Some Dirichlet Forms and Pseudo Differential
Operators with Conditionally Exponential Convex
Function
Fatma. M. Kandil
Department of Mathematics; Faculty of Science; King Abdulaziz University Jeddah, Saudi Arabia
Email: [email protected]
Abstract
--
In this paper, we present a concept of conditionally
exponential convex functions and we prove that under suitable assumptions it is possible to define a Dirichlet forms constructed by a class of pseudo differential operators with symbols defined in terms of conditionally exponential convex functions.Index Term
--
Pseudo differential operators –Dirichlet forms-conditionally exponential convex function..1- CONDITIONALLY EXPONENTIAL CONVEX FUNCTIONS. Definition 1.1. A real valued function
:
R
n
R
is said to be conditionally exponential convex if for anyn
n
R
x
x
1,
,
andc
1,
,
c
n
R
We have
(
)
(
)
(
)
0
1 ,
k j n
k j
k j k
j
x
x
x
c
c
x
(1.1)
Theorem 1.1.
([8] Theorem 3.7). A continuous function
:
R
n
R
is conditionally exponential convex onR
n if and only if it can be represented in the form:)
(
1
1
)
;
(
)
;
exp(
1
)
(
)
(
22
} 0 \{
2
d
x
x
x
x
x
x
C
n
R
(1.2)
where
C
0
is a constant,
:
R
n
R
is a continuous negative quadratic form onR
n and
is a positive bounded measure onR
n\
0
.Moreover, the triple
C
;
,
is uniquely determined by the function
.
Lemma 1.1.
Let
,
1,
2 be continuous conditionally exponential convex functions. Then(i)
(
)
(
0
)
0
;
(
)
(
)
(
R
n)
(ii)
(
)
C
(
1
2)
for some constantC
0
.(iii)
C
1
1
C
2
2 is a continuous conditionally exponential convex function.For the proof see [1,2,3].
(iv)
1l2 is a continuous conditionally convex function and the inequality)
(
)
(
)
(
12 122
1
l
l
l
holds for all
;
R
n.For the proof of (iv) we have the following:
Lemma 1.2.
Let
2:
R
n
R
be a continuous conditionally exponential convex function and
be its square root then forn
R
,
we have)
(
)
(
)
(
(1.3)
Proof.
By definition of conditionally exponential convex function we have
0
)
0
(
)
(
)
(
)
(
)
(
)
(
)
(
-)
(
)
(
)
0
(
)
(
)
(
det
2 2
2 2
2 2
2 2
2 2
2 2
And therefore by (i) and (ii)
2 2
2 2
2 2
2 2
2 2
))
(
)
(
)
(
(
))
0
(
)
(
2
))(
0
(
)
(
2
(
)
(
2
)
(
2
Or
)
(
)
(
)
(
)
(
)
(
2
2
2
2
Thus we find)
(
)
(
)
(
)
(
2
)
(
))
(
)
(
(
2
2
2
2
Which prove the lemma.
n
R n n
l
d
u
R
L
u
R
H
12,(
)
2(
)
:
(
1
(
))
~
(
)
2
(1.4)
Where
n
R x
dx
x
u
e
u
~
(
)
(
)
on
H
1l2,(
R
n)
we have the norm
u
d
u
n
R l
2 ,
2 1 2
)
(
~
))
(
1
(
For
(
)
2s we obtain the usual sobole v spaces. For more details see [4,5], with this norm the spaceH
1l2,(
R
n)
is a Hilbert space and 0(
)
n
R
C
is a dense subspace of)
(
, 2 1l n
R
H
and for allu
H
1l2,(
R
n)
,2
0 2 1 2
0 ,
2 1 2
)
(
D
u
u
u
l
lWhere
n
R
l x l
d
u
e
x
u
D
)
(
~
)
(
)
(
)
(
122 1
Using Lemma 1.1, we can construct the chain
)
(
)
(
)
(
12, 21 n l n n
R
L
R
H
R
H
(1.5)
Proposition 1.1.
Suppose
1 and
2 are two continuous conditionally exponential convex function and that
2(
)
1(
)
(1.6)Holds for all
R
n.
Then
it
follows
that
1
2 1 ,
2
1
,
2
C
u
lu
(1.7)
For all
u
H
1l2,(
R
n)
. In particular 12, 1( )n
R l
H
is continuously embedded into the spaceH
1l2,2. Since the function,
(
1
2)
1l2 is a continuous conditionally convex function then for any]
2
,
0
(
t
it follows that]
2
,
0
(
),
(
1
)
1
(
2 12
t
C
t
l
(1.8)
A Dirichlet form on
L
2(
R
n)
is a closed symmetric non negative bilinear formB
with domainD
(
B
)
such that)
(
B
D
u
impliesu
1
D
(
B
)
and)
,
(
)
1
,
1
(
u
u
B
u
u
B
The pair
(
D
(
B
),
B
)
is called Dirichlet space [9,10,11] 2. A CLASS OF CLOSED BILINEAR FORMSLet
m
N
;
n
j
N
for1
j
m
,
letR
R
njj
:
be a continuous conditionally exponential convex function
0 \
)
cosh(
1
(
)
(
nj
R
j j j j
j
x
dx
(2.1)
j
n nj j
R
where
j is a positive finite symmetricmeasure on nj
\
0
.
R
We denote by
n
1
2
m the image of
junder the mapping
)
0
,
,
0
,
,
0
,
,
0
(
;
:
j
j
j
R
R
x
x
T
j
is denoted by
. j
Letb
jL
(
R
)
be independent ofx
j.
we denote by),
,
,
;
,
,
(
'
jx
1x
j 1x
j 1x
mx
x
k
x
k,
1
k
m
,
and we identify
m j
j
R
R
R
R
R
j
1
1 1
'
with a subspaceof
R
;
b
j
b
j(
x
'
j).
Let
:
R
n
R
be continuous conditionally exponential convex function
mj
j j 1
)
(
)
(
in this case
n
R
d
u
R
L
u
R
H
1/2,(
)
2(
)
:
(
1
(
))
~
(
)
2
And
H
1/2,(
R
);
(.
,.)
1/2,
is a Dirichlet space. For)
(
, 2 /
1
R
H
u
the function(
u
1
)
H
1/2,(
R
).
For 0
(
)
C
R
we define),
(
)
'
(
)
(
1
x
A
x
b
x
L
j jm
j
j
Where
1
j
m
,
R
j j x
j
x
e
d
A
(
)
.(
)
ˆ
(
)
ˆ
d
x
b
e
x
L
R
m
j
j j j j x
)
(
~
)
(
)
'
(
)
(
1
Since
b
j is independent ofx
j we can associate withL
a symmetric bilinear form defined on 0(
)
R
C
by,
)
(
)
(
)
'
(
)
,
(
)
,
(
1/21
2 / 1
0
b
x
A
x
A
x
dx
L
B
jm
j R
j j j
n
Where
A
1j/2 has the symbola
1j/2,
1
j
m
.
Proposition 2.1.
For all
,
0(
)
C
R
we have
, 2 / 1 , 2 / 1
)
,
(
C
B
(2.2)
Therefore,
B
has a continuous extension toH
1/2,(
R
n)
.Proof.
Since
C
0(
R
n)
is dense inH
1/2;(
R
n)
it is sufficient to prove (2.2) for all test functions. For,
0(
)
n
R
C
itfollows that
mj
j j m
j
j j
j
j
x
A
x
A
x
C
A
A
b
B
1 0
2 / 1
0 2 / 1
1
0 2 / 1 2 / 1
))
(
);
(
)
'
(
(
)
,
(
Since
, 2 / 1 0
2 / 1
C
A
j
for all 0(
)
n
R
C
it followsthat
B
can be regarded as a continuous bilinear form on)
(
, 2 /
1 n
R
H
.Proposition 2.2.
Let
B
as in Proposition 2.1 and suppose that there exists a constantd
0
0
such thatn
j
d
x
b
j(
j)
0for
all
1
,
,
'
(2.3)
Then we have for all
H
1/2,(
R
n)
0
)
,
(
B
(2.4)
And 2
0 0 2
, 2 / 1 0
)
,
(
d
d
B
(2.5) Proof.
For
H
1/2,(
R
n),
we have
0
)
(
)
(
)
(
,
)
(
)
,
(
2
1 2 / 1 0
1
2 2 / 1 '
0 1
2 / 1 2 / 1 '
dx
x
A
d
dx
x
A
x
b
A
A
x
b
B
m
j R j m
j R
j j j m
j
j j j j
n n
(2.6) Which gives (2.4)
Note that (2.5) is a Carding-inequality [9,10] and we get by (2.6) and (2.3)
2
0 0 , 2 / 1 0
2 0
2
1 0
2
1 0
)
(
|
)
(
~
|
)
(
1
)
(
~
)
(
)
,
(
d
d
d
d
d
d
d
d
B
n n
n
R R
m
j
j j R
m
j
j j
From proposition 2.1 and 2.2 it follows that
B
with domain)
(
)
(
B
H
1/2,R
nD
is a closed symmetric bilinear form onL
2(
R
n).
Theorem 2.1.
Let
B
as in Proposition 2.2 For
,
H
1/2,(
R
n)
we have
n n
R R
dx
dy
x
J
x
y
x
x
y
x
B
(
,
)
(
(
)
(
))(
(
)
(
))
(
,
)
(2.7)
Where
mj
j j
j
x
dy
b
dy
x
J
1 '
)
(
)
(
2
1
)
,
(
(2.8) Proof.
Using the notations introduced above, (2.1), we write
)
,
(
)
(
x
x
jx
'j
And
R
j j j x j
j
j
x
e
x
x
dx
F
j)
,
(
)
,
)(
(
'
'(2.9)
n j R j j j j j n j R j j R j j j j j j j j j j mj R R
j j j j j j n j R j j j j n n n n
x
d
x
I
x
b
x
d
d
x
F
x
F
A
x
b
x
d
dx
x
A
x
A
x
b
dx
x
A
x
A
x
b
B
1 1 ' 1 2 / 1 2 / 1 ' 1 2 / 1 2 / 1 ' 1 1 1~
)
~
(
)
~
(
~
)
~
,
)(
(
~
,
)(
)(
(
)
(
~
)
(
)
(
)
(
)
(
)
(
)
(
)
,
(
Where
R j j j j j j j j j jj
x
A
F
x
F
x
d
I
(
~
)
(
)(
)(
~
)
(
)(
,
~
)
For
f
,
g
L
2(
R
n)
let
R R R R
z y x
dxdydz
x
g
x
f
e
t
g
t
f
(
)
(
)
( )(
)
(
)
It follows that
R R j j j j j j j j j j j jj
x
y
F
x
F
x
dy
d
I
(
~
)
(
1
cosh(
))(
)(
,
~
)
(
)(
,
~
)
~
(
)
j j R R R R j j j j j j j j j j j j j j j j j j j j R R R R
y x j j j y x R R j j j j j j j j j y y j j
dx
dy
x
y
x
x
y
x
x
d
dy
x
x
x
y
x
x
x
x
y
x
dx
dy
d
d
x
e
e
x
F
e
e
dy
d
x
F
x
F
e
e
x
I
j j j j j j j j j j j j)
(
))
(
)
(
))(
(
)
(
(
2
1
~
)
(
~
)
~
,
(
)
~
,
(
))(
~
,
(
)
~
,
(
(
2
1
)
(
~
)
~
,
)(
(F
)
1
(
)
~
,
)(
)(
1
(
2
1
)
(
~
)
~
;
)(
)(
~
;
)(
)(
1
)(
1
(
2
1
)
~
(
j
Which finally gives
n j j j R R j j j nj R R R
j j
dx
dy
x
b
x
y
x
x
y
x
x
d
dx
dy
x
y
x
x
y
x
x
b
B
n n n n 1 1,
)
(
)
~
(
2
1
)).
(
)
(
))(
(
)
(
(
~
)
(
))
(
)
(
).(
(
)
(
(
)
~
(
2
1
)
,
(
1
And therefore the theorem is proved. Now it is easily seen that
Theorem 2.2.
Suppose that
B
satisfies the assumptions of Proposition 2.2 and Theorem 2.1. Then(
H
1/2,(
R
n);
B
)
is a Dirichlet space.Proof.
REFERENCES
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convolution Semi-groups’’ Proc. Pakistan Acad Sci., 24(3), (1987), 249-259..
[3] C. Berg, and G. Forst, Potential theory on locally compact abelian group. Berlin, Heidelberg, New Yark, Springer Verlag, 1975 [4] E. Popescu, A note on Feller semi groups Potential Analysis
14(2001),207-209.
[7] H.A. Ali, Pseudo Differential operator with conditionally exponential Convex function and Feller semigroups, A. M. S. E. Vol 40,, No 3 (2003), 31-43.
[8] M. S. Elshazly, Ph. D. Thesis, Al-Azhar University Cairo. Egypt, 1991.
[9] N. Jacob, A G
aˆ
rding inequality for certain anisotropic Pseudo differential operators with non-smooth symbols, Osaka J. Math., T.26, 1989, p.857-879.[10] N. Jacob, Pseudo-differential operators and Markov processes vol.1: Fourier analysis and semi groups. Imperial College Press, London, 2001.
