 Open Access
 Total Downloads : 136
 Authors : Kahar ElHussein
 Paper ID : IJERTV2IS110338
 Volume & Issue : Volume 02, Issue 11 (November 2013)
 Published (First Online): 15112013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Abstract Harmonic Analysis and Ideals of Banach Algebra on 3Step Nilpotent Lie Groups
Abstract Harmonic Analysis and Ideals of Banach Algebra on 3Step Nilpotent Lie Groups.
Kahar ElHussein
Department of Mathematics, Faculty of Science, AlFourat University, DeirElZore, Syria
Department of Mathematics, Faculty of Arts and Science, Al Quryyat Al Jouf University, Kingdom of Saudi Arabia
November 1, 2013
Abstract
9
9
Let N be the 6dimensional nilpotent Lie group and let be its vector group. we construct a 9dimensional new group that contains the two groups N and 9. We will define the Fourier transform on N, in order to obtain the Plancherel theorem. Moreover, we show how F. Treves and

Atiyah methods can be used to obtain the division of distributions on

To this end, a classification of all ideals of the Banach algebra Ll(N)
of N will be obtained.
–
–
Keywords: 3 Step Nilpotent Lie Groups, Plancherel Formula, Partial Dif ferential Equations, Ideals of Banach Algebra Ll(N).
AMS 2000 Subject Classification: 43A30, 35D 05, &43A25

Introduction.

The abstract harmonic analysis, is a powerful area of pure mathematics that has connections to analysis, algebra, geometry, theoretical physics and solving problems in robotics, image analysis, mechanics,engineering. Abstract harmonic analysis is the field in which results from Fourier analysis are extended to topological groups which are not commutativ This analysis is generally a hard theory and its difficulty makes the noncommutative version of the problem very challenging. The main task is therefore the case of Lie groups which is locally compact, not compact and not commutative. If the Lie group N is assumed to be noncommutative, it is not possible anymore to consider the dual group N . Recently, this problem found a satisfactory solution with my papers [4, 5, 7].
Here are some interesting examples of these groups. Let G5 be the real group consisting of all matrices of the form
I 1 x1
x
x
2
1 x4
2
2
0 0 0 0 \
0 1 x1 x3 0 0 0 0
2
0 0 0 1 0 0 0 0
2
0 0 0 1 0 0 0 0
x
x
0 0 1 x
0 0 0 0
0 0 1 x
0 0 0 0
– 2
– 2
2 x
2 x
0 0 0 0 1 x
2 x 1x
( )
0 0 0 0 0 1 x2 x3 – x1x2
\ 0 0 0 0 0 0 1 x1
0 0 0 0 0 0 0 1
0 0 0 0 0 1 x2 x3 – x1x2
\ 0 0 0 0 0 0 1 x1
0 0 0 0 0 0 0 1
2 2 5 2
G5 (is called the Cartan group G5 or the generalized Dido problem), where
5 5
5 5
(x5, x4, x3, x2, x1) E JR . Let K = JR be the group with law
1
1
1
1
(x5, x4, x3, x2, x1)(y5, y4, y3, y2, y1)
5
5
= (x + y
+ x y2 – x y
+ x x y , x
+ y + x2y
– x y , y
+ x – x y , x
+ y , x
+ y )
5
5
2
2
1
1
2
2
2
2
3
3
1
1
2
2
2
2
4
4
4
4
2
2
1
1
2
2
1
1
3
3
3
3
3
3
1
1
2
2
2
2
2
2
1
1
1
1
5 5
5 5
for any (x5, x4, x3, x2, x1) E JR and (y5, y4, y3, y2, y1) E JR . The inverse of an element (x5, x4, x3, x2, x1) is
(x5, x4, x3, x2, x1)1
x1 x2
2 1
2
2
2
2
2
2
= (x5 – x – x2x3, x4 – x2 – x1x3, x3 – x1x2, x2, x1) (2)
Dixmier had proved in [2, p.331] that there is a group isomorphism between G5 and K. Thanks to this isomorphism, the group K can be shown as a semi direct product JR3 v JR vJR of the real vector groups JR, JR and JR3 see [4], where
p2 p1
3
3
p2 is the group homomorphism p2 : JR — Aut(JR ), which is defined by
p2(x2 )(y5, y4, y3) = (y5 – x2y3, y4, y3) (3)
3
3
p2
p2
and p1 is the group homomorphism p1 : JR — Aut(JR v JR), which is given by
p1(x1 )(y5, y4, y3, y2) = (y5 +
x1 2
x
x
2
2
2
2
2
2
y2, y4 + 1 y2 – x1y3, y3 – x1y2, y2) (4)
where Aut(JR3) (resp.Aut(JR3 v JR)) is the group of all automorphisms of (JR3)
p2
(resp.(JR3 v JR)). Let N be the real group consisting of all matrices of the form
p2
1
x1
x3
x
0
1
x2
x5
0
0
1
x4
0
0
0
1
1
x1
x3
x
0
1
x2
x5
0
0
1
x4
0
0
0
1
I \
\( )
6
6
E
E
where (x6, x5, x4, x3, x2, x1) JR . The group can be identified with the group
(JR3 >< JR2 ) >< JR be the semidirect product of the real vector groups JR, JR2 and
p2 p1
JR3, where p2 is the group homomorphism p2 : JR2 — Aut(JR3), which is defined
by
by
by
by
3 2
3 2
p2
p2
p2(x3, x2 )(y6, y5, y4) = (y6 + x3y4, y5 + x2y4, y4) (6) and p1 is the group homomorphism p1 : JR — Aut(JR >< JR ), which is given by
p1(x1 )(y6, y5, y4, y3, y2) = (y6 + x1y5, y5, y4, y3 + x1y2, y2) (7) where Aut(JR3) (resp.Aut(JR3 >< JR2)) is the group of all automorphisms of (JR3)
p2
(resp.(JR3 JR2)), see [4]. and the inverse of an element
p2
(x6, x5, x4, x3, x2, x1) = (x6x3x4x1x5x1x2x4, x5x2x4, x4, x3x1x2, x2, x1)
(8)
(8)
–
–

Using the technique in [4, 6, 7] as a guideline, our goal in this paper is to generalize the classical Fourier transform on 3 step nilpotent Lie groups, to obtain the following results

Plancherel formula on these groups theorem 2.1

Theorems 3.1 and 3.2 for the division of distributions, and classification of all left ideals of Banach algebra L1(N) theorem 4.1 and corollary 4.2
The point I wish to make in this paper that the Fourier transform is ex actly the classical Fourier transform on JRn. Therefore, I do hope this paper will be intended to draw the attention of Analysts and Algebraist to this new way. Due to the analogues structure of two groups, it suffices to study the non commutative Fourier analysis on one of the them, for example N.



Plancherel Formula on N.

In the following we supply JR9 a new structure of group by defining on
JR9 = JR3 x JR2 x JR2xJR x JR a new multiplication as:
X.Y = (x6, x5, x4, x3, x2, t3, t2, x1, t1)(y6, y5, y4, y3, y2, s3, s2, y1, s1)
= ((x6, x5, x4, x3, x2, t3, t2)(p1(t1)(y6, y5, y4, y3, y2, s3, s2), y1 + x1, s1 + t1)
= ((x6, x5, x4, x3)p2(t3, t2)(y6 + t1y5, y5, y4, y3, s3 + t1s2, s2), x2 + y2, y1 + x1, s1 + t1)
= (x6, x5, x4) + (y6 + t1y5 + t3y4, y5 + t2y4, y4), x3 + y3, s3 + t1s2, s2 + t2, x2 + y2, y1 + x1, s1 + t1)
= (x6 + y6 + t1y5 + t3y4, x5 + y5 + t2y4, x4 + y4, x3 + y3, t3 + s3 + t1s2,
y2 + x2, s2 + t2, y1 + x1, s1 + t1) (9)
9 9
9 9
for all (X, Y ) E JR x JR . In this case the group N can be identified with the closed subgroup JR3 x{O}>< JR2x {O} >< JR of JR9 and M with the closed subgroup
p1 p1
JR3 x JR2 x {O}xJR x {O} = JR6 of L.
U
U
E U
E U

Let CCX(N) be the space of CCX functions on N. Let be the com plexified universal enveloping algebra of the real Lie algebra n of N; which is canonically isomorphic to the algebra of all distributions on N supported by the identity element O of N. If v , then we can associate a right invariant differential operator (ID) on N by:
r
r
(ID'f?)(X) = v * 'f?(X) =
N
'f?(Y 1X)v(Y )dY (10)
E
E
— U
— U
for any 'f? CCX(G), where dY = dydy3dy2dy1 is the Haar measure on N which is the Lebesgue measure on IR6, Y = (y, y3, y2, y1), X = (x, x3, x2, x1), y = (y6, y5, y4), x = (x6, x5, x4) and * denotes the convolution product on G . The mapping v IDv is an algebra isomorphism of onto the algebra of all invariant differential operators on N . For more details see [9].
E U
E U
U
U
x x
x x
Let M = IR3 IR2 IR be the vector group of N which is the direct product of IR3, IR2 and IR. we denote also by the complexified universal enveloping algebra of the real Lie algebra m of M. For every v , we can associate a differential operator CD on M as follows
r
r
CDf(X) = 1. *c v(X) = v *c 1.(X) =
M
1.(X – Y )v(Y )dY (11)
U
U
1—
1—
for any 1. E CCX(M), X E M, Y E M. where *c signify the convolution product on the real vector group M. The mapping v CDv is an algebra isomorphism of onto the algebra of all invariant differential operators (with constant coef ficients)on M.
S
S
S S
S S
E – E
E – E
As in [4], we will define the Fourier transform on N. Therefore let (N) be the Schwartz space of N which can be considered as the Schwartz space of (M), and let 1(N) be the space of all tempered distributions on N.We denote by L1(N) the Banach algebra that consists of all complex valued functions on the group N, which are integrable with respect to the Haar measure of N and multiplication is defined by convolution on N, and we denote by L2(N) the Hilbert space of N.
Definition 2.1. For every f CCX(N), one can define function f CCX(IR9)
as follows:
–
–
f(x, x3, x2, t3, t2, x1, t1) = f((p1(x1)(p2(x3, x2)(x), t3 + x3, t2 + x2)), t1) (12)
–
–
9 3
9 3
for all (x, x3, x2, t3, t2, x1, t1) E IR , where x = (x6, x5, x4) E IR .
It results from this definition that the function f is invariant in the following sense:
–
–
f((p1(h)(p2(r, k)(x), x3 – r, x2 – k, t3 + r, t2 + k)), x1 – h, t1 + h)
= f(x, x3, x2, t3, t2, x1, t1) (13)
3
3
–
–
9 2
9 2
for any (x, x3, x2, t3, t2, x1, t1) E IR , h E IR and (r, k) E IR , where x = (x6, x5, x4) E IR . So every function W(x, x3, x2, x1) on N extends uniquely as an invariant function W(x, x3, x2, t3, t2, x1, t1) on IR9.
Theorem 2.1. For every function F E CCX(L) invariant in sense (13) and
for every u E U, we have
u * F (x, x3, x2, t3, t2, x1, t1) = u *C F (x, x3, x2, t3, t2, x1, t1) (14)
for every (x, x3, x2, t3, t2, x1, t1) E L, where * signifies the convolution product on G with respect the variables (x, t3, t2, t1) and *Csignifies the commutative convolution product on B with respect the variables (x, x3, x2, x1).
Proof : In fact we have
u * F (x, x3, x2, t3, t2, x1, t1)
= r F (y, y3, y2, s)1(X, x3, x2, t3, t2, x1, t1)l u(y, y3, y2, s)dydy3dy2ds
G
G
r
r
= r F (p1(s1)(y, y3, y2)1, s)(x, x3, x2, t3, t2, x1, t1)l u(y, y3, y2, s)dydy3dy2ds
G
G
= F [(p1(s1)((p2(y3, y2)1((y) + (x))), x3, x2, t3 – y3, t2 – y2), x1, t1 – s)]
G
u(y, y3, y2, s)dydy3dy2ds
E
E
Since F is invariant in sense (13), then for every (x, x3, x2, t3, t2, x1, t1) L we get
r
r
PuF (x, x3, x2, t3, t2, x1, t1) = u * F (x, x3, x2, t3, t2, x1, t1)
= F [(p1(s1)(p2(y3, y2)1(y + x), x3, x2, t3 – y3, t2 – y2), x1, t1 – s)]
G
r
r
u(y, y3, y2, s)dydy3dy2ds
= F [x – y, x3 – y3, x2 – y2, t3, t2, x1 – s, t] u(y, y3, y2, s)dydy3dy2ds
G
= u *C F (x, x3, x2, t3, t2, x1, t1) = QuF (x, x3, x2, t3, t2, x1, t1)
where Pu and Qu are the invariant differential operators on G and B respectively.
E S
E S
Definition 2.2. If f (N), we define the Fourier transform of f as follows
r
r
.F f(, 3, 2, 1) =
N
f(X, x1)e i ((<,<3,<2,<1),(X,x1))dXdx1 (15)
where X = (x, x3, x2) E IR5, = (6, 5, 4) and dX = dxdx3dx2
Definition 2.3. If f E S(N), we define the Fourier transform of its invariant f as follows
r
r
F f (, 3, 2, 0, 0, 1, 0)
= f(X, t3, t2, x1, t1)e i ((,3, ,1),(X,x1))e i ((Âµ,,>,v),(t3,t ,t1))
IR9
dXdx1dt3dt2dt1dÂµd>dv (16)
where ((Âµ, >, v), (t3, t2, t1)) = Âµt3 + >t2 + vt1
r
r
Lemma 2.1 For every u E S(N), and f E S(N), we have
r
r
IR3
=
IR3
F(u * f )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
F(u)(Ff )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
r
r
and
= F(f )(, 3, 2, 0, 0, 1, 0)F(u)(, 3, 2, 1)
IR3
F(uV * f )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
= F(f )(, 3, 2, 0, 0, 1, 0)F(u)(, 3, 2, 1) (17)
for any (, 3, 2, 1, Âµ, >, v) JR , where uV(y, y3, y2, s) = u(y, y3, y2, s) .
9 1
9 1
E
E
Proof: By (17) we have
r
r
r
r
IR3
=
r
r
IR3
=
IR3
F(u * f )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
F(u *c f )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
F(u)(Ff )(, 3, 2, Âµ, >, 1, v)dÂµd>dv
= F(f )(, 3, 2, 0, 0, 1, 0)F(u)(, 3, 2, 1)
So
V V
F(u * f)(, 3, 2, 0, 0, 1, 0) = F(u *c f)(, 3, 2, 0, 0, 1, 0)
= F(f )(, 3, 2, 0, 0, 1, 0)F(u)(, 3, 2, 1)
Theorem 2.1 (Plancherel formula). For any f E L1(N)n L2(N), we
2
2
get
f * f (0, 0, 0, 0, 0, 0, 0, 0, 0) = If(x, x3, x2, x1)I dxdx3dx2dx1
N
N
N
N
2
2
= IFf((, ( , ( , ( )I d(d( d( d(
(18)
where
IR6
3 2 1
3 2 1
f (x, x3, x2, t3, t2, x1, t1) = f ((p1(x1)(p2(x3, x2)(x)), t3, t2), x1 + t1)
= f((p1(x1)(p2(x3, x2)(x)), t3, t2), x1 + t1)1) (19)
Proof : If f E S(N). Then we have
f * f (0, 0, 0, 0, 0, 0, 0, 0, 0))
= f (x, x3, x2, x1)1(0, 0, 0, 0, 0, 0, 0, 0, 0)l f(x, x3, x2, x1)dxdx3dx2dx1
N
N
= f p1(x11)((x, x3, x2)1(0, 0, 0, 0, 0, 0, 0, 0)), 0 – x1l f(x, x3, x2, x1)dxdx3dx2dx1
N
N
= f (p(x11)((p2(x3, x2)1((x) + (0, 0, 0))), 0, 0, 0 – x3, 0 – x2), 0, x1l
N
N
N
N
f(x, x3, x2, x1)dxdx3dx2dx1
= f (p(x11)(p2(x3, x2)1(x)), 0, 0, x3, x2), 0, x1l f(x, x3, x2, x1)dxdx3dx2dx1
N
N
= f (p(x11)(p2(x3, x2)1(x)), x3 – x2), x1l f(x6, x5, x4, x3, x2, x1)dxdx3dx2dx1
N
N
= f (x, x3x2, x1)1l f(x, x3, x2, x1)dxdx3dx2dx1
N
N
N
N
= f(x, x3, x2, x1)f(x, x3, x2, x1)dxdx3dx2dx1 = If(x, x3, x2, x1)I dxdx3dx2dx1
2
N IR6
Now equation (17), give us
r
r
f * fv(o, o, o, o, o, o, o, o, o)
=
r
r
IR9
=
r
r
IR9
=
r
r
IR6
=
IR18
F(f * fv)(, 3, 2, Âµ, >., 1, v)dd3d2d1dÂµd>.dv
F(f *c fv)(, 3, 2, Âµ, >., 1, v)dd3d2d1dÂµd>.dv
F[fv](, 3, 2, o, o, 1, o)F(f) (, 3, 2, 1)dd3d2d1
fv(y, y3, y2, o, o, y1, o)e i ( ,Y )f (x, x3, x2, x1)e i ( ,X)
r
r
dydy3dy2dy1dxdx3dx2dx1dd3d2d1
= fv((p(y1)(p2(y3, y2)(y)), y3, y2), y1)e i ( ,Y )f (x, x3, x2, x1)e i ( ,X)
IR18
r
r
dydy3dy2dy1dxdx3dx2dx1dd3d2d1
=
IR18
f( y, y3, y2, y1)e i ( 3 1 , 3 1 )f (x, x3, x2, x1)e i ( 3 1 , 3 1 )
r
r
dydy3dy2dy1dxdx3dx2dx1dd3d2d1
= F(f) (, 3, 2, 1)F(f) (, 3, 2, 1)dd3d2d1
IR6
2
2
2
2
= r IFf(, , , )I dd d d = r If(x, x3, x2, x1)I dxdx3dx2dx1
IR6
3 2 1
3 2 1
which is the Plancheral's formula on N.
Corollary 2. 1. In equation (18), replace the second f by g we obtain the Parseval formula on N
r
r
r
=
IR6
f(x, x3, x2, x1)g(x, x3, x2, x1)dxdx3dx2dx1
Ff(, 3, 2, 1)Fg(, 3, 2, 1)dd3d2d1


Division of Distributions on N.
–
–
If we consider the group N as a subgroup of L, then f E S(N) for x1, x2 and
x3 are fixed, and if we consider M as a subgroup of IR9, then f E S(M) for
9
9
t1, t2 and t3 fixed. This being so; denote by SE(JR ) the space of all functions
9
9
<I(x, x3, x2, t3, t2, x1, t1) E CCXl(JR9) such that <I(x, x3, x2, t3, t2, x1, t1) E S(N) for x1, x2 and x3 are fixed, and <I(x, x3, x2, t3, t2, x1, t1) E S(M) for t1, t2and t3 fixed. We equip SE(JR ) with the natural topology defined by the seminormas:
—
—
<I sup
(x,x3,x2,x1)EM
IQ(x, x3, x2, t3, t2, x1, t1)P (D)<I(x, x3, x2, t3, t2, x1, t1)I t3, t 2, t1 fixed
—
—
<I sup
(x,t3,t2,t1)EN
IR(x, x3, x2, t3, t2, x1, t1)S(D)<I(x, x3, x2, t3, t2, x1, t1)I x3, x2, x1 fixed
(20)
I 9
I 9
S E S
S E S
where P, Q, R and S run over the family of all complex polynomials in 9 variables. Let E(L) be the subspace of all functions 'lj E(JR ), which are invariant in sense (13), then we have the following result.
E
E
–
–
Lemma 3. 1. Let u E U and CDu be the invariant di.fferential operator on the group M, which is associated to u, acts on the variables (x, x3, x2, x1) M, then we have
E
E

The mapping f 1— f is a topological isomorphism of S(N) onto SI (JR9).
I 9
I 9

The mapping <I 1— CDu<I is a topological isomorphism of SE(JR ) onto
its image
Proof : (i) In fact rv is continuous and the restriction mapping <I 1— R<I
I 9
I 9
on N is continuous from SE(JR ) into S(N) that satisfies Ro rv= IdS(N) and
rv oR = IdSI (IR9), where IdS(N) (resp. IdSI (IR9)) is the identity mapping of
E E
I 9 9
S S
S S
(N) (resp. E(JR )) and N is considered as a subgroup of JR .
–
–
To prove(ii) we refer to [12, P.313 315] and his famous result that is: "Any
invariant di.fferential operator on M, is a topological isomorphism of S(M)
onto its image" From this result, we obtain that
CDu : SE(JR9) — SE(JR9) (2l)
E
E
E
E
S
S
S
S
is a topological isomorphism and its restriction on I (JR9) is a topological iso morphism of I (JR9) onto its image. Hence the theorem is proved.
x x { }x x { }
x x { }x x { }
x { } x x { } x
x { } x x { } x
In the following we will prove that every invariant differential operator on N has a tempered fundamental solution. As in the introduction, we will consider the two invariant differential operators IDu and CDu, the first on the group N = JR3 O JR2 O JR, and the second on the abelian vector group M = JR3 JR2 O JR O . Our main result is:
Theorem 3.1. Every nonzero invariant di.fferential operator on N has a tempered fundamental solution
Proof : For every function 'lj E CCXl(JR9) invariant in sensen (13) and for every
u E U, we have
u * 'lj(x, x3, x2, t3, t2, x1, t1) = u *C 'lj(x, x3, x2, t3, t2, x1, t1) (22)
for every (x, x3, x2, t3, t2, x1, t1) E L, where * signifies the convolution product on N with respect the variables (x, t3, t2, t1) and *Csignifies the commutative
convolution product on M with respect the variables (x, x3, x2, x1). In fact we have
IDu n 'lj(x, x3, x2, t3, t2, x1, t1)
= r 'lj (y, y3, y2, s)1(X, x3, x2, t3, t2, x1, t1)l u(y, y3, y2, s)dydy3dy2ds
N
N
r
r
= r 'lj (p1(s1)(y, y3, y2)1, s)(x, x3, x2, t3, t2, x1, t1)l u(y, y3, y2, s)dydy3dy2ds
N
N
= 'lj[(p1(s1)((p2(y3, y2)1((y) + (x))), x3, x2, t3 – y3, t2 – y2), x1), t1 – s]
N
r
r
u(y, y3, y2, s)dydy3dy2ds
= 'lj[((p1(s1)(p2(y3, y2)1(y + x), x3, x2, t3 – y3, t2 – y2), x1), t1 – s)]
N
r
r
u(y, y3, y2, s)dydy3dy2ds
= 'lj [x – y, x3 – y3, x2 – y2, t3, t2, x1 – s, t] u(y, y3, y2, s)dydy3dy2ds
M
= u nc 'lj(x, x3, x2, t3, t2, x1, t1) = CDu'lj(x, x3, x2, t3, t2, x1, t1) (23)
I 9
I 9
I 9
I 9
9
9
for all (x, x3, x2, t3, t2, x1, t1) E IR . By Lemma 2.1, the mapping 'lj ,— CDu'lj is a topological isomorphism of SE(IR ) onto its image, then the mapping 'lj ,— IDu'lj is a topological isomorphism of SE(IR ) onto its image. Since
R(IDu'lj)(x, x3, x2, t3, t2, x1, t1) = IDu(R'lj)(x, x3, x2, t3, t2, x1, t1) (24) so the following diagram is commutative:
I 9 I 9
SE(IR ) IDu
—
PuSE(IR )
R R
S(N) IDu
—
PuS(N)
S
S
S
S
,— S
,— S
Hence the mapping 'lj IDu'lj is a topological isomorphism of (N) onto its image. So the transpose tIDu of IDu is a continuous mapping of !(N) onto !(N). This means that for every tempered distribution T on N there is a tempered distribution E on N such that
IDuF = T (25)
Indeed the Dirac measure c belongs to S!(N).fundamental solution on N for any element u E U.
As in [6] we show how Atiyah method [1], can be generalized for our group
N to obtain a tempered solution by the following theorem.
Theorem 3.2. Every invariant differential operator on N which is not identically 0 has a tempered fundamental solution.
Proof : For each complex number s with positive real part, we can define a distribution T s on L by:
(T
(T
, f)
, f)
IF(u)(, 3, 2, 1)I
IF(u)(, 3, 2, 1)I
s f 1
IR6
IR6
2 s
IR6
IR6
F f(, 3, 2, 1)dd3d2d1 (26)
F f(, 3, 2, 1)dd3d2d1 (26)
s
s
9
9
1—
1—
E S
E S
for each f (IR ), where (6, 5, 4) and d d6d5d4. By Atiyah theorem [1], the function s T has a meromorphic continuation in the whole complex plan, which is analytic at s 0 and its value at this point is the Dirac measure on the group N . Now we can define another distribution T s as follows.
ITs, f\ IT s, f \
IF
IF
3
3
2
2
1 I
1 I
F
F
3
3
2
2
1
1
3
3
2
p>2
1
1
f 1 (u)(, , , ) 2 s (f )(, , , 0, 0, , 0)dd d d(27)
IR9
IR9
IR9
IR9
9
9
for any f E S(IR ) and s is a complex number, with real(s) is positive. Note that the distribution Ts is invariant in sense (13), so we have
Iu uV C T s, f\ Iu uV C T s, f )\ IT s, u C uVC f )\
IF
IF
3
3
2
2
1 I
1 I
F
F
3
3
2
2
1
1
3
3
2
2
1
1
f 1 (u)(, , , ) 2 s 1 (f ))(, , , 0, 0, , 0)dd d d
L
L
L
L
where uV(y, y3, y2, y1) u(y, y3, y2, s)1 and
f
f
u C f(x, x3, x2, x1)
f(x – y, x3 – y3, x2 – y2, x1 – y1)u(y, y3, y2, y1)dydy3dy2dy1
G
is the commutative convolution product on M. By Lemma 2.1, we get:
Iu uV C T s, f\
f V 21s 1
F
F
Hence
(u)(, 3, 2, 1)
IR9
F(f )(, 3, 2, 0, 0, 1, 0)dd3d2d1
u uV C T s
Ts
(28)
*
*
ln view of the invariance (13), the restriction of the distributions u —uV *c T s
Ts+1 on the subgroup IR3 x {0} >< IR2x {0} >< IR "'" N are nothing but the
distributions
p2 p1
u * uV *c T s T s+1 (29)
The distribution T s can be expanded a round s 1 in the form
CX)
T s f3j(s + 1)j (30)
j=6
O
O
where each f3j is a distribution on N . But u * uV *c T s T s+1 can not have a pole at s 1 (since T c5N ) and so we must have:
u * uV *c f3j 0 for j < 0
u * uV *c f3O c5N (31)
Whence the theorem.


Ideals Algebra L1(N).
We refer here to [5] for the characterization all left ideals in the Banach algebra
L1(N)
Lemma 4.1. (i)The mapping 8 from L—1(N )1M to L—1(N )1N defined by
Â¢ 1M (x, x3, x2, 0, 0, x1, 0) — 8(Â¢ 1M )(x, 0, 0, x3, x2, 0, x1)
is a topological isomorphism
Â¢ 1N (x, 0, 0, x3, x2, 0, x1) (32)
1 1
1 1
(ii) For every w E L (N) and Â¢ E L (N), we obtain
r(w *c Â¢1M )(x, 0, 0, x3, x2, 0, x1) w * Â¢1N (x, 0, 0, x3, x2, 0, x1)
w * Â¢(x, x3, x2, x1) (33)
where x (x6, x5, x4), y (y6, y5, y4), and
r
r
(w *c Â¢ 1M )(x, x3, x2, 0, 0, x1, 0)
Â¢ [x – y, x3 – y3, x2 – y2, 0, 0, x1 – y1, 0] w(y, y3, y2, y1)
M
1
1
dydy3dy2dy1 , Â¢ E L (N) (34)
Proof : (i) The mapping 8 is continuous and has an inverse 81 given by
Â¢ 1N (x, 0, 0, x3, x2, 0, x1) — 81(Â¢ 1N )(x, x3, x2, 0, 0, x1, 0)
Â¢ 1N (x, x3, x2, 0, 0, x1, 0) (35)
1
1
(ii) It is enough to see for every Â¢ E L (N)
r
r
e(w *c Â¢ IM )(x, 0, 0, x3, x2, 0, x1)
= Â¢IM [x – y, y3, y2, x3, x2, y1, x1] w(y, y3, y2, 0, y1)
M
r
r
dydy3dy2dy1
Â¢IM [x – y, y2, x3, x2, y1, x1] w(y, y3, y2, 0, y1)
M
r
r
dydy3dy2dy1
= Â¢[p1(x1)(p2(y3, y2)1(x – y), x3 – y3, x2 – y2), x1 – y1]
N
w(x, y3, y2, s)dydy3dy2dy1
1
1
= w * Â¢(x, x3, x2, x1), Â¢ E L (N) (36)
I
I
If I is a subalgebra of L1(N), we denote by I its image by the mapping "'Â· Let J = I M . Our main result is:
Theorem 4.1. Let I be a subalgebra of L1(N), then the following conditions
I
I
are equivalents.

J = I M is an ideal in the Banach algebra L1(M).

I is a left ideal in the Banach algebra L1(N).
Proof: (i) implies (ii) Let I be a subspace of the space L1(M) such that
J = IIM is an ideal in L1(M), then we have:
w *c I IM (x, x3, x2, 0, 0, x1, 0) I IM (x, x3, x2, 0, 0, x1, 0) (37)
1
1
for any w E L (M) and (x, x3, x2, x1) E M, where
* I
* I
w c I M (x, x3, x2, 0, 0, x1, 0)
M
M
= ( [ Â¢ [x – y, x2 – y2, 0, 0, x1 – y1, 0] w(y, y3, y2, y1)
1
1
It shows that
dydy3dy2dy1, Â¢ E L (N)
w *c Â¢ IM (x, x3, x2, 0, 0, x1, 0) E I IM (x, x3, x2, 0, 0, x1, 0) (38)
for any Â¢ E I. Then we get
e(w *c Â¢IM )(x, 0, 0, x3, x2, 0, x1)
= u * F (x, 0, 0, x3, x2, 0, x1) E e(I IM )(x, 0, 0, x3, x2, 0, x1)
= I IN (x, 0, 0, x3, x2, 0, x1) = I(x, x3, x2, x1) (39)
It is clear that (ii) implies (i).
It is clear that (ii) implies (i).
It is clear that (ii) implies (i).
It is clear that (ii) implies (i).
–
–
I
I
corollary 4.1. Let I be a subalgebra of the space L1(N) and I its image by the mapping ,.. such that J = I M is an ideal in L1(N), then the following conditions are verified.

J is a closed ideal in the algebra L1(M) if and only if I is a left closed ideal in the algebra L1(N).

J is a maximal ideal in the algebra L1(M) if and only if I is a left maximal ideal in the algebra L1(N).

J is a prime ideal in the algebra L1(M) if and only if I is a left prime ideal in the algebra L1(N).

J is a dense ideal in the algebra L1(M) if and only if I is a left dense ideal in the algebra L1(N).
The proof of this corollary results immediately from theorem 4.1


conclusion.

The Fourier transform has a natural generalization to our groups and many of the classical results can be extended. In fact our results obtained in this paper show the beauty of the natural extension of the Fourier transform to the non commutative and non compact Lie groups. Its powerful can be seen also through the following astonishing results.
–
–
Let H be the 3 dimensional Heisenberg group consisting of all matrices of
the form
1
x1
0
x
0
1
0
x5
0
0
1
0
0
0
0
1
1
x1
0
x
0
1
0
x5
0
0
1
0
0
0
0
1
H = I \
40
It is easy to show that the group H is a normal subgroup of the group N. The most interesting result that can be deduced from my work [4, 8] for this group is: "Any invariant di.fferential operator on H is globally solvable".


Acknowledgment

This work is not supported by any university and not supported by any research center.
References

M. F. Atiyah., 1970, Resolution of Singularities and Division of Distribu tions, Comm, on Pure and App. Math, Vol. 23. 145150.

J. Dixmier., 1958, Sur les repr'esentations unitaries des groupes de Lie nilpotents, III, Canad.]. Math. 10, pp. 321348.

G. S, Chirikjian and A. Kyatkin, 2000, Engineering Applications in Non commutative Harmonic Analysis, Johns Hopkins University, Baltimore, Maryland, CRC Press.

K. El Hussein., 2009, A Fundamental Solution of an Invariant Differential Operator on the Heisenberg Group, International Mathematical Forum, 4, no. 12, 601 612.

K. El Hussein., 2011, On the Left Ideals of Group Algebra of the Affine Group. International Mathematical Forum, Vol 6. no. 4, 193204.

K. El Hussein., Eigendistributions for the Invariant Differential Operators on the Affine Group, Int. Journal of Mat. Analysis, Vol. 3, no. 9 (2009), 419429.

K. El Hussein., 2013, Non Commutative Fourier Transform on Some Lie Groups and Its Application to Harmonic Analysis, International Journal of Engineering Research & Technology (IJERT) Vol. 2 Issue 10, 2429 2442.

K. El Hussein, 2010, Note on the Solvability of Lewy Operator, Int. Math. Forum, 5, no. 8, 389 – 393.

H. Helgason., 1984, Groups and Geometric Analysis, Academic Press.

S. Majid., 2006, What is the Quantum Group" Notices of the AMS.

W.Rudin., 1962, Fourier Analysis on Groups, Interscience Publishers, New York, NY.

F. Treves., 1962, Linear Partial Differential Equations with Constant Co etfficiens, Gordon and Breach.

