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
6
L
the c k - s y m b o l belongs to the contact structure so it can be
transfered to \any Examples: (a) Let
f (
S 2 ) E OhY
Everywhere
DF1
htc
M is a strongly pseudoconvex boundary in &
be the Kohn sub
- Laplacien on the space of
Then
o , E
a,, ~
The operator
( M and I
a' + 't - at. -
%
(PI)
f
- --ara ax +
(Oh ) - (
(0, q)
-forms.
r > * - ~ ~ 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 <-elliptic
f
(2 5) E 0,
h
:
operator.
b) The aprior$ estimate
i s valid in Stein norms for a (and therefore f o r any) c)
for a
f
(2.
s)
is a Fredholm operator from
(and therefore f o r any) k 6
Remark
IR -
k(
S k(M) P
E , k'
to
Jjk -m
< k (M)
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
AN INDEX FORMULA FOR E L L I P T I C BOUNDARY PROBLEMS
A. DYNIN
C o r s o tenuto a B r e s s a n o n e d a l 1 6 a1 24 giugno 1977
I give an analytical formula for index of elliptic boundary problems for s c a l a r differential operator and f o r some systems of differential operators of even o r d e r in bounded domains with smooth boundaries in euclidean space. 1. Notation. x = ( X 1'
n * 2,. ..-.,En) E (a).
.
.... , x ~ ) c R I~ = (
The t e r m "smooth" always means
cCO.
Let U be an open bounded domain in Points of Y a r e denoted y denoted
Y
.
-
The (2n-3)
.
/Rn
with smooth boundary Y
.
Cotangent vectors at y with length 1 a r e
manifold
S (Y) of all such
*
Y
is supplied
with canonical orientation: the manifold T ( Y ) of all cotangent vectors of Y i s
n-1 lf ) locally.
R ~ x (-R ~
coordinates on
( Rn-1
nates on (tRn-I)*.
R = (dylndql) I \ .
Let (yy
. . . . ,yn-l)
be any sfstem of
and ( T1, . . , , 9n-1 ) the dual system of coordi-
.
Then the orienting (2n-2) - form
... A
(dy n d q ) does not depend on the choice of l e local n n
coordinates and therefore gives an orientation of (
v
:
T* (Y) *R be the euclidean metric, so that S (Y) = z
Now let
-~ (1).
Then
the orienting (2n-3)- form u on the S (Y) is defined by i t s property
2. Elliptic Boundary Problems. Let A be a s c a l a r differential operator of o r d e r
2 rn with
smooth
coefficients oi
(X)D
A : k(x)+ la(l(tm
&(XI
.,%Ec*
(5)
-
I t s principal symbol is the function on TY (-U ) = U x ( lR,n )
d ( A ) (x.
f
=
=
*
a .(4
14,= 2-
The operator A is assumed elliptic i. e. A
) $0,v x
x,
c
c. Y ~ E ( R " ) *
Consider ;\ -polynomial with coefficients from
cQ)( S (Y)) of order 3 m
is the inward unit conormal at y. Y F o r each 7 this polynomial has no root with zero imaginary part. where 3
If
n
Yu#
> 2 therelexactly m roots with positive imaginary part (this is an
easy consequence of the connectivity of S (Y) )
.
If
n = 2 then we
assume this property especially. We can factorize the A -polynomial into the product of two polynomials with smooth coefficients
scn, (zJ,;\ A
d f(
where a l l roots of and all roots of
+
1 = 6
(
t3,a
1 r - (Z$,,A 1,
(5.2 ) a r e in the upper complex )
a r e in the lower
1 - halfplane
1 - halfplane.
Consider a boundary problem
Here B. a r e boundary differential operators of order m J
j
with smooth
coefficients
L
Bj : U +
bN(y) D ' ~ / Y ,
I 4 < mj B
j
: C*
(q
9
cW(Y)
We suppose that the boundary problem satisfies the Shapiro-Lopatinsky condition of ellipticity which we take in the Agmon-Douglis-Nirenberg version / I / (cf. lectures by F. TrC ves): Consider l-polynomials of degree m . with smooth coeffients on S ( Y ) J A
6(
J
B
.
,
= C ( B j ) (Y,$,+A()J)
5
The Agmon-Douglis-Nirenberg condition is the linear independence of these polynomials modulo )I -polynomial
r + ( 3 ,A ) f o r every t9'
We can represent this condition in an equivalent form, Let 2.(tJ 3' be the remainder from division of )1 -polynomial
9 -polynomial 6+ A
(
T3,C))
6(Bj)(Z5,2) =
7. J
where than m
j
.
A
6 (Bj)
Let
)
(E8, a ) by
:
t
(t '1) 6
S
( t @ , a) b + y j ( s , A )
and Z: a r e ;\ -polynomials and the degree of J
1
'E: ( 2,;)) is l e s s J 3
Consider the square ( m x m ) - m a t r i x valued function on S (Y)
The Agmon-Douglis-Nirenberg condition is obviously equivalent to non
-
degeneracy condition
+
a)i t ,
(ADN)
det .t(
0,
V r , E s (Y)
3. The Index Formula As usual the elliptic boundary problem
(a) leads
to a linear conti-
nuous operator
which is a Fredholm operator and therefore has a finite index ind
Q=
( s e e e. g.
/
dim Ker 2
/
and
a-
/
8
dim
coker 4,
/).
It is known that the index depends only on the symbol
6(Q) =
( w(A),
6 (B1),
We express it by means of the (1)
ind
=
In -(-1
-
- 3
b(Bm)
T(a)which (n-2)'
(2~i)~''
/
1-
a)
is defined by b(
~p [ r ( ~ L ) - l d ~ ( f Z 2n-3 g
J
The intengrand is the t r a c e of (an-3)-power of ( m x m ) -matrix valued differential 1 - f o r m y (
(2.)
-1
dp! Q )
in the exterior algebra of matrix
valued differential forms. So we integrate (2n-3)-form over the (oriented ) manifold
S (Y).
In particular the Index Formila shows that if m < n-1 then ind 'The prof. oE (1) -involves a special homotopy of of symbols of elliptic boundary problems f o r the A
-
a=
0
6(CL) in the space with pseudo-differen-
(By the way, this is the &st place where pseudo-
tial boundary operators.
differential operators of positive o r d e r were introduced a s early a s in
/
1961: s e e
4
/
and
/
5
1.)
The homotopy is P(t, (Bj)(2$'1) = (1-t)qj(r$,)i Here '1)s t
6
+
a- ( Z $ , I ) + r ; ( r 8 , a1
1 and
This homotopy may be covered by homotopy of boundary value problems (
at ) f o r the same operator
A which all a r e elliptic
b3
Z
C%~=W@
and the condition (ADN) is satisfied (cf. /5/ and lectures by F. Treves). Stability o s Index under homotopies implies (2)
ind
a=
ind Q, ,
The ( l x ( m + l ) -matrix (3)
F(Gl) = c (
6(
) ( 1
can be factorized
@r[ar)
where
6(9) =
(<(A),
l,9,
.... ; Y m - l )
is the symbol of the (elliptic ) ~ i r t c h l e problem t f o r the operator A. We consider
~ [ a a s the )
symbol C ( R) ~of a system of pseudo-
differential operators R (Strictly speaking the R
a4 in ( C*
( Y ) ) which ~ is elliptic by (ADN).
a is elliptic in the
Douglis-Nirenberg sense only,
otherwise we have to modify the Dirichlet problem, cf. / 5 / and lectures by F. Treves. ) Now by algebraic properties of Index the equality ( 3 ) irdplies
Q Wt
D =O
(cf. 1 2 1 ) . Therefore (
9 (2))
Finally the Index Formula (1) coincides with the Index Formula discovered by A. Dynin and B. Fedosov (cf. /6/ ) for the elliptic pseudodifferential system R
a
on the manifold without boundary
Y
formula can be derived f r o m the famous Atiyah
.
Of course such
- Singer formula and
actually this was accomplished by the author (Proceedings of the Conferen c e on the Mathematical Methods in Physics, Dubna, 1964) and by B.Fedosov / 6 / . Nowadays B. Fedosov /7/ has found a completely analytical proof ofYhe formula. Therefore we have an elementary proof of our f o r mula (1).
4. The Index Formula for Systems.
Consider now an elliptic (NxN)-system
A
of order 2m.
Suppose that we can again factorize i t s principal symbol
+
&3c-=
m.
By a theorem of Lopatinsky this factorization exists if and only if the rank of the (NxmN) -matrix
4r 2
c~~#~I-'[~~,A~~>--.#
+
is maximal for any T and any curve
3
A
t embracing the spectra of the
F ( Z3,). ) in the upper complex A -halfplane (cf.
/ 2 / ).
We m y reproduce all above considerations f o r boundary problem = (A, B1,
. ... ,Bm)
where B. a r e now (NxN) -systems of boundary J
differential operators. The (ADN)-condition means that the corresponding (mN x mN)-matrix
r b ) is non-singular
f o r any
3
€
S (Y)
As above we have
~d
(4
Q,
F
hat
a+k d
Ra
where
is t h e operator of the Dirichlet problem f o r the elliptic system A
.
However i t s index can be non-trivial. In / 2 / M. Agranovich proved that the
ind 0 is equal to the index of an
elliptic system of pseudocdifferential operators on a manifold without boundary
.
Applying tha analytic Index formula to the Agranovich system
we can derive the following result: Consider the space of cotangent balls of the
~ ( 5= )B X D , The boundary of
D
D={[;
15153
1 .
(c) is
&D(C)= (YrD)
U s(;),
~ ( 52 ) U M B D
.
The symbol a A ) is defined on the S (E)
Extend it artificialfy on the
L
rest Y g D
Let
. Note that
F(A)
S
Y % D h a s a representation
(c) =
<(A)
S
(E),
hr
This is the d e s i r e d extension of #(A) to a mapping 6tA) of a l l
GL ( N , & ) Now if g~ (c)is supplied with canonie orientation then ( - 1) n+l ind 3 = ! ~p [ g (A) -?(;(A)
8~ (c)
into
(2 IZ iln
Note that if N c n then ind
(zn-I)!
b
=
1
2n-1
6oiu')
.
Thus we obtained the Index F o r m u l a ind
a=
At l a s t t h e r e exists the general Index F o r m u l a of Atiyah and Bott. It w a s proved by the a u t h o r (unpublished) and Boutet de Monvel /3/.
References
/1/ Agmon, S. ,.Douglis,
A., Nirenberg, L . ,
and Applied Mathematics,
v . 12 (1959),
Communications on Pure 623
- 727.
/ 2 / Agranovich, M. S., Uspekhi, v. 20 (1965), no. 5 ,
3-120.
/ 3 / Boutet de Monvel, Acta Math. , v. 126 (1971), no. l r 2 ,
/4/ D~nin, A-S., /5/
Donlady V.
11-51.
141 (1961), 21-23.
Dynin, A. S., Doklady, V. 141 (1961),
285
- 287.
/ 6 / Fedosov , B. V. , Funkzionalny Analiz i ego Prilojeniya,
v. 4 (1970)
no 4, 83 -84. / 7 / Fedosov,
/ 8/
B,V., Trudy, v. 30 (1974), 159-241
Hormander, L. , Linear Partial Differential Operators, Berlin, 1963.
Of c o u r s e every 6 - e l l i p t i c operator is hypoelliptic. It is i n t e r e s t i n g that hypoelliptic b-homogeneous I;
and B e a l s ( /I/, /6/ ) a r e
l e f t i n v a r i a n t o p e r a t o r s of Rockland
<-elliptic.
References
/ 1/
B e a l s , R. , Seminaire Goulaic-Schwartz, exp. XIX ,1976-1 977
/2/ Dynin,
A. ,Soviet Math. Doklady, v. 16 (1975) No
€,
1608-1612.
/3/ Dynin, A. Soviet Math. Doklady , v. 17 (1976) No 2, 508
/4/ Folland, G.B. , Stein E. M. Math.
V.
Communications on P u r e and Appl.
27 (1974) , n. 4; 429
- 522
/5/ Shubin, M. A. Soviet Math. Doklady, v.
/ 6/ Rockland,
- 513.
12 (1971), No 1, 147
Ch , Hypoellipticity on the Heisenberg group,
- 151
Preprint
CENTRO INTERN'AZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
GENERAL MIXED BOUNDARY PROBLEMS FOR E L L I P T I C D I F F E R E N T I A L EQUATIONS
G.
ESKIN
Hebrew U n i v e r s i t y o f Jerusalem
Corso t e n u t o a Bressanone d a l 1 6 a1 24 giungo 1977
General mixed boundary problems f o r e l l i p t i c d i f f e r e n t i a l equations Gregory Eskin Hebrew U n i v e r s i t y of Jerusalem
0. I n t r o d u c t i o n Let
G
be a bounded domain i n
I'
.
Assume t h a t
rl
on two p a r t s Consider i n
G
r and
R~
w i t h a smooth
n-1-dimensional
boundary
i s divided by a smooth n-2 dimensional manifold
T2 , s o t h a t
Tl u F2
=
r
, Tl n i;,
=
rO.
a second order e l l i p t i c equation
w i t h a mixed boundary conditions
where
B ( x , ~ ) a r e d i f f e r e n t i a l o p e r a t o r s of o r d e r
k
nj,
k = 1,2.
A
c l a s s i c a l example of a mixed e l l i p t i c boundary problem i s t h e following
where
-aan
i s t h e normal d e r i v a t i v e .
Some o t h e r
examples
r0
of mixed problems w i l l be given l a t e r . The mixed problem f o r e l l i p t i c equations of second order was considered by Lienard [16], Fichera [ l l ] , Miranda [ l a ] , Magenes and Stamplccia [17], Skampaccia [25].
The general case of mixed problems f o r higher order
e l l i p t i c equations withtwo independent v a r i a b l e s was considered by 3. P e e t r e The L -theory of such problems was given by E. Shamir [Zl].
6201.
P
I n 1962 t h e author has studied a general mixed problem f o r a multidimensional These i n v e s t i g a t i o n s were continued i n a
equation of the.second order. s e r i e s of joint papers with
M.
~ i igk ,
where a general theory of
pseudodifferential equations on a manifold withaboundary was developed. The present l e c t u r e s a r e based on the mongraph [6] where t h i s theory was completed and s i m p l i c i f i e d (see a l s o paper [5])
I n [5] and 161 one can f i n d
a l s o more d e t a i l e d references. A s usually i n the theory of t h e e l l i p t i c boundary problems we s h a l l consider
a t f i r s t t h e case when
n
G = R+ = {(x',x,)
5 -- Rn-l + rOa
,xn n-1
r2 = R-
= I ( X ' I ' X ~ - ~ ) xn-l , >OI, n-2 r 1 l G a R and L, B1, B2
1,
> 0 =
X'
= ( ~ ~ ~ , x ~ - ~ ) ~
I (x",xn-l) , xn,l
< 0)
,
a r e homogeneous d i f f e r e n t i a l operators
with constant c o e f f i c i e n t s :
The study of the problem polynomials
(0.6),
L(S), B ~ ( < ) B2(5) ,
(0.7) i s not convenient, because t h e a r e vanished f o r
5
= 0.
Denote
Then
i(5)
w i l l be polynomial on
and
6
and t h e order of
i-L
w i l l be less than t h e order of L(E) # 0
for
6#
0
we have
L(S).
Since
1 5 1 )5 ~
+
C1(l
L(5)
i s e l l i p t i c , i.e. li(S)l
( C(1
+
.
I E I ) ~
Analogously we s e t
Instead of t h e problem
(0.6),(0.7)
i t is more convenient t o consider the
f o l l o k i n g problem
A
where
i(D), ^B1(D), B2(D)
i ( g , ^B1(<), g2(€,).
a r e pseudodifferential operators with symbols
We note t h a t t h e problem
(0.8).
(0.7) only by the terms of lower order.
(0.6),
Therefore t h e conditions of
the normal s o l v a b i l i t y f o r t h e e l l i p t i c problem depends only on t h e p r i n c i p a l p a r t s of
(0.9) d i f f e r from.
(0.1),
L(x,D), Bi(x,D),
(0.2),
(0.3) which
i = 1,2, w i l l be
t h e same a s t h e conditions of t h e s o l v a b i l i t y and the uniqueness f o r t h e problem
(0.8),
(0.9).
1. Sobolev's spaces with weights Even f o r i n £i n i t e l y d i f f e r e n t i a b l e f (x) , gl (x) , &L (x) t h e s o l u t i o n of the problem
(0.8),
(0.9) has,in general, s i n g u l a r i t i e s f o r
Outside of t h e hyperplane w
C -function.
x = x = 0 n n-1
the s o l u t i o n
u(x)
Let
s
IR
xn = x = 0. n-1
is a
Therefore i t i s n a t u r a l t o study t h e problen (0.8),
i n t h e functional spaces with t h e weights which vanish f o r be a r b i t r a r y .
distributions
u(x)
u(x)
(0.9)
xn = x ~ =- 0.~
The Sobolev space H ( R ~ ) i s a space of tempered
with t h e norm
S
where
-
;(<)
distribution
w
1 u(x)
is t h e Fourier transform of t h e
ei(x'S)dx
iCO
u(x)
A s usually we define t h e space
H~@$)
a s t h e r e s t r i c t i o n of
~ ~ ( 8 3 t3o
wFth t h e following norm
where
n
i s an a r b i t r a r y function i n HS(/Rn) which r e s t r i c t i o n t o R+
&(x)
i s equal t o
u(x).
Let
N
20
be an integer.
space of tempered d i s t r i b u t i o n s
The space
that
H
n s,N
+
u(x) with t h e following f i n i t e norm:
i s defined i n t h e same way a s n
Hs,N(~n) we denote t h e
H~
a:)
).
We note
n = Hs(R+).
= Hs(R ), Hs,O(~:) Denote by H ~ ( I R ~ - I )t h e n-1 Sobolev space i n R We s h a l l denote t h e norm i n H:(IRn-l) by H
s , o (IRn)
By
.
1~11:.
so t h a t
Let
H
1
s,N
(IRn-l)
be t h e space of tempered d i s t r i b u t i o n s with t h e f i n i t e
norm
1 n-1 1 n-1 P(R+ ) , RSpN(R+ )
The spaces
-
a r e defined i n t h e same way a s H (IRn) ,Hs,N(~Q s
We note some simple properties of, the spaces Lemma 1.1:
I 5
Let
Ck(l+
H
s,N
and
+
HASN.
1<'I) a-k, v k l O
and l e t
acn-l A(D') i.e. -
a pseudodifferential operator
d . 0 .
with t h e symbol A(<'),
-
A(D)v =
-
(2vln-l
s
Then f o r a r b i t r a r y
and
w
J A(E')i(S')e-i(x"E')d~
for v
E
c;@tn-)'
N
The proof of Lemma 1.1 follows immediately from t h e f a c t t h a t t h e norm i n
Denote by p;
(pl)
(xnml
p'
t h e r e s t r i c t i o n operator t o t h e hyperplane
t h e r e s t r i c t i o n operator t o t h e half-space
< 0).
n
xn = 0,
= 0 and by
x n-1
> 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. ,
i s the Fourier transform.
operator i n Lemma 3.1:
tl(Rn-') Operators
and a r e unbounded f o r H; (IRn-')
flf
a r e bounded i n
6 =
+-
1 7
e+
under t h e Fourier transf o m .
&
N
L ~ ( R ~ - ~ ) 161 <
1 7
i s t h e image of
= FHJ
, ~
Here
O(X~-~)=~
n+ i s a bounded
is bounded i n
zkyN(IRn-l)
where I?' 6
where
Then
It is obvious t h a t
H ~ ~ R ~ - ' )because ,
3
B(X,,~)
20
is arbitrary
.
The proof of t h e Lema 3.1 i s given i n [ 6 ] . The following lemma i s a consequence of t h e Lemma 3.1. ~ e m ~ n3.2: a 3
every function
% 161 < +
2~
can be represented i n
unique way i n t h e following form
-f = -f -
2- E
where
+ T+
%,N, -f+ +E -+HSSN
&
=n%.
4. F a c t o r i z a t i o n of an e l l i p t i c symbol. Lcit
Symbol b(c ') is c a l l e d e l l i p t i c i f
b ( ') ~ f 0
homogeneous f a c t o r i z a t i o n of an e l l i p t i c symbol
5 n- 1 we mean a representation of b(6 ')
where :3)
b-(t')
b+((')
, b ( ~ ' )+E dP
be a homogeneous function of order a
b(E ')
and
(b-(6'))
b+(c') for
for
5 ' f 0.
for
5
0.
By t h e
b(c') with respect t o
i n t h e following form
s a t i s f y the following conditions :
5' # 0 has an a n a l y t i c continu;ltion with respect
t o t h e upper (inner) h a l f -plane
to
b) b+(c",cn-,+ 'C
-
iT) (b (Sf',Sn5ia))
1~"1
-
> 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",
where
L,t(
= L (5",&-l)L+(S'*r5n-1)
a r e polynomials i n
5n-1
r of t h e order
m.
5. Solution of t h e mixed problem i n t h e half-space (continuation). Now we a r e a b l e t o solve t h e equation (2.7). Let (5.1)
bl(S1) b;l(S')
= b-(S1)b+(S')
be t h e homogeneous f a c t o r i z a t i o n of Dividing (2.7) by
Let 2
;-(
b l ( ~ l ) b ~ l ( ~,' )
we obtain
-1 blb2
be t h e index of f a c t o r i z a t i o n of
b+(S1) with respect t o
, i.e.
t h e order of
5'.
1 so - R ~ A< 1
I t follows from t h e lemma 3.2 t h a t i f
has a unique s o l u t i o n
then equation (5.2)
(C-l)
,
where
Therefore i f Is-m
2
- ~ e % l<
1
then
s o l u t i o n of t h e equation (2.2) i n
v(xl)
given by (2.6) i s t h e unique
H' 1 ( 8 - l ) : s- pN
-
-
The formula (5.5) can be w r i t t e n i n a more symmetric form.
Welave
blbil
= b-b+.
SO t h a t
blbrl=
b2b+
Further
Theref o r e
Thus we have proved t h e following theorem: Theorem 5 . 1 : Let -
Let t h e Shapiro-Lopatinskii c o n d i t i o n (2.3) b e s a t i s f i e d .
Z b e the indexof factorizationof
bl(~')b;l(~').
& e &
s
&
N 2 0
be such t h a t
(5.7)
0
S
< s
- m2 -
+ N 2 max
i=1,2
(mi
1
WZ(
+ T),
i s non i n t e g e r i f s
s
Then t h e r e e x i s t s a unique s o l u t i o n
(0.8), g;
(0.9)
's-m2-
for everi
f
u(x)
5 mnx
i=1,2
(mi
1
+ jl).
( R ~ ) of t h e mixed problem s,N + F H ~ - ~ , ~ (g1Q F~ H s - m l - 2 ~1 (IlF-l), + H
1 (C-l). 1.N
6. Examples of t h e mixed problems i n t h e half-space. 6.1)
where
Consider t h e equation
A
A =
aL 2 axn
+
" T n-1
o p e r a t o r w i t h t h e symbol
(I
15" 1
D' I + 112 , 1
.
Consider t h e following mixed problem
is a pseudodifferential
Here
J5:-1
bl(Eq) =
+ 15"1~ ,
b2(E1) = 1
The f a c t o r i z a t i o n of t h e symbol bl(Et) b i 1 ( ~ ' )
so t h a t
1 E = p. Therefore i f
-12 < s < -32
(6.4)
has t h e following form
then f o r every
f
E
and
s
-3
+N
> 2
Hs-2,N(c) * g1e Hs-
t h e r e e x i s t s a unique s o l u t i o n
$,N(c-l)
u 6 Hs,N($)
g2 F
p 1N tPt'-l> -
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<
Let s
-
is noninteger
Bk(x,D)
& N
s
s a t i s f y t h e ~ h a p i r o - ~ o p a t i n s k icondition i on s a t i s f y t h e fol!.owing
-
V.t+"'r0, 1 (% + ?).
F@Z(X~*) m2 < 1, if
s
5 max
k=1,2 -
inequalities:
s f N > m a x (% + k=1,2
$1 p"d
Then mixed problem (0.1). u E H ~ , ~ ( G )gk.
(0.3)
i s normally solvable f o r
E HS-%- Z,N 1 ( r k 1, k = 1.2,
I n general 2 ( x 1 ) (8.1)
(0.2),
c
f
i s an a r b i t r a r y function on
can be not f u l f i l l e d .
H,-~,~(G).
r0
SO
t h a t t h e condition
Then we s h a l l introduce a space of pfecely
constant order of smoothness i n a following way: Let
{,.(x) J .such t h a t
u
1%
j
1 be s
J
6
t h e p a r t i t i o n of u n i t y i n
6
1 if
6 #
.
and
1s
j
be r e a l numbers
Then we denote by
H ~ s j ?, N ( ~ )
a functional. space with t h e following norm
where
i s t h e norm i n t h e space
IIVju1Es
N(G). jY I n t h e same way we introduce spaces H' (Tk). k = 1.2, with t h e norms ISjI.~ ]Ivllis.) , N = where pVmj i s t h e r e s t r i c t i o n of ,j to rkJ J j' It i s c l e a r t h a t p'q, = 0 i f n Tk = @ j j Now l e t 0 < s Re;e(xrl) m2 < 1 f o r x" E n r0 , s + N > max (m + -)1 j j j k=1.2 k 2 1 and s i s noninteger i f s < max (mk+ Then t h e problem (0. I), (0.2), 1 j k=l.2 (0.3) is normally solvable f o r u HISj1 ,.(G), Hs
j'
IIP'~VII;
.
-
-
.
-
' '{s~I-~,H
(GI, gk
E
fI{ sj)-
%- 7 1 (rk),
k = 1.2
,
9 . Generalized mixed problem , It was shown i n t h e s e c t i o n 2 t h a t t h e s o l u t i o n of t h e mixed problem (0.8).
(0.9) If
t
(5.4)
can be reduced t o t h e s o l u t i o n of equation (2.7).
- Rex = 6>161< -1 2
1 - m 2- -2 (5.3),
Now we s h a l l study equation (2.7) f o r
t.
Let (9.1)
t = s
then equation (2.7) has a unique s o l u t i o n
( s e e t h e s e c t i o n 5).
arbitrary
Set
t-ReB=-a+6
.
.
m
1 t h e equation (2.7) f o r h ( t l ) = r b-
<$.
> O i s a n i n t e g e r . 161 (G ;-I; ... hl) HL,,+~,~
-
mn-l>
Consider
Let
m,
-
A
end we look f o r
-+
b+v+
E
,N
>
,
-
E
*-
.
,
m bk ~ ( c " , c ~ , ~=) C qk(el') be a homogeneous function of t h e order k=O cWf o r 5" f 0 which is a polynomial with respect t o 5n-1'
..
-
Let
Q(511,fn-l) # 0
T
equation
(2.7)
f 0.
= (5",5n-1)
n+ $ + 0- 8
so t h a t
=
6'
for
Then
%,N(~n-l)
a s it follows from Lemma 3 -2.
,
Theref o r e t h e
can be rewritten i n t h e following form
So t h a t t h e L i o q v i l l e theorem gives t h a t
Therefore i n t h e c a s e ( 9 . 1 ) t h e equation c2.7) has s o l u t i o n but it i s not " 4 1 ((Rn-2) unique and depend on m a r b i t r a r y functions ck(S1) F H 'k Consider now t h e case
-
(9.4)
t
c(c')
c
- Rel=m+
-
H&6,N@
n-1
-
i f the solutions
6
,
m
-
> 0 i s a n h t e g e r and n-1
) c.HAnN(R
v+(E1)
to
,
m
> 0.
Since
we have a s i n t h e section 5 t h a t
)
c ~d,Ny
..
v-(c1)
H > ~ , ~e x i s t then they must
+-
be given by t h e formulas (5.4).
-'+
161
But, i n general, n - h
does not belong
There i s a following simple formula f o r t h e
Cauchy i n t e g r a l :
m (9.5)n+< = I I~ k=l % where (9.6) 1 It i s obvious that
-+ nm+6, N
A+
,,
+,an%,
L fm. nk-1 (~",E,-,)~(s".E,,)~s,_, -m
n
Am
-
...+
E H
A+
only i f f o r atmost a l l
4
5"
~
we have
but t h e sum i n (9.5) belones t o ~
,
~
Therefore i n t h e c a s e (9.4)
t h e r e i s t h e uniqueness of t h e s o l u t i o n of
equation (2.7) but f o r t h e e x i s t e n c e it i s necessary' t o s a t i s f y t h e c o n d i t i o n (9.7). We n o t e t h a t i f
t-Reg=
1 7 (mod
k),
k
i s i n t e g e r t h e n t h e image of t h e
o p e r a t o r corresponding t o t h e equation (2.7)
I t-~o;el < T , t 1
Therefore only i f (0.8),
(0.9)
has a
= s
unique s o l u t i o n .
is not close.
- m2 - 71, t h e n If
t h e mixed problem
t-Rex= - m + 6 ,
t h e n t h e mixed problem has an i n f i n i t e l y dimenstonal k e r n e l . t-Reie= m
+ 6,
m >
0,161<
1
then t h e mixed problem (0.8),
rn>
1 0, 1 6 1 < 7
If (0.9)
has an
i n f i n i t e l y dimensional cokernel. Therefore f o r
I t- ex/
mixed problem (0.8),
1 >we s h a l l introduce some g e n e r a l i z a t i o n of t h e 2
(0.9)
i n u d e r t o o b t a i n a normally s o l v a b l e problem.
Consider a t f i r s t t h e c a s e (9.1).
Then i n o r d e r t o g e t r i d of t h e k e r n e l
of equation (2.7) we s h a l l impose a d d i t i o n a l boundary c o n d i t i o n s on
n-2
.
Thus we s h a l l consider t h e following problem
where
B-(D) k
operator on
en
for
p''i$u
a r e p s e u d o d i f f e r e n t i a l o p e r a t o r s and IRn-2
Im 5, <
. 0
We suppose t h a t . t h e symbols and
B;(c1,cn)
Bk < s
-
(9.11)
is t h e restriction
Bi(<',Sn)
are analytic i n
1 , so that the restriction
e x i s t s and does n o t depend on a choice of
It may be shown t h a t we can choose
(9.10),
=
p"
eu.
such t h a t t h e problem (9.9), .!I 1 s = m2 2 Rex m u 6 Hs,N($),
B;(<' ,en)
has a unique s o l u t i o n
+ -
- +6
for arbitrary
e
'
1 5j
g j 0 E Hs-$i-l
gk E
H~-~,N@*
\-%-
T1N ( & tj , k = l , 2 ,
zm.
J
I n t h i s c a s e we s h a l l add t o t h e b o u n d a r y
Consider now t h e c a s e (9.4).
c o n d i t i o n s (0.9) m o p e r a t o r s of t h e type
i n order t o compensate f o r t h e cokernel of t h e problem (0.8),
(0.9).
Thus we s h a l l consider t h e following problem
where = m
i
%,N
+ yk
(nn), v k ( x l )
+
< -1,
We can choose
Sk=
Gki(Cv)
E
RY
S+yk-
( R ~ - ~ )a r e unknown,
ordSGki (S1',
=
-2'
such t h a t t h i s problem has a unique s o l u t i o n .
W e n o t e t h a t f o r second order e l l i p t i c equations t h e generalized mixed
problems a s (9.9), (9.10),
(9.11)
and
(9.12),
(9.13),
n a t u r a l because we can choose t h e f u n c t i o n a l space t h a t t h e u s u a l mixed problem (0.8),(0.9)
(9.14)
H S N (
a r e not i n such a way
has a unique s o l u t i o n .
But i n t h e
c a s e of mixed problem f o r higher order e l l i p t i c equations and systems of e l l i p t i c equations i t i s i m p o s s i b l e i n general t o choose such f u n c t i o n a l space. problem.
So t h a t i n t h i s c a s e we a r e imposed t o consider generalized mixed I n g e n e r a l , f o r higher order e l l i p t i c equations and systems of
equations we need a t t h e same time both a d d i t i o n a l boundary c o n d i t i o n s a s i n (9.11) and a d d i t i o n a l p o t e n t i a l s a s i n (9.131,
(9.14).
10. The Riemann-Hilbert problem f o r t h e system of f u n c t i o n s As i n t-he c a s e of second o r d e r e l l i p t i c e q u a t i o n s t h e mixed problem f o r system of e l l i p t i c e q u a t i o n s o r f o r a s i n g l e h i g h e r o r d e r e l l i p t i c equation i s connected w i t h t h e s o l u t i o n of t h e Riemann-Hilbert problem
where
A(5') m
matrix, C Let
~ l a ~ ~ ( ~ ' fis an e l l i p t i c homogeneous of o r d e r a 29 j=l for 5' # 0 and v+ E HtaNYV- E Ht-a.N.
=
5"
U =
-
+
be f i x e d .
Consider t h e following Riemann-Hilbert problem
i n one v a r i a b l e
-
where
- -
-+
1
-
-
We s h a l l show l a t e r t h a t t h e study of t h e Riemann-Hilbert problem (10.1) i s based on a n i n v e s t i g a t i o n of t h e family of Riemann-Hilbert problem
(10.2) depending on t h e parameter The Riemann-Hilbert
.
u=,&
(10.2) can be w r i t t e n a s a system of p s e u d o d i f f e r e n t i a l
o p e r a t o r s on t h e h a l f - l i n e :
P+' A ( U , D ~ - ~ ) V + ( X ~=-h~ + 'p )
(10.3)
,
where we have t a k e n t h e i n v e r s e F o u r i e r transform f o r (10.2) and then t h e r e s t r i c t i o n on t h e h a l f - l i n e
-
~ e t - -n + ( E ~ =
-
t-a"
hl = A-
where
h.
En-1
1
JR+ ; We n o t e t h a t p+v'-
;+ ='+t(5
5. We put
n-1
)
= 0,
.. , u--
V+
(t-a)-
= A-
V-,
Then we o b t a i n t h e f o l l o w i n g - s y s t e m which is e q u i v a l e n t t o
-t t-a ~ ~ (E 0~ -, ~=I A ( ~ , E ~.A - ~ ) n-
+
-
, u+
-+ 6
~
~
-
u-
c
,
--
H
~
~
hls , 6 'o,N("l) ~ ~
is equivalent t o t h e s o l u t i o n of t h e
The s o l u t i o n of t h e system (10.4) system
,.
where
h+ =
n+ -hl. I f we t a k e t h e k v e r s e Fourier transform we s h a l l o b t a i n from
where
AO(w,Dn-l)
U+ =
1
-
m
S
AO(u, Sn-l)
u+(Sn-,)
+
-OD
the operator of m u l t i p l i c a t i o n on
-ixn-l'n-~ e
O(xn),
U+
'0,~
'
h+
dkl,e'
+
H
~ ',
is
~
The f i r s t problem which we s h a l l consider w i l l b e t o f i n d t h e condition
+
f o r 0 A0
t o b e a Fredholm operator i n
+
H (El1). O,N al, a
...,
lim Ao(w,
Let
a, =
+
Theorem 10.1:
The necessary and s u f f i c i e n t condition f o r /J*A
t o be a Fredholm operator i n (10.7)
-"+ H
O,N
0
(w,S
n-1 )
is t h e following:
There a r e no r e a l n e ~ a t i v enumbers among t h e eigenvalues
~ I ~ L P Proof: -
( s e e [S], 163): Set
I 5 j 5 p, be t h e eigenvalues of t h e matrix b. By using t h e j' condition (10.7) w e can choose t h e branch of t h e logarithm such t h a t
Let
b
(10.9)
-
1 &2 < Re b3 < 2
,
15 j z p .
It i s easy t o s e e t h a t t h e matrix
-1 a+b- ( 5 )b ( 5 ) n-1 + n-1
has t h e same l i m i t s
a s t h e matrix
En-l
when
AO(u, En-,)
+
f
OD
-
Set
Lemma 10.2: P*
( s e e [5], [6])
a r e bounded i n
Now using t h e L&
where
...-
C1
I f conditions (10.9) holds then t h e operators
.
H 6RL) O,N 10.2 and t h e following r e l a t i o n
i s a Hilbert-Schmidt operator,we can p r w e t h a t t h e operator
-
-1 P- a+ g+
+ = P+A-I o a+
~g
i s a l e f t and r i g h t r e g u l a r i r e r of
fl+A.
( s e e [5],[6] f o r t h e d e t a i l s ) :
where
T1,T2
a r e compact operators i n
that
n +Ao
i s a Fredholm operator i n
-
HO,N.
It follows
from (10.11)
-+
H ~ , ~ '
It i s easy t o r e w r i t e t h e condition (10.7) f o r t h e case of t h e equation
(10.3) Let
o r (10.2).
a (0) j
Set A(') ( E " , E ~ - ~=) A(E",En-l)
, 1 5 j 5 p,
(En
- 5 I c " ~ )-a
be t h e eigenvalues of t h e matrix
~ ( O ) ( o , - i ) [A(') (o,+i) I-l. Then t h e necessary and s u f f i c i e n t conditions f o r t h e operator (10.2) t o be a Fredholm operator from 1
H+,
to
1 # 7 (mod k)
(10.12) t
- Re
where
i s an a r b i t r a r y integer
k
l n a(') j
Ht,@:)
.
a r e t h e following:
,
11.Normal s o l v a b i l i t y of p s e u d o d i f f e r e n t i a l o p e r a t o r of t h e h a l f - l i n e (continuation) I n t h i s s e c t i o n we g i v e two o t h e r s methods t o prove t h e normal s o l v a b i l i t y of t h e e q u a t i o n (10.6). 11.1) S e t X(X,~)
E
,D
= AO(~,Sn-l)
x
E n
dd)
1
b+(Sn-l)
.
b + ( ~ , - ~ )1
and l e t
W e note t h a t
and t h a t t h e o p e r a t o r s
x(xne1) Al(~,Dn-l),
AO( W , D ~ - I~, )
[x(xn,,),
a r e Hilbert-Schmidt o p e r a t o r s .
Ro t h e following o p e r a t o r
Taking i n t o account t h a t t h e o p e r a t o r of
5
Ixn-ll
and t h e commutators
-1 [ x ( x ~ - ~ ) a+ , b- (Dn,l) Denote by
- a+ brl(Sn-l)
1 C o r n ), X(X,-~) = 1 f o r
1 @ (x,,)
(l-x(x,,) A
A1(~,Sn-ll
n+
a+ b l l b+
-1 b+ (Sn-l)n
+ b-
-1 a+
is the inverse
we have
a r e compact o p e r a t o r s i n
+ .
where
T1,T2
11.2)
The p r e v i o u s two methods of c o n s t r u c t i n g r e g u l a r i z e r f o r t h e
e q u a t i o n (10.6) has t h e form f
O A
0
(w,D,
-
show t h a t i f
+ -1 0 A. (W,D,_~).
a+ = aIf
H
+
(IR~) ~ , ~
t h e n t h e r e g u l a r i z e r of
a+ f a-
6 '%(W,D,-,)
t h e n t h e r e g u l a r i z e r of
is n o t a n o p e r a t o r of t h e sane c l a s s .
I n t h i s s e c t i o n we
extend t h e c l a s s of p s e u d o d i f f e r e n t i a l o p e r a t o r s of o r d e r z e r o on t h e h a l f - l i n e i n such a way t h a t t h e extended e l a s s se7:tains t h e r e g u l a r i z e r s of the operators Let
M(t)
+
O AO(w,Dn-l).
be a m a t r i x which belongs t o
C-
for t
> 0
and s a t i s f i e s t h e
estimates
(11.3)
t
c
1-1
k d%(tl
<
C
E
I n (11.3), by
A
M (z)
6
i s fixed,
t
E t
-S- E
-1+t+ € 3. 7
0 < 6 <
t h e Mellin transform of
< t < 1, O ; k < -
for
0
for
1 L t < m
and
E
1
, O L k < m
is arbitrary.
> 0
Denote
M(t) :
m
(11.4)
P(r) = J ' M(t) tz-'dt.
A
Then M(z)
6 < Rez < 1
is analytic i n the s t r i p
-6
and it s a t i s f i e s
t h e estimates
I z%($l
(11.5)
Conversely i f
5 A
M(z)
s a t i s f i e s (11.51,
CEn
$6 +
E ( Rez
1 1-
6
- E , pw 2 0,
VE
i s a n a l y t i c i n the s t r i p 6 < Rez < 1 - 6
then
M(t)
E cm
for
> 0,
t
> 0 and
and t h e estimates (11.3)
holds. Let
be t h e i n t e g r a l operator defined by
M
A
Then Mu+ = A
where
u+(z)
M(z)-u+(z) , A
A
i s t h e Mellin transform of
the Mellin operator and
h
M(z)
We s h a l l c a l l M
w i l l be c a l l e d t h e symbol of t h e operator
The c l a s s of t h e operators (11.6),where (11.31, w i l l be denoted by
U+(X,-~).
M(t)
M.
s a t i s f i e s t h e estimates
m.
1 We s h a l l consider the following c l a s s of operators on t h e half-linefR+:
where AO(w,Dn-l)
i s t h e pseudodif f e r e n t i a l operator
with t h e symbol A o ( w , n l ) ,
and
T
M
($.d. 0)
1 on R
i s an t n t e g r a l operator of the c l a s s
i s a compact operator i n H (lR1) O,N
.
m
Since
A(F,",€,-~)
limit
a+ =
-
A 0
p % O D
(1l:~f
operator '(11.8)
is a homogeneous matrix of c l a s s always e x i s t s .
cOD f o r 5' f
0
the
By t h e symbol of t h e
we mean t h e p a i r of matrices.
Ao(w,En)
-
on t h e closed l i n e
-
5.5,5.+OD
and
+
1 2niz 1-0
(11.9) a
where
a,
a
- a-
2 n i z e 2niz 1-e
AO(w,Sn)
lim
A
+ M(z)
on t h e closed l i n e
Rez =
1 9
•
*;S, Theorem 11.1:
+
(see 151, [61)
I&&l
+ O+X(X,~)
&&2
= 0 A2 (w,~,,~)
(11.7)
with t h e symbols
M2
+ x ( x ~ - ~ ) M+~T1+ Q
+
= O % ( w , D ~ - ~ )tO
+ O+T2
be two operators of t h e form
and -
- - a2-
1 ~ ~ ( w , " p . ~a+ ). 2niz 2 1-e where
a:
=
li. en-1-
A
+ M2(z),
~ ( w , S ~ - ~a:) , =
Then t h e i r product
and *3 -
2siz Zniz
e
cad be pYt i n t h e form (11.7):
has t h e symbol:
which i s equal t o t h e product of fir symbols of
8
gnJi
A 2; that is
where -
i
a? = l i m
En-l+*w
A3(~,En-1)
.
f + = al a;
As a consequence of t h e Theorem 11.1 we o b t a i n t h e following condition f o r
&
t h e normal s o l v a b i l i t y of an operator For & -
t o be a Fredhdlm operator i t i s Tiecessary and s u f f i c i e n t t h a t t h e
determinant of i t s symbol (11.8), (11.12)
d e t Ao(w,Snji # 0,
(11.13)
d e t -(
a
+
1-e
Zaiz
I f conditions (11.121,
i s now zero:
-< Edi+ +
,
det
a+ - # 0,
1 f i ( z ) ) b ~ , ~ e z7 =
(11.13) a r e f u l f i l l e d we can construct a r e g u l a r i z e r
.6?
In particular, l e t
a=O+
+ O AO(~,Dn-l)
-w
(11.9)
2aiz a-e2aiz 1-e
-
R of t h e operator
(11.16)
of t h e f o m (11.7) :
i n t h e following form:
t h a t is, M = 0, T = 0.
AO(~,Dn-l),
i s a Fredholm operator i f
d e t AO(~,Sn-l) # 0,
-
En < a
,
d e t a -+ # O ,
Then
(11.17)
d
Since
e
a+
t ( 1-e
,we have
-1 + iT 2
z =
2riz %e 2niz 1-e
-
x
e
)
2Tiz
=
.
1 # 0, Rez = T
- e-271~
Condition (11.17) is equivalent t o t h e following det(e2"
(11.18)
1
+ a+-1 a_)
-w
# 0 for
<
T
<+
.
+..
Hence t h e conditions (10.7) and (11.17) a r e equivalent. Let
1 . l+e
t =
-271T
Then
t
0
5
-
1 when
-< T
(
.
+
The condition (11.17) can be w r i t t e n i n t h e following form (c.f. (11.19)
connects
ta+
.
+
(1-t)a-)
# 0 for
+
(1-t)a-
i s t h e segment i n t h e space ofmatriceswhich
det(ta+
We note t h a t
Lt 5 1
1131)
a+ = AO(~,*)
with
0
a- = AO(w,a).
Now we s h a l l f i n d t h e index of a Fredholm operator Theorem 11.2: When (11.121,
&
of t h e form (11.7).
(4-1.13) hold then t h e index of t h e operator
i s given by
ind
&=
(11.20)
1
5 Aarg
1 + A 271
det
A~ (w,
En-1
a arg d e t (
+x
Z
1-e
,15-,-~ - -- +
En-1=
-
+
OD
a2riz 1-e
I n p a r t i c u l a r t h e index of t h e operator
(11.21)
ind
+ 2a
=:
1 2iT
-
Aarg d e t (ta+
bar3 A. ( 4 Sn-l)
+
(1-t)a-)
- + im
z= 1
-2 -
+
1 4 0
O AO(~,Dn-l) i s given by t h e
formula
+ O A.
z= 1 2
+~(z))l
l n l t=l
=
+
O5
+
OD
,OZtL1.
12. The factorization of matrices arid the construction of regularizer. In two previous sections we have constructed such regularizers of the
O+. A
operators
that
(11.2) holds where T1,T2
are compact operators.
Now we shall construct more precise left and right regularizers of
+ 0. A
such that
+
(12.1)
RpAo=I+K1,
(12.2)
8AOR2= I + K 2 ,
+
where %,K2
are operators of finite rank that is the images of ~~(i=1,2)
are finite dimensional.
In the scalar case (see sections 5 and 9) the
solution of pseudodifferential equation on the half-line or the solution of the corresponding Riemann-Hilbert problem (2.7) was based on the factorization of an elliptic symbol. A factorization is also possible for elliptic matrices
<",
for a fixed
but in general the factors in the factorization are
discontinuous functions of
5".
At first we consider subclass of elliptic matrices for which the factors in the factorization depend smoothly on the parameters.
heo or em
12.1: (see [5], [6]) Let AO(~,
be the same matrix as in the
sections 10 and 11. Let the condition (10.7) be fulfilled. Assume that for + + 1 all o E s ~ -the ~ operator 8 AO(w,~n-l) has a unique inverse in H 6R ) . 0 Then the matrix AO(~,
-
(12.3) AO(~, = A-(w,Sn-l)A+(~,
where -
-1
( 12-41 A+
b+(En-l) -
-1
(a,Enel) = b+
(En,1) + c+(w, Errl) ,
are the mirtrices in (10.8).
for -0 0, C -
m
--
estimates (12.6) / $ D ~
T-n-1
with respect to -
and c+(b),< n-1+%T) ,isanalytic in <,-l+i~ (w,<,-~ + i ~ ) for TL 0 and satisfies the
5
'pk (1+lCn- I+lr1)&6~+k =
c+(w,E_~ + IT)~
7
0,
S i m i l a r l y t h e m a t r i x C-(~I,S,~+~T) i s a n a l y t i c i n of c l a s s for
-
T
5
C~
c,-l
for T
'E.n-l+f~
< 0,
r 5 0, and s a t i s f i e s t h e e s t i m a t e s (12.6)
-l+ir) for
In addition
q.
(12.8)
in
d e t A+(W,<,-~
+
ir) # 0
for
C'
20
The c o n d i t i o n s of t h e Theorem 1 2 . 1 a r e f u l f i l l e d f o r s t r o n g l y e l l i p t i c matrix
AO(~,
matrix
AR(~.En,l>
t h a t i s f o r such e l l i p t i c m a t r i x
A~+A:
a
2
For example, i f t h e matrix
I
-
AO(~,En-l)
11 <
AO(~y
.
is c l o s e t o t h e u n i t m a t r i x
AO(~,
,
E
E
is s m a l l ,
then i t i s a s t r o n g l y e l l i p t i c matrix and s o it has a f a c t o r i z a t i o n of t h e 'form (12.13). Conversely i f a matrix then t h e equation
AO(~,
has a f a c t o r i z a t i o n of t h e form (12.3)
. fl + ~ ~ ( w , < ~- - ~ ) .g+ u + i s uniquely s o l v a b l e i n =
O,N
d.1)
and i t s s o l u t i o n can be w r i t t e n i n t h e following form:
We now consider t h e g e n e r a l c a s e of a n e l l i p t i c matrix
which
AO(w,
s a t i s f i e s t h e c o n d i t i o n (10.7). Theorem 12.2
(see [5], [ 6 ] ) :
There a r e an i n t e g e r
where
QN(w,
i n (12.4), degree 2N,
a d matrices
N
~ ~ ( w , < ~ P2N(wy
have t h e same p r o p e r t i e s a s t h e m a t r i c e s
(12.5)
and
P
2N
P2N(uySn-1) =
u
n
2N
c
k=O
1
i s polynomial matrix h
,
k P ~ ( ~ ) S ~ - ~
such t h a t
A+ (a,
-
of
W e s h a l l u s e t h e Theorem 12.2 f o r c o n s t r u c t i n g t h e l e f t r e g u l a r i z e r of t h e
operator
fl +A 0 ( ~ , 5 ~ -.~l e) t i1 denote
I f we apply
..
Rl
t h e o p e r a t o r defined by
t o t h e l e f t i n t h e equation
. n +A0u+-. = ..g+
we o b t a i n
(see 151, [61)
where
+
Ck(wY
a r e some symbols and
We n o t e t h a t t h e operator
+
2N 2 Ck(w,6n-1) k-1
1 ?;i
OJ
;
k-1 p-l 2N
+
-03
i s an o p e r a t o r of f i n i t e rank. The r i g h t r e g u l a r i z e i
-
R2
such t h a t
(12.2) holds can be constructed i n
a s i m i l a r way by using t h e following f a c t o r i z a t i o n of t h e matrix Ao(w,cn-l):
where
+ QilY ~:f)have t h e
same p r o p e r t i e s as Q* P N' 2N
i n (12.11).
13. General boundary problem f o r system of p s e u d o d i f f e r e n t i a l equation on a half-line. Let
A ( w , S ~ - ~ ) be t h e same matrix a s i n (10.3) and l e t t h e condition (10.12)
be fulfilled.
Then t h e operator
P:~(w,~n-l)
i s a Fredholm operator from
1 I R ) f o r every w E s"-~. We denote t h e dimension ( ~ 2 ) into t,N Ht-ay~ $. of t h e k e r n e l of t h e operator p:~(w,~n-l) by n+(w) and t h e dimension
H+
of i t s cokernel by
n-(w).
By analogy w i t h t h e s c a l a r case, t o make t h e
1 boundary v a l u e problem on R+ c o r r e c t l y posed we add t o t h e equation (10.3) a t l e a s t max
n-( w) p o t e n t i a l s s o a s t o e l i m i n a t e t h e cokernel of t h e
operator at
(A(U,D ) n-1
x
m+ 2 max n+(w)
be any i n t e g e r s such t h a t -n-(u)
= m,
15j
m+,
5' #
where 15 k
m
is t h e index of
2 m-,
,
boundary conditions
w
m-
< A ( ~ , n-l). D
2 max w
(13.1)
ord
Let
(E")
jk
cm f o r 5''
Bj(5')
5'
n- (w) and
~ A ( ' J J , D ~ - , ) .Let
m
+
and
m-
m+-m-=n+(w)
-
B j ( S t ) , Ck(c'),
be homogeneous p-dimensional vectors,
cm
for
=
ordS,Ck(E1) =
Bj,
bk. Re
Bj <
- p1
t
Retk < a
Bjk =Bj + r k
be homogeneous functions of order
- t - -1 2
- a + 1,
f 0.
C onsider i n
IRf
t h e f o l l a r i n g problem
+
where
1
pk. hj a r e complex numbers, V + ( X ~ - ~E) Ht,NW+). Ide suppose t h a t t h e following condition holds: (13.4)
Let
Suppose t h a t
0.
E
max n+(w)
s o as t o e l i m i n a t e t h e kernel of
= 0
n-1
and we impose a t l e a s t
The system of equation (13.2),
(13.3)
g E Hta,N~l).
i s uniquely solvable f o r any
w € sn-? I f t h e condition (13.4) i s f u l f i l l e d then themethods of t h e s e c t i o n 12 gives p o s s i b i l i t y t o c o n s t r u c t t h e l e f t and t h e r i g h t i n v e r s e of t h e o p e r a t o r
(13.2)
(13.3). We s h a l l consider t h e c o n s t r u c t i o n of t h e l e f t inverse. the right inverse i s similar. (13.5)
Rg =
Let
be t h e operator s i m i l a r t o (12.13):
R
+-I+ - - I . t?r P 2N (w,nn-l)(QN) 0 (Q,) A (D~-,) l g
-~i~'~~
W e note t h a t t h e kernel of t h e operator
Rg = 0. g-
Then
EH~c'),
The construction of
-
o+(~;)-' 161
At-dlg = 0. 3
and
1g = 9;
R
is
Therefore
zero,
Indeed, l e t
( ~ i 1 - l t-a l g g-
= g- )where
-
1 has t h e support i n IR-.
'l'hus
g = p i l g = 0.
t By applying A+R
t o t h e equation (13.2) we obtain (see [5], [6]):
+ A:%, where
and
Cki +
a r e some symbols.
The system (13.6) i s equivalent t o t h e system (13.2) s i n c e t h e kernel of
R
is zero. N t-N The equation A- A+ v+
~ ' ' . A Icp+ ~ = 0
for
--
15 k
cp+
5N
has a s o l u t i o n v+ and then
-t
v+ = A+
E
+
1
Ht@Z )
0+4N-N~ -cp+.
equation (13.6) i s equivalent t o the equation
with t h e following a d d i t i o n a l conditions
Now we s u b s t i t u t e (13.8) i n the boundary conditions (13.3): W e obtain
i f and only i f Therefore the
i s reduced i n an
(13.3)
Therefore t h e s o l u t i o n of the problem (13.21,
equivalent way t o t h e s o l u t i o n of the system (13.8), where
wk,
1f k
5
j-1
p"~,,~
-1 t ~ y . Then we obtain
P 2N
(13.3) i s equivalent t o the s o l u t i o n
Thus t h e s o l u t i o n of t h e problem (13.2). (13.101,
(13.10),
I n order t o solve t h i s system
2N, have t h e form(13.7).
we apply t o (13.8) t h e operator
(13.9),
(13.11) ,where
1 5 k L 2N,
of the system (13.8),
(13.91,
a r e given by (13.7).
Moreover t h e s o l u t i o n of t h e problem (13.2), (133)
wk,
i s reduced t o t h e s o l u t i o n of t h e algebraic system (13.9),(13.10),
which i s a system of Pjy
1 2 j 2 m-.
system (13.9), w ( ' ) ,I i
+ m+
3N
equations with
(13.10),
5 m-
Then the
Indeed l e t
be a n o n t r i v i a l s o l u t i o n of t h e (13.11).
is given by (13.8) with
Then
g = 0,
v
(0)
(xn
,p(0),l
-
1
5 j 5 m-,
i s a w n t r i v i a l s o l u t i o n of t h e
(13.3) and t h i s i s a contradiction with t h e
homogeneous problem (13.2), condltian (13.4).
unknowns wk' 1 - k<2N,
is fulfilled.
(13.11) has a unique solution.
(0) 1 2 k ( 2N, pj , 15 j
v(O)
+ in-
Suppose t h a t t h e condition (13.4)
iiomogeneous system (13.91, (13.10), where
2N
(13.111,
We note t h a t i f not a l l
w p ) , 1 5 k 5 28, a r e zero then
v (0) ( x ~ - ~ is ) nonzero a s i t follows from t h e homogeneous system (13.11)) which i s a consequence of the system (13.8). Therefore t h e matrix
M(u)
maximal rank f o r a l l w the l e f t inverse t o (13.111,
wefind
E
M(w)
of t h e system (13.9), (L3.10),(13.11) n-3
S
.
.
Thus t h e r e e x i s t s a matrix
Applying L(u)
L(w)
has a which i s
t o the system (13.91, (13.10),
w 1 & k ~ 2 N pk, , l ( k ~ m - . k'
then t h e formula (13.8) gives the expression f o r
When wk,pk
a r e known
V + ( X ~ -'~ )
Thus we s h a l l construct the l e f t inverse t o t h e operator defined by the
equations (13.21,
(13.3).
We n o t e t h a t t h e system (13.8) i s a system of i n t e g r a l equations w i t h a degenerated k e r n e l and we have repeated above t h e u s u a l procedure f o r t h e s o l u t i o n of such i n t e g r a l equations.
I n canclusion we n o t e t h a t t h e method
of t h i s s e c t i o n can b e used f o r t h e c o n s t r u c t i o n of t h e parametrix f o r t h e g e n e r a l boundary problems f o r t h e system of e l l i p t i c p s e u d o d i f f e r e n t i a l equations ( s e e [S], s e c t i o n 6). 14. Necessary and s u f f i c i e n t conditions f o r t h e e x i s t e n c e of t h e uniquely s o l v a b l e boundary problem. I n t h i s s e c t i o n we s h a l l f i n d out when t h e c o n d i t i o n (13.4) i s f u l f i l l e d . We s h a l l formulate t h e following two simple lemmas: Lemma 14.1:
Let
be a continuous family of t h e Fredholm o p e r a t o r s
A(w)
,
E snm3,which map t h e Banach space B1 on t h e Banach space =2* Then t h e r e e x i s t s v e c t o r s gk E B2, 1 5 k L m - such t h a t t h e o p e r a t o r & . I )(u, cl,.
..,c
) = A(W)U+ "'-
has no cokernel f o r a l l w Lemma 14.2:
Let &(o)
z
k=1
n-3
w E S
.
Then f o r each
wo E S
5 m-,
v+
E
+
Ht,N,
pl,
...,Pm-
f o r every
Now we s h a l l f i n d o u t when t h e r e e x i s t symbols t h e problem (13.12), 14.3:
n-3
the kernel
Ck(c",F;n-l),
such t h a t t h e system
has a solution
L-
,
n-3 U(w ) c S 0 I -where U(U ) is some
I t follows from t h e Lemma 14.1 t h a t t h e r e e x i s t symbols 1( k
B2
0-
.
w,
B~ x E
s~-~.
has a continuous b a s i s i n
neighbourhood of
m-
ekgk, which maps
b e a continuous family of t h e Fredholm operator which
have no cokernel f o r a l l of .@(o) -
6
m-
(13.13)
g 6 H
t-a,N
(R')
,
B j ( 6 ' ) . ~ ~ ~ ( t such " ) , that
i s uniquely s o l v a b l e .
Suppose t h a t t h e system (14.1) has a s o l u t i o n ( V + ( X ~ -,P,, ~)
..p
m-
f o r every
w6
s ~ and - ~ g(xn-1)
'
d+) . There acist
Ht-a
boundary
conditions
such t h a t t h e condition (13.4)
- s"'~
continuous b a s i s on -- - -
i s f u l f i l l e d i f and only i f t h e r e e x i s t a
of t h e s o l u t i o n s of t h e homageneous system (14.1).
It follows from t h e Lemma 14.2 t h a t t h e kernel of t h e operator defined by
(14.1)
is a v e c t o r bundle.
It follows from Lemma 14.3 t h a t t h i s bundle
must b e t r i v i a l bundle f o r t h e e x i s t e n c e of (14.2)
such t h a t t h e c o n d f t i o n
(13.4) is f u l f i l l e d . I n order t o formulate t h e conditions of t h e e x i s t e n c e of a uniquely s o l v a b l e problem (14.1),
(14.2) i n more i n v a r i a n t terms we need t h e d e f i n i t i o n of t h e
index of a family of Fredholm operators. c l a s s e s of-isomorphic v e c t o r bundles on bundle.
Let
differences vector bundle
Q
and
fir
.
n-3
)
be t h e s e t of a l l
We denote by g t h e t r i v i a l
E'-F'
a r e c a l l e d equivalent i f t h e r e e x i s t s a
F @Q
where Q t h e d i r e c t sum of t h e vector bundles
means t h e isomorphism of t h e v e c t o r bundles.
The s e t of t h e c l a s s e s
of t h e equivalent formal d i f f e r e n c e s w i l l be denoted by K ( S " - ~ ) . K ( s ~ - ~ ) is an a b e l i a n group i f we d e f i n e and
-(E-P)
3s
[El
(E-F)
+
= E $ E'
- F $ F'
E d e f i n e s an element of ~ C ( S ~ - which ~) we s h a l l denote
I t i s obvious t h a t an a r b i t r a r y element of
-
(E1-F')
The s e t
= F -E.
Each v e c t o r bundle
by [ E l .
Two
such t h a t
E $ F' @ Q w E'
and
Sn-3
Vect(S
be t h e formal d i f f e r e n c e of two v e c t o r bundles.
E-F E-F
Let
[I?]
,
where
E and F a r e vector b n d l e s .
(s"-~) Let
A(w)
A(@) ...
can be w r i t t e n be a
continuous family of Fredholm operators and l e t (u,cl, ,C ) = m m= d(w) u + CCk gk be t h e operator which has no cokernel f o r each k=l w E s ~ ( s- e e~Lemma 14.1). Then t h e k e r n e l of *(w) i s a v e c t o r bundle
( s e e Lemma 14.2). equal t o e
The element of K ( s n - ) ) of t h e family A(u)
IC(A(U))
I m- ,
m-
Bl
X
m-
E
on
gl,
..., gm-
B2.
(m+-m- = m
sufficiently large
is equal
B (5'1, Ck(C1), E (c"), 1 2 k j jk such t h a t t h e condition (13.4) i s f u l f i l l e d i f and only i f
t o t h e index of 1( j
.
does not depend on t h e choice of
such t h a t t h e operators 2 ( ~ )maps
For m+
i s c a l l e d t h e index
K(A(w))
and it w i l l be denoted by
It can be shown t h a t
Theorem 14.1:
- m-
r
~A(U,D~-~ t h)e)r e e x i s t
I p(U,Dn
t h e index of t h e family
-
i s equal t o
- - [m-] -
[m+]
=
2 m+,
[m].
The next theorem which i s based on t h e r e s u l t s of Atiyah and Bott find out when
-
K(
[d.
Theorem 14.2: I n order t o sufficient that for
N
Let
IN
be t h e u n i t matrix i n
G ( - ~ A ( U , D ~ -=~ [ml ))
cN.
i t i s necessary and
s u f f i c i e n t l y l a r g e t h e matrix
(IAo(c",cn-l)O
(1
II
II
-
0
i s homotopic t o t h e matrix T
i n t h e c l a s s of e l l i p t i c matrix s a t i s f y i n g t h e condition (10.7). 15.
Generalized mixedroblem i n t h e half-space.
Consider i n
n
P(+
an e l l i p t i c system of d i f f e r e n t i a l equations:
with mixed boundary conditions
IN
where t h e symbols have t h e following orders
Bij
P
P
+
C tj i s equal t o t h e order of i=l j-1 ( 5 ) a r e a n a l y t i c with r e s p e c t t o for
2m =
l r.
en
s
+N
> max
1 7). Re Bi <
(mki+
i,k
s is non i n t e g e r i f s
5 max
(mki
i, k
We suppose t h a t
and we choose
?++N(rR+)n
+
1
)
s
,
det$~~~(Oli:,~=~
b 5, < 0, and
- 1,
Re
yj <
-
with (15.1)
P;(
vj
eE
--
n-3 Lc)ES
2 R+
be
connected
(15.4):
P C Bllj j=l
where vj, hi
9
s + 2'
.
Hi+Rey ~ ( I R ~ ' ~Let ) . P 3 2 f i x e d and consider t h e following family of mixed problem i n
E
1
-
,xn) (%Dn-l~Dn) l u (x j n-1
a r e t h e complex numbers,
m-
+ 1 vj G l i j ( W ~ D ~ - 6, ) ( j=1
~ ~ - ~ ) ) =
Suppose t h a t t h e following c o n d i t i o n i s f u l f i l l e d : (15.9) (15.5)
For given
-
s ~ t-h e ~boundary
and f o r a i b i t r a r y
s
problem
i s uniquely solvable.
(15.8)
Then t h e following theorem holds: Theorem 15.1:
Let t h e c o n d i t i o n (15.9) b e h l f i l l e d .
-
mixed problem (15.1)
(15.4)
Then t h e generalized
i s uniquely s o l v a b l e .
+-
To prove t h i s theorem t a k e t h e F o u r i e r transform w i t h r e s p e c t t o
x" and
'n-1 then we make the-change of v a r i a b l e s x ~ = - l ~+ 5 1 jn= When we put = ( 1 + (~111)?n-l, 5, = ( 1 + I E ~ ~ I we o b t a i n t h a t En-1
7
+
11,
t h e operator corresponding t o t h e problem (15.1) n-2 a s a p s e u d o d i f f e r e n t i a l operator i n K
&(w),
(15..4) can be considered
with t h e o p e r a t o r valued symbol
which i s t h e o p e r a t o r corresponding t o t h e system (15.5)
t h e symbol
&(a)
uniquely s o l v a b l e .
(15.8)
i s i n v e r t i b l e and so t h a t t h e problem (15.1)-(15.4)
is
Now we s h a l l discuss t h e condition (15.9).
Suppose t h a t t h e boundary o p e r a t o r s B
lij
(D)
E snm3 t h e Shapiro-Lopatinskii condition.
and
B2ij(D)
e x i s t s a smooth b a s i s
ep(<',xn) = ( e l r ( ~ ' , x n ) , . . . , e
of t h e s o l u t i o n s of t h e system
x
n
-+
+
.
f u l f i l f o r each
I t follows from t h e Lemma
14.3 applied t o t h e system of d i f f e r e n t i a l equations (15.5)
which d e c r e a s e f o r
-
Therefore t h e c o n d i t i o n (15.9) g i v e s t h a t
(see [ b ] for further details).
LIJ
-
5,,1 .
Pr
that there
( c 1 , x n ) > , (6' # 0)
.(<',<,) be t h e Fourier transform of k~ ekj = 0 f o r xn < 0.
Let
ekj(St,xn)
E
, where we put
The general s o l u t i o n of t h e equation (15.5) has the following form:
where
6") i
L:~)(w,D~-,,D~) I f j , 1 5 i ( p,
=
j-1
11
i s a p a r t i c u l a r s o l u t i o n of (15.5)., . 1 1 ~ ( ~ ) ( 5 ) i s t h e matrix i n v e r s e t o ij
Illij
(5)
11,
W ~ ( X , ~ )E Hi- L Y N d ) a r e a r b i t r a r y functions. 2 I f we s u b s t i t u t e (15.11) i n t h e boundary equations (15.6), (15.7), (15.8) we s h a l l o b t a i n a f t e r easy transformations s i m i l a r t o those i n s e c t i o n 2 t h a t the s o l u t i o n of t h e problem (15.5)of some boundary problem (13.2),
(15.8) i s equivalent t o t h e s o l u t i o n
(13 -3)
.
Therefore the condition (15.9) can be reformulated a s a condition of t h e uniquely s o l v a b i l i t y of
operator
u.
1
E
H
s+ti,N
an)
+
and
s
i s s u f f i c i e n t l y l a r g e then t h e
$ A ( u , D ~ - ~ ) has no kernel, s o t h a t the mixed problem (15.1) ,(15.2), t
(15.3) i s uniquely solvable without d d i t i o n a l boundary conditions (15.44. If
s
i s negative and
has no cokernel.
Is1 i s s u f f i c i e n t l y l a r g e then t h e operator
So t h a t t h e mixed problem (15.1)-(15.4)
GA(u,D~-]
i s uniquely
(D') (vj x 6 ) . It i s obvious what i s t h e kij formulation of t h e mixed problem f o r bounded domain G w i t h a smooth boundary solvable without p o t e n t i a l s
r , so
G
we don't g i v e i t here.
I n conclusion we s h a l l mention two boundary problems t h e s o l u t i o n o f w h i c h i s s i m i l a r t o t h e s o l u t i o n of mixed problems, considered above. The f i r s t is t h e transmission problem f o r two domains with smooth boundaries, which have a common p a r t .
16. Lienard A. Probleme p l a n de l a derive'e oblique dans l a the'orie du p o t e n t i e l , Jour. Ecole P o l i t . 5-7(1938), 35-158 and 177-226. 17. Magenes E. and stampac&a G. I problem6 a1 contorno p e r l e equazioni d i f f e r e n z i a l i d i t i p 0 e l l i t t i c o . Ann. S c . b r m a l e Sup. P i s a 7(1958), 247-357. 18. Miranda C. Su a l c u n i a s p e t t i d e l l a t e o r i a d e l l e equazioni e l l i t t i c k , Bull. Soc. Math. France 86(1958), 331-354. 19. Nirenberg. L. P s e u d o d i f f e r e n t i a l operators, Proc. Symposia Pure Math. V. 16 Amer. Math. Soc. 1970, pp. 147-168. 20. Peetre. J. Mixed problems f o r higher order e l l i p t i c equations i n two variables, I Ann. Scnola Norm. Super P i s a , 15(1961),337-353.
-
21. Shamir E. Mixed boundary v a l u e problems f o r e l l i p t i c equations i n t h e plane, t h e Lp-theory, Ann Scgola Norm. Slper 17 (1963), 117-139. 23.
, E l l i p t i c systems of s i n g u l a r i n t e g r a l o p e r a t o r s I Math. Soc. 127(1967), 107-124.
-
Trans Amer.
24.
, Boundary v a l u e problems f o r e l l i p t i c convolution systems " p s e u d o d i f f e r e n t i a l operatars" CIME, Edizione Cremonese Roma, 1969.
24.
, Regularity of mixed second ~ r d e r ~ e l l i p t problems, ic Israel Journ. of Math. 6(1968), 350-168.
25. Stampacchia G. Problemi a 1 contorno e l l i p t i c i ccm d a t o d i s c o n t i d , d o t a t e de soluzione holderiane, Ann. Math pure e app1.51(1960), 1-38.
.
26. Simonenko I, A new g e n e r a l method of i n v e s t i g a t i n g l i n e a r operator equations of s i n g u l a r i n t e g r a l equation type I,II,Izv. Akad, Nauk. SSSR s e r . Mat. 29(1965), 567-586, 757-782fRussian) 27. &bin M., F a c t o r i z a t i o n of matrices depending on a parameter and e l l i p t i c equations i n a half-space,Math. USSR sb. 14(1971), 65-84. 28. ~ i i i M. k and Eskin G.,Elliptic equations i n mnvolution i n a bounded domain and t h e i r applicatjons,Russian Math. Surveys 22(1967), 303-332. 29.
, Normally s o l v a b l e problems f o r e l l i p t i c systems of equations i n convolution ,Math. USS% Sb. 4(1967), 303-332.
30.
, The Sobolev-Slobodetsky spaces of v a r i a b a e order with weights norms and t h e i r a p p l i c a t i o n s t o t h e mixed problem, S i b i r s k i Math. Journ. v.9.N.5 (1968).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M~.E.)
HYPOELLIPTICITE POUR DES OPERATEURS DIFFERENTIELS SUR D E S GROUPES DE L I E NILPOTENTS
B . HELFFER
(centre de M a t h 6 m a t i q u e s de I t E c o l e P o l y t e c h n i q u e 91128
-
Palaiseau
- France)
C o r s o t e n u t o a B r e s s a n o n e d a l 1 6 a1 2 4 giugno 1 9 7 7
HYPOELLIPTICITE POUR DES OPERATEURS DIFFERENTIELS SUR DES GROUPES DE LIE NILPOTENTS. par
B. HELFFER
(Centre de MathCmatiques de lvEcole Polytechnique 91128
- Palaiseau - France)
On se propose dc montrer dans cet expos&, comment lcs
rbsultats de Boutet de Monvel en particulier
- Grigis - Ilelffer
[3] slappliquent
116tude de 11hypoellipticit6 pour des opCrateur8
diffhrentiels invariants sur un groupe de Lie nilpotent de rang de nilpotence 2.
I1 svagit essenticllement de rbinterprbter les condi-
tions n6cessaires et suffisantes dvhypoellipticit6 donnbes duns [ 3 ] en termes de reprCsentations de groupe.
$ 0
Enonce des r6sultats
On consid&re un groupe de Lie G de dimension n, simplemeat connexe, connexe, nilpotent, dont lvalg&bre de Lie "ddmet une
76 d6composition de la forme
:
avec
On suppose que $ 2 est de dimension 2 1 (le cas o; q2 = 0 est Bvident) On munit cette alg6bre de Lie de la famille de dilatations suivante : bt (Y)
=
tj Y
pour Y dans ~j ( j = 1.2)
On sait que llapplication exponentielle
q
et t > 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
.
Pour m o n t r e r l ' h y p o e l l i p t i c i t 6 au v o i s i n a g e d e
:
l l o r i g i n e , il s u f f i t d e v b r i f i e r l a c o n d i t i o n aux p o i n t s ( 0 , 5 ) de
x.
E t u d e de l a condition ( i i ) La c l b d e 1 8 C t u d e de l a c o n d i t i o n ( i i ) r e s i d e d a n s l a forme p a r t i c u -
l i i r e qu'a a
(x,l)
P
c
(P) n e depend en f a i t que de
f 1.7.
p o p b r a t e u r s d i f f i r e n t i e l e du l e r o r d r e
i ( i = 1, que nous n o t e r o n s d a n s l a s u i t e L'X Avac c e s n o t a t i o n s ,
X est d 6 s o r m a i s
Ln donnte d e s p
on b c r i t a
fix0.
(x,S)
:
a
,..
1
,(ail
=, *
( i z 1,
..,PI
, p ) avec X = (x,S).
(PI e o u s l a forme
:
il l a s p b o l e d e L 3 ( j - l , . . p ) . On diteigne P a r lx
.
X
*
j ( j = 1 , . , p ) d6f i n i t dana gZn un p- p l a n que nous
la
Y
S
~
noterons CX.
A ce plan p - p l a n le rang de la 2
5 est
attach6 un invariant symplectique qui est 4e
- forme symplectique sur RZn
On pose
restreint
Ch '
:
et on v6rifie que
:
2 q ( ~ ) = rang [tui'uj'(X)]
2 q ( ~ )est done le rang au point X de la 2-forme symplectique sur T''R~\o
restreinte
A
C.
Par une transformation symplectique dans R 2n , on peut trouver une forme canonique pour 1: (en fonction de 2q(X) 1. I1 existe un opha 2 n rateur unitaire tie L (R (continu aef(lIn) dans et de
SJi*,
5
Y?R") d s n e 3 < R",) associ6
On a pris
cette transformation synplectique tel que:
ici comme nouvelles coordonnbes sur 3Rn :
(g,y,z)
avec X
r
XI,..,
X
qtx,
Y = Xq(X)+l,-,
Xp
On dira que aa(x,y,Dx)
- .q(h) est une forme r6duit.e de
aL,x.
Dans sa forme
r i d u i t e , on c o n s i d 4 r e r a
h
(x, y,D ) comme un o p i r a t e u r s u r ~ ( R Q )
x
d i p e n d a n t du p a r a m t h e y d a n s Rp-lq(X)
I1 r k s u l t a a l o r s du thGor6me 3.1 d e [3], l a p r o p o s i t i o n s u i v a n t e
:
P r o p o s i t i o n 1.3 Sous l e s hypothGses du !l'hior6me 1.1, l e s deux c o n d i t i o n s suivant e s sont 6quivalentes (ii)
I(
k %(PI
iii,
:
eat injectif d a n s f ( ~ " )
E B P - ~ ~ ( ~a)x, ( ~ , y , ~ xest ) i n j e c t i f dans Y ~ B ~ ( ~ )
En c o n c l u s i o n , nous venons de c o n s t r u i r e e n t o u t p o i n t I d e
pour t o u t y d a n s R p-2q(X) une a p p l i c a t i o n n dhfinie par
t e l l e que
X¶Y
et
d e 3m'kd a n s A ~ ( ~ ) ( C )
:
:
si P e e t dane
amp
e t Q e s t d a n s !Jlm''k'
On r 6 i c r i t le th6or6me (1.1) s o u s l a forme s u i v a n t e
:
ThPor6me 1 . 4 S o i t P un o p k r a t e u r p s e u d o d i f f k r e n t i e l d a n s !Rrnsk,
traneversalement
elliptique le long de 2, et A un point de
x.
Les assertions suivantes sont kquivalentes
:
5C n I1 existe un voisinage conique de A dans T R \o, dans lequel
(i)
P est hypoelliptique avec perte de k/2 dkrivkes.
(P) est injectif dans
%,Y
5
2.
y(R~") ).
Application i lV6tude de lfhypoellipticitk pour des opkrateurs diffhrentiels invariants
gauche sur des groupes
nilpotenfs de ranR de nilpotence 2. Soit (XI,. de
.
G2
..
(Zpcl,, Zn) une base
X ) une base de
P
et soit P un opkrateur diffkrentiel invariant homogGne dlordre
Soit u.(x,5) J
soit
.,
le symbole principal de iX. (j=l,.,,p) et J
lC
le c6ne lisse dans T G\O dbfini par
u1(x,5)
= O,..,
:
u (x,5) = 0. P +C
est un sous-fibrb vectoriel de T G\O. On peut identifier G A lZn en prenant lea coordonnkes exponentielles; lV616ment neutre du groupe est l1origine dans 33". (2.1) est dans la classe Remarquons enfin que
q1
dans
16
.
On voit aisbment que P defini par
am'm(~n,~).
n i 0)x.&n slidentifie b:Q 1 ~ennulateurde
On peut aussi 11 identifier
gC 2'
Nous allons montrer que la condition du thkorbme (0.1) est suffisante, la nbcessitk rksultant dfun thkorbme de Beals [I].
Ellipticit& Transversale
soit 5 C Rn\O, on consid&re la representation de
5
suivante
Cqest une representation scalaire de 3 qui correspond
:
la repr6-
N
~entatibnnl de G suivante
:
Si on choisit des coordonn6es sur G (x,z) telles que
:
<2.
n, est une representation irreductible unitaire, non triviale s i . S f 0 .
La condition d'injectivit6 de n
S
(PI s'6crit alors simplement
I".")
:
pour I; f 0
Cqest la condition dqellipticit8 transversale pour P.
1
Si on identifie RP et F2
ltannulateur de
*, lea represen-
dans $
tations construitea sont celles qui sont aesoci8ee par la mhthode dea
I
orbites (cf. [4] p 154) aux Clbments d e a
P cst done un element de
2'
ampktransversalement elllptique
aLlona appliquer le th6orhme
et nous
(1.4).
D6monstration du thSor6me 0.1 On doit v6rifier la condition du theorArne (1.4) pour tout point X
dans
xn
I
{ 0j x lRn (autrement dit, X dans GI).
En effet 1 'hypoellip-
ticit6 dans un voisinage de llorigine en r6sultera (Remarque 1.2), et, 110p6rateur 6tant invariant
gauche, il sera hypoelliptique
dans G. On d6finit les repr6sentations de et y dans R~-2q(X) ;
?,Y
G
suivantes; pour X dans
est d6fini par
:
.
appartient i 9 "' (i=l,. ,p) i et que 2 appartient i 91r0 (i=p+l,. ,n) i Le thior6me (0.1) r6sulte de la virification du point suivant : On a simplement utilis6 le fait que X
. . IV
est associie A une repr6sentation x irrbductible, unitaire, '=x,Y X,Y non triviale de G. N
On va construire explicitement x
X,Y.
11 r6sulte de lf6tude faite au $ 1 (et de [J]) que l1on peut trouver, X 6tant fix&, une nouvelle base de Xi [i=l,. .q(~))
telle que
, xi
.
(i=l?. ,q(~))
Q;. :
,
Yi [i=l,
#p-2q(~)
:
def
4,y(zj)
=
U
J
. z i
J
.
(j=p+l,. .,n-2p)
('x, ,y, N
Soicnt
N
N
x
z ) l e s coordonn6es e x p o n e n t i e l l e s s u r G a s s o c i b e s
l a base c h o i s i e ci-dessus
On pose pour v d a n s
:
(R~")) N
On v f r i f i e que c t e s t une r e p r b s e n t a t i o n d e G d a n s
s a n t l a formltle d e Campbell-Hausdorff
Cette reprt5sentation e s t u n i t a i r e ; sc.ritation x
X,Y'
.
en
utili-
p o u r l e s g r o u p e s n i l p o t e n t s de
e l l e donne b i e n s u r
e l l e e s t non t r i v i a l e c a r h f 0.
ment q u ' e l l e e s t i r r i d u c t i b l e ( c f . [ 4 ] , p 7 2 ) .
l a repri-
On montre f a c i l e , -
Le t h & o r & n e(0.1) e s t
a i n s i compl&tement d6montr6.
Rcinarclue 2.1
On dhuigna p a r
yj
:
G 0I
l i e n avec l a m6thode d e s o r b i t e s (reap.
Q=I,.., p - ~ q ( h ? ) r e a p .
S o i t lh t
suivante
Y
1 ql)
:
( c f . [4])
l l e s p a c e v e c t o r i e l engendrh p a r l e e
l e e X ! (j=l,.., q(X0). J
l a forme l i n b a i r e s u r
3
N
a e s o c i i & (X,y)
On p0.e
d e l a rnani&re
:
A llaide du thbor&me de [4] p 154, on peut verifier que, si GO est
le groupe
associe A
Go,
representation induite T
N
la reprhsentation x
(q 0'1JeX,Y )
X,Y
est Cquivalente
la
= ind X(%,~) GoPG
3e x1 r Y est la representation xX,Y (exp X) = exp i (ly,X,X) pour X dans Go. On peut ainsi montrer qu'on a obtenu avec les represents-
oi
N
tions
N
toutes les classes dq6quivalence de reprbsentations 5 et 5 ? Y , 3e 46 irreductibles, si l1on remarque hue deux elements 1 et 1' qui x
coincident sur q2 sont sur la m6me orbite de la reprbsentation coadjointe de G , si et seulement si, elles coincident sur
Go1'
BIBLIOGRAPHIE
i 1]
R. Beals
-
[2]
R. Beals
-
[3]
L. Boutet de Monvel, A. Grigis, B. Helffer
Seminaire Goulaouic
- Schwartz (1977)
Comptes rendus du colloque de St-Jean de Monts (Juin 1977)
-
Parametrixes d*opbrateursp~eudodiff6rentiel~ i caracthristiques multiples Aetbrisque 43-35 (1976), p 93 121
-
-
[4]
L. Pukanszky
[5]
C. Rockland Hypoell ipticity on the Heisenberg.Group. Representation Theoretic criteria, (preprint)
[ 61
L. P. Rothschild and E.M. Stein
Leeon sur les representations des groupes Dunod (1967)
-
operators and nilpotent groups.
-
Hypoelliptic differential Acta Math. (1977).
CENTRO IN TERN AZIONALE MATENATICO ESTIVO
(c.I.M.E.)
LECTURES ON DEGENERATE E L L I P T I C PROBLEMS
J . J . KOHN
Princeton University
-
Prirlcetor~ N J , U . S . A .
Corso tenuto a Bressanone dal 1 6 a1 24 g i u g n o 1977
Lecture 1. These lectures will be concerned with equations (and systems of equations) which a r e
" close"
t o elliptic.
By this we mean that they a r e
limits of elliptic ones; such as, for example, the bat equation can be expressed a s the following limit:
a 27 + =a= ax
a2 -
lim ( - 6-0 ax
a2
a
6 7 - + = ) .
at
We will call this phenomenon degenerate ellipticity.
The emphasis here will
be on L -methods, we will be studying our equations by means of the 2 Let Q be a domain in IRn and let
following variational problem.
b e a s e t of m-tuples of functions on 52
.
Let Q be a quadratic form on
given by:
where
...,um
1 u = (u ,
),
v = (vl,
and ( , ) denotes the L2-inner product on Q
...,fm
f = (f' ,
)
of functions on Q,
variational problem, t o find u c
(3)
8
...,v m )
. Given an m-tuple
w e associate with Q the following such that we have
~ ( uV), = (f, V) f o r all v c %(
.
a
Let P be the map of
W/
then, if
into m-tuples on S2 defined by:
contains all m-tuples with compact support we have
whenever u satisfies (3). In t h i s way, t h e above variational problem can be used t o study the operator P on the space
8.
In many applications it is also useful t o
set Q(u, v) = (Lu, Lv) where L is an operator of o r d e r m; in this case P = L'L,
where L' denotes the formal adjoint of L,
problem can be used t o study the operators L and L'
and our variational
.
The following is a standard result in Hilbert space theory.
6.
Theorem.
Q(u,
(7)
V)
Suppose 6: satisfies the conditions:
-
= Q(v, U)
T h e r e exists c > 0 such that: (9)
Further, if
2
11 u 11 -< CQ(u, U)
f o r all u
.
is dense in the space ( ~ ~ ( ' 2 )then ) ~ f. o r each f
t h e r e exists a unique u in closure of
(10)
0
~ ( uV), = (f, V)
for all
a
under the norm Q, vr
8.
t
(L~(Q))~
such that
Proof:
Let
8
which sends v
denote-the closure of
ea
under (2. Then the functional
into (f, v) is bounded in the Q-norm, since by (9):
Hence this functional h a s a unique representative u
E
3
which satisfies (10).
It is shown in [l] that the hypotheses (7) and (8) can be weakened considerably, thus giving this method m r e flexibility; h e r e we will not enter into these m a t t e r s since we want t o concentrate on the main ideas. Our attention in these lectures will be focussed on the smoothness of the solution u.
F o r this purpose we will need a - p r i o r i estimates which
a r e stronger then (9). To illustrate the method which will be developed we will give a proof of t h e hypoellipticity of the heat equation.
This is probably
the most difficult way of studying the heat equation but i t has the advantage that it can be used t o analyze much more general equations. differential operator defined on function in IR
12. Theorem.
If u, f r L2(Q),
the equation Lu = f and if f is
cmon u K.
by:
and if u is a weak
cmo r anaopen subset
U of
solution of then u is
.
Let of
QcR 2
2
Let L be the
P, P'
Let u
t
t
cm(n) such that b' 0
= 1 on a neighborhood of the support
cW(SZ) then f r o m (11) we have
by integration by p a r t s we obtain:
hence
and s o
f r o m (12) we deduce
Substituting Dk-'u
for u
X
with k > 2,
/IY Dk~ U/I2 5 (:D:-
(17)
and integrating by parts, we have:
k ' ~ u , S D ~ U+) const.
11 K
1
11
X
.
and using the Schwarz inequality we get:
Applying (18) with
r
replaced by: Y'
and k by k
-
1 t o the l a s t t e r m of
(18) we have:
// C Dk~ U/I2 -< const. ( /IY D X~ - ~ L U 11 '+ // C ' D:-~LU 1) '+ 11 K"
(19)
where
c
C;(Q) and
c"
D:-u'
= 1 on a neighborhood of the support of
T h u s , by induction and by dmosing a n increasing sequence of
Cl
=
r
and
ck = < '
// 2, I'
.
r . c crn(n) with
J 0 and 5. = 1 on a neighborhood of t h e support of 6 . J J-1
we obtain
and the c a s e k = 1 is estimated by (16). F r o m (11) we obtain
k- 2
Applying Dx
t o this equation combining with (20) we deduce inductively
that:
Finally, differentiating (21) above with respect to t and combining with the above, we obtain:
Up t o now we have assumed that u r based on that assumption.
c*,
and obtained the estimates (23)
Now assume that u is a weak solution of Lu = f
and that f is C* in an open s e t U.
We want t o show that if P
u is C* a neighborhood of P (and hence in all of U). function
r,
t' E
00
CO SO that
r
neighborhood of the support of
For u
E
L 2 define S6u by:
r c;(lEtn)
with
U then
So choose the
= 1 in a neighborhood of P and
I. Let
c
t'
= 1 in a
Then, for 6 $ 0,
we have S6u e cm(Rn). Wrthermore we have
and this converges to
r' ~
hood of the support of C f that for each us
@ for f every cu (since f is in
.
cWin a neighbor-
Hence, replacing u with S u in (23) we conclude
11 (Das6u 11
6
is bounded independently of 6 .
! , D ~ U6 L 2 for all r and so u is in
Therefore,
on the interior of the set defined
by !, = 1. Which concludes the proof of Theorem 12. Now we will formulate a Dirichlet problem for the heat equation which will serve a s a model for the more general degenerate elliptic case. Let n be a bounded region in R a
L
with a smooth boundary bS2,
crnfunction defined in a neighborhood of MZ
outside of
and since
we have (28)
and r
in S2.
Then if u
6
such that,
~ ~ (we5have )
dr
let r be
$ 0, r > 0
hence
Following Fichera ( s e e 1211, we divide bn into t h r e e subsets L1, C 2 and Cg consists of those points where r
C3.
W2
- I;3
X
$
0, L
2
is the subset of
in which r < 0 and C is the remainder (i. e . C is given by t 1 1
r = 0 and rt > 0). If we s e t u = 0 on the interior of C U C we have X 2 3
f r o m this we can deduce that if Lu = 0 then u = 0 since u is constant on the line segments, t = const. points in C3.
and dense s e t of these segments have end
Thus, given f r
cW(E)and
v in
cmdefined on the interior
of Z3 U C 2 t h e r e exists a t most one smooth function u on Lu = f and u = v on the interior of
6
such that
Lg U C 2 .
Suppose Lu = 0 and u = v on bQ and suppose u
E
c*(E). This
implies that at certain points of bn, v may have t o satisfy compatibility conditions.
To s e e this suppose that the origin is in W1 and that r is of
the f o r m r(x, t) = t (31)
-
g(x) =
g(x) further let' s suppose that
-
2 Mx t higher o r d e r terms.
The boundary near the origin is 'flattened" by choosing the local coordinates
h e r e the prime denotes differentiation of g ( f ) with respect t o f u(6, 0) = v ( f ) and g' (0) = 0,
. Since
we s e e that a t the origin
Thus we obtain a s a necessary condition vfS(0) = 0 in case M =
(35)
-21 '
1 Now suppose M $ -2 ; differentiating (33) with respect to
5
we
find, with the aid of (34). that a t the origin
Hence we obtain the compatibility condition: at the origin
Next, assuming M
$ 71 , T1 ,
s o that u
T
and u r
f
a r e known at
the origin (from (34) and (36)), we .differentiate (33). in turn, with respect
to
T
and twice with respect t o
5,
and find that at the origin the following
expressions a r e determined, in t e r m s of derivatives of v:
Thus again we shall obtain a compatibility condition on the derivatives of v a t the origin in c a s e the determinant of the coefficients vanishes, i. e.
The general situation is given i n the following theorem, which is proved
i n [I]. Theorem 36.
Let v be the boundary values of a C*
equation in t 5 -Mx
+ ...
2
solution of the heat
(in a neighborhood of the origin). T h e r e is a
of positive numbers such that, if M is equal t o one
sequence M ,M 1 2'"'
of these values, then v and i t s derivatives satisfy necessary compatibility conditions a t the origin. conditions.
The distinguished values M. a r e obtained a s follows: the 1
numbers c. = 1
a =
1 -+2-'
F o r any other value of M there a r e no compatibility
4Mi
k = 1,2,
a r e roots of the Lequerre polynomials L ( ~ with ) k
... . These polynomials a r e
defined by
Lecture 2. The
'I
physical" reason behind the compatibility conditions f o r the
heat equation can be found in the uniqueness of solutions given by (26). Take the region fl t o be bounded by the a segment of the x-axis and the graph of the formation t = g(x) where g(x) < 0 in the interior of the segment and 0 a t the end points.
(x,g(x)) with g' (XI
$
Then the segment is in L1,
the priority
0 a r e in Cg an t h e r e s t of the boundary is in C
2'
Now translating let u s place the origin at a point with g(x) of the form (31). Then the value of u at the origin is completely determined by the values of u on (x,g(x)) with x < 0 and g(x) < 0 and also by the values of u o r the points with x > 0, g(x) < 0.
-
This is seen by looking at the Dirichlet
problem in the regions { (x,t) 1 g(x) < t < 0, x < 0) and {(x, t ) 1 g(x)
-
x 2 0)
. Thus, one should expect to have compatibility conditions. W e will now give a rough description of the results contained in [I].
Consider the equation: ij Lu = a u..
(38)
13
with
+ biu.1 + cu = f
~a~jc.c.>O. 13-
Here u(x) is a r e a l function defined in a compact domain i2C IRn ( o r on a manifold), with C* boundary; x = (xl.
...,xm )
represent the
coordinates, and we have used subscripts to denote differentiation; we have also used the summation convention.
.
ctO in
The coefficients a r e .real and of c l a s s
The f i r s t boundary value problem consists of prescribing the
values of u o r a certain portion of the boundary bi2.
We wish to obtain
unique solutions of the problem which a r e smooth up to and including the boundary.
If the leading part is elliptic, we have the usual Dirichlet
problem otherwise we proceed following the definitions of Fichera in [2]. Let v = (v 1'
...,vn)
denote the exterior normal vector at point of bn,
we
divide bS2 into t h r e e portions a s follows:
L3
= the set of non-characteristic boundary points, i. e., those
ij where a V.V.> 0, 1 J
2
=
i the s e t of characteristic boundary points f o r which (b -alJ)v. >O, "
3
1
The
I'
Dirichlet problem" is that of finding a solution of (38) with
prescribed values on C U C 2 3'
After subtraction of a function with the
same values, we may suppose that the given boundary values on C U C 2 3 a r e zero.
We now improve the following conditions:
LZ U Z3 is closed.
(a)
(b) - c is very large positive compared to the other coefficients and their derivatives of second order, and to the derivatives of o r d e r < 3 of the ij
r , a s well a s t o y
-1
defined below.
To formulate (c) we define certain invariants.
Let xo be a point
of Z2 and let r be a function which vanishes identically on bS2 with dr
$0
and r < 0 in S2 ; then, on C 2 the values of
a r e independent of the particular coordinate system.
p > 0 on C Z and, since the vector r v. a/ax.(r r a 1 1 k m
km
)
-c
i
By hypothesis,
is parallel to v
i
and
0, it follows that o < 0. Furthermore, CY/@'
-
invariant under change of the function r o r under multiplication of the operator L by a positive factor.
To formulate the l a s t condition set
The third'condition is:
is
then a vanishes on ZZ. Let LO = a''
..
a2
then: axiax
j (d) For e constant o O depending only on n we require
42.
Theorem.
If f r HZm(Q) (i.e. all derivatives of f up to order Zm
a r e square integrable) and if conditions (a), (b), (c) and (d) a r e satisfied with o
0
2 n -1 = (100n 3 ) then the unique weak solution of (38) is in H m'
The proof of this theorem is rather intricate and most of [3] is devoted to it.
Here we will only make a few remarks about this theorem.
First, the choice of
E
0
is much too small and u i s much smoother then
just in H however, t o describe the smoothness of u precisely we m' would have to discuss some rather complicated weight functions- -which would take us too f a r afield.
Second, (b) can very often be bypassed by
changing the unknown function; for example the equation u - u = f is x x t equivalent to the equation vxx if we set v =
- vt
At
and g = e f.
- AV = g
for an arbitrary constant A
Finally, in case of the boundary point
for the heat equation discussed above we show in [I] that at these points the solution u i s not
COO,
even i f all the compatibility conditions a r e
ratisfied, i t s smoothness depends on the size of M (in (31)) which controls the size of y.
Furthermore, we should remark that this theorem is global
in nature, in other words the smoothness of f at one point has influence
on the smoothness of u at every point.
In the rest of this lecture we will
discuss some general global results concerning the smoothness of solutions in the variational problems introduced in the f i r s t lecture.
W e will deal
with the case when Q is of f i r s t order, i. e.
the higher o r d e r case can be reduced to this (see also [4] for a direct treatment of the higher o r d e r case). Let Q C Fin be a bounded domain with a smooth boundary bSZ defined by r = 0,
where r r
and grad(r) $ 0 near HZ.
COO
Anel c cm(lJ)
there is a neighborhood U of P and functions tI# such that tl, Now if
0 (a)
...,tn-l' r
form a coordinate system on U with origin P.
is the domain of Q we s e t
BI is dense in
m
=
@ , ( ~ ~ ( and 5 )assume ) ~
n
and ( C ; ( Q ) ) ~ C,@.
(LZ@))
is complete under Q.
(b)
(c) If f (d)
t
b. q = (cpl, ..., m
~ ~ (and 5 q) t ,&
r CO(U),
OD 6
a
0
with
then f q t
a ...,ati (5q))
(Ki(M),
(el Suppose
VC I C U
c;(uoS~)
44.
a, where
T
and h = (hl,.
tr =t
such that KThu = ThKu for
hr.
1h 1
..,hn -
..
,&f
...,n - 1.
s o that (tth, r ) t U
with supp(cp)(_V
fl
Furthermore if K : c * ( M ~ ) 0 sufficiently small then Kq c f i
Q is non-characteristic at P c
Definition.
) then
f o r i = 1.
t
whenever (t, r) c V then we assume that if cp c then Thcp c
If P r M ,
HZ if, relative to the
coordinates introduced above alJ (P) is non-singular. hn 45.
Under the assumption of Theorem 6 , if G: is non-characteris-
Theorem.
tic for each P
c
m,
if
satisfies the conditions (a) to (e) and if Q is
compact relative to L
(i.e. w sequence in
2
bounded with respect to
the Q-norm has a convergent silbsequence in L ). 2 there exists a unique cp 6 and then cp
c
a.
a such that
Definition.
E
( ~ ~ ( 5 ) ) ~
Q((P,J, ) = (a,+) for all J, e
Furthermore, the operator that takes q t o cp is completely
continuous and its eigenspaces a r e in 46.
Then if n
If u
t
c*(Q)
&.
we define
IIu 1,
for a non-negative integer
m by
i f m < 0 we define ltu 11
by
m
W e denote by H the completion of m
cm(z)under the norm 11 1) m '
As in the discussion of the last equation, to prove the smoothing of the solution cp if we proceed by first establishing a priori estimates for the derivatives. 49.
This is done with the help of the following.
Lemma. Suppose that Q satisfies ( 7 ) , (8) and (9).
compact with respect to L
2
if and only if for each
r
Then Q is
> 0 there exists a
constant C(c) > 0 such that:
for all cp t
a.
Here we set llcp
2
= Z
j
llcpj 11
2
.
The above lemma follows by a standard argument from the fact
that the L2-norm is compact with respect to the -1 norm. To estimate the mth derivatives of cp we proceed a s follows. We f i r s t choose a covering of
E o b = @.Let this covering. J
51.
-
{(O..
c
..,5N 1 be a partition of unity on
> 1, let Now on a fixed Uv with v -
. We apply
Lemma.
where 5;
...,U~
UV, if v 2 1, admits a boundary coordinates system (t, r) and
such that
ZP.< m
by neighborhoods Uo, U1,
P
P
P
=
5
(PI,
subordinate to
...,PnVl)
with
P
(50)to Dt(Svq) = (Dt3,,ql....,DtSvqk).
Under the above assumptions:
~;(u,,nii), (1= 1
on the support of 5, and the constant C
depends on m. If @ is the solution to the variational problem in Theorem 45 then
Since Q is non-characteristic we have:
Furthermore q satisfies the equations:
ij
Iff 11
C a ~~~~~.+lowerorderterms=~.. P 1 I~l=l J ff
Hence, by the non-characteristicity, we have
Replacing
$,$J in (53) by D:~P
and using Lemma 51 and (52) we obtain
Differentiating (55) repeatedly and multiplying by
Gt,
taking L -norms 2
and combining with the above we obtain the appriori estimate,
The same estimate holds for v = 0, the proof being simpler since we do not have normal derivatives.
Summing over y choosing
r
small we
obtain:
and since
llqll
const.
110 11
we obtain by induction on m:
Now we have to show that the derivatives o f , 9 a r e actually square integrable.
It would be hard to apply the smoothing operator directly a s in
the first lecture.
Instead we use the method of
" elliptic
regularization".
Suppose
5
f o r all cp r
is a f o r m defined over
&
which satisfies
.
B. Then by standard elliptic theory it is shown that if
G(('~19) = (a, 9) f o r all 9 r 8 then cp is smooth whenever
CY
is smooth
(this can in fact be shown by the same techniques a s in the f i r s t lecture). 6 Now we s e t Q t o be the form on
e
defined by:
with 6> 0. Then, given a
c
(Cw-(52)) m there exists a unique q~6 r ' 6
6
Q (cp ,9 = (a, 9) f o r all 9 r
(62)
6
such that
.
Furthermore, examining the proof of (59) we s e e that the estimate
holds with a constant that is independent of 6 (for 6 small and 6 > 0). Hence a s 6
-
-.
0 the cp6 have a subsequence that goes to a limit qJ0 t
but now f o r 9 c
dJ
we have
0 Therefore cp = cp , by uniqueness.
cm(G)
F r o m compactness it follows that t h e r e is a discrete s e t of { A } such that
.
f o r some q r &. p f 0 and all 9 such p and l e t E
$0
and a
E
nA
L e t SA denote the space of all
-.SA denote orthogonal projection.
:
L (Q) t h e r e exists a unique 2
Q(% 9)
(66)
-
9)
(PE
-E
F o r fixed X,
such that
9 ) = (a,JI),
To s e e this ,note that the left hand side cannot be z e r o f o r all $ unless
cp = 0 ; if it were s e t 9 =
n A cp
we conclude l2 p = 0 s o that cp E S but A A
then cp = 11 cp = 0. Hence by Fredholm theory a unique cp e A
(Y
H
t
m ,
(i. e.
/Im
IJa
<
W)
0
exists.
If
then arguing a s above we can show that cp t Hm
and
We wish t o show that S A Cfl
element 0 s (Y
c
...,r 9
with k < q.
8 such that (0, u.) =
0 for j = 1 , .
cQ)(E)such that
J
(a,u . ) = 0 f o r j = 1,.
J
CrF be the solution of (661, then pE E 3 such that
I/$' /Im
be an orthonormal basis of
..,uk c B ,
and a s s u m e that ul,
S
let rl.
.
..,k
We will construct an
..,k
and 110
and (a,cr
p+l
)
11 = 1.
$ C
.
We can find a sequence
Let
Let E
.
+
0
J
llcpEJ 11 -- m for if such a sequence did not exist then by (67) would be bounded for all rn and hence would have a covergent
subsequence whose limit
$
would satisfy the equation
which is impossible since a is not orthogonal t o SX. Let
ev
=
converges in H
dv -
m'
then by (67) we can find a subsequence which
s o the limit 8 is C* and hence 8
f o r every m,
E
.
Further, by (66) we have
hence 8
c
S
X'
Now setting $ = a
and hence (8, a .) = 0 for j = 1, J Theorem 45.
in (66) with j < k we obtain
j
-
...,k
which concludes the proof of
Lecture 3
In this lecture we wish to discuss local regularity. example a bounded domain
.QCR 2
exists a unique solution u of u
c
C*(Z).
ax
. Given
= f
neighborhood of this p a r t of kQ.
2 0,
c
Cm(Z) it is c l e a r that there
with u = 0 on M , furthermore
This problem, however, does not have local regularity; suppose
that part of the boundary is given by,
p
f
Consider, f o r
y = 0 and let f(x, y) = p(x)a(y) in a
Suppose further that p(0) *- C: that
that p(x) > 0 f o r 0 < x < a and p(x) = 0 f o r a <x< b.
Then if a
is not differentiable it is c l e a r that the solution u will not be differentiable in a rectangle which has (a, b) even though f = 0 on this rectangle.
Consider the operator
where the X. a r e real C* vector fields defined in an open subset U of 3
lRn.
68.
In [5] Hdrmander proves the following. Theorem.
If the Lie algebra generated by Xo,
functions, when evaluated at P
, contains
5,...,%
over C*
all tangent vectors at P and if
this condition i s satisfied for each P c U then L is hypoelliptic. Here we will outline a proof of this theorem along the lines of [6] organized in such a way that it will introduce the methods described in F o r fixed P
(141.
c
U we want to show that, under the hypotheses of
the theorem, if f is C* in a neighborhood of P then every solution is of Lu = f is C* in a neighborhood of P.
For this it suffices to show
that there exists a neighborhood V of P and positive constants
E
> 0,
c > 0 such that
where
llu 11
denotes the usual Sobolev norm.
F i r s t we want to point out
that if L were an arbitrary second order operator the inequality (69) would not imply hypoellipticity, in fact, the wave operator (Lu = u satisfies this inequality with 70.
Lemma.
E
xx
-u ) YY
= 1.
If L is given by (67) and if (63) holds then L i s hypoelliptic.
The main point is the following inequality, which is easily proven by
integration by parts:
this holds for all u r
c;(v)
with some fixed c > 0, . where
VC
U.
Let P be a pseudo-differential operator of order s, the following gives a control of [L,P]. Lemma.
72.
If P is a pseudo-differential operator of order s then
there exists a constant c > 0 such that
for aU u r Proof:
c:(v).
F i r s t we show that
k (74)
j=l
for all u r
c:(v).
11 2 5 1 ( LPu, Pu) 1 t const. 11 PU 11 2 ,
IIxjPu
Choose 5
neighborhood of sup(5) with C)
from functions in
c:(v'
).
6
c ~ ( u with )
7' C U;
5 = 1 on V.
Let V' be a
then (71) is also valid (with a different
Thus we can apply (71) to 5Pu, we observe
that
for all u
6
c ~ ( v ) .Furthermore
the support of the total symbol of
since the support of u is disjoint from
15, PI we have, for every m.
Thus (74) follows.. Next observe that
[L,PI = 2
C [X PIX. t C [X., [X.,
j=1
j'
J
J
J
= Z T.X. t S = Z X.T.
+ S'
k
k
1
J J
j-1 where the T ., S and S' J
J
PI t [Xo, PI
-
[P, ]
J
a r e pseudo-differential operators of degree s.
Thus we s e e that
for all u
c
c:(v).
W e denote by
symbol ( I t 15 IZ)S/Z.
Then
k (79)
s A the pseudo-differential operator with
k
Z IIX~UII:= Z
1
1
k
1 1 ~ ~ ~2 .< ~Z 1 R1 X . A ~2 Ut c~o n~s t . J
-1
J
2
IIuIIs
2 < I (I..A"U, ~ " u 1) + const. 11. 11 S
c -
RLU 11,2 t I ([L,A'IU,
A'U)
I
IIu 11
t const.
:
it then follows that
11 x .u 1) 2 5 const. ( 11 LU il 2S+
k . Z
l
J
!j u 11 ):
for a l l u
e
c:(v)
.
and (73) follows by substituting this into (78). Arguing a s above we can replace u by course, requires increasing C appropriately).
A
kc
u in (69), (this, of
Then applying (73) we
obtain
and hence
f o r all u
6
c:(v)
.
Now suppose u Let 5' r
c;(v)
c
c&(u) and
such that
c'
5E
crn(v) we then replace 0
= 1 on the support of 5,
u by 5u.
then t u = 53' u
and we have replacing P by 5 in (77) and observing that then s u p p ( ~ ~supp(5) )z and s u p p ( ~ ) C supp(5). Thus, instead of (81) w e obtain: (83)
5u (ht1)c < const. (IISLullkE + llS1ullkr +
II"II)
by induction on k (and choosing an increasing sequence of 5' e
c~(v),
each of which equals one o r the support of the previous one) we obtain: (84)
11 5u 11 (ktl). -< const.
( I 1 5 ' ~ u l l ~+ , Ilk'uII).
To obtain hypoellipticity f r o m these estimates we proceed either by using a smoothing operator, a s in the f i r s t lecture (replacing P in (77) by the smoothing operator) o r by using elliptic regularization a s in t h e second lecture (deriving the estimates of L 6 = L t 6 4 with constants
independent of 5 ).
This proves Lemma 70.
To prove Theorem 78 we w i l l prove (69) under the hypotheses of the theorem.
F i r s t let Y. be defined by Y i . a d Y+ ~,...,i 1 P
.iP-11.
IXi *il.. P
-.
=
P
From the Jacobi identity and the hypothesis it follows
that there a r e a finite number of the Y's whichspan all vector fields bi a neighborhood of
T.
@
Now let
be the set of all pseudo-differential
operators of order zero such that if P
c
@,
then there exists
c
'
.
> 0 and
c > 0 such that:
i o r ali u
e
c:(v).
The set satisfies the following properties: ( A
@ is ideal (i. e.
or' order
if P r
@
0 then PQ s -1
(Dl X . A s J
P
for j =.O
and Q is a pseudo differential operator
and QP
6
p.
Further if
P a @ then P* e f f .
,...,m.
(C) If P 4 @ then [X., PI J
t
@
for j = 0,.
..,rn.
It then follows inductively from (C)that Y.
A
every 0 order pseudo-differential operator in
P
8
and hence by
'l***'~
theorem.
which would prove the
It remains to establish the above properties.
(A) i s immediate.
For j > 0 w e have
io (71) implies that x.A-'
2
6
P
for j > 0.
.
Now consider
-1 2 llxOA u < (XOu,Tu) -
11-
2
+ const. 11 u 11
< (Lu, Tu) -
L
C (X. u, T u ) t const. j=l J
where T is a z e r o - o r d e r pseudo-differential operator.
11 u 11 ,
Then
2 2 2 J ( x ~ u , T u2 ) ~const. (IIx.uII +IIu/l ) J
(88)
and hence we obtain (again using (71)) that X ~ A " 6 T o establish (c) suppose P c
(89)
@
@.
s o (85) holds, now consider:
26 11 [xj, fl u 11 2 = (XjPu, T ~ ' u ) - (PX.u. T u), J
when T~~ is a pseudo-differential o p e r a t o r of o r d e r 26.
It then follows
that f o r j > 0
s o that taking .26 < min(1,
E)
we obtain [X., PI 3
(89) with j = 0, then
F r o m (79) and (77) it follows that
c
@
f o r j > 0.
Consider
Combining this and (73) with (91) we obtain
Similarly we estimate l a s t t e r m of (89) by
Hence we conclude that
x@-' e @
which establishes the theorem.
For t h e most precise estimates of these operators we r e f e r t o 181, where a n approximate parametrix f o r L is constructed.
Lecture 4. We will now consider the more general situation of a f i r s t order rlundratic f o r m Q defined on
~8by
(43) and we assume that
pt.0perti.e~( a ) to ( c ) of the second lecture.
Suppose P c bC2 we say that
Q i s subelliptic a t P if t h e r e exists a neighborhood
--
cot~tcnts c
rl~crcl
U of P and positive
and C such that
flu consists
Theorem.
3h.
satisfies
of all p
t
d
with supp(p)C U flc
.
If Q satisfies the conditions of Theorem 6,
satisfies
properties ( a ) to ( e ) and if Q is subelliptic and non-characteristic a t P, then the following local regularity holds. I; fI
i?
If
cr
6
L (SZ) and cr is cm on 2
t h e n the unique solution p of Q(q,4) = (a,$) is also
crn on
U
n R.
c;(u)
In fact if 5, 5'
and if 5' = 1 on the support of 5 then
Furthermore, the solution cp of Q((P,+) = h(cp, +) a r e also
cooin
U fliZ
.
The proof of this proposition is derived by f i r s t obtaining a-priori
c;(u)
estimates f o r derivatives of cp.
Let (,G1
support of 5 then substituting.
Suppose (t, r ) boundary coordinates on
U.
the partial Fourier transform ;(r, r )
We define f o r u
e
c:(u*)
e
with 5' = 1 on the
by:
where t
n-1
r = Ztr
i i
and dt = dtl.
..dtn-l.
Let
/\ts
denote the tangential
pseudo-differential operator defined by:
Then if cp c
8,
we have
S ' A ~ ~e M d
. Replacing
rp in (95) by this we
have:
f o r all rp e
q.
The following is proved in [I] '(Lemma 3.1). 101.
Lemma.
If 5" c (c;(u)
and 5" = 1 on the support of &
then
The non-characteristicity of Q implies that
The theorem is then obtained by following the same scheme a s in the proof of Theorem 45. The following result gives an important class of Q that a r e subelliptic with s =
-21 '
103.
E
Theorem.
k
Q is elliptic on ( ~ ~ ( 5 2 ) )i. e. there exists C > 0 suc 0
that
for all q
6
(~:(52))
k
and if there exists a constant C1 > 0 such that:
where d s denotes the volume element on M2
.
Then there exists C" > 0
such that
This result is proved in [I], (Theorem 5) the idea of the proof is t o
--21
apply (104) to/\
(p,p0) where sup(p) C VnE and where
$
is defined by:
then
The result follows by using standard arguments in elliptic theory. The type of variational problem that we have been discussing a r i s e s f r o m the study of systems of equations a s follows.
Suppose A : ( ~ ~ ( 5 ) ) ~
w- k (C (a)) is a f i r s t o r d e r operator given by
where a? r cw(Z). Given la
CY t
w-
(C
(Q))
k
k (or i n (Lp(Q)) ) we wish t o solve
the equation: (110)
Au=a. Let B : (c*(E))~ ( c ~ ( E ) )given ~ by +
such that (U2)
6
Then, clearly, a necessary condition for the solvability of (110) is
*
*
Let 9 c Dom(A ) with A $ = 0 then
*
(Au, 9) = (u,A $1 = 0
(114)
.
Hence, another necessary condition for the solvability of (ll0) is that a is
*
orthogonal t o the null space of A 115. Theorem.
&
Let
*
(which we denote by ??(A )).
*
.
= Dom(A ) 0 Dom(B) Suppose that there exists
c > 0 such that
k
Then if cu c (L (a)) with Ba = 0 and a 2
1q(A*)
there exists a u
r
Dom(A
satisfying (110). Outline of proof:
fl
= {cp
EBIA*(P = 0,
Bcp = 0 )
.
Let the
c
Dorn(A))
I'
Laplacian"
L be defined by
with Dom(L) = {cp e
0 lBcp
Dom(A
*
I.( since clearly Z/C q(L) and if cp
*
and A cP c
Horn(L) then
. Then
q(L) =
Observe that L is self-adjoint and that (U6) together with (US) imply that the range of L is closed.
Therefore, the necessary and sufficient
condition for solving
is that a is orthogonal to 9
to q(A ),
x.
Now suppose Bcu = 0 and cu is orthogonal
then a i s orthogonal to
3/ and hence we can solve (119) so,
we have
applying B to this equation we obtain
taking inner products with B(P we have
so that (120) implies that:
*
hence u = A q is a solution of (110). Observe that this i s the unique solution which i s orthogonal to Let
-)Z (A).
B = Dom(A*) n Dom(B)
define the form Q by:
in the sense of L2, now on
B we
[lo]). zs
Suppose
...,zn
SZC Cn is a bounded domain with a smooth boundary. Let
denote the coordinates on C" and x = Re(z.), y. = Im(z.); J J J J
then, a s usual, we define t h e operators:
a = - (l ". J
- i-) a
a ax.J
ayj
a
i
a
a
and - - - - ( - + i - ) . J ax.J aYj.
Consider the system of partial differential equations given by:
where the a . satisfy the compatability conditions: J
Given a1,.
..,an
on Q satisfying (130) we wish t o find u satisfying (129).
More generally, in the standard notation of differential forms w e let
aP' denote all
4
and, a s usual we define
e,xpressed by:
:
aP' -dy
The problem then i s given a
r
aP''(12)
by:
satisfying:
to find
g' '-'(a)
4r
such that
Here we a r e mainly interested i n the regularity properties of
4.
F o r simplicity most of o u r discussion will concern the c a s e p = 0, q = 1. Suppose that
52C
C
2
and that the origin l i e s on bSZ.
Further,
suppose that t h e r e is a neighborhood U of the origin s o that U strictly convex and suppose further that U
{ (zl,z2) I Re(z2) < - 0)
n5
is contained in the s e t
Then the function z-I is holornorphic on U 2
and is smooth on U fl
-
{ (0, O)},
is
n SZ
but it cannot be extended a s holomorphic
function t o any neighborhood of (0,O).
Now suppose that on 52 we can
always solve the system (129) provided a satisfies (130) and suppose also that when a is in p
E
c*(u) 0
c*(E)
then there is a solution u
E
cCO(E). Let
such that p = 1 o r a neighborhood V of the origin.
then, since
zp= 0
t h e r e exists u
E
in V,
c*(E)
if the compliment of
with
we s e e that a
Zu = a
E
cm(E); so,
Set
by assumption,
(observe also that Za = 0).
Now,
has a bounded component, denoted by O then
Hartog7s theorem s t a t e s that every holomorphic function on Q can be extended t o a holomorphic function on 52 U O. In particular, if (0.0) then every holomorphic function in Q can be extended past (0,O).
r O
Let
then h is holomorphic on S2 but clearly h cannot be extended past (0,O) since u is smooth.on
5. Thus we s e e that (129) cannot be solved (with
smooth solutions) on a domain with "holes". If CZ is convex ( o r more generally psuego-convex, a s will be explained below) then the equations (129) always have a solution (see [ll]). The question that we will discuss h e r e is the regularity of solution.
Suppose
SZCen is convex and that in a neighborhood U1 of the origin we h:tve:
Choose a subneighborhood U of (0,O) such that p
a s above.
Again we define
ct
U1 and a function
by (135). Now suppose that there i s
some solution u of (129) which is regular where a is regular.
in particular u is regular in (U1-U) n
and in V
n5 .
Thcn,
Let h k t h e
holomorphic function defined by (136). F o r 6 positive, we evaluate h on the line z
2
=
-
6
s o that
F o r given positive numbers 6 0 and R we define the set S by:
We assume that U1 and U were chosen s o that regular in V fI
SC
U T - U.
Since u i s
and in (U1-U) n il! we know that, in particular,
n is
bounded independently of 6 o r S and the s e t ((0, - 6 ) 10 < 6 < 6 O). evaluating at (0, - 6 ) we s e e that it behaves like c i r c l e (z - 6 ) with 1' of 6
.
1 z1I
=
R with 0 < 6 < 6 O,
1
t h e r e and that on the
h is bounded independently
This is impossible since h(0, - 6 ) is the average of h(z
1'
I z1 1 = R.
NOW
-6)
with
Hence, in this example, we s e e that no solution of (129)
sing s u p p ( u ) c sing ~ u p p ( a ) . H e r e by sing supp stands f o r singular support and sing supp(u) is the subset of
5
such that if
(zl, z2) ) sing supp(u)
then t h e r e exists a neighborhood U of (zl+z2) such that u is C" li
n5 .
13!3.
or
The s a m e type of argument a s above proves the following result.
Proposition.
If
QC (En
is convex and if there is a germ of a
iioLornorpli c curve contained i n bS2 then t h e r e exists
-
?Q = 0 such that whenever 3 u =
ct
ct
we have sing supp(u)
r
q
d' with sing supp(ct).
Here, a s usual, a g e r m of a holomorphic curve means a g e r m of n complex analytic variety of dimension one.
-
In studying existence and regularity f o r a on domains with boundary
Given a point xo c b ZL we
a crucial role is played by the Levi form. define an (n-1)-dimensional subspace of
n
(E
by
T h e Levi f o r m is the quadratic form which operates on this subspace,
d f f i ~ l r dby:
is pseudo-convex a t x
We s a y that
0
if the Levi f o r m at xo is now
negative and Q is strictly pseudo-convex a t xo if the Levi form a t xo is T o define the Levi-form invariantly we define a sub-
pbsitive definite.
bundle of the complex tangent bundle to bn,
.
1 0 denoted by Ti
1 0 of Ti at xO consists of all the vector at xo of the f o r m such that (140) holds.
L =
3
a
L n 5 , where
, h e r e < , > denotes contraction. It follows f r o m a
alge&a identity, due t o Cartan, that if L and L ' a r e two vector fields
Lie
1, 0
with values in Tb
where [L,L?] = Given x
, then
LC' c
0
- L-T L .
bQ in a neighborhood U of x
0
we wish t o define
a special basis f o r the 1-form, in t e r m s of which the quadratic form Q, n that corresponds t o 5, is e a s i e r t o analyze. L e t w = gar, where g is the function chosen s o that I w
Let
w
of U.
LIJ
J
Another way of expressing this is by L ( r ) = 0. Then
t h e Levi form a c t s on l? O ' by sending L into
a
The fiber
1
...,w n
,
Then
...,Ln
-1 w
n
I=1
a t each point of U, i. e.
be a n orthonormal basis of the (1, 0)-forms a t each point
...,-n
,
w
is an orthonormal basis of (0,l)-forms.
be the dual b a s i s of w
1
..., n. Observe that:
,
w
Let
T h u s , on b R the vector fields L1,
similarly
I#
a,
...,Ln-l
have values in T1' O and b
Ln-l have values in T":.
We define the vector field T
on LJ by:
...,Ln-l'
It follows f r o m (144) that T ( r ) = 0 and hence the restrictions of L1,
5,..-,'n-1'
T f r o m a local basis of the tangent space of bS2.
then the vector field [L i'
If,
i, j < n
] is tangent to bS2 and s o on bS2 it can be
j
expressed by:
From (142) it then follows that the Levi f o r m in t e r m s of this basis is given by g-lc.
.. We will renormalize
r s o that g = 1 on b a then the
1J
L e v i form is e.. with respect to the basis L 1J
.
,Ln-l-
Relative to this basis we represent a (0,l)-form by cP =
zq.GJ, 3
then we have:
and
when the dots represent t e r m s which a r e linear combinations of the cp. ' s J (they do not involve any derivatives of the q.1. 1
Lecture 6. The following theorem is proven in [lZ] and [13]. 149.
Theorem.
If Q is pseudo-convex and if cr is a (0,l)-form with
components in cCQ(E)and if 20 = 0 then t h e r e exists a u
e
c*(E)
such
that Tu = a. This is a theorem of global regularity it is proven by using a of Q with appropriate "weight functions". local. regularity
Here we will concentrate on
by means of establishing subelliptic estimates.
From
proposition 139 we know that local regularity sequences additional conditions. We will now consider the form Q on ( 0 , l )-forms associated with
3.
We have:
f o r q,$
t
3
and
8=
(Dom 5) fl (Dom
z*).
Integration by p a r t s shows
that
when u, v r c ~ ( u E ) . Hence, on U
0
and
n E,
we have
154.
Proposition.
If xo
c
bQ
and if the Levi-form is non-negative at xo
then there exists a neighborhood U of xo and constant c > 0 such that
for all q
r
Here
flu= I 9 6 d 1 supp(9)C U 5)
and llqt
11 '=
The following is a consequence of theorem 103. 156.
Corollary.
If the Levi-form is strictly positive definite at xo
-.
1 then Q is subelliptic at x with s. = 0 2
The inequality (155) is derived from the following identity:
which i s valid for all q
c
aU without any assumptions on the Levi form.
We will give a brief outline of the proof of (157). From (148) we have:
Now by integrating twice by parts we have: n-l
Since L. for j < n and tangential and since q = 0 on bS2, 3 n t e r m s appear in the above. for
no boundary
Combining (158) and (159) with the expression
Z* in (152) we obtain the. desired identity (157). To investigate the subellipticity of Q we follow the method
described i n 1141. 160. Definition. at x. o -
If x
0
denoted by I to be the set of g e r m s of functions at x0 which satisfy f r I if and only if t h e r e exists a neighborhood U
the following condition. of xO and constants
162. Definition. x
0
bS2 we define the set of subelliptic multipliers
c
E
If xo
> 0, c > 0 such that
c
b52 we define the set of subelliptic functionals at
denoted by F to be the set of n-tuples of g e r m s of functions at x 0
which satisfy the following. a neighborhood U of x
164. Theorem.
0
v = (v 1'
...,vn) c F
and constants
E
if and only if there exists
> 0, c > 0 such that
Lf 52 is pseudo-convex a t x0 e b a then I and F have
the following properties: (a) I is an ideal. (b) I =
when
defined by h r
R 2/-r
denotes the r e a l radical of I and is if and only if there exists and integer
and f c I such that 1hlm < Ifl.
(c) If f
(d) Lf v
6
I then (Llf, L2f,.
(1).
,
...,v(") ...,ci
(f) (ci ,c , 1 i2
-
..,Lnf)
e
F.
(i) c I J
a F then det v.
, O ) s F for i = 1,.
n-1
. ..,n-1.
To prove that Q i s subelliptic at xo is equivalent to proving that 1 r I.
Property (a) follows.from
which i s immediate.
To prove (b) let h r
w,
then it suffices to show
that for some 6 we have:
since then the fact that bS2 is non-characteristic will imply that
11 hV 11
( C 'QW, V).
We have
when 626 is a tangential pseudo-differential operator of order 26.
k
if 2 S < E
andif
-
wehave
-
1 h l Z k < If1 then
Then,
which concludes the proof of (b). Property (c) follows from the observation that ll~*cpll < Q(q, q) and thus :
By non-characteristicity we have
Then (168)
//At6Z (Lif)qi 11 2 = (Z ( L ~ ~ WS2~6d, = - (
t const. llq
t const.
26
s
f L~OP, ~ rp)
-
Z(fcpi,
s26~iq)
12
llcp 11
2
1 So choosing 6 5 7 E proves (c). Property (dl is a consequence of the inequality
(169)
ck)
(det A ~ ~ ) L J < const. E A . z.Z. 13 1 J
.
whenever A.. is non-negative, in this case we have A.. = Z vi(kk(k) v. 1J 1 J k J ' Property (ei follows immediately by replacing cp by np in (257); in
fact, we obtain
Ilrqll12 c CQ((P, (PI.
To prove property (f) we first establish the following inequality:
f o r all q,,J, E
8u ; where
D is any tangential vector field on U (i.e. a t
vectorfield such that D r = 0 on b 4 and SO is a tangential pseudo-
t
differential operator of order 0 on U.
The inequality (169) is established
by expressing Dt a s a combination of L1,.
..,Ln-l, L1,. ..,Ln-l
and
T and reasoning a s in the proof of proposition 154. To prove property (f) we set
+.1 = kC c.I k(Pk a
and we prove that Q(+, +) < const. Q ( a (P).
let D = - where t l,. k atk
..,t2n-1 ,
-
We then
r i s a boundary coordinate system
and apply (170) to Dt = Dk and S ' =
hi^
then summing over k we
have
and since
This establishes property (f).
Lecture 7
W e will now use the properties of I given in the above theorem to
construct an increasing sequence of ideals contained in I.
If G is a set
of n-vectors we will denote by det G the set consisting of all determinants of n x n matrices whose rows a r e vectors in G. Let Fo consist of the
...,cin- 1' 0) for
vectors (c. , 1,
e ( r , det
;Oi,
(:r , det Fo}
.
...,n-1
..,0,1).
and (0,.
i=l,
Set
$=
where (r, det Fo) denotes the ideal generated by the set
Inductively, we let F - Fk-l U ((Llg.
•
..,Lng) I g c %)
and
it
. We then havi k c ktlC I. Now note that F k = Fo U { (Llg, ...,Lng) 1 g k}. We denote by % I ~ )the variety of I*, that =
E
is all x near x0 such that f(x) = 0 for all f
E
To get a geometrical
Ik.
picture of the process of the process of passing from
to Ikfl we define
for x r V(%) the Zariski tangent space of Ik at x denoted by Z: O($), 1 a s follows:
Further we define the null space of the Levi form
175.
)irX
If x is sufficiently near x then x 0 1 0 V(Ik) and ZZ; ($1 ll %x $ {O}
Proposition.
only if x 6
by:
t
V ( I ~ +if~ and )
.
This proposition is an immediate consequence of the definitions. Since x
.V($tl)
is equivalent to x
'V(Ik) and all determinants in Fk vanish at x; this means that there exists (5, 3,) $ 0 which satisfies E
c
....
whenever (al,
...,an)
E
Fk and this means that
L c..5.
i
1.l 1
j=l,
= 0
...,n-1
and
C (Lig)$
= 0
which is equivalent to Z $Li r Z : '(IQ fl
for
gc
xx.
If we now assume that r i s real analytic in a neighborhood of x 0 then we can apply the theory of ideals of real analytic functions to the situation,
We restrict the definitions of all the ideals introduced above to
ideals of germs of real analytic functions at xo.
Then we can apply a very
important result of Lojasiewic z (see [15]) which gives the "Nullstellen Satz" for r e a l analytic functions. real analytic functions at x
0
Namely, let J be an ideal of germ of
and let
1 (V(J)) be the ideal of germs of Xo
analytic functions at x that vanish on V(J), then we have 0
Another important theorem that is useful here i s the coherence theorem of Cartan. analytic variety at x
This theorem says that if V i s a germ of real0
and if
7X (V)
denotes the ideal of germs of real
analytic functions at x which vanish on V, hood U of xo and a set V
nU
of
IX
SC V
then there exists a neighbor
U such that S is open and dense in
and such that for each x r S there exists finitely many elements (V) that generate 0
Yx(v).
177. Definition.
VC bQ
If V is a germ of a r e a l analytic variety a t x
0
4
bS2 and
we define the holomorphic dimension of V by hol dim (V) =
(178)
. gig dim(Zx1,0 ox (v))nwx) .
0
179. Theorem.
If xo
4
bS2,
if Q is pseudo-convex, if r is analytic in
a neighborhood of x and if t h e r e exists a neighborhood of U of x 0 0 such that U
n bSt
does not contain any g e r m s of r e a l analytic varieties
of positive holomorphic dimension than Q is subelliptic a t x Let S be the s e t of all x r e a l analytic variety V with x
4
4
0'
bS2 such that t h e r e is a g e r m of a
vC
b S2 and hol dim(V) > 0.
We will
outline a proof of the fact that if r is r e a l analytic in a neighborhood of
x and if S2 is pseudo- convex then t h e r e exists a neighborhood U of x 0 0 such that
The theorem then follows from (180) since it shows that when S =
V(I) =
I#
and hence 1 4 I.
T O prove (180) we will show that if U
I#
then '~'(1)
k
-3
is not empty then it has an open dense subset which is not contained in V(Ikll).
Suppose x is a simple point of >qI ) and suppose that _/' (V(1 ) ) k x k
is generated by a finite number of elements in hood
WC V ( I ~ )s o that
that W
n s = 4.
k
then x h a s a neighbor-
each point in W has the s a m e properties and s o
Now suppose that W has an cpen subset W1 such that
wlC 7 / ( 1 ~ + ~then ) for
y
E
W1 we have
for any y 0
E
hence hol dim W' > 0 and s o
W
w'C
S which is .a
contradiction. The following theorem has recently been proved by Diedrich and Fornaess (see [16]). 181. Theorem.
If 52 is pseudo-convex, if xo c bn , if r is real analytic
in a neighborhood U of xo and if
VC U
U bS2 is a germ of a real
analytic variety then there exists a germ of complex analytic variety
WC U
U bS2 such that hol dim V = dim W. This theorem is very deep, in fact W is not in general a subset
1, 0(I(v))fI of V and the structure defined by Zx
X
is not integrable.
Combining Theorems 181 and 179 we obtain the following. 182. Theorem.
If $2 i s pseudo-convex,
xo
E
bS2,
r i s real analytic
in a neighborhood of x and if there exist a neighborhood U of x 0 0 such that U
n bS2
does not contain any germs of holomorphic curves
then Q i s subelliptic at xo.
In view of Proposition 139 we have 183. Corollary.
If S2 i s convex, xo
neighborhood of x
0
bS2,
and r is real analytic in a
then Q is subelliptic at xo if and only if there
exists a neighborhood U of xo such that U germs of holomorphic curves. Now consider the operator
7 :aP'
-
n bn
does not contain
&' . This leads to a
ap''C
quadratic form Q defined in a space (150). In t e r m s of the basis o
then
B P'
1
...,w n
,
is characterized by p e
&*
again by formula
if we express rp 6
if and only if
flP'
&'
q5..
'
.
by
=0
lgw-jq
on bQ when j
= n. The procedure for studying subellipticity of Q can 9 be generalized to this case. Given x bS2 we define the ideal Iq for 0
Q on (p, q)-forms the same way a s I was defined for (0, 1)-forms (it
is clear that 1, is independent of p).
Similarly we define the set F,
of subelliptic functianals to be vectors (gl,
for all
fit
...,gn)
such that
a s in previous definitions, where A,B and C range over
ordered p-tuples, q-tuples and (q-1)-tuples, respectively and
Then there a r e properties of Theorem 164 with exception of (d) hold a s rotated (i. e. replacing I with Iq and F with F ~ ) .The property
(d) is generalized by
If v (1), . . . , v (n-qtl)
(d)q det
(n-qtlf
' F
then det
(v (n-qtl) J
c I ~ ,when
(i)) denotes the set of determinants of all (n-qtl)C (n-q+l) (v. J
minors of the metrix (v.(i)). J Theorem 182 then generalizes TO the following statement. 186. Theorem.
SZ is pseudo-convex,
If
x 0
c
bQ, o r real analytic in a
neighborhood of xo and if there exists a neighborhood U of xo such that
U
n bSZ
does not contain any germs of complex analytic varieties of dimen-
sion greater o r equal to q then Q on
sPs is subelliptic at xo.
For the case q = n-1 the following criterior is easy to see. Suppose Q i s pseudo-convex, x r bQ and r analytic 0 n- 1 in a neighborhood of x Let A = C c.. be the trace of the Levi-form. 0' 1 l1 187. Proposition.
Then there exists a neighborhood U of x
0
such that no germs of (n-1)-
dimensional complex-analytic varieties lie in U exists germs of vector fields A 1' such that AlS
....Ak(A) $ 0.
...,Ak
n k&2
if and only if there
at x with values in T 0 b
+
In fact in this case one can also obtain necessary and sufficient conditions for subellipticity with r r
188.
Theorem.
A1..
..,Ak
where A
at x
0
.
fl i s pseudo-convex and if xo c bQ then Q on
If
is subelliptic at x
COO
0
,d
PI
n-1
if and only if there exist germs of vector fields
with values in : T O t:T
such that A1,.
..,Ak(A) $ 0
is trace of the Levi form.
The proof of this theorem i s along the lines of [17] for sufficiency and [18] for necessity.
Lecture 8 One of the important consequences of subelliptic estimates is that the Bergman projection operator is pseudo-local.
Let
34
denote the
space of holomorphic functions on Q which a r e square integrable and
B : L2(Q)
-
denote orthogonal projection.
Now consider the operator
on (0, 1)-forms-
with Dom(D) = {rp 413q e ~ o r n ( ~ *end ) ~ * r pt Dorn(T))
. This operator
f o r general systems was discussed in Lecture 5. We denote the null space of
by
3''
and a s in Lecture 5 we
observe that
If we denote by H :' : L
l(S2)
integrable (0, 1)-forms onto 191.
Proposition.
xO' the orthogonal projection of squarexO'' then we have the following result.
+
0
If the range of
bounded operator N ::L
'(Q) -Lo*'(a) 2
is closed then t h e r e exists a unique such that
(192) and (193) Furthermore, if f
e
LZ(Q) and f
E
Dorn(5) then
The operator N was constructed in Lecture 5. a =
Applying (192) t o
Tf we have f r o m the dimension in Lecture 6 that
and s o
B B =~ 0 , thus Bf
c
H.
F u r t h e r if f r )f then Bf = f and f
1)4
then for h r
we have
s o that Bf = 0.
This shows that (194) defines the Bergman projections
operator. F r o m (194) it follows that if N ,is pseudo-local then B is pseudolocal. Now we take up operators induced on the boundary. the forms of type (p, q) whose components a r e in C*
tP''C aP'
aPadenote
(z).We define
by:
')c pP'
It then follows that B ( F ~ *
where
Let
fi ''
is well defined.
is the quotient
s o that we have:
aP8 q/Tp'
In c a s e p = q = 0,
and it is easy t o check that
O =
cm(bn) and in a
b
neighbarhood
..., n
U with a special basis o1,
o
a s before we can express cp P
boa
as
n-1
q = Z cp.~', l
J
with q. " ~ ~ ( ~ n b S 2 ) J
and
a
(200)
At each point in MI
u =
n-1
z E.(u)GJ l
with u
t
gPs
.
we have an inner product on the (p, q)-forms evaluated
at that point, this induces an inner cp,$
cco(unbn)
J
we have cp
$
P
product on
6 ' '
s o that if
cm(bS2) and we define
where dS is the volume element in bQ. On
6
consider the quadratic
form
The following theorem then holds (see 1191 and [ZO]). 202.
Theorem.
The estimate
holds if and only if the Levi f o r m has a t each point of bQ a t least max(qt1, n-q) non- zero eigen-values of the s a m e sign. Let
q b C L2(bR)
be t h e null space of
-
ab
(i. e. the L2-closure of
-
ab).
T o each element of
b
this corresponds a unique holomorphic function
on 52 whose ?'boundary value" it is. We define the Szego operator S.
L (bn) 2
-
b
t o be the orthogonal projection.
Now if (202) holds then the
operator
h a s a closed range and hence t h e r e exists a pseudo-local operator Nb : L :
'(bn) --L,;'(bS2)
have for f
E
defined analogously with (192). (193). Then we
L2(bQ) and f e ~ o r n ( 5) b
Let P : c W ( b n )-.c r n ( b n ) be a differential operator with the property that
gbis contained i n t h e
of P is orthogonal t o
xb
null space of P
*.
Then, clearly, the range
and hence
We wish t o study the local solvability of the equation
that is if f is a function defined i n a neighborhood U of x 0
UC bQ)
E
bSZ (we take
we wish t o find u defined on a neighborhood U1 of xo s o that
(206) is satisfied on U1. 208.
Proposition.
loc If f c LZ (U) and if
--1
a weak solution u r L
of (206) then Sf is Tfsmooth"on U", f o r any "neighborhood
loc 2 (U1)
U" such that
-
u"C ul. Here by "smooth" on U" we mean that it belongs t o the class of
functions on U" which a r e restrictions of Sg t o U" with g r L2(bn) and g = 0 on a neighborhood of "smooth" means at least
? t L2(bn)
on U'
suchthat
cm(u1). .
6
cW(uv) with 0
7=f
smoothness of
6"
SF on
5 = 1 on a neighborhood of E".
on a neighborhood of
and v = 0 outside of U'.
on a neighborhood of
Thus, since S is pseudo local,
in
To prove 208 take 5 Let
U".
CY
Consider g = f
hence. Sg =
ST
Gf1.
- Pv,
is "smooth".
Let v = 5u
clearly g = 0
Observe that the
U" is independent of the values of
T
outside of U".
Recently F. Treves proved the 'following result (see [Zl]). 209.
Theorem (Treves).
If
satisfies the hypothesis of Theorem 201,
if the Levi-form is non-degenerate at every point of b n and if further the function r is r e a l analytic then the operator
nb is analytic hypoelliptic.
This then implies that the operator N is pseudo-local in the b N q is analytic on any open set on which 9 is analytic. .b
analytic sense, i. e:
In particular, if the conditions a r e satisfied for q = 1 (i. e. at least max(2, n-1) eigenvalues of the same sign and non-degeneracy of Levi form) thus the operator S is analytic pseudo local. Now take f
E
34b
to be the boundary value of a holomorphic function
which is not r e a l analytic in a neighborhood of xo then the equation (207) does not have a solution in any neighborhood of x 0' The following a r e two examples of P that satisfy (206).
First l e t
P be the differential operator whose adjoints can be locally expressed by:
As a second example we take P t o be
In this example we have the following result. 212.
Proposition.
If N
b
on (0,l) f o r m s is analytic pseudo-local,
if P is given by (211) and if f is a function defined in a neighborhood of
x
0
r
b R then the condition Sf is analytic in some neighborhood of x
0
i s necessary and sufficient for the solvability of (207). Proof:
The necessity h a s been proven above.
f i r s t solving the gloval problems. in a neighborhood of x
0
That is, take
The sufficiency follows by
1. L2(b12)
s o that
=
f
we wish t o find v satisfying
it is easy t o check that if v is given by
t h e n v satisfies (213).
Since Sf is analytic then, by use of the theorem of Cauchy-Kowalevski, tt~er-eexists an analytic function w defined in a neighborhood of x
that Pw = Sf.
0
such
Then the solution of (207) is u = v t w.
The f i r s t example of the phenomenon of non-solvability is the famous Lewy equation ( s e e [22]). This equation is given by the operator
P :c*(lR3) +Cm(IFi3) defined by:
where z = x
+ iy
3
and x, y, t then coordinates on l R
corresponds t o the boundary of the domain (216)
QC C
2
.
This operator
given by:
2 2 Q = { ( z 1 , z 2 ) e C 1 1 m z 2 > lzll 1
then 2 2 b~ = {(zlJ z2) e C 11m z2 = *Izll 1
a
t h e r e is only one operator of the form a - + a z1 which is tangent t o b Q , this operator is given by:
Up t o multiples on b n b-
a
az2
.
Consider the correspondence between Et3 and bQ given by ( z z ) c b a 1' 2 (218)
z1 = z and z 2 = t
Under t h i s correspondence P = L.
* -c
given by dx dy dt then L =
+ ilzl 2
If we let volume element on bQ be
and hence condition (206) is satisfied in
this c a s e we can easily check that that the operator given in (211) is
-LE.
Hence if we can show that Nb for this domain is analytic pseudo-local then we would obtain that the necessary and sufficient condition f o r the local solvability of Lu = f is the analyticity of Sf.
In this c a s e however,
the general theorems do not apply since the Levi form is 1 X 1 and thus cannot have two non- vanishing eigenvalues. In this case, however, the boundary b a is a group and the operators L, S and Nb a r e invariant under the action of this group s o that they can
be computed explicitly ( s e e [23]). If (z, t). (z ' ,t ' )
(z, t)
(219)
IR3 the group operation is given by: (z', t') = (ztz', t + t V t 2Im(z*:'))
Multiplication by an element of this group can be extended t o a holomorphic automorphism of
where
*
onto
5
.
The operator S is then given by
denotes convolution under the group.
-*
&
The operator a N 8 b b b
( s e e (214)) is then given by:
-* a ~ 2-a
b b b
1 f = f s 2T2( z '-it)
I I
1zl2-it log (
IZI
tit
.
The analysis given in [23] and [24] then prove the desired result. Finally vve r e m a r k that the problem of non-solvability f o r equations with simple characteristics has been completely settled by Xirenberg and T r e v e s ( s e e [25]). In that c a s e non-solvability can only occur i f the symbol of the operator is complex.
In general however, non-solvability
can occur also for operators with r e a l symbol, take f o r example the operator ,.
--2
1 I-' P where P is given by (215) and
a P =a;
a
iz -
at
References [l] Kohn, J. J. and Nirenburg, L. : "Noncoercive boundary value p r o b l e m s ,
"
CPAM, vol. XVIII No. 3, p. 443-492(1965). [Z] F i c h e r a , A., ordine,
"
"Sulle equazioni l i n e a r i ellittico-paraboliche del second0 Acc. Naz. L i n c e i Mem. Ser. 8, Vol. 5, p. 1-30(195 ).
[3] Kohn, J. J. and Nirenberg, L. : "Degenerate E l l i p t i c - P a r a b o l i c Equations of Second O r d e r ,
It
CPAM, Vol. XX, 797- 872(1967).
[4] Sweeney, W. J., "The D-Neumann problem,
" Acta
Math. 120, 223- 277
(1968). [5] Hllrmander, L. : "Hypoelliptic second o r d e r differential equations,
"
Acta Math. 119, 147-171(1967). [6] Kohn, J. J. : "Pseudo- differential o p e r a t o r s and hypoellipticity,
If
P r o c . Conf. P a r t i a l Diff. Eq., pp. 61-99, AMS Pprvidence, R.I. (1971). [7] Radkevitch, E. V. : "Hypoelliptic o p e r a t o r s with multiple c h a r a c t e r i s t i c s . Mat. Sb. 79(121), 193-216(1969). [7] Kohn, J. J. : "Pseudo-differential o p e r a t o r s and non-elliptic p r o b l e m s ,
"
CIME Conf. S t r e s a 1968 Ed. C r e m o n e z e Rome, 157-165(1969). [8] Stein, E. and Rothchild, L., "Hypoelliptic differential o p e r a t o r s and nilpotent groups,
" Acta
Math. 137, 247-320(1976).
[9] Folland, G. B. and Kohn, J. J. : "The Neumann p r o b l e m f o r t h e CauchyRiemann complex,
"
Ann. Math. Studies
#
75(1972).
[lo] Kohn, J. J. : "Propagation of s i n g u l a r i t i e s f o r t h e Cauchy-Riemann equations, " P r o c . C. I. M. E. Conf. on Complex A n a l y s i s of J u n e 1973, 179- 280.
"
Ill] HBrmander, L., "L 2-Estimates and existence theorems for the tor,
" Acta
Math., vol. 113, 89-151(1965).
[12] Kohn, J. J., "Global regularity for folds,
3 -opera-
" Trans.
3
on weakly pseudo-convex mani-
AMS, vol. 181, 273-291(1973).
[13] Kohn, J. J., "Methods of partial differential equations in complex analysis,
of Symp. in P u r e Math. vol. XXX p a r t 1; AMS,
" Proc.
213- 237(1977). [14] Kohn, J. J., "Sufficient conditions for subellipticity on weakly pseudoconvex domains,
" Proc.
N. A. S. vol. 74 no. 7, 2214-2216(1977).
I151 Lojasieuricz, S., "Sur l e probleme de l a division,
" Studia
Math. 87-
137(1959). [16] Diedrich, K. and Fornaess, J. E., "Complex manifolds in real-analytic pseudo-convex hypersurfaces,
"
Proc. N. A. S. (to appear).
1171 Kohn, J. J., "Boundary behaviour of
manifolds of dimension two,
" J.
3
on weakly pseudo-convex
Diff. Geom. 6, 523-542(1972).
[IS] Greiner, P. , "On subelliptic estimates of the
c2," J.
7-~ e u m a n nproblem in
Diff. Geom. 9, 239- 250(1974).
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Ann. of Math., vol. 81
NO. 3, 451-472(1965). [ 201 Kohn, J. J. , "Boundaries of complex manifolds, " Proc. Conf. in
Complex manifolds, Minneapolis 1964, 81- 94. [21] Trevs, F. J., lectures in this volume. [ 2 2 ] Lewy, H.
, "An example of a smooth linear partial differential equation
without solution,
l1
Ann. Math. 66, 155-158(1957).
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" Proc.
N. A. S.
72, NO. 9, 3287-3289(1975).
1241 Folland, G . B. and Stein, E., "Estimates for the analysis on the Heisenberg group,
" CPAM
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Conditions necessaires e t suffisantes pour l ' e d s t e n c e e t l f a n i c i t k des solutions du Foblbe de l a d6riv6e oblique
Ka&
TAIRA
Department of Mathematics, Tokyo I n s t i t u t e of Technology, Japan Departemat de Mathernatiques, Univerais de Paris-W, France
-
IntrodEuction.
Dans c e t t e Note on considere l e problbme de l a dkrive'e
oblique pour l e laplacien donnant l i e u au p r i n c i ~ e
maximum e t on donne des
con6itiorrs n6cessaires e t suffisantes pour l f e x i s t e n c e e t l ' u n i c i t 4 des solut i o n s avec perte d
' derivke ~ en cornparaison du cas'coercif dans le cadre des
espaces de Sobolev (TGorkme 2). de (8) (cf
1.
.
Dans l a demonstration on u t i l i s e des r6sultate
(7) ) e t l e theorbme 5.9 de H"0rmander (4).
Formulation
euclidien R ~ ,
-
a
& problbme.
-
Soit
un domaine b r n k d'un espace
&taut m e varidt.6 compacte
bord
? i
r
de classe C m
, de
dimension n. On consi&&rel e problbme de l a derivee oblique d v a n t : Pour deux fonctiona f
at
9
d6finies dans
e t sur
r
respectivement, trouver m e
Ici
e s t l e laplaciert sur
r, I!.
r k e l l e s e t C w sur
e s t M champ de vecteurs sur
3,
a
et
b
sont des fonctions
e s t l a n o r m d e u n i t a i r e extgrienre
r.
B B
vdeurs
r, e t
O(
On e'tudie l e moblkme de llexistence e t l ~ n n i c i t 6des solutions &e (+) dans
l e cadre 3es espaces de Sobolev.
Pour 6 t a b l i r des th6orhmes dlunici% a e s solu-
tions, on u t i l i s e l e princive du maximum p o s i t i f au bord cornme dans (8) (cf.
(1 ))
. Ch
a3
suppose donc que l a c o n a t i o n aux lfmites
s a t i s f a i t an principe
du m a x i m p o s i t i f an bord :
D 'aprhs l e thhorbme X
de Bony-Courrkge-Priouret (1 )
, on v o i t
que l a propri6t-6
(PMB) esl: s a t i a f a i t e si e t seulement si
On obtient a l o r s l a
% a alors t
tieinaraue 1.
-
Dms l e cas oii a(x)
>
0 BUT
rest* v d a b l e en remplaqant llhypothhse ( ~ . f) par :
r , l a proposition I
2.
-
R6duction au b o d .
e x i s t e m e solution unique
v E Ht(Q )
7 E H*t'-
(
r)
avec
t E R , il
du prohlbme
sans S Z ,
A V =o
I
=
=
Gn 86fi n i t l e n o n n de Poisson De mdme, pour toute nsiqae
Pow toute
v E Iis(n )
r* 1:H (Q)
f E He2
~( avec
r/ ) -~+ Ht (n) par
v =
3 7.
s 2 2, il e x i s t e une solution
du prohlhme
-
A v = f
dans
On d 6 f i n i t l e nomu e
a, r.
Green
b a alors l a
- -Pour deux fonctions
P r o ~ o s i t i o n2.
f E H'-~(Q)
avec s L> 2, il e x i s t e une solution u E nt(Q ) ( t et sedement s i il e x i s t e une solution Cp E Ht-lI2( -
&
€
r
H ~ - ~ / ~) (
& problbme (+) pj ) & lldouation s)
On rambne donc lle'tu6e du problhme (+) I c e l l e de l ~ d q u a t i o n(+t) 8ur
Rappelon8 que 110p6rateur T f 787 :
e s t un opkrateur pseudo-diffdrentiel du premier ordre sur
3. entiel
i t u d e de T. T
, on
Lemme 1
-
(3)).
Pour c a l c u l e r l e symbole de 110p6rateur pseudo-diffdr-
a besoin du
.-
r (cf.
r.
Posons r
Ce lemme se d6montre en u t i l i s a n t l a fornmle de Green.
Soit
(x,3 ) = (x.,,%,
l o c d e s du f i b &cotangent
. . . ,x*,
~ * rs t
,S l , X 2,
. . ,5
(gij (x)),
n-1
=-,
) des coordom6es l a mdtrique rie-
mannieme de f' induite par l a d t r i q u e naturelle de R ~ . On dt5signe par cu x (
,
rC
Rn
la detui8me forme fondamentale en un point x de lfhypersurface
)
e t par M(r) Is ccmrbure moyenne en un point
x de
r
respective-
mend;,.
m
u t i l i s a n t un r d s n l t a t de Fujivara-Uchiyama (2) e t l e l a m e 1
Lemme 2.
+
--
Le -bole
& l'op6ratepr
, on peut
T =a
pseudo4iff6rentiel
t
o( t b e s t donne var t A
n
2
e s t l a matrice inverse
ds
(gij'(x))
D'aprhs l e lemme 2, on v o i t que l8op8rateur T
ment si la fonction
e s t e l l i ~ t i a n esi e t seule-
a ne afannule en aucun point de
r , i.e.,
% u t i l i s a t une varfante dfune m6thode de A@uo~-lirenbergc-3
on obtient l a
si e t seulement
(81,
(A.0)
>
a(x)
s
a o r s , pour t o u t
r,
sur
0
2
312, l f o d r a t e m T : H*'/~(
+ H *3/2 ( \-
)
)
e s t d1incTice &, Bromg2.q-t
que { T p s ~ s - ' / 2 ( r )
:
TT
= O ) C C ~ ( ~ o)n o, b t i e n t
dlapr'es l a remarque 1 e t l a proposition 2 l e
.- -
Corollaire 1 (H.I)
On s u m s e
a(x)z
S o r s , pour t o u t e t topoI.oei~ue -
&
O s
2
HS-'/~(
>
(A-0)
a(x)
(B-0)'
b(x) Z#E 0
b(x)_L O 3/2, l'o$rateur
T e s t un isomorphisme alaebriaue
r) sar
r)
He3l2(
& sedement
r. r.
snr
0
r
Pour Qtu&ier d e s conditions necessaires e t s u f f i s a n t e s pour qua, pour t o u t
s
2 3/2,
1lop8rateur
T : H ~ - ~( /I-'~ )
alg6brique avec perte dl=
ot
a(x)
T
: R ~ - ~( / ~)
a)
Tq
E ~
>
0
r
sur
+ Hs'3/2
(
r)
s o i t un iaomorphisa
d6riv6e en comparaison du c a s coercif, i.e.,
du c a s
r,introdulsons un op6rateur l i n e a i r e non born6 e t fermt:
-+ H8-3/2
(
r)
Le domaine de d 6 f i n i t i o n de
de l a manikre suivanh r
T
est
8 (T )
)}
{
H "3b(
r) z
~ - ~ / ~ ( r
b ) T . p = ~ . p
pourtoute
~Q('c).
En raisonnant comme dans l a d h o n s t r a t i o n du c o r o l l a i r e 4.4 de (6), on ddduit
du lemme 2 e t du th6ori.me 3.1 de Melin ( 5 ) l a
Ici d i v 0(
e s t l a divergence de 0( n-1
e s t l a crochet de Poisson de f
et
par rapport B
(gij(x) )
et
gg,
D'apAs l e lemme de SoboLev, la proposifion 1 e t l a proposiuim 2, on a l e
De plus, en u t i l i s a n t l e thdorhme 5.9 &e (4)) om peut d b o n t r e r l e
Carollaire 3.
-On guvpose r
(H.1)
2
(R.2)
a(x3
0
&
b(x)
r-
0
Il. existe m e constante Co
0 telle
Ik=lG d k ( ~ $ k ~ s ~ o a (ST*^. ~ ~ f ~
m,pour t o u t
s
2
3/2, ;L'o~drateur
est un isomorvhisme ala6brioue -(B.I)
4. 2 avec
~(XI
>
o
gyy
Conititiom ngcessaires
: H *3 2'
(
r1
& seulement &
-
r
H ~ - ~( / ~ 1
r o = [ x E ar( 4r= 0 ] . & snffisantes.
-
uiiilisant l a proposition
t = s e t l e corollaire 1 , on obtient .d*abordle
(H.1)
a(x)
Plors, tontes f € (A.0)
2 o &
2
O
sur
r
p r o b l h e (+) admet une solution m i m e as-*(Q)
& (B.0)'
g
9E H ~ - ~ / r~ ()
H s ( G ) .pour
u
si e t seulement si l e s h m o t h h
gg& s a t i s f a i t e s t
~
4 a(x)
>
o
(B-0)
b(x)
$
0
(
b(x)
sur
r.
r.
De &me, on obtient d'aprks l a proposition 2 avec
t
a1
e t le corollaire
2 le
~hkorhme1,
-Soit s 2
@.I)
a(z)
2 -
o &
(8.1)
b(x)
>
0
cCls 2b(x)
En
-
pour tout
sir did +
* 1 E TxP
C.2)
+ 4. & s u u ~ o s e:
b(x) 1_ 0
ror xEro,
( X E r s
(s-3/21
a ( ~ ) ~ o f -
\ I T I ~ >~ 0~ ~ ~ ~ )
OBl?l II*
Alors, & problBme (t) admet une solution aniaue u HS" (Q) h u t e s r E H ~ - ~ ( Q ) , + 9 ~ a ~ - ~r ' 1.~ ( Remarque 2. - S i llhypoth'ese (C ?, ) &ans l e th6oSme 1 de (8) g
De plus, d'apr8s l e corollaire 3, on obtient l e
(Ii.2)
51; exis& une constante Co 7 0 t e l l e
aae
J.M.
Bony, P. Courrk~gee t P. h i o u r s t , , Semi-groupes de Feller sur une
vari6te 8 bor& compacte e t problbmes aux l i m i t e s int6gro-diff e'rentiels
du second ordre dormant l i e u au principe &urncdmau (Ann, Bst. Fourier, 901.
18, 1968, p. 369-521).
D. Fnjiuara and K. Uchiyama, On some dissipative boundary value problems
f o r the Laplacian (J. Math, Soc. Japan, vol. 27, 15'1, p. 625-635). L. HZrmander, Pseudo-differential operators and non-elliptic born-
problems (Ann. of Math., L
vol. 83, 1966, p, 129-209).
. Hi6mander, A c l a s s of hypoelliptic pseudodifferential w h l e characteristics (Math. Arm,,
operators with
vol. 217, 1V5, p, 165-188).
A. Melin, Lower bounds f o r pseudo-differentid operators (Ark. f%r Mat., VO~.
p. 117-140).
9,
K. T e a , GII some non-coercive boundary value problems f o r the Laplacian
. .
(J F ~ CSei. Mv.Tokyo, Set. IA, vole 23 p. 343-367)
.
K. Taira, Snr l e p r o b l h e de l a d6riv6e oblique (C. R. Acad, SC. Paris,
t. 284, se'rie A, 1977, p. 1511-1513). K. T&a,
Sur l e problene de l a d6riv6e oblique
(a
~ardtre).
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
(c.I.M.E.)
BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS
F. TREVES
R u t g e r s U n i v e r s i t y , U . S . A.
C o r s o tenuto a B r e s s a n o n e d a l 16 a 1 24 giugno 1977
BOL%DARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS
F!canpois Treves Rutgers University, L;. S. A.
Introduction This s e r i e s of l e c t u r e s p e s e n t s a systematic treatment of boundary value problems f o r e l l i p t i c equations
- without
priori distinguishing
between t h e coercive problems ( a l s o c a l l e d of t h e Lopatinski-Shapiro type) and t h e noncoercive ones. The property we a r e concerned with here is t h a t of r e g u l a r i t y
U J
t o t h e boundary. Icow new r e s u l t s w i l l be pro-
vedr what is shown is t h z t known r e s u l t s can be "integrated" i n a unif i e d approach. N o r is t h e approzch very novel: it i s e s s e n t i a l l y t h a t of t r a n s f e r i n g the problem t o t h e boundary, where it t u r n s i n t o an int e r i o r problem. I n one version or another it has been much used i n t h e p a s t , perhaps first by Calderon, s h o r t l y afterwards by Seeley, Htirnan-
der, Vishik, e t c . If t h e r e could be any claim of novelty i n t h e present treatment (aqd the author i s not even s u r e of t h a t ) it would l l e i n the manner i n which t h e t r a n s f e r t o t h e boundary is effected: I n the neighborhood of t h e boundary t h e e l l i p t i c equation (or system) Ew = f i s decomposed i n t o a couple of equations
a5
(1) where
+ A (t)
- A-(t)u = g,
(2)
dv
- A+( t ) v
P
h
,
{resp., A - ( t ) ) i s a pseudodifferential operator on t h e boun-
dary, depending smoothly on t h e v a r i a b l e t t r a n s v e r s a l t o t h e boundary, e l l i p t i c negative (resp., e l l i p t i c p o s i t i v e ) of order
one ( i n
t h e case
we discuss; it could be of d i f f e r e n t order i n other cases). A s f o r g
azzd h they m e d i s t r i b u t i o n s on the boundary, depending on
f u n c t i o n a l of v, usually g
a
t; G is a
Jv, where J i s a simple operator. The bo-
undary conditions bearing on w t r a n s l a t e i n t o a s i n g l e boundary condlt i o n beaxing on the s o l u t i o n of (1):
( 3)
&
u!o)
= U*
,
with uo a d i s t r i b u t i o n on the boundary, which may a l s o depend on t h e s o l u t i o n v of (2) (an&, of course, on t h e d a t a ) . Then one solves EQ.
( 2 ) "backward"
, that
is, from a c e r t a i n value T of t which defines
a hypersurface p a r a l l e l t o t h e boundary i n s i d e t h e domain, t o t = 0 (which d e f i n e s t h e boundaxy), and a f t e r t h i s , one puts t h e value of
v i n (1)and (3), and s o l v e s this Cauchy problem forward, with i n i t i a l datum ~ ( 0 ) . The main advantage of such an approach is t h a t we g e t a l l we want without introducing any t o o l more s o p h i s t i c a t e d than standard pseudod i f f e r e n t i a l operators on the boundary, those c a l l e d of type (1,0) by Hbrmander (see
[6
). Indeed not only is
&
of t h a t type ( i n most ap-
p l i c a t i o n s it w i l l be even b e t t e r s it xi11 be a c l a s s i c a l pseudodiffer e n t i d l operator, t h a t is, an asymptotic s e r i e s of operators with homogeneous symbols) but s o w i l l be t h e
parametrices of t h e forward
Cauchy problem f a r (1) and of the backwaxd Cauchy problem f o r (2). I n p a r t i c u l a r they decrease t h e wave-front s e t s (of g, h, u(0)). Thus ever y t h i n g is reduced t o deducing t h e r e g u l a r i t y of t h e i n i t i a l value u ( 0 ) from t h a t of t h e datum uo i n (3). This reduction is established i n Chapters 11, 111. I n Chapter I V we show t h a t t h e problems of LopatinskiSha2iro type
are characterized by t h e property t h a t
is e l l i p t i c ( t h i s hsd a l r e a -
1 3 J ). Thus t h e theorems of rcgJ193) reduce t o t h e a s s e r t i o n t h a t
djj been pointed out bj Calderon; s e e
l a z i t y up t o t.he boundary ([I],
pseudodifferentidl operators a r e pseudolocal, and a c t i n a c e r t a l n manner on the Sobolev spaces (by usicg t h e p r c p s r t i e s of t h e param .-.t r i c e s of ( I ) & (2) we d e r i v e t h e coercive estimates). Although t h i s xi11 not be discussed here the same observation a p p l i e s t o a n a l y t i c r e g u l a r i t y : t h ~~ a x a n e t r i c e sof (1) & (2) s l s o decrease the a n a l y t i c wave-front
+
s e t s (keep i n mind t h a t A-(t) have order one: j.
I n Ch. V we describe t h e 7-deumann problem a d determine, t o a cert a i n extent, t h e corresponding boundary operator
&
r we d e t e r n i n e
i t s p r i n c i p a l symbol, and a l s o its subprincipal symbol at the p o i n t s of its c h a r a c t e r i s t i c s e t (which i s a smooth conic manifold and on which t h e p r i n c i p a l symbol vanishes e x a c t l y of order two). Availing ourselves of t h i s information we show how the known necessmy and suff i c i e n t conditions f o r t h e v a l i d i t y of t h e s u b e l l i p t i c %-estimate a r e
a p r t i c u l a x case of r e c e n t r e s u l t s on c e r t a i n c l a s s e s of pseudodiffer e n t i a l operators with double c h a r a c t e r i s t i c s (see
23 , [73 ). I n Ch. V I we d i s c u s s r a p i d l y the boundaxy value problems we c a l l of
principal
m-8 & is then a pseudodifferential
operator of principal.
type ( ~ e f .~1.1). Sere a s i n t h e preceding chapters we do not prove any "theorem" : we s t a t e two r e s u l t s , one of t h e author e s s e n t i a l l y charact e r i z i n g the h y p o - e l l i p t i c i t y of d i f f e r e n t i a l operators of p r i n c i p a l ty-pe, t h e other one of Yu. Egorov, c h a r a c t e r i z i n g the subellip&
pseu-
d o d i f f e r e n t i a l operators. Ch. I gives a quick overview, without proofs, of pseudodifferential operator theory. One of t h e motivations f o r such a sm7ey i s t o "fix" tine terminology and l a t e r avoid confusion. It is important t h a t we k n ~ w what we axe dealing with, he it t r u e operators o r c l a s s e s of such modulo r e g u l a r i z i n g ones. It should be admitted, a t t h i s point, t h a t t h e deconp o s i t i o n (1)-(2)-(3)
of t h e o r i g i n a l boundary value problem is only va-
l i d modulo regulaxizing operators, and s o is the e n t i r e armment i n thes e l e c t u r e s . But because of our s p e c i a l point of view look at r e g u l a x i t y theorems
- t h i s does not
- because
we only
endanger our conclusions.
CONTEXTS Introduction Chapter I. Backsound on pseudodifferential operators. Wave-fYont sets Chapter 11. The generalized heat equation and its parm.etrix Chapter 111. Application t o boundaxy value problems f o r e l l i p t i c equations Chapter I V . Chapter V.
Coercive boundary value problems
K The b-lieummn problem i n subdomains of C
Chapter V I . Boundary value problems of p r i n c i p a l type Bibliographical references
I. SACKGIICUND ON PSEUDODIFFERENTIAL OPERATORS. WAVE-FRONT SETS
Ye s h a l l systematically use standard notations I f xl,
..., xn a r e
coordinates i n Euclidean space R ~ o, r l o c a l coordinates i n a c h a r t of some (always C
00
, 1. =., smooth)
manifold X, by
2;
.
convex topological v e c t o r space, by :C compactly supported C
CO
we mean the mono-
. ~f E is a l o c a l l y
a m i a ( d / d ~ 1 ) ~ 1 . .( d / $ ~ ~ ),~by n Dx we mean i-la'
(x;E) we denote the space of
functions i n X with values i n E, by
the s,pace of E-valued.distributions i n X, by
cs
a *(x;E)
( X ~ E )the subspace of
& * < X ; E ) c o n s i s t i n g of t h e compactly supported d i s t r i b u t i o n s . A l l thes e spaces, and all o t h e r standard spaces we might use (such as t h e space of
cC0 mappings
X
+ E)
ern
(x;E),
are endowed with t h e i r standaxd l o -
c a l l y convex topologies. Whenever E = @, the complex plane, we s h a l l omit mentioning it and w r i t e , 2. g., :C of
8
(x) instead of
:C
(x;c),
$ ' (x)
instead
s ( ~ r C ) e, t c . I n p r a c t i c e E w i l l almost always be a f i n i t e dimension-
a l v e c t o r space over t h e complex numbers. But sometimes I t w i l l be a Hilbert space'or t h e Banach space of bounded l i n e a r operators a c t i n g on it ( i f H is t h e i l i l b e r t space, we denote by L(H) t h e space of continuous endomorphisms of H). Let X, Y be two smooth manifolds, E, F two l o c a l l y convex spaces. Ve r e c a l l t h e Schwaxtz k e r n e l s theorems any continuous l i n e a r map of c~!x;E) into
a
* ( Y ; F ) is of the form
where ~ ( x , is ~ ) a d i s t r i b u t i o n i n X X Y valued i n L(E;F), t h e space of continuous l i n e a r operatoss E
--b
F s u i t a b l y topologized (E and F must
be "reasonable" spaces, which is d e f i n i t e l y the case f u r Banach spaces,
.j?rechet spaces, spaces l i k e
@*
, & ' , :C
, etc.;
a l s o the d i s t i n c -
t i o n between d i s t r i b u t i o n s as currents e i t h e r of degree zero o r of maximum degree should be maintained, but we s h a l l disregard it here. )
L e t u s denote by K t h e continuous l i n e a r mapping (1.1). t h e kernel ~ ( x , is ~ )w a t e l y =@lax continuous l i n e a r of: C nuous l i n e a r map of
i n x and i n y i f K
We say t h a t defines a
(x;E) i n t o Cm (Y;F) and i f K extends as a conti-
& ' (X;E)
into
a * (Y
f ~ ) .
Interms of completed topo-
logical. t e n s o r products t h i s is equivalent with t h e property t h a t
If E and F axe both r e f l e x i v e , the separate r e g u l a r i t y of ~ ( x , is ~ )
equivalent with the f a c t t h a t K maps c:(x;E) transpose ,
t ~ maps ,
i n t o c ~ ( Y ; F )and its
( l i n e a r l y and continuously) :C
(Y
) into
(E', F' a r e the r e s p e c t i v e topological d u a l s of E and F). f a c t t h a t K extends as a continuous l i n e a r map
cm (x;L* )
with the
* ( x ~ E )-+8 * (Yj ~ )
and t h a t its transpose t~ extends as a continuous l i n e a r map & @ ( Y !F')
-+
~'(x~E'). Suppose momentarily t h a t Y = X. The mapping (1.1)
is s a i d t o be very
r e g u l a r i f it is s e p a r a t e l y r e g u l a r i n x and i n y and i f , moreover, the kernel ~ ( xy),
a
f& ' (X)C
X ~ L ( E ; F ) )is a
cm
function i n the complement of
the diagonal of X >C X (valued i n L(E;F)). Let us go back t o the general case, with Y not necessarily equal. t o X. Xe s h a l l say t h a t the operator (1.1) continuous l i n e a r map of
&@ (x;E)
is ~ u l a r i z i n gi f it extends as a
i n t o C* (YSF). I n order t h a t t h i s be
the case it is necessary and s u f f i c i e n t t h a t ~ ( xy) , be a
cm
function i n
X X Y with v a l u e s i n L(E;F). Notice t h a t , then, the transpose t~ is a continuous l i n e a r nap of
6' (Y;F' ) i n t o
cOD (x;E'
)
.
The last of t h i s list of d e f i n i t i o n s concerning mappings (1.1) is t h a t of properly supported: the kernel ~ ( x , y )( o r the mapping K ) is s a i d t o be properly supported i f K naps: C
its transpose t~ maps: C
(x;E) i n t o
(Y$F') i n t o
& ' (YIF) and
& '(xJE').
if
.
noreover,
Thin is equivalent with
saying t h a t t h e kernel ~ ( x , y )has the following property;
Given any compact subset %of X t h e r e is a compact subset
(1.3)
of Y -
such t h a t , f o r any
support of
,d),
E cOD(x) with supp ,d C'x(supp ,d =
we have s u p p ( b ( x ) ~ ( x , y ) ) C X X
any c o n n c t subset
ofY
Yt'
; and given
t h e r e is a compact subset $$
cm (Y)
such t h a t . f o r any p o r t of
b
r
f l y ) ~ ( x , y ) l i e s i n 3$ %
B supp )Y
C J(; , the
of X sup
rC; .
If ~ ( xy,) is properly supported, K extends as a continuous l i n e a r map
t
C ~ ( X I E4 ) 8 1 ( ~ ; ~ and ) * K as one from c ~ ( Y ; F ' ) i n t o ~ * ( x ; E * ) . If we combine t h i s with the d e f i n i t i o n of s e p a r a t e l y regulax operators
we s e e t h a t 4
If K(X,Y)is
properly supported. and s e p a r a t e l y r e g u l a r i n x
K d e f i n e s a continuoas l i n e a r m a p A :C ,
resp..
,
&I(XIE),
(x;E) (-.
~ * ( x : E ) )into
E * ( Y ; F ) , reap., $ ' ( Y ; F ) )
03 cC (YIF) T I.-
, C~
y,
(x:E),
, c*(Y;F),
.
Ble now r e c a l l some p r o p e r t i e s of very re&w,l*lsrkernels. Tius we assume, throughout t h e remainder, t h a t X
Y.
The singular support of a d i s t r i b u t i o n u, t h a t is t o say, the smallest closed subset in the complement of which u is a C*
functions, w i l l be de-
noted by s i n g supp u. The following r e s u l t is c l a s s i c a l (cf. [lfl
, Ih.
52.1)s t h a t the kernel K ( x , ~ )is v e r y ~ ~ l a Then x .
Theorem 1.1.- %pose (1.5)
(I.g . ,
s i n g dupp
KU
C
s i n g supp u
,
Ij
€
& ' (XSE)
Ku is an F-valued c m f u n c t i o n i n every open subset of X i n which
u is an P-valued
cW function).
Th. 1.1 is often r e s t a t e d by saying t h a t a very r e g u l a r operator E i s pseudodlocal. A standard application is t h a t , i f t h e r e is a very regular, properly supported l i n e a r operator P r
KP
= I d e n t i t y of
& ' (x;F),
& '(%;F) --* &
(x;E) such t h a t
then
V
silig supp PU = s i n g supp u,
u E
& *(x;F).
Another property of very regulax kernels is the followingr (1.6)
~ ( xy,) be a very r e g u l a r kernel i n X% X, valued i n L(6;F). r e is a p r o p r l y g p o r t e d kernel %(x, Y ) E t h a t K - K1 -
E
2-
3 ' ( X)t X;L(E;F) )
c*(xxx;L(E$F)).
R o o f : Kl(xrY) = b(x, Y)K(X,Y ) with
b 6 cm (XX X) properly
supported (cf.
(1.3)) and equal t o one ine a neighborhood of t h e diagonal. Pseudodifferential operators a r e a s p e c i a l case of very. r e g u l a r operat o r s of the kind (1.1).
We now r e c a l l t h e i r d e f i n i t i o n s and main properties.
F i r s t we define pseudodifferential operators i n an open subset U of a Buclidean space H"
..,xn) ; t h e dual coordinaI1,. ..,Xn and we write x. 5 - 13 ..+ x s n). For
(where the v a r i a b l e is x
t e s a r e denoted by
*
(xl,.
the moment we r e s t r i c t ourselves t o d i s t r i b u t i o n s and functions with comglex values; the case of more general values, say l y i n g i n a Banach s p a c e ,
is gotten by straightforward
t h e Fourier
We s h a l l systematically use
transform,
-
~ ( 3 )J e - i x * 3
(1.7)
U(X) dx.
and the Fourier inversion formula,
say f o r u i n
4 ( E i n ) (or generalized i n t h e various standard manners).
For any r e a l m w e denote by s ~ ( u , u ) t h e space of
a(*, y,
3 ) in UX U$R"
(I. 9)
o ore
having t h e following p r o p r t y t
of U X U , t o every t r i p l e of elements
To everycompact subset
a, 6 ,
n of z+ there
generally, the space
-p1 1 takes 0
is C
-
c(.&;a,l5,1
srn (u,u) p,b
)PO
such t h a t
is defined by s u b s t i t u t i n g m+
8 la+^ 1
- 13 1 i n t h e exponent at t h e r i g h t , i n (I. 10) ; usually 6 5 ;_< p _< 1 . ) The space sm(U,b) i s topologized i n t h e
for m
5
functions
one ob-
vious manner ( i t is then a Frechet space). Since a ( x , y , S )
E sm(IJ,U) is tempered 6
its Fourier transform, a(x, , z ) ~ ( x , y )= (2n )-n $(k,y,y-x),
with r e s p e c t t o
, we
may form
. We introduce then t h e kern'el-distribution
t h a t is,
Me denote by O P S ~ ( U , Uthe ) space of l i n e a r operators defined by kernels of t h e kind (1.11)
I
Notice t h a t i f a(x, y,
If u is any element of C?(U)
5 ) = a(x, 5 )
,
is independent of y, (I. 12) r e a d s
Definition 1.1.- We say t h a t a continuous l i n e a r operator: C
is a pseudodifferentid. operator of order m A
€ OFSm(U,u) such that P
Y
i n t e r s e c t i o n by in
ym(u), m
* (u)
The space of pseudo-
y) m ( ~ ) .
U w i l l be denoted by
w i l l be denoted by *(U);
( R,
--+a
i f t h e r e is an operator
- A is r e g u l a r i z i n g ( i n u).
d i f f e r e n t i a l overators of order m The union of the
in U
(u)
their
-OO(U) ( i t s elements are the r e g u l s r i z i n g operators
u).
The continuity p r o p e r t i e s of pseudodifferential operators axe b e s t described by means of the Sobolev spaces* $ = #(IR")
is the space of tempe-
r e d d i s t r i b u t i o n s u i n R" such t h a t
s can be any real number. We r e c a l l t h a t
I\ 11
is a Hilbert norm on H ~ ,
L2 ((wn ) for t h e ob-
which t h u s is a Hilbert space, a c t u a l l y a copy of II0 vious isomorphism u
(1
(1.15)
-x
(1
- A~)'%
s 2 u x = (2n 1
(isomorphism from
-
lie have t h e i n j e c t i o n s with dense images
e
1 +
x
dC
H8
t i2 S/Z
8 6
d'
.
onto I2 i
f i ( s ) d~ u
.)
By transpo-
s i n g the first one we obtain the n a t u r a l i n j e c t i o n of t h e dual ($)* of I? into
8 *.
whose image is e x a c t l y H-s
another, a d ( 1 H)'.
If s
p
s*
- 4)'is the , 8 is
is l a r g e r than t h e norm
f
(thus
canonical isometry of H'
8'. and
contained and dense i n
11
IIs,
on
H8
. Etc.
(cf.
Now, i f U is any open subset of lRn, by $,=(u) d i s t r i b u t i o n s i n U such t h a t bu t? space of elements o f
8
and H-s
di
f o r every
[13]
axe d u a l of one onto its &dual. the nora 11 ]Is
, Ch.
24 & 25).
one denotes, t h e space of
4 c. c:(u),
by %(u) t h e
compactly supported i n U. These spaces can be
e q u i p p d with n a t u r a l l o c a l l y convex topologies (then s O c ( u ) is a r s f l e xive h ' c h e t space); q O C ( ~and ) $(u)
can be i d e n t i f i e d t o t h e dual of
each other. The following r e l a t i o n s a r e t r u e setwise (the f i r s t one is a l s o t r u e topologically) a
(3
lF
means " d i s t r i b u t i o n s of f i n i t e orderw.) The proof of the r e s u l t s
we now s t a t e can be found i n many a t e x t on pseudodifferential operator theory (see e. g. Theorem 1.2.- Any operator P ii:!~)
~ ; Z ( U ) ,whatever t h e real number s.
--p
From Th. 1.2 and (1.16) Corollary 1.1.- P
c~!u),
V m ( u ) induces a continuous l i n e a r map
&*( u )
G
--t
one derives
( u ) induces a continuous l i n e a r map
$ ' (u)
cOD C
(u)
-P
( i n other words, t h e kernel d i s t r i b u t i o n
associated with P is separately regular). Theorem 1.3.- Every pseudodifferential operator i n TJ is pseudolocal ( i t s associated kernel is very regular). Theorem 1.4.-
If
P 8
= t ~ ,a l s o belong*
If Q is -
9 m ( ~ t)h e 9m(~).
transp%
an operator belonging t o
the c o m p E PoQ belongs t o
Y
tP
of P
and its adj&,
*
P
m' (u) and is P r o p e r l y ~ p o r t e d ,
*'(u).
The requirement t h a t Q be properly supported insures t h a t the compose
PoQ wakes sense. If we want t o define pseudodifferential operators on a manifold we
need t o know t b a t t h e concept of pseudodifferential ope-
in open
subsets of Euclidean space is i n v a r i a n t imder coordinates changes. Thus l e t V denote another open subset of R",
a diffeomorphism of U onto V ,
and consider the diagram
-1
$ is defined
as the compose ~ , O P O ~ , (b, is the d i r e c t image -1 map on d i s t r i b u t i o n s from U t o V-, its inverse). I n (1.17)
8,
Theorem 1.5.-
P E Ym(u)
pb
c
9 m(v).
If we do not require t h a t V, i n (1.17),
be an open subset of kin, but
only t h a t it be a cCO manifold diffeomorphic t o such a subset, we may use t h a t diagram t o define
# a s a pseudodifferential operator
(of order
m) i n t h e manifold V. It is then easy t o extend t h e d e f i n i t i o n t o an axbit r a r y (smooth) manifold X a Let P be a continuous l i n e a r operator
E *(x) -+$ '(x).
If V is
any open subset of X, we denote by PV t h e composition
where the f i r s t arrow stands f o r the n a t u r a l i n j e c t i o n mapping, the t h i r d one f o r t h e n a t u r a l r e s t r i c t i o n mapping (of d i s t r i b u t i o n s from X t o v). Definition 1.2.- W t o r (resp., -
e t h a t t h e operator P ~ s e u d o d i f f e r e n t i a lo p e -
of order m)
& X i f g i v e n a n y s n subset
phic t o an open subset of a Euclidean
sws,
V o f X diffeomor-
the "induced" operator PV
~ s e u d o d i f f e r e n t i a loperator (resp., of order m)
V.
Y m ( ~ t)h e
We s h a l l denote by of order m i n X, by
Y ) (X)
space of pseudodifferential operators
(resp.,
(x)) t h e union (resp., the in-
t e r s e c t i o n ) of t h e spaces y m ( x ) , m E.@. It follows from Th. 1.2 t h a t -O0
(x) i s t h e space of r e g u l a r i z i n g operators i n X.
The composition of two elements, P, Q of V(X), Po&, might not be d e f i ned u n l e s s we s p e c i f y t h a t Q is properly supported (cf. Th.
1.5).
This
r e s t r i c t i o n is removed if we d e a l with =quivalence c l a s s e s of pseudodiff e r e n t i a l operators modulo,-& g-er
&.
e., with
elements of
t h e q u o t i e n t spaces
Indeed, by (I. 6 ) , w e know t h a t every c l a s s i n
..
t i v e which is ~ o p e r l ysupported. Thus, i f P, Q
..
(X) contains a representas
Y (X),
t h e i r compose
PoQ is defined as t h e c l a s s of an operator PoQ where P E P, Q 6 Q and &
.
i s properly supported.
Another advantage i n dealing with c l a s s e s mod
-*(x)
is t h a t we
can make them a c t on a r b i t r a r y d i s t r i b u t i o n s , whether they have compact support o r n o t , provided we axe w i l l i n g t o reason m&ulo Y(X) and i f P
E
is properly supported, we may define
ment fi of &*(x)/cm(X)
cm
I
k for
If
6
any e l e -
as t h e c l a s s of Pu, where u is any r e p r e s e n t a t i -
ve of t ( r e c a l l i n g (1.4)). I n p r a c t i c e , often without r e c a l l i n g it, one d e a l s with equivalence c l a s s e s of pseudodifferential operators modulo r e g u l a r i z i n g ones, r a t h e r than with t h e pseudodifferential operators themselves. We come now t o the symbolic calculus of pseudodifferential operators. n For t h i s we go back t o t h e open s e t U i n Euclidean space R Let u s r e f e r
.
t o t h e elements of s"(u,u)
as amplitudes. By a symbol of degree m we then
mean an amplitude belonging t o sm(U,U) a ( x , ~ , 8) = a ( x , 3 ) m e n d e n t of t h e second v a r i a b l e , y. These symbols form a subspace sm(u) of s"(u,u), topologized i n an obvious manner. ( s i m i l a r l y one can define the symbol spaces
sm&(u). )
P,
A t t h i s . p o i n t a s l i g h t complication a r i s e s . In p r i n c i p l e we want t o as-
s i g n a unique symbol t o each pseudodifferential operator in'U. a(x,
3)
sm(u) and i f A is the operator defined by a ( x ,
But i f
3) v i a
(1.13)
(operators such as A make up a space which we denote by O P S ~ ( U ) ) ,any pseudodifferential operator which d i f f e r s from A by a r e g u l a r i z i n g opera-
tor (cf. Def. 1.1) must be assigned t h e same symbol a ( x . 3 ). And conversely i f we assume t h a t A (given by (1.13))
i s i t s e l f r e g u l a r i z i n g , then any re-
g u l a r i z i n g operator w i l l have a ( x , E ) as symbol
- as well
as any other
symbol which s i m i l a r l y d e f i n e s a r e g u l a r i z i n g operator ( v i a formulae of the kind (1.13)).
This makes it c l e a r t h a t we must reason modulo symbols which
define r e g u l a r i z i n g operators. These form a space which we denote by S-*(u)
and which can be shown t o be i d e n t i c a l with the i n t e r s e c t i o n of
, Th.
the spaces sm(u) as m ranges over IH ( [ 6 ) 1
2.8).
We s h a l l write
*m Definition 1.3.- Equivalence c l a s s e s belonging t o S ( u ) w i l l be c a l l e d w t o t i c symbols of d e g r r m
& U.
In a sense t h e equivalence c l a s s modulo S-* (u) of a symbol a(x, c h a r a c t e r i z e s i t s "growth" as
-
0 for large
1~
1
, its
\ 1
class mcd
3
)
+m. I n p a r t i c u l a r , i f a(x, 5
s - (u) ~
is zero.
The most common method of constructing asymptotic symbols is based on use of formal symbols; Definition 1.4.
- JB
a formal symbol of d e g r m
U we mean a sequence
>
U, of respective degP sm bols a .( x , r ) 5 J t h a t mo m, the m . decrease and tend t o oo J
- .
IL
{a. J
A formal symbol
f
..
j=O,l,
J
is usually denoted as a s e r i e s
2
( a c t u a l l y , a "formal s e r i e s " ) ,
..
m . ( j = 0, 1,. ),
a
j-o
j
. It can e a s i l y be shown, Xj (x, 5 )
then, t h a t t h e r e exJst cut-off functions
R ~ such )
E C? (U
t h a t t h e modified s e r i e s
a c t u a l l y converges i n s m ( u ) (commonly one s e l e c t s a sequence of compact s u b s e t s K . of U, with K . contained i n the i n t e r i o r of K J J j+l
U , and a sequence of numbers R t o have
%(x,$)
= 0 for x
j
e Ky1
7
+a
ar
, and
, exhausting
chooses the
131 < F t j ,
j
Xj(x,X)
-
s o as
1for
151 >
X.
x i n K . and Rg+l ). Of course, the choice of the cut-offs J J can be v a r i e d i n many ways; the important f a c t is t h a t the corresponding
"true" symbols (1.21) d i f f e r by elements of
S-O)(U)
. Thus we have
the
"transitions" (1.22)
formal symbols
--+
asymptotic symbols
I-+
symbols
( t h e last axrow c o n s i s t i n g of s e l e c t i n g a r b i t r a r i l y a r e p r e s e n t a t i v e of any given asymptotic symbol). An important c l a s s of formal symbols is t h e one i n t h e following Definition 1.5.- J E symbol
s ern
a c l a s s i c a l symbol of de,gee m we mean a formal
aj(x,S)
,..., i i3. ( x , z ) &
such t h a t , f o r each j - 0, 1
j function of ( x , z )
&
U X R ~ ,positive-homogeneous of d e g x m
-j
with r e s p e e
lase
f
(1.e., a j ( x . p j )
-
p > 0)
pm-j a j ( x , S ),
f,
IS/
It is customary t o extend by homogeneity t o a l l values of every individual term a.(x, f ) i n t h e c l a s s i c a l symbol
J
therefore r e l i n q u i s h t h e demand t h a t these terms be of U & IRn 8 they now are Definition 1.6.
- If j
COO
aj
cW
5f
0
a , and j j i n the whole
i n U X (lRn \ $03). is a c l a s s i c a l symbol i n U, and if P Is a
pseudodifferential operator in U having it as its (formal) symbol, term of degree m, ao(x, denote it by
z), is c a l l e d the p r i n c i p a l symbol of P.
We s h a l l
C(P).
The notion of p r i n c i p a l symbol can be generalized t o symbols other than the c l a s s i c a l ones, but we s h a l l not do s o here. So f a r we know t h a t c e r t a i n pseudodifferential operatars have symbols and t h a t , i n t h i s case (and provided we deal with asymptotic symbols), ;heir symbols a r e uniquely defined (thanks t o Th. 2.8 of [ 6 ] ). We come now t o t h e question of assigning a symbol t o every pseudodifferential oper a t o ~ .This is e a s i l y done t I f P
Y m ( u ) , P is congruent mod
t o an element A b OPS~(U,U);l e t then a(x,y,
5)
d e f i n i n g A as i n (1.12). Let u s write, f o r any h'
I n t e g r a t i o n by p a s t s y i e l d s a t once t h a t
-ca ( u )
Sm(u,u) be an amplitude
ea
+
,
1 SO
This i n t u r n implies t h a t
where (1.25)
rN(x,y,
5
( - ~ ) a a a ( x . ~ ), ~ E Sm-N-l lai = ~ + 1
I
(u,u>
.
fiom t h i s we deduce e a s i l y t h a t
is a formal symbol for 2. This i n t u r n defines an asymptotic symbol ( ~ e f
1.3)
-
symbol of P. We conclude t h i s p a r t of the present s e c t i o n by
r e c a l l i n g how t h e algebraic operations on pseudodifferential operators r e f l e c t on t h e i r symbols (we s h a l l s o l e l y deal with asymptotic symbols, while omitting the a d j e c t i v e asymptotic: when we c a l l symbol a formal s e r i e s of symbols, w e mean t h e asymptotic symbol defined by the formal symbol i n question). Theorem1.6.-L&a(x,r) t h a t of P '
Let Q 6 -
&
thesymbolof P E y m ( u ) .
Then
a: ( - 1 ) ' a ' ( a 2 ~ ) ( x . - ~ ) , and t h a t of t h e a d j m
y m * ( u )with s y m m b ( x , z ) , The symbol of PoQ is eqcal t o
Corollary 1.2.PoQ
a
denote
- QoP . m
c(x, n c(x.5 )
3 ) be
t h e symbol of the cammtatm [pea]
- $la.bl
(x,S)
E p*'-2(~).
-
Me have denoted by b , b f
the Poisson bracket of a and b
Let 'us now r e t u r n t o t h e diagram (1.17) bian matrix of the diffeomorphism
b at
I
b
the point x. Let us denote by y the
t -1 vaxiable point i n V , and by J (Y) t h e transpose of t h e inverse -1 b the contragredient) of J ( b (y)). Theorem 1.7.of $E -
=.
(2. ,
Ci) m(U), a'(y,
a ( x . 5 ) denote the symbol of P E
4m(~).
Jaco-
and denote by J (x) the
& ) I$
Ue have
The f u l l significance of Th. 1.7 is b e s t understood when defining the symbols of a pseudodifferential operator i n a manif old X, P. Let (V,xl,. x ) be an a r b i t r a r y l o c a l c h a r t i n X (thus n n (1.18).
By means of t h e l o c a l coordinates (xl,.
from V t o an open subset U of R"; rator
-
r(: has a symbol
dim X),
..
,X
.. ,
PV the compose
n ) we may t r a n s f e r PV
t h e t r a n s f e r r e d pseudodifferential ope-
(belonging t o
6 ( ~) )
symbol of P i n t h e l o c a l c h a r t (v,xl,.
which
.. ,xn).
we
define
as
%
It follows a t once from
Th. 1.7 t h a t t h e property t h a t the symbol of P i n any l o c a l chast i s c l a s s i c a l (Def. 1.5) is independent of the choice of the l o c a l coordinat e s . We may therefore t a l k of c l a s s i c a l pseudodifferential operators i n a manifold. If P is a c l a s s i c a l pseudodifferential operator of order m i n X, i n each l o c a l c h a r t (\I,% (Def. 1.6),
,...,x,),
it has a p r i n c i p a symbol
w ( P ) ( x , ~). I f we change coordinates i n V and c a l l yl,.
the new coordinates, and r i v e from (1.20) t h a t
jl,...,3nthe
.., Yns
associated cocoordinates, we de-
where y = b(x). The meaning o f (1.30) is t h a t
*
.
e P ) d e f i n e s a bona f i d e
function on t h e cotangent bundle T X over X or, more accurately, on t h e
*
complement of the z e r o section, T X
0
cause of the assumed positive-homogeneity
3
~ i t rhe s p e c t t o
*
, we
may view it as a
, in
t h a t bundle. Actually, be-
(of degree m) of
cm
tT(P)(x,3)
function on t h e cosphere
bundle S X over X. One of t h e g r e a t advantages of pseudodifferential operators is t h a t
they enable u s t o microlocalize. Let
*
P
be a conic open subset of
0 : here conic r e f e r s t o t h e f a c t t h a t , i f (x,
TX
s o does (x,
p 5)
whatever
> 0. We
3 ) belongs
s h a l l c a l l base of
*
r
,
to
its canoni-
z(r)the
c a l image i n the cosphere bundle S X. We s h a l l denote by
space of pseudodifferential operators i n X whose asymptotic symbol i n any l o c a l c h a r t has its support i n an "open cone" whose base is r e l a t i v e l y compact i n
p
( t h i s means t h a t some t r u e symbol r e p r e s e n t i n g the'asymptotic
syntbol has t h i s property). On t h e other hand we s h a l l say t h a t two operat o r s P, .Q
y
(x)
/' , and
ate ~ q u i v d e n ti n
Write P
t h e i r asymptotic symbols i n any l o c a l c h a r t (V,%, portion of gularizing*
r
which l i e s above V. I f P r J O in
f.
&
Q
in p , i f
...,xn ) are equal i n t h e , we
say t h a t P is
re-
It is c l e a r t h a t one may d e f i n e t h e sheaf of germs of
*
pseudodifferential operators over T X \ 0 by means of the presheaf
1 -L.
(P
T X\ 0
)I ,
where
p
ranges over all possible conic open s u b s e t s of
, and i k ) ( f ) denotes
r e l a t i o n P ruQ i n Definition I. 7.
'
4
t h e space of eguivdlence c l a s s e s modulo t h e
f.
- We s h a l l say t h a t
c r n ( P )(g., to g o c ( P ) )
a distrihiltion u
GPu
E
X belongs t o
(rp.,'- EqOc(x))
xhatever t h e (properly supported) pseudodifferential gperator P
X
Definition 1.8.-
te byF(u)
3 * (x).
Let u
We c a l l wave-front s e t of u and deno-
of T X \ O -
such t h a t u
ca(p
).
Thus WF(U)is a conic closed subset of u (1.31)
P
the complement of the union of all the conic open subsets
*
The projection of WF(u)
*
T X
0
. i n t o t h e base manifold, X,
is e x a c t l y ~ q u a lt o s i n g supp u. Theorem 1.8.bution u
in x
t o cW( -
)
SO
Corollary 1.4.
m ( ~ be ) properly supported. Given any d i s t r i -
Let P E
such t h a t u E q O c ( r ) we have PU does Pu
-
u
Let us denote by
e q i E ( p ). g u
belongs
.
a*
(XI,
~ ~ ( pC uW ) F(~).
* (P) the quotient of
4
(X) (viewed a s a l i n e a r
space without topology) modulo the subspace farmed by t h e d i s t r i b u t i o n s i n
X which belong t o
*
(over T X \ O
cW( p ).
The sheaf defined by the presheaf
1
*(
*
)
1
or, more correctly, over t h e cosphere bundle S X) is some-
times c a l l e d t h e sheaf' of microfunctions i n X. Lastly a few wards about e l l i p t i c pseudodifferential operatars. First l e t U be an open subset of R". A symbol a ( x , T ) € s"(u) is s a i d . t o be e l l i p t i c i f t h e r e i s a symbol b(x, b(x,
5) '1 i n U )C R".
5)E
s'~(u) such t h a t a ( x , S
)a
An asymptotic symbol of degree m w i l l be c a l l e d
e l l i p t i c if it h a s a representative which is an e l l i p t i c "true" symbol of degree m. These d e f i n i t i o n s t r a n s f e r at once t o m y l o c a l c h a r t (V ,xl,.
.,xn)
.
i n t h e n-dimensional manif old X.
Definition 1.8.- A s e u d o d i f f e r e n t i a l operator P of order m be e l l i p t i c i f its symbol i n every l o c a l c h a r t . i s ellip=.
&X
is s a i d t o
If t h e symbol of P i n every l o c a l c h a r t is c l a s s i c a l
(A. e.,
i f P is a clas-
s i c a l pseudodifferential operator) we may make use of i t s p r i n c i p a l symbol
*
C(P)which is defined in the whole of T X N 0 (and is positive-homogene@(P) does not vanish
ous of degree m). Then P is e l l i p t i c i f ard onlyif
*
a t any point of T X
'0
.
Theorem 1.9.- I) $ V m ( x ) is e l l i p t i c i f and only i f t h e r e is Q C such t h a t PQ a QF
I mod
ifm(~)
-Ob(x).
I n order t o construct t h e "approximate inverse" Q of Th. 1.9 it s u f f i ces t o construct its asymptotic symbol i n each l o c a l chart of X. Thus it s u f f i c e s t o reason i n the open s e t U of Ktn. The asymptotic symbol of Q is then constructed by f i r s t associating w i t h Q a formal symbol, as followsz Let &(x,
5 ) denote t h e aymbol of P
s e n t a t i v e of it such t h a t a ( x , l )-I b(x.5)
-
E V ) m ( ~ ) , a(x,
3 )E
sm(U) a repre-
~ ~ ~ ( U8e )then . seek a formal symbol
b j ( x , f ) such as t o have a 0 b = 1 (see (1.27)). This is j-0
achieved by takings
*om
Cor.
Corollary 1.5. (1.33)
-
1.4 and Th. I. 9 we derive at once t
- ~fP € 9 m ( ~ i)s ellip%, WF(FU) = UF(U)
Of course (see (1.31))
(1.33)
,
b'
U€$~(X).
has t h e implication t h a t s i n g supp Eu
s i n g supp u f o r all d i s t r i b u t i o n s u i n X. It should be noted t h a t (1.33)
is not t h e exclusive w o p r t y of . e l @ t i c p s e u t t d l f f e r e n t i a l operatars;
c e r t a i n n o n e l l i p t i c operators a l s o enjoy it.
I n p r a c t i c e we shall want t o d e a l with functions and d i s t r i b u t i o n s whose values l i e i n Banach spaces (most often, i n f i n i t e dimensional ones) and therefore d e a l with pseudodifferential operators valued i n spaces of t h e ~ i h dL(E;F), t h e ( ~ a n a c hspace) of bounded l i n e a r operators E
-+ F.
Most
statement t h a t precede extend r o u t i n e l y , but some do not. We s h a l l i n d i c a t e some points t h a t r e q u l r e case. First of a l l , when t h e d i s t r i b u t i o n s a r e valued i n a Banach space E which i s not a Hilbert space, t h e d e f i n i t i o n of -Sobolev spaces H'(R"$E) same as t h a t i n t h e s c a l z c case, but does not make out
b e r t space
- only a M a c h space,
is t h e
of. ~ ( H " I E )a H i l -
0 n and although H (R ;E) i s i d e n t i c a l t o
L2 (R n ;E) a s a topological v e c t a r space, t h i s might not be s o as a normed space, since Plancherel's formula i s i n general not v a l i d ( i t is v a l i d whenever E is a Hilbert space). Unless E i t s e l f is reflexive, I ~ ( R ~ ; E ) x i l l not be, n m dl1 q O c ( x s ~ )nor ~ ( X I Ebe ) reflexive. In dealing with pseudodifferential operators valued i n L(E) pay particulax a t t e n t i o n t o t h e implication of noncommutativity
,
one must
- not only
of t h e s c a l a r pseudodifferential operators, b u t of t h e elements of L(E) For instance, i f dim E
> 1, C a r .
.
1.3 x i l l not be any mare v a l i d ( i n gene-
r a l ) , t h i s is due Co t h e f a c t t h a t , i f a and b a r e two symbols valued i n L(E), of degrees m and m' respectively, t h e commutator a Q b
(1.27))
has degree m + m' and not ( i n general) m
f a c t t h a t t h e i r usual commutator, a(x,
+ m1
-baa
(see
- 1, due t o the
3 )b(x, 5 ) - b(x, ?$)a(x,.j
), does
not necessarily vanish i d e n t i c a l l y . Finally we r e c a l l t h a t a, pseudodifferential operatar with values i n L(E) is s a i d t o be e l l i p t i c when i t s p r i n c i p a l symbol is i n v e r t i b l e a t
*
every point of T X \ 0
.
11. THE GENERALIZED HEAT EQUATION AND ITS PARANEXRIX
Throughout t h i s chapter X w i l l be, as before, a a t infinity8 n
cW manifold
countable
dim X ; t w i l l be the v a r i a b l e i n the r e a l l i n e &? (most
often, i n t h e closed h a l f - l i n e
E+).
By m we denote some number
which, i n t h e most s i g n i f i c a n t applications, i s equal e i t h e r t o
>0 , one or
to
two; T w i l l be some number > 0. We s h a l l deal with functions and d i s t r i b u t i o n s valued i n a Hilbert space H (over C). I n t h e application H w i l l be f i n i t e dimensional but there is no added complication by not r e q u i r i n g dim H noted by
I
, whereas ,
The norm i n H w i l l be de-
t h e operatcn. nacm i n L(H), the space of bounded
l i n e a r operators i n H, w i l l be denoted by w i l l be denoted by (
< +oo.
)H
.
I\
I\
. The inner product i n H
Our b a s i c ingredient is a pseudodifferential operator of order m i n X,
A(+.),
valued i n L(H), depending smoothly on t i n
i n every l o c a l c h a r t ( a x l , .
. This means t h a t
..,xn) ~ ( t is) congruent modulo r e g u l a r i z i n g
p r a t a c s which a r e cWfunctions of t (what we s h a l l always shorten i n t o "equivalent" and symbolize by
h, ;
here t h e r e g u l a r i z i n g operators must be
valued i n L(H)) t o an operatar r
where
(11.2)
x,t,J )
is cWfunction
of t ~;CO,T[valued i n
srn(fi;~(ii)).
I n the forthcoming we s h a l l o f t e n drop the s u b s c r i p t (r and r e f e r t o a(x,t,
5 ) as t h e
symbol of ~ ( t i) n the char%
(n,xl ,...,x,)
t h i s name should be reserved f o r t h e c l a s s of a ( x , t ,
3 ) mod
although S-CD(a;~(~)).
We are i n t e r e s t e d i n solving t h e following i n i t i a l value pcoblemr
ult=,
(11.4)
N
I, t h e identity&
Iq X.
H,
I n p r i n c i p l e t h e s o l u t i o n ~ ( t should ) be an equivalence c l a s s , modulo r e g u l a r i z i n g operators i n X depending smoothly on t, of continuous l i n e a s operators
&' (x;H)
--+
$ ' (X;H)
depending smoothly on t ( i n (o,T[).
But
without a d d i t i o n a l hypotheses about ~ ( t t)h e r e is no reason why such a sol u t i o n should e x i s t . We are going t o make a hypothesis on ~ ( t which ) not only w i l l insure t h a t U does e x i s t , but a l s o t h a t it possesses a conven i e n t i n t e g r a l representation. Observe t h a t when X = lRn and ~ ( t =) the Laplace operator i n n v a r i a b l e s , i n which case t h e symbol of
- 15 \ , Eqq.
(11.3)-(11.4)
poblem f o r t h e heat equation
Ax ,
~ ( tis)
define t h e parametrix i n t h e forward Cauchy
2 - Axl
= 0.
In general we s h a l l make
the following hypothesis r (11.5)
j&
(n,xl,. . .,xn) be l o c a l
,z )
a(x,t tor A
c h a r t i n X. There is a s y w
s a t i s f y i n g ( I I . ~ ) , and defining-
(11.1)
the opera-
( t ) congruent t o ~ ( t modulo ) r e g u l a r i z i n g operators i n
- fL
, such
t 8 [o,T(
depending smoothly=
moreover t h a t
toeverycompactsubsetKI?f~~@,~(thereisacomp&
(11.6)
subset K 1 of t h e open half-pthat (11.7)
e1
- a ( x , t , 5 )/(I
+
a-
1x12)m/2
s H
+
i s a b i j e c t i o n (hence a l s o a homeomorphism), --
5
~
R
~
.
Z
&
E
; Re
6 lC
=
K1.
We may now s t a t e the main r e s u l t of t h i s chapter;
z
< 03 &
H
whatever ( x , t )
& K,
Theorem 11.1.
- Und.er
Hypothesis (11.5)
t h e problem (11.3)-(11.4)
s o l u t i o n ~ ( t which ) is a function of t E[O,T( ( c f . (1.19)).
valued i n
has a
O(X;L(H))
There is a representative of t h e equivalence c l a s s ~ ( t with )
t h e following_property: I n each l o c a l c h a t i s equivalent -
u,
(11.10)
has t h e f o l l o w i n g ~ o m r t i e s r
R
n
cm n
n-tuples a, P €
+
t h e r e is a constant C
~
w function of t
in
+ $*(x;H)
L(H) ;
-+
n
X
(o.T( , t o
every p a i r of
>/
and t o e v e r y g a i r of i n t e g r~, N
> 0 such t h a t ,
[o,T(
9 Eg* ( xz ~ H )
(O.T(X
of
1axa d5E ld rt S ( x . t . 5 )I[
&C ~
of OPS (R;L(H)) given by
t o e v e r y m c t subset
(11.11)
i n question
0
t o an element u R ( t )
whose s y m m (11.9)
(R,%, ...,x n ) Of X t h e representative
whatever ( x , t )
C t-N( 1 +
5
0
inx,3 E R",
151)rm-'P'-Nm
.
valued i n t h e space of continuous l i n e a r which s a t i s f i e s (11.3)-(11.4)
belongs t o
t h e equivalence c l a s s u ( t ) . -
It follows from (11.11) S-*(~;L(H)),
&.
e.
that, far t
> 0, U
t h e operator (11.8)
rr(x,t,J
) belongs t o
i s regularizing, i n other words,
t h e equivalence c l a s s ~ ( t is ) zero. This generalizes t h e well-known gcoperty of t h e parametrix of t h e heat equation. Proof of Th. 11.1% A. Existence of the parametrlx ~ ( t )
It s u f f i c e s t o reason i n t h e (generic) l o c a l c h a r t and patch t h e U
n( t ) together
(n .. ,xl,.
,xn)
afterwards, by Keans of a sinooth p a r t i t i o n
of u n i t y i n X. Thus we cons+;ruct t h e symbol
UR
a c t u a l l y we cons-
t r u c t a f o r m a l symbol ( ~ e f .1.4)
out of which a t r u e symbol can 1 a t e r . o n be constructed, by using cut-offs as indicated i n Ch. I. Ue take t h e operator ( 8 1 ) t o be the operator A ( t ) i n (11.5) and omit t h e subscripts between ~ ( t and ) ASL(t),
n
J.
; we do not d i s t i n g u i s h any more
which we a l s o denote by a(x,t,Dx) ( a ( x , t , j ) is
it symbol). Reasoning formally we write
and r e q u i r e , f o r 0
_< t
< T,
which may be rewritten, with t h e notation (1.27), (11.15)
'aE
This eq. (11.15)
- a ( x , t , ~ ) ~ PMO ,
O < t < T *
i s the "translation" of (11.3);
as f o r (11.4) it trans-
lates into (11.16)
U ( X , O , ~I )(the
i d e n t i t y of H).
By a v a i l i n g ourselves of t h e b a s i c hypothesis, ( I I . ~ ) , we axe going t o obtain 2 ( ( x , t , > ) i n t h e farm
r
where k i s a s u i t a b l e formal symbol of d e g e e zero, valued i n L(H),
de-
pending holomorphically on the complex v a r i a b l e z i n an open neighborhood of t h e i n t e g r a t i o n contour subset of
n % (O,T(.
provided ( x , t ) remains i n a g i v e n compact
Ye have used the n o t a t i o n p -
and s h a l l continue t o use it henceforth.
p( 5 )
-
(l+ &12)&,
Ue s e l e c t a x b i t r a x i l y a r e l a t i v e l y compact open subset a number T
0
closure of i n (11.6)
'0 (9
-
(To ( T ,
noof n ,
and take t h e compact s e t K i n (11.6) t o be t h e
n 0 x @,T~<.
Ye take the compact subset K' of E-
accordingly, and denote by M t h e maximum narm of t h e inverse of
t h e mapping (11.7) as ( x , t ) ranges over K , closed smooth curve Since a
0 ( e Ptzk)
(11.18) k
over IRn and z over a simple
winding mound K' i n C* e Ptz(a
eptz
(11.19)
5
\ K'
.
0 k ) we may r e w r i t e Eq. (11.15)
~ k ( ~ , t , dzP ~ ~ z0 ),
as
where*
%+ prk-a(x.t,F)Ok.
W e a r e going t o solve ( i n t h e sense of farmal symbbls) the equation
whlch implies at once (11.18). synb01 k s a t i s f y i n g (11.20) (2si)-'
(11.21)
It w i l l t u r n out t h a t the (unique) farmal
w i l l also satisfy
9if
k ( x , t , T ;z)dz
-
I.
\d ( x . t ) ~0, 5 E R,.
which, far t = 0, is nothing e l s e but (11.16).
-----------
Solution of (11.20) t Ye r e w r i t e (11.20) as follows r
setting E =
[ZI
- p - ' a ( x , t . s g " (inverse i n I@)). we note t h a t ?-'(a 0 k - ak) - L i:P $ . a#o +OD and deg k - 1. W e solve (11.22) by t a k i n g k - L k j==o m
f11.23) has degree requiringr
By induction on j we see' e a s i l y that
m
(11.25)
j
C
-
- j inf(l,m),
6
deg k j
indeed defines a formal symbol (since m
>o).
F'urthernore
,
(11.26)
Ifj>(l, k.isafinitesumoftermsoftheformEblE J
...
with r
), 2,
which implies
k
j
vaxying from t e r m t o term but always remaining
and with each bi 5
ern
function of t
(o,T(
valued i n
d S I(Q;L(H))
independent of z ( i = I,..., r ) , such moreover
t h a t dl +...+ -
dl ,( m
According t o (11.26),
3 '
therefare,
noin
To every r e l a t i v e l y compact open subset
(11.27)
brE
r y number TO
, 0
such t h a t , f o r each j t i o n of ( t , z )
-
9 (o,T~[.)( m
and t o eve-
t h e r e i s a compact subset K'
,..., kj(x,t,Y
0, 1
;z)
of
E-
crnfunc-
(IE \ Kg), holworphic with r e s p e c t t o
z , valued i n S ' ( n O ; ~ ( H ) ) .
@
--------
R o o f of (11.21) r Fix a r b i t r a r i l y ( x , t ) i n p-la(x,t,j) (11.26).
and
is a bounded l i n e ~ operator H -P
-
- aO)-l
Writing ~ ( z ) (ZI
5
i n fin; then a. =
H and s o axe t h e bi i n
and keeping i n mind t h a t
winds
around the spectrum of a. we g e t once: (217i)-~,fE(z)dz
'd
-
I
,
$E(z)blE(z)
d
...brE(e)
d z - 0 ( i f r)].),
whence (11.21) (by using (11.26)).
------------------ -----
Estimates of the symbols
aj(x,t,? )
e ptz kj(x,t,
( 2 1 i)-'f
J By (11.27) we see t h a t t h i s formula defines [O,T(
r indeed, it does f o r ( x , t ) i n
a larger o p n s e t
o1
we might replace
then r e s t r i c t back :x,t) t o
0
Uj(x,t,
6
.
5) f o r
5 ;z)dz (x,t) i n
but i f we replace
0 by
by a d i f f e r e n t contour. If we
it follows fr.om t h e Cauchy i n t e g r a l theo-
rem t h a t we recover t h e same value as before. Ye note t h a t , i f z E (11.28)
13;
dt(eetz)l
( const. (I+
<
const. (I+
151)-
pra t - N
131)-'1
5 const. t" ( 1 + jx~)-le'
&
!ill +x
(11.29)
11 4; $!$ f k j ( x , t , ~ ;z)ll 5
((pt)'e-~O~t
pr-N
(we have a v a i l e d ourselves of t h e f a c t t h a t Re z
On t h e other hand we derive from (11.27);
)( ,
5 -
co
far ( x , t ) i n
const. ( 1
+
< 0
mj
151)
0 on and z i n
-/@I
).
6
By combining (11.28)
and (11.29)~ and applying Leibniz formula, we get
for all (x,t) i n
,
5
i n R".
*
This implies (11.11).
B. Uniqueness of t h e paxametsix The uniqueness of t h e paxametrix, needless t o say i n t h e sense of equivalence c l a s s modulo r e g u l a r i z i n g operators, follows from various stvldard considerations which axe of i n t e r e s t i n t h e i r own r i g h t , and which we now go i n t o rapidly.
First of a l l we d i d not have t o solve Equation (11.3) while prescribing t h e value of t h e s o l u t i o n at time t
where t' is any number such -that 0
0. We could have solved
,( t'
< T.
By the same procedure as i n
? a r t A we can f i n d a s o l u t i o n u ( t , t l ) having a representative which, i n any l o c a l c h a r t defined by
(h,%,
..., a ) ,
i s equivalent t o a n operator ~ ~ t'() t , P
where k n
is t h e same symbol as i n (11.17)
dent of t'
. This is due t o t h e v a l i d i t y of
The contour of i n t e g r a t i o n
$ may
t
i n p a r t i c u l a r it is indepen(11.21) where we may take t = t ' .
a l s o be taken t o be the same as i n P a r t A.
The s o l u t i o n of (11.31) enables u s t o solve t h e inhomogeneous Cauchy prrblem 2 -
at - A
(11.34)
) = f
&XX
(o,T(
,
u\t_O
Here f i s an H-valued function o r d i s t r i b u t i o n i n X X ment of
-
u0
[o,T(
8 * (XIH) ( i n all r i g o r we must reason modulo C*
2X
.
, uo an e l e -
(XI). ~f f is
s u f f i c i e n t l y r e g u l a r with r e s p e c t .to t, say continuous, we may write
Next we look at t h e backward Cauchy tnW ioFor each t i n [o,T]
equation.
*
we denote by A ( t ) the a d j o i n t of t h e operator ~ ( t )
as an L(H)-valued pseudodifferential operator i n X ( i n order t o define t h e a d j o i n t of ~ ( t we ) make use of a ( s t r i c t l y p o s i t i v e ) d e n s i t y X. Then, i n any l o c a l c h a r t (f2,x1,.
i s a ( x , t , s ) , a formal symbol
where a ( x , t ,
5)* stands
*
..,xn),
in
i n which t h e symbol of ~ ( t )
m A ( t ) is given by
f o r the a d j o i n t of a ( x , t , r ) as
operator on H. If we assume t h a t a ( x , t , J )
8
bounded l i n e a r
s a t i s f i e s the b a s i c hypothe-
sis (11.6) s o w i l l any reasonable t r u e symbol constructed out of (11.36). By d u p l i c a t i n g t h e construction i n P a r t A we can now construct an opera-
t o r v ( t , t l ) solution t o
where t' is any number such t h a t 0 < t ' < T. A s a matter of f a c t , our hypot h e s i s t h a t ~ ( t i)s smooth up t o t = 0 implies t h a t w e can construct a r e p r e s e n t a t i v e of v ( t , t e ) which is
cW
i n t h e closed i n t e r v a l (O,tl>.
This
I s important i n t h e applications.
L e t then v * ( t , t l ) denote t h e a d j o i n t of ~ ( t , t ~m ), r a t h e r , a properly supported r e p r e s e n t a t i v e of t h i s equivdence c l a s s . By transposing (11.37) we g e t (11.38)
W,*
( t * . t ) + v*(t*,t)o*(tl)
-
~ ( t , t * ) 11x
x
(0.t.
,
with R, R o r e g u l a s i z i n g (and depending smoothly on t, t', If we therefore d e f i n e the operator G by the farmula
- it
~f(t)
(11.40)
and suppose t h a t (11.34)
v*(te,t) f(t1)dt'
,
holds we obtain, by i n t e g r a t i o n by p a s t s ,
(At t h i s p o i n t it is advisable t o use a properly supported r e p r e s e n t a t i v e of ~ ( t s)o t h a t both R and Ro i n (11.38),
Eq. (11.4.1)
(11.39j a r e properly supported).
can be r e w r i t t e n as
We may now e a s i l y prove t h e uniqueness of the parametrix. L a t i l ( t ) be
a r e p r e s e n t a t i v e of t h e paxametrix constructed i n P a r t A, u l ( t ) ole of another s o l u t i o n of (11.3)-(11.4); t i o n of t i n
(o,T(
we assume t h a t ~ ' ~ ( its ) a
cW func-
valued i n t h e space of continuous l i n e a r mappings
&*(x;H)
a* . (x;H)
-+
E * (x;H) a r b i t r a r y ,
(11.42), with wo
a
cCO
-
~t s u f f i c e s t o put ~ ( t ) [ ~ ( t )
function of t i n (0,~[valued
- ul(tgwO
in
-
t o conclude t h a t ~ ( t ) u l ( t ) is
.
in
'w(~;~(~))
The proof of Th. 11.1 is complete. Remark 11.1.- The uniqueness of the parametrix ~ ( t and ) the s i m i l a r propert y f o r u ( t , t l ) have t h e consequence t h a t
(11.43)
u ( t , t W ) N u ( t , t * ) u ( t 8,t")
(11.44)
u ( t , t 9 ) N v*(tl,t)
If
if 0 5 t*'C t1d 0
t
< TI
St*$ t.
Thus Formula (11.42) can be r e w r i t t e n i n the farm
f o r every u E
cW(CO,T[;
the &Po-ellipticity Theoren 11.2.valued i n %pose
8' (x;H)) . I n t u r n t h e
expression (11.45) implies
of t h e Cauchy woblem (11.34) r
le+ Y
be an open subset of X, u
a C00 function
of t iq [o,T(
3 ' (x; H). t h a t u(0)
Then u 6 cW(Y Y -
E
C* ( Y I H )and t h a t
hl? - * ( t ) u € cm (Yx
dt
(o,T(IH).
[o,T($H).
Since pseudodifferential operators decrease wave-front s e t s we could have replaced t h e open s e t Y
*
s u b s e t of T X k 0
.
c X,
or. 1.4)
i n Th. 11.2, by any conic open
We s h a l l now r a p i d l y describe some p r o p e r t i e s of t h e paxametrix ~ ( t ) with very few h i n t s about t h e i r proofs.
r,llWEh,-4he-&p1,1tce,-tr,a"sf~m Let k ( x , t ,
3 IZ)
be t h e symbol i n (11.17) and s e t
(11.47)
Ud(x,t,f
)
-
(2111)-I
e
d
where -$is t h e same contour a s i n (11.17).
tz
# k (x,T;z)dz
,
# and
Note t h a t k
# are bona
f i d e formal symbols. It is easy t o check t h a t k# is t h e Laplace transform of
#
, i n the
following senses +a,
(11.48)
-Pzt
5
&#(x,t,s
0
) dt
.
0 For u s t h e important t h i n g i s t h a t (11.49)
x(x,t,F)
-
~#(x,t,5)
C*
function of t ip (o,T(
(n;L(H)).
with values i n
-a
The motivation f o r t h i s is t h a t (reasoning formally)
(
xz )
by i n t e g r a t i n g by parts with r e s p e c t t o z.
# One could have gone d i r e c t l y t o k by performing a Laplace transform with
a -
,
U#
$: A(j)(O) and obtained by j J* # we obtain a r e p r e s e n t a t i v e inverse Laplace transform. By means of
r e s p e c t t o t on the operatar
U
U# ( t ) i n t h e l o c a l c h a r t (
fL
R .%,.
..,xn) of t h e p a r m e t r i x ~ ( t which )
happens t o be a smooth function of t f o r
p o s i t i v e values of t, not
- t h i s is not s o s u r g r i s i n g s i n c e we know we c o ~ l dextend ~ ( t t)o such values - j u s t by s e t t i n g ~ ( t , ) 0 ( a s axequivalence j u s t those
<
T
=
class) f a r all t
> 0.
The symbol (11.47) X
t h e study of t h e operator U U
.
is no% going t o be used, I n
*
The operator U U
---------------> 0 and use a p o p e r l y
We assume t h a t X is equipped with a density
supported repcesentative of ~ ( t ) ,which we may and s h a l l assume defined far a l l t
> 0,
i n accardance with t h e last remark. We then regard uo
u ( t ) u 0 as a continuous l i n e a r mapping which we c a l l U. It has an adoint, U 00
l i n e a r map Cc (11.50)
it
B * (Xi H) , which
(R+ ;8 * (XIH)) -+ $ * (x; H),
(u*v>(x> =
where ( f a r each t
3 0)
jm 0
u(t)*v(n,t)
cW (G+;$* (X; H))
we may regard as a continuous
and which i s given by
at,
:c
v e
(xxR+;H),
*
~ ( t )i s t h e a d j o i n t of t h e pseudodifferential ope-
r a t m i n X, ~ ( t ) ( i t is a l s o properly supported). Let then < E :C g(t)
-
1 far t < R
(%),
(R>o) and l e t us s e t
.+a
F"rom t h e f a c t t h a t ~ ( t is ) r e g u l a r i z i n g i n X whatever t once t h a t i f difference K
Cl E :C (E+),
r - Kcl
lence c l a s s of K
5
- m in X,
5 , ( t ) = 1for t
E
R1 (
> 0 we derive a t R >~ 0 a l s o ) t h e *
is r e g u l a r i z i n g i n X. Ue denote by U U t h e equiva-
modulo regulaxizing operatars i n X.
Theorem 11.3.- The corn>= of order
,
*
U U i s an e l l i p ~ s e u d o d i f f e r e n t i a loperatar
valued i n L(H), formally self-& j * .
Actually, i n guoving Th. 11.3, one can obtain d e t a i l e d information
*
about t h e symbol of U U i n an a r b i t r a x y l o c a l c h a r t It i s convenient t o use t h e symbol (11.47)
(n,%, ...,a)of X.
(we s h a l l however omit t h e
s u p e r s c r i p t s #). A formal symbol for u(t)*U(t)
O(XIL(H)) i n t h e
l o c a l c h a r t under consideration is obtained by taking (see
Th. '1.6j
. *
and a farmal symbol f a r U U is then obtained by i n t e g r a t i n g (11.52) 0 t o +a,
. This shows e a s i l y t h a t U*U is formally s e l f - a d j o i n t
(A.
from
=.,
its
asymptotic symbol is s e l f - m i n t , i n an obvious sense). By taking i n t o account (11.46)
*
and (11.47) we see t h e symbol of U U we have j u s t described
d i f f e r s from t h e following symbol
by a symbol of arder
< - m.
Thus (11.53) plays a r o l e analogous t o t h a t
of p r i n c i p a l symbol ( a t l e a s t i n the l o c a l c h a r t
(n,%, ...,xn)).
Notice
that (11.54)
x(x.
5 ) is e l l i p =
(of order
- m)
positive-definite,
mare p r e c i s e l y , i f h is an a r b i t r a r y element of H, x g
Furthermore,
if a(x,O, 3 )
a(x.0,
5 )* commute
R, 5
E Rn,
i n L(H),
It should be underlined t h a t if a ( x , 0 , 5 ) and a ( x , O , S )
*
do not commute,
t h e e q u a l i t y (11.56) might not hold and a s a matter of f a c t , t h e r i g h t hand s i d e night not even e x i s t , as t h e example H
(
)
- c2 ,
a(x,t,
5 )=
shows. Instead of dealLng with ~ ( t =) u ( ~ , o )we could
have d e a l t with U ( t , t' ), 0 $ t'
<
.
+a We could have thus proved t h a t
t h e c l a s s of pseudodifferential o p e r a t a s i n X, which can be denoted by
,+a Jt
* u ( t , t' ). u ( t , t '
exploited.
) d t , is e l l i p t i c of ~ d e r-me These r e s u l t s w i l l now be
Let us denote by ~ ( t ) u , ( t , t ' ) properly supported r e p r e s e n t a t i v e s of t h e equivalence c l a s s e s s o denoted above; we may a l s o assume t h a t U(t)
is defined f o r a l l t & O , of a p o s i t i v e d e n s i t y i n X chart.
(n., ..,xn)
denote t h e
and ~ ( t , t ' )f o r a l l t
), t'
(t'
5 0).
By means
- or e l s e by r e s t r i c t i n g t h e mgument t o a l o c a l
- we may use Sobolcv nmms on X. For each s i n R we corresponding norm \I ]Is , and t h e inner product by ( , )s.
I n what follows
0
w i l l denote a r e l a t i v e l y compact open subset of X.
F i r s t of all, i f u0 is an a r b i t r a r y element of: C (11.57)
we have
(&;H),
T
0 46
But of course KT is a r e p r e s e n t a t i v e of t h e c l a s s U U , and t h u s by Th.
11.3
we derive t h a t rn
Next, l e t f be an a r b i t r a r y element of: C
This follows from the extension of Th. operators which we have denoted by constants, i n (11.58)
(03 :+;H)
11.3 t o t h e equivalence c l a s s of
C"
~ ( t , t ' ) * u ( t , t ' )dt. Note t h a t t h e
and ( I I . ~ ? ) , depend on the choice of T
>0
course not on those of uo and f ) . By a v a i l i n g ourselves of (11.58)
(but of Er
(11.50) i n conjunction with Formula (11.45) we obtain: Themen 1I.b.- To e v e r y r e l a t i v e l y compact open subset
< T' < T, that, fm all u & c ~ ( O E+!H), X every number T* such t h a t 0
8
f X and t o
t h e r e i s a constant C
>0
A l l other estimates one might wish follow *om
t h i s b a s i c one, (11.60)
For instance, i n s t e a d of estimating t h e zero-narm of
u(t)
, one
r
can e s t i -
mate i t s s-th norm; one can a l s o estimate t h e s-th norm of t h e t - d e r i v a t i -
<s
ves of u ( t ) (now, below, s* is any number
; C depends on i t s choice):
Suitable estimates a r e a l s o v a l i d f o r t h e t - d e r i v a t i v e s of negative order
(L. g.,
i n t e g r a l s from 0 t o t, i t e r a t e d a number of times).
"Orthogonal projections on the kernel and cokernel" ................................................... ~ ( t ) ,~ ( t ) ,U ( t , t ' )
Here w e l e t
*
arid re l e t U U a c t on
'(x;H)
-
a c t on t h e space of "microfunctions"
$*(X;H)/C*(X;H)
(by using properly
supported r e p r e s e n t a t i v e s of a l l these c l a s s e s of operators). Let us then
*
the inverse of U U I by Th. 11.3 it i s an ( e l l i p t i c )
denote by (u;)"
p s e u d d i f f e r e n t i a l operator of order m. W e then form
(11.6,)
Po
Theorem 11.5.-
1.6)
-
U(LJ*lJ)-'u*
a c t i n g on
cm ((G,T[;
I n the sense of o p r a t c r s on the s
P o = F:-?:
8
.$ * (x;H)). p (11.62) ~
we have:
.
; Po= I = ~ e r [ ~ - A ( t $ ;[ & - A ( ~ ~ O P ~ = O
One may paraphrase t h e equations (11.64)
by saying t h a t Po is an approxi-
mate arthogonal projection on t h e kernel of
a 6t
- ~ ( t i)n t h e space of
microfunctions, (11.62). Lastly, for any g i n (11.62), l e t u s c a l l Egg t h e c l a s s i n (11.62) of f o r some g i n t h e c l a s s g.
the distribution Theorem 11.6.-
With t h e preceding notation s e t E = (I
- P ~ ) E. Then, ~
i n t h e sense of l i n e a r operatars a c t i n g on the s p a 3 (11.62),
we have:
111. APPLICATION TO BOUNDAFtY VALUE PROBLENS FOR ELLIPTIC EQUATIONS
From now on we suppose H finite-dimensional. Ifloreover the order of the operatar ~ ( t )of Ch. I1 w i l l be
one; and
the l e t t e r m w i l l be used t o de-
note- t h e degree of t h e l i n e a r p a r t i a l d i f f e r e n t i a l operator i n
I(
X [0,?[
under study, namely
where, f o r each j
l,...,
m, P . ( x , ~ , D ~is ) a linear partial differential
J operator of order j i n X whose c o e f f i c i e n t s axe C*
i n X X (o,T[
functions of ( x , t )
valued i n L(H) (thus E is a determined system of l i n e a r PDO).
Actually we do not l o s e anything by taking t h e P . t o be c l a s s i c a l pseudoJ d i f f e r e n t i a l operators of order j r e s p e c t i v e l y , i n X, with values i n L(H)
- which we then denote by PJ. ( t ) . can be regarded as a
cm
For each j t h e p r i n c i p a l symbol of P . ( t ) J function of ((x, p ) , t ) i n (T*X \ 0) % [ 0 , ~ ( v a -
lued i n L(H), positive-homogeneous of degree j with r e s p e c t t o t h e f i b r e
3
variable
. Me axe now going t o make a t r u l y r e s t r i c t i v e hypothesis
about our operator P, namely t h a t its p r i n c i p a l paxt i s s c a l a r ; t h i s w i l l cover t h e main cases we s h a l l be i n t e r e s t e d i n ;
.., m,
For every j = 1,.
111.2)
t h e p r i n c i p a l symbol P
0 0 -equal t o p . ( x , t , f )I, where P . ( x , t ,
J
J
5)
(x,t,3) j,O is complex valued.
is
Let u s then s e l e c t a s c a l a x pseudodifferential operator of order j i n X, 0 with p r i n c i p a l symbol, P .(x, t, 3 j S m ) and s e t
depending smoothly on t E (o,T(, 0 c l a s s i c a l # l e t it be P . (1 _<
J
(111.3)
P-IP0+P1,
3), moreover
j-1
< m - 1 (with r e s p e c t
where P1 is n o t n e c e s s a r i l y s c a l a r , but has order
t o both x and t). Of course, i f P is a d i f f e r e n t i a l operator, we may take 0 each P .(t) t o be one. Note t h a t
m
J
(111.4)
P1- xPj(t))?-j j-1
,
d e g ~ ~ . ( t ) $ j - l , l ~ j ~ m . J
We now s t a t e t h e e l l i p t i c i t y hypothesis; it concerns the p r i n c i p a l symbol of P, or r a t h e r t h a t of PO r
+ and
There axe two i n t e g ~ m,
(111.61
m 5 1, such t h a t , polynomial
m-,
whatever (x,
++
such t h a t m
5)E
T*X
\
0, t
m- = m
t (o,T(,
the
0 + C(P ) with r e s p c t t o z has e x a c t l y m r o o t s with
r e a l part ' ,0
and m-
with r e a l p
e
<0
.
Remark 111.1.- When P is a d i f f e r e n t i a l operatar o r , more generally, an antipodal p s e u d o d i f f e r e n t i a l operator, which means t h a t C(P 0 =
+-
0
@ ( P ) ( x , t , - 4 ;-z), and provided t h a t dim X
l y have m+
= m = m/2
(and then m is n e c e s s a r i l y
)(x,t,3
;z)
> 1, we must necessarie). This follows by
a standaxd c o n t i n u i t y argument and the f a c t t h a t t h e complement of the o r i -
gin in
3-space i s connected.
We s h a l l use t h e e l l i p t i c i t y hypothesis (111.6)
t o f a c t o r i z e the prin-
0 c i p a l symbol of P as foLlowss
+
where t h e zk (resp.,
t h e z i ) a r e the r o o t s with r e a l p a r t
>0
( r e s p a ,( 0).
l e a d s t o a f a c t o r i z a t i o n of
We wish t o s h m t h a t t h e f a c t o r i z a t i o n (111.7) the operatca: E i t s e l f , of t h e kind
+
with
c(K-)
?o = I = (PI
)
. I f m+
= 0 t h i s is obvious r we may take
+3
Thus, i n t h e remainder, we s h a l l assume m
:41
=
I,
1. Let us write
+
Although t h e r o o t s ZT; cannot, i n general, be represented as continuous functions of ( x , t , I ) ,
f a r each j the c o e f f i c i e n t
xjo(x, t , 5 ) ) i s a cW function i n ( T * of degree j with r e s p e c t t o
3 . This
0) X
(o,T(
+o
Mj ( x , t , l )
(rasp.,
, positive-honogeneaus
follows &om t h e f a c t t h a t these i
c o e f f i c i e n t s a r e symmetric functions of the r o o t s zk ( r e s p a , z i ) , and
+
t h a t t h e two s e t s of r o o t s zk and z i s t a y a p a r t , as ( x , t , J ) v a r i e s . Now, f o r each j, we may s e l e c t a c l a s s i c a l pseudodifferential operator i n X, s c a l a r , depending smoothly on t d (o,T(, bol ~ Z o ( x , t , j ) , and we s e t
J
(111.10)
+ j=1
we have
with P" of t h e same type as P' i n (111.3).
-k
lkO(t), with p r i n c i p a l syn-
J
W e s h a l l reason by induction. Let us assume t h a t we have found, f a r
& 1, operators
N
2
>I and RN such t h a t (14) +
t
-
(111.12)
M(N)
(111.13)
dnilj(t))
6;
+
+
-
C...,
m-
j=l,
f.
2
+
'(N)j(t)
%
atm - j
9
.., m2)
and t h e
I M j ( x , t , ~ ) ( j = 1,.
2
( t ) axe c l a s s i c a l pseudodif f e r e n t i a l operators i n X, .M(~)j valued i n L(H), depending s m o o t h l y 2 t (o,T( ;
We s h a l l obtain a l l t h i s with N + 1 i n t h e place of N. I n order t o do t h i s we regard
We seek
2
+$ i n
2 AN = Mt(8+1) - M(lq) +
as unknouns and write t h a t
+t h e farm
.( t )
j=l,...,m-
c l a s s i c a l pseudodifferential operatca: of arder j-N depending smoothly on t E
CO,T(.
-
+
with
q,j ( t )
a
i n X, valued i n L(H),
We proceed a s followsr We m i t e t h a t
She p r i n c i p a l symbol of the right-hand s i d e i n ( I I I . ~ ~ ) ,regarded a s an
operator of order m
+ A
~ is A
+ <_ ~ m +
m-
- N,
vanishes i d e n t i c a l l y . Observe t h a t the order of
- 2li 4 m - N.
Consequently we r e q u i r e t h a t
t h a t is t o say, by v i r t u e of ( I I I . ~ ~ ) , 1111.1~)
c(N+O)
~*;)
+
c(N")
By t h e i r d e f i n i t i o n t h e polynomials
Q(&
d4) o 3) ( x , t , g
.
( ~ ( ~ 1 ) ,e)
(with r e s p e c t
t o z ) a r e coprime, ,and uniformly s o as (x, 5 , t ) remains i n compact subsets of (T*X \ 0)X Q,T< pect t o
( 5 ,z)).
(observe t h a t Eq. (111.19) is homogeneous with r e s -
By t h e c l a s s i c a l Beeout's theorem,
~ (+ 4 )O(G) and
axe uniquely determined .polynomials i n z , of degrees
m -1, m--1
res-
pectively, with c o e f f i c i e n t s which axe smooth functions of ( x , j , t ) ( t h e with r e s p e c t t o
degree of homogeneity of these symbols and m--1
r e s p e c t i v e l y ) . It only remains t o s e l e c t
(3,z)
far-+.2 ,J-. ( t )
+
i s m -1
classical
pseudodifferential operatars I n X, valued i n L(H), depending smoothly on
t , with t h e appropriate p r i n c i p a l symbols ( s p e c i f i c a l l y i f ( I I I . ~ ? ) is taken as d e f i n i t i o n of R t i e s from (111.12) t o (111.16)
+
1. J. d z j @ ( ~ ) \ ~ = ~ ) r
we s h a l l have a l l the proper(~+1)
s a t i s f i e d with h+l i n s t e a d of 14. Since we
know, by ( I I I . ~ ~ ) t, h a t they a r e s a t i s f i e d when N = 1, we can go t o the l i m i t f a r W a r b i t r a r i l y l a r g e , namely N = + w (indeed, t h e operators
+ M
obviously converge). A t t h e l i m i t , r a t h e r than w r i t i n g N = + w , we
(N 1
s h a l l omit the s u b s c r i p t s N. Thus (111.8) is s a t i s f i e d . Actually we s h a l l go one s t e p f u r t h e r and divide R by PI- r (111.20)
,
R-QM-+R'
R'
- C.. j=l,.
,m
-
A3(t)d;'j
;
Tnis poses no ~ o b l e m rit i s j u s t a question of replacing a l a r g e eerlough n u d e r of times dropping the primes
+ W +
'
by K-
-
j
,
,
m
((t ) d
f-'. Finally +
i n the notation f a r R ' and s u b s t i t u t i n g W
(after for
Q) we obtain the decomposition
+ 20 with N- of the same kind as M (see (111.10)) (111.20)-(111.21).
and R the same as R ' i n
Now suppose t h a t we a r e dealing with t h e equation
By (111.22) we see t h a t it is equivalent t o t h e system of (two) equations
Let u s consider a t y p i c a l equation of the kind
,
(111.26)
II=~;+
C...,
j31,
,
cj(t)a;-j
r
where, f o r each j, C . ( t ) is a pseudodifferential operator i n X of order
J
-
j with values i n L(H), depending smoothly on t. Let u s s e l e c t once and f o r
A
a l l an e l l i p t i c pseudodifferential operator we take
of order
one i n
X, s c a l a r ;
t o be c l a s s i c a l , and properly supported. We s h a l 1 , f u r t h e r -
more r e q u i r e t h a t t h e r e be a ( c l a s s i c a l e l l i p t i c ) pseudodifferential oper a t o r of order such t h a t
- 1 i n X,
A A-'
properly supported, which we denote by
= I d e n t i t y of
$ '(x)
.
A - ~and
Let us not worry whether such
an operator does e x i s t r When t h e manifold X is compact, which is t h e onl y case t h a t t r u l y i n t e r e s t s us, it c e r t a i n l y does; Equip X with a Riemanniam metric, l e t
Ax denote
t h e Laplace-Beltrami operator on X for t h a t
A x is
metric (we s h a l l suppose here t h a t
h=
(1
- A,
)'
. (when X i s compact,
i s void, and B ' ( x ; H ) =
,( 0) ; a possible choice i s
the requirement " p o p e r i y supported"
&*(x;H).)
,..., r, wJ
Let us then s e t , f o r each j = 1
+
denote by W
t h e r-vector with components w
a smooth function of t with values i n a function but valued i n
' (x;H@
kT).
1.
=
A'-' d"w !
., ...,w
r
; we s h a l l
. Each component i s a
' (x;H), hence k' i t s e l f i,iote t h a t we have r
is such
, we
If we multiply both members i n Eq. (111.26)~ by
may r e w r i t e
it in t h e following manner t
We observe t h a t t h e "coefficients*
#
cj(t) =
pseudodifferential operators of order
one i n
me
X, valued i n L(H), depen-
ding smoothly on t. We gather t h e equations (111.27) & (111.28) together i n a s i n g l e system,
dt?
(111.29) with
TI
equal t o
-b 4 - m(t)~ = G ,
t h e r-vector with compqnents a l l zero, except the r - t h one,
nlerg,
and where
O f course we view
m ( t ) is t h e r X r matrix
( t ) as pseudodifferential operator of order
one
i n X with values i n L ( H @ ~ ) . A standard and important remark i s t h a t
i t s p i n c i p a l symbol
where we view smoothly on ( ( x ,
@ ( M ( t ) ) is such t h a t
d ( W ( t ) ) as a matrix over the r a g L(H)
5 ), t )
*
Q (T X \ 0 )
of degree one with r e s p e c t t o
, and
X (o,T(
, &epenc?ing
positive-homogeneous
5 . Thus t h e determinant d e t i s computed
i n t h a t r i n g (although the r i n g L(H) is not commutative i f dim H
> 1,
t h e computation of t h a t determinant i s made easy by the f a c t t h a t a l l rows,
except possibly the last one, i n
6'
(m( t ) ) ,
a r e scalax mul-
t i p l e s of t h e i d e n t i t y of H ; a t any r a t e , i n t h e application of what a l s o the last row w i l l be a sca-
precedes t o EQq. (111.24) and (111.25)
lar .multiple of t h e i d e n t i t y ) . F i r s t we apply . t h e preceding transfarmation t o Eq. (111.24); i n t h i s A 1-j j - t j 1,. , m and s h a l l denote case r = m g we s e t uj =
-
-a
by u t h e m--vectar
. .
AlmJdtJ-lv,
,
dt
.. -,
with components uJ. Let us s e t r i g h t away vJ =
j ==, I,.
.., m+, and c a l l -+v t h e vector with components v j . .
whose components a r e a l l zero, except
Ye then denote by Jv t h e m--vector
A
t h e l a s t one, equal t o
1-m-
v. With t h i s notation Eq. (111.24) reads
According t o (111.31) we have (cf. (111.10) d e t Zz1
& (111.22))
t
- ~[~-(t)ll
= C(N-~)(X,~,~,Z)
where t h e determinant is now computed i n the complex f i e l d (we have t a ken advantage of t h e f a c t t h a t t h e p r i n c i p a l symbols of the operators ~:!t)
"J
a r e s c a l a r multiples of t h e i d e n t i t y of H).
On t h e other hand we note t h a t , according t o (111.20),
Ue may then denote by
s?
+
the m -vectar with components a l l zero,
+
except t h e last one, equal t o 1 1 1 )
Ru. It i s c l e a r t h a t
3 i s a r e g u l a r i z i n g ~ r a t o-ri n X,
-
on t E (o,T(, -C
valued i n L(H@C" 8H 8
depending smoothly
+
ern
).
+
We now denote by g t h e m -vector with all components equal t o zero, except t h e last one, equal t o
+
Aa-".f.
Thus Eq. (111.25) reads r
(111.37)
d e t {ZI
- tY[~+(tg 1
-
5
6 ( ~ + ~ ) ( x , t ,,z)
.
The r e l a t i o n s (111.36) and (111.37) show t h a t t h e eigenvalues of A-(t) and those of roots
Z;
, and
the
- ~ ' ( t ) s t a y i n the + of opposites - r k'
open half-plme C- (they a r e t h e polynomial c(E'O)(x,t,j ,z)).
+
It ought t o be underlined t h a t we axe viewing, here,
+
matrices of s i z e m-X
+
m-
d (AA(t)) a s
. To have them as symbols valued
2
of l i n e a r mappings H @ cm
2 -
4
H @
, one must
i n t h e space
then tensor them (on
t h e l e f t ) with the i d e n t i t y of H. We may thus s t a t e t h a t (111.38)
t h e basic hypothesis, ( I I . ~ ) , of Ch. 11, is v e r i f i e d by A-(t) and by
- A+(%).
The next s t e p is t o adjoin " i n i t i a l conditions" t o Bq. (111.23);
Here h .
J
(111.40)
o$*(x;H)
f o r each j, and
3
B ~ ( X , D ~t ), =
where, f o r each choice of j
-
+o,.
..,dj
.. , k
1,. , J
-
a pseudodifferential operator i n X, valued i n
The
.., d
1,.
, Bj, k ( x , ~ x )is
L(H).
a
first t h i n g we do is t o divide each B = B j ( x , ~ , ,) by
3
P, using the f a c t t h a t E i s a monic polynomial with r e s p e c t t o
Note t h a t t h e degree of B! a s a polynomial i n J We then replace t h e conditions (111.39) by
at
:
at does not exceed m - 1.
Xext w e divide
E'. J
by 3- r
Xere not only i s the degree of now
degQj
<
deg B'.
J
$!J < m--1
(as polynomial i n a t ) but a l s o
- deg N- zm - m- - 1 = n+ - 1. By v i r t u e
of (111.24) we nay r e p l a c e t h e conditions (111.42) by
+
Let u s denote by h# t h e
..,Y ),
1,.
by
9 -vectmwith
-
( Q j f ) ) t=O ( j = j ~ ( 0 t)h e one whose components are (Q .v QO The f a c t
3+
J
t h a t degree ( i n a t ) of Q . is
J
regarded as a
components h
3
)(
,< m+
)I
- 1 implies t h a t 2
+
m matrix, s i n c e
.
may indeed be
+
F i n a l l y we may r e w r i t e t h e i n i t i a l conditions (111.39) i n t h e manner
63 $0)
(111.46) uhere of
d3
-$- 2
30)
,
is t h e pseudodifferentidl operator i n X, valued i r i the space
V X m- matrices with e n t r i e s i n L(H) defined as follows: I f one
mites
-
we have
Consequently, i f
8.~ * (kj a l , . .., V ,kl,...,m-)
i s a generic e n t r y of
we have
Let us summarize what we have done so far i n t h i s chapter:
3
The system of equations (111.23).
(*>
PU
-
(111.39)
t
B ~ ( x , D8~t,) ~ l + o = hj
t.
(Xjs9
has been t r a n s f armed i n t o t h e system. (111.32).
(**I
at? - A-(t)? = J?
(**I
JtV A + ( t ) b = +
-
,
a0
1,
(111.36)~ (111. '6)1
- 20
=
i
2-Rh.
Let u s emphasize t h e f a c t t h a t these equations a r e exact ( i n s o far as
r\ A
= I exactly, which is possible t o achieve when X is compact).
Nevertheless we have n o t q u i t e succeeded i n lltriangulazing" t h e problem we have only approximately done so, s i n c e Eq. (***) s t i l l contains
*;*. (*)
t
But we s h a l l s e e t h a t the above transformation still enables us
t o analyze some impcrrtant aspects of (*). Flrst of a l l l e t u s point out t h a t t h e r e is a one-to-one
corresponden-
c e between s o l u t i o n s of (*) and s o l u t i o n s of (**)-(***): t h e argument i n t h e preceding pages has shown how t o go from u to
d E cm ((o,T[;.&(xIHBE~
-
e
cCd((O,T[;
4 * (x;H))
+
)), ?C c ~ ( ( o , T ( ; B * ( x ; H ~ P P)). Conver-
+
s e l y one can go from .u t o u simply by taking t h e l a t t e r t o be t h e f i r s t component of t h e fcrrmer. However,it might be as d i f f i c u l t t o s o l v e (*)-(***)
exactly as it
i s t o solve ( d i r e c t l y and e x a c t l y ) (*). W e s h a l l r e p l a c e (**) by the following approximation:
and modify a l s o (*) (111.51) (111.52)
-
accardingly: A
= J
X
x
(O,T(
,
-*
~ $ ( O ) - h ~ - 2 ~ ( 0g) X .
This modified problem we can easily solve
- provided we strengthen
+
a l i t t l e b i t our hypotheses on A-(t) and on f , and by way of consequence on
: s p e c i f i c a l l y t h a t we assume t h a t
all t h e s e functions a r e smooth,
with r e s p e c t t o t , i n t h e closed i n t e r v a l (0,~)
. This is of no s e a t i m -
p o r t i n t h e a p p l i c a t i o n s ( i t can be achieved by s l i g h t l y decreasing T). Since
- ~ * ( t )s a t i s f i e s t h e b a s i c assumption
(11.5) we can solve t h e
backward Cauchy problem f a r Eq. ( I I I . ~ ~ ) ,s t a r t i n g at t
v r i t e ( f o r an a r b i t r a y choice of G ( T ) c
8' (x;H))
-
T, and thus
-4.
where U ( t , t ' ) is t h e r e l e v a n t parametrix. We can then put t h e s o l u t i o n -b
v# of (111.50)
thus obtained i n t o (111.51)-(III;~~)
and solve the farward
Cauchy problem, s t a r t i n g a t t = 0 (cf. (11.45));
where U - ( t , t' ) is t h e p a a m e t r i x for (111.51)-(111.52).
I n doing, t h i s w e
have taken advantage of t h e f a c t t h a t A-(t) s a t i s f i e s (11.5). O f course, once we use t h e pasametrices f o r t h e s e various Cauchy pro-
blems, we only obtain approximate s o l u t i o n s
- modulo e r r o r s t h a t involve
r e g u l a r i z i n g operators a c t i n g on the various data. The question is then t o go from t h e s e s o l u t i o n s t o s o l u t i o n s of (*)
&
(***), a d from t h e r e
t o s o l u t i o n s of our o r i g i n a l problem (*). Here we s h a l l focus on t h e r e g u l a r i t y of t h e s e s o l u t i o n s , and show how t h e r e g u l a z i t y of one s e t of s o l u t i o n s determines t h a t of the other.
- Let u, -b
Lemma 111.1.
-*
v be s o l u t i o n s of (+*), <+-) where we assune t h a t 4
4
g belongs t o C * ( C O , T ) ; ~ ~ ( X ; H @1). C ~and l e t u#
Then
(111.53)-(111.54). (111.55)
-*
- v# -*
E c*(xF@,T<;
+
ii@gm )
.
a
v#
.
be defined by
-C
If moreover we assume t h a t u (0)
#
- ?(o)
-
f
C*
-
.
d- 5 E C ~ ( X X [ O , T ( ; Had" )
(111.56)
(x: H @em) , then
- u#
- * - *
-*
fioofr It is very simpler it s u f f i c e s t o observe t h a t ul = u 4 -+ v s a t i s f y t h e equations v#
-b
, vl
=
-
where
++-*
a r e l i n e a r combinations of u, v , ul
and
, vl*
with c o e f f i c i e n t s
which a r e matrices, of t h e a p p r o p i a t e s i z e , with r e g u l a r i z i n g operators
as e n t r i e s . We apply then Formula (11.45)
-*
t o ul
, and
t h e analogous
where the various R's s t a n d for r e g u l a r i z i n g operators, depending smoothly on t or on ( t , t l ). F i r s t (111.55) follows at once *om
+
c a l l t h a t U ( t , ~ )is regulaxizing &s soon as t account i n (111.59)
< T.
(111.60)
i f we r e -
Taking t h i s f a c t i n t o
-+
we deduce (111.56) i f we assume t h a t ul(0) is c m i n X.
Definition III.1.- The problem (**)-(***) i s s a i d t o be hypo-elliptic i f given any o p n subset Y
of X,
any d a t a
3 ' (X;Z@E J ) whose r e s t r i c t i o n s t o Y (111.61)
+
+
ZEC ~ ( ~ O . ;T~)' ( X ; H@ ern )),
a r e smooth,
& cm ( Y X < O , T ~ : X B E ~) ,
5E
i.
=.,
C ~ ( Y ;@ H c Y ).
(;,a of (**)-(***),
every s o l u t i o n (111.62)
?E
-
cm ( ~ o , T ) ; ~ ' ( x ; H @ c)),~ f f
i g i n f a c t smooth i n Y 111.6
such t h a t
far t
T, &,
e.,
+
ern ( ( o , T ) ; ~ ' ( x ; H B c ~)), +
?EC~(Y;H@E~-), ?EC~(Y;H@C~).
%ark
111.2.- There would be no gain i n g e n e r a l i t y i n r e l a x i n g Condition
(111.62)
and allowing
and
?t o
be d i s t r i b u t i o n s i n t
. Indeed,
by a stan-
-*
dard argument, our hypotheses on g and the equations (**)-(***) themselves would automatically imply (111.62)
(one could allow g not be smooth with
r e s p e c t t o t outside of Y , but what counts is its smoothness i n Y ) . Remark
111.3. - The h y p o - e l l i p t i c i t y of Problem (**)-(***) is obviously
equivalent t o t h a t of our o r i g i n a l problem (*) (defined i n evident manner). F i n a l l y we r e c a l l t h a t the operator
i n X is 'said t o be hypo-ellip-
t i c i f it preserves t h e s i n g u l a r supports. Remark 111.4.-
I f one p r e f e r s t o consider "wave-front s e t s " hypo-elliptici-
t y ( c f . Def. 1.8), namely t h e property t h a t t h e operator under study preserves t h e wave-front s e t s , and n o t merely t h e singular supports, it should be s a i d t h a t a l l t h e statements i n t h i s last p a r t of Ch. 111, i n p a r t i c u l a r Th. 111.1 below, have t h e i r counterpaxts v a l i d f o r t h i s concept. 'Theorem 111.1.- The problem (**)-(***) i s hypo-elliptic i f and o n l y 3
& &X
t h e s e u d o d i f f e r e n t i a l operatw
-i-' T ~ c f :
Suppose f i r s t t h a t
@
-PO-elliptic.
I s hypo-elliptic. By the f a c t -that gseudo-
+
. l i f f e r e r . t i a l o p e r a t a r s , here I; ( t ,t ' ) , w e pseudolocal, we d e r i v s from (111.54)
-+
t h a t vd
rizing for t
< T).
i s smooth i n 1
* (o,T(
+
(remem5er t h a t U ( t , ~ )is regula-
By t h e f i r s t p a r t of Lercma IX.1 we conclude t h a t a l s o
? i s srr.ooth i n Y X (o,T[
-t
; i n particular v(0)
+
6
C * ( Y ; X @ C ~). We take t h i s
a
i n t o account i n the r e l a t i o n 30)G
-
cCO(Y;H @ern ).
-
with ;#(O)
0
- 2 ?o),
It s u f f i c e s then t o apply t h e last p a r t of Lemma 111.1
&
i n X, valued i n H @
not hypo-elliptic:
t h e r e i s then a d i s t s i b u t i o n
, whose r e s t r i c t i o n
t o sone open s e t Y C X is not
is
ern
-
63 6 is cm , S e t ?(to
c m b u t such t h a t the one of
Note t h a t v? (111.65)
--
atwJ
-
urn = w
-
m
/\wjfl
<
if j
m-
. Ict
-
i n X X (0, T) with values i n H @ Cm
X
denote t h e l a s t
-
and define v = s e t t i n g vj =
Am
i
= U - ( t , 0 ) u 0 ; then:
define inductively
US
,
By (descending) induction on j and by (111.64)
Let then
and d e r i v e t h a t
= u(0).
Suppose now t h a t
+ u
?(0) = h#
- A-(t)u,
m-th) component of dtu
-'x, an6 the m+-vector
-
- 3w is cW
. We have r
(2.e . , t h e
A'-' a J-lVv. ~ q .(111.32)
3
we s e e t h a t u
?=
(v
+ ,... ,v m )
1
as before, by
is automatically s a t i s f i e d . Since
[at - ~ - ( t ) l ? v", + [d, - ~-(t)_j(l: - W) , -$
+
we s e e t h a t ?C c ~ ( x % ( o , O ; H @P ). We then s e t
we have
Z GcW(X X (o,T>;H
b u t (191.63)
-+
H @C
i s not.
v
- A+(tj?+R;';
3
3
@ E' ). On the other hand, z ( 0 ) = x(0)
By v i r t u e of our hypothesis about mapping Y
;=[dt
. Thus
8
(111.61)
<
, * h# = @ % +
2?(0)
and, of course, (111.62)
3
= uo
is
.
acW
ase t r u e ,
3 . CUEX3CIVE BOUIUJARY VALUE FROBLENS
Coercive problems a r e a p a r t i c u l a r case of t h e type of problems (*):
I n t h e present chapter P w i l l be tine same e l l i p t i c operator a s i n Ch. 111. 3ut f o r t h e sake of s i m p l i c i t y we s h a l l assume t h a t dim H
=
+ 1, i n other
words t h e operatar P w i l l be s c a l a r . Coercivity is c h a r a c t e r i z e by two conditionsr
3
( 1 ~ ~ 1 )
, the
number of r o o t s of t h e p u n o m i a l i n z , real p x x ~ I V 2. )
<0
3
q ' ( 9 j ) ( ~ , , a ) of the boundary operators
the p r i n c i ~ a lsymbols
a
d ( ~ ( x ,)t , 3 ,z), uith
;
3 ( x , ~ ~ t ,) ( j
-
I,.
.. , 9.1 a r e l i n e a r l y independent modulo
5,~)
~ ~ H - ~ ) ( X , O , (see ( I I I . ~ ) ) , whatever
It should be noted t h a t t h e p r i n c i p a l symbols homogeneous functions of
-a p-r i a r f ,
at)
J
by
t y degree with r e s p e c t t o
(x.3) & T
* . .L
\ 0
.
6 ( 3 j ) ( x , 3 ,z) are positive
( 5 ,z) of
t o t h e i r degrees, d
ply each B .(x,P,
, the
number of boundary conditions, a q u a 1 t o m-
j'
r e s p e c t i v e degrees m (with no r e l a t i o n , j as polynomials i n z ) . Actually we may multi-
/\ -'"j and
( 5 ,z)
assume henceforth t h a t t h e i r homogenei-
i s equal t o zero
- whatever
j. This i s
u h a t we s h a l l do.
s i n c e , by ( I V . ~ ) , nem bns.is --
9
degz d(h'O},
we s e e t h a t t h e
~
( .) 3form a
J
3-
of t h e space of polynomials i n the v a r i a b l e z modulo ~ ( K - ' ( o ) ) .
ue ;nay e f f e c t t h e d i v i s i o n of
r i t h degz b .
J
m
~ ( B ~ j ( x ,. z3) by
v ( l . i - O ) ( x , ~ ,,~z j
(as
PO-
- - 1. The p o l j c o m i d s b . form a b a s i s of t h e v e c t a r space J
of polynomials i n z of degree
.
< m-
Now, i f we go back t o t h e division formulas (111.41)
8 ( I I I . ~ ~ ) ,we see
. 5
t h a t , for each j, b (x, , z ) is the p r i n c i p a l symbol of $?(0), J J say, according t o (111.LC?),
t h a t is t o
-
(IV .4)
I-'
b j ( x , S ,z)
k-0
5
d<,k)(x.O,
k
Because of t h e homogeneity of degree zero of b . with respect t o
J
see t h a t pect t o
5
d ( < , k ) ( x ; ~ , ~ ) is positive-homogeneous of degree
&(@j.,k)(xs
we obtain t h a t with r e s p e c t t o
we
- k with r e s -
-
therefore, i f we s e t , a a i n ( I I I . ~ ~ ) ,(BjPk
, and
(5 , z )
$,k-l(~)~k-l,
) i s positive-homogeneous of degree zero
3 . It is then c l e a r t h a t t h e coercivity condition
(1v.2)
is equivalent t o t h e property t h a t (1~-5)
The vectors
P J.(x,
J)=
( 6 ( 6 3 j , k ) ( x p 5 ))k=l,,,,,r)
are linearly
independent, whatever (x, o r , i n other words,
m-sm-
(1v.6)
6(d3)(~,3) = ( b ( @ j , k ) ( ~ ,)~) j , k l , , ., whatever (x, 3 ) T*X 0 .
matrix
is i n v e r t i b l e ,
Since we can r e t r a c e our s t e p s and c l e a r l y go back kom (1v.6) t o (1v.2) we may s t a t e l Proposition I V . l . if3 -
= m-
- The boundary value problem
and t h e boundary operator
ce of t h e operators B j in
,63
@
(*) i s coercive i f and only
is ellip=
(with t h e present choi-
is a pseudodifferential operator of order zero
x).
By combining t h i s with Th. 111.1 (and Cor.
1.5
& renark following it)
we obtain:
- If
Theorem I V . 1. Qpo-elli3tic.
t h e boundary value problem (*) is coercive, it is
I n t h i s s e c t i o n we i n d i c a t e b i e f l y how t o d e r i v e t h e standaxd coercive estimates from Th. 11.4. We assume throughout t h a t t h e pseudodifferential operators B
j
(jil,.
..,m-)
and
& axe
of order
(
& is
e l l i p t i c ) . Ue
continue t o reason under the same hypotheses as i n the first p a r t of t h i s
+
-
chapter, i n p a r t i c u l a x we s h a l l assume t h a t a l l data, A-(t), i n the closed i n t e r v a l (0,~).
g ( t ) axe
cCD
We s h a l l use Sobolev norms on d i s t r i b u t i o n s
i n X without d i s t i n g u i s h i n g where t h e values of those d i s t r i b u t i o n s l i e . , Ne s h a l l consider an a r b i t r a r y function belonging t o: C
(xx(o,T()
(thus
> To where To is some number < T) which we denote by u, and d e f i n e f = Eu, u = B ~ ( ) , (j = 1,...,m-). We a l s o define it vanishes for t
+
-3+
the vector-valued functions u, v as indicated i n Ch. 111. I n what follows C w i l l be a constant independent of u (and t h e r e f o r e of To< T).
by applying (11.61)
(keeping i n mind t h a t what was denoted by m i n Ch. I1
i s now equal t o one, whereas m denotes now t h e degree of
Actually, when k = 0, ( I V . ~ ) follows from (11.60). t h a t J? =
at? - ~ - ( t ) ? . Bow,
s e r o , we g e t
Gn the other hand.
We begin
using t h e f a c t t h a t
I?)%
k'e have used t h e f a c t
@
is e l l i p t i c of order
We apply (111.46):
-*
+ - 3 ?(o)
u(0) = hd
. By
(111.41)
and (111.43)
we
s e e t h a t B: and E f axe pseudodifferential operators i n ( x , t ) o f d e s e e zero,
J
J
like B
. Therefore the j
Q . a r e of order
J
- m-,
and s o i s
n i t i o n of s ' we dkrive from t h i s , and from (IV.8),
3 . After
a redefi-
(1V.9)~
We apply the andogue of (IV.'~) with ~ ( 0 s) u b s t i t u t e d f a ~ ( 0 and ) s' =
s
+
k
-
m- -1/2.
W e get
We may d s o apply t h e analogua of (TV.7)
-*
3
for v instead of u , keeping i n + 1 mind t h a t we must then exchange 0 and T and t h a t V(T) = 0 (since u fT) = 0 and v1 = X-ul):
- m- f o r s i n conjunction 2 - 3 ? , a ~ tdh a t 311 compo-
We f i r s t apply (1V.12) where we s u b s t i t u t e s + k with (1V.11). 3
lie note t h a t
dt? -
~~'(t);*=
nents of g a r e zero, except the l a s t one, equal t o
~ l - ~ + fU ' e. obtain8
We put t h i s back i n t o (1V.10) and from t h e r e i n t o ( I V . ~ ) , where we now
s = m
-k
!and r e d e f i n e s ' )
t
+
We have used t h e f a c t t h a t a l l components of Jv a r e zero, except t h e l a s t one, equal t o
-
r\l-m vl,
and a l s o t h e f a c t t h a t m
a$1
t h e property t h a t , i f j 4 mf. k remain
+
<_ m + 1, and j s k
We apply once again (1v.12)
= ='vj+'.
- m - = m+ . Now we use
We obtain, provided t h a t
- 1,
with
2 = 0,
1. We see t h a t t h e right-hand s i d e
i n (IY.15) does not exceed a constant times m
Finally,
we r e c a l l t h e d e f i n i t i o n of
2 and t h e f a c t t h a t
for a s u i t a b l e s" (which can be a s close as we wish t o -a i f we choose
- s'
l a r g e enough). W e &so use t h e f a c t t h a t
t h a t if k v a r i e s from 0 t o m
++
1, and j *om
4~4 j+l = 0 to m
~j dlc+jul t
- - 1, then
j+ k
v a r i ~ sfrom 0 t o m, Ue obtain t h e coercive estimate3
fro^ this we
can e a s i l y derive t h e standard versions of t h e coercive
estimates. For instance, i f we replace u , by
A Su
we e a s i l y derive t h e
r"
' )($tu
estimate of
d t (which we could a l s o have derived direc-
0
t l y by s u b s t i t u t i n g s
+
m for s
+
k i n (IV.~)). W e could a l s o derive t h e
"local" coercive estimates, by taking T > 0 small enough and using the standaxd inequality
( x X [o,T()*
The l o c a l estimate is t h e v a l i d f o r all u E: C
--C
One may a l s o want t o "unravel" t h e composite vector hff,
. The most favora-
b l e circumstances i n which one can do t h i s is when t h e degree, as a polynomial i n (111.41))
at
-
, of
every B . ( X , D ~8, ) does r?ot exceed m 1. Then (cf. J t -* t h e Q. a r e i d e n t i c a l l y zero, and h# a (ul, ,u,, ). We might a l s o
...
J
want t o allow t h e orders of t h e 33
j
, as pseudodifferential
operators i n
( x , t ) , . not t o be necessarily zero; t h i s is usually the case when they axe d i f f e r e n t i a l operators. Let m . be t h e order of B ; then what has been deJ j noted by B. should now be denoted by A'mj~ Under these hypotheses
J
5
.
Estimate (1v.17) can be r e w r i t t e n
A s befure s' i s a r e a l number a s close t o -ooas one wishes (C depends on t h e choice of s' ). One might a l s o have estimates f o r d e r i v a t i v e s of negative order with r e s p e c t t o t. Etc, e t c . On t h e subject of coercive estimates and t h e i r use t o r e f i n e t h e measurement of hypo-ellipticity we r e f e r t o [9],
Ch. 2.
A s a f i r s t example of noncoercive boundaxy value problem we s h a l l descri-
f2 of eN, which we assume
be now t h e b-Neumann problem i n an open subset
(I.=., c*)
t o be bounded and have a smooth
boundary, X. Our aim is t o ob-
t a i n some information about t h e "boundary operator"
6 (see
(111.49)).
This information w i l l enable us t o apply Th. 111.1 and derive from r e c e n t r e s u l t s about pseudodifferential operators with double c h a r a c t e r i s t i c s
n
t h a t , under a s u i t a b l e hypothesis, t h e x-ileumann problem i n
i s hypo-el-
l i p t i c . However we s h a l l not motivate t h e i n t e r e s t i n t h i s boundary value problem; we s h a l l not i n d i c a t e how its hypo-ellipticity can be used t o d e r i ve various r e s u l t s about holomarphic functions of s e v e r a l complex variables. On t h i s s u b j e c t we r e f e r t o [8
].
Let us describe t h e notation used i n t h e present chapter. The r e a l coord i n a t e s i n El2* dl1 be denoted by
5,...,%,yl,.
-
fl
..,yN
; we s h a l l m i t e
-
..,,
= xj + y j ( 1 ~ j s N ) . and e (el, z N ) , x = Re z , y I m z (note j t h a t x is a point of lRN, i d e n t i f i e d with t h e subspace yl =. .= yN 0 of
z
ZY R ' ). W e s h a l l denote by
a
-
.
C
aJ dzJ a f o r m of type ( o , ~ ) , say with \J\==q smooth c o e f f i c i e n t s : t h i s means t h a t t h e aj a r e functions i n R~ (or =
ern
i n some subset of R
2N
) with values i n E, J ranges over t h e s e t of ordered
multi-indices of length q: 0 ( q i f q & l , .T=
,,..., jQ ) with
(j
; i f q = 0,
l<jl<
...
a is a complex function;
< ' j (N.
Q
pis a
If
I? of s c a l a r functions and Y(tsome$ subset of 4: we denote .by
t h e space of t h e f m s a a s above with c o e f f i c i e n t s
.
$ = cm, c:, 9 cO, e t c . ; w e
aj E.
space
3 (M) 3 ( ~e.)g.:,
may even extend t h i s t o d i s t r i b u t i o n s
i f i t makes sense (depending on t h e nature of
;
can be
n,
its
, the -
closure
3
operator
i
H , e t c . ) . We r e c a l l t h e d e f i n i t i o n of the
whole space C
(m) -P
AO*~C*
A ~ ' ~ ' ~ c (~q <( N~) )r
We may use t h e canonical b a s i s dx E, dyJ (I, J a r e ordered multi-indices I of various lengths) i n t h e e x t e r i o r algebra over t h e dual of R' 2
t h e hermitian p o d u c t and norm of L (R
z
2N
C
t o extend
) t o d i f f e r e n t i a l forms on R~
r
their^'
,
fI,JdxInd~J. 9 d5ndyJ P I*J crr * IpJ inner product is (f ,g)O = (we assume t h a t t h e f I , gI, dxdy I'J c o e f f i c i e n t s f I , , gI, axe,squaxe integrable).. With t h i s d e f i n i t i o n , i f f =
-
J
i f a is a form of type ( o , ~ )as above, and p =
-
C
P J t dzJ.
15'1 =q'
is a
form of type ( o , ~ ' ) , both with square-integrable c o e f f i c i e n t s , we see that
= 0 if q
p
q'
, whereas,
( i n all t h i s dxdy = dxl.. .d5,,dy1.. .dy
Id
2i.j is t h e Lebesgue measure i n Ei ).
Xe may then define t h e formal a d j o i n t of
~t i s a l i n e a r operator '$'
0.3) where U =
S a - - 2
7 for
2 inner product (v.2).
the L
,/\O*q~m(m)
I:
IJ1-q-1
( X I
t
C..
kl,. ,I4 k #.?*
E 2~ J * &
d; J '
*
itz,
. In t h e summation a t t h e r i g h t i n (1.3) J is junction of & J ' t o J * and reordering; &ye + 1 or - 1
IJ 1=p
obtained by ad
i
,
i f q = p'
a, dFJ
k
=
a c c o r d b g t o t h e p x i t y of t h e permutation t h a t brings t h e s e t
fk]
CI J '
t o i t s ordered form, J ( t h i s notation w i l l systematically be used from now on). Ths d i f f e r e n t i a l operator
a
=
a+ 67
i s c a l l e d the
I$ complex Laplacean i n (E ; an easy computation shows t h a t (with a a s above)
('5
o a - - L+ aaJazJ,
.b)
\J\=q
A
where
A
is t h e or&inaxy h p l a c e operator i n R",
= I*
rh2 ) 7.
j-1 a z .3z
J
r e a l valued function r i n C'
Ve s e l e c t a C"
such t h a t X is defined by
r = 0 , w i t h d r f 0 i n a tubulas neighbarhood of X, and r The pxincipal symbol
G ($)(z,
5 ) of 9. is
ern X
ti2')
' ~ ' ~ 9+ 1 ~ ~/\O'q-l~N
a s d r ( z ) , we obtain a C*
r\O'q~q(~2N) i n t o
a C*
with
(AO'~E'
i s t h e obvious subspace of t h e complex e x t e r i o r algebra w e r R f o r e , i f we s u b s t i t u t e f o r
n.
<0 in
a smooth function i n t h e
cotangent bundle over %i2N(which may be i d e n t i f i e d with values i n t h e space of l i n e a r mappings
3
2N
). There-
s e c t i o n of t h e cotangent bundle, such
section of t h e bundle of homomorphisms of
A ~ ' ~ - ~ T * ' ( S ~(T ~*')
stands f o r t h e complexified co-
tangent bundle). Since all bundles d e a l t with here a r e t r i v i a l we may identify
b!-&)(z;dr)
with a matrix (0. ) J'
Vl-q, \J'\ q-1
, which,
by ( ~ . 3 ) , is
seen t o be t h e following one:
Definition V . l . given a form f
!v.$)
The 7 - ~ e u m a n n problem 6 /\O'PC~ (11)~ another
n u s finn. = 0
in
fl is the
problem of finding,
farm u 6 A O ' % ~ ( ~ v e r i f y i n g
on 3R ,
('5.7)
u(&>(z,*)u
(V.8)
~ ( Q ' ) ( Z , & ) ~=Uo O"
an.
We could have vaxied t h e r e g u l a r i t y requirezents. on f , m d for instance
look at t h e case where f f , i s C~
i n t h e closure
res that
(v.?)& (v.8) w i l l
A ~ * ~ L ~ ( Q5 u) t. we shall assume, i n f a c t , t h a t of
n. The requirenlent t h a t u i s c2 i n
insu-
have.a reasonable meaning. Me s h a l l a c t u a l l y
seek t o sbow t h a t u is cm up t o pieces of the boundary
an where t h i s
i s s o f o r f (we s h a l l have t o make some hypotheses i n order t o obtain t h i s ) .
BY v i r t u e of
(v. 5 )
we may r e w r i t e (V. 7) as follows (when q ? 0) 8
where J is t h e nulti-index o3tained by adjoining j
to J'
reordering (uJ is the corresponding c o e f f i c i e n t of t h e form u). A s f u r (v.8) it can be r e w r i t t e n as follows, provided q is ) 0,
(v.10)
When r
= 0, f o r a1 multi-indices J ,
J = q , we have
where J* is derived from J by adjoininp; k end reordering,
*
K = J \Ijf.
When q = 0 ( i n which case u is a scalax function), (v.8) r e a d s
with
(v. 12)
- If q = N ,
Condition
R e n ; z k V. 1.
t o every J ' of length N
- 1 there
(v. 8)
i s of course void. Furthermore,
is a unique j
reads
Since dr f 0 when r
0, t h i s is equivalent with
J'
, and
thus
(v,?)
Since i%q. (v. 6) according t o (v.4) reads
we s e e t h a t ,
when
q = N , the'--ijeumann problem reduces t o t h e D i r i c h l e t
problem, A t t h i s s t a g e we l o c a l i z e t h e analysis near an a r b i t r a r y point of t h e
boundary of
f).; which we s h a l l
N
take t o be t h e o r i g i n i n 4:
, and
choose
N t h e coordinates i n C s o a s t o have, near t h e o r i g i n ,
(v. 151
r(z)
-
N-1
~e rN +
z
Re
a2, o
C
j,k=l
J
k
zjek + 2 S 0 ( r ; z )
+
0(\z\3),
where
is c a l l e d t h e Zevi form of
fl ) a t t h e point
b n (cC of
0.
By a v a i l i n g ourselves of t h e expression (V.15) we axe going t o r e w r i t e i n a more p r a c t i c a l l y u s e f u l fashion t h e boundary conditions ( ~ . 9 ) , (v.10). We s h a l l s k i p a l l computations, which axe simple manipulations of indices. We regard t h e f o r m u
-
\JI-s
as a vector, with components u
uJ dEJ
J *
Actually we r e p l a c e it by ,one of its l i n e a r transformationsr
defined a s followsr (v.18)
When q = 0, S ( z )
-
where J'
-with.
J 1
N-1
u J g u 2 -(-I)'(&)
d z ~ k-1
I,
If 6 J, -
-1
#
If HEJ,
= 0 . When q > 0,
J i#
uJ
EN) and K uJ
~BJ
~ 3
~
'
K ~ Z :
i s obtained by adjoining k
- 4\dr\-' j I$K,~$!J~
= J ~ f j {K=\[kj.
E
EjJ' CkJ'
&&
Gjdek
Jv ; K
'
~
I n f a c t l e t u s use t h e d i r e c t sum decomposition U#
(v. 20)
-
u'
where u' = 0 i f N
EJ
Proposition V . l . -
Condition (v.9) (which presumes q > 0) is equivalent t o :
J
and u j
+ uR ,
3
0 if N ~ J .
I n our reformulation of (v.10) we s h a l l use the d i f f e r e n t i a l operatcn. L defined by (v.12).
We s h a l l a l s o use t h e matrix
, a c t i n g on vectors
of t h e kind u' (and transfarming them i n t o l i k e vectors), defined by
where N $$J; J ' Noting t h a t u# m o s i t i o n V. 2.
- J\U]-
K~fk].
u' when q = 0 ( t h e r e is no u" then) we s t a t e r
- Condition
(v. 10) (which presumes q > 0) -equivalent
X)U* = 0
(v.23)
(i;+
m
to
r = 0.
We may a l s o t a k e (v.23) as expressing t h e boundary condition (v.11) (which presumes q = 0 ) i f we agree t h a t
f
0 when q
-
0.
One may say t h a t t h e a-Neumann boundary conditions (v. 7)-(v. 8) axe equivalent t o t h e conjunction of D i r i c h l e t conditions onl.the cpmponent u" of u# with "Weumann-like" boundary conditions on the component u' # Lastly we wish t o r e w r i t e 4. (v.6) i n terms of u
.
.
We look c l o s e l y
a t (v.18)-(v.19)
and take i n t o account t h e f a c t , fcllowing from ( ~ . 1 5 ) , t h a t
(V.24)
=o((z\)
3 zj
if
j < ~ ,2 &
dznl
(we have already used above t h e f a c t t h a t
= 1 * 0 ( 1 z \ ~ ), z d 0 .
br f: 0 i n a neighbarhood of 0.) bzN
- ~(z)
According t o t h i s we s e e t h a t ~ ( 0 ) = 0 and t h a t , consequently, I
i s i n v e r t i b l e i n a s u i t a b l e neighborhood of t h e a r i g i n . We then s e t u = (I
- ~(z))-1u# i n
(v. 6).
This equation becomes
where
It is now convenient t o switch coordinates, kom Re z ,
.., 14)
j
, Im
z . ( j = 1, J
to
can
Note t h a t i n t h e s e coardinates (near t h e a r i g i n ) t h e boundaxy X = be i d e n t i f i e d t o a piece of t h e hyperplane of t h e coordinates x j ! j < ~ k,
N)
tv.29)
2-
. Let u s s e t
'br
3Fj
= p j + i q j (pj
.
-
q j real.; j
.., N)
1,.
*
Yk
- I&); - A
;R
h
ay v i r t u e of (V.24) we haves
Let u s introduce the a d d i t i o n a l notations
An easy computation shows t h a t
A = B2 za22 + 2 N 0 - B+
(v.32) where
dr
&
a
dr
h-+At
4 1 - R 2 3 + Mo + iH1 3 -1 (32 b2
dr
is t h e Laplace operatar on X l
h
@
- (g5&?I j-1
+
J
+
,
r.
From (v.26) and '(l7.32) we s e e t h a t
L+T,,
TP%br
(v.33)
where hl is a cWmatrix-valued function near 0 and T' a f i r s t - o r d e r d i f f e r e n t i a l operator, with cWmatrix-valued c o e f f i c i e n t s , whose p r i n c i p a l p a r t
is t a n g e n t i a l t o X. We f a c t o r i z e d i r e c t l y Eq. (v.25)r
Here a s i n t h e sequel we completely disregard t h e e i r o r terms coming from ~ g u l a r i a i n goperators. They can be handled exactly as i n Ch. 111, Ch. IV. And they have no e f f e c t on t h e reasonings nor on t h e i r conclusions. We g e t s A~
0.35)
-
RAR-'
+
+
( 2 ~ - ' 1 ~ ~-'h
- E)I+ ~
-
,~
h
~
Eq. (v.36) shows t h a t A i s a pseudodifferential o p r a t o r of order one on b ). Ye may and s h a l l X, depending smoothly on r ( t h i s implies [G,A] take t h e p3.ncipal symbol of A t o be
It w i l l be shown below t h a t Re b(A) ) 0 i n t h e complement of t h e zero )F
s e c t i o n i n T X. Since Mo is a r e a l vectar f i e l d on X, its p r i n c i p a l symbol
&ro)
is purely imaginary; we s h a l l see t h a t
(v.37) is
The square-root i n
6 (- 0.) + t?(Plo)/~
t h e p o s i t i v e one. From
(v.35)
&
(v.37)
2 0. we g e t
We decompose t h e equation (v.25) i n t o t h e system1
#
(v*39)
G au
- AUir * V# ,
# (v.40) t o which we must adjoin the boundary conditions (~.21),(~.23), which we r e w r i t e here r
R
(v.41)
z+
h{xo
- 15+ 4 r ) u * = o
,
u" =
O-whenr
Let us then d e f i n e the pseudodifferential operator on X, A.
= O.
, with
values
i n t h e space of matrices t h a t transform v e c t o r s of t h e kind u' i n t o l i k e ones, as follows: A ~ U (0) *
(v.42) I f ue e x t r a c t R
2'
.
#
= (AU )*IFO
from (v. 37) and put it i n t o
(v.41),
r e nay r e u r i t e t h e
l a t t e r as
where u1 ( 0 ) , u W ( 0 ) , v ' ( 0 ) a r e t h e values of u ' , u", v' at r = 0 (these val u e s a r e functions i n X) and where
(v. 44)
(8.
-
A.
-i
+ Re]. [1(tt0
.
~ +)kr])FO
The l o c a l r e p r e s e n t a t i o n of t h e a-Neumann problem (v. 21)-[V . 23)-(V . 25) provided by (v. 39)-(v. 40)-(v.43)
i s t h e analogue of t h e decomposition of
(*) i n t o (**)-(***) i n Ch. 111. It should indeed be not.jd t h a t t h e equat i o n s (v.39)-(v.40)
must be s a t i s f i e d i n t h e portion r
<0
of a neighbor-
hood of t h e o r i g i n , and t h a t t h e r o l e of t h e v a r i a b l e t . i n Ch. I11 w i l l here be played by Remark V.2.-
- r.
When q = 0, a l l t h e matrices ~ ( e ) , .
t i c a l l y , and t h e pseudodifferential operators A ,
, T, T ' , hl A. ,(B ' a r e
vanish idenscalar
- as
they should be. The p r i n c i p a l symbol of the houndary operator
We s h a l l w e t h e notation
cI (v.45)
17
. Then we
j
=
fj*
1)?
(j
-
@'
.... W-l),
1.
have
u ( -A * )
- \512,
@(K,
- irk) - 5 * a r ,
We d e r i v e at once from (v.37).
In all t h i s
3r is
we observe t h a t
B~
(v.47)
.., 1i);
t h e "vector" with components a r / d e . (j = 1,.
RIrz0
~ ( 8 3 ' >) 0 . Let
(v.44) and (v.45):
= 2
J b rl
u s multiply
lPo .
J
By v i r t u e of (V.46) we see t h a t
6 ( a 8 ) by
yh+ 1m
-1(512 - (
~ ~ e . ) r / l a r l ) ~
cshr/\2r\
,
and s e t
(v. 48)
F = )3rI2 B~ b ( @ ' )
The c h a r a c t e r i s t i c
set of
A'
- 151' 16rI2 -
, 2. e . , t h e
.
)~:*ar)~
zero-set of
d
(a' ) , i s con-
tained i n the s e t
( c . a r 1 - 1511arl
(v. 49)
I~
q.6 r
2
The f i r s t - ~ n e of t h e s e of these conditions r e q u i r e s complex function c c =
iqN/&
ca ill{
. By the f a c t t h a t
o
.
5 = c Fr
f o r some
t h i s i n turn requires
=
( s f . ( v . z ~ ) ) , and therefore (v.49) implies N
(V.50)
+
I
- 0 ,
jEl.
..., N-1.
We . a l s o observe t h a t t h e second condition (V.49) can be r e w r i t t e n Im c > O which, for z small, i s equivalent with implies
l?, 30
.
gj = lj= 0 f o r a l l j = 1,...
. But i f 7 II
= 0,
(v.50)
h-1, hence i f we r e s t r i c t t h e
concept of c h a r a c t e r i s t i c s e t t o the cosplement of t h e zero s e c t i o n i n t h e cotangent bundle (here over a portion of X) we see t h a t w e must complement (v.50) with (v.51) I n t h e reg& region (v. 51). Bo
<0
63'
i s ellip=.
Note a l s o t h a t , i n the
> 0. Recalling t h a t dim T*X = 2(2h - 1 ) we s t a t e
i
? r o p o s i t ion V. 2.-' Chax
If z 0 Char
am
@'
submmifold of dimension 213
is s u f f i c i e n t l y c l o s e t o t h e o r i &
9'with
t h e f i b r e T:
l -
=
iihen 8 = 1 t h e r e a r e no equations (v.50) ; Lhar 5y t h e i n e q u a l i t y (v.51).
' is d e f i n e d simply
I n t h e remainder of t h i s s e c t i o n we suppose N
3 2.
Let u s i n t r o d u c e t h e following complex v e c t o r f i e l d s :
The complex conjugates
,... , z ~ - dle f i n e
what is c a l l e d t h e induced
Cauchy-Hiemann o p r a t o r on X ; f o r more information on t h i s important t o p i c we r e f e r t o Ch. V of
181 .
It is seen at once t h a t Eqq.
(v. 53)
,
@(Zj)=O
(v.50) can be r e w r i t t e n : j-1,
..., id-1.
Cn t h e o t h e r hand l e t u s denote by w t h e (N-1)-vector w
j
- dr /
d z j dzfi
(V.54)
, and (2
Thus, i n t h e r e g i o n
wi.lere ( c
jk
bj. Q (z) t h e one with components
I hrl)-'~ q N> 0 ,
\d(Z)I2
-
w i t h components
f ( Z .). Then J
I w . . ~ 1 ~ / (+ 1
1.1~)
.
we have ( c f . ( v . 4 7 ) ) ~
) is a s e l f - a d j o i n t p o s i t i v e s e n i d e f i n i t e (11-1)
(i'i-1) n a t r i x
6ependending smoothly on t h e v a r i a b l e p o i n t i n a s u i t a b l e neighbarhood of t h e o r i g i n , i n t h e base X. S i n c e , by (v.24), wj(0) = 3 far a l l j have :
.
2j = ~ 3 ;, 7 =P?:
X is a sing-
JNaP>O.
1
X-0
( i n x ) , t h e i n t e r s e c t i o n of
0
j
*
of T
< ;;, w e
We d e r i v e from a l l t h i s : m o s i t i o n V.4.
- The p r i n c i p a l
two on the (smooth) manifold Char
V.3.-
Remark
Let u s show t h a t
d(@' )
symbol
vanishes exactly of order
.
*
C(A) i s
> 0 on the ,portion of T*X
0
which l i e s over a s u f f i c i e n t l y small neighborhood of the o r i g i n i n X (2nd f o r s m a l l enough values of
- r). We derive
from (v.37) (using t h e notation
of (v.45));
==!l3l2 -
6 [ ~ )
(v.57)
5.ar/larl
( ~ e
+ i Re
~ . /I ha x \
Thus Re G(A)>/ 0 near 0 and Re @(A) = 0 only i f Re
$ .br
therefore I m plies
.
bar = 'fCl\arl
and
= 0. But we have j u s t seen t h a t t h i s conjunction i m -
5 = O.
Q.
E. D.
The subprincipal symbol of the "boundary operator".
&'
We continue t o r e p r e s e n t t h e v a r i a b l e point i n the cotangent bundle over X 6y .,I(
. ., X N , ~ , Y..~,yN, , . 5 1 ~ " ' 9J,-l, l l , . . . . , ~ , ) .
since we
a r e d e a l i n g here with c l a s s i c a l pseudodifferential operatars we can cons i d e r t h e i r t o t a l symbol which i s a formal s e r i e s of symbols t h a t a r e pos i t i v e homogeneous of i n t e g r a l degreees with r e s p e c t t o t h e f i b r e v a r i a ble
(5,T). Thus t h e
1 - j, and the one of
.. .
t o t a l symbol of A is a. + al +
a' is b o + bl
+... with deg b
..., with deg a J. j
= 1
f j = 0, 1,. ) By d e f i n i t i o n , the subprincipal symbol of
-j
@'
=
also
is the
quantity r
The subprincipal symbol is e a s i l y shown t o be i n v a r i a n t under coordinates change
- provided
it is r e s t r i c t e d t o t h e s e t of zeros where t h e principal
syfikoi a n d a l l its f i r s t d e r i v a t i v e s vanish. By Frop. V.4 t h e l a t t e r i s e x a c t l y what happens i n t h e case of of
therefme the restriction
* is invariant.
~ ~ ( 6) 3t o' Char The computation of
.&' , and
is r o u t i n e t it i s based on t h e informa-
el(@')
&',
t i o n provided by ( ~ . 3 0 ) , (v.36) and, of course, t h e d e f i n i t i o n of
(V .44). Let us here content ourselves with giving its value at a point
E Char (B * which l i e s d f r e c t l y above t h e m i g i n (i.c., i n t h e cotangent space t o X at 0). By Prop. V . 3 such a point is completely determined by r e q u i r i n g t h a t
7.
=
5 ( B s ) is positive-homogeneous of
1 (since
degree zero, it does not r e a l l y matter what value of 3ecalling t h a t h =
A r,
one e a s i l y f i n d s
We r e c a l l t h a t f i s t h e matrix defined i n (v.22). is a scalar
8;
Its generic e n t r y
where J , K axe multi-indices of length q such t h a t
li @ J * N @ K. We can compute of t h i s , w ~ dof
7~ > 0 we choose).
bl(a')
8
:
by using (v.22).
But t h e computation
is made e a s i e r i f we assume t h a t t h e Levi f o r m
(iJ.16) has been diagonalieed a t t h e m i g i n r
which is always possible by' a l i n e a r change of coordinates i n gN
9 o s i t i on V. 5.
- Suppose t h a t
t o t h e i n t e r s e c t i o n of Char
sein
.
Then
(v. 60) holds. The r e s t r i c t i o n of e l ( @ ' )
aswith t h e cotangent space t o X a t t h e
i s a diagonal matrix with diagonal eritries equal t o
(v .62)
C
lj-
Aj) j@J where J E g e s over t h e c o l l e c t i o n of-multi-indices with l e n g t t q which
2(
j€J
-
&o not contain N.
Hypo-ellipticity with l o s s of one d e r i v a t i v e . Condition
z(q)
_-*---------------------------------------------------------
Bop. V.3, V.4, V . 5 enable us t o use r e c e n t r e s u l t s of various authors (mainly s e e
@'
[z],
171) t o
obtain necessary and s u f f i c i e n t conditions f o r
t o be w p o - e l l i p t i c with l o s s of one d e r i v a t i v e , which means t h a t
f o r any open subset
of t h e boundary X, any d i s t r i b u t i o n u' i n
0
and any r e a l number s,
(we have denoted by V ' t h e space of v e c t o r s of t h e kind u').
Property
(v.63) indeed evidences t h e l o s s of one order of smoothness, s i n c e
i s of order ~r
if
no d e r i v a t i v e -
loss,
&'
@'
were e l l i p t i c , which i s taken t o be t h e case of
& 'u'
in
qocwould imply u'
in
qz: . B o p e r t y
(v.63) i s equivalent with l o c a l estimates ( v a l i d provided t h a t t h e open set
6 is
small enough) r
\\ u1
(V.64)
const. f i d a u '
11
,
u v €: C
( & ;v') .
If (v.64) holds one can show t h a t t h e so-called +-estimate holds f o r t h e
8-Neumann
problems a l l one has t o do is t o d u p l i c a t e t h e argument i n
Ch. I V ( " ~ o e r c i v eestimates"), using now t h e operator = dj'ul
& (u' + u")
(and, as i n Ch. I V . taking advantage of t h e estimates of Ch. 11
f o r t h e parametrix of t h e heat equation; here t h e r o l e of t h e operator .
P of Ch. IV i s played by
A+T
.
; s e e (v.25)).
k'e continue t o use t h e notation of t h e preceding section. I n p a r t i c u l a r
d ois t h e p o i n t i n Char&'
-
defined by z
*
Eo t h e tangent space t o T X a t
Uo
, and
0.
7
a
1. We s h a l l denote by
(I:
by Eo its complexification.
b(@') about dobegins with t h e quadratic
The Taylor expansion of form (11.65 ) where, f m each j (V. 66)
J
< N, q5
=
cj -
Nzl
is t h e l i n e a r p a r t of
2iZjzj
&(o) - k=l GjGk
$v 2
at do (cf. ( v . 5 2 ) ) ~
; k *
ble denote by Q(Q,Q') t h e b i l i n e a r farm on Eo defined by t h e quadratic f o r m (V.
6 5 ) ; we extend it b i l i n e a r l y (not sesquilinearly: ) t o :E
We introduce now t h e carionical symplectic form on Eo N
N-1 =
(v. 68)
j-1
dzj~dx+ j
C
j-1
d [C
and a l s o extend it ak a b i l i n e a r form t o Eo (I:
t h e r e is an endomarphism of Eo
(v. 69)
ara,e).= i
, Q , such
X(U,W)
,
qj n dyj
. Ye see t h a t
,
.
. Since
. i s nondegenerate
that [C
a, B 6 E~
.
The following i s not d i f f i c u l t t o prover m o s i t i o n V.6,- The eigenvalues of t h e endomurphism P axe t h e r e a l numbers 2
hj *
- 2Aj
!j
-
.., 3-1)
1..
(where, we r e c a l l , t h e
3J. axe
t h e eigenvalues of t h e Levi form of X a t t h e o r i g i n ) . iie coma now t o t h e r e s u l t s i n [2], open s e t
0 contains t h e
t h e followingz
[7].
They t o l l us t h a t i f the.
m i g i n and i f (v.64) holds. then we nust have
(v.70)
let
Ij ( J
-
..
1,. ,r) be t h e p o s i t i v e eigenvalues of D.
whatever t h e eigenvalue
p of dl (&
d such t h a t 1 Q = 0 f o r some d ) 0,
) t h e vector 8 ml,
the r-tuple
m,
in Eo(C
of ..., mr -
nennegative integers, -#.
According t o Prop. V.6 we may take bly a f t e r changing t h e i n d i c e s of t h e if r
<j
- 1. &om
rj 2 laj I , j - 1,. .., r ( ~ o s s i 2 J.);
of course r
V.5 w e deriver
Prop.
< N,
and
2j
= 0
-
If then we a l s o take i n t o account (v.67) we s e e t h a t Condition (v.70) is equivalent t o (v. 72)
r
Whatever t h e eigenvalue
/A
of ~ ~ ( a ' ) , + 5-1
5 > 0.
It i s obvious t h a t Condition ( ~ . 7 2 ) , i f it holds a t do , w i l l a l s o hold i n a f u l l ' neighborhood of holds i n a neighborhood of
u0 ( i n
uo , we
Char
& * ).
Since (V.67) a l s o
derive t h a t Condition (v.70) w i l l
.
hold i n a f u l l neighborhood of W, as soon as (v.72) holds a t The r e s u l t s i n 121, [7] the s e t
then t e l l us t h a t t h e estimates (V.64) holds i f
0 is smdll enough.
NOW, according t o l?cop.
z Aj-
(v. 73)
jGJ
V.5,
(v.72) can be r e s t a t e d by saying t h a t li-1
C aj+zpj12
J~!J
0
j-1
whatever the multi-index J , of length q, such t h a t N
$ J.
Property (v. 73) can, i n -turn, be rephrased a s follows8 z ( ~ ) ~The Levi farm of X at t h e a r i g i n has a t l e a s t N which n e
> 0 or a t l e a s t q + 1 which axe < 0.
- q eigenvalues
Thus, accarding t o t h e main r e s u l t i n 123,
[?I, we may s t a t e s
& ' be hypo-elliptic
Thearem V. 1.- In order t h a t
with l o s s of one d e r i -
v a t i v e i n some open neighbcrhood of t h e o r i g i n i n X it is necessary& - s u f f i c i e n t t h a t Condition Z(q)O
-
A s we have s a i d t h e estimate
u. (v. 64)
implies t h e $-estimate
for the
b -Neumann problem. Another proof of t h e +-estimate can be found i n
1 8 3 , ch. III. Remark V.4.
- When q = N Condition z ( ~ )is t r i v i a l l y s a t i s f i e d .
We know t h a t
i n t h i s case t h e 7 - ~ e u m a n n problem reduces t o t h e D i r i c h l e t problem. Remaxk V.5.-
fl (or its boundaxy X)
The apen s e t
is s a i d t o be strongly
pseudoconvex at t h e point z = 0 i f t h e Levi form a t t h a t point, ( ~ . 1 6 ) ,
2. e . ,
is positive-deflnite,
every one of its eigenvalues
I n t h i s case it is c l e a r t h a t z ( ~ holds ) ~ provided q does not: t h e operator l o s s ! ) when q = 0 and
&*
n
is
not hypo-elliptic
>/
lj i s , >
0.
1. For q = 0 it
(with any r e g u l a r i t y
i s s t r o n g l y pseudoconvex.
BOUNDARY VALUE PROBLEMS OF PRINCIPAL TYPE
VI.
Th. V .1 shows t h a t , i n t h e 7 - ~ e u m a n n problem, t h e hypo-ellipticity i s determined by t h e subprincipal symbo1,of t h e boundary operator
&.
There a r e problems i n which it is dependent only on properties of t h e p r i n c i p a l symbol of
6
. This is t h e case when
@
is of p r i n c i p a l
YE. kt u s describe what t h i s means. W e s h a l l assume, as i n Ch. I V , t h a t
Y
=
m-
, i, e.,
t h e number of boundary conditiozs is equal t o t h e num-
ber of r o o t s of t h e polynomial i n z, b ( ~ ) ( xt,,
3 ,z), with r e a l part > 0.
Since we a l s o assume t h a t dim H
< +co
we s h a l l regard
@
as a pseudodiffe-
r e n t i a l operator i n X with values i n the space of r g r matrices, f o r some integer r ), 1 (these axe matrices with s c a l a r entries). Definition VI.1.-
We say t h a t t h e boundary value problem (*) ( ~ h .111)
of principal type i f (VI.1)
2
)I = m- and i f the following holds a
d ( x , ~ ) ~ ~ * ~ ' - ~ , d e t ~ ( @ ) ( x , implies ~ ) = O
a The c h a a c t e r i s t i c
det ~ ( 6 3 ) ( x , T )
5
set
of
fo
.
& , Char a , i s . t h e subset
it
of T X 4.0 where
a(bJ)'(x,3 ) is not invertible, I. e.,
When ( ~ 1 . 1 ) holds Char
63
is a subset of a smooth submanifold of T*X
of codimension one, of 'course conic, When
\
0
6(@)i s r e a l , it is such a
submanifold, otherwise it is ( i n general) a proper subset of such a m a n i fold. Let bP(x, $ ) denote t h e cofactor matrix of 6(63)(x,
5 ), 63 ' any clas-
s i c a l pseudodifferentidl operator i n X with principal symbol be, B any such operator with principal. symbol I d e t
&
and
deg B of
'
@(a).It is clear t h a t ae63
d i f f e r from B by a (matrix-valued) operator of degree
- 1. Since t h e hypo-ellipticity
d3ae) implies
t h a t of
(resp., t h e s o l v a b i l i t y
of
4, i n m
3
y questions one may assume t h a t
t h e ~ i n c i p a symbol l of t h e pseudodifferentidl operator under study is a s c a l a r multiple of t h e i d e n t i t y matrix. Thus the statements below concern t h e operator B and we write d ( B ) to
d3
we take b = d e t
b(x,f
)I. When applying these r e s u l t s
&(a).Note t h a t t h i s axgment,
applied t o most
e l l i p t i c systems, shows t h a t they a r e reducible t o t h e type (.111.1). Yhen
' i s of principal type (which t h e n e c e d i n g asgument does @I
one can reduce, a t l e a s t microloca;lly, t h e study of
63 t o
t h a t of
a n e n t i r e l y s c a l a x operatar, a l s o of p r i n c i p a l type. Indeed, t h e rank of t h e
r X r matrix 6 (63)at any ( x , ? )
a
Chaz
f i n d an e l l i ~ t i cmatrix-valued operator hood of (x,
4 ),
63 I
(~1.3)
is exactly r
- 1 and one can
such t h a t , i n a conic neighbcr-
is t r i a n g u l a r , with r-1 diagonal e n t r i e s of t h e form
+ S,
S G
M
9 'l(x),
t h e r - t h one being of p r i n c i p a l type. This remains t r u e even when " p i n c i p a l type" i s taken t o mean
Y
Let a ( x . 3 ) be a ~ * c o r n ~ l e x function on T X of a is t h e (complex) vectar f i e l d
* on T X
0
By a b i c h a r a c t e r i s t i c strip of a we mean a C' T"X \ 0 such t h a t , for a l l O < t
g ( t ) is
.
0
The Hamiltonian f i e l d
, curve ]0,l(3
t
Y(t) 6
t-#
0 and proportional t o t h e
vector f i e l d Ha a t t h e point ~ ( t( i)n p a r t i c u l a r ,
I(
is a "true" curve).
5i.nce a is complex t h e r e w i l l not be, i n general, b i c h a x a c t e r i s t i c s t r i p s of a (even i f d a f 0). But i f a i s r e a l t h e r e is a unique b i c h a x a c t e r i s t i c s t r i p of a through every point (xO,
3"
Whenever such a s t r i p e x i s t we have a Definition VI.2.- &J
Condition
@
q
(3be
-
ET*X \ 0 iuch t h a t da(?O,
*
B
&
6
\
0
. Ye say t h a t
if there exists a
COO
func-tion
such t h a t t h e following i s t r u e * b(x.3)
-
V(x,5)60,
I
along any b i c h a r a c t e r i s t i c s t r i p s ~ e ( q b )contained i n
0 Pr
d
~ e ( q b ) ( x , Z )# 0 ;
(v1.6)
)
0.
const. along it, s i n c e Haa = 0.
an open subset of T X
(q*) i s s a t i s f i e d by
so)#
0 on
which Re(qb) = 0,
if 1m(qb) > 0
a t some point, ~ m ( ~ b>,) 0 &.
g later p
s (for t h e a r i e n t a t i o n of HR e(qb))* We say t h a t B s a t i s f i e s t h e s t r i c t condition (CI)') in if t h e function g
&
can be s e l e c t e d s o a s t o s a t i s f y , i n addition t o ( ~ 1 . 6 )& ( v I . ~ ) , the f o l lowing : (~1.8)
b does not vanish i d e n t i c a l l y on any b i c h a r a c t e r i s t i c s t r i p A Re (qb) contained i n
0 on which Re(qb)
We say t h a t B s a t s i f i e s t h e condition (
(y'))a t a point of (xO, j"). -
(xO,
5')
It can be shorn (see [ll],
T*X
[15])
\
0
= 0
.
') (resp., t h e s t r i c t condition
i f it does i n some open neighborhood
t h a t i f ( ~ 1 . 6 ) 65 ( ~ 1 . 7 ) (resp., &
( ~ 1 . 8 ) ) hold f a r some function q , then ( ~ 1 . 7 ) (resp., & ( ~ 1 . 8 ) )w i l l hold f o any other function which s a t i s f i e s ( ~ 1 . 6 ) .
b bout
( ~ 1 . 8 ) s e e [14] .)
It is reasonable t o conjecture t h a t ( ~ 1 . 9 ) i f t h e operatar B s a t i s f i e s t h e following condition;
then, -
i n order t h a t B -PO-elliptic
i n X, it i s necessaxy-
s u f f i c i e n t t h a t B s a t i s f y t h e s t x i c t condition ( y! ') at every point of T*X \
o
.
Observe t h a t Property (V1.10)
is weaker than t h e property of being of
p r i n c i p a l type, which reads
o or
db stands for t h e d i f f e r e n t i a l of b with r e s p e c t t o (x,
z).) Let us
i n d i c a t e b r i e f l y t o what extent t h e conjecture (VI. 9) has been proved.
rnf?:~enti%?~iatars_ In t h i s s e c t i o n we suppose t h a t B is a ' d i f f e r e n t i a l operator; t h i s i s d i f f i c u l t t o r e c o n c i l e with t h e f a c t t h a t B is of order zero, but what we mew is t h a t B becomes a d i f f e r e n t i a l operatar a f t e r multiplication by a s u i t a b l e e l l i p t i c operatac (of t h e kind
/\
for some i n t e g e r m>, 0). We
assilme t h e m u l t i p l i c a t i o n effected. Since then b ( x , - 5 ) i f we assume, as we s h a l l , t h a t t h e open s e t
0 in
-
(-l)%(x,3)
Def. VI.2 contains
*
every f i k e TxX which it i n t e r s e c t s , Condition ( ~ 1 . 7 ) must be replaced by (~1.11)
along any b i c h a r a c t e r i s t i c s t r i p A ~ e ( q b )contained i n on which ~ e ( ~ =b 0, ) 1m(qb) does not change sign.
Definition YI.3.-
We say t h a t B (assumed now t o be a c l a s s i c a l pseudo-dif-
f e r c n t i a l operatar i n X, not necessarily a d i f f e r e n t i a l operator) s a t i s -
b cf T*X
f i e s Condition (P) i n t h e open subset
& such t h a t
function q
( ~ 1 . 6 ) d ('41.11)
\
0 if t h e r e i s a Cm
holdhold
Now it is c l e a r t h a t i f both ( ~ 1 . 6 ) - ( ~ 1 . 1 1 ) and ( ~ 1 . 8 ) hold we have; (~1.12)
~ m ( ~ keeps b) t h e same s i g n on each connected component of t h e zero-set of Re(qb)
Theorem V I . l . -
Let B
0.
&
be a d i f f e r e n t i a l operator i n X. Suppose t h a t each
point i n T*X \ 0 has an open neighbarhood function q such t h a t d
5
~ e ( q b )f 0
on Char
( ~ 1 . 8 )@ (~1.12) hold. Then B -PO-elliptic %pose
0i n which there is a smooth B, and such t h a t furthermore i n X.
(2.e., (~1.10) holds) and tkit ~d a function q 0 such
t h a t B is of p r m u p g
t h e r e is an open s e t
6C
T*X \ 0
t h a t e i t h e r ( ~ 1 . 8 )i s not t r u e or t h e r e is z b i c h a r a c t e r i s t i c s t r i p s ~ e ( ~& b )
0 , on which ~ e ( ~ =b 0) and
a t some point of which 1m(qb)
has a zero -of .-odd order (along t h e b i c h a s a c t e r i s t i c i n question). T ~J B is not hypo-elliptic i n X.
Th. VI..l is. proved i n
1143
.
S u b e l l i p t i c operat'ors
This is an i n t e r e s t i n g subclass of hypo-elliptic operators. Definition VI.4.- Qseudodifferential
operator A of order m
t o be s u b e l l i p t i c i f every point x0
X has an open neighbarhood
in X
is s a i d
fi
such t h a t , far some number 6 s 6 (xO) 20, t h e following is t r u e r (~1.13)
II"
Il,-l+s,
I.
const*
[IAU (IO *
\/
U E
~ r ( h ) .
(We suppose made some choice of t h e Sobolev norms i n X.) It is easy t o s e e t h a t (~1.13) implies t h a t , whatever s r e a l , t h e r e is an open neighborhood
ns
(VI .14)
I
of xO and a constant Cs>
0 such t h a t
V U Ecm(Rs)
I
or t h a t , whatever t h e compact s e t K CX, t h e rea.l,numbers a,
8'
lls+m-l+6
5 Cs
is a constant C = C(S,S' ,K) (~1.15)
11 A"
> 0 such
*
,there
that
I \ u I ~ ~ ~ - C ~ ~+( ~\ \ sA U \ \ fi")Isg), ~ +
vu
f
c ~ ( K,)
It is a l s o easy t o check t h a t i f A is s u b e l l i p t i c i n X, we have (VI .16)
WAU)
-
WP(U) ,
d
u
c 8' (XI,
hence A is, i n p a x t i c u l ~ ,hypo-elliptic. I n applying t h i s notion of s u b e l l i p t i c i t y t o our operator B we should consider d i s t r i b u t i o n s with values i n 0?
t
t h e extension is routine.
'41.5.- Let k be a n i n t e g e r & 0. We s a y t h a t B s a t i s f i e s t h e
iiefini t i m condition
(y;) i n
t h a t ( ~ 1 . 6 )& (Vi.17)
t h e open s e t
&C
T*X \ O
t h e r e is q
E
crn(O) such
( ~ 1 . 7 )bold, and a l s o t h e following;
a l o n g e v e r y b i c h a r a c t e r i s t i c s t r i p s ~ e ( q b )contained i n
&,
on which ~ e ( ~ =b 0, ) ~m(~b has ) z e r o of order a t most k.
(9;) implies
Note t h a t
the s t r i c t condition
ved t h e following important r e s u l t ( s e e 1 4 Theorem VI.2.-
( y e ) Yu. .
Egorov has pro-
3 , a l s o [lo]):
I n order t h a t t h e c l a s s i c a l pseudodifferential operatm B
i n X be subellip&, of T*X \ 0 have an -
i t is necessary and s u f f i c i e n t t h a t o p m g h b o r h o o d i n which
(q;)i s
every p o i n t
s a t i s f i e d by B
some i n t e g k~ > , 0, depending on t h e p o i n t i n question.
-
I f t h i s i n t e g z k can be chosen independently o f t h e p o i n t , -then * whatever xo
in X,
Estimate ( ~ 1 . 1 3 ) w i l l hold f o r
f a r a s u f f i c i e n t l y s m a l l open neighburhood
R
6
= (k
+ I)-' (and
of xo).
The examples of boundary v a l u e problems f o r e l l i p t i c equations where
o f t h e kind considered i n t h e p r e s e n t chapter
boundary o p e r a t o r s occur a r e well-known!
t h e s t a n d a r d ones a r e t h e so-called "oblique d e r i -
v a t i v e - problems ( s e e
[ 2 3 , 1 7 1 ).
[?.I :b:w.on,
S.. 9ouglis.A. & Nirenberg. I.- E s t i ~ a t e snear t h e S o u n d u y
f3r solutions of e l l l p ~ a r t i a dl i l f s r e n t i a l equations s a t i s -
Qing-general
boundaxy c o n d i t i o n s I, Comm. Pure Ap?l. Piath.,
( 1 ~ 5 0 )623-727; ~
11, Comm. Pure Appl. Math., l J (1%4),
2
35-02.
b4]
Treves, F.- m o e l l i p t i c p a r t i a l d i f f e r e n t i a l equations of principal m e . S u f f i c i e n t conditions 3 Necessary conditions, Comm. Pure ~ p p l .m t h .
F5J
24 (1971), -631-620..
h e v e s , F.- Minding numbers and t h e s o l v a b i l i t y condition Journal Diff. &om.
10 (1974).
135-149.
Rressanone ( ~ o l z a n o ,ltdly) June 1977
(Y)).
