f vadx. Lower bounds for pseudo-differential operators

Lower bounds for pseudo-differential operators

ANDERS MELIN

Institut Mittag-Leffler, Djursholm, Sweden

O. Introduction

L e t A be a classical scalar 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 o f o r d e r m (cf. K o h n - -

~qirenberg [7]) in a n o p e n subset f2 o f R". W e are i n t e r e s t e d in e s t i m a t e s o f A f r o m below o f t h e f o r m

R e ( A u , u) ~ CluJ~,) , u e C~(K) (0.1) w h e r e K is a c o m p a c t subset o f f2 a n d ]u](s) is t h e n o r m o f u in t h e space H(s ) of f u n c t i o n s w i t h d e r i v a t i v e s o f o r d e r s in L 2 a n d ( v ,

u ) = f vadx.

^{I f}

s ~ m/2 t h e e s t i m a t e is always t r u e for some C since ( A u , u) is c o n t i n u o u s in H(,,/2). On t h e o t h e r h a n d , if s < m/2 it is e a s y t o see t h a t (0.1) implies

Re am(x,

~) >_

0 (0.2)

w h e r e a m is t h e principal s y m b o l o f A. I n t h e o p p o s i t e direction GSrding [3]

p r o v e d t h a t if (0.2) is valid, t h e n we can for e v e r y e > 0 a n d e v e r y s f i n d a c o n s t a n t C - - - - C ( K , e , s ) such t h a t

R e (Au , u) + elu]~,) ~ Clu](:,) , u e C~(K) (0.3) i f tt = m/2. A simple m o d i f i c a t i o n o f t h e p r o o f gives t h e same r e s u l t for a n y

# > ( m - - 1)/2. I n f a c t if A satisfies (0.2) a n d m / 2 > t t > ( m - - 1)/2 t h e n we c a n write

(A -4- A*)/2 + e(1 + [D[2) ~ = P * P + Q

w h e r e P a n d Q are 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 in K2 a n d t h e o r d e r of Q does n o t e x c e e d m - - 1.

H o w e v e r t h e s i t u a t i o n b e c o m e s m o r e c o m p l e x w h e n # - ~ ( m - 1)/2. I t was p r o v e d b y H O r m a n d e r [5] t h a t (0.2) does i m p l y t h a t (0.3) is v a l i d for some e > 0, b u t t o h a v e (0.3) for e v e r y e > 0 we m u s t clearly in a d d i t i o n t o (0.2) place a r e s t r i c t i o n on t h e t e r m s in A o f o r d e r m - - 1. I n this p a p e r we shall s t u d y n e c e s s a r y a n d sufficient conditions on A for (0.3) t o be v a l i d for e v e r y e > 0 w h e n

I18 ARKIV FOR MATElVIATIK. VO1. 9 :No. 1

# = ( m - 1)/2. The p r o o f depends on t h e localization t e c h n i q u e i n t r o d u c e d in H S r m a n d e r [5, 6] b u t requires more careful estimates of r e m a i n d e r terms. I n a d d i t i o n we h a v e t o m a k e a complete s t u d y of inequalities o f t h e following kind,

f v(y) ~ (1/~x!fl!)a~y~D ~ v(y)dy > O, v 6 C~(R")

(0.4) Re

J I~+~l _<2 where a~ are complex numbers~

This will be done in section 2 a n d t h e s t u d y of (0.3) when # = (m -- 1)/2 will be carried o u t in section 3. I n section 4 we shall use our results to prove a t h e o r e m d u e to Radkevi5 [10] a b o u t hypoellipticity for a certain class of classical pseudo- differential operators w i t h non-negative principal symbol. W e also m e n t i o n t h a t t h e i n e q u a l i t y (0.3) w i t h # = ( m - 1)/2 has been t r e a t e d in t h e vector v a l u e d case b y Lax-Nirenberg [9J. (See also [2] a n d [11].)

F i n a l l y I w a n t to t h a n k m y t e a c h e r prof. L. H S r m a n d e r for his k i n d i n t e r e s t in this work a n d m a n y suggestions for improvements.

1. N o t a t i o n and preliminaries

W e shall m a k e use of t h e familiar n o t a t i o n

D i = -- i ~/axi,

where i = % / - - 1 a n d if D ~ (D 1 . . . . , Dn) is t h e g r a d i e n t vector a n d a ---- (a 1 . . . an) is a multi- :index w i t h t h e a i non-negative integers t h e n D ~ denotes the differential operator D ~ I . . . D ~ n . W e set [al ~--~aj" a n d a! = a l ! . . . a n ! . I f y~-- ( Y l , " - , Y n ) we define y~ in a similar way. B y S or 5 ( R n) we d e n o t e t h e set o f all functions

E C ~(R ~) such t h a t

sup lx~D~r <

~ (1.1)

x

for all multi-indices a a n d ft. H(~) is the completetion of 3 in t h e n o r m

=

(2=)-n f

(1 ~- I~[2)~1~(~)12 d~ (1.2)

luJ~,)

where 4 denotes t h e Fourier t r a n s f o r m of u

~(~) = f e -~<~' ~>u(x) dx.

I f X is a n open subset of R" a n d a belongs to C ~ (X • R") a n d satisfies t h e i n e q u a l i t y

]D~D~a(x,

~)] < C~,~,K(1 -}- 1~[) m-l"l , x E K , ~ C R n (1.3) w h e n K is a c o m p a c t subset of X, t h e n

a(x, D)

will denote t h e corresponding pseudo-differential operator (of order m)

L O W E R B O U N D S F O R S P 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 119

a(x , D)u(x)

= (2~) -" t f

e~<"'~>a(x ' ~)4(~)d~ , u E C~(R") .

I

I n particular, ff ~ and u belong to 3 t h e n

~p(D)u

will denote the function whose Fourier transform equals ~ .

I f A is a classical pseudo-differential operator of order m with symbol a, then

a i

(with j = m, m -- 1 . . . . ) will denote the part of a t h a t is homogeneous of degree j with respect to ~. Finally with a as in (1.3) we shall use the n o t a t i o n

a s~ = (iD,,)~(in~)~ a.

2. A study of the inequality (0.4)

L e t a~ denote complex numbers and consider the i n e q u a l i t y

Re

f v(y) ~ (1/cdfl!)a~y~D%(y)dy >

0 ,

v e C~(R n)

(2.1) J

where k is a non-negative integer. Of course the form defined in (2.1) does n o t determine the coefficients a~ uniquely, b u t we claim t h a t there exist uniquely determined real coefficients by such t h a t

l~e / v ( y l ~ + ~ t <_l,(1/cdfl!)b~y~D%(y)dy ' v E C;(R n)

(2.2) defines the same form. For the existence of such b E it is enough to prove t h a t

Im / v(y)y~D~v(y)dy = Re / v(y){ (y~D~ -- D~y~)/2i} v(y)dy

can be written in this form, but this follows easily from the fact t h a t

yjD k -- Dky i -~ i~jk

(2.3)

To prove the uniqueness when the coefficients are real we replace

v(y)

^{in (2.1)}

b y

q(y -- tx)e ~'<y'~>

^where

f I~(y)[2dy

= 1 and get the expression Re

f qv(y) ~ (1/~!fl!)a~(y + tx)~(D + t~)~'q~(y)dy

J

la+~l -< k

which is a polynomial in t with the leading coefficient la+~l =k

Hence by letting t tend to infinity we can conclude t h a t two sets of real coefficients defining the same form must coincide and t h a t the validity of (2.1) implies t h a t the form h is positive semi-definite. W h e n k -~ 2 and there are no lower terms the converse s t a t e m e n t is valid for b y formula (2.3)

120 ARKIV FSR MATEMAT~:. Vol. 9 No. 1

v(y)yiDkv(y)dy / f + J f [v(y)]~dy , e C~(R") (2.5)

v(y)nkyiv(y)dy iSik v

so when taking real parts in (2.1) y and D can be treated as commutating operators.

I n particular, if h ( x , ~) is a real quadratic form on R n 9 R", then we have a well defined form

/ v(y) h ( y , D ) v ( y ) d y , v

Re E C~(R")

a n d by a diagonalization of h one immediately sees t h a t the expression is non- negative if h is positive semi-definite. Also, when k ~- 2,

When ]c ~- 2 we get the correspondence between the coefficients in (2.1) and (2.2) b y the equations

= 0 a /2. (2.7)

b~ ~ R e a ~ if a ~ - f l = f i 0 , b~--~Rea 0 - ~ Im l~l =1

We shall now confine ourselves to the case when ]c equals 2 in formula (2.1) and let h and f denote the quadratic respectively the linear part of the corresponding polynomial with real coefficients on R " | R n. I f ( u , v) denotes the standard Euclidean scalar product on R" O R n, then there is a unique symmetric trans- formation H such t h a t

h(u) = ( H u , u ) , u C R" | R" .

B y i we shall also denote multiplication by the imaginary unit in R" @ R" when this space is identified with C" as a real vector space under the isomorphism

.R" @ R " ~ (x , ~ ) --> x -+ i ~ e C" .

We shall examine the invariance of (2.1) when the symbol ( y , D) is transformed as a vector in R" @ R" and f i n d t h a t our inequality is invariant under the symplectic group of /~n 9 R n.

Definition 2.1. We define the sympleetic bilinear form a on R" 9 R" b y the equation

a ( u , v ) = I m ( u , v ) = ( - - i u , v ~ - ( u , iv~ , u , v e R" O R~ (2.8) where ( u , v) denotes the Hermitian scalar product on C n. The corresponding linear transformations on R " Q R" under which a ( u , v) remains unaltered are called canonical transformations. They form the symplectic group S p ( R ~ 9 R").

I f H is strictly positive, then ( H u , v} defines a scalar product b on R" | R"

and since

( H i H u , v } -~ ( H u , ( - - i H ) v } , u , v e R" O R "

L O W E R B O U N D S F O R 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 121 i H is s k e w - s y m m e t r i c w i t h r e s p e c t t o b a n d it follows t h a t t h e s p e c t r u m o f i H is s i t u a t e d on t h e i m a g i n a r y axis, s y m m e t r i c a l l y a r o u n d t h e origin. T h e last s t a t e m e n t is clearly v a l i d e v e n if H is o n l y p o s i t i v e semi-definite since t h e eigenvalues o f a linear t r a n s f o r m a t i o n d e p e n d c o n t i n u o u s l y on t h e t r a n s f o r m a t i o n a n d H = lim~_~0 H + eI, w h e r e I is t h e i d e n t i t y t r a n s f o r m a t i o n .

Definition 2.2. B y T r H (Tr h) we shall m e a n t h e s u m o f t h e positive e l e m e n t s in i . S p e c t r u m (ill) w h e r e each e i g e n v a l u e is c o u n t e d w i t h its m u l t i p l i c i t y .

I t follows f r o m t h e a r g u m e n t a b o v e t h a t T r H d e p e n d s c o n t i n u o u s l y on H . Remark 2.3. I n o r d e r to i l l u s t r a t e t h e n a t u r a l role o f T r H we shall p r o v e t h a t it is i n v a r i a n t u n d e r s y m p l e c t i c t r a n s f o r m a t i o n s . I f Z E Sp(R" | R") t h e n t h e p o l y n o m i a l h(zu ) c o r r e s p o n d s to t h e s y m m e t r i c t r a n s f o r m a t i o n g ' H z , w h e r e Z ~ is t h e a d j o i n t of X w i t h r e s p e c t t o t h e s t a n d a r d E u c l i d e a n s t r u c t u r e o f / P | R", so we h a v e to show t h a t

T r Z' H ~ = T r H . (2.9)

N o w b y t h e d e f i n i t i o n o f S p ( R n Q R")

i X t = Z - 1 i

h e n c e

i Z ~ H Z = 2 -1 i H g

w h i c h implies t h a t iz'H Z a n d i H h a v e t h e same c h a r a c t e r i s t i c p o l y n o m i a l . THEOREM 2.4. The inequality (2.1) with k = 2 is valid i f and only i f

(i)

h(x , ^{= -} a J / !flt >_ 0 (ii) f ( x , ~) = ~. [~+~1 =1 R e a~x~ ~

vanishes in the null space of h, and

(iii) R e a o ~ - - ~ . l ~ l = ~ I m a~/2 - - ( H - ~ f , f ) / 4 + T~rH >_ O.

Remark 2.5. A l t h o u g h H -1 does n o t h a v e t o exist t h e expression ( H - i f , f } is well d e f i n e d b y (ii) a n d as is easily seen

( H - ~ f , f } = sup ( u , f } 2 . (2.11) h(~) < 1

I n view o f (2.7) we m a y a s s u m e t h a t t h e coefficients a~ are real in t h e p r o o f o f T h e o r e m 2.4. Of course we shall m~ke use o f t h e trivial f a c t t h a t we could as well let v r u n t h r o u g h 3 in (2.1).

Definition 2.6. W e shall s a y t h a t t w o p o l y n o m i a l s p~ a n d P2 on R n G R"

are s y m p l e c t i c a l l y e q u i v a l e n t if t h e r e is a canonical t r a n s f o r m a t i o n Z such t h a t

p l ( Z ( V ) ) = p 2 ( V ) , V e R n @ R a .

B y a s y m p l e c t i c basis for R" | R" we shall m e a n a basis {e 1 . . . . , e , , f l . . . f , } w i t h t h e p r o p e r t y

~ ( e j , e ~ ) = ~ ( % . , h ) = 0 , ~ ( e ~ , A ) = - - ~j~.

122 ARKIV FOR MATEMATIK. Vo1. 9 NO. 1

LEMMA 2.7. Let P l a n d p~ be symplectically equivalent p o l y n o m i a l s of the second degree with real co.efficients. T h e n the inequality

Re / v(y)pl(y , D) v(y) d y >_ 0 , v e ~ (2.12) is valid for Pl i f a n d only i f it is valid for p~.

We have already proved the necessity of condition (i) in Theorem 2.4. Using L e m m a 2.7 it will be easy to finish the proof of Theorem 2.4 in the case where h is positive definite. We observe t h a t

~(u , v) = ( H S u , v> = b ( S u , v)

where S = ( i l l ) -1 is skew-symmetric with respect to b. B y a standard result concerning normal transformations on a Hilbert space we shall be able to reduce our problem to the case n = 1. In fact, we have the following

I~EMMA 2.8. L e t { ~1 . . . . , 2n} be the positive elements (counted with m u l t i p l i c i t y ) i n i . S p e c t r u m (S). T h e n R n 0 R n can be written as a direct s u m

R o + R o = 2 + 5

w h i c h is orthogonal w i t h respect to b a n d where V~ has an orthonormal basis {e i , fi}

i n the same sense such that

Sej -= - - ~ J i , Sfi = Xiei , J = 1 . . . n . (2.13) Notice t h a t this condition implies t h a t {El . . . E n , F 1 , . . . , Fn} with E j

= ~T1/2ej, ~j

^{~ - -} ~j--1/2fj is a symplectic basis for R n | R". I f {E ~ . . . . , E ~ is the standard orthonormal basis for C', then {E ~ . . . E ~ F ~ , . . . , F ~ with

~F ] = i E ~ is a symplectic basis for R" O R n. Let Z denote the canonical trans- formation which maps E ] on Ej and F ~ on Fj for j = 1 , . . . , n . I f

H - i f = t i E 1 ~- . . . -~ t,E~ Jr "~1F1 -+- . . . -+- ~,F~

a n d

v = x~E ~ ~ - . . . + Xn E ~ -~- ~1 F~ ~ - . . . -~ ~ F ~ t h e n an easy computation yields

a~ -[- <f ,

zv> + h(xv)

= a~ ~- ~

).7'(tjxj -]-

Tj~j -4- X2 -~ ~2) . (2.14) j = l

According to L e m m a 2.7 we have to s t u d y the inequality Re / v(y){a ~

Replacing v b y

+ ;l(tjyj + jnj + y} + n})} v(y) dy >_ 0, v e S . (2.15)

j = 1

e i(z~JYJ)/2v(y)

L O W E R B O U N D S F O R P S E U D O - D I F I ~ E R E N T I A L O P E R A T O R S 123 in (2.15) and repeating the same argument after a Fourier transformation we get

~he equivalent inequality,

s

(a~ ( H - i f , f > ~ 4 ) J

lv+)l,dy +

iv+) ~j--l(y2 + n2>v(y)dy ~ O,

Here we have used the equation

L e t

u E C~(R).

~j (tj q- T~) b(n-~f, H-~f)---- ( H - ~ f , f } .

B y expanding the right-hand side of the inequality

0 < f I(O --

2iy)ul 2 dy , ]

v e 3 . (2.16)

we get for real 2

f (t(D 2 + ).2y2)u dy ~ ). / ^s

^{lul ~}

^dy.

(This is essentially the proof of the u n c e r t a i n t y relation.) tIence we obtain by choosing 2 ~- 1

f +

^>

iv+l=

a n d choosing

v(y) ~-- e -lyl'12

we see t h a t the inequality cannot be improved.

Since Tr H ~ ~ 27 a this will complete the proof of Theorem 2.4 in case H -1 exists. The general case follows from a continuity argument. We replace the coeffi- cients a~ b y a~(e) in such a w a y t h a t H will be replaced by

H ---- H ~- sI

while

0 are conserved. F r o m (2.6) we get (for s ~ 0) f and a 0

f v(y) X (~l~)~y~D~v(y) ~y < Re f v(y)

t~e (1/~!fl!)a~ (e)y~D~ v(y) dy

Y

a n d since

g(s) = a~ -- (H~-lf,f}/4 -~-

Tr(H~) tends to g(0) when e tends to zero the necessity of (ii) and (iii) follows.

Conversely, if (ii) and (iii) are fulfilled we can choose a function ~(s) tending to zero when s tends to zero such t h a t

_ _ H - 1

a~ q- cp(e)

( ~ f , f } / 4 q- Tr (H ) ~ 0

a n d then the sufficiency of our conditions follows since for fixed v the left h a n d side of (2.1) depends continuously on the coefficients a~.

Proof of Lemma 2.7.

The idea is to subject v in formula (2.12) to isomorphisms of S which lead to linear transformations of ( y , D) generating

Sp ~ Sp(R ~ 9 R~).

We shall introduce three subsets

G1; G 2

and G 3 of

Sp:

G 1 is the group of all transformations of the form T -1 G T' where T is an isomorphism in R ~ and T' denotes the adjoint transformation. This is the group

124 A I ~ K I V F61~ M A T E M A T I K . V o l . 9 : N o . 1

o f linear t r a n s f o r m a t i o n s in R" | R n o b t a i n e d b y regarding t h e second copy as t h e dual space of t h e first.

G~ is t h e set of all e l e m e n t a r y canonical t r a n s f o r m a t i o n s , i.e. t r a n s f o r m a t i o n s o f t h e form

C n ~ ( z I . . .

Zn) --->-

(iw(1)Z 1 . . .

iU'('OZ,,)

(2.18)

where ~ is t h e characteristic f u n c t i o n of a subset of { 1 , . . . , n}.

G a is t h e group of t r a n s f o r m a t i o n s

- R~ 9 R ~ D ( x , ~) --> ( x , ~ - - S x ) (2.19) where S is a s y m m e t r i c linear t r a n s f o r m a t i o n on t h e E u c l i d e a n space B n.

I f t h e v a l i d i t y of (2.12) for Pl implies t h e same for p~ ---- Pl ~ Z w h e n

Z C G i,

t h e n we shall s a y t h a t (2.12) is Gi-invariant.

To prove t h a t (2.12) is G 1 -- i n v a r i a n t we replace

v(y)

in (2.12) b y

vT(y) v(Ty)

where T is a n isomorphism in /~=. B y t h e chain rule

/)a(VT ) __--

((TtD)av) w

where

(T'D~)e i<x"

~> = (T'~)e ~<~' ~>. A s u b s t i t u t i o n of y b y

T - %

in t h e integral yields us t h e e q u a t i o n

/ /

R e

vr(y)lh(y, D)vr(y)

dy = Idet T 1-1 Re v ( x ) p l ( T - i x ,

T'D~)v(x) dx

from which t h e G~-invariance follows.

W h e n proving the G~dnvariance we shall construct t h e partial F o u r i e r t r a n s f o r m t h a t corresponds to ~ in (2.18),

w,(j)= 1

where

=

Replacing v in

/

e - i < x j ' - v J > U ( ~ l , . 9 ~ j ~ l , x j , ~ j + l , 9 9 9 ,

Sn)dxj,

(2.12) b y ~w(v) a n d using t h e formulas

Y S ~ ---- ~,,YJ,

D J ~ , , :

~ D i

if ~v(j) = 0

f % v(y) v dy = ] v(y)u(y) ^{d y ,}

we see t h a t (2.12) is also G2-invariant.

I n t h e p r o o f of the Ga-invariance we replace v in (2.12) b y w i t h S as in (2.19). N o w

D.(e-i < sy, Y >/:v(y) )

= e - i < Sy~ y > / 2 ( D i _

Si(y) )v(y )

u E 3 .

(2:20)

e-i<sy'Y>/2 v(y)

LOWER BOUNDS FOR PSEUDO-DIFFERENTIAL OPERATORS 125 w h e r e Sj(y) is t h e j t h c o o r d i n a t e o f S ( y ) a n d since y a n d D c o m m u t e in (2.1) w h e n /c = 2 a n d t h e coefficients are real we get

R e / e-~<sx'y>/2v(y)D~(e-~<SY'Y>/2v(y)) dy = (2.22)

/

= R e v ( y ) ( D - - S ( y ) ) ~ v ( y ) d y , [~1 ~ 2 .

T h e Ga-invariance n o w follows i m m e d i a t e l y f r o m (2.21) a n d (2.22). I n o r d e r t o c o m p l e t e t h e p r o o f of L e m m a 2,7 we h a v e to show

LEMMA 2.9. T h e sets G1, G~ a n d G a generate together the symplectic group S p ( R n + Rn).

W e shall use t h e s y m b o l A ( n ) to d e n o t e t h e set o f all subspaces 2 o f R n G R"

w h i c h are isotropic in t h e s y m p l e c t i c g e o m e t r y a n d o f m a x i m a l dimension, i.e.

2 E A ( n ) if a n d o n l y i f dim ( 2 ) = n a n d

a ( u , v ) = 0 w h e n u , v C 2 . (2.23) R n = [ E ~ . . . . , E ~ a n d i R n : [El ~ . . . . , F ~ belong t o A ( n ) . I t is e a s y t o see t h a t if 20 E A ( n ) is t r a n s v e r s a l to i R ~, t h a t is, t h e p r o j e c t i o n ~o--> R" along i R n is surjeetive, t h e n t h e r e is a s y m m e t r i c linear t r a n s f o r m a t i o n S on R" such t h a t

~o = { ( x , S x ) ; x e R n } .

L e t 2 o E A ( n ) . B y looking at t h e image o f t h e p r o j e c t i o n 20D ( x , ~ ) - - > x E R ~

we realize t h a t t h e r e exists a n e l e m e n t a r y canonical t r a n s f o r m a t i o n Z such t h a t Z(~o) is t r a n s v e r s a l t o i R ~. (Cf. [1, w 96]).

P r o o f of L e m m a 2.9. L e t Z E S p . B y t h e r e m a r k a b o v e t h e r e is a Z1 in G 2 a n d a s y m m e t r i c linear t r a n s f o r m a t i o n S such t h a t

~1 ^o y,(iR ~) = { (x , S x ) ; x e R~} . Choosing :~2 as in (2.19) we get

iy,2 o Zl o z ( i R ~) = i R ~ .

Since R ~ is t r a n s v e r s a l t o i R ~ it follows t h a t x ' R ~ is t r a n s v e r s a l to g ' i R ~ for a n y linear b i j e c t i v e Z'. H e n c e t h e r e is a s y m m e t r i c linear t r a n s f o r m a t i o n S 1 such t h a t

iZ2 o ) ~ 1 o z ( R n) : { ( x , S i x ) ; x ~ . R n } .

I f Xa d e n o t e s t h e c o r r e s p o n d i n g m a p d e f i n e d in (2.19) a n d Za = Xa ~ i o X2 ~ Zl o g we h a v e

z , ( - R ~) = R ~ , z ~ ( i R " ) = i R ~ .

126 ARKIV :FOR 1VIATEMATIK. Vol. 9 No. 1

T h u s Z4 ~ A 9 B w h e r e A a n d B a r e l i n e a r t r a n s f o r m a t i o n s on R n. Since Z4 E S p we m u s t h a v e Z4 E G 1 a n d we conclude t h a t Z b e l o n g s t o t h e g r o u p g e n e r a t e d b y Gx, G 2 a n d G 3.

Remark 2.10.

I t is e a s y to describe t h o s e p o s i t i v e s e m i - d e f i n i t e linear t r a n s - f o r m a t i o n s H o f R ' O R" for w h i c h T r v a n i s h . T h e following s t a t e m e n t s a r e e q u i v a l e n t

1 ~ T r H - ~ 0

2 ~

H ( R n | n) C 2

f o r s o m e 2 C A ( n ) 3 ~ H - I ( 0 ) ~ for s o m e 2 C A ( n ) .

T h e e q u i v a l e n c e b e t w e e n 2 ~ a n d 3 ~ follows f r o m t h e f a c t t h a t 2 z ---- i2, w h e r e 4 • d e n o t e s t h e o r t h o g o n a l c o m p l e m e n t of 2 w i t h r e s p e c t t o t h e s t a n d a r d E u c l i d e a n s t r u c t u r e . L e t G d e n o t e t h e p o s i t i v e s q u a r e r o o t o f H . Since

GiG

is s k e w - s y m m e t r i c t h e e q u i v a l e n c e b e t w e e n 1 ~ a n d 2 ~ will follow f r o m t h e following c h a i n of i m p l i c a t i o n s :

1 ~ < : > i l l is n i l p o t e n t

~ GiG

is n i l p o t e n t

~ G i G = O ~ H i H =

0 r ~ . W e shall e n d t h i s section w i t h a n a p p l i c a t i o n of o u r r e s u l t s t o t h e i n e q u a l i t y

[ A~y~v(y) + ~ Bfl)~v(y)l ~ dy ~ c Iv(y) l 2 d y , v E C~(Rn).

(2.24)

1 1

H e r e A~ a n d B~ a r e v e c t o r s in a (complex) H i l b e r t s p a c e 9 ( a n d since t h e r e is no a m b i g u i t y we use t h e s y m b o l s (A , B) for t h e s c a l a r p r o d u c t a n d [A[ f o r t h e n o r m in 9(.

B y using (2.5) we g e t

I A,y,v(y) + ~ B~D~v(y)12 dy ~-

(2.25)

1 1

+ 2 ~. R e ( A , ,

B~)y~,D~v(y) + ~

R e ( B , ,

B~)D,D~v(y)} d y .

W e n o w i n t r o d u c e t h e G r a m i a n o f h~ . . . h , C c)d as t h e m a t r i x

V(h~ . h~) = ((hi, h ~

N o t i c e t h a t R e

G(h 1 . . . h~)

is a real p o s i t i v e s e m i - d e f i n i t e m a t r i x since R e ( g , h) is a n E u c l i d e a n s c a l a r p r o d u c t in t h e real s u b s p a c e of c)d s p a n n e d b y {h 1 . . . h~} o v e r R. L e t

H = R e G ( A ~ . . . A n , B ~ , . . , B , ) ,

t h e n b y {2.25) t h e i n e q u a l i t y (2.24) is e q u i v a l e n t w i t h

f Iv(y)l'dy, v e C ~ ( R ' ) .

>

J

L O W E I ~ : B O U N D S F O R 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 127 Theorem 2.4 then gives us

PROPOSITION 2.11. The inequality (2.24) is valid i f and only i f c _< Re (-- i (A,, B~)) + Tr Re G(A1 . . . . , A n , B~ . . . B~).

1

(2.27) I n particular, the best constant in (2.24) is a continuous function of the vectors A v and B~.

_Remark 2.12. H6rmander [5] has given necessary and sufficient conditions for (2.24) to hold with a positive constant c. We have however not managed to derive an easy criterion for this directly from (2.27).

R e m a r k 2.13. The validity of the inequality (2.1) when k ---- 2 a n d conditions (i) and (iii) of Theorem 2.4 are fulfilled (with h positive definite) can also be proved in the following more general way. For the sake of simplicity we assume t h a t the coefficients are real. Let X~ (i = 1 , . . . , 2n) be symmetric linear operators defined in a dense set F of a complex Hilbert space c)~ and suppose t h a t X~(F) is contained in F for all X/ and t h a t the following relations hold (eft (2.3)),

X j X k - - X k X j = [ X j , Xk] -=- - - % , / - - 1Jjkl. (2.28) Here J = (Jjk) denotes the matrix for i, the multiplication b y the imaginary unit in R n @ R n, and I is the identity transformation in 9(. We shall prove the inequality

R e ( p ( X ) v , v) > 0 , v E_F , (2.29) where X = (X 1 . . . . , X2n) and p(u) = ( H u , u } -~- ( f , u> ~- a~ is defined as in Theorem 2.4. I t follows from (2.28) t h a t the left hand side of (2.29) is well defined.

B y the computations after L e m m a 2.8 we can find a symplectic matrix Z such t h a t

2 84

( H z - l u , Z-lu> = 2/(u 2 -F- %+i) ; 25 = Tr H .

i = l i ~ l

We now introduce new operators X/ b y the equations

2 n

X i = E xi~X~ + gj I ,

where g = 1 / 2 z H - % I t is t h e n easy to verify t h a t the operators X/ will satisfy (2.28). Since we have

p ( X ) = ~ 2~(X~ q- X2,+,) - - ~ ( H - l f , f > I § a~

i = l

the inequality (2.29) will follow from the u n c e r t a i n t y relation.

128 ARKIV FOR MATEMATIK. Vol. 9 No. 1

3. L o c a l i z a t i o n of l o w e r b o u n d s for 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

W e shall n o w a p p l y t h e results o f section 2 to a localization o f t h e i n e q u a l i t y (0.3). L e t ~ be a twice c o n t i n u o u s l y differentiable f u n c t i o n d e f i n e d in a n o p e n set in R M. W e t h e n define t h e H e s s i a n of ~ as t h e m a t r i x

H ~(x) ---- ((iDj)(ink)~(x))~,k=l . (3.1)

THEOREM 3.1. A s s u m e that a,~ >_ O. T h e n the inequality (0.3) with # ~-- (m - - 1)/2 holds f o r every s ~ 0 and every s with suitable C when K is a n y compact set in

~2 i f and only i f

R e am_l ~- 1/2 T r Ham > 0 at the zeros of am in 1 2 • n-1.

R e m a r k 3.2. T h e c o n d i t i o n am >~ 0 is a c o n s e q u e n c e o f (0.3) if we a s s u m e t h a t A is self-adjoint, w h i c h is no restriction. I f we do n o t i n t r o d u c e a n y nor- m a l i z a t i o n o u r conditions are:

]~e am >__ 0 (3.2)

n r ~

a n d R e a~_ 1 -[- (1/2) I m . ~ a~lj) ~- (1/2) T r HRoa~ > 0 (3.3)

j - - ~ _ -

a t t h e zeros o f l ~ e a ~ in ~ 2 • n-~.

R e m a r k 3.3. F r o m (2.9) a n d t h e i d e n t i t y r m-1 t H ~x

H . m ( X , r~) = Z ~ ' ~ ) Z r > O ,

w h e r e Z = r~/2E 9 r -1/2 E a n d E is t h e u n i t m a t r i x o f o r d e r n, we conclude

r ~ J

t h a t T r H , ~ in T h e o r e m 3.1 is h o m o g e n e o u s o f degree ( m - 1) in ~.

I n t h e p r o o f o f T h e o r e m 3.1 we shall r e d u c e ourselves t o t h e case m ---- 1. L e t R e d e n o t e p r o p e r l y s u p p o r t e d 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 in f2 w i t h s y m b o l (1 ~ 1512)e/2 a n d set A e -~ R - o A R -e. Recall t h a t a 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 A in f2 is called p r o p e r l y s u p p o r t e d if b o t h p r o j e c t i o n s supp K A --> ~2 are p r o p e r , w h e r e K A is t h e d i s t r i b u t i o n kernel o f A. See also [6], p. 148. T h e n A satisfies (0.3) w i t h # - - - - ( m - - 1 ) / 2 for e v e r y e > 0 a n d e v e r y s w h e n C is s u i t a b l y chosen if a n d o n l y if t h e same holds for A o w i t h /~ = (m - - 2~o - - 1)/2. Since a,,,(x, ~ ) - ~ 0 implies grad(~,~) a,n(x, ~) -~ 0 we also f i n d t h a t t h e conditions in T h e o r e m 3.1 are left i n v a r i a n t if we replace A b y A~. H e n c e f o r t h we shall t h e r e f o r e a s s u m e t h a t m = 1.

P r o o f that the condition in Theorem 3.1 is necessary

A s s u m e t h a t a l ( x ~ ~ ----0 w i t h ( x ~ ~ in / 2 X S n-1. Choose a n y E > 0 a n d let K be a c o m p a c t set in ~2 c o n t a i n i n g x ~ as a n i n t e r i o r p o i n t . W e w a n t t o show t h a t (0.3) implies

L O V ~ E R B O U N D S F O R P S E U D O - D I F F E R E N T I A L O P E R & T O R S 129

R e

ao(x ~ , ~o) @

1/2 Tr

H,~(x ~ , o ) > O .

(3.4) I n order to get (3.4) we shall construct a class of functions with s u p p o r t near x ~ a n d with Fourier t r a n s f o r m c o n c e n t r a t e d close to the half-ray g e n e r a t e d b y

~0 in R". F o r t h e sake of simplicity we assume t h a t x ~ 0. L e t

r E C k ( R " )

a n d set

u~(x) = 4"/2v(~x)e ~<~''r

T h e n ]uz[(0 ) = Ivl(o) and

uxEC~(K)

for large 4. A simple calculation yields

(Au~.,ux)

= ( 2 ~ ) -'~

ff+<.,,>a(x/4, ^{42~~ +}

^(3.5)

Since t h e first-order derivatives of a I a t (0, Vo) vanish a n d since ([42~ ~ @ 4~:[ @ 1) -1 = 0(1)4-1(1 + I~l)

a Taylor expansion of a a b o u t ( 0 , ~o) will give

a(x/4, 42~ ~ -~ 4~) = ~ a~a(O , V~163 +

ao(O , ~o) ~_ 0(1)4_1(1 @ [~1)4 . (3.6) la+~l=2

Then applying the Fourier inversion formula a n d using (0.3) with s = - - ¹ we get w i t h some c o n s t a n t C t h a t does n o t d e p e n d on 4

R e f v(x) ~ (l/~!fi!)a~(0,

V~ §

J la+~ 1=2

(~ + l~e ado, ~ 0 ) ) j Iv(x)? dx + 4=100) >_ ~lu~l~_x) 9 +

N o w Iu~, ](_l) - + 0 w h e n 2 - + oo. H e n c e b y first letting 2 t e n d to infinity a n d ² t h e n letting e t e n d to zero we arrive at the i n e q u a l i t y

l~e f v(x) ~. (1/~,fl,)a~(0,

~~ @ Re ao(O, ~o) f

[v(x)l~

dx ~ 0

(3.7)

]aA-fl[=2

w h e n v belongs to

C~(R").

The i n e q u a l i t y (3.4) n o w follows b y T h e o r e m 2.4.

Proof that the condition in Theorem 3.1. is sufficient

Following H S r m a n d e r [5] we shall m a k e a localization of t h e estimates b y m e a n s of p a r t i t i o n s of u n i t y in t h e variables x a n d $. There exist sequences (%)if=0 a n d (~vk)ff= 0 of non-negative functions belonging to

C~(R n)

such t h a t the following conditions are fulfilled (See [5], p. 141--142):

= . %(x) = ~vk(~ ) = 1 (3.8)

0 0

1 3 0 AI~XIV ]~6R 1V~ATEMATIK. Vol. 9 No. 1 (and a t m o s t

a n d finally

2" supports overlap)

I x - - y ] ~<C if x , y e s u p p ( ~ k ) (3.9)

l D ~ ( x ) l ~ < C~ (3.10)

0

]$ - - ~1 ~ C[~1112 if ~ , V e supp (~p~) a n d k ~= 0 (3.11)

(~j(}) -

~(v))" _<

c 1 } - - vl" I}1-1 i f It - v l _< 1}1/2 9 ( 3 . 1 2 ) W h e n proving this p a r t of T h e o r e m 3.1 we m a k e t h e convention t h a t constants depending o n l y on A a n d t h e chosen p a r t i t i o n s of u n i t y are to be d e n o t e d b y t h e same s y m b o l C. Of course, we m a y restrict ourselves to t h e case s = -- 1/4 in (0.3), so we h a v e t o show t h a t if e > 0 is given, a n d K is a c o m p a c t set in /2 t h e n t h e r e exists a c o n s t a n t C such t h a t

ge (Au, u) § ~lul~0) >_ Clul~_l~), u r C?(K).

(3.13) F u r t h e r m o r e , since t h e c o n t r i b u t i o n to Re

( A u , u)

t h a t comes f r o m I m a 0 a n d t e r m s of n e g a t i v e degree is continuous in

H(_1/2),

we m a y also assume t h a t

A = a ( x , D )

w i t h a = a l § o a n d a o real valued.

F i n a l l y we notice t h a t t h e v a l i d i t y of (3.13) is n o t affected if the s y m b o l of A is c h a n g e d outside a n e i g h b o u r h o o d of K . H e n c e replacing a b y

( 1 - - 9)(1 § [~21)1/2 §

Fa(x, ~),

where ~o belongs to C~(/2) a n d 0 ~ F(x) < 1

w i t h e q u a l i t y to t h e right in a n e i g h b o u r h o o d of K , we m a y assume t h a t A is defined in /2 = R" a n d t h a t t h e symbol of A equals r ( ~ ) = (1-}-I~]2) 112 outside a c o m p a c t set in R". W e shall prove (3.13) w i t h K replaced b y 1~.

To begin w i t h we shall split up u b y its s p e c t r u m a n d m a k e t h e corresponding a p p r o x i m a t i o n s of t h e operator. L e t 0 4= ~i belong to t h e support of ~v i a n d let

be a n u m b e r w i t h 0 < 8 < 1. W e introduce t h e differential operators

A]v(x)

= y~ ( l f ~ ! ) ~ ( x , ~Jla)(D - -

~t~)~v(x) , v e C~(R~ .

lal < 2 W e shall prove t h e i n e q u a l i t y

, g 2

I(Au u) -- ~, (Ay~vj(cSD)u,

Wj(~D)u)I < Cdlul~0) § Cal 1(-114) for u e

Cg(Rn).

(3.14) Of course (3.14) is valid if it holds for A = (1 §

IDI2) 112

a n d w h e n A is replaced b y B =

b ( x , D )

where B satisfies ( 1 . 3 ) w i t h m = 1 a n d vanishes w h e n x is outside a c o m p a c t subset o f / ~ " . W h e n A = (1 § iDIP) ~/2 we get b y u s i n g Parseval's f o r m u l a a n d (3.8)

( A u , u) -- ~ (A]~vj(aD)u, ~j(c~D)u)

= ( 2 a ) - " / - where w i t h r(~) ~- (1 § I~12) I/~

L O W E R B O U N D S F O R P S E U D O - D I F F E I ~ E N T I A L OIqBI%ATOI%S 131 g(~) : ~ ~(c~)(r(~) -- ~ (1/~!)r(~)(~i/6)(~ -- ~J/(~)~).

J I~1 <- 2

F r o m (3.11) a n d Taylor's formula we deduce t h e existence of a c o n s t a n t C such t h a t [q(~)l --< C~-3/~1~1-'/2 if 1~t >-- C/O

a n d using this inequality we easily prove (3.14) in this ease.

We shall now prove (3.14) for B. I f b(~, ~) denotes t h e Fourier t r a n s f o r m of b w i t h respect to t h e variables x t h e n

//

( B u , u) - - ~ (B~fj(~SD)u , Wi(SD)u) ~- (2n) -2" H~(~, V)4(~)E(v)d~dv where

H,~($, V) = 1/2 ~ (Vj(6~ e) -- Vj(6V)) 2 b(v -- ~, ~) + (3.15)

J

+ ~ Vj(~)Vj(~v){b(,7 - - ~ , ~) - - ~ (1/o,!)b(~)(V - - ~ , ~/~)(~ - - ~/~)=} 9 Our a p p r o x i m a t i o n will be good for small 6 since we h a v e

L E ~ 3.4. F o r large N there are constants C N inde2oendent of 6 and f u n c t i o n s

% , N ~ C ~ ( R ~) such that

I/L~(#, n)l ~< c ~ ( 1 + I# - n]) - ~ + ~ . ~ ( ~ ) ~ , ~ ( n ) 9 (3.16) Proof. B y carrying out partial integrations one obtains

ID~-b(v, ~)l --< C~,~(1 + tVl)-u(1 -~- I~l) ~-l~l (3.17) a n d Taylor's formula t h e n gives us an estimate for t h e second t e r m in t h e expression for H~ w h e n ~([~1 + IV]) is large b y

C~(1 -[- IV - - ~ I) -~-~ ~ Y~j((5~)Wj(~SV)]~ - - ~J/O]3qj, ~(~) (3.18)

j ~ 0

where

qj,~(#) = sup (1 + [~ § t ( ~ i / ~ - # ) E ) - ~ . O < t < l

F r o m (3.11) we obtain

1~ -- Vl ~ C6-1/2[VI~/2 ~ ]Vl/2 if j~0j(O~)~oj.(6V) # 0 a n d IV[ > 4C26-1, so if j~j(8~)Fj(@) @ 0 a n d [~1 -4- IV! > 12CZ6-1 t h e n we m u s t h a v e ]~1 > 4C26-a.

B y replacing ~ b y ~i/~ a n d ~ by ~ in the inequality above we get

I~ -- ~J/6] <_ Cb-1/2[~I~/2 <_ I$1/2 if [~] + ]~[ > 12C26 -1 a n d

j~)j((5~)y)j(eSV) =~

O . T h e n t h e sum in (3.18) can be e s t i m a t e d b y

{c~-,~-~/~( 1 + Iv - ~!)-~]~1-'/~}( 1 + Iv ^- ~ [ ) - ~

132 ARKIV FOR ~IATEMATIK. Vol. 9 No. 1

if 6([~/] § I~t) is large enough, and since the first factor tends to zero when ([~l § [V]) -~ ~ we realize t h a t the second sum in (3.15) can be estimated b y the right h a n d side of (3.16).

I t remains to estimate the first sum in (3.15). Now (3.12) together with (3.17) gives us

(~j(~) - ~(~))21/~(~ - ~ , ~)l _< c ~ ( 1 § I~ - ~l) - N

if 1 2 - - V l <

1~1/2

(and [ ~ I § IT] is large). Finally if [~--~11 ~ I~]/2 and N ~ 2 then

lb(~ - - ~ , ~)l ~ C~(1 § I~I)(i § I~ - - ~I) =2~ ~ 8(1 § 17 - - ~I) -N if [~l § l~l is large enough. This completes t h e proof of the lemma.

The validity of (3.14) for B now follows from L e m m a 3.4 and the inequality

We shM1 also carry out a partition of u in the variables x. I f ~ = ~(x) is a real valued function in

C~(R")

then

ge ( / cy(A~v)cy~dx-- / A~(q~v)cp~dx)=

=- Re / ~ ~

(1/~!)aa(x,

~1/6){~(D --

~s16)% -- (D -- ~s/~)~(q~v)}dx . ..I Io~1 =2

We shall replace ~ b y

cyk(xI~Jl 1/2)

and v by

yJi(~D)u(x).

I n doing so we note t h a t when the differentiations on the right h a n d side are carried out we obtain in addition to the term

- ~Ivl. ~ ~ ( ~ , ~/~)D~/~!

1~1=2

only terms containing a factor ~D~0. Since ~ ~ D ~ - - - - 0 these will drop out when we sum over ]c which gives

~ e {(Ay~j(~D)u ~j(~D).) - - E @ o / ~ , , ~ = ~ , ~ ) } = ( 3 . 1 9 ) k

= -

Here we have introduced

u{~(x) = ~ ( x I~ s l~/2)~s( ~D)u(x ) .

We shall use the symbol 0(1) to denote functions t h a t can be estimated uni formly when it and ~ vary. Summing over j in (3.19) and using (3.14) we get

L O W E I ~ B O U N D S FOI~ I ~ S E U D O - D I F F E I ~ E N T I A L O P E R A T O I ~ S 133

l~e ( A ~ , ,~) = Z l~e (A]~{ ~ , , ~ ) -- (3.20)

j , k

- j , k I~I=2

~ ~ f (1/a!)a~(x , ~/a)l$Jl(~kn%)(xl~J1112) IWJ(aD)ul~dx +

+ aO(1)lul~o) + CoO(1)[ul~_114), u r

C:(Rn).

The fact t h a t a satisfies (1.3) with m = 1 uniformly when x E R ~ and t h a t

~ f [v,i(~D)updx=f Iu(x)pdx

implies t h a t the second sum in (3.20) can be written as O(1)alul~o). Hence

U 2

~ e ( A ~ , ~) = ~ ~ e ( ~ u ~ ~ , ~ ) + a0(1)[~l~o) + 0,0(1)1 I(-1~,) (3.21)

j , k

We shall now make a change of variables and introduce the functions v~ ~ by the equations

~(x)

= ~ ' < "

~Jl~>v{~((x - x ~) t~

^11/2)

where a n d

xJk= I~J]-ll2x k

x k = ~ 1 ~ 1 - 1 / 2 e s u p p (~k) 9 The last relation follows from the definition of ~k a n d since

lv~k(y)l

= ~k(z k + y)ly~i((~D)u((x k -t- y)1~i]-112)i

we also have by (3.9)

supp(v{ k) c { y e R n ; l y ] ~ C } .

This fact together with a change of variables in the integrals

(A~u{ k , u•)

and a Taylor expansion of a at

(xik, ~i/(~)

with respect to the variable x gives us

I~ j In/2(A~u~ ~ , u ~ k) = ( 3 . 2 2 )

: ~, f (X/a!)a~(x jk -}- yI~J[ -1/~ , ~i/5)I~'liall2(D~ :

[otl <__ 2

J

= ~, (1/~x!fl!)a~(xik, ~i/,~)i~Jl(l~l-l~l)/2 f v~k(y)y~D~v~k(y)dy +

la+~l < 2 J

-4- o(1)a-ll~Y1-11~ Z /

ID%~(y) 12dy .

lal < 2 J

I n order to estimate the remainder in (3.22) we introduce the functions v~ b y the equations,

y,J( aD)u(x) = e i <0'' ~J/~ >vg(xI~i] '12)

^.

134 ARKIV :FOtt )IATE1KATIK. Vol. 9 No. 1 B y (3.11) there is a constant C such that

[ ~ - - ~J[ ~ CIVil 112 if ~ e Spectrum

(~i(~D)u)

and hence

l~] <-- C/~

if $ e Spectrum (v~). (3.23) Now v~ ~ differs from %v j,~ only b y a translation and using (3.10) and (3.23) together with Parseval's formula we get since the v/~ ~ have their supports in a fixed compact set in / P

[By first summing over all indices j with 18Ji > d-12 we get

(~-5

lcn - ~

f IWj(dD)ul:dx

^~-- O(1)(~]U](0) @

2

O(1)Cd]U]~-xI4 )

and using this together with (3.21), (3.22) and (3.24) (with N = 2) we get

I~e (Au,

u) = (3.25)

= ~. l~JI - ~ ~ e ~ (1/~!~!)a$(x ~ , ~J/o)I~Jl (~I-B~I)/'~

[ ~(y)y~D~v~(y)dy +

j , k [a+~l -< 2 d

§ DO(1)lul~0) +

QO(1)lul~_~/4), u ~ C~(R").

We have now reduced our problem to an estimate of the individual terms in (3.25). In the rest of the proof d will be kept fixed so t h a t the remainder term

~O(1)]ul~0) in

(3.25)

is greater than

--(e/a)lu](2o).

In order to complete the proof it is enough to find a sequence 5i tending to zero when j tends to infinity such t h a t

l~e ~ (1/~!fi!)a~(x ~ ,

~J/~)l~il ([~I-I~1)/2

f vJ~(y)y~D%~(y)dy -~

(3.26) d

-~ (e/2),flv~k(Y)12dy

>-ejf

For multiplying b o t h sides of (3.26) b y using (3.24) and t h e equality

~, ID%J~k(y)12dy.

]$il-n/2 after a summation over k and

we realize that (3.13) follows from (3.25) if the resulting terms are summed over j.

Remark 3.5.

Since

~o~ (ojk , $1/(~)[$J i(l~E-l~l)/2 f vJk(y)y~D%Jk(y)dy

can be estimated b y a constant times

L O W E R B O U N D S F O R : 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 135

lr I lD v (y) I dY

w h e n [~ + fil--< 2, a n d since I~[ t e n d s to infinity w h e n j tends to infinity we m a y replace a~ in (3.26) b y a ~ if we a d d the t e r m

a0(x , f iv (y)I:dy

to the left h a n d side of the formula.

All t h a t we n e e d is furnished b y t h e following lemma.

L ~ A 3.6.

Let K o be a compact subset of

R " ~ { 0 } ,

let y be a real valued continuous function on 1 ~ • Ko, such that'y is constant when x is outside some compact set in t~ ~, let C be a fixed positive number and suppose that

implies

(x , 7) E R ~ • Ko , a~(x , 7) = O

1/2 Tr

Hal(X,

7) ~- ~ ( x , 7) > 0 . (3.27)

Then there is a function ~(~)--~ 0 when 2--> ~ such that

l~e ~. (1/~!fl!)a~(x, V)22-1~+~l f v(y)y~D~v(y)dy ~-

(3.28)

la+~[ <- ² Y

+ y ( x , 7) f lv(Y)E2dY + ~(~) f ~ ID~v(Y)I2dY ~ 0

J J

lal < 4

for every ( x , 7 ) C R " • o and every v in C~(R n) with support in {y; [y] < C}.

I n order 4o see t h a t L e m m a 3.6 implies the existence of a sequence ~j tending to zero in (3.26) we change (3.26) in accordance with R e m a r k 3.5 a n d set K0 = ( ~ - - I s n - - i ' ~ --~ Jr1 : ]~Ji 1/2,

ej ~- e()'j), (X, 7) = ( xjk '

~J~--ll~ji--1)' T --~

@ / 2 ) + a 0 a n d v = - v { k.

Proof of Lemma 3.6.

Our p r o o f will be indirect. W e shall assume t h a t it is not possible to choose e(~) in such a w a y t h a t (3.28) holds a n d e(X) ~ 0 w h e n ;t - + a n d see t h a t this leads to a contradiction. Our a s s u m p t i o n means precisely t h a t there exist a e > 0 , sequences X j ~ w h e n j - - ~ a n d ( x j , v j ) E / ~ " X K 0 , a n d functions vj belonging to

C~(R ~)

with s u p p o r t s in {y ; ]y[ < C} such t h a t

]vj(y)l:dy : 1

a n d

(1/~!fi[)a~(xj,

ni)~]-g~+~t

f vj(y)y~D~vj(y)dy +

(3.29) R e

la+BI -< ² J

+ Y ( x i , , i ) / ]vi(Y)'2dY ~-0/,~1<_ 4 [D~vj(y)]~dY < 0 .

136 A R K I V FOR I~IATElgATIK. V o l . 9 N O . 1

W e shall use some c o n v e x i t y properties for t h e derivatives of a positive function.

I f g is a n o n - n e g a t i v e f u n c t i o n in

C~(R n)

t h e n there is a c o n s t a n t Cg such t h a t [gradx

g(y)]2

<

Cgg(y).

I t follows t h a t

Igrad,.,~

a~(x , ri)] 2 <_ Cgoa~(x , ,q) , (x , V) q. R" X Ko ,

(3.30) since t h e i n e q u a l i t y is trivially fulfilled w h e n al(x , ~ ) = l~]l.

H e n c e we get for [c~[ = ]fi] : 1

[a~(xi , Vy)Aj ./vj(y)D%j(y)dy] < Ay]a~(xj , "qi)] [vjI(o)lvi[(1)

^<

(3.31a)

, 2

< (1/3n)a~(xi , "qj)).~ + C

]vj](~)

la~(xj , Vj)Aj f y~[vj(y)i2dyl <_ C,~j]a~(xj ,

Vj)I ~<

(1/3n)a~(xj , rli))~ ~ ,+- C'

^(3.31b) where C' is a c o n s t a n t i n d e p e n d e n t of j .

Using (3.29) a n d (3.31) t o g e t h e r w i t h t h e facts t h a t ~ is a b o u n d e d f u n c t i o n a n d t h a t for some c o n s t a n t C ~

] ~ (1/~!fl!)a~(xj, "r vi(y)y~Davj(y)dy] < C"lvjl~2 )

(3.32)

l a + ~ I = 2

we o b t a i n t h e i n e q u a l i t y

where C is a n e w constant.

F r o m t h e i n e q u a l i t y

we conclude t h a t t h e sequence

V 2

I j[(~) ~< Clvjl~)/~ (3.33)

[vi~z) ~< Ivl(0)Ivl(4)

(vi)

is b o u n d e d in t h e Hd-topology. B y passing to a subsequence i f necessary we m a y therefore assume t h a t

(vi)

converges to a f u n c t i o n v0 in t h e H3-topology.

Using (3.31), (3.32) a n d t h e boundedness of t h e sequence [vj](2) we realize t h a t t h e sequence

22al(xi, ~i)

m u s t be b o u n d e d (and b y (3.30) t h e same s t a t e m e n t is v a l i d for t h e sequence ~j g r a d

al(xj,

Vi)). H e n c e

al(xj, Vj)

t e n d s to zero w h e n j t e n d s t o i n f i n i t y a n d since

al(xj,

Vi) is greater t h a n some positive c o n s t a n t w h e n x is outside some c o m p a c t set in R n we conclude t h a t it is no restriction to assume t h a t there is a (xo, %) E R n X K0 such t h a t

(xj, ~i) --> (x0, %) w h e n

j--> oo , al(xo , %) = O .

B y choosing p smaller in (3.29) we m a y replace

v i

b y v0 (for large j). T h e n a p p r o x i m a t i n g v0 b y a f u n c t i o n in

C~(R n)

a n d using t h e c o n t i n u i t y o f ~ we get t h e existence of a f u n c t i o n v in

C ~ ( R n)

a n d a small positive n u m b e r a such t h a t

L O W E R B O U N D S F O R 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 137

f lv(y)lUdy 1 and

l~,e ~ (1/~!fl!)a~(xi, .rljJ,~ j y~n~v(y)dy -]- (3.34)

Y

§ (42)

]v(y) L2dy < o .

j ~ l J J

To simplify notations we introduce

f~ = grad(~.,)al(xj, ~i) , Hj = H,~(xj , Vj) , H I ---- Hj + (~E

where E denotes t h e unit matrix. Since HI tends to the positive definite matrix H~ we m a y assume t h a t H I is positive definite for all j. Then an application of Theorem 2.4 to (3.34) gives us

~2{al(xj, 1~j) -- <(H;)-lfj ,fj>/2} -~- ~,(xo, ~]0) -]- e/2 ~- 1/2 T r H ; < 0 . (3.35) The proof will be finished if we can prove t h a t (3.35) is in contradiction with (3.27).

A Taylor expansion gives us

a l ( ( x j , T]j) -31- h) -~. al(Xj , T]j) + <fj , h> .2[_ <Hjh , h>l 2

§

O(1)lh] 3 when h e R 2" and Ihl _ c . Since f / : 0(1)Z71 and H i < H ; we get

0 ~_~ al((Xj~, T]j)

- - ( H ; ) - l f $ . ) =

al(xj, T]j)

- -

<fj,

(H/)-l.f/> -~- -Jr- <Hi(H;)-lf j , (H;)-Ifj>/2 @"

o(1);.7~

<

al(x 1 , ~]j) -- <fj , (H;)-lfj>/2 ~- O(1)/~j -3 . Hence

0 <_ ~.(al(xj, ~j) -- <fj, (H~)-lfj>12) + 0/4 for large j. Then it follows from (3.35) t h a t

y(x0, ~0) + QI4 ~- 1/2 Tr H i ~ 0

and b y letting j t e n d to infinity and a t e n d to zero we get using the continuity of Tr

gx0, + 014 + 112 Tr

Ha,(Xo, <-- 0

which is in contradiction with our assumptions. This completes the proof of the lemma.

4. Applications of the results to hypoellipticity

Let to be an open set in R n. We shall consider the class of classical pseudo- differential operators A in tO satisfying the following conditions:

138 AI~I~IV r61~ MATEMATIK. Vo]. 9 :No. 1

a ~ ( x , ~ ) > 0 when ( x , ~ ) E f 2 X S "-1 (4.1) a ~ _ l ( x , ~) = - a , ~ _ l ( x , ~) when a m ( x , ~) = O . (4.2) Here m denotes the order of A. On the set N of zeros for am in ~Q• we can define the function

I ( x ~) -~ I ~ a(J) (~ ~) - - 2Jam ,(X ~)l + , m ( j ) ~ , - , (1/2) T~ H,,,(x ~) , 9

j ~ l

The following result, due to l~adkevi5 [10], gives a sufficient condition for A to be hypoelliptic in zQ.

TtI~OREM 4.1. I f A is properly supported and satisfies (4.1) and (4.2) and i f I ( x , ~) > 0 on N , then for every compact set K in ~2 and every real number m' there exists a constant C such that the following estimate is valid

U 2 --(s) 2 2

[ + {

A U (1]2)

+

IA(8)ul(-lp)}

_< c{

IAul~o) ~-

lU1 m')}

,

U

r C ~ ( K ) . (4.3)

s=l

Here A (s) and A(~) denote properly supported pseudo-differential operators in ~ with symbols a (~) respectively a(,).

~ e m a r k 4.2. B y using the results of l~emark 2.10 we can easily get equivalent formulations of the condition I ( x , ~) > 0 on N.

Proof of Theorem 4.1. Let / ~ denote properly supported (self-adjoint) pseudo- differential operators in Q with the symbols

a,~(x, ~) = (1 -~- ]~12) #2 .

We shall apply Theorem 3.1 to the pseudo-differentiM operator

T~ : ~ m - l j - - (~ ~ (R1/2A(S))*(R'/2A(s)) - - ~ ~ (R-1/2A(s))*(R-1/2A(s)) ~-

s~l s~l

+ (A* - - A ) * ( A * -- A ) + 4(A*A -- A A * ) - - ~R 2m-~

where ~ is a small positive number.

We have to examine the symbol aT~ of T,. We shall make use of the following formulas (see [6], p. 147 and p. 149.) where A and B denote properly supported pseudo-differentiM operators in ~:

a , . ( x , ~) ~ ~ (iD~)an~(~z(x , ~)/fi! .

Then we see that the principal symbol of A * A - - A A * is homogeneous of degree 2m -- 2 in ~ and vanishes on N. The principal symbol of (A* -- A ) * ( A * - - A ) is homogeneous of the same degree and equals

[2 I m am_ , --~ 2 a~'~J) [2

LO~VEI~ BOUNDS FOI~ :PSEUDO-DY~YEI~ENTIAL OI~EI~ATOI:tS 139 on N. The symbol of (RI/2A('))*(R1/2A (')) can be written as

(Oh(ms))2(1 ~_ ]~[2)112 _~ 2a~)_la~)( 1 -4- 1~12) 1]2 - ~ ib

where the leading part of b is homogeneous of degree 2 m - 2 in ~ and real valued. Of course the symbol of

(R-1/2A(~))*(R-1]2A(8))

can be written in a similar w a y with a (') r e p l a c e d b y a(~) and (1-4- ]~[2)1]2 replaced b y (1 + ]~12) -1t2 .

Let K be a compact set in ~2. Then for some constant C o [grad(x,~ ) a n ( x , ~)12 < Coa~(x , ~) , (x , ~) e K • S "-1 ,

(Cf. (3.30)), and it follows from the homogeneity of am, that if 8 is chosen small enough then the principal symbol of T~ is non-negative and has the same zeros as am when x belongs to a neighbourhood of K.

I f the symbol of T~ is written

O'T~ ~ t2m_l "~- t2m_ 2 - ~ . . . .

then b y our assumptions and computations above

~ e t2m_2(X , ~) "J- 1/2 Tr Htzm_l(X , ~) = (4.4)

~ ] ~ a(i),n(j)t Ix , ~) - - 2iam_l(x , ~)12 -4- 1/2 T r

Ht2m_l(X

, ~) -- 8 when (x , ~) e N . Since Tr H~2,~_1(x, ~) tends to Tr H ~ ( x , ~) when 8 tends to zero (with uniform convergence on every compact subset of N) the expression above in (4.4) is non- negative for small 8 on the zeros of t2m--X in Y20 • S ~-1, where ~20 is some neigh- bourhood of K. Hence b y choosing 8 small and applying Theorem 3.1 with s ~ 8/2 we get

~ e ( T ~ u , u) + (8/2)]U]~m_l) > Q]UI(~,) , U C C ~ ( K ) . (4.5) B y expanding the left hand side of (4.5) and using the inequalities:

IAv ^-- A * v l 2 < 2(IA*v! 2 -4- ]Avl 2) , v e C ~ ( ~ )

~:~e ( ~ m - l a y ' v ) = ]:~e (Av, .~m--lV) ~ 4 8 _ I I A v I~o) + 84-~ Iv l~-~) + 8C' i v I~,), v e C~ (K) we get with some constant C the following inequality

~t 2

I I(~-1) + z,

21 U (112) -~- ~ ²

]A(s)Ul2(1]2) -~ IA*ui~0) _ C(iAul(o) § lUl~m')), (4.6)

²

s = l s : l

u e C ~ ( K ) . This completes the proof of the theorem.

140 Alex:IV ~'i51~ :lVlATEMATI:K. Vol. 9 No. 1

References

1. CARATH~ODOI~Y, C., Variationsrechnung und partieUe Differentialgleichungen erster Ordnung.

Berlin, Teubner 1935.

2. }~RmI)mCHS, K., Pseudo-differential operators. New Y o r k University, 1968.

3. G)~I~DI~G, L., Dirichlet's p r o b l e m for linear elliptic p a r t i a l differential equations. Math.

Scand., 1 (1953), 55--72,

4. I-Ib~MA~DEI~, L., F o u r i e r I n t e g r a l Operators I . Acta Math., 127 (1971), 79--183.

5. - - ~ - - Pseudo-differential operators a n d non-elliptic b o u n d a r y problems. Ann. of Math., 83 (1966), 129--209.

6. --~)-- Pseudo-differential operators a n d hypoelliptic equations. Amer. Math. Soc. Symp.

Pure Math. 10 (1966), Singular integral operators, 138--183.

7. K e l l y , J. J. & ~I~ENBEI~G, L., A n algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18 (1965), 269--305.

8. KURANISHI, M., On estimate II(A(D) + B(x))u(x)H > c[]u H. Amer. Math. See. Lecture notes, Summer Institute, Global analysis, 1968.

9. LAx, P . D. & IqmE~BERG, L., On s t a b i l i t y for difference schemes; a sharp form of G&rding's inequality. Comm. Pure Appl. Math., 19 (1966), 473--492.

10. RADx~w~, E . V . , A p r i o r i estimates a n d hypoelliptie operators with multiple characteristics.

Dokl. Akad. lVauk S S S R , 187 (1969), 274--277. Also in Soviet Math. Dokl., 10 (1969), 849--853.

11. VAILLX~CO~Y~, R., A simple p r o o f of L a x - - l q i r e n b e r g theorems. Comm. Pure Appl.

Math., 23 (1970), 151--163.

Received October 20, 1970 Anders Melin

I n s t i t u t Mittag-Leffler Aurav/~gen 17

182 62 D j u r s h o l m Sweden