
u
∈
L
loc
1
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
, we write
u
∈
loc
Li
p
k
(
Ω
,
Λ
l
)
,
0
≤
k
≤
1
, if
(6)
∥
u
∥
loc
Li
p
k
,
Ω
=
sup
σ
Q
⊂
Ω

Q


(
n
+
k
)
/
n
∥
u

u
Q
∥
1
,
Q
<
∞
for some
σ
>
1
.
For
u
∈
L
loc
1
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
, we say
u
∈
BMO
(
Ω
,
Λ
l
)
, if
(7)
∥
u
∥
*
,
Ω
=
sup
σ
Q
⊂
Ω

Q


1
∥
u

u
Q
∥
1
,
Q
<
∞
for some
σ
>
1
.
HardyLittlewood maximal operator is defined by
(8)
𝕄
s
(
u
)
=
𝕄
s
u
(
x
)
=
sup
r
>
0
(
1

B
(
x
,
r
)

∫
B
(
x
,
r
)

u
(
y
)

s
d
y
)
1
/
s
,
where
1
≤
s
<
∞
,
u
is a locally
L
s
integrable form and
B
(
x
,
r
)
is a ball with radius
r
.
Potential operator is first extended to differential form by Bi in [8], which is defined as follows:
(9)
P
u
=
P
u
(
x
)
=
∑
I
∫
Ω
K
(
x
,
y
)
u
I
(
y
)
d
y
d
x
I
,
where the kernel
K
(
x
,
y
)
is a nonnegative measurable function defined for
x
≠
y
,
u
is a differential
l
form, and the summation is over all ordered
l
tuples
I
.
We need the following two lemmas for the HardyLittlewood maximal operator and potential operator to prove Theorem 7.
Lemma 1 (see [<xref reftype="bibr" rid="B8">9</xref>]).
Let
u
∈
L
t
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
, be a differential form in a domain
Ω
and
𝕄
s
be the HardyLittlewood maximal operator,
1
≤
s
<
t
<
∞
. Then,
𝕄
s
(
u
)
∈
L
t
(
Ω
)
and there exists a constant
C
, independent of
u
, such that
(10)
∥
𝕄
s
(
u
)
∥
t
,
Ω
≤
C
∥
u
∥
t
,
Ω
.
Lemma 2 (see [<xref reftype="bibr" rid="B9">8</xref>]).
Let
u
∈
D
′
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
, be a differential form in a domain
Ω
and let
P
be the potential operator. Then, there exists a constant
C
, independent of
u
, such that
(11)
∥
P
(
u
)
∥
s
,
Ω
≤
C
∥
u
∥
s
,
Ω
,
where
1
<
s
<
∞
.
The following lemma was established by Iwaniec and Lutoborski in [10].
Lemma 3.
Let
u
∈
L
s
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
,
1
<
s
<
∞
, be a differential form in a domain
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(12)
∥
u
Ω
∥
s
,
Ω
≤
C
∥
u
∥
s
,
Ω
,
where
u
Ω
is a closed form.
We will use the following generalized Hölder inequality repeatedly in this paper.
Lemma 4.
Let
0
<
q
<
∞
,
0
<
p
<
∞
, and
s

1
=
q

1
+
p

1
. If
f
and
g
are measurable functions on
ℝ
n
, then
(13)
∥
f
g
∥
s
,
Ω
≤
∥
f
∥
q
,
Ω
∥
g
∥
p
,
Ω
for any
Ω
⊂
ℝ
n
.
We also need the Caccioppoli inequality for differential form.
Lemma 5 (see [<xref reftype="bibr" rid="B11">11</xref>]).
Let
u
∈
D
′
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(14)
∥
d
u
∥
s
,
B
≤
C
diam
(
B
)

1
∥
u

c
∥
s
,
σ
B
for all balls with
σ
B
⊂
Ω
and any closed form
c
, where
1
<
s
<
∞
, and
σ
>
1
.
The following Poincaré inequality appears in [5].
Lemma 6.
Let
u
∈
D
′
(
Ω
,
Λ
l
)
,
l
=
0,1
,
…
,
n

1
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(15)
∥
u

u
B
∥
s
,
B
≤
C

B

diam
(
B
)
∥
d
u
∥
s
,
B
for any ball
B
in
Ω
.
First, we establish a Poincarétype inequality for the composite operator
𝕄
s
∘
P
.
Theorem 7.
Let
u
∈
L
t
(
Ω
,
Λ
l
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
t
<
∞
, be a differential form in a domain
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(16)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
Ω
∥
t
,
Ω
≤
C
∥
u
∥
t
,
Ω
.
Proof.
From Lemma 3 and Minkowski inequality, we have
(17)
∥
u

u
Ω
∥
t
,
Ω
=
(
∫
Ω

u
(
x
)

u
Ω
(
x
)

t
d
x
)
1
/
t
≤
(
∫
Ω

u
(
x
)

t
d
x
)
1
/
t
+
(
∫
Ω

u
Ω
(
x
)

t
d
x
)
1
/
t
≤
(
∫
Ω

u
(
x
)

t
d
x
)
1
/
t
+
C
1
(
∫
Ω

u
(
x
)

t
d
x
)
1
/
t
≤
C
2
∥
u
∥
t
,
Ω
.
Replacing
u
with
𝕄
s
P
(
u
)
and combining Lemmas 1 and 2, we obtain
(18)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
Ω
∥
t
,
Ω
≤
C
2
∥
𝕄
s
P
(
u
)
∥
t
,
Ω
≤
C
3
∥
P
(
u
)
∥
t
,
Ω
≤
C
4
∥
u
∥
t
,
Ω
.
Theorem 7 has been completed.
Then, we estimate the Lipschitz norm of
𝕄
s
P
(
u
)
in terms of
L
t
norm.
Theorem 8.
Let
u
∈
L
t
(
Ω
,
Λ
l
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
t
<
∞
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(19)
∥
𝕄
s
P
(
u
)
∥
loc
Lip
k
,
Ω
≤
C
∥
d
𝕄
s
P
(
u
)
∥
t
,
Ω
,
where
k
is a constant with
0
≤
k
≤
1
.
Proof.
Setting
u
=
𝕄
s
P
(
u
)
in Lemma 6 we have
(20)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
t
,
B
≤
C
1

B

diam
(
B
)
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
for all balls
B
with
B
⊂
Ω
. Using Hölder inequality, we have
(21)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

t
d
x
)
1
/
t
×
(
∫
B
1
t
/
(
t

1
)
d
x
)
(
t

1
)
/
t
≤

B

(
t

1
)
/
t
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
t
,
B
=

B

1

1
/
t
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
t
,
B
≤

B

1

1
/
t
C
1

B

diam
(
B
)
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
≤
C
2

B

2

1
/
t
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
.
From the definition of Lipschitz norm and (21), it follows that
(22)
∥
𝕄
s
P
(
u
)
∥
loc
Li
p
k
,
Ω
=
sup
σ
B
⊂
Ω

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
=
sup
σ
B
⊂
Ω

B


1

k
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
sup
σ
B
⊂
Ω

B


1

k
/
n
C
2

B

2

1
/
t
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
=
sup
σ
B
⊂
Ω
C
2

B

1

k
/
n

1
/
t
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
≤
sup
σ
B
⊂
Ω
C
2

Ω

1

k
/
n

1
/
t
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
≤
C
3
sup
σ
B
⊂
Ω
∥
d
𝕄
s
P
(
u
)
∥
t
,
B
≤
C
3
∥
d
𝕄
s
P
(
u
)
∥
t
,
Ω
.
Theorem 8 has been completed.
Now, we establish the norm comparison theorems of the composite operator between Lipschitz norm and BMO norm.
Theorem 9.
Let
u
∈
L
s
(
Ω
,
Λ
l
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
∞
, be a differential form in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(23)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
≤
C
∥
𝕄
s
P
(
u
)
∥
loc
Lip
k
,
Ω
,
where
k
is a constant with
0
≤
k
≤
1
.
Proof.
From the definition of BMO norm, we have
(24)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
=
sup
σ
B
⊂
Ω

B


1
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
=
sup
σ
B
⊂
Ω

B

k
/
n

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
sup
σ
B
⊂
Ω

Ω

k
/
n

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤

Ω

k
/
n
sup
σ
B
⊂
Ω

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
C
sup
σ
B
⊂
Ω

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
C
∥
𝕄
s
P
(
u
)
∥
loc
Li
p
k
,
Ω
.
Theorem 9 has been completed.
Theorem 10.
Let
u
∈
L
s
(
Ω
,
Λ
l
)
,
l
=
1,2
,
…
,
n
,
1
<
s
<
∞
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(25)
∥
𝕄
s
P
(
u
)
∥
loc
Lip
k
,
Ω
≤
C
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
,
where
k
is a constant with
0
≤
k
≤
1
.
Proof.
From (21), we obtain
(26)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
C
1

B

2

1
/
s
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
s
,
B
.
Based on Lemma 5, we get
(27)
∥
d
𝕄
s
P
(
u
)
∥
s
,
B
≤
C
2
diam
(
B
)

1
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
s
,
σ
B
.
From the weak reverse Hölder inequality, it follows that
(28)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
s
,
σ
B
≤
C
3

B

(
1

s
)
/
s
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
σ
′
B
,
where
σ
′
>
σ
>
1
. So, we get
(29)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
C
1

B

2

1
/
s
+
1
/
n
∥
d
𝕄
s
P
(
u
)
∥
s
,
B
≤
C
4

B

∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
σ
′
B
.
Let
σ
′′
>
σ
′
; we obtain
(30)
∥
𝕄
s
P
(
u
)
∥
loc
Li
p
k
,
Ω
=
sup
σ
′′
B
⊂
Ω

B


(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
≤
sup
σ
′′
B
⊂
Ω
C
4

B

1

(
k
/
n
)

B


1
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
σ
′
B
≤
C
5
sup
σ
′′
B
⊂
Ω
C
4

B


1
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
σ
′
B
≤
C
6
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
.
Theorem 10 has been completed.
Combining Theorems 8 and 9, we obtain the following result easily.
Corollary 11.
Let
u
∈
L
t
(
Ω
,
Λ
l
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
t
<
∞
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Then, there exists a constant
C
, independent of
u
, such that
(31)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
≤
C
∥
d
𝕄
s
P
(
u
)
∥
t
,
Ω
.
3. The Weighted Lipschitz and BMO Norm Inequalities
In this section, we obtain some weighted inequalities for the composition operator
𝕄
s
∘
P
.
The weight function we use here is
A
(
α
,
β
,
γ
,
Ω
)
weight. The
A
(
α
,
β
,
γ
,
Ω
)
class is a new weight class which was first proposed by Xing in [7]. It contains the wellknown
A
r
(
Ω
)
weight as a proper subset.
Definition 12.
One says that a measurable function
ω
(
x
)
defined on a subset
Ω
⊂
ℝ
n
satisfies the
A
(
α
,
β
,
γ
,
Ω
)
condition for some positive constants
α
,
β
,
γ
, if
ω
(
x
)
>
0
a.e., and
(32)
sup
B
⊂
Ω
(
1

B

∫
B
ω
α
d
x
)
(
1

B

∫
B
ω

β
d
x
)
γ
/
β
<
∞
.
Let
u
∈
L
loc
1
(
Ω
,
Λ
l
,
ω
)
,
l
=
0,1
,
2
,
…
,
n
. We say
u
∈
loc
Li
p
k
(
Ω
,
Λ
l
,
ω
)
,
0
≤
k
≤
1
, if
(33)
∥
u
∥
loc
Li
p
k
,
Ω
,
ω
=
sup
σ
Q
⊂
Ω
(
μ
(
Q
)
)

(
n
+
k
)
/
n
∥
u

u
Q
∥
1
,
Q
,
ω
<
∞
for some
σ
>
1
, where
Ω
is bounded in
ℝ
n
,
ω
is a weight, and
μ
is Radon measure defined by
d
μ
=
ω
(
x
)
d
x
.
Let
u
∈
L
loc
1
(
Ω
,
Λ
l
,
ω
)
,
l
=
0,1
,
2
,
…
,
n
. We say
u
∈
BMO
(
Ω
,
Λ
l
,
ω
)
, if
(34)
∥
u
∥
*
,
Ω
,
ω
=
sup
σ
Q
⊂
Ω
(
μ
(
Q
)
)

1
∥
u

u
Q
∥
1
,
Q
,
ω
<
∞
for some
σ
>
1
, where
Ω
is bounded in
ℝ
n
,
ω
is a weight, and
μ
is Radon measure defined by
d
μ
=
ω
(
x
)
d
x
.
Now, we estimate the weighted Lipschitz and BMO norm for the composition operator
𝕄
s
∘
P
.
Theorem 13.
Let
u
∈
L
q
(
Ω
,
Λ
l
,
μ
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
q
<
∞
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Radon measure
μ
is defined by
ω
(
x
)
d
x
=
d
μ
, and
ω
(
x
)
∈
A
(
α
,
β
,
γ
,
Ω
)
for some
α
>
1
, where
1
<
p
<
∞
,
β
=
α
q
/
(
α
p

p

α
q
)
,
γ
=
α
q
/
p
, and
α
p

p

α
q
>
0
. Then, there exists a constant
C
, independent of
u
, such that
(35)
∥
𝕄
s
P
(
u
)
∥
loc
Lip
k
,
Ω
,
ω
≤
C
∥
d
𝕄
s
P
(
u
)
∥
p
,
Ω
,
ω
.
Proof.
Using Hölder inequality with
1
/
q
=
1
/
α
q
+
(
α

1
)
/
α
q
, we have
(36)
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

q
ω
(
x
)
d
x
)
1
/
q
=
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
ω
(
x
)
1
/
q

q
d
x
)
1
/
q
≤
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

α
q
/
(
α

1
)
d
x
)
(
α

1
)
/
α
q
×
(
∫
B
(
ω
(
x
)
1
/
q
)
α
q
d
x
)
1
/
α
q
≤
C
1

B

diam
(
B
)
(
∫
B

d
𝕄
s
P
(
u
)

α
q
/
(
α

1
)
d
x
)
(
α

1
)
/
α
q
×
(
∫
B
ω
(
x
)
α
d
x
)
1
/
α
q
.
Using Hölder inequality with
(
α

1
)
/
α
q
=
(
1
/
p
)
+
(
α
p

p

α
q
)
/
α
q
p
, we obtain
(37)
(
∫
B

d
𝕄
s
P
(
u
)

α
q
/
(
α

1
)
d
x
)
(
α

1
)
/
α
q
=
(
∫
B

d
𝕄
s
P
(
u
)
ω
(
x
)
1
/
p
ω
(
x
)

1
/
p

α
q
/
(
α

1
)
d
x
)
(
α

1
)
/
α
q
≤
(
∫
B

d
𝕄
s
P
(
u
)
ω
(
x
)
1
/
p

p
d
x
)
1
/
p
×
(
∫
B
(
ω
(
x
)

1
/
p
)
α
q
p
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
α
q
p
=
(
∫
B

d
𝕄
s
P
(
u
)

p
ω
(
x
)
d
x
)
1
/
p
×
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
α
q
p
.
Since
ω
(
x
)
∈
A
(
α
,
α
q
/
(
α
p

p

α
q
)
,
α
q
/
p
,
Ω
)
, we get
(38)
(
∫
B
ω
(
x
)
α
d
x
)
1
/
α
q
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
α
q
p
=
(
×
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
p
(
∫
B
ω
(
x
)
α
d
x
)
×
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
p
)
1
/
α
q
=

B

2
/
α
q
×
(
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
p
(
1

B

(
ω
(
x
)
)
α
d
x
)
×
(
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
1

B

×
∫
B
(
1
ω
(
x
)
)
α
q
/
(
α
p

p

α
q
)
d
x
)
(
α
p

p

α
q
)
/
p
)
1
/
α
q
≤
C
2
.
From (36), (37), and (38), we know that
(39)
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

q
ω
(
x
)
d
x
)
1
/
q
≤
C
3

B

diam
(
B
)
(
∫
B

d
𝕄
s
P
(
u
)

p
ω
(
x
)
d
x
)
1
/
p
.
So, we obtain
(40)
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
=
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

d
μ
≤
(
∫
B

𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B

q
ω
(
x
)
d
x
)
1
/
q
×
(
∫
B
1
q
/
(
q

1
)
d
μ
)
(
q

1
)
/
q
=
(
μ
(
B
)
)
(
q

1
)
/
q
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
q
,
B
,
ω
≤
C
4
(
μ
(
B
)
)
(
q

1
)
/
q

B

diam
(
B
)
∥
d
𝕄
s
P
(
u
)
∥
p
,
B
,
ω
.
Based on the definition of the weighted Lipschitz norm and (40), we have
(41)
∥
𝕄
s
P
(
u
)
∥
loc
Li
p
k
,
Ω
,
ω
=
sup
σ
B
⊂
Ω
(
μ
(
B
)
)

(
n
+
k
)
/
n
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
≤
C
5
sup
σ
B
⊂
Ω
(
μ
(
B
)
)

(
n
+
k
)
/
n
+
(
q

1
)
/
q
+
1
+
1
/
n
×
∥
d
𝕄
s
P
(
u
)
∥
p
,
B
,
ω
≤
C
5
sup
σ
B
⊂
Ω
(
μ
(
Ω
)
)

(
n
+
k
)
/
n
+
(
q

1
)
/
q
+
1
+
1
/
n
×
∥
d
𝕄
s
P
(
u
)
∥
p
,
B
,
ω
≤
C
∥
d
𝕄
s
P
(
u
)
∥
p
,
Ω
,
ω
.
Theorem 13 has been completed.
Theorem 14.
Let
u
∈
L
s
(
Ω
,
Λ
l
,
μ
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
∞
, be a differential form in
Ω
. Radon measure
μ
is defined by
ω
(
x
)
d
x
=
d
μ
, and
ω
(
x
)
∈
A
(
α
,
β
,
γ
,
Ω
)
. Then, there exists a constant
C
, independent of
u
, such that
(42)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
,
ω
≤
C
∥
𝕄
s
P
(
u
)
∥
loc
Lip
k
,
Ω
,
ω
,
where
α
,
β
,
γ
are some positive constants.
Proof.
From the definition of the weighted BMO norm, we have
(43)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
,
ω
=
sup
σ
B
⊂
Ω
(
μ
(
B
)
)

1
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
=
sup
σ
B
⊂
Ω
(
μ
(
B
)
)
k
/
n
(
μ
(
B
)
)

(
n
+
k
)
/
n
×
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
≤
sup
σ
B
⊂
Ω
(
μ
(
Ω
)
)
k
/
n
(
μ
(
B
)
)

(
n
+
k
)
/
n
×
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
≤
(
μ
(
Ω
)
)
k
/
n
sup
σ
B
⊂
Ω
(
μ
(
B
)
)

(
n
+
k
)
/
n
×
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
≤
C
sup
σ
B
⊂
Ω
(
μ
(
B
)
)

(
n
+
k
)
/
n
×
∥
𝕄
s
P
(
u
)

(
𝕄
s
P
(
u
)
)
B
∥
1
,
B
,
ω
≤
C
∥
𝕄
s
P
(
u
)
∥
loc
Li
p
k
,
Ω
,
ω
.
Theorem 14 has been completed.
Combining Theorems 13 and 14, we obtain the following corollary.
Corollary 15.
Let
u
∈
L
q
(
Ω
,
Λ
l
,
μ
)
,
l
=
1,2
,
…
,
n
,
1
≤
s
<
q
<
∞
, be a solution of the nonhomogeneous Aharmonic equation (4) in
Ω
. Radon measure
μ
is defined by
ω
(
x
)
d
x
=
d
μ
, and
ω
(
x
)
∈
A
(
α
,
β
,
γ
,
Ω
)
for some
α
>
1
, where
1
<
p
<
∞
,
β
=
α
q
/
(
α
p

p

α
q
)
,
γ
=
α
q
/
p
, and
α
p

p

α
q
>
0
. Then, there exists a constant
C
, independent of
u
, such that
(44)
∥
𝕄
s
P
(
u
)
∥
*
,
Ω
,
ω
≤
C
∥
d
𝕄
s
P
(
u
)
∥
p
,
Ω
,
ω
.
4. Applications
In this section, we use the theorems we obtain to estimate the norms of Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.
Example 16.
Let
u
=
(
u
1
,
…
,
u
n
)
be a map from
Ω
to
ℝ
n
.
J
(
x
,
u
)
is the Jacobian determinant. Now choosing the subdeterminant of the Jacobian determinant,
(45)
J
(
x
j
1
,
x
j
2
;
u
i
1
,
u
i
2
)
=

∂
u
i
1
∂
x
j
1
∂
u
i
1
∂
x
j
2
∂
u
i
2
∂
x
j
1
∂
u
i
2
∂
x
j
2

.
We know that
J
(
x
j
1
,
x
j
2
;
u
i
1
,
u
i
2
)
d
x
j
1
∧
d
x
j
2
is a
2
form. Let
v
=
J
(
x
j
1
,
x
j
2
;
u
i
1
,
u
i
2
)
d
x
j
1
∧
d
x
j
2
. If
v
∈
L
s
(
Ω
,
Λ
2
)
,
1
≤
s
<
∞
, from Theorems 9 and 14, we obtain the following results:
(46)
∥
𝕄
s
P
(
v
)
∥
*
,
Ω
≤
C
∥
𝕄
s
P
(
v
)
∥
loc
Li
p
k
,
Ω
,
∥
𝕄
s
P
(
v
)
∥
*
,
Ω
,
ω
≤
C
∥
𝕄
s
P
(
v
)
∥
loc
Li
p
k
,
Ω
,
ω
,
where
C
is a constant,
0
≤
k
≤
1
,
𝕄
s
is the HardyLittlewood maximal operator,
P
is the potential operator, and
ω
is a
A
(
α
,
β
,
γ
,
Ω
)
weight satisfying
α
>
1
,
β
=
α
q
/
(
α
p

p

α
q
)
,
γ
=
α
q
/
p
, and
α
p

p

α
q
>
0
.
Example 17.
Let
u
(
x
)
=
(
u
1
,
u
2
,
…
,
u
n
)
:
Ω
→
ℝ
n
be a
K
quasiregular mapping,
K
≥
1
. Then,
(47)
v
=
u
i
(
x
)
(
i
=
1,2
,
…
,
n
)
or
(48)
v
=
log

u
(
x
)

is a generalized solution of the following equation:
(49)
div
A
(
x
,
∇
v
)
=
0
,
A
=
(
A
1
,
A
2
,
…
,
A
n
)
.
Here,
A
i
(
x
,
η
)
can be expressed as
(50)
A
i
(
x
,
η
)
=
∂
∂
η
i
(
∑
i
,
j
=
1
n
α
i
,
j
(
x
)
η
i
η
j
)
n
/
2
.
α
i
,
j
in the formula above are some functions, which can be expressed in terms of the differential matrix
D
u
(
x
)
, and satisfy
(51)
C
1
(
K
)

η

2
≤
∑
i
,
j
=
1
n
α
i
,
j
η
i
η
j
≤
C
2
(
K
)

η

2
,
where
C
1
(
K
)
,
C
2
(
K
)
>
0
. If we assume that
𝕄
s
is the HardyLittlewood maximal operator,
P
is the potential operator,
0
≤
k
≤
1
, and
ω
is a
A
(
α
,
β
,
γ
,
Ω
)
weight satisfying
α
>
1
,
β
=
α
q
/
(
α
p

p

α
q
)
,
γ
=
α
q
/
p
, and
α
p

p

α
q
>
0
, according to Theorem 13 and Corollary 15, we obtain the following inequalities:
(52)
∥
𝕄
s
P
(
v
)
∥
loc
Li
p
k
,
Ω
,
ω
≤
C
∥
d
𝕄
s
P
(
v
)
∥
p
,
Ω
,
ω
,
∥
𝕄
s
P
(
v
)
∥
*
,
Ω
,
ω
≤
C
∥
𝕄
s
P
(
v
)
∥
p
,
Ω
,
ω
,
where
C
is a constant.