RESEARC H Open Access
Some Orlicz norms inequalities for the composite
operator T ∘ d ∘ H
Zhimin Dai
*
, Yong Wang and Gejun Bao
* Correspondence: zmdai@yahoo.
cn
Department of mathematics,
Harbin Institute of Technology,
Harbin, 150001, China
Abstract
In this article, we first establish the local inequality for the composite operator T ∘ d ∘
H with Orlicz norms. Then, we extend the local result to the global case in the L

(μ)-
averaging domains.
Keywords: composite operator, Orlicz norms, L
?φ?
(?μ?)-averaging domains
1 Introduction
Recently as generalizations of the functions, differential forms have been widely used in
many fields, such as potential theory, partial differential equations, quasiconformal
mappings, and nonlinear analysis; see [1-4]. With the development of the theory of
quasiconformal mappings and other relevant theories, a series of results about the
solutions to different versions of the A-harmonic equation have been found; see [5-9].
Especially, the research on the inequalities of the various operators and their composi-
tions applied to the solutions to different sorts of the A-harmonic equation has made
great pro gress [5]. The inequalities equipped with the L
p
-norm for differential forms

1
loc
(R
n
)
and ν >0a.e;see
[17]. |D| is used t o denote the Lebesgue measure of a set D ⊂ ℝ
n
,andthemeasureμ
Dai et al. Journal of Inequalities and Applications 2011, 2011:105
/>© 2011 Dai et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution
License ( which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
is defined by dμ = ν(x)dx.Weuse||f||
s,O
for
(

O
|f |
s
dx)
1
s
and ||f||
s,O,ν
for
(

O

2
···j

(x)dx
j
1
∧ dx
j
2
···∧dx
j

in ℝ
n
,whereJ =(j
1
, j
2
, , j
ℓ
), 1 ≤
j
1
<j
2
< <j
ℓ
≤ n, ℓ = 0, 1, , n, are the ordered ℓ-tuples. The Grassman algebra Λ
ℓ
is a

J
with sum-
mation over all ℓ-tuples J =(j
1
, j
2
, , j
ℓ
), ℓ = 0, 1, , n. Let C
∞
(Θ, ∧
ℓ
) be the set of infi-
nitely differentiable ℓ-forms on Θ ⊂ ℝ
n
, D’(Θ, Λ
ℓ
) the space of all differential ℓ-forms
in Θ and L
s
(Θ, Λ
ℓ
)thesetoftheℓ-forms in Θ satisfying

Θ
(
J
|ω
J
(x)|

i
dx
i
∧ dx
j
1
∧ dx
j
2
···∧dx
j

(1:1)
for all ħ Î D’( Θ, Λ
ℓ
), and the Hodge codifferential operator d
⋆
is defined as d
⋆
= (-1)
nℓ+1
⋆ d⋆ : D’(Θ, Λ
ℓ+1
) ® D’(Θ, Λ
ℓ
), where ⋆ is the Hodge star operator.
With respect to the nonhomogeneous A-harmonic equation for differential forms, we
indicate its general form as follows:
d
∗

|A(x, h)| ≤ a|h|
s-1
, A(x, h)·h ≥ | h|
s
,and|B(x, h)| ≤ b|h|
s-1
for almost every x Î Θ
and all h Î Λ
ℓ
(ℝ
n
). Here a, b >0aresomeconstants,and1<s < ∞ is a fixed expo-
nent associated with (1.2). A solution to (1.2) is an element of the Sobolev spa ce
W
1
,s
loc
(Θ, Λ
−1
)
such that

Θ
A(x, d
¯
h) · dψ + B(x, d
¯
h) · ψ =
0
(1:3)

x, d
¯
h
)
=0
,
(1:4)
which is called the (homogeneous) A-harmonic equation.
In [15], Iwaniec and Lutoborski gave the linear operator K
y
: C
∞
(Θ, Λ
ℓ
) ® C
∞
(Θ, Λ
ℓ-
1
)as
(K
y
¯
h)(x; θ
1
, , θ
−1
)=

1

(1:5)
where
υ ∈ C
∞
0
(Θ
)
is normalized so that

Θ
υ(y)dy =
1
.Theℓ-form ħ
Θ
Î D’(Θ, Λ
ℓ
)is
given by
¯
h
Θ
= |Θ|
−1

Θ
¯
h(y)dy( =0
)
, ħ
Θ

that has generalized gradie nt. We define the harmonic ℓ-fields by
H
(
Θ, Λ

)
= {Θ ∈ W
(
Θ, Λ

)
: d
¯
h = d

¯
h =0,
¯
h ∈ L
s
(
Θ, Λ

)
for some 1 < s < ∞
}
and the
orthogonal complement of
H
(

h − G
(
¯
h
),
(1:6)
where ħ is in C
∞
(Θ, Λ
ℓ
), Δ = dd
⋆
+ d
⋆
d is the Laplace-Beltrami operator, and
G : C
∞
(
Θ, Λ

)
→ H
⊥
∩ C
∞
(
Θ, Λ

)
is the Green operator.

|f |
χ

dμ ≤ 1

.
(2:1)
The following class G(p, q, C) is introdu ced in [19], which is a special pro perty of a
Young function.
Definition 2.2. Let f and g be correspondingly a convex increasing function and a
concave increasing function on [0, ∞). Then, we call a Young function  belongs to the
class G(p, q, C), 1 ≤ p <q < ∞, C ≥ 1, if
(i)
1
C
≤
ϕ(t
1
p
)
f
(
t
)
≤ C, (ii)
1
C
≤
ϕ(t
1

3
t
q
,
(
ii
)
C
2
t
p
≤ f
−1
(
ϕ
(
t
))
≤ C
3
t
p
,
(2:3)
where C
1
, C
2
, and C
3

dx

γ
β
< ∞
,
(2:4)
where the supremum is over all balls O with O ⊂ Θ. We write ν(x) Î A(a, b, g; Θ).
Remark.NotethattheA(a , b, g; Θ)-class is an extension of some existing classes of
weights, such as
A
Λ
r
(Θ
)
-weights, A
r
(l, Θ)-w eights, and A
r
(Θ)-weights. Taking the
A
Λ
r
(Θ
)
-weights for example, if
α
=1,β =
1
r

)
, and dμ = ν(x)dx,
where ν(x) Î A(a, b, a, Θ) for a >1and b >0with ν(x) ≥ ε >0for any × Î Θ. Then,
there exists a constant C, independent of v, such that
 T(d(H(v))) − (T(d(H(v))))
O

ϕ
(
O,μ
)
≤ C  v
ϕ
(
ρO,μ
)
(2:5)
for all balls O with rO ⊂ Θ, where r > 1 is a constant.
The proof of Theorem 2.4 depends upon the following two arguments, that is,
Lemma 2.5 and Theorem 2.6.
In [9], Xing and Ding proved the following lemma, which is a weighted version of
weak reverse inequality.
Lemma 2.5. Let v be a solution of the nonhomogeneous A-harmonic equation (1.2) in
adomainΘ an d 0<s, t < ∞. Then, there exists a constant C, independent of v, such
that


O
|v|
s

ρ
O
.
(2:7)
For the composite operator T ∘ d ∘ H, we have the following inequa lity with A(a, b,
a; Θ)-weight.
Theorem 2.6. Let us assume, in addition to the definitions of the homotopy operator
T, the exterior derivative d, the projection operator H, and the measure μ in Theorem
2.4, that q is any integer satisfying 1<q<∞, v Î C
∞
(Θ, Λ
ℓ
), ℓ = 1, 2, , n, be a solution
of the nonhomogeneous A-harmonic equation (1.2) in a bounded convex domain Θ and
Dai et al. Journal of Inequalities and Applications 2011, 2011:105
/>Page 4 of 12


O
|T(d(H(v))) − (T(d(H(v))))
O
|
q
dμ

1
q
≤ Cdia m( O) |O|



1
q
(2:8)
for all balls O with rO ⊂ Θ for some r >1.
For the purpose of Theorem 2.6, we will need the following Lemmas 2.7 (the general
Hölder inequality) and 2.8 that were proved in [5].
Lemma 2.7. Let f and g are two measurable functions on ℝ
n
, a, b , g are any three
positive constants with g
-1
= a
-1
+ b
-1
. Then, there exists the inequality such that
 fg
γ
,Θ
≤ f 
α,Θ
 g
β
,
Θ
(2:9)
for any Θ ⊂ ℝ
n
.
Lemma 2.8. Let us assume, in addition to the definitions of the homotopy operator T,

p
log
α
+
t
belongs to G(p
1
, p
2
, C), 1 ≤ p
1
<p <p
2
, t >0,anda Î ℝ.Herelog
+
t is
a cutoff function such that log
+
t = 1 for t ≤ e otherwise log
+
t =logt. Moreover, if a
= 0, one verifies easily that (t)=t
p
is as well in the class G(p
1
, p
2
, C), 1 ≤ p
1
<p

)
. Then, there exists a constant C, independent of v, such that

O
|T(d(H(v))) − (T(d(H(v))))
O
|
p
log
α
+

|T(d(H(v))) − (T(d(H(v))))
O
|

d
μ
≤ C

ρ
O
|v|
p
log
α
+
|v|dμ
(2:11)
for all balls O with rO ⊂ Θ for some r > 1. The following definition of the L

h
O
|)d
μ
(2:12)
for all Θ such that
ϕ(|Θ|) ∈ L
1
loc
(Θ; μ
)
, where the measure μ is d efi ned by d μ = ν(x)
dx, ν(x) i s a we ight, and τ , s are constants with 0<τ, s ≤ 1, and the supremum is over
all balls O with 4O ⊂ Θ.
By Definition 2.10, we arrive at the following global case of Theorem 2.4.
Theorem 2.11. Let us assume, in addition to the definitions of the homotopy operator
T, the exterior derivative d, the projection operator H, the measure μ, and the Young
function  in Theorem 2.4, that ν Î C
∞
(Θ, Λ
k
), k=1, 2, , n, be a solution of the non-
homogeneous A-harmonic equatio n (1.2) in a bounded L

(μ)-a veraging domains Θ and
(|ν|) Î L
1
(Θ; μ). Then, there is a constant C, independent of ν, such that
 T(d(H(v))) − (T(d(H(v))))
Θ

(
Θ,μ
)
≤ C  v
ϕ
(
Θ,μ
)
.
(2:14)
Remark. Note that the L
s
-averaging domains and L
s
(μ)-averaging domains are also
special L

(μ)-averaging domains. Thus, Theorem 2.11 also holds for the L
s
-averaging
domains and L
s
(μ)-averaging domains, respectively.
3 The proof of main results
In this section, we will give the proof of several theorems mentioned in the previous
section.
Proof of Theorem 2.6.Let
t =
αq
α

q
=


O
(


T(d(H ( v))) − (T(d(H(v))))
O


ν(x)
1
q
)
q
dx

1
q
≤


O


T(d(H ( v))) − (T(d(H(v))))
O


αq
≤ C
2
diam(O) |O|
1+
r−t
rt

v

r,ρ
2
O


O
(ν(x))
α
dx

1
αq
,
(3:1)
where r
2
, r
1
are two constants satisfying r
2

dx

1
r
=


ρ
2
O
(|v|(ν(x))
1
q
· (ν(x))
−1
q
)
r
dx

1
r
≤


ρ
2
O
|v|
q

ρ
2
O
(ν(x))
−β
dx

1
βq
.
(3:2)
Observe that v(x) Î A(a, b, a, Θ), hence


O
(ν(x))
α
dx

1
αq


ρ
2
O
(ν(x))
−β
dx


O|
1+
α
β

1
|ρ
2
O|

ρ
2
O
(ν(x))
α
dx

1
|ρ
2
O|

ρ
2
O
(ν(x))
−β
dx

α

4
diam(O) |O|
1+
r−t
rt
|ρ
2
O|
1
αq
+
1
βq


ρ
2
O
|v|
q
ν(x)dx

1
q
≤ C
5
diam(O) |O|


ρ

≤
C
2
|O|
(3:5)
for all balls O ⊂ Θ.
Dai et al. Journal of Inequalities and Applications 2011, 2011:105
/>Page 7 of 12
We obtain from Theorem 2.6 and Lemma 2.5 that


O
|T(d(H(v))) − (T(d(H(v))))
O
|
q
dμ

1
q
≤ C
1
diam(O) |O|


ρ
1
O
|v|
q

>r
1
> 1 are two constants. Note that  is an increasing func-
tion, and f is an increasing convex funct ion in [0, ∞), by Jensen’sinequalityforf,we
obtain that
ϕ
⎛
⎝
1
χ


O


T(d(H(v))) − (T(d(H(v))))
O


q
dμ

1
q
⎞
⎠
≤ ϕ
⎛
⎝
1


1
χ
p
C
p
2
|
O
|
p
(diam(O))
p
(μ(ρ
1
O))
(p−q)
q

ρ
2
O
|
v
|
p
dμ

1
p

|
p
dμ

= C
3
f


ρ
2
O
1
χ
p
C
p
2
|
O
|
p
(diam(O))
p
(μ(ρ
1
O))
(p−q)
q
|

|
v
|
p

dμ.
(3:7)
Since 1 ≤ p <q < ∞, we have
1+
p−q
pq
=1+
1
q
−
1
p
>
0
, which yields
diam(O) |O|μ(ρ
1
O)
p−q
pq
≤ C
4
diam(Θ)|O||ρ
1
O|

(
t
1
p
)
. Thus,

ρ
2
O
f

1
χ
p
C
p
2
|O|
p
(diam( O))
p
(μ(ρ
1
O))
p−q
q
|v|
p


1
χ
C
9
|v|

dμ
≤ C
10

ρ
2
O
ϕ

1
χ
|v|

dμ.
(3:9)
Dai et al. Journal of Inequalities and Applications 2011, 2011:105
/>Page 8 of 12
Combining (3.7) and (3.9), we obtain that
ϕ
⎛
⎝
1
χ


(μ(ρ
1
O))
(p−q)
q
|v|
p

d
μ
≤ C
11

ρ
2
O
ϕ

1
χ
|v|

dμ.
(3:10)
Applying Jensen’s inequality to g
-1
and considering that  and g are doubling, we
obtain that

O


ϕ

|T(d(H(v))) − (T(d(H(v))))
O
|
χ

dμ

≤ g

C
12

O

|T(d(H(v))) − (T(d(H(v))))
O
|
χ

q
dμ

≤ C
13
ϕ




1
q

≤ C
15

ρ
2
O
ϕ

|v|
χ

dμ.
(3:11)
Therefore,
1
μ(O)

O
ϕ

|T(d(H(v))) − (T(d(H(v))))
O
|
χ

d

χ

dμ.
(3:12)
By Definition 2.1 and (3.12), we achieve the desired result
||T(d(H ( v))) − (T(d(H(v))))
O
||
ϕ
(
O,μ
)
≤ C||v||
ϕ
(
ρO,μ
)
.
(3:13)
With the aid of Definition 2.10, We proceed now to derive Theorem 2.11.
ProofofTheorem2.11.NotethatΘ is a L

(μ)-averaging domains, and  is dou-
bling, from Definition 2.10 and (3.12), we have
1
μ(Θ)

Θ
ϕ


1
1
μ(Θ)
sup
4O⊂Θ

C
2

ρO
ϕ

|
v
|
χ

dμ

≤ C
3
1
μ(Θ)
sup
4O⊂Θ

Θ
ϕ

|

(
Θ,μ
)
≤ C  v
ϕ
(
Θ,μ
)
.
(3:15)
4 Applications
If we choose A to be a special opera tor, for exampl e, A(x, dħ)=dħ|dħ|
s-2
,then(1.4)
reduces to the following s-harmonic equation:
d

(
d
¯
h|d
¯
h|
s−2
)
=0
.
(4:1)
In particular, we may let s =2,ifħ is a function (0-form), then Equation 4.1 is
equivalent to the well-known Laplace’sequationΔħ =0.Thefunctionħ satisfying

log
+
t, p >1,
v =(

n
i
=1
(x
i
− y
i
)
2
)
2
−n
2
and O
={x =(x
1
, , x
n
)| : ε
2
≤ (x
1
- y
1
)

,
(4:2)
v
x
i
x
i
=(2− n)

n

i=1
(x
i
− y
i
)
2

−(n+2)
2

n

i=1
(x
i
− y
i
)

n
denotes the volume of a unit ball in ℝ
n
(n >2),
and
1 <
1
r
n−2
≤|v| = |(

n
i
=1
(x
i
− y
i
)
2
)
2−n
2
|≤
1
ε
n−2
, applying (3.11) with c =1,dμ = dx,
we obtain
Dai et al. Journal of Inequalities and Applications 2011, 2011:105



)d
x
≤ C


ρO
|
v
|
p
log
+
|
v
|
dx

≤ C

1
ε
(n−2)

p
log
1
ε
(n−2)

log
1
ε
(n−2)
.
(4:5)
Example 4.2. Let us assume, in addition t o the definitions of ε, r,  of Example 4.1,
that y =(y
1
, y
2
) be a fixed point in ℝ
2
,
v =log(

2
i
=1
(x
i
− y
i
)
2
)
1
2
and O ={x =(x
1

i
=1
(x
i
− y
i
)
2
,
(4:6)
v
x
i
x
i
=

2
i=1
(x
i
− y
i
)
2
− 2(x
i
− y
i
)

which implies the function v is harmonic.
With respect to the estim ation of

O
ϕ(|T(d(H(v))) − (T(d(H(v))))
O
|)d
x
,Example
4.2 proceeds in much the same way after replacing |O|=s
n
r
n
and
1 < |v|≤
1
ε
n−2
with |
O|=πr
2
and |log ε|<|v| ≤ |log r| < 1, respectively. Here we omit the reminder
process.
Acknowledgements
The authors wish to thank the anonymous referees for their time and thoughtful suggestions.
Authors’ contributions
ZD finished the proof and the writing work. YW gave ZD some excellent advices in the proof and writing. GB gave
ZD lots of help in selecting the examples as applications. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.

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Cite this article as: Dai et al.: Some Orlicz norms inequalities for the composite operator T ∘ d ∘ H. Journal of
Inequalities and Applications 2011 2011:105.
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