(3.12) h ( 0 ) = g ( 0 ) = 0 , d g ( 0 ) = 0 ,
Oh(O)/Ox~O.
Let
~ECN,o, O~/Ozl=O,
and let u be a holomorphic representative of ~. AsOu/Ozl EOo,
solvingOv/Ozl=OU/OZl
withVEOo,
replacing u byu - v ,
we see t h a t we m a y assume t h a tOu/Ozl
= 0 . In this situation, it is easy to show, see [31], [13], t h a tu=vo5
for some holomorphic function v in a "local projection" of ~t under the m a p(3.13)
, cLocalizing everything near 0 E C n, we define
~ : = { z : l x l l < c , l y l l ~ C ,
IXnI<C, Izttl~-Iynl<V, f(xl+iyl,z')<O},
N:={z:IxlI<C, lyll<s IXnl<C, IzHl§ f(xl§
On the solvability of linear partial differential equations in spaces of hyperfunctions 65
In the sequel, we shall assume that 0<r/<<c<<l are chosen small enough. T h e n it is represented by a function u=vo5 where v E O ( A ) and A c C n-1 is the open set defined by
A : = { Z : I x n l
<s Izttl-~lynl <?], -Xn--~-G(zt",yn)
< 0 } ,where G(z", yn)=minlml<_~ 9(yl, z", yn). We note that A is a supergraph of a Lip- schitz function since, with ~ = ( z " , y,~) and an obvious notation, we may write
G(~') - a(~) = g(Yl (~'), ~') - g ( Y l (~), ~) <_ g(Yl (~), ~') - g ( Y l (~), ~) ~
Cl(- l.
W h e t h e r 0 belongs to the support of fi or not depends, on one hand, on the po- sition of the fiber 5 -1(5(0)) of 0 with respect to ft, on the other, on the holomorphie convexity of A at 5 ( 0 ) = 0 E C n-1. We have the following lemma.
L e m m a 3.13. The following properties hold for 0<r/<<e<<l.
(1) If 5(0) EA, or if 5(0) E 0 A and 5(0) belongs to the envelope of holomorphy of A, then O does not belong to the support of it.
(2) If 6(O)EOA, then either the support of it contains the set NN5-1(5(0)), or it does not meet it.
(3) If S(O)EOA and A: D---~fX ( D c C denotes the open unit disc) is an analytic disc with A(0)=0, the following property holds: 0 does not belong to the support of it irA(D) intersects A, while if A ( D ) c O A , then the projection of the support of
either contains or does not meet A(D).
Proof. Properties (1) and (2) are self-evident. To prove property (3), we argue as in [32, L e m m a 3.2]. Let us assume that v is holomorphic in the union of A and a small ball around A(~-0) (TOED\{0}, A(~'0) may belong to A or not) but cannot be holomorphieally extended near 0 E C n-1. Introducing (as in
[32,
L e m m a 1.2]) the envelope of holomorphy A of the union of A and this small ball, this means t h a t 0 ~ A. We shall reach a contradiction. Set e = (0, ... , 0, 1)E C '~- 1 and let d(z) denote the euclidean distance of z E A to the boundary of fX. Since A ( D ) C A and A is a supergraph of a Lipsehitz function, A ( r ) + i t e E A , for small t > 0 and every r E D , and d(A(~-)+te)>_t/C. On the other hand, d(te)<_t, since 0E0~x. T h e function u t ( T ) = - l o g d(A(T)+te), t > 0 , is subharmonic in D, uniformly bounded in some disc of center T0, radius 0>0, Using the mean value property we obtain_ [ [ ut (7) dr -< 1 (Qr-TrO 2) ( - log t/C)
- l o g t <ut(O) < J JD +Tro2M).
Letting t-+0 +, we reach a contradiction. []
66 Paulo D. Cordaro and Jean-Marie Tr4preau
T h e differential operator O/OZ 1 is of principal type, with principal symbol
~1'
To emphasize the link with Section 3.3, we shall use the notation P = ~1 IT<, c " .
Taking (3.7), (3.11), (3.12) into account, we have
of k_(2h Oh k (2 h oh
p = k ~ z i , R e P = 2 \ Oxi/I, I m p = - ~ \ ~
T h e characteristic points over N are thus defined by h = 0 , Og/Oyi =0. Our task is now to establish a link between the properties of the symbol p and its Hamiltonian field, which occur in the statements of Theorems 3.10-3.12, and the properties in L e m m a 3.13, which are stated in terms of the fibers and the image of the map 5: (zl, z')~-*z'. For example, we would like to recognize a bicharacteristic interval, or a characteristic point of positive type, by looking at the behaviour of the function yl~--~Og(yi, z", yn)/Oyi. This will be possible, thanks to the following lemma.
L e r n m a 3.14. Let OET~vC n be a characteristic point with d p A d p A # N =0 at 0.
In the coordinate system (xl,Yi, x", y', k ) , we have i 0 . _ 0 0
H N = ~ z l - - z a ~ z i + 8 - ~ at O, with a=(ah/O21)(Oh/Ozi) -1, /3EC.
Hence, 7r.H N is tangent to the zero set E = { h = 0 } of R e p at 7r(0), and if we
I t !
use (Yl, x , y ) as a coordinate system on E, we have 7r, H N p = a - - 0 at Tr(O),
Oyl
with / I /,I I > 0 . Proof. By our assumption we have the relation
(3.14) d~l = a d ~ l + i b s at 0,
" * n
on T ) C , for some a, bEC. Using only formula (3.3) we get ( H ~ , d z j } - = { ~ l , z j } N = i { ~ l , z j } = i ~ l j , j = l , . . . ,n.
O n t h e solvability of linear partial differential equations in spaces of hyperfunctions 67
Using (3.14) and formula (3.3) we c o m p u t e
7%
o ' : 1
=-igd~j at0, j = l , . . . ~n.
To c o m p u t e a, we rewrite (3.14),
and chase the coefficients of dzl and d21 in this equality, in our coordinate system.
For the sake of simplicity, we shall write f~l for Of/Ozl, etc. We note t h a t the t e r m ibkOf gives no contribution, since at 0, f ~ = 0 and dx,~=dg with g~ = 0 , ge~ = 0 . As f ~ , f ~ do not depend on xn, we obtain
f ~ ,~ = afz~ ~ , f ~ e~ = a f ~ ~.
By addition we get fe~ ~ =afzl ~ . As g does not depend on xl and h = 0 at 0, we obtain a=hel/h~ 1 and the first p a r t of the lemma.
T h e second part follows, since the formula for H f f gives H~h=O at 0 and
N __1
Hp y l - ~ ( l + ~ ) with real p a r t c~=h2zl/41hz~l 2. []
We are now in a position to prove the s t a t e m e n t s of Section 3.3.
Proof of Theorem 3.10. If b~0 is a bicharacteristic interval of O/Ozl, then b is the projection of an integral curve of HNep along which p = 0 , and the condition in L e m m a 3.14 is satisfied. Hence h = 0 , Og/Oyl = 0 along b, and by L e m m a 3.14, z", Yn are constant = 0 along b; hence also
9(Yl, z", yn)=g(yl, 0, 0)
is constant = 0 along b.So z~=0 on b and b = N D ~ - I ( 0 ) , and L e m m a 3.13, p r o p e r t y (2) applies. []
From now on we assume t h a t O/Ozl satifies condition (P) on S~v+C ~, though this is p e r h a p s not necessary for the next result (we shall avoid proving an analogue of the Hanges-SjSstrand T h e o r e m [7] in our context, condition (P) simplifies things).
Proof of Theorem 3.11. T h e o r e m 3.10 applies to one dimensional bicharacter- isties, so we consider the case of a complete two dimensional bicharacteristic. We must be careful since its definition is global while our reduction is local. How- ever, this is not i m p o r t a n t for the following reason: if ~ belongs to a two dimen- sional bicharacteristic, and also to a bicharacteristic interval, as p r o p a g a t i o n along a bicharacteristic interval has already been established, we m a y localize our s t u d y close to an endpoint of it. Hence we m a y assume without loss of generality, t h a t
68 Paulo D. Cordaro and Jean-Marie Tr6preau
v~ does not belong to, or is an endpoint of, a bicharacteristic interval. T h e n it is easily seen t h a t the complex structure of the reduced two dimensional bicharac- teristic t h r o u g h 0 is locally determined and the following argument is meaningful.
By T h e o r e m 3.7, we m a y assume t h a t there exists a germ of a two dimensional manifold 0 C B c N such t h a t the space obtained by shrinking any bicharacteristic interval in B to a point has a complex structure, the holomorphic functions of which are induced by solutions of HNu=O. We note t h a t the m a p 6 is well defined on the reduced space, since a bicharacteristic interval projects in a fiber of 6 by the proof of T h e o r e m 3.10. It is holomorphic since Hp~'Zj=i{C~, z j } = 0 if j > l . Hence 5: B--~C n-1 m a y be locally considered as an analytic disc, as in L e m m a 3.13, prop- erty (3). T h e l e m m a applies (actually A is pseudoconvex [31] and 6 ( B ) c O A but we do not need these facts.) [~
Proof of Theorem 3.12. It is an immediate consequence of L e m m a 3.13 and the following result.
L e m m a 3.15. With the notation in Lemma 3.13, ~ ( 0 ) E A if OEI; ~ is a point of positive type.
It was proved in [31] that, if Imp=-kOg/Oyl does not change sign from - to + along any bicharacteristic of Re p, it does not change sign from - to + along any integral curve of O/Oyl. No such result holds for sign changes from + to - . T h e m a i n ingredient in the proof of L e m m a 3.15 is t h a t this is however the case on the p a r t / ; ~ of the characteristic variety, when condition (P) is satisfied.
Proof of Lemma 3.15. We identify S~v+C n with N and we p a r a m e t r i z e the zero set E={zcN:h(z)=O} of R e p by Yl, x", y'. T h e vector field HN~p induces a vector field 7r.HN~p on E (because p is homogeneous of degree 1) and by condition (P), Og/Oyl does not change sign along its integral curves. We shall use the h o m o t o p y
X t = ( 1 - t ) T r . H N e p + t ~ 0 < t < l . /)Yl ' - -
between the vector fields 7r.HNep and O/Oyl on E. By the definition (3.5) of l; ~ and L e m m a 3.14, we have near 0,
0 Og
Xt = at ~ oYl with cq > 0, if Oyl = O.
It is t h e n a consequence of the Bony-Br6zis lemma, see [13, L e m m a 7.3.4], applied to :hOg/Oyl, t h a t Og/Oyl does not change sign along the integral curves of Xt, 0 < t < l . We m a y assign a sign s(t) to t: s(t)=-l,O, +1, depending on if Og/Oyl
On the solvability of linear partial differentia[ equations in spaces of hyperfunctions 69
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Received January 7, 1997 Paulo D. Cordaro
Instituto de Matems e Estatistica Universidade de Ss Paulo
Caixa Postal 66281, CEP 05315-970 S&o Paulo, SP.
Brazil
email: cordaro@ime.usp.br
Jean-Marie Tr~preau I n s t i t u t de Math~matiques Universit~ Paris VI Case 247, 4 Place Jussieu F-75252 Paris
France
email: jmt@ccr.jussieu.fr