P(D) x") x" = O. Rn/W P(D) sx ~ ty P(D) P(D) On fundamental solutions supported by a convex cone

(1)

On fundamental solutions supported by a convex cone

ARNE ENQVIST

1. Introduction

L e t P(D) be a p a r t i a l differential o p e r a t o r in R n w i t h c o n s t a n t coefficients a n d /" a closed c o n v e x cone in R ' . T h u s we a s s u m e t h a t x, y E / ~ a n d s , t > O implies t h a t sx ~ ty E F. T h e p r o b l e m discussed here is to decide w h e n P(D) has a f u n d a m e n t a l solution w i t h s u p p o r t in F.

W h e n F is a p r o p e r cone, t h a t is, w h e n /~ contains n o s t r a i g h t line, this c o n . d i t i o n m e a n s precisely t h a t P(D) is h y p e r b o l i c w i t h r e s p e c t t o t h e s u p p o r t i n g planes of F which m e e t F o n l y a t t h e origin (Gs [4], see also H S r m a n d e r [5, T h e o r e m 5.6.2]). I n t h e o t h e r e x t r e m e case w h e r e /" is a h a l f space s u f f i c i e n t conditions were g i v e n long ago b y P e t r o w s k y (see G e l f a n d - - Shilov [2]), a n d a c o m p l e t e a n s w e r t o t h e q u e s t i o n was o b t a i n e d b y H S r m a n d e r [6]:

I n general t h e i n t e r s e c t i o n F gl ( , _ / ' ) -~ W is a linear subspace a n d x E F implies x ~- y E / 7 for e v e r y y E W. This shows t h a t _/1 is t h e inverse i m a g e in R n of t h e i m a g e V o f 2" in Rn/W u n d e r t h e q u o t i e n t m a p . I t is clear t h a t V is a p r o p e r cone. W e shall use t h e n o t a t i o n s n ' = d i m W , n" ---- n - - n' a n d c o o r d i n a t e s x ~ (x', x") such t h a t W is d e f i n e d b y x" = O. Also for n' > 0 a n d n" > 1 sufficient conditions for t h e existence o f a f u n d a m e n t a l solution o f P(D) w i t h s u p p o r t in F, analogous to t h o s e of P e t r o w s k y for n " ~ - 1, h a v e b e e n g i v e n b y Gindikin [3]. W e shall e x t e n d t h e s e in t h e d i r e c t i o n suggested b y t h e t e c h n i q u e used b y H S r m a n d e r [6]. H o w e v e r , w h e n n " > 1 t h e r e are polynomials such t h a t P ( ~ ' , D") is n o t h y p e r b o l i c for a n y ~'. This i n t r o d u c e s a new d i f f i c u l t y a n d in c o n s e q u e n c e o f this t h e r e s u l t is f a r f r o m complete.

I n Section 2 we i n v e s t i g a t e t h e general n e c e s s a r y conditions. T h e m e t h o d s u s e d are v e r y close to t h o s e of H 6 r m a n d e r [6]. I n t h e h y p e r b o l i c case t h e principal p a r t plays a v e r y i m p o r t a n t role. (See L. Svensson [9].) H e r e t h e principal p a r t does n o t give so m u c h i n f o r m a t i o n a b o u t = th e p o l y n o m i a l , a n d we h a v e n o t b e e n able t o f i n d a n y s u b s t i t u t e . H o w e v e r , in S e c t i o n 3 we s t u d y some s t a b i l i t y p r o p e r - ties o f t h e n e c e s s a r y conditions w h i c h allow us to c a r r y t h e m o v e r t o v a r i o u s polynomials r e l a t e d t o t h e b e h a v i o r of P s t i n f i n i t y .

(2)

2 A B N E E N Q V I S T

I n Section 4 we reduce t h e existence theorems to a priori estimates. To be able to p r o v e these estimates, we are forced to assume t h a t P(~', D") in some w a y can be w r i t t e n as a p r o d u c t of hyperbolic polynomials. I n Section 5 we give general mfficient conditions of t h a t t y p e a n d in Section 6 we investigate polynomials t h a t are p a r t i a l l y homogeneous.

W h e n deg P = 2, we solve our problem completely. (Section 7.) W e h a v e also tried to f i n d new m e t h o d s to prov e t h e existence of f u n d a m e n t a l solutions.

I n Section 8 we p r e s e n t a constructive m e t h o d in t h e case Of two variables.

The subject of this p~per was suggested to me b y I~rofessor L a r s t t 6 r m a n d e r , whose c o n s t a n t criticism a n d e n c o u r a g e m e n t h a v e been invaluable. I a m also v e r y grateful to h i m for m a n y valuable suggestions.

2. G e n e r a l n e c e s s a r y c o n d i t i o n s

L e t P(D) be a p a r t i a l differential operator in R" w i t h c o n s t a n t coefficients a n d let V be a proper, closed a n d convex cone in R"". Set / ~ = R~'• V, x : (x',x") 6 R " • R"" a n d Ix[ : maxl_<j_<,[xi]. F o r e v e r y real n u m b e r A we define t h e set V* as follows

V* = {v" e R~ <v", z"> ~ A, Ix"[ = 1, x" e V}.

THEORE~r 2.1. Assume that P(D) has a fundamental solution with support in 1".

Then the following condition is satisfied.

(2.1) There is a constant C such that the following is true: Let ~'----> ~"(~') be any analytic function such that P($', ~"(~')) = 0, I m ~"(~') 6 -- V* and [(~', ~"(~'))] < M for all ~' 6 Y2(~ 0, _R) : {~' 6 C ' ; I~' -- $~1 < R} where }~ s R" and R > O. Then m i n ( R , A ) ~ C l o g ( 2 + M ) .

W e need some l e m m a s for t h e proof.

LEMMA 2.2. I f P(D) has a fundamental solution with support in F, then for every compact neighbourhood K of zero there are constants C and # such that

(2.2) ]u(0){ ~

C ~

sup [D~P(D)u[, u E Cg(K).

[at -<~ -1'

Proof. I f P ( D ) E = 8, supp E C / ~ t h e n u(0) = E(P(D)u). Since t h e s u p p o r t of E is a subset of - - / ' , a n d /7 is regular in t h e sense of W h i t n e y , we conclude t h a t (2.2) holds for a r b i t r a r y K . (See Schwartz [8, p. 98].)

LEMMA 2.3. Let P(D) satisfy condition (2,2) and let g 6 C~(K) be equal to 1

(3)

O N F U N D A M E N T A L S O L U T I O N S S U P P O R T E D B Y A C O N V E X C O N E

i n a neighbourhood of zero. Set K ' = (--I") f'l supp (dz). T h e n there exist constants C and #, such that for all u e C~(R ~) with P ( D ) u = 0

(2.3) lu(0)t < C ~ sup ID~ul .

lal ~ # K'

Proof. We apply (2.2) to the function v = Zu, noting that P ( D ) v =

~ ~:o (P~(D)u)D~z/a!.

LE:~MA 2.4. Let ZN(t) ~- 1~ when ltl < 1/2N and zN(t) = 0 otherwise, t E R.

I f q N is the convolution of N factors ZN we have (i) supp ~N C (__ -2,1 1),.

>_0, f

~Ndt = 1,

(ii) dk~lv(t)/dt k is a measure with total mass < (2N) a when 0 < k < 57.

Proof. See the proof Of Lemma 2.2 in HSrmander [6].

In the following lemma we use the notation

~ ) N ( ~ , ) : j ~ - n " N ( ~ I l R ) . . " ~oN(~n, I R ) .

L~MMA 2.5. Let ~o E R ", and let F disc s = Y2(~0, R ). p Set

Then

(2.4)

be a function that is analytic in the poly-

ix'lkiu~(x')I

^<( 4 N / R ) k sup IF(C)l, 0 < k < N.

Proof. See the proof of L e m m a 2.3 in HSrmander [6].

Proof of Theorem 2.1. Consider

I t is clear t h a t P ( D ) u N = 0 and t h a t uP(0) = 1. Let K ' be the set in L e m m a 2.3.

We have to estimate u p and its derivatives in K'. B y hypothesis <x", I m $"(~')>

0 for x E K ' and $' E tg(~ 0,/~), and furthermore there is a constant 6 > 0, t

such that Ix' l > 6 or < x " , I m ~ " ( ~ ' ) > ~ A 6 for all ~ ' e z g ( ~ , / t ) . Set K 1 = { x e K ' ; I x ' l > 6} and g 2 = { x e K ' ; ( x ' , I m ~ " ( $ ' ) > ~ 6 A for all ~ ' e g ( ~ ' 0,R)}.

Using L e m m a 2.5 with k = iV we obtain

[uN(x) I ~ (4-1~/6R) N, X E K r

(4)

AI~NE E N Q V I S T

Differentiation of u ~ gives factors, which are coordinates of ~' or $"(~').

these are bounded by M we obtain

Similarly we obtain

Since sup

ID=u~l ~

C(1 + M)~(4N/aR) ~.

lal -< ~* Kt

sup [Dau~l ~ C(1 @M)*~e -aA.

lal-<~ K,

I f we use this in L e m m a 2.3 and observe t h a t u~r = 1 it follows t h a t (2.5) 1 __< C(1 + M)."((4N/bR) ~v + e-aA).

We choose the integer 32 so t h a t

(~/~)/(8e)

< N < (dR)/(4e) which is possible if R > (Se)/d. Then we obtain t h a t

1 < C(1 + M ) ' e -q~i~(R'~), where ~ 1 = min (1, d/(8e)), and this completes the proof.

The conditions in Theorem 2.1 are hard to apply since t h e y involve r a t h e r general polydiscs in P-l(0). However, when the function $' -~ $"(~') is algebraic of bounded degree and 1" is semialgebraic, it is possible to sharpen these conditions.

(A semiMgebraic set is a set t h a t can be defined by finitely m a n y real polynomial equations and inequalities.)

TI~EOR:EM 2.6. Let qo be an integer and assume that P(D) has a fundamental solution with support in the semialgebraic, closed and convex cone I ~. Then there are constants A o and R o such that i f the function C"" ~ ~' --~ ~"(() E C n" is analytic and algebraic of order ~ q 0 i n Q(~;, /?0) (t0 e R ' ) f and i f P(~', ~"(~')) = 0 in

! r

~(~o, Ro), then there is at least one point ~, C 0($o, -Ro) such that I m $"(~') ~ -- V~o.

We need some preliminaries before the proof. The function Theorem 2.6 shall satisfy the following conditions:

(2.6)

~' --> g"(~') in

~"(~') is analytic in D(~;,R). There are polynomials pj = p i ( $ ' , T), j = n ' q - 1 , : . . , n with ,degpj < q 0 , s u c h t h a t p j ( ~ ' , $ i ( $ ' ) ) = 0 in D(~;, R) and aj(()Aj(r 4 : 0 in D(~0,/~ ). Here aj and Aj denote respectively the coefficient of the t e r m of highest degree and the dis- criminant of pj as a polynomial in T.

Let U, be the set of all ($~,R,A) C R "+2 with If'0[ < t , R < t , such t h a t there exists a function $"(~'), satisfying (2.6) with [~"($')] < t, I m ~"(~') E -- V]

and P($', ~ " ( $ ' ) ) : - 0 for all $' in ~(~0, R), t

LEM~A 2.7. The function

(5)

f ( t ) : s u p { s ; ~ ( ~ o , R , A ) ~ U , , O < s < A , O < s < R } !

is algebraic for large t.

Proof. I f t h e condition (~, R, A) E U, can be built u p from equations a n d inequalities involving polynomials in real variables, t h e n t h e l e m m a follows f r o m t h e Tarski -- Seidenberg theorem. However, t h e condition (~, R, A) E U, can be expressed in t h e following way:

I~'ol < t, R < t. I n a d d i t i o n there are polynomials pj w i t h d e g p / _ < q0 a n d laj(~')Aj(~')l 2 4= 0 w h e n I~' -- $'01 < R, such t h a t for some ~j0 w i t h ]P~(~0, ~j0)I ~ = 0, j = n ' ~- 1 . . . n, we have I m 9 = I m ( v , , + l , . . . , ~ ) E - - V*, IP(~',~)I 2 ~ 0 a n d I~1 G t for all ~' w i t h

I$' -- ~'01 --< R, if ~i is t h e value for 0 = ~' of t h e unique continuous solution ~ of pi(O, ( r ) = 0 d e f i n e d on the line segment between ~ a n d ~', such t h a t a ^~

Tjo

for 0 = ~0, !

B y L e m m a A.9 in H S r m a n d e r [6] this condition can be expressed in t h e required algebraic form.

Proof of Theorem 2.6. T h e o r e m 2.1 shows t h a t f(t) ~ C log (2 + t). Since f is increasing a n d algebraic for large t, lim~_~o~f(t) exists. L3t A o > lim~_~ f(t) a n d set R o = Ao/~, where y is t h e c o n s t a n t we get in L e m m a A2 in [6] for v = n' a n d M ~ deg (-[-[:,+l(ajAi)), whenever Pi are irreducible polynomials of degree q0. This proves t h e theorem, for a polydisc 9(~0, R) w i t h this radius con- !

tains one w i t h radius A o a n d real centre where a i a n d A i do n o t vanish.

COROLLARY 2.8. I f P ( D ) has a f u n d a m e n t a l solution with support in the closed, convex and semialgebraic cone I ~, then P satisfies the following condition:

(2.7) There are constants A o and R o such that i f ~'o E R ~', ~", ~" E C n~, ~" # 0 and f2(~' 0, R0) ~ ~' - ~ ~(~') E C is an analytic function satisfying the equation P(~', ~" ~ ~(~')~") ~ 0 when ~' C ~(~'o, Re), then there is at least one point ~' E ~2(~ o, Re) such that I m (~" ~- ~(~')~") ~ -- V*.

- - n X

R e m a r k 2 . 9 . A s s u m e t h a t P ( D ) E = 5, supp E ~ / " = {x C R ~ xn,+l > ~,+21 jI}.

I f ~(~,,+2 . . . ~,) E %' (R "+1) is t h e Fourier t r a n s f o r m of E w i t h respect to x ~ , + 2 , . . . , x~, we o b t a i n t h a t P ( D 1 , . . . , Dn,+v ~,,+2, 9 9 9 , $n)/~(~,,+2 . . . ~n)--

~ ( x l , . . . , x ~ , + l ) a n d /~ is a n a n a l y t i c function of ~ , + 2 , . . . , ~ w i t h values in ~'(R~'+~). N o w it follows from TrOves [10] (see also Appendix) t h a t the operator P ( D 1 , - . . , D~,+v ~ , , + 2 , - - . , $,) has c o n s t a n t strength. I t is n o t clear if this follows from t h e conditions in Theorem 2.1.

Remark 2.10. I n t h e p r o o f of Theorem 2.1 (and Theorem 2.6) we h a v e o n l y

(6)

6 ARNE ]~NQWST

u s e d t h e e x i s t e n c e of a local f u n d a m e n t a l solution, i.e., a d i s t r i b u t i o n E s u c h t h a t P ( D ) E = ~ in a n e i g h b o u r h o o d of zero a n d s u p p E ~ 1".

W e f i n i s h t h i s s e c t i o n b y p r o v i n g a t h e o r e m w h i c h shows t h a t t h e r e is no re- s t r i c t i o n in a s s u m i n g t h a t F h a s i n t e r i o r p o i n t s . T h u s , we shall a s s u m e this l a t e r on.

THEOI~E~ 2.11. The operator P(D) has a fundamental solution with support in the hyperplane <x, N> --- 0 i f and only if P(~ ~ tN) ~ P(~) for all t E R, ~ E C ~.

Proof. W e c a n a s s u m e t h a t N - - ~ (0 . . . 0, 1) a n d P(D) = ~,~ a1(D')D..i L e t E be t h e f u n d a m e n t a l solution. L o c a l l y we c a n w r i t e E ~-- ~,o E2(x') | D~, ~(x,).

F r o m P ( D ) E = 6 we o b t a i n t h a t

ai(D,)Ek={(~o(X') for i ~ - O

+~=~ for i > O.

H e n c e , ao(D')E o = 8(x') a n d , if m > 1, am(D')E ~ = O, a,,(D')E~_ 1 ~- a,~_I(D')E, ~-

j - i - 1 ,

~- 0 . . . B y i n d u c t i o n it follows t h a t a a (D)E~_i = 0. T h u s 0 = (a,,(D'))~+lao(D')E o = (am(D'))~+l~(x'), w h i c h implies t h a t a~ = 0 for m > 1.

3. Stability of the necessary conditions

W e shall h e r e p r o v e a t h e o r e m w h i c h gives s o m e i n f o r m a t i o n a b o u t P a t i n f i n i t y .

THEOREm 3.1. Let P(D) satisfy (2.1) with respect to 1" ~ R ~" • V. Assume that ~j E R ~, s 1 ,t i e R, a i E C,, where sj and tj -+ ~ , j - + ~ and that there is a constant N such that 1~1] ~ t~, tj ~ s~, sj < t~ for all j. Furthermore, assume that

Qi( ) = + s/', + t/") j +

I f ~' E R ~" and Qo(~', ~") # 0 for some ~" E R ~", then Qo@, ~") ~ 0 for all ~" E C ~"

with I m ~" E - - i n t V*.

Proof. L e t ~' E R n' be s u c h t h a t deg~, Q0(~0,' ~") > 1 a n d a s s u m e t h a t ~" E {3n', I m $ " E - - i n t V* a n d Q0(~, $") - - 0.

L e t v E C a n d t a k e N " E R ~" s u c h t h a t d e g ~ Q 0 ( ~ , ~ " ~ ~:N") > 1 a n d let bi(~' ) b e t h e coefficient of t h e t e r m o f h i g h e s t degree w i t h r e s p e c t t o 9 of t h e p o l y n o m i a l Qy(~', ~" -~ zN"). N o w , consider Qj(~', ~" ~- zN") as a p o l y n o m i a l o f (~', ~) a n d w r i t e it as a p r o d u c t o f i r r e d u c i b l e f a c t o r s . Set d i : t h e p r o d u c t o f t h e d i s c r i m i n a n t s o f t h e s e f a c t o r s , c o n s i d e r e d as p o l y n o m i a l s of T. W e h a v e t h a t A 1 : Ay(~' ) : bi(~')di(~' ) ~ 0 is a p o l y n o m i a l of $' a n d if ~ : ~(~') is a c o n t i n -

(7)

uous solution of t h e e q u a t i o n Qj(~', ~" + ~N") = 0, t h e n z(~') is a n a l y t i c w h e n Aj(~') 4=- o.

L e t B c r be a ball w i t h centre ~ e I l"' a n d radius r > 0. Then b y L e m m a A2 in [6] there is a b a l l B o c B w i t h centre ~ ' e R " a n d radius y r > 0, such t h a t Ao(~' ) 4= 0 in B o. F u r t h e r m o r e , for e v e r y j there is a ball Bj C B 0 w i t h centre ~ E R ~' a n d radius y2r > 0 such t h a t Aj(~')4= 0 in 17i. This implies t h a t there is a ball /~ c B 0 w i t h real centre a n d radius y2r/2 = ylr > 0 a n d a subsequence Aik of A i such t h a t Aik(~' ) 4= 0 in /3 for all k. I n order n o t to complicate t h e n o t a t i o n s we assume t h a t this is t r u e for t h e whole sequence, i.e., Aj(~') ~ 0 in /~ for all j > 0 .

Now, consider t h e solutions of t h e e q u a t i o n Q0(~:, ~" -t- "oN") = 0, $' E/~.

These are a n a l y t i c in ~t a n d if the radius r > 0 above is small enough, t h e r e is a c o n s t a n t e > 0 such t h a t some of t h e solutions, s a y %, satisfies t h e condition I m (~" + Zo($')N ~) e -- V* for all ~' e B. I f ~0($') has t h e m u l t i p l i c i t y /, w h e n

$' E/~ a n d if U c 13 is a small n e i g h b o u r h o o d of zero t h e n Rouch6's t h e o r e m shows t h a t t h e e q u a t i o n Qj(~', ~ " + T N " ) = 0, ~ ' E / ~ has e x a c t l y /z solutions in z0(~') + U for large j . L e t Tj be one of these. T h e n Tj --> z0 u n i f o r m l y on /3 when j --> oo.

if j

a n d

This implies t h a t

I m (~" + zj(~')N") E -- V~2 for all is large. Thus, for large j

P(~ + sj~', ~: + tj(( +

~ / ( ) N " ) ) = o,

~'eB

for all ~ ' e / ~

I m (~: + tj(U + ,j(U)N")) e -- V*~/2 for all ~' 6 / ] .

ff

I t follows t h a t (2.1) is n o t valid since t h e n we would h a v e rain (tje/2, sj~lr) < C log (2 + [$j[ + Cl(sj + tj)) =

= 0 ( m i n (log tj, log sj)), j ---> 0%

which implies t h a t

rain (s, r) = O.

This completes t h e proof.

COROLLARY 3.2. Let P(D) have a fundamental solution with support in 1" =

= lt" X V and let p be the principal part of P. Then for every ~' e tl n" we have that p(~', D") is either hyperbolic with respect to V or zero.

(8)

A R I~TE :E]~Qu

4. Reduction of existence theorems to a priori estimates

I n this section we shall assume t h a t P satisfies t h e n e c e s s a r y conditions of T h e o r e m 2.1. L e t k E ~)/(R n) so t h a t for some c o n s t a n t s C a n d N

k(~ § V) --< k(O(1 + C l~I) ~.

(See p. 34 in [5].) I f 1 < p ~ oo we shall use t h e n o r m

( f

IluH~,~ = ( 2 ~ y " l k ( ~ ) a ( ~ ) l P d ~ , u e S ( R ~

As beibre, let /~ ~ R n" • V be a c o n v e x cone w i t h i n t e r i o r points. (Cf. T h e o r e m 2.11.) T h e n we are i n t e r e s t e d in t h e q u o t i e n t n o r m s d e f i n e d b y

l]ullpr.~k=inf{llvllp, k; v = u on / ' _ = - - / ' , v e ~ ( R " ) } , u e ~ ( R " ) .

Tm~o]cEM 4.1. Let P be a p o l y n o m i a l and a an element of c)s such that for every compact set K there is a constant C for which

U F

]l ]]1.1 <_ C ][P(n)u]l[-1/,, for all u C C : ( K ) .

Let 1 < pj .< ~ .a n d .kj e Oi(R"), . j = 1, 2, . T h e n for every f E B~,kj(R"~176 ~), j = 1, 2 . . . . , with s u p p f c F there is a solution u to P ( D ) u ~ f such that

E B l~ / ~ ' ~ . . . .

s u p p u C / ~ a n d u ~j,~ki~.,/, j--~ 1 , 2 ,

T h e p r o o f is r a t h e r long so we f i r s t p r o v e t h e following local version.

T ~ E o ~ E ~ 4.2. Let a e =X(R'), ~ c c R" and let P be a p o l y n o m i a l for which there is a constant C such that

U E

]] Ii~,~ ~ C I]P(D)u]]~-~/, for M1 u e C : ( - - / 2 ) .

T h e n there is a u C B| with s u p p u c I ~ such that P ( D ) u = ~ in ~ .

Proof. T h e e q u a t i o n P ( D ) u = (~ in Q m e a n s t h a t ~(P(D)v) -~ v(O) for all v e C ~ ( - - 9 ) . W e h a v e t h a t

Iv(0)] < HvHr-~ < C I]P(D)vH~-~/,, v e C : ( - - Y 2 ) . T h e n t h e linear f o r m

P ( D ) v - + v(O) on P ( D ) C ~ ( - - t 2 )

can be e x t e n d e d b y t h e H a h n - B a n a c h t h e o r e m to a linear f o r m g on C~(R~) such t h a t

I?~(W)] ~ CllWl]FI,-1/a

T h u s u is a d i s t r i b u t i o n w i t h s u p p o r t in /~, such t h a t P ( D ) u ~- 5 on .Q n a p u C Bo~,..

(9)

O ~ ~ U ~ D A M E ~ T A L S O L U T I O N S S U P P O R T E D B Y A C O N V E ~ C O N E

Using local f u n d a m e n t a l solutions of t h e t y p e i n t r o d u c e d in T h e o r e m 4.2 we can p r o v e t h e following a p p r o x i m a t i o n t h e o r e m .

T~EOREM 4.3. Let [22 C [22 be bounded open sets in R ~ and let P satisfy the conditions of Theorem 4.2 for every [2 c c R ~. Set _ F ~ int T' and denote by N 1 the set of solutions u C C+(12i) of the equation P ( D ) u = O, such that supp u c D 1 [3 F. Furthermore, let N i have the topology induced by C~ I f

# E ~g'(R"), F ~ 13 supp/~ ~ ~ , 1. ~ [3 s u p p P ( - - D ) # c ~ [22 ~ 1"~ Cl supp # c ~ [22, then the restriction to I-21 of the elements in N 2 f o r m a dense subset N': of N~.

Proof. I f we p r o v e t h a t e v e r y v C c~)'(~(~1) which is o r t h o g o n a l to N~ is also o r t h o g o n a l t o N 2, t h e s t a t e m e n t will follow f r o m t h e H a h n - B a n a c h t h e o r e m .

L e t [2a C D~ be o p e n b o u n d e d sets such t h a t [22 C C [2a a n d G [24 -t- [2a c 0 /~a. F u r t h e r m o r e , let E be a local f u n d a m e n t a l solution in [2a, i.e. P ( D ) E ~-- in [ 2 a , s u p p E C F . Set u = ~ 0 * E w h e r e ~0 E C~(1. [3 (0122) [3123). T h e n P ( D ) u ~--0 in s a n d s u p p u c F. T h u s

0 = v(~v 9 E ) = qp 9 E * ~(0) ~ (E 9 v)(~v), q~ E C F ( F N (C[22) Ct [23)"

L e t Z C C~~ be i in a n e i g h b o u r h o o d of ~ 2 a n d set # ~ Z ( ] ~ . v). T h e n F ~ gl supp # C ~ , F ~ Cl supp P ( - - D ) # C C ~2"

T h u s b y h y p o t h e s i s

/~~ I3 supp # C C [22.

H e n c e we can choose ~o E C~([21) such t h a t ~0 ~ 1 in a n e i g h b o u r h o o d of F ~ I f we set #2 = ~v# ~ yJ(/~*v) a n d v 1 ~ - P ( - - D ) # 2 we o b t a i n t h a t #1 E ~8'([21), v - - v 2 E ~'([21) a n d _P~ [3 supp (v - - v2) = O, which implies

( v - - v 2 ) ( u ) = 0 a n d v2(u ) = # 2 ( P ( D ) u ) ~-- 0 for all u E N 1. T h u s v(u)--~ 0.

LEMMA 4.4. There is a set [2 C R ~ such that i f D r ~ v[2 then the p a i r of sets D r and [2+: satisfies the hypotheses in Theorem 4.3 for all v > 1.

Proof. L e t p be t h e principal p a r t of P a n d let c o > 0. Set N 0 = (0, No),

,, . . ,, . r

N 2 ~ (Ni, coN1), . , N , = (N',,~oN~), w h e r e N o , . . , N ~ E i n t V * a n d p , 0 for j--~ 1, 2 . . . . , n. F u r t h e r m o r e , we choose t h e v e c t o r s Nj so t h a t e v e r y ~ E R n can be w r i t t e n as a linear c o m b i n a t i o n o f - - N o , N 1 , . . . , N , w i t h n o n - n e g a t i v e coefficients. (Observe t h a t this c o n d i t i o n is i n d e p e n d e n t o f co > 0.) Set

Q - - ~ { x E R " ; <x, N j > < ] , j : l , 2 , . . . , n, <x, N 0 > > - - I }

a n d D~ = v[2. W e shall p r o v e t h a t t h e s e sets will do if co > 0 is large enough.

(10)

l 0 A R N E ENQVIST

I t is clear t h a t s is a b o u n d e d n e i g h b o u r h o o d o f zero. W e n o w w a n t to s h o w t h a t

# e ~3'(R~), f ~ N s u p p # c D~+2, F ~ 13 s u p p

P(--D)/~ c c ~9~ ==>

f o 13 s u p p # C C ~ .

Set

Uj(s)

: {x ^{6 R ~}; x" ⁶i n t V, <x, N i > > s},

j

= 1, 2, . . . , n. T h e n t h e s t a t e m e n t a b o v e is a c o n s e q u e n c e of t h e following:

# 6 % " ( R ~ ) , # : 0 in

Uj(s), P ( - - D ) / ~ - O

in

Uj(t)=>#---- 0

in

Ui(t )

(j = 1 , 2 , . . . , n).

F r o m T h e o r e m 5.3.3 in H h r m a n d e r [5] we n o w get t h a t it is s u f f i c i e n t t o p r o v e t h a t e v e r y c h a r a c t e r i s t i c p l a n e t h a t i n t e r s e c t s

Ui(t )

also m e e t s Uj(s). A p l a n e t h a t does n o t m e e t

Ui(s )

h a s a n o r m a l t h a t lies in t h e d u a l cone U~ o f

Ui(O ).

H o w e v e r , U* is t h e c o n v e x hull of

{hNj ; h >

0} tJ F * , since t h i s is closed.

T h u s we h a v e t o p r o v e t h a t if

N = ( O , N " ) C F *

t h e n

p ( h N j - k N ) 4:0

for all h > 0. ( N o t e t h a t a p l a n e w i t h n o r m a l in F * is d e f i n e d b y a n e q u a t i o n in t h e x"

v a r i a b l e s , so it m e e t s

Uj(s)

if a n d o n l y if it m e e t s

Ui(t). )

H o w e v e r (4.1)

p ( h N j ~ - N ) = h m p ( N ~ , c o N j ' + h - ~ N ") =#0

f o r large o~ a n d h > 0.

To see t h i s we f i r s t o b s e r v e t h a t

h-iN" 6

V*, Nj' 6 V* a n d

p(N~, D")

is h y p e r - bolic w i t h r e s p e c t t o V b y R e m a r k 2.10 a n d T h e o r e m 3.1. I t follows f r o m T h e o - r e m 1.3 in [9] t h a t also

p(N~, --iD")

is h y p e r b o l i c w i t h r e s p e c t t o V, w h i c h b y T h e o r e m 5.5.4 in [5] implies (4.1).

Proof of Theorem

4.1. L e t ~O b e t h e sets d e f i n e d in L e m m a 4.4 a n d l e t

~% 6 C~~ be 1 in a n e i g h b o u r h o o d o f ~9~_ r T h e n it will be s u f f i c i e n t t o p r o v e t h a t t h e r e e x i s t % 6 [-]~o Blo~ pj, akj s u c h t h a t

II~,~(u~+, - u~)ll~j,o~ _< 2 -~, ~ _< ~, j _< ~,

P(D)u~+ x = f

in f 2 + 2 a n d s u p p u ~ + 1 c F.

I n f a c t , f o r s u c h u~ we o b t a i n t h a t u - - > u in

B~aki

as v - + oo, w h e r e

P(D)u ~ f

a n d s u p p u C / ' .

S e t u 1 =- E 9 (~3f), w h e r e E is a local f u n d a m e n t a l s o l u t i o n in a large neigh- b o u r h o o d of zero. T h e n

ui 6 rl~ Bpi, ak i, P(D)ul = f

in /2 2 a n d s u p p u 1 C /7.

W h e n u~ . . . . , u~ are chosen, we w a n t t o c o n s t r u c t u~+x. T h e n t h e r e is a dis- t r i b u t i o n

Vo 6 ['1~ Bpi,,k j

s u c h t h a t

P(D)v o ~ q~+af ~ f

in ~ + 2 , s u p p v 0 c F.

I f v = v 0 - - %, t h e n

P(D)v =- P(D)v o -- P(D)u~

= 0 in D~+I- Choose ~ 6

C~(F)

so t h a t s u p p to c --~62 1 a n d H~,(to * v - - v)[[vj, akj ~< 2-~-~ for # _< v, j < v. I t follows f r o m T h e o r e m 4.3 t h a t t h e r e is a C ~ f u n c t i o n w s u c h t h a t

P(D)w ---- 0

in f 2 + : , s u p p w c F a n d I I ~ , ( w - t o * v ) ] l e i , , k

i _ < 2

-~-1 f o r # < ~ , j < v . Set u~+ l : v o - w . T h e n

(11)

O~ F U N D A M E N T A L S O L U T I O N S S U P P O R T E D B ~ A C O ~ V E X C O ~ E

II~,(U.+l - u.)ll,j..kj -< 2-". /~ _< ~, j < ~, a n d

supp %+~ C F.

This proves t h e theorem.

P(D)%+~ = f in ~9+ 2

11

5. Sufficient conditions

I f t h e operator P(~', D") is hyperbolic w i t h respect to V for sufficiently m a n y

~' E r t h e n we can prove t h e existence of a f u n d a m e n t a l solution w i t h support in F = R =" • V. The following definition helps us to describe sets which suffice.

Definition 5.1. L e t 0 < ~ < 1 a n d let B C C " be a ball w i t h radius R a n d centre ~ . W e s a y t h a t a subset S of B is of type N~ if for an a r b i t r a r y , logarithmicMly plurisubh~rmonic f u n c t i o n g > 0 t h e following i n e q u a l i t y is t r u e

g((o) <-- (sup g)~-~(sup g)~

B S

(Cf. L e m m a 3.2 in [6].)

THEOREM 5.2. Let P be a polynomial in n = n' ~- n" variables and let N =

= (O,N#), where N " E V * = V*. Furthermore, let a($') ~ 0 be the coefficient of the term of highest degree of P($ + "vN) as a polynomial in T. Then the operator

l o c ~ a n "

P(D) has a fundamental solution E E B~o,a with s u p p E ~ / ' = x V if P satisfies the following condition:

(5.1) There are constants R > O, A and 8, 0 < (5 < 1, such that in every ball B with radius R and real centre ~0, t there is a subset S~o, of type N~ for which it is true that ~' E S~o,, I m p " E - - V~ ~ P ( $ ) ~ : 0 . I t is sufficient to prove t h e t h e o r e m ibr A = --1. (Cf. page 349 in [6].) F o r 1 < p < ~ , kE~)~(R ~') we set

Hul{pv-~ = inf{llvllp, k; v = u on V_ = - V , v eS(R~")}, u e 3 ( R ~ ' ) , where

\l/p

LEMMA 5.3. Let Q be a polynomial in n" variables with Q(~") # 0 when I m ~" E -- V*I and let ]c E ~?~(R~'). Then there is a constant C depending only on A, n" and deg Q such that

V _

Ilullp,~k < CllQ(D")ull~-k for all u E C~(Rn~).

(12)

12 A R N E E N Q V I S T

P r o @ Set

= o f )

Then s u p p E ~ V, Q(D")E = ~ and

,E(~,, _< C f *~(e")]/0(~") de",

where C only depends on A, n" and deg Q, t h a t is IIEIl~.g~ < C. I f g = Q(D")u on V_, then E . g = u on V_ so t h a t

Ilu]l~-~ _< liE * gig, ~ --< C Ilgllv, which shows t h a t

v_ < C "

IIullp, ~ _ IIQ(D )ullv.~ for all u ~ C~(R~").

LE~MA 5.4: I f h(~',x") is analytic in ~' and k C%~(R ~') then ]lh(~', ")llev,-~

is logarithmically plurisubharmonic.

Proof. Ilh($', ")llp~ is the norm of an analytic function with values in a Banaeh space, hence it is logarithmically plurisubharmonic.

LEMMA 5.5.

If

the polynomial P satisfies condition (5.1) with A = --1 and K c C R ~, then there is a constant C such that

sup II~(U, .)lIpv-~ < C sup [[P(~', D")~(U, ")llp~-~

R n t R n '

for all u e C : ( K ) , k e r Here ~(~', .) is the Fourier transform of u with respect to x'.

Proof. For ~'E St,, we have

ia(U)I II~(~', ")I[pv,-~ ~ C lIP(U, D")~(U, ")[[p,v-k < C G

where G = supRn, [IP(~ ', D")~(~', ")[Ip, k. The first estimate follows from L e m m a v_

5.3 and the second from the fact t h a t the function

~'-~ HP(r D")~(r .)[/,~-~

is logarithmically plurisubharmonic and of exponential type. The properties of the set S~0. now show t h a t

,, _ _ "'lr[V- "11--~/'~,~ <

la(~;)[ Ilu(~0, ")]LF,r~ < C ( s u p ra(~')l I1~(~', ,,~p,k, - _

]~'-$o'] _<R

_ < C ( s u p la(~')]

I1'~(~', ~,,~.k~

. ~ ' l V - ~ l - ~ G a , 0 < 6 < 1, I I n t

(13)

ON FUNDAMENTAL SOLUTIONS SUPPORTED B ~ A CO:XW-EX CONE 13 which implies t h a t

la(~')I I1~(~', .)11~-~ _ CO, ~' r R ' . H o w e v e r , b y L e m m a A 1 in [6] t h e r e is a 0 E A such t h a t

a(~') <_ 6' a(~' + zo) < C'

[a(~' + z0)I, z e C, [zl = 1.

T h u s

Since Ilu(~' +

zo,

.)11~ is s u b h a r m o n i e as a f u n c t i o n o f z we n o w o b t a i n t h e re- q u i r e d e s t i m a t e .

LEM~rA 5.6. Let P be a polynomial and b C ~ ( R " " ) . I f for every compact set K, there is a constant C 1 such that

(5.2) sup/1~(~', ")tip, v_ bk ~ C1 s u p I l P ( ~ t , D")~(~', 9 )II~,~, v_

i~.n" Rrt"

for all u e C~(K), k C ~)s then for every k C 7s

n)

there is a constant C such that (5.3)

Ilull~,

r_ ^{~ <_}c llP(D)ul]~- k for all u e C~o(g).

Proof. See t h e p r o o f o f T h e o r e m 3.10 in H S r m a n d e r [6].

Proof of Theorem 5.2. T h e t h e o r e m follows i m m e d i a t e l y f r o m L e m m a 5.5, L e m m a 5.6 a n d T h e o r e m 4.1.

6. Partially homogeneous operators L e t P be h o m o g e n e o u s in t h e ~" variables, t h a t is

P(~) = ~ a ~(~ )~ 9 ' "

lal ==

I f P(D) has a f u n d a m e n t a l solution w i t h s u p p o r t in A = {x ~ R n ; x n , + l >

c ~ , + 2 IxjI}, c > 0, t h e n b y R e m a r k 2.9 we k n o w t h a t t h e o p e r a t o r P ( D 1 .. . . . D~,+> ~,,+2, 9 9 9 , ~ ) has c o n s t a n t s t r e n g t h . This implies t h a t b(~') = a(~,0 ... 0)(~') is s t r o n g e r t h a n % for all ~, i.e. t h e r e is a c o n s t a n t C such t h a t

~ ( ~ ' ) < C b(~') for all ~ a n d all ~ ' C R ~'.

(6.U Set

= • = {Q ; Q(~") = limj.+~ (P(~j, ~")/b(~j)) for some sequence ~j E R "' such t h a t d ( ~ , b-l(0)) --~ m a s j - ~ m},

(14)

14 A R N E : E I ~ Q V I S T

w h e r e d(~j, b-a(O)) is t h e distance in C" f r o m ~ t o t h e zeros o f b. W e a s s u m e here t h a t b is n o t a c o n s t a n t , for this w o u l d i m p l y t h a t P is a p o l y n o m i a l in ~"

which is a trivial case. T h e set 2; is a c o m p a c t subset of t h e set o f all p o l y n o m i a l s o f degree m. F u r t h e r m o r e , we h a v e

THEORElV[ 6.1. I f the polynomial P(~) : ~1~[:0~ a~(~ )~ ' "~ satisfies (2.1) and ao~

is weaker than b for every a, then Q(D") is hyperbolic with respect to V for every Q in X~,.

Proof.

(Cf. the p r o o f of T h e o r e m 2.2 in H S r m a n d e r [7].) If P ( ~ ,

~")/b(~]) -+ Q(~"}

as j - + oo, t h e n w e c a n a s s u m e that e v e r y coordinate of ~i has fixed sign. I n p

order n o t to complicate the notations w e a s s u m e that all these coordinates are n o n - negative. It follows f r o m the Tarski-Seidenberg t h e o r e m that if Q ( ~ " ) = Z q ~ "~

t h e n

a ! ! . ,

(6.2) i n f { v I ;27 ] ~(V )/b(~7 ) -- q~L ~ <= 1/t, d(v', b-l(0)) ~ t, V~ ~ O , . . ~], ~ 0}

is an algebraic f u n c t i o n of t for large t. B y r e p e a t e d use of t h e same t h e o r e m we get t h a t t h e i n f i m u m o f n o n - n e g a t i v e ~2, w h e n t h e i n f i m u m in (6.2) is a t t a i n e d , is a n algebraic f u n c t i o n o f t for large t, a n d so on. F o r large t we h a v e t h e P u i s e u x series e x p a n s i o n

k~

v ' ( t ) =

0 9

--oo

a n d P(~'(t),~")/b(~'(t))-+Q(~") as t - + ~ . L e t s ~ O a n d consider P(#'(t) ~- t ~ ', ~")/b(~'(t)) = ~ . ( ~ a~)(#'(t))t~!zl~'~/fi!)~"~/b(~'(t)).

a t~

Since a s is w e a k e r t h a n b a n d d(v'(t), b-l(0)) ~ t it follows f r o m T h e o r e m 3.3.2 a n d L e m m a 4.1.1 in [5] t h a t a~)(#'(t))/b(~l'(t)) : O(t-I~l), t - + ~ , if fi 4: 0. T h u s , if 0 ~ s ~ 1 we h a v e

P(~l'(t) -~- t ~ ', ~")/b(~'(t)) :

= P(~'(t) + t ~ ', t~")/t"b(~'(t)) --+ Q@'), t -+ ~ . N o w t h e t h e o r e m follows f r o m T h e o r e m 3.1.

W e are going to s t u d y t w o special cases of partially h o m o g e n e o u s operators.

First w e will consider the case n" = 2 a n d later o n operators, s u c h that

Q(D")

is

strictly

hyperbolic for all Q c Xp. T h u s , let n" = 2 a n d P(~) = ~r~1=~ a~(~')~"%

First w e reformulate the necessary conditions.

T~EORE~ 6.2. Let P ( r Z a ( ~ )r where b : a(m,o ) is stronger than aa for all ~. Assume that P satisfies the necessary conditions of Theorem 2.6 with

(15)

ON ~ U N D A M E N T A L S O L U T I O N S S U P P O R T E D B Y k C O N V E X C O N E 15 respect to R " • W and let V = {x" ~ R ~ ; x++~ ~ Co ~ be a cone ~ W such that every Q ~ Z is hyperbolic with respect to V. T h e n for every c ~ c o there are constants R o and t such that i f P(~', ~(~'), 1 ) = 0 in a ball B with real centre ~' and radius R o, where ~ is analytic, then we have I m ~(~') = 0 for some ~' ~ B . Furthermore, l~(O')l ~ c i f d(O',b-~(O))~_t and O ' e B .

Proof. Assume t h a t d(~:, b-~(0)) - + ~ a n d P ( ~ , ~")/b(~:) --+ Q(~") E Z as v - + zc. I f P(~: + $', ~,(~'), 1) = 0 for ]~'] ~ R 0, t h e n we o b t a i n from Rouchd's t h e o r e m (cf. t h e p r o o f of T h e o r e m 3.1) t h a t there is a subsequence of %(~'), which converges to 2 as ~ - + 0o, u n i f o r m l y for I~'] ~ no, where Q(2, l) = 0, so t h a t

12[ ~ c 0. This proves the last s t a t e m e n t . W r i t e P as a p r o d u c t P(~) = b(~') I ~ (;,,,+~ -

~(;')~~

1

F r o m T h e o r e m 2.6 w e n o w obtain that there are constants A a n d R o s u c h that for e v e r y ~,,+2 E R there is a point ~' w i t h I~' -- ~'i ~ R o s u c h that I m ~,+2T(~') ~_ A . If w e let ~++2--> + oe w e see that there is a point ~' w i t h

I~'-- ~'I ~ R o s u c h that I m ~ ( ~ ' ) ~ 0.

THEOREM 6.3. Let P ( D ) satisfy the conclusions of Theorem 6.2 and set W ---- {x" C R 2", x,,+x ~_ c 4 ix,,+21}, where c is the constant in Theorem 6.2. T h e n the

nlor with support in A R " • W.

operator P ( D ) has a f u n d a m e n t a l solution E E - +, b

The p r o o f is similar to t h e p r o o f of Theorem 1.1 in H S r m a n d e r [6]. W r i t e P($', $,,+1,1) as a p r o d u c t of irreducible factors a n d let A be t h e p r o d u c t of their dis- criminants, w h e n t h e y are r e g a r d e d as polynomials of $++v Set R ~ ( b ( ~ ' ) A ( ~ ' ) ) M

where M is a large integer a n d set for v C C ~ ( R ~)

N~(v, ~') : m a x IR(C')bC)I [ l ~ (D++I -- Ti(C')D~'+2)v(C,^ ' ")l[pS,~

1

where t h e m a x i m u m is t a k e n over all labellings of t h e zeros ~i" (k E ~)s LnMMA 6.4. The f u n c t i o n N~(v, ~') is logarithmically plurisubharmonic.

Proof. See t h e p r o o f o f L e m m a 3.6 in H 6 r m a n d e r [6].

LEMM~_ 6.5. Let v C C~~ n) with v(x) = 0 for Ix' I ~ H . T h e n N ( V , ~') ~ e I-/lIm ~'1 SUp ~ 7 ( v , ~ ' ) , ~ --- O, 1 . . . m .

a n '

Proof. See t h e p r o o f of L e m m a 3.7 in H S r m a n d e r [6].

(16)

16 AI~NE ENQVIST

L~MMA 6.6. Let P(D) be as in Theorem 6.2. Then for every compact set K there is a constant C such that

s u p N , ( v , ~ ' ) < C s u p N ~ + l ( v , ~ ' ) ,

vEC~(K),

^{v =} O, 1 , . . . , m - - 1.

i~.rd Rn ~

Proof. B y L e m m a A1 in H 6 r m a n d e r [6] we first choose a finite set A c R"' for polynomials of degree < d e g P . L e t B = R 0-~ 3. F o r e v e r y ~ ' C R n" a n d B > 0 we can f i n d a 0 E A such t h a t t h e distance from ~'~-zO to the zeros of R is a t least B w h e n [ z I = B . L e t

0 • = { ~ ' @ z O ; I z l = B , I m z ~ 0 }

a n d denote b y zg~, j = 0, l, 2, t h e points a t a distance < R o -~ j from O • I n w h a t follows we shall work in t h e sets /2+; t h e same a r g u m e n t s can be applied in t h e sets Y2j .

F r o m L e m m a 5.3 above a n d L e m m a 3.2 in H S r m a n d e r [6] we o b t a i n t h e following estimates

sup IR(~')b(~')[ II ]z~ ( O - % 1 _ "vJ(~')Dn'+ 2)~)(~', ")Hp, w-k <~

D+ 1

(sup N v ( V ,

~t))X--d( SUp

[R(g')b(g')l ][ ~ (D,,+I -- ~j(~')Dn,+2)v(~ , ^ ' ")llp,~)~ ~ ~<

Im rv+ 1 ($') =0

v + l

__< (sup N(V, ~,))l-a( SUp ]R(~')b(~')] ]1 ~-~ (D,r -- 5(~')Dn'+2)v(g,~ ' ")[11,,~)~ a

Im vv+ 1 (~')=0

< (SUp Nv(V, ~'))l-~(sup N.+x(V , ~'))~ < C (sup N ( v , ~'))l-~(sup N~+l(V , ~'))~.

H e r e 0 < 6 < 1; t h e first i n e q u a l i t y follows from L e m m a 3.2 in H S r m a n d e r [6], t h e second is a consequence of L e m m a 5.3, a n d the f o u r t h follows f r o m L e m m a 6.5.

I f we n o w t a k e t h e m a x i m u m of t h e left side over all labellings of t h e zeros we o b t a i n t h a t

sup N~(v, ~') < C (sup ~V(v, ~'))~-~(sup N~+~(v, ~'))~.

DO+ R n' Rn t

The same e s t i m a t e holds for /2 o a n d since the f u n c t i o n N~(v, ~') is plurisubharmonic, we can use the m a x i m u m principle for the f u n c t i o n N~(v, ~ ' + zO).

We o b t a i n t h a t

sup N~(v, ~') < C (sup/V~(v, ~'))~-~(sup ~+~(v, ~'))~,

Rr~P R n ' R n '

i.e.,

(17)

O N F U N D A ~ I ] ~ N T A L S O L U T I O N S S U P P O R T E D B Y 2~ C O N V E X C O N E 17 sup

N,(v, ~') < C

sup N +~(v, ~').

L ] ~ M ~ 6.7.

Let P(D) be as in Theorem

6.2.

Then for every compact set K there is a constant C such that

sup _R(~')g(~') I1~(~ ', .)H~.W~ ~ C sup ]7(~') IIP(~', D")~(~', .) H~,W~, if

v ~. C~(K).

R n ' B n '

Proof.

R e p e a t e d use of L e m m a 6.6 gives t h a t

sup IR(~')b(~')i Ii~(~', ")Jl~,~ = sup

No(V, ~') <

R n t R n '

< C sup

N,,(v, ~') = C

sup IR(~')I liP@,

D");;(~',

.)][p wZ <

l l n ' l t n '

_~ C sup _R (~') HP(~ ',

D")~(~',

.)[I,,WT~ = C G.

R n #

B y L e m m a 6.5 we h a v e

IR(~')b(C')I

II~(~', ")lip,w% _~

C G

for jim ~'l --~ const.

L e t A be t h e set we get f r o m L e m m a A1 in H S r m a n d e r [6] w h e n we a p p l y it to polynomials of degree _~ deg

(Rb).

F o r e v e r y ~' there is a 0 C A such t h a t

/~(~')g(~')

<_ C ~(~' -~ zO)g($' + zO) <_ C' ]R(~' ~- zO)b($' @ zO) l

for Izl = 1. This gives t h a t

~(~')b(~') II~(~' @

zO,

.)Hv,Wz < C G, lz[ = 1, which implies t h a t

where

~(~,)g(~,) ,A , jjv(~, ")11~,r < c G,

C is i n d e p e n d e n t of ~'.

Proof of Theorem

6.3. L e m m a 6.7 a n d L e m m a 5.6 show t h a t for e v e r y c o m p a c t set K a n d e v e r y k E ~ ( R n) there is a constant C such t h a t

[lullp,~gk ~ C IlPKn)ull~.~k, A_

u C C~(K),

so the t h e o r e m follows f r o m T h e o r e m 4.1.

We h a v e not been able to prove a n y analogues of T h e o r e m s 6.2 a n d 6.3 for a r b i t r a r y n", so we m u s t p u t some e x t r a conditions on t h e polynomials w h e n n" > 3. Therefore, we shall now s t u d y polynomials satisfying t h e following condition:

(18)

1 8 A R N E E N Q V I S T

(S) P(~)--~ ~.1~1-~ a~(~') ~'~ has real coefficients,

b--~ a(~.0 ... 0) is stronger t h a n a~ for all er a n d Q(D") is strictly hyperbolic w i t h respect to V for e v e r y Q E 2:.

L e t W be a convex cone such t h a t int W D V ~ {0} a n d set A --~ R"' • W. T h e n we can prove

THEOR]~M 6.8. I f P satisfies condition (S) then the operator P(D) has a fun- damental solution E E B 1~ oo, Y, with support in A.

F o r technical reasons we will consider Q o ( E ) - - P ( ~ ' , ~ " - - i N " ) where N " E i n t W*.

TH:EOR]~ 6.9. Let K C R ~ be a compact set and k E ~)<(R").

polynomial introduced above, then there is a constant C such that IlulI~,~ok ~-- C tiQo(D)uI[~,-k, u E C~(K).

I f Qo is the

Proof of Theorem 6.8. Set k(~) ~- 1/Q0(~ ). T h e n it follows f r o m T h e o r e m 6.9 a n d T h e o r e m 4.1 t h a t Qo(D) has a f u n d a m e n t a l solution E 0 E Bl~ w i t h s u p p o r t hi~ w i t h sup- in A. This implies t h a t P(D) has a f u n d a m e n t a l solution E E ~|

port in A. (Cf. page 349 in H S r m a n d e r [6].)

L~MMA 6.10. Assume that Q(D") is strictly hyperbolic with respect to V for every Q in X. Then there is a neighbourhood of Z, in the set of all real polynomials of degree m, such that all polynomials in this neighbourhood are strictly hyperbolic with respect to the cone W i f i n t W ~ V ~ { 0 } .

Proof. W e h a v e to prove t h a t there is a n e i g h b o u r h o o d of 27, in t h e set of all real homogeneous polynomials of degree m, such t h a t e v e r y p o l y n o m i a l Q in this n e i g h b o u r h o o d satisfies t h e following conditions:

Q(N") # 0 for all N " E W * ~ { 0 } .

The e q u a t i o n Q(E" + -oN") = 0, where N" E W*,

JN"] ~-- 1, ~" E R'", ]~"] = 1 a n d N" _1_ ~", has o n l y real simple zeros.

However, this is obvious for e v e r y p o l y n o m i a l in 27 satisfies these conditions a n d 27 is compact. (Cf. D e f i n i t i o n 5.5.1 in H S r m a n d e r [5].)

L e t P satisfy condition (S) a n d set Q~(~) = I(P(E', ~" - iN") + P(~)(~', ~" -- iN")) where N" E i n t W* a n d a - - ~ ( a ' , 0). The first step in t h e p r o o f of T h e o r e m 6.9 is t h e following lemma.

(19)

O ~ F U n D A M E n T A L S O L U T I O N S S U P P O R T E D B Y A C O ~ V E X C O ~ E 19 LEMMA 6.11. There is a constant C such that

9 I ~ _ #

I1~(~', )11~, ~0,~ ~< c ~ IIQ~<(8, D"),7(~',

.)ll,,~~

~=(~', o)

for all ~' ~ R ~', u~C~~ ~) and k~%(R~').

To prove this we need a l e m m a due to L. H S r m a n d e r .

LEMMA 6.12. Let s C C" be a convex set and let Q1 . . . Qm of degree < 2 5 , such that Qj(~) =4= o, j : 1 , . . . , m, when ~ ~ ~ . f u n c t i o n s gl . . . . , g,,, that are analytic in ~ , such that

and

1 = Z'Qj(~)gj(~), ~ C ~,

m

lm(~)l < C l ~ tQx(~)l, ~ex~.

1

Here C is a constant that depends only on m and 25.

be p o l y n o m i a l s T h e n there are

Proof. The v a r i a t i o n of t h e a r g u m e n t of Q1 is a t m o s t ~(N 1) w h e n ~ E 12, for i f $1, $ 2 e a 9 t h e n Qj(t~lJr- ( 1 - t ) ~ 2 ) = a I I ( t - & ) , where & ~[O, 1]. Since .(2 is s i m p l y connected we can choose a n a n a l y t i c b r a n c h of 01IN wj in ~ . W e o b t a i n t h a t t h e v a r i a t i o n of t h e a r g u m e n t of O1./N _-t/# is < az(N ~ 1)/N < ~. T h u s t h e r e are constants aj C (3,. tajl ~- 1, such t h a t Iarg (ajQ)/N)] < u ( N ~ 1)/2N < ~ / 2 w h e n ~ E ~9. This implies t h a t there is a c o n s t a n t c > 0 such t h a t

c QIlN

I ; I < l~e (a~Qjn,) < tQSI, ~ e ~.

Set q : ~ "~ aj~gj ,~IlN . T h e n

(6.3) c Z IQjl 1IN < R e q ~_ lqi ~ X

IQjl 'IN

a n d

q=N (E

ajQ) N) = Qjhj.

1 1

T h u s

1 : ~ Qj(~)gj(~), ~ E D where gj(~) = hj(~)fl(~) -m~.

T h e f u n c t i o n hj is a s u m of t e r m s of t h e f o r m II(akQ~lN), where t h e p r o d u c t s consist of N m - N factors. Thus, according to (6.3)

lhji ~ C'xlq[ Nm-N,

m

Igj(~)l ~ C~lq(,~)l -~ ~ Cl(~, IQjl),

1

which shows t h a t

where t h e c o n s t a n t C depends o n l y on N a n d m.

(20)

20 ATONE ENQVIST

Proof of Lemma

6.11. We are going to a p p l y L e m m a 6.12 to t h e polynomials Qa"

Since

(a(fl(~')/b@)) -+ 0

if I~[ 4= 0 a n d

d(~',

b-~(O)) - + oo, we o b t a i n from L e m m a 6.10 t h a t there exists a c o n s t a n t t > 0 such t h a t Qa(~', ~") g= 0 if I m ~" E N" --

- - i n t

W*

a n d d @ , b-~(O)) >_ t. L e m m a 6.12 shows t h a t if d(~', b'~(O)) > t t h e n there are a n a l y t i c functions ga, ~', such t h a t

1 = SQa(~', ~")go,.~,($"), I m U' ~ - - W * , a n d

lga,~'(~")l <-- C/X

]Qa(~', r ~

C~/Oo(~', ~"),

I m ~" E - - W * . (Cf. t h e p r o o f of T h e o r e m 5.5.7 in [5].) Set

f"

^" ^" ^"

T h e n we see i m m e d i a t e l y t h a t Ea, ~, e Boo, ~0, w i t h [lEa, ~,l],, ~o b o u n d e d b y a con- s t a n t t h a t does n o t d e p e n d on ~'. B y changing t h e i n t e g r a t i o n contour we o b t a i n t h a t supp Ea, ~, ~ W. Moreover,

~ ( ~ ' , x " ) =

~, Ea,~,(Qa(~',D")~@,')),*

u E C : ( R " ) .

a = ( a ' , o)

I f ha. ~, e S a n d ha, r =

Qa(~', D")a(~', x")

on W_ t h e n

h~,(x") = X(Ea, ~, .ha,~,)(x" )

= a(~', x") on W_. This implies t h a t

ll~(~', 9 )1[~,~0~ _< IIh/ll, ~;k ~ - < 211Ea,~, * ha, ~;111, ~ok <: _

C X h a,~'[I,,k"

'h

T h u s

a=(a', 0)

for all ~' E t l ~' w i t h

d@,

b-l(0)) > t. W h e n ~' C t l ~' is a r b i t r a r y , t h e n b y L e m - m a A 2 in H S r m a n d e r [6] there is a point 0' = 0'@) C R ~' w i t h [0'[ =<

t/7

a n d d(~' + 0', b-l(0)) > t. This implies t h a t

IJ~(~', 9 )1/2, ~o<~,+o,, .)~ -< 6' z ][Qa(~' + 0 , D")~(~', ~ - ' ")[[~, ~. ~ - Thus

~ a 2 03 u E C~~ ~) a n d k E ~2((Rn'), for all ~ ' E R '+,

N o w we w a n t to f i n d a b o u n d for t h e r i g h t - h a n d side of the estfmate in L e m m a 6.11. L e t t be t h e c o n s t a n t i n t r o d u c e d in t h e p r o o f of t h a t lemma. Thus we h a v e Qa(~', ~") :V 0 for all ~ if

d(~',

b-1(0)) ~ t a n d I m ~" C _N" -- i n t W*. F u r t h e r - more, let 0 ~ z E C ~ ( { x ' C R n ' ; Ix'I < 1}) a n d set Zo'(~')= Z ( ~ ' - - 0 ' ) . I f r

(21)

O N F U N D A M E N T A L S O L U T I O N S S U P P O R T E D B Y h C O N V E X C O N E 21 we define )/~,(~') b y setting Zo'($') = Xo,( I~e ~'). W h e n 0' 6. T = {~' 6. R~';

d ( ~ ' , b - l ( 0 ) ) > t + 2} and f i = (fi',0), Ifil = 1 we define

~(v) = ~(~, ~, o'; v) = ( ~ ) - " f xo,(~')Q~)(~)~(~)/Q~(~)d~, v 6.

C ~ ( R " ) .

J

LEMMA 6.13. Let E be defined as above. T h e n E 6. B 1~ and supp E C A = It ~' X W. Furthermore, i f q~ 6. C~~ ") then Ilq)EIl.,l is bounded uniformly with respect to 0'.

Proof. B y changing the integration contour in the immediately t h a t supp E C A.

Let K be a compact set and assume t h a t f l = (1,0 . . . . ,0).

then

~" variables, we obtain I f v 6. C ~ ( K )

f ~ '~

/~(v) = (2~)-"'

zo,(~ )v(~)/(~l- ~j(~~ =

rj

+

J

~j

where ~ o = ( $ e , . . . , ~ , ) and y j ~ ' R + i if I m v j < 0 and y j ~ R - - i if I m D > 0. Furthermore, ~j denotes the support of 0Zo,/0~ 1 between Y1 and R.

We get the estimate

I (v/1 _< f ,o f +

r1

f f

+ C . s u p I~(~1, ~~ d~ ~ < 6'1 I~(~)1 d~,

where C 1 is independent of 0', for fl~(~ + i~)l d~ _< e ~'~/l~(~)l d~.

We also need the equivalence between two norms t h a t we are going to use.

The following lemma, which is inspired b y Beurling [1], proves this.

LEMMA 6.14. Let K ' C R " b e a compact set and let ~v C C~(R"') be 1 on K ' . I f Xo, is the f u n c t i o n defined above then there is a constant C such that

supli~(~', 9 )lh,z _< c sup sup [l(v5 9 ^~

(xo,u))(~,

^~ ^' ")lh,~

R n z l t n r o' E T

for all u e C ~ ( K ' X l t " ) and lce ~)~(R"").

(22)

22 ARNE ENQVlS~

Proof. L e t g be an element in the unit ball of the dual space of C~~ " ) e q u i p p e d w i t h the norm II " [ll.w~ a n d set v(x') = (u(x', "), g>. I f we p r o v e t h a t

(6.4) sup I~(~')t _<< C sup sup l(v~ 9 (Zr v 9 C~(K'),

R n~ R n~ O' ~ T

t h e s t a t e m e n t will follow as g varies.

A s s u m e t h a t (6.4) is false. Then there is a sequence v i 9 C~(K') such t h a t s u p Iv~(~')l = 1

R n ~

and

[(~ 9 (;Co.~j))(~')[ __< 1/j for all ~' 9 R": a n d 0' 9 T.

T a k e ~ 9 R " such t h a t 1~i(~)[ = 1 a n d set ~vi(~' ) = ~i(~' 3- ~ ) . T h e n we h a v e t h a t

I~/(~')[ < l~vi(O)[ = 1 for all ~ ' 9 R ~' a n d

] f ~(~' -- ~l')Z(~' 3- ~i -- O');vi(~')d~' <-- 1/j for all ~ ' 9 R '+ a n d 0 ' 9 T.

W e can n o w choose a s u b s e q u e n e e ~oik of wi such t h a t ~vik converge u n i f o r m l y t o h on e v e r y c o m p a c t set. Then h is analytic, ]h(~')] < [h(0)[ --~ 1 for all ~' 9 R ' . F u r t h e r m o r e , b y L e m m a A2 in H S r m a n d e r [6] we can choose a sequence 0~ 9 T such t h a t ~ ; - O; is b o u n d e d . Finally, choose a s u b s e q u e n e e of ~ ; k - O;k which converges t o --0' 0. T h e n

f w(~' ~')Z(~' Oo)hOl')d~l ' ~ 0 for all ~' 9 R"'

This implies t h a t ~ 9 ~Y-~(Zoo,h ) ~ O, so Xoo,h - - O, for ~-l(;G,h ) is analytic.

H o w e v e r , h is also analytic a n d ;~o0, ~ 0, so h---0, which contradicts t h a t lh(0)l = 1. This proves the lemma.

N o t e t h a t we h a v e only used t h a t T is defined b y some polynomial b 4= O, w i t h given degree. Thus, the constant C in the l e m m a depends only on the degree o f the polynomial defining T and not on the polynomial itself.

I f k 9 t h e n there are constants U a n d N such t h a t k(~ 3 - ~ ) (1 3-Cl~[)N/c(~) for all ~ , ~ 9 R n. F o r given C a n d N we shall here use the n o t a t i o n ~X(R n, C, N ) for all k 9 Oi(R ~) such t h a t k(~ 3- ~) < (1 3- C l~})Nk(~) for all ~, ~ / 9 R". W e shall n o w use L e m m a 6.13 a n d L e m m a 6.14 to p r o v e

LEMMA 6.15. Let K G R n be a compact set, C o, N C R+ and let Q~ be as before.

Then there is a constant C such that

(23)

O~N : F U I ~ D A M E ~ T A ~ L S O L I ) ' T I O ~ S S U P X : ' O t t T E D B Y A C O I N ~ V E X C O k N E 2 3

liQ~(D)ull~,-k <_ CllQo(D)ull~_k for all u C C~(K) and k E ~ ( R ' , C

O , N ) .

Proof.

L e t

u C C~(K)

a n d ~v E C ~ ( R ' ) , where ~ = 1 in a n e i g h b o u r h o o d of K.

F u r t h e r m o r e , let ~ E C~(R") be 1 in a n e i g h b o u r h o o d U of zero, such t h a t ( ( C U ) ~ K ) f l s u p p ~ v - ~ O. I f E = E ( ~ , fi, 0') is the distribution in L e m m a 6.13, Ifl] = 1, a n d S D g =

Q~(D)u

in A_, t h e n

~v((~vE) 9 g) = ~vZo,(D')Q~)(D)u

in A_.

Thus

II~y.o,(n')Q~)(D)ull~:-k < ]]~v((~E)

9 g)lll,~ _< Ilwlll, ~ llqEll~,x HgIi~, ~ ~< C [iglll,~, which implies t h a t

D' (~) D u

z_

livzo,( )Q~ ( ) ill,

~ <_ c

liQ~(D)ulI~,-~.

N o t e t h a t C is i n d e p e n d e n t of 0 ' ~ T. H o w e v e r , with Q ~ ) = Q~)(~', D"), sup I1(~ * (~o,Q~)~))(~', ")11~,~% _< c~ llv Zo,(n')Q~)(D)ull~,-~ <-- C2

llQ~(n)ull~-~.

I n view of L e m m a 6.14, this implies t h a t

sup [IQ~)(~ ', n")t~($',-)i11,~% < C IiQ=(n)ull~,%

1~. n~

i . e . ,

/ .

I1 (8) , *- ] ,

sup

Q~ (2, D")~(~',

.)Ill, k < C

IIQ~(~, D")R(~', ")l[~%d~'.

R n p J

B y using t h e technique in the p r o o f of T h e o r e m 3.10 in t t S r m a n d e r [6] we o b t a i n from this t h a t

H e n c e

[lQ~)(D)ullzl._k <_ C [IQ~(D)u[l~,-k.

IIQ~+~(D)ull~-k = 1

II(Qo(D)

-4-

2Q~)(D)

-- Qo@(D))uII1.A-k <---

<_ lIQo(n)uli~-k d-

IlQ~)(n)ulllA_k d- ilQ~')(D)UlilA,-k ~<

<~ C(llQ~(n)ull~-~ §

[IQ0(D)ull~-k).

This proves the lemma, for ~ = (~', 0) and fi = (fi', 0), lfil = 1 are arbitrary.

Proof of Theorem

6.9. B y L e m m a 6.11 we have

~Y' t

, , ")Ill, k,, < ~ z

fIQAD)ulr~-~,,

li~(~', ) i l l , ok" _< c liQ~(~, n")4(~',

^{~ -}

(24)

2 4 AI~ l~7:E ENQVIST

w h e r e k , / ( ~ " ) = k(~', ~"). H o w e v e r , t h e r e are c o n s t a n t s C o a n d N such t h a t kr, E ~ ( ( R ~', C 0, N ) for all V'. T h u s L e m m a 6.15 implies t h a t for all V'

~ =

)11 , Q0 + _

By

using t h e t e c h n i q u e in t h e p r o o f of T h e o r e m 3.10 in [6] we o b t a i n f r o m this t h a t

Iju]l~:5, k < C IjQo(D)u[]~-k, u E C : ( K ) .

W e shall n o w i n v e s t i g a t e in detail t h e n e c e s s a r y conditions for p o l y n o m i a l s P s a t i s f y i n g t h e following condition:

( 6 . 5 ) - - a~(~ )~ , ' w h e r e b@) ~- a(m ' 0 .. . . . 0) (~') is real a n d

s t r o n g e r t h a n a~ for all ~. F u r t h e r m o r e , a s s u m e t h a t Q(~") is s t r i c t l y h y p e r b o l i c w i t h r e s p e c t to e v e r y v e c t o r in t h e i n t e r i o r o f t h e p r o p e r , closed a n d c o n v e x cone V* for all Q E 27, a n d t h a t N " --~ (1, 0, . . . , 0) E i n t V*.

N o t e t h a t we can m a k e b real b y m u l t i p l y i n g t h e p o l y n o m i a l b y t h e c o m p l e x c o n j u g a t e o f b. Observe also t h a t , since 2: is c o m p a c t , t h e r e is a smallest, p r o p e r , closed a n d c o n v e x cone V* D/V" s a t i s f y i n g t h e c o n d i t i o n in (6.5).

T~]~o~]~M 6.16. Let P be a polynomial satisfying (6.5). Then the following con- ditions are equivalent:

(i) P ( D ) is an evolution operator with respect to every half space containing R n' • (V ~ {O}) in its interior.

(ii) I m P(~) is dominated by P(~).

(iii) P satisfies the necessary conditions of Theorem 2.6 with respect to I " - ~ R n" • W for every cone W such that i n t W D V ~ ( 0 } .

W e are going t o p r o v e t h e t h e o r e m a f t e r some lemmas.

LENMA 6.17. Let a and b be polynomials such that a(~j)/b(~j)-+O, j--> oo, for all sequences ~j E R n such that d(~j, b-l(0)) --> ~ , j -+ oo. Then a is dominated

by b.

Proof. Set e(t) = sup {la(~)/b(~)l ; d(~, b=l(0)) ~ t > 0}. I t follows f r o m t h e a s s u m p t i o n t h a t e(t)-+ 0 as t--> oo. F r o m L e m m a A2 in [6] we o b t a i n t h a t t h e r e is a c o n s t a n t 7 > 0 such t h a t for e v e r y ~ E R n t h e r e i s a 0 E Rn,.TJOI < 1, such t h a t t h e d i s t a n c e f r o m ~ ~- tO t o t h e zeros o f ab is a t least t. N o w we h a v e t h a t