A. Avantaggiati ( E d.)
Pseudodifferential Operators with Applications Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 16-24, 1977
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11091-7 e-ISBN: 978-3-642-11092-4 DOI:10.1007/978-3-642-11092-4 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Liguori, Napoli 1978 With kind permission of C.I.M.E.
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CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.
1
PSEUDO-DIFFERENTIAL OPERATORS ON HEISENBERG GROUPS
Corso t e n u t o a Bressanone dal 16 a1 24 giugno
1977
Introduction The convalution
operators on the euclidean spaces a r e only a particular
case of convolution operators on a r b i t r a r y L i e groups.
Pseudo-differential
operators a r e roughly speaking convolution operators with variable coef
-
ficients. In classical theory of such operators it is important to have standard dilations on a euclidean space. So we pay attention only to Lie groups with dilation i. e. with
1 -dimensional group of automorphisms
converging to infinity when r e a l parameter increases to infinity. It is well known that all Lie groups with dilations a r e nilpotent. In this seminar I consider only Heisenberg Groups. These groups a r e the simplest non-abelian nilpotent groups, sis
They appear in Complex Analy -
and such operators on strictly pseudoconvex boundaries a s J. Kohn
sub-Laplacian, induced Cauchy-Riemann operators,
singular integral
operators of Cauchy-Henkintype can be locally considered a s convolution operators wtih variable coefficients on Heisenberg groups.
The characte -
ristic feature of all these operators is an anisotropy of their singula~ities tight with complex tangent directions. Actually this is a contact structure. Generally a contact structure is given on 2n+l -manifold by 1-form w such that the (2n+l) -form w Ad..wn
.. . .
#
~w d
0
If a strictly pseudoconvex boundary is defined by r e a l function
g
with
df f
0 then we m a y take
o=
1 ( 2f i
)
By D-arbour theorem
Heisenberg groups a r e local models- f o r any contact manifolds. I construct a theory of pseudo-differential operators which belong to the
contact s t r u c t u r e a s classical pseudo-differential operators belong to the smooth structurev It is impossible to derive a theory of pseudo-differential operatoi-s without an elliptic accompaniment. I introduce a kind of ellipticity which a s I hope can elucidate some striking analogies between non-elliptic in usual s e n s e
-complex and elliptic complexes.
Note that the problem of pseudo-differential operators on homogeneous Lie groups was put forward by E. Stein at the Nice congress (cf 141). Our r e s u l t s were mainly announced in /2/
and 131. Here we exposed
them in m o r e precise form. 1
Heisenberg Lie groups and algebras Heisenberg algebra H n
euclidean space if
X
4
a%
R 2n+1
(xOf X ,
if we
X l1
= (1; 0, 0,)s
.n>
can be obtain f r o m the standard
0,
supply it with such commutators:
)cdcZn+l
m 2n+l
where
x.6
R
.
xl, xll E
IR"
then
This Heisenberg a 1 g e b r a . i ~a Lie algebra of the step 2 all [x,
b,t g a r e
zero. Let (H,be a corresponding simply-connected Lie group. A s mani 2n+l ; the m ~ t i p l i c a t i o nis fold i t is identified with )(I x x y = (x0
+
J"
+ 2 ( < XI,
y">
<XI(,
Therefore 0 s e r v e s a s unit and x-' = Heisenberg group.
yl>! , x1 I. y' , 1'1
-x
+ yl*)
The group #,,is
called the
The identity mapping x
cj
x
coincides with the exponential
lHn
exp: Hn 3 There a r e dilations
Jt x
= (t2xo, t x l , txu)
Hn . Of
in the Heisenberg group the Let
6t
t > 0,
,
course
$
and
a r e automorphisms of the L i e structures in
Hn
and H
be correspolding spaces on R 2n+l
s ( Mn) , s' ( H ) n
The operators of the left and of the right shifts by elements
n'
. y O DIn
a r e continuous in this spaces. Makeover the Lebesgue measure is b i l a t e r a l ly invariant on the
jH
n
W e may identify the Heisenberg algebra H with the Lie algebra of n
left -invariant 1 -st o r d e r differential operators. Pick out generators of the complexification of this algebra
Adopt the notation
x1 = (x;.....,
I
Xn) ,
x
II
rt
= (Xl.
u
.... x n
t
The contact structure on the M is defined by a left-invariant 1 -form n u (XI = A x o + 2 <x', Axti)
-
2 <xti
,drt>
o
1, X = (XO.X. X 1
H. Weyl quantization.
3.
The modern theory of pseudo-dif&~ntialoperators took i t s shape in the sixties however we can find i.ts origin a s ear*
a s in t h e beginning of the
thirties: t h e r e was a probrem of' quantization in the Quantum Mechanics and H. Weyl gave a general solution.
The problem is to construct of non-cum_
muting operators of multiplication : A
X :
H X
t&
k
5
X=(X,,%~,-*.,%,~
and of differentiation
This operators generate a 2 -step Lie subalgebra W in the L i e algebra n h 3 ( h% ) ) of all contirmous linear operators in the d ( Rn ).
q(
Let
\W91 be the
symple-connected Lie group of this W
n
. Then the
w,
can be realized a s a Lie group of continuous linear operators m the +I
4(R
We have the exponential map exp : Wn
)
Therefore f o r a
W
n
~ d ( ) ~we kcan define ~ ~ an operator
I
x, P f
where
A
A
(x, p ) =
,.,
iC+]
is the Fourier transform. The expression ( r ) is similar to the inverse
Fourier transform of the
fh (i .q ) .
Let us justify this definition.
It is easy to s e e that the integral ( & )
converges in the space
a 15 ( 1 ~ * ) 1
under the strong topology and actually is an integrhl operator with the Schwartz kernel
We can rewrite this formula a s follows
where3
b72
is t h e inverse Fourier transformation (from the p to the z)
In this form the definition ofXf f o r e v e r y f E R~~ ) is valid. x, P 2n then f p? is a Continuous operator from d t ( R to 4 f € ~(IL,,
4;
(2
If f c
.sf(fl:Pl
then
f
x p) is a cont*uous r,c
Chose moderate operators (cf / 5 / ): Let
We say that such
f a r e symbols of order
y~')
* operators f r o m 4 ( R ) td,(/lR) t m
mE
IR.
Take
(cf. [ 5 7 )
m
let
~r
f
I .- 4fR41-+ ~
6 w*{R~ then
n7,.j?~~/44
and these operators a r e called Weyl operators. In particular it i s possible to t ake a product of Weyl operators If then there exists a f t
~I*'"Z
W
(nn)such
that
fjC Oh w?;
j
4 2 ,
fi
A
x , P
= fl
$1
,
f
*
/r
p)
f2 (I,
and f
(x. p ) =
order.
{f } ( x
( f l f 2 ) ( X , P) +
p) + t e r m s of the lower
This a simple consequence of the Hausdorff -Campbell formula
f o r the product of exponents. Moreover f
Let
(x^,i;)*
m
W
n
(R
0
-
=
f
P
fi
x> P)
(
) be the subspace of
wm ( R ~whose )
have the following property: there exists f € 0
(i) f o (tx , tp) = tm f o (x. p) (ii) f
m-1 W
- f0
We s a y that such
(
,Y t> 1,I x 1
+
Ip I >
liptic (cf order m
5
t-
)
EPe w*- * fxr !PI > +
Example - :
is called elliptic
)(:
operator from
if
fo (a, p)
f
0
-I
with principal part
ro (r , p ) = (fo (x , p)) f o r
1
the Hermite operator
.
.
A
E =
% +xL C - a3xL
E'[R*) = [ ~ r d i : ~u k~nrc Z ) (x^
0
hr
Let
Then f
1
f&!w, be the c l a s s of a l l elliptic operators of m p) E E?L! % then t h e r e exists i t s parametrix 'C(2,F)
Let
If f ( x,
E
such that
1 ; it is known that the elliptic operators a r e hypoel-
A
.
Ip1>
f a r e symbols with principal part f l-
1 xl
+
")
(
R")
An operator f ( x. p ) E 0 Wm P 0 for
wrn
elements
[&
42 ~
E
2
-
-1
s,
L k( & *)
if and only if
to
c'm
f (
$) is a Fredholm
3 ( & ) f o r some (and therefore f o r g
any) k
E IIQ.
- differential operators
Pseudo
4.
on
#(
Apply the Weyl quantization for construction of pseudo-differential ope-
01 n'
r a t o r s on
F i r s t of all the operators of multiplication by coordinate functions. /s X
:k(x)
-+
xU(x),
I
( X ~ , X , X ~ ~ )
X '
and left- invariant operators
generate a 3-step L i e subalgebra in Lie algebra this
If !j
z(+(#*I/.
Treating
Lie algebra a s above enables to define
f 4 (R
ew+t
*,X
)then
(
*t )
is an integral operator with the Schwartz
Kernel
This formula is valid f o r every definition of If
#t?
f (
d(~'~'') *,X
A
$. X
)
then
f
1f f ~ d y & : y ) t h e n Let
f
{ 6 ?J '($.+(land
we can use it for
a,*
(E$ : 4 ' ( kf,,1 J 4 (HI,,) , ) : 4 ( 1 4 ' (N,)
lY] be a 7 - homogeneous function on the t
y
Define
(klas
such that
(k) (9G ) 4
lr+
J .
p / ( x l ~ l I g c ~ , p
We s a y that such
a r e symbols of o r d e r
f
m
Indicate the matn formulas of the Symbolic Calculus: (I)
J 6 y mfpla)then
If
{.3 6 Or
(XI) Let
f
(
2)
2,
'(Hh1,j: ( 2 ,
=
- (2 2) 9Y f
m.
Then t h e r e exists
6
{ 6 y'
3
i
""'
such that
and
(111) Assume that
f E
Consider a diffeomorfism that ( i. e.
(
X
HI,
) has the c o r n p e t suppart with respat to x.
of a neighborhood of the support. Assume
conserves the contact form
2
w
up to a functional multiplier
i s a contact mapping).
Let
Y
f
Then
(n,y) = f ( Z(x)
. I d x ( x ) l WY 1 .
and
The last property permits the standard extension of manbfold
M.
3*
P
3.1
0
Thus we have
0
to any contact
(M) with a symbolic Calculus
a s above. lw
Now we introduce a subclass f
every f t h e r e exists a (i)
cii,
fO(xJ
r
x)
$ - 4.
6
(ff/-] of [€
EyZ(glh) such
0
= t-{G,x),
r ' l ( l ~ , )such
that f o r
that
Vt>r,
~Ix/>I.
ym-'CM, 1
We say that the symbols
f
have principal parts
f
0
The formula (11) of the Symbolic Calculus shows that in principal parts the product of pseudo-differential operators i s t k p r ~ d u c tof left-invariant operators depending on
x
a s a parameter. F o r study of.
such products
it is very convenient to use the Heisenberg-Fourier transform on the It is defined by means of the non-degenerate s e r i e s of unitary represen
Nk. -
tations of the group. By the Stone -von Neumann theorem they a r e equivalent to the representations Zp depending on non -zero r e a l parameter in the space
z l ( ~ I:
r (x,]
r
such that
= r l ,s
( ~ ' ) . 2 ~ r3P, ( x o ~ =a~ i.
r' he Fourier-Heisenberg transform is by definition
This i s an integral operator with Schwartz kernel
,-
s o that
We s e e that this formula ( SO
*
is valid f o r any distribution f r o m
we can extend the definition of
pp
dfw
4 '(6.
to
$ %%
This leads to a representation of left-invariant operators by operators.
Moreover
and the principal part of
is given r vb?~)
2- 1
~ x r t ~ ~=~fo~ (o,+G, ) ) o
2ia t
b -homogeneous
Let foo (XI be the
t
coincide with
f o (X)
f a r from origin
By
by means of the principal part
function on the d H k h i c h
. Consider the operators
t -homogenity we have -I
where Therefore t h e r e a r e significant only Finally we define a
6m -symbol
r
=:
4
k&)=
A
"
(x. X))W(X)
XI 1 6 Op
of MI operator
A
=
f00
t
1
a s an operator valued function on the manifold of contact directions
6h, (f
k(L)
a'
( x , l , z t . , 2 at. t-)
t
"I
The
6 -symbol reveals usual properties : k
*'lh%] then
(iiii)
*,* '')[ { , ( ~ * , $ L # , ( ~ ~ ~ J ~ f~('~')] ~~#+( * - ~ ~ i
is the energy operator of the harmonic escil -
lator of Quantum Mechanics. Actually this example has appeared in / 4 / and by the way it served an origin of our study. (b) The induced Cauchy-Riemann operator
belongs to
0,.y " l " )
-
am on the
(0,
q) -forms
witll
(c) The Cauchy-Henkin integral can be considered a s operators M i f we take their boundary values; they belong to
.
0,
S on
7 "(MI
and their symbol is
b, (
s ] orthogonal ~ ~ ~ projector ~ on the linear span of e
= zero -wt3 We say t k t an operator f
- .yz
r,(S)
(Ell-2) The operators
(g,
A
60
r ~ "is'
X)
A
h*({ it, x 1)
Let M be a compact contact
6 - e l l i p t i c if
lJ DIla r e invertible in
manifold.
If
(cfi
s k-m P
then
P
A
f (r.X)
6,
f
M
isboun&dfrom
s;(M)
( M ) f o r any k
The following properties a r e equivalent f o r a)
)
/4/).
f ( ; , R ) G % Y ~
to
fie
We can intmduce a scale of
anisotropic spaces of functions and distributions on
of B. Stein
4
is a 0
It follows from t h e well-known p r o p e r t i e s of t h e Sobolev
I. CIIUII,,~
spaces t h a t
Lemma1.2:
x
If
s(
1
if
u
c
Hs.N(~n) and
s
.
>
The
t r i v i a l ( see [6]):
5,
s
iIp:~\ ;-
isnonintegerand
$,N
s+N
5 cllul!s,~
2. Solution of the mixed problem i n t h e half-space. Consider t h e mixed problem
(0.81,
(0.9) assuming t h a t
's-2,N('3
3
n-1
H
chosen l a t e r . We suppose t h a t
5
s
+N
L
.
> max
i=1,2
(mi
+ $)
and
s
is noninteger i f
max (mi + ) i=l, 2 A p a r t i c u l a r s o l u t i o n of t h e equation (0.8) i n h e half-space s
taken I n t h e following form
can be
where
ef 6 Hs-2yN@n)
i s an a r b i t r a r y extension of
f
to
Etn
and
F
-1
i s t h e inverse Fourier transform. Let
fi (S ')
be t h e root of equation
imaginary p a r t .
i ( 5 ' .En) = 0
Then t h e general s o l u t i o n i n)@ N $ Sq
with t h e negative of t h e equation
(0.8) has t h e form
where
v ( x l ) is an a r b i t r a r y function belonging t o
.
(if?-')
I n order
t o s a t i s f y t h e boundary conditions (0.9) we obtain t h e following system of equations f o r
where
v(x'):
bi(E1) = B ~ ( S ' . X E . ) ) , ; i ( ~ l ) = bi((l+15'l)w.~n-1).
bi(D')
are
A
3r.d.o.
with t h e symbols
bi(S1), i = 1,2.
(see Lemma 1.2). We assume t h a t t h e boundary operators B1(D)
and
B2(D)
satisfy the
Shapiro-Lopatinskii condition what means t h a t (2.3)
Let
bi(e1) $ 0 H'
Set
(2.4)
5' #
&,-+ , N (n-1~
Rgi(xl)
v-(x')
for
=
v+(x') =
0. i = 1,2. )
be extensions of
- P ' B ~ ( D ) U- ~b l ( ~ 0 v , ,. t g 2 - P * B ~ ( D )-Ub~2 ( ~ ~ ) . v Rgl
h
gl and
g2
~o
n-1
R
.
Then
+
v+ E H
(IRn-'1,
i s t h e subspace of
where
t H s,N
w i t h supports i n
e-l .
Taking t h e F o u r i e r transform i n (2.4) we o b t a i n
where
hi(xl) = Rgi(xt)
G(6'')
By exluding
- P ' B ~ ( Dlug,
i = 1,2
from (2.5) we o b t a i n
So t h a t
Therefore t h e s o l u t i o n of t h e equation (2.2) i s reduced t o t h e s o l u t i o n of t h e The equation (2.7) i s c a l l e d Wiener-Hopf equation o r Riemann-
equation (2.7).
H i l b e r t equation and the way t o s o l v e t h i s equation i s t h e f a c t o r i z a t i o n of .
t h e function
.
il(5');i1(5').
For t h e f a c t o r i z a t i o n we s h a l l need some p r o p e r t i e s of t h e c a u c h y i n t e g r a l .
3. The Cauchy i n t e g r a l Let Denote by
f (x')
nt
E
c;(iRn-l)
and l e t
f (El) b e t h e F o u r i e r transform of
t h e following operators:
f (x').
that
SO
+
Let
0
for
x
~I-Z f o r a11 f ( x V ) E
C;(R~-~).
be t h e operator of t h e m u l t i p l i c a t i o n on
n-1
where
n+z +
a
F
> 0 and O(X,-~) = 0 f o r x ~ - 0 -
(T < 01,
+
C) b+(S",En-l (St',cnZ;),
T = Im
en-,>
i s continuous i n
en-:
O(T = Im
(c",cn-l,~)
0).
for
1 ~ 1
+ l ~ ~ -+ ~ 1 > 0 -
+ i~) )
i ~ (b-(5",~n-l )
2 (ord b, = a
ord b+ =
i s a homogeneous f u n c t i o n i n
- g),
i s a complex number, i n
general.
Without l o s s of g e n e r a l i t y we can assume t h a t (4.2) b+(O,+l) = 1. The order of homogenuity of b+(St) i s c a l l e d t h e index of t h e f a c t o r i z a t i o n of, b ( S t ) Lemma 4.1: Let b(E1) b e an e l l i p t i c symbol. Then b ( c ' ) admit a unique homogeneous f a c t o r i z a t i o n condition
(4.1) assuming t h a t t h e normalization
(4.2) i s f u l f i l l e d .
There i s t h e following formula f o r t h e index of f a c t o r i z a t i o n
a =
4 . 3
1 2a
+ -
a r g b(c91,cn-l)
5.1
-d
1;
r - m
=
- -iI n 2n
+*
1
b(0,-1) b(0,+1)
1.
The proof of t h e Lemma 4.1 i s given i n [ 6 ] . Examples of t h e computation of t h e index of f a c t o r i z a t i o n : 1 ) Let where
b(cl)
be a s t r o n g l y e l l i p t i c symbol, i . e
bk(c') a r e r e a l l k=1,2, and
bl(ct) f 0
for
follows from (4.3) t h a t
"'a
1 + 2ri
i n particular, 2) Let
b(0,-1) b(0,+1)
In
B
=
a2
b(e') = b(-c')
if and
' b(0,-1) n-1
2 3.
= b(0,+1)
Then
-
b(5')
5' f
= bl(E1) 0.
+ fb2(c1) ,
Then i t
3) Let Then
r
~ ( 5 ' ) be an e l l i p t i c polynomial of t h e order
and
n-1 >_ 3.
r = 2m and L(S", -
of t h e mixed problem (6.1),
(6.2). 6.2)
Consider t h e following mixed problem f o r t h e equation (6.1)
p
2
Here
n
a
where
a
=
k" k=l 1 is an integer. bl(E1)
a
2 LP(-IE',$~-~
f a c t o r i z a t i o n of
x
, ak
+
-
1
7
...,n,an
= 1 and
1 ~ ~, ~ b2(Cv) 1 ~ =) 1 and t h e index of
is equal t o
bl(Eq) bil(E1)
1 = p(y
a r e r e a l , k = I,
arc-
-
an,l)
So t h a t t h e i n e q u a l i t i e s (5.7) have t h e following form
1 1 P ( ~ -arc-
-
71
W e note t h a t i f
-3 < s 2
a
n-1
)
< s
p = 1 and
3 < -
-
C:
- -~i' a r c t g a n-l) + I ,
p (=3.
a,-l
1 % arctg
S + N > P +
< 0 t h a t we can take N = 0 an-1
' But:
if
an-l
,
> 0 then i t is
1
necessary t o take 6.3)
N
> 0.
Consider t h e following mixed conditions f o r equation (6.1):
a r e real,
..,n..
k = 1,.
i = 2
Then it follows from t h e formula (4.3) t h a t t h e index of f a c t o r i z a t i o n f o r this case equal t o
2';
1
(2) a r c t g an-1
1
-
So t h a t t h e i n e q u a l i t i e s
(1) a r c t g an-1
(5.7) gives
7. Asymptotic behaviour of t h e solution of t h e mixed problem near t h e hyperplane
.
xn = x = 0 n-1
The formulas (2.2) and(5.6) give an e x p l i c i t representation f o r t h e unique s o l u t i o n of t h e problem (0.81,
(0.9).
Suppose t h a t
f,glg2
are
m
C
functions equal t o zero i n t h e neighbourhood of t h e i n f i n i t y . Then taking an expansion of and f o r for
x
+
2
+x
where H
+
0 (see 6
is arbitrary,
b2(c1), b+(c')
+
for
we can obtain t h e following expansion f o r f o r t h e details):
c
(x,,~ + Xn X(0,+1)), z2 =
I -
1
-m
bl(c'),
~ Xn-l
~ ($1, +
-
Xn
X(O,-l),
[
~
~
u(xl,xn)
and a l s o can be calculated e x p l i c i t l y . Above we have supposed t h a t X noninteger. replace
z i s an i n t e g e r and z + m2 2 0 then it i s necessary t o
If
z?
by
z p n =i i n t h e two f i r s t t e r n s i n (7.1).
Therefore it follows from (7.1) xn = x = 0 n-1
t h e hyperplane if
8.
z
+ m2 2 0
that
u(xl,xn)
t h e same smoothness a s
r-2
E+o12 In r
r
or
r =-
a? is an integer,
and
has i n t h e neighbourhood of
Mixed problem f o r second order e l l i p t i c equation i n a bounded domain. Consider now s mixed problem (0.11, (0.21, (0.3).
on
is
Fk
(k=1,2)
Let
Bk(x,
DP
t h e Shapiro-Lopatinskii condition i n a corresponding l o c a l
system of coordinates.
Let
To
x" 0
be a r b i t r a r y and l e t x(x;)
be t h e
index of t h e f a c t o r i z a t i o n (4.1) i n a l o c a l system of coordinates. be proven t h a t z ( x " ) 0
H
HiSN(Tk), k = 1 , 2
(GI,
s,N corresponding t o
It can
does not depend on a choice of a svstem of coordinates
is a smooth function on
and t h a t .X;x") Bnote by
satisfies
ro'
Sobolev's spaces with weights
Suppose t h a t (8.1)
max Fk? &xu) xTfErO
- min k d x " ) ~''~r~
< 1
.
Then on t h e base of t h e Theorem 5.1 t h e following theorem can be proved by t h e usual technique of "faozen" c o e f f i c i e n t s .
Let
Theorem 8.1: Fk(k=1,2). (8.2) s
o
0.
, exp
d6finit un diffihnorphisme.
On identifie
vecteurs r6els invariants
gauche sur G en associant
:5
*
G ,
avec les champs de
champ de vecteurs, encore not6 X, d&fini par
X dans
le
:
Cette identification s16tend de maniire unique en un isomorphisme entre 1 'alg6bre enveloppante (complexifiie) U (3) tous les op6rateurs diffirentiels invariants
et llalg&bre de
gauche sur G ( A coef-
ficients complexes). Tout opbrateur diffbrentiel invariant
31. ( X 1 , . . , X
n
forrnent une base ordonn&e, le d6veloppement cet unique;
Uu ophrateur diffirentiel invariant
d'ordre m si
gauche sur G est de la forme:
:
& gauche sur G est dit homog6ne
Pour s i m p l i f i e r , d a n s c e t expose, nous n e donnerons un Cnoncb que d a n s l e c a s d'op6rateui-s
d i f f e r e n t i e l s homogbes.
On a
d e s thbor&mes a n a l o g u e s d a n s l e c a s non-homog&ne ( e n remplavant d a n s 1'Cnonce h y p o e l l i p f i q u e p a r h y p o e l l i p t i q u e avec p e r t e de m/2 dbriv6es). Dans l a s u i t e , on a p p e l l e r a groupe n i l p o t e n t d e r a n g 2, un groupe ayant l e s p r o p r i b t b s ci-dessus. On dhmontre l e thborime s u i v a n t
:
S o i t G un groupe n i l p o t e n t d e r a n g 2, s o i t P un o p h r a t e u r d i f f e
-
r e n t i e l homog&ne d e & g r b m. s u r G, a l o r s P e s t h y p o e l l i p t i q u e si, e t seulement si, p o u r t o u t e r e p r e s e n t a t i o n x, u n i t a i r c , i r r b d u c t i b l c , ~ l o n
I I-
t r i v i a l e , x (PI UJ
vecteurs C
e a t i n j e c t i f dans fz ( o i yx d f s i g n e l t r s p a c e d e s
de l a reprbsentation).
La f o r m u l a t i o n du theorhme e s t due l e th60r;me
;Rockla.nd
[ 5 ] , q u i a d6montr0
d a n s l e c a e du groupe d e Heisenberg Ifn.
montre que l a c o n d i t i o n du th6or;me
Dans [ 1 ] , 8 e a l s
e e t n h c e s s a i r e pour t o u s l e e
groupee n i l p o t e n t s (munie d'une d i l a t a t i o n ) e t il d h n o n t r e l a cond i t i o n s u f f i s a n t e pour l e s g r o u p e s H
n
x 8k.
Ultkrieurement
i notre
t r a v a i l , B e a l s donne d a n s [2] une n o u v e l l e d t m o n s t r a t i o n du thbor&me 0.1 e n u t i l i s a n t ees c l a e s e s
5
1.
S"Y.
Rappels s u r l e a r e s u l t a t s d e [3] Soit P
de degrii m.
=
p(x,D) un o p b r a t e u r p s e u d o - d i f f b r e n t i e l s u r R~
Nous supposerons que P e s t r b g u l i e r , i . e que son symbole
t o t a l p admet un dCveloppement asymptotique.
o i p j est homog&ne d e d e e m - J
( j e s t un e n t i e r ou un d e m i - e n t i e r
positif). Soit
x
3t
un c a n e l i s s e d e codimension p d a n s T Rn\ 0.
que P e s t nu1 d v o r d r e k s u r p(x.5)
E !nmfk),
:
~i pour t o u t j i n f b r i e u r ou Q g a l
& lvordrek -
2j.
Soit
un p o i n t d e
(x,S)
e t nous Q c r i r o n s
Nous d i r o n s
P E 9tmsk (OU
i. k I 2 , p j s v a n n u l e
2 , on d Q f i n i t , p o u r un 616ment P d a n s 3m,k ,
(P) e s t un QlCment d e An ( C ) (algAbre d e Veyl d e s o p h r a t e u r s (x,S) d i f f Q r e n t i e l s 4 c o e f f i c i e n t s polynomiaux s u r f ( R n ) ) a
.
Pour un hlbrnent d e !nmyO( o p 6 r a t e u r s p s e u d o - d i f f 6 r e n t i e l s r h g u l i e r s ) , 0
o x , = (P) e a t simplement l e ~ y m b o l ep r i n c i p a l d e P a u p o i n t ( x , S ) .
On a l e a p r o p r i 6 t 6 s s u i v a n t e s Pour P d e n s
Eli p a r t i c u l i e r si
:
e t Q dane 9m!sk'
k = k* = I
CoordonnCes adaptges Au voisinage d t u n point ( ~ ~ , g , ~ )x, d e on introduit un syst6me
de
m
sont C , homog&nes de degr6 0 (resp. 11, non o; les u i (resp. v.) 1 toutes nulles, de sorte que, au voisinage de ( x ~ , ~ ~ ) est , d6finie par l e syst6me d'iquations: ui u .(r,D) (dans 3
Bcrire
aosl), .lor.
.
,o(i = 1,. ,p).
Soit
uj
l'op8rateur
si p est dans imyk,on peut toujoura
:
o; Aa est un opbrateur pseudodiff8rrentiel dqordre m-k/Z + 1a1/2
:
On a, en tout point (x,S) de
Construction de parametrixes
On dira que P dans
amrkeet
traneversalement elliptiqua de long de
2 , s'il eat elliptique d'ordre m en dehors de de 2 poss&de un voisinage conique
,
,
r
x,
et si tout point
dann lequel on a
15 1 2 1
On a alors le th6orime suivant
(cf Th. 5.2.1 de [3] >
:
3
C
> 0,
am' t r a n s v e r s a l e m e n t
S o i t P un o p h r a t e u r pseudodif f 6 r e n t i e l d a n s e l l i p t i q u e l e l o n g de
x,
e t s o i t ( x o,go) un p o i n t d e
l ~ e as s s e r t i o n s s u i v a n t e s a o n t C q u i v a l e n t e s
x.
:
dans ( i ) 11 e x i s t e un v o i s i n a g e conique d e ( x0' s o ) d a n s T'R"\o, hypo l e q u e l P e s t microlocalement e l l i p t i q u e a v e c p e r t e d e k/2 d i r i v h e s .
k
Rcmarque 1.2
(P)
e s t i n j e e t i f dans f
