Applied Mathematics, 2011, 2, 694-698
doi:10.4236/am.2011.26091 Published Online June 2011 (http://www.SciRP.org/journal/am)
On Riesz Mean Inequalities for Subelliptic Laplacian
Gao Jia, Jianming Wang, Ya Xiong
College of Science , U ni versi t y of Sh ang h ai fo r Scie nce an d Tech nology, Shanghai, China
Received March 23, 20 1 1; revised April 6, 2011; accepted A p r il 13, 2011
Abstract
In this paper, we mainly focus on the Riesz means of eigenvalues of the subelliptic Laplacian on the Heisen-
berg group . We establish a trace formula of associated eigenvalues, then we prove differential inequali-
ties, difference inequalities and monotonicity formulas for the Riesz means of eigenvalues of the subelliptic
Laplacian.
n
H
Keywords: Heisenberg Group, Riesz Mean, Subelliptic Laplacian
1. Introduction
Until now, the eigenvalue estimations of Laplacian on the
bounded Euclidean domain have been extensively studied
(see [1-5]). In recent years, some academics have already
started to pay attention to the Heisenberg group n
H
, such
as P. Levy-Bmhl [6], D. Müller [7], P. C. Niu [8], G. Jia [9]
and so on.
The Heisenberg group plays an important role in several
branches of mathematics such as representation theory,
harmonic analysis, several complex variables, partial dif-
ferential equations and quantum mechanics. In the past
decades research on Heisenberg sub-Laplacian has
achieved considerable progress. But the problem of the
invariant differential operator eigenvalue for the Heisen-
berg group, did not be studied deeply.
In this paper, the Riesz mean inequalities of eigenvalues
for the subelliptic Laplacian is treated. And some differen-
tial inequalities and difference inequaliti es are established.
The outline of the paper is as follows. In Section 2, we
first recall some definitions and the lemmas that will be
used in the following, and then establish the trace formula
of eigenvalues. Main results and their proofs will be given
in Section 3.
2. Preliminaries and Trace Formula
Let n
H
denote Heisenberg group which is a Lie group
that has algebra , with a nonabelian group law
21n
gR


 
 
1112 22
121212 2121
,, ,,
,, 2
xyt xyt
x
xyyt tyxxy

  (2.1)
For every
 
11112 222
,, ,,,n uxytuxytH
The Lie algebra is generated by the left invariant vector
fields
2 2,
1, 2,,,
iiii
ii
XyYx
x
ty
in
t


 



(2.2)
And Tt



. We set






11
,,, ,,
nnn
HuXuXuYu Yu .
Remark 2.1
It is easy to see that i
X
, , are skew symmetr ic
operators, and i
YT




,4, ,,
ijij ii
XYTXT YT 
 0,


 
where


,
X
Y denotes the standard commutator
X
YYX

.
Definiti on 2.1 [10].
The subelliptic Laplacian is defined as


22 2
22
22
11
22
22
2
4
44
n
nn
ii i
Hii i
ii
iii
i
XY x
yt xy
yxy
xt t

 
 







 

(2.3)
By the definitions and properties of i
X
and i Y, it is
easy to see that n
H

is invariant with respect to left-
translations.
Let us concern with the eigenvalue problem
, in ,
0, on .
n
Huu
u


 






 (2.4)
where

is a bounded domain of the Heisenberg group


G. JIA ET AL.695
n
H
with smooth boundary. By [8], we see that the
Dirichlet problem (2.4) has a discrete spectrum on a Hil-
bert space with Inner product denoted , , and its ei-
genvalues by 12
0k

  with
lim
kk


12
uu

1,
and orthonormalize its eigenfunctions

2
0,, S so that
,ddd
ij ijij
uuuu xyt 


,,1ij.
Here, denotes the Hilbert space of the func-
tions such that

1,2
S

2
uL

i
X
u,



2
i
Yu L



,
and denotes the closure of .
1,2
0
S

C
L0 For the sake of simplicity, let be a form


22
1
n
n
H
LX



 
Y

.
There will be a distinguish ed subset

12
,,,
j
j
J

 

,,,

c
of the spectrum of , L

11
\ ,,

j
jj
J

    is the complement of
j
J
, and
j
J
P, c
j
J
P will be the corresponding spectral
projections. We shall be interested in traces of


j
J
P
fL ,
where

j
f

is any function defined on the spectrum
of .
L
Definition 2. 2.
If is an increasing sequence of real numbers,
for , the Riesz mean of order

1
kk



0z0

 of


k

can be defined as [11 ,12]
 
1k k
Rz z






 (2.5)
where
 
:max0,
k
z

k
z



  is the ram p function.
Definition 2. 3.
Two symmetric operators ,VW



1,,n
are defined as
. , ,xu Wuyu

 Vu
Remark 2.2.
,
,0,
I
XV









 
,
,
,0,
I
YW









 

,0 , XW YV
 



0
,
here
I
is an identity operator. In fact, we have
  
,2
2
X
Vuxu yxu
xt
uu
x
yu
xt
 













i.e.


,
X
V

I, similarly


,YW I

, and so on.
Theorem 2.1.
Let
j

and
j
u be eigenvalues and -normalized
eigenfunctions of the subelliptic Laplacian. Let
2
L
22
:, ,
jm jmjm
TXuuYuu
 

for ,1,2,jm

 and 1,, n


. Then for each fixed

,
 


,
2
jj jmj
jm
jj
c
qj
jm
j
jm
JJ
jm
j
jq
Jqj
J
ff
fT
fT















 



(2.6)
Specialized to




j
fz
j




 , we can obtain





,:
,:
2
jm
jq
j
m
jm mj
jq
jq zqj
jm
j
R
zT
T
zz
z






















(2.7)
To derive out Theorem 2.1, we need the following
lemma.
Lemma 2.1 [6].
Let 0
x
y

 and 0

. Then

11
yxCy x
yx
 





(2.8)
where
,if 01 or 2
:2
1, if 12
C











.
Proof of Theorem 2.1.
Observe that because

1
j
j
u
 is a complete or-
thonormal system,
22
11 11
,,
nn
jjm jmj
mm
TT XuuYuu



 
 
 m
According to [10] the formal commutator identity






,,,
A
BCBA CABC, we have




,LVXXYY V,X2

 
 

. Similarly, we
get


,2LW Y


 .
Thus
 
22
11
2
1,,, ,
4
n
n
jjm
m
jj
H
TLVuuLWu
u







 


jm
u
(2.9)
By [2], we obtain



 


,
jj
JJ
tr PfLtr PfLXV

, and

G. JIA ET AL. 696






2
,
2
2,
4,
j
jm j
jm
jj
c
qj
jm
Jjm
Jjm
j
jq
Jqj
J
ff
trPfLX uu
fXuu















 




(2.10)
And similarly ,


 

,
jj
JJ
trP fLtrP fL Y W


then






2
,
2
2,
4,
j
jm j
jm
jj
c
qj
jm
Jjm
Jjm
j
jq
Jqj
J
ff
trPfLY uu
fYu u















 



(2.11)
Summing of the (2.10) and (2.11), we obtain





,
22
4
j
jm j
jm
jj
c
qj
jm
J
jm
Jjm
j
jq
Jqj
J
ff
tr PfLT
fT















 



Since



j
jj
jJ
J

f
tr PfL




, we get (2.6), and
the proof of the Theorem is completed.
3. Riesz Means Inequalities
In this section, we derive differential inequalities and
difference inequalities for the Riesz means


1k k
Rz z








. Here

are ordered eigen-
1
kk



values of the sub elliptic Laplacian on a bounded domain.
Theorem 3.1.
For 02

 and 1 z

, then we have
 
11
12
n
Rz Rz
z
 





, (3.1)
 
12
n
Rz Rz
z







, (3.2)
and


12
n
Rz
z






is a n ondecreasing function with r espect to
z.
For 2

 and 1 z

, then we have
 
11
1n Rz Rz
z








(3.3)
 
1
RznRz
z




(3.4)
and


n
Rz
z

  is a nondecreasing function with respect to
z.
Proof. Let the first term on the right of (2.7) be




,: ,
,, :
jm
jm
jm
j
m
jm zmj
zzz
GT














By Lemma 2.1, the expression can be simplified to






,: ,
1
,: 1
1
,:
,, :
2
2
jm
jm
j
jm
jm
j
m
jm zmj
jjm
jm zm
jjm
jm z
zzz
GT
CzT
CzT
































By symmetr y in jm

, extending the sum to all
subtracting the same quantity from the final term in (2.7),
we find
m
 

1
,: 1
2
2,,
j
j
jm
jm zm
Rz CzT
Rz












(3.5)
where



  1
,:
,,
.
jq
j
qj
jq j
jq zqj
Rz
zC
Tz










 
 

We average over 1,,n


 in (3.5),
  
1
11
22
n
jj
j
znRC zRz

 
 






,,

(3.6)
Since
 
1k k
Rz z









, and

RzRz 
1
 
 
RzRz
,

1
zz




1
1jj j
 


(3.7)
By (3.6) and (3.7),
  
1
1
22,,
n
Rz R
CzRzR zz
nn









and then

G. JIA ET AL.697
 
11
22 2
1,
n
RzRzR
CCzz
nnn








 




,
2
(3.8)
Now we separate into three cases.
Case 1. 1

. In this case and 1C




  1
,:
,,
0
jq
jqj
jq j
jq zqj
z
Rz
z
T


















then
 
1
22
10
z
RzR z
nn
 



 
1
which is (3.1).
Since , substituting it to (3.1), we
can get (3.2).
 
1
RzRz




Case 2. 0

. Since the sum defining


,,Rz 
runs over
j
q
z

  ,
 

 
110
2
j
qj qjqj
qj qj
zC C
C




 
 

  



Therefore
  
  

1
11
1
,, 1
1
n
jj
j
Rz Cz
CzRzRz



 
 




 
 

Substituting this into (3.8), we have
 
1
22
1
zRz Rz
nn
 




which is equivalent to (3.1), also we can get (3.2).
On the other hand
 
11
22
121
22
12
0
nn
nn
n
RzzRzz
Rz
zz





 

 
 
 

 
 


 
 





1 
thus

12
n
Rz
z






is a nondecreasin g function with r espect to
z.
Case 3. 2

. Similar to case 2, we obtain



1
,, 1
jq
jq j
z
Rz CTz



 



 

but now 11
2
C


0, then

,, 0Rz 

. Sub-
stituting it into (3.8),
 
1
22
10
CCz
RzR z
nn

 





Then we have
 
11
1n Rz Rz
z






 , which is
(3.3).
Since




1
RzRz



, we have
 
1
RznRz
z


 , which is (3.4).
Similarly
 

1
2
0
nn
nn
RzRzz Rznz
zz

 











thus


n
Rz
z

  is a nondecreasin g function with respect to
z. This completes the proof of the theorem.
4. Acknowledgements
This work was supported by Shanghai Leading Aca-
demic Discipline Project (S30501) and Innovation Pro-
gramm of Shanghai Municipal Education Commission
(10ZZ93).
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