Two-Weight Poincarè Inequalities for Differential Forms in Averaging
Domains
David G. Saunders
Department of Mathematics
College of Science and Engineering
Seattle University
900 Broadway
Seattle, WA 98122. USA
Abstract
In this paper we first prove some local two-weight Poincarè inequalities for differential forms then discuss
s
global two-weight Poincarè inequalities for differential forms in a new kind of domain, the L
averaging domain, which was introduced by S. Ding and C. Nolder in their recent paper [1].
( ) -
1. Introduction
Poincarè inequalities are a mathematical tool that help us make integral estimates by using derivatives.
This is extremely helpful when we don’t have the function itself, but we do have its derivative. An
example of this would be a partial differential equation. In this case, du would be available, but u would
not be. Poincarè inequalities are an extremely widely used tool, the American Mathematical Society had
about 129 papers regarding these inequalities on their website, http://ams.org.
Poincarè inequalities only let us estimate our functions in certain domains, or open subsets of  . John
n
domains are very commonly used. These domains are relatively smooth, like a sphere. L (  ) - averaging
domains let us use our inequality over domains that have sharp corners. An example of a domain with a
sharp corner is the equation f ( x)  1 x rotated about the x-axis.
s
Differential forms are generalizations of functions in  . For example,
n
u ( x, y, z )dx  v( x, y, z )dy  w( x, y, z )dz is a differential -form in  3 if u , v, w are differentiable.
Also, f ( x, y , z ) is a differential 0-form in  if f is differentiable. In  , a differential
3
form:
u ( x) 
u
i1i2 ..ik
n
( x ) dxi1  dxi2  ...  dxik
k -form has the
. Differential forms have many
applications in different fields, including potential theory, partial differential equations, nonlinear analysis,
differential geometry and control theory, etc. In recent years, many interesting results regarding differential
forms have been found (see [1-4] for more information). Many of these applications involve integral
estimates for differential forms (Vector fields, which are widely used in physics and electromagnetism, are
special cases of differential forms).
A weight is a locally integrable nonnegative function in  . In Physics you assign density to an object,
this is a type of weight. Weights in Poincarè inequalities let you integrate over other types of measures (i.e.
volume to density).
n
1
Definition 1.1: Let w be a locally integrable nonnegative function in  and assume that 0  w  
almost everywhere. We say that w belongs to the Ar  class 1  r   and 1     , or that w is an
n
 
Ar   -weight and write w Ar   if
 
1 ( r 1)
 1
 1

1
sup 
w dx 
dx 


w
B | B | B
 | B | B

n
for all balls B   .
( r 1)

Definition 1.2: We call a proper subdomain    an L
n
and if there exists a constant
s
(  ) -averaging domain, s  1, if    
C such that
1s
1s
 1

 1

s
s

  C sup 

|
u

u
|
d

|
u

u
|
d

B
B


0
  (B )

  ( B)

2
B


0

B




s
l
for some ball B0   and all u  Lloc (, ). Here the measure  is defined by d  w( x)dx ,
where w( x )  0 almost everywhere, and the supremum is over all balls B  .
 l to mean all l  forms, the space of differential l  forms by D, l  , and
Lp , l  as the l  forms w( x)   wI ( x)dxI   wi1i2 .... il ( x)dxi1  dxi2  ...  dxil with
We will denote
I
wI  L (, ) for all ordered l  tuples I. Note that the definitions above can be found in [3].
p
2. Local Two-Weight Poincarè Inequalities for Differential Forms
In this section we will prove two local two-weight Poincarè inequalities.
Theorem 2.1: Let


u  D' ( B, l ) and du  Lp B, l 1 , where u B 
1
udu and l  0,1,2,...n.
| B | B
For 1  s  n,   0, and ( w1 , w2 )  A1 ns s n s  , there exists a constant C independent of
u and
du such that
1
 1 1

s

s
  | u  u B | w1 dx   C | B | n s


B

1

n
  | du |n w2 dx 


B

(2.1)
for all balls B   .
n
We use the notation
1

p
|| u || p ,,w    | u | p w( x)dx 


which means we can write (2.1) as
 1 1
|| u  uB ||s , B, w1  C | B |
n

s
|| du ||n, B, w2
In order to prove this theorem we need a few lemmas. The first is the generalized Hölder inequality that
can be found in [3]. The second Lemma can also be found in [3].
2
Lemma 2.2 (Hölder’s Inequality): Let 0 < α < ∞ , 0 < β < ∞ , and s-1 = α-1 + β-1. If f and g are measurable
functions on Rn, then
|| fg || s , E || f || , E  || g ||  , E .
(2.2)
for any set E   .
n
Lemma 2.3: Let

u  D' ( B, ) and du  L B,
l
np


  | u  u B | n p dx 
B


p
( n p )
np
l 1
 . Then u  u
np
n p
Q
is in L
Q,  and
l
1/ p


 C1   | du | p dx 
B

(2.3)
where Q is a cube or a ball in  , with l  0,1,2,...n and 1 < p < n.
n
We will now prove the following version of the two-weight Poincarè inequality.
Theorem 2.4: Let


u  D' ( B, l ) and du  Lp B, l 1 , where u B  1
| B | B
udu and l  0,1,2,...n.
For 1  s  n, 0   , and w1 , w2   An s 1 , there exists a constant C independent of
u and du such
that
1s
1/ n
1
s




  | u  u B | w1 dx   C | B | s   | du |n w2 dx 




B

B

n
for all balls B   .
Proof: Let (w1 , w2 )  Ar ( ) where r = ns and   1 . Using (2.2) and (2.3) we get
1s
s


1/ s 

   | u  u B |w1  dx 



 B

1s
s


  | u  u B | w1 dx 


B

1n


n/s
   w1 dx 
B

ns
sn

 ns
  | u  u B | ns dx   C1 || w11/ s || n,B || du || s ,B


B

Applying Hölder’s inequality again, we find that
1/ s


1/ s




1/ n
1 / n s
|| du || s ,B    | du |s dx  =   | du | w2 w2
dx 
B

B

n s


1/ n

1/ n


1/ n n
   | du | w2
dx 
B

 1  ns

 ns
 
  1   n  n s dx 
   w2 

B

n s

(2.4)


1/ n n
   | du | w2
dx 
B

s

 ns
  1  ns dx 
   w2 

B

3
(2.5)

1/ n



1/ n n
   | du | w2
dx 
B

|| 1
w2
||1s//(nns ), B
(2.6)
Combining (2.5) and (2.6) yields
1s
s


  | u  u B | w1 dx   C1 || w11/ s || n,B || 1 ||1s//(nns ),B


w2
B

Since w2  An / s (1) , we obtain
1/ n


1/ s
n/s
|| w1 || n, B || 1 ||1s//(nns ), B    w1 dx 
w2
B

1/ n


  | du |n w2 dx 


B

(2.7)
ns
  1  s /(ns )  ns

dx 
  w2 
B

1
n
1  n

1
s




( n / s ) 1
n/s
   w1 dx   1 
dx  
  w2 

B
B
 


n
1 

1
s
1






(
n
/
s
)

1
1
1
n/s
1 
| B | s 
w1 dx 
dx  






w2 
 | B | B 
 | B | B
 


1/ s
 C2 | B |
1/ n
(2.8)
From (2.7) and (2.8) we find
1s
s


  | u  u B | w1 dx 


B

1/ n


 C | B |   | du |n w2 dx 
B

1
s
This completes the proof of theorem (2.4).
Proof of Theorem 2.1:
(w1 , w2 )  Ar ( ) where r = 1 
1s

n s
s

1s
( ns ) ns
1n


  w1n / s dx 


B

1n

 

n / s
 C1   | du |s dx    w1 dx  .
B
 B

(2.9)
We know:
1s
.
1 1 ns
 
and Lemma 2.3 we get
s n
ns
ns /( ns )


   | u  u B |
dx 
B



  | du | s dx 


B

 sn . Then we have
1s
s




s
  | u  u B | w1 dx     | u  u B |s w1 / s dx 




B

B

By Hölder’s inequality with
and 

1s



 /n
 / n
   | du | s w2 w2
dx 
B

4




 
n  ns 
 /n n
   | du | w2
dx     1 
dx 
w2 

B
  B 

ns


s
1n
ns

    1  ns  

n

   | du | w2 dx    
dx
  w2 
 

B
 B
 



  ns 
1n

( n s )
ns
(2.10)
Combining (2.8), (2.9), and (2.10) we get,
1s
s


  | u  u B | w1 dx 


B

Since
1n
n / s


   w1 dx 
B

 
 
   1  ns  
    w2  dx 
 
B
s
ns
ns
1n


  | du |n w2 dx  .


B

(w1 , w2 )  Ar ( ) we obtain
1n
n / s


  w1 dx 


B

n / s



  w1 dx 
B

=
n

 n
s


  1  ns dx 


   w2 

B

ns
ns
ns
ns
 
n / s

 
ns

   w1 dx     1  dx 


w
2


B
 B  
 
s
s
ns
ns
ns


s
ns

n / s
  1  ns  
   w1 dx   
dx

 
w2 
 B 
 B
 


ns


s
ns


 1
n / s
 1  1  ns  

 w1 dx  | B | B  w2  dx  
 | B | B

 


 1 1
| B |
 
 
   1  ns  
    w2  dx 
 
B
s
 /n
 /n
 1 1
 C2 | B |
n

s
So then we have
1
 1 1

s

s
  | u  u B | w1 dx   C | B | n s


B

1

n
  | du |n w2 dx 


B

This completes the proof of theorem (2.1).
3. Global Two-Weight Poincarè Inequalities for Differential Forms
The following definition of John Domains appears in many papers. See [4] for an example.
  John domain,   0 , if there exists a point
x   by a continuous curve    so that
Definition 3.1: We call  , a proper subdomain of  , a
n
x0   which can be joined with any other point
d  ,    | x   |
5
for each
   . Here d  ,  is the Euclidean distance between  and  .
The following Lemma was proved by C. Nolder in [4].
Lemma 3.2: Let    be a δ-John domain. Then there exists a covering  of  consisting of open
cubes such that:
n

 Q ( x)  N   ( x) ,where x   n .
ii) There is a distinguished cube Q0  , called the central cube, which can be connected with every
cube Q  by a chain of cubes Q0 , Q1 ,, Qk  Q from  such that for each i  0,1,, k  1,
i)
Q
Q  NQi .
Also, there is a cube Ri   , (which does not need to be a member of ) such that
n
Ri  Qi  Qi1 , and Qi  Qi 1  NRi .


u  D' (,l ) and du  Lp ,l 1 , where l  0,1,2,...n. For 1  s  n, 0   ,
and w1 , w2   A1( n s ) s n s  , there exists a constant C independent of u and du such that
Theorem 3.3: Let
1
1
 1
s

n

s

  C   | du |n w2 dx 
|
u

u
|
w
dx
B
1

0
|  |








where  is any
(3.3)
Ls    averaging domain and B0 is some ball in the covering  of  appearing in
Lemma 3.2. It should be noted that all results true for balls are also true for cubes.


u  D' (,l ) and du  Lp ,l 1 , where l  0,1,2,...n. For 1  s  n, 0   ,
and ( w1 , w2 )  An s 1 , there exists a constant C independent of u and du such that
Theorem 3.4: Let
1
1
 1
s

n
s
n




|
u

u
|
w
dx

C
|
du
|
w
dx
B0
1
2
|  | 








s

Where
is any L    averaging domain and B0 is the central cube in the covering  of  .
(3.4)
4. Application
Now we will discuss some more applications of differential forms.
Let D be any domain in  with
n  2. Also, let f : D   n , where f  ( f 1 , f 2 , f 3 ,..., f n ) is a
mapping. The Jacobian matrix Df (x) is defined by
n
 f 1

 x1
Df ( x)   
 f n

 x1

f n
xn


f n

xn




,




which is also called the differential matrix of f . The Jacobian determinant J  J ( x, f ) of f is
defined by
6
J  J ( x, f )  det Df ( x)
From the paper by T. Iwaniec and A. Verdo’s, [5], we know that the Jacobian can be written as
J ( x, f )dx  df 1  df 2  ...  df n
which is an n-form. Our local and global results are true for any number of forms. Thus, replace u by
J ( x, f ) in inequalities (2.1), (2.4), (3.3), and (3.4) and we get the following estimates for the Jacobian:
1
ns
1
s

s
  | J ( x, f )  J ( x, f ) B | w1 dx   C | B | s
0


B

1s
s


  | J ( x, f )  J ( x, f ) B | w1 dx 


B

1

n
  | dJ ( x, f ) |n w2 dx 


B

1/ n


 C | B |   | dJ ( x, f ) |n w2 dx 
B

1
s
1
s
 1





s
n




 |  |  | J ( x, f )  J ( x, f ) B0 | w1 dx   C   | dJ ( x, f ) | w2 dx 





1
(4.1)
(4.2)
1
n
(4.3)
1
 1
s

n
s
n




|
J
(
x
,
f
)

J
(
x
,
f
)
|
w
dx

C
|
dJ
(
x
,
f
)
|
w
dx
B0
1
2
|| 








(4.4)
6. References
[1] S. Ding, C. Nolder (2003), Ls(μ)-Averaging Domains, Journal of Mathematical Analysis and
Applications, 283, 85-99
[2] S. Ding (1998), Some Examples of Conjugate P-Harmonic Differential Forms, Journal of Mathematical
Analysis and Applications, 227, 251-270
[3] S. Ding, P. Shi (1998), Weighted Poincarè-Type Inequalities for Differential Forms in Ls(μ)-Averaging
Domains, Journal of Mathematical Analysis and Applications, 227, 200-215
[4] S. Ding, Yun Ying Gai (2001), Ar -Weighted Poincaré-Type Inequalities for Differential Forms in
Some Domains, Acta Mathematica Sinica, English Series, 287-294
[5] T. Iwaniec and A. Verdo (1999), A Study of Jacobians in Hardy-Orlicz space, Proc. Of the Royal
Society of Edinburg, 128A, 539-579
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