Asymptotic estimates for spectral functions connected with hypoelliptic differential operators

ARKIV FOR MATEMATIK Band 5 nr 35

1.64 04 4 C o m m u n i c a t e d 26 F e b r u a r y 1964 A. PLEIJEL a n d L . G~RDING

Asymptotic estimates for spectral functions connected with hypoelliptic differential operators

By

N I L S 1NIILSS051

1. Introduction

L e t x = @1 ... xn) be c o o r d i n a t e s in R n a n d p u t D k = (2~i) -1 ~/~xk a n d D ~ = D~ ' ...D~n.

W e shall consider a h y p o e l l i p t i c d i f f e r e n t i a l o p e r a t o r

M(D)= ~M~D ~

w i t h c o n s t a n t coefficients. L e t us a s s u m e t h a t t h e coefficients M~ a r e real, so t h a t

M(D)

is f o r m a l l y s e l f - a d j o i n t . Moreover, we s u p p o s e t h a t M ( ~ ) - > + co w h e n ] ~ ] - + oo, ~ ER n.

I f S is a n o p e n s u b s e t of R ~ a n d we define

M(D)

on

C~~

we g e t a s y m m e t r i c l i n e a r o p e r a t o r a 0 in t h e H i l b e r t space

L2(S).

W e l e t A be a s e l f - a d j o i n t e x t e n s i o n of

%. T h e n b y t h e s p e c t r a l t h e o r e m A h a s a s p e c t r a l r e s o l u t i o n E ( 2 ) o f c o m m u t i n g p r o j e c t i o n o p e r a t o r s i n c r e a s i n g w i t h 2 ( N a g y [7]). T h e o p e r a t o r E(2) is g i v e n b y a k e r n e l

e~(x,y),

t h e s p e c t r a l f u n c t i o n of A :

E(2)u(x)=~sex(x,y)u(y)dy,

w h e r e ex is i n f i n i t e l y d i f f e r e n t i a b l e in S • S. T h i s is p r o v e d in H S r m a n d e r [5] in t h e case w h e r e A is s e m i - b o u n d e d a n d in t h i s p a p e r in t h e g e n e r a l case.

I f in p a r t i c u l a r

S = R ~,

t h e r e is a u n i q u e s e l f - a d j o i n t e x t e n s i o n A 0 of a 0 w i t h a c o r r e s p o n d i n g s p e c t r a l f u n c t i o n e0.~ w h i c h is e a s i l y c o m p u t e d b y a F o u r i e r t r a n s f o r - m a t i o n :

eo, x(x,y)= f~, e x p ( 2 z i ( x - y , ~ } ) d ~ .

(~) <

W e are going t o give a r e s u l t on t h e b e h a v i o u r of e(2) =

eo.~(x, x)

when2--> + oo. W e shall s h o w ( T h e o r e m 1) t h a t t h e r e a r e r e a l n u m b e r s a a n d

t,

a > 0 a n d t a n i n t e g e r ~> 0 such t h a t for some n u m b e r k > 0

a n d

k-12a(log 2) t ~< e(2) <~ k2a(log 2) t (2 large) e ' ( 2 ) = 0 ( 1 ) 2 a l ( l o g 2 ) t (2-->+ c~).

A n a n a l o g o u s r e s u l t h o l d s for t h e d e r i v a t i v e s of e0.a w i t h r e s p e c t to x, y:

e(o~'~)(x,

x) = JMf(~)<~ ~2~d~"

I f n = 2 , we shall p r o v e a s h a r p e r r e s u l t , n a m e l y e ( 2 ) = k(1 + o ( 1 ) ) 2 a ( l o g 2) t for a p o s i t i v e n u m b e r k, a n d w h e r e t = 0 o r t = 1.

T h e p r o o f of T h e o r e m 1 uses a n a l y t i c c o n t i n u a t i o n p r o p e r t i e s of t h e f u n c t i o n

~. rC~LSSOrr

Asymptotic estimates for spectral functions

e(~), which follow f r o m results in the a u t h o r ' s paper [8]. I n p a r t i c u l a r cases t h e as- y m p t o t i c b e h a v i o u r of e(~t) has been investigated b y G o r t j a k o v [3], w h o t h e n also c o m p u t e d t h e n u m b e r s a a n d t. F u r t h e r we prove an a s y m p t o t i c result for

e;~(x, y)

w h e n S is arbitrary. F o r this we show an estimate for a f u n d a m e n t a l solution of

(M(D)-~)

when ~ - - > - c~, a n d a p p l y a T a u b e r i a n t h e o r e m of Ganelius for the Stieltjes t r a n s f o r m a t i o n . W e get the following result (Theorem 2), valid also in t h e n o t semi-bounded case,

I e~(x,

x) - e0, ~(x, x) [ = 0 ( 1 ) ~a-b(1og ~t) t (~---> + c~ ),

where a, t correspond to the p o l y n o m i a l M(~) as above a n d b > 0 is the largest n u m b e r such t h a t

[grad M(~)[ ~< C([M(~)[ + 1) ~-~ (u e R") for some n u m b e r C. W h e n 2 - - > - ~ , we h a v e with some c > 0

e~(x,y)

= 0 ( 1 ) e x p ( - c [2 [b).

2. N o t a t i o n s . T h e spectral f u n c t i o n

W e i n t r o d u c e t h e following c u s t o m a r y notations. I f a is a multi-index (~1 ... an), where t h e a, are n o n - n e g a t i v e integers, we p u t [a] = al + ... + zr a n d ~ = ~ ' . . . ~ with ~=(~1 ... ~n). W e write D j = ( 2 z

i)-l~/~xj(]=l ... n)

a n d D = ( D 1 . . . nn). L e t M($) = ~ M ~ ~ be a complex p o l y n o m i a l in ~1 . . . ~n, n ~> 2. T h e n it corresponds to a differential operator

M(D)=~M~D ~

with c o n s t a n t coefficients. W e shall assume t h a t it is hypoelliptic, i.e. ( H 6 r m a n d e r [5]) t h a t

M(~)(~)/M(~)-->O

( [ ~ [ - ~ , ~ r e a l ) , (1) where M(~)(~)=D~M(~), a n d the relation (1) holds for all a with [a[ > 0 . Moreover, we suppose t h a t M is real. T h e n it follows f r o m (1) t h a t either M(~)--~ + ~ or M(~)--> - co when [~:[ -->c~ (~ real). L e t us choose t h e sign of M so t h a t M(~)--> + ~ . L e t S be an open subset of R n. W e shall t h e n work in the Hilbert spaceL2(S) with inner p r o d u c t

(u,v)=~zu(x)v(x)dx

a n d n o r m [[u[I = ( u , u ) ~. I f we define

M(D)

on t h e set

C~(S)

of all infinitely differentiable functions, which vanish outside c o m p a c t subsets of S, we get a linear o p e r a t o r a 0 in

L2(S),

which is also symmetric, since M is real so t h a t

M(D)

is formally self-adjoint. L e t us assume t h a t A is a self-adjoint extension of a 0 a n d t h a t A is b o u n d e d f r o m below, A >~ ~0I, say, where I is t h e i d e n t i t y o p e r a t o r in

L2(S).

I n t h e sense of N a g y [7], to A there corresponds a spectral resolution E(~), which is a projection-valued, non-decreasing f u n c t i o n on t h e real line. W e h a v e E ( 2 ) = 0 for 2 < 20. Since M is hypoelliptic, the following s t a t e m e n t holds ( H 6 r m a n d e r [5]).

To e v e r y multi-index :r there is a positive integer r such t h a t

D~u

is continuous (i.e. there is a continuous function v such t h a t

D~'u =v

in t h e distributional sense) for e v e r y distribution u such t h a t

M(D)ru

is locally square integrable, a n d we h a v e a n inequality

sup

ID~u(x)] < C([[i(n)vu][ +

Hull), (2)

x e K

where K is a n y c o m p a c t subset of S, a n d C is i n d e p e n d e n t of u b u t m a y d e p e n d on

ARKIV FOR MATiEMATIK. B d 5 nr 3 5

~, r, K, a n d S. Of course, sup~,K

ID~u(x) l

m e a n s supz~K Iv(x)], where v is the con- tinuous function equivalent to

D~u.

B y (2) it m a y be shown ( H 6 r m a n d e r [5]) t h a t E(~) is given b y a kernel

e4(x,y),

the spectral function of A, such t h a t

E(2) u(x) = f s e4(x,y)u(y)dy (u E L~(S)),

where e4 is defined a n d infinitely differentiable in S • S. F u r t h e r

e4(x, y ) = e4(y, x)

for all

x, y C S.

W e also h a v e an estimate

e(~'~)(x, y)--il~+~lD~ D~ e~(x,

y) = 0(1)/~p+ ~(l~+~l) (3) when )~-~ + co, u n i f o r m l y on c o m p a c t subsets of S • S, with some positive n u m b e r s p a n d k a n d all g,fl. F o r e4 we also h a v e t h e following lemma.

L e m m a 1. F o r a n y ~ a n d a n y

x E S , e(~'~)(x,x)

is an increasing f u n c t i o n of 4, a n d t h e v a r i a t i o n w i t h respect to ). on a n y real interval A satisfies the i n e q u a l i t y

v a r e~ ~'~) (x, y) ~ (var e(~ ~' ~) (x, x). v a r e(~ ~'z) (y, y))89

A A A

for all x, y E S a n d all a, ft.

Proo/.

F o r the proof we refer to Bergendal [1], t h e L e m m a s 1.2.2 a n d 1.2.1. There t h e l e m m a is p r o v e d for the spectral f u n c t i o n of a n elliptic operator, b u t the proof only uses t h a t ( e 4 - % ) is the kernel of an o r t h o g o n a l projection if 2>/~, a n d so it works as well in our case.

I n particular it follows f r o m the l e m m a t h a t for a n y x, y, ~,fl t h e function

e(~'~)(x,y)

is locally of b o u n d e d variation. I f ~t <)~0, t h e n G(~)= ( A - X I ) -1 exists as a b o u n d e d operator in

L2(S),

a n d ]](A ~I)-~ll ~< (~0-~)-~- I f the integral of a real function with respect to a spectral measure is defined as in N a g y [7], t h e n G(~t) = ~+0~r (#

- ~ ) - l d E ( # ) .

I f in (3) t h e n u m b e r p is smaller t h a n 1, t h e n

G4 (x, y) = (l~ - ~.) ld% (x, y)

(4)

4.

is defined as a continuous function in S • S (this is seen e.g. b y a n integration b y parts). F r o m t h e definition of the integral with respect to a spectral measure ( N a g y [7]) it follows t h a t on C~(S) (and also on L2(S)), G4 is t h e kernel of (A -~t) -1. W e shall call G~ Green's function corresponding to A. W e see t h a t for ~ e C g ' ( S ) t h e function ~0(x)-(G4(x,-),q)) is continuous (no eorrecl:ion is needed). I f in (3) also (P + / c l ~ +/~1) < 1, we get f r o m (3) t h a t

G~ ~'~)

is continuous in S • S, a n d

a p ,~)

(x,

y) = (~

- ~.)-~de~ ~' ~) (x, y).

(5) If instead of A we consider t h e o p e r a t o r

B = A r

with a positive integer r, t h e n B is self-adjoint a n d b o u n d e d f r o m below, a n d B is f u r t h e r an extension of M(D)~, defined on C~'(S). Since 21/(D) ~ is hypoelliptie, B has a spectral f u n c t i o n

er, x(x,y),

a n d for large ~ we h a v e

er.4

=e2~,. Hence, t a k i n g r large enough, we m a y m a k e t h e e x p o n e n t in (3) smaller t h a n 1, if we have e~,~ instead of e4. Hence, for a n y M, ~ a n d fi, (5) holds for the Green's functio n a n d the spectral f u n c t i o n of A ~ if we take r large enough.

r~. NILSSOr,r,

Asymptotic estimates for spectral functions

W e h a v e a p a r t i c u l a r l y simple case when the set S is the whole of R n. T h e n t h e Fourier t r a n s f o r m

~t/(~) = j" exp( -

2zi(x, ~))/(x) dx

t a k e n in the sense of Schwartz [11] is a u n i t a r y m a p p i n g of

L2(S)

into

L~(R~),

a n d :~a0ff-1 is multiplication b y M(~). Hence a 0 has a unique self-adjoint extension A0, a n d since the spectral resolution/~r(2) of .~g = ffA~9 t-1 is multiplication b y t h e charac- teristic function of the set (~IM(~) ~ ~<2} a n d the operator

(A r - 2 )

-2 is multiplication b y (M(~) r --2) -1, we h a v e

eo,~,a (x, y) = | exp ( 2 ~ i ( x - y,

~))

d~

JM

a n d Go, r,a (x, y) =

2)-1

exp (2zd(x - y, ~)) d~. (6)

The integral is absolutely c o n v e r g e n t for large negative 2 if r is large enough, since for a hypoelliptie p o l y n o m i a l i ( ~ ) we have I M(~)I ~> C]~I ~ for all large real ~ with some positive constants c a n d C ( H S r m a n d e r [5]).

W e n o w give a result on t h e a s y m p t o t i c b e h a v i o u r of

eo.a(x, x),

w h e n 2 tends t o -~c~.

Theorem l. L e t P(~I ... ~ ) be a real polynomial such t h a t P ( ~ l ... ~n)-+ + c~when I~:1 -+ co (~e real) a n d let ~ be a multi-index. I f

t h e n there are positive n u m b e r s

c, C,

a n d

a,

a n d a non-negative integer t such t h a t C-~2a(log 2) t ~< e(2) < C2a(log 2) t (2 > c)

a n d e'(2)=0(1)2 ~ X(log 2) t (2--> + ~ ) .

I f n = 2 , t h e n t = 0 o r t = l a n d

e(2) =(kq-o(1))2a(log 2) t (2"-> A- c~)

with some positive c o n s t a n t k.

Remark.

I t is clear t h a t the n u m b e r s a a n d t are u n i q u e l y d e t e r m i n e d b y P a n d ~.

W e shall call a =

a(P, ~)

a n d

t - t(P, o~)

the E - n u m b e r s of the pair (P, ~).

T h e proof of T h e o r e m 1 depends on the following l e m m a which is a p~rticular case of results in the a u t h o r ' s paper [8] (Theorems 1 a n d 2 a n d L e m m a 2).

L e m m a 2. Consider a real algebraic manifold V(2): p(2,~) - 0 in R n depending on 2 E R . Here p(2,~) is a real p o l y n o m i a l in 2 C R a n d ~ E R L Suppose that, for some 20, V(20) is n o t e m p t y a n d t h a t grad e p(20, ~) =4= 0 for all ~ E V(2o). F u r t h e r assume t h a t there is a b o u n d e d subset ~2 of R " such t h a t V(2) c ~ for all 2 in a n e i g h b o u r h o o d of 2 o. F o r 2 in a n e i g h b o u r h o o d of 20, let coa(~) be a differential ( n - 1)-form on V(2) such

ARKIV FOR MATEMATIK. B d 5 n r 35

t h a t in a n y local coordinate system on V(20) with coordinates ~' picked among the

~:~ (also defining a local coordinate system on V(~t) for ~ in a neighbourhood of 20) the coefficients of o)~(~) are regular analytic algebraic functions of (;t,~'). Define the function

~(~) = f,(~) ~ (~)

in a (sufficiently small) neighbourhood of 20. Let G(~) be a primitive function of g(2).

Then there is a finite set W of points ~1 ... s~E C such t h a t G() 0 m a y be continued analytically along a n y p a t h in C not passing through a n y point of W. Moreover, all the determinations of G(2) in the neighbourhood of a n y ;t E

(C-W)

span a finite dimen- sional linear space over C. P u t ~ = 2. maxj]~jl. Then, if 21 > ~ and if G1(2 ) is a function element of G() 0 at 21, there is a real n u m b e r c and to every positive integer N a n u m b e r K such t h a t

] G(2)] ~<KI2] ~

for all ~ with ]21 >~ and for all determinations of G(2) t h a t m a y be obtained from GI(~ ) b y analytic continuation at most N rounds in the region 121 >~. These proper- ties hold also for the function g(2) itself. I f n = 2 , then there is a positive integer N such t h a t for [~ I >

T~g(2)=g(2)+h(2), T~h()O=h(2),

where T is analytic continuation one round in the positive sense along circles ]~t I = constant.

For the proof of Theorem 1 we shall also need the following lemma.

L e m m a 3. Let q(~l ... ~n) be a complex polynomial. Then there is a n u m b e r a such t h a t when ]21 > a we have grad q(~)#0 for all ~E V(2): q(~) =2.

Proo/.

Consider the algebraic manifold grad(q(~)) = 0. I t consists of a finite n u m b e r of connected components

F 1 ... Fs,

and q(~) is constant =~.j on every Fj. Then for ]~[ > m a x ( ] 2 j l ) we have t h a t grad q(~) is different from zero for all ~E V(2). The lemma is proved.

Now let us turn to the proof of Theorem 1. I t follows from l e m m a 3 t h a t e(~) is real analytic for ~t greater t h a n some a. Consider the derivative

/(~)=e'(2).

We m a y write ](2) as an integral over V(2): P(~I ... ~n) =~,

I t is clear t h a t the differential ( n - 1)-form (o~(~)=

(d~/dP(~))v(x>

on V(2) has regular algebraic coefficients in a n y local coordinate system with coordinates a m o n g the

~. Hence e(2) has the properties stated in L e m m a 2.

F r o m the fact t h a t all the determinations of e(~) span a finite dimensional linear space over the complex numbers it follows (see e.g. Goursat [4], p. 4 4 7 - 4 6 0 ) t h a t in a neighbourhood of infinity e(;t) is a finite sum of terms of the t y p e 2#(log ~t)~H().), where fi is a complex number, v a non-negative integer, and H(2) is analytic a n d single- valued in a neighbourhood of ~ . Hence every such function H(;t) m a y be developed into a L a u r e n t series ~-~_~oak~t z, convergent in a neighbourhood of oo. F u r t h e r all the 531

r~. rqLSSOrr

Asymptotic estimates for spectral functions

functions H(2) are linear c o m b i n a t i o n s of functions of the f o r m 2V(log 2)/'h(2), where F is a complex n u m b e r , /t an integer, a n d h(2) some b r a n c h of e(2). Because of the estimate of e(2) o b t a i n e d in l e m m a 2 the L a u r e n t series of a n y H(2) contains only a finite n u m b e r of non-zero terms with a positive exponent.

L e t us write e v e r y t e r m 2r 2)"H(2) so t h a t H(2) = ~o =- ~ ak2~ with a 0 # 0 , which we can always do, choosing fl conveniently. T h e n a m o n g the terms of e we select t h e 'largest' ones, first t a k i n g those h a v i n g Re({/) maximal, = a , say, a n d a m o n g these keep those who h a v e v maximal, = t , say. Then, in e v e r y such ' m a x i m a l ' term we replace H(2) b y the c o n s t a n t t e r m in t h e L a u r e n t expansion. T h e s u m of the selected terms is t h e n a function

~(2) = 2a(log 2)~(cl 2 ~' + . . . + c 12 i/~') = 2a(log 2) t" O(log 2),

where the c~ a n d k~ are constants a n d the k~ real, a n d we m a y suppose t h a t (I) is n o t identically zero. B y our m e t h o d of picking the terms in ~0 we h a v e

e(2) - ~(2) = o(1) 2~(log 2) t (4-+ + o~). (7)

We h a v e (I)(#)=(I)a(#)+ i(P2(/z), where (1) a a n d qb~ are functions of the t y p e

g(#) = A lsin(d~kt + e~) + . . . + A qsin

(dqkt + eq),

(8) where the A j, dj a n d ej are real c o n s t a n t s a n d q some positive integer. I t is well k n o w n (see e.g. Besicovitch [2], p. 5, Th. 12) t h a t a function g of t h e t y p e (8) has the following p r o p e r t y . I f to 0 is in the range of g, t h e n there is to e v e r y e > 0 an increasing sequence

#1,/z2,... of real n u m b e r s a n d a positive n u m b e r

K,

such t h a t / t ~ - + + c~when 7"

-+ -4- c ~ , ~ t j + l - / ~ j < K for all 7" a n d

Ig( J)- ol . . .

B y this p r o p e r t y we get t h a t @2--- 0. For, if there were a n u m b e r Y 0 4 0 in the range of (P2, t h e n there would be a sequence (~j), tending to + ~ , such t h a t

IIm(~(2j))l > ly012?(log b)t/2,

a n d f r o m (7) it would t h e n follow t h a t e(2j) is non-real, if j is sufficiently large, which is a contradiction, since e is real. H e n c e (I)2 ~ 0 , a n d (I)=(I) I. A n analogous a r g u m e n t shows t h a t (I)~>0, as a consequence of the inequality e(2)~>0. N o w let us consider

e'(2).

F r o m t h e w a y of picking the terms in ~ we find

e'(2) - a U - a ( l o g 2)t(P(log 4) +2~-a(log 2)t(I)'(log 4) (9) + o ( 1 ) 2 " - ' ( l o g 2 ) ~ (2-+ + ~ ) .

B y the same t y p e of a r g u m e n t s as a b o v e for r a n d (I) 2 we get f r o m (9) a n d e'(2) >~ 0 (e(2) is e v i d e n t l y increasing)

(I)'(/~) >~ - a(I)(kt ). (10)

Since (I) is n o t identically zero, a n d (I) >~ 0, there is an increasing s e q u e n c e / % #2, ... such t h a t / x j-+ + ~ w h e n 7"-+ + ~ , a n d with two positive n u m b e r s C a n d K

( ~ j + I - # j ) < K , dp(~j)>C, for all j.

ARKIV FOR MATEMATIK. Bd 5 nr 35 L e t # be an a r b i t r a r y n u m b e r

:>/.tl,

a n d let ~uz be the largest juj which is ~</~. T h e solution of the differential e q u a t i o n u ' = - a u which passes t h r o u g h t h e p o i n t (#l, (I)(#z)) is

u(x)

= (I) (/~l) exp ( -

a(x -/~l)).

F r o m (10) it t h e n follows (I)(x)>~u(x) for x ~ # z , a n d so a m u s t be non-negative, since otherwise (I) would grow exponentially, b u t we k n o w t h a t it is bounded. F u r t h e r x =~u gives

(I)(#) ~> C e x p ( - a K ) > 0 .

B y (7) we n o w get the s t a t e m e n t a b o u t e(~t) in the theorem. T h a t a b o u t e'(~) follows f r o m (9).

I t remains to show the stronger assertions in the case n = 2. B y L e m m a 2 there is an integer N > 0 such t h a t in a n e i g h b o u r h o o d of c~ we h a v e

TNe '=

e ' +

h, TNh =h.

Hence h(~) is in a n e i g h b o u r h o o d of ~ a single-valued f u n c t i o n of

~I/N,

a n d so is F(~) = e ' ( 2 ) - (2N~i)-lh(2)log ~. T h u s h a n d F m a y be developed into Puiseux series in a n e i g h b o u r h o o d of ~ . F r o m the estimate b y L e m m a 2 holding for e'(~) it follows t h a t h a n d F are of p o l y n o m i a l growth, a n d so their Puiseux expansions contain only a finite n u m b e r of non-zero terms with a positive exponent. H e n c e in a n e i g h b o u r h o o d of ~ we h a v e

k 0 k0 t

e'(~)= ~

ak~k/N+(1og~) ~

bk~ kIN

k = o v k = - ~

B y integration we find

e(2) = F~(2) + (log 2) F2(2 ) + b N (log 2)2/2,

where F 1 a n d F~ are Puiseux series, convergent in a n e i g h b o u r h o o d of ~ a n d contain- ing only a finite n u m b e r of terms with a positive exponent. Since e(2) grows faster t h a n some positive power of 2, t h e t e r m

b_N(lOg2)2/2

is n o t the leading one, a n d the particular s t a t e m e n t for n = 2 follows. (It m a y be shown t h a t a c t u a l l y

b_N:O. )

The t h e o r e m is proved.

N o w we r e t u r n to our hypoelliptie p o l y n o m i a l M(~) a n d t h e unique self-adjoint extension A 0 in

L2(R ~)

of

M(D),

defined on

C~(R'~),

a n d the spectral function

eo.~(x,

y ) = ~M(~)<~ exp

(27d(x--y,

~})d~. F o r an a r b i t r a r y multi-index ~ we h a v e

Z ) O, 1 ( ~ ,

JM ^(~)~2

Hence T h e o r e m 1 gives a result on t h e b e h a v i o u r of

e(~'~))(x,x)

when x--> + ~ , a n d to the pair

(M, ~)

we h a v e a pair of E - n u m b e r s

a(M, o:)

a n d

t(M, oQ.

3. A n e s t i m a t e for a c e r t a i n f u n d a m e n t a l s o l u t i o n

We consider ( M ( ~ ) r - ~ ) with a positive integer r a n d ]t large a n d negative. T h e o p e r a t o r (M(D)r-~t) has a t e m p e r a t e f u n d a m e n t a l solution with pole zero which is the inverse Fourier t r a n s f o r m of

(M(~) r-~)-l.

H e n c e t h e f u n d a m e n t a l solution with pole x is

hr.z(x,y) =~ (i(~) r _~)-1

exp

(2zi(y

- x,~}) d$,

z~, z~ILssorr

Asymptotic estimates for spectral functions

w h e r e the integral is absolutely c o n v e r g e n t if r is large enough. I t is clear t h a t h~,a is t h e c o m p l e x c o n j u g a t e of t h e G r e e n ' s function of A~ given b y (6). W e are going to show t h a t outside the pole t h e f u n d a m e n t a l solution t e n d s e x p o n e n t i a l l y to zero, w h e n ~--> - ~ . F o r t h a t we shall need the following l e m m a .

L e m m a 4. I f M(S) is a hypoelliptie p o l y n o m i a l of degree m, t h e n t h e r e is a largest n u m b e r

b=b(M)

such t h a t

O<b<~l/m

a n d

I/<=)(S)[ < r + 1) ~ ~=' (1U

f o r some n u m b e r C a n d all real S a n d all ~. I f r is a positive integer, t h e n

b(M ~) = b(M)/r.

Proo/.

F o r a proof we refer to H 6 r m a n d e r [6], T h e o r e m 3.2, e x c e p t for t h e last s t a t e m e n t , b u t this is easily checked using t h a t it is p r o v e d in H 6 r m a n d e r [6] t h a t if b is the largest n u m b e r such t h a t (11) holds for all ~ with I~l = 1, t h e n (11) holds for all ~ with the s a m e b.

W e also h a v e

L e m m a 5. L e t N be a hypoelliptic p o l y n o m i a l a n d p u t b =b(N). T h e n

IN<=>(s + zs0)

^-

N+(s)I < cIg(lsl + 1)-~ + ljo)

for some c o n s t a n t C, all ~, all real S a n d ~ ~> I a n d all c o m p l e x z w i t h I z[ ~< 1. H e r e S0 is a r b i t r a r y in R n a n d C a n d c are positive a n d i n d e p e n d e n t of S, z, a n d z.

Proo/.

B y T a y l o r ' s f o r m u l a

N(=)(S + ~zS0) - N(=)(S) = ~ (~z)Wj (S),

]=1

w h e r e m is t h e degree of N a n d N j is a linear c o m b i n a t i o n of d e r i v a t i v e s of N of order ( I ~ I + J)" B y L e m m a 4 we h a v e

I 'z Nj(S)I < C@I'(IN(S)I + 1) (12)

w i t h some c o n s t a n t C. F r o m the well-known inequality

xay l-a<< - x + y

for x, y > 0 a n d - - ' ~ =(IN(S)[ + 1 ) a n d

a=]b,

we get

0~<a~<l, then, f r o m (12), p u t t i n g x - e j , y

Iv~zJNj(S)l <

CNJ(IN(S)[ +

1)-o'=q[N(S)I + 1 +

7~llb).

Since IN(S) l >~ IS[ k with some positive k for large I Sl t h e proof is complete.

F r o m L e m m a 4 we also get

L e m m a 6. I f N(S) is hypoelliptic, t h e n

IN+(S)l < c(ISl + +

1)

for all cr a n d all real S, where c a n d C are positive constants. W e m a y n o w give a n e s t i m a t e for the f u n d a m e n t a l solution considered.

L e m m a 7. L e t N(S) be real a n d hypoelliptic a n d let N ( S ) - > + cr w h e n IS]--> c~.

P u t b = b ( N ) . T h e n 534

ARKIV FOR MATEMATIK. Bd 5 nr 35

hz(x,

y) = f e x p (2~i(y - x, ~>) (N(~) - 2)-~d~

:is (with respect to y) a t e m p e r a t e f u n d a m e n t a l solution with pole x of

( N ( D ) - 2 )

when 2 is large and negative, and

D~h~(x,y)

= O(1) exp (-cl21 ~) ( 2 - * - ~ )

~or x =4=y, all :r and some c > 0 . T h e estimate is uniform on compact subsets of the region x 4 y .

Proo/.

Take an a r b i t r a r y ~oER ~, let z be a complex n u m b e r and p u t H~(~,z)=

N ( ~ + [21~z$0)-2. B y L e m m a 5 it follows, taking ~ = 12[ ~, t h a t there are positive n u m b e r s C', C and ( - 4 1 ) such t h a t

C-I(IN(~)] + ] h i ) <

IH ( ,z)l <C(IN( )I + Ihl) (2<21) (13)

f o r all real $ and all z with I zl ~< C'.

Now, for I I m (z) l ~< c' the inverse Fourier transform of 1/H~(~, z) (with respect to

~) is equal to exp

(2~iz]21b(y,

~0})h~(0, y). I n fact, a translation b y z ]2 I~o corresponds b y the Fourier t r a n s f o r m a t i o n to multiplication b y

exp(2giz]2]b(y,~o}),

since

H~(~,z)

keeps a w a y from zero when Izl <~c' (see Nilsson [8], p. 114).

L e t

B(y)

be a positive definite homogeneous polynomial of degree /. T h e n B ( y ) e x p (2~i [2[~(y,~=o})h~(0,y) is (as a function of y) the inverse F o u r i e r t r a n s f o r m of

B(D~)(1/Ha(~,z)).

F r o m the rules of differentiation we see t h a t

B(D~)(1/H~(~,z))

is a linear combination of terms (H~')(~,z)-... -H~J')(~,z))/Ha(~,z)

r+l,

where 5 1 ~ ~1 =]"

Now it follows from (13) and L e m m a 5 t h a t

]H (~) (~,

z)/H~($,

z)] 4 C(]~ I + 1) -c I~l

for all real $, all 2<21 and all z with ]z] <c', and where C is a constant. So we m a y conclude t h a t if [ is sufficiently large we have

IB(D~)

(1/H~(~, z)) I ~< C(I~ [ + l ) - n - 1

with some n u m b e r C, i n d e p e n d e n t of ~, 2 and z for ]z I

~c'

and 2<21. B u t t h e n w e get, p u t t i n g

z - ic' :

IB(y)

exp

(27~c'Nb(y,

~0))h~(0, y)[ ~< C (2 < 2~) for all y, where C is some number, i n d e p e n d e n t of 2 and y.

Since ~o is arbitrary, the lemma follows in the case a = 0. To get it for a r b i t r a r y a we need only notice t h a t for y ~:0 we have

N(Dy)Sh;.(O,y)=2~ha(O,y)

and t h e n use (2).

4. Asymptotic estimates for the spectral function w h e n the domain S is arbitrary

First we are going to establish a relation between the Green's functions of A r and A~, Gr.~(x, y) and

Go r.~(x, y) = hr.~

(x, y), respectively.

L e m m a 8 (see Odhnoff [10]). I n

L2(S)

one has the following identity.

535

~. I~IILSSON,

Asymptotic estimates for spectral functions

Gr, x(x, ") =~vhr.~(x," ) + ( B - k ) lk~.~(x," ),

(14) where x is a r b i t r a r y in

S,

~oEC~(S), ~v real a n d ~v(y) = 1 in a n e i g h b o u r h o o d of x.

F u r t h e r B = A r, a n d

kr.~(x, y) = (~v(y) By - B~p(y) )hr.~(x, y).

(In particular

kr,~(x,. ) E C8r

Proo/.

L e t us denote t h e right side of (14) by/~,~(x,. ) a n d prove t h a t

( ( B - ,~) u,b.~(x,. )) = u(x) (15)

when u E O ( B ~) = f l ~ ~ ( B j) c

C~176

(the last relation b y (2)). I n fact, we h a v e seen t h a t

((B-~)u,G~,~(x,.))=u(x)

for all u such t h a t

( B - ~ ) u E C ~ ( S ) ,

and, ( B - X ) -1 being bounded, we should t h e n h a v e

(v,/r.~(x,') - G~,~(x,.

)) = 0 for all v E C~(S), a n d the l e m m a would follow. To verify (15) we first consider (with u E D(BOO))

( ( B - X)upfhT.~(x,. )) = (~v(B-~)u,h~.~(x,. ))

- ( ( B ~)~u, hr,~(x,'))+((y~B- B~f)u, hr.~(x,'))

- u(x) + ((y~B - B~f) u,h~.a(x,

")), (16) where in the last step we h a v e used t h a t

hr.~(x,')

is a f u n d a m e n t a l solution of

(M(D) ~-4)

with pole x. N o w we consider

((B - , ~ ) u , ( B -;t)-~k~.~(x,. )) = (u, kr. ~(x,. ))

=(u,(~vB-B~f)hr, x(x,')) =((ByJ-yJB)u,h~.~(x,')),

(17) where t h e last step is p e r m i t t e d since the differential o p e r a t o r (B~v-y~B) vanishes outside a c o m p a c t subset of 8 - {x}. The l e m m a n o w follows f r o m (16) a n d (17).

N e x t we are going to estimate the term

(B-2)-lk~,~(x,.)

in (14). B y L e m m a 7 we h a v e

where c is a positive c o n s t a n t a n d b corresponds to M b y L e m m a 2 a n d the estimate is uniform in the n e i g h b o u r h o o d of a n y point in S. I t follows t h a t

II(B-A)-ikr,~(x, ")ll =o(1)exp(-cl l

L e t ~ be an a r b i t r a r y multi-index, a n d let us consider/)~(B-~t)-~kr,~(x,"

). By

^{(2) we}

t h e n get, if r is large enough,

D~(B--~)-~kr,~(x,y)--O(1)exp(--C']),ib/r) (~-->--oz)

with a positive c o n s t a n t c', a n d t h e estimate is uniform on c o m p a c t subsets of eo • S, where ~ is a n e i g h b o u r h o o d of a n a r b i t r a r y point in S. B y L e m m a 8 it is t h e n easy t o see t h a t

D~(G~.x(x,y) -Go,,,x(x,y))

= O ( 1 ) e x p ( - k ] X [ ~/~) (~-+ - ~ ) u n i f o r m l y on c o m p a c t subsets of S > S, where k is a positive constant.

ARKIV FOR MATEMATIK. Bd 5 nr 35 Since (Gr,~(y, x) - Go, ,,~(y, x)) =

(Gr,~(x, y) - Go,~,s(x, y)),

it also follows t h a t

D~(G~,~(x,y)-Go,~,~(x,y))

= O ( 1 ) e x p ( - k l 2 p l*) (2-+ - co)

u n i f o r m l y on c o m p a c t subsets of S • S. Hence, w i t h a n a r b i t r a r y positive integer s, if r is large enough

(A~ + A~)

(Gt,~(x, y) - ao,~,~(x,

y)) = 0(1) e x p ( - kl21 b~ ) (~-+ - o~)

uniformty~on- c o m p a c t subsets of S • S. B y well-known e s t i m a t e s for elliptic o p e r a t o r s (of the t y p e (2)) it t h e n follows t h a t for a n y p a i r (a,fl) of multi-indices

( ~ ( ~ , f l ) ,

i x , y ) - y ) ) = o ( 1 ) e x p ( - klml - co), (18)

if r is large enough.

I f we a s s u m e A > 0 , A 0 > 0 , we h a v e b y (6) a n d (18)

- - ~0. r,,k~, y)) = 0(1) exp ( --

d o

w h e n ~ - + - c~. To get i n f o r m a t i o n for (er.~-eo.r,x) f r o m this e s t i m a t e we shall use a T a u b e r i a n t h e o r e m b y Ganelius. T h e t h e o r e m to be q u o t e d is u n p u b l i s h e d b u t will a p p e a r in t h e M a t h e m a t i e a Scandinaviea; t h e corresponding t h e o r e m for the La- place t r a n s f o r m a t i o n has been a n n o u n c e d in [12]. (If we are c o n t e n t with the result

ez(x,x) =

(1 + o(1))

eo.~(x,x)

we can use a T a u b e r i a n t h e o r e m b y Keldish [13], where t h e T a u b e r i a n condition is

I t follows f r o m T h e o r e m 1 t h a t this condition is satisfied, if r is large enough.) First we define a slowly oscillating f u n c t i o n as a positive, continuous f u n c t i o n L o n t h e positive real line such t h a t

L(cco)/L(co)--->l

w h e n (o--> + c~ for e v e r y c > 0 . T h e n we h a v e

L e m m a 9. L e t the f u n c t i o n a(#) be locally of b o u n d e d v a r i a t i o n for # > 0 . Suppose t h a t ff~ ~ (/z + co)-ida(/z) is c o n v e r g e n t for co = some x o > 0. (and hence for e v e r y co n o t o n t h e n e g a t i v e real axis). L e t c, n, a n d v be real n u m b e r s , c > 0 , 0 < n ~ < 8 9 a n d v < l . L e t L(co) be a slowly oscillating function. Then, if

f o ~ ( # + c o ) - i d a ( / z ) = O ( 1 ) e x p (-c[co[ ~) (co--> + ~ ) (19)

a n d s u p (

f da(#))<.O(1)coV-*L(co)(~o-->+~),

(2O)

a~

w~<~l~<a) + o ) 1 - n

t h e n a(co) = 0(1) o ' - * L ( o ) (co--> + co).

N o w we are going to a p p l y this T a u b e r i a n t h e o r e m to t h e f u n c t i o n

(~(~) = (eo, r~.(x, x) e(/'2)(x, x))

1'r ~ILSSON,

Asymptotic estimates for spectral functions

with ~ a r b i t r a r y a n d

x E S .

W e let (a,t) be the E - n u m b e r s of (M,~). W i t h c = t h e n u m b e r k of ( 1 8 ) , u -

b/r, v =a/r

a n d L ( a ) ) = (log co) t we h a v e t h a t

c, ~, v,

a n d L satisfy t h e conditions of the lemma, if r is large enough, a n d also the other conditions a r e

eo.r.~(x,x)

in satisfied, (19) because of (18) a n d (20)because of the estimate for ( ~ / ~ ) (~'~) T h e o r e m 1 a n d the fact t h a t

e(/ff)(x,x)

is a non-decreasing function of ~, which w a s s t a t e d in L e m m a 1. H e n c e we get the conclusion of L e m m a 9:

(e(o~'/)~(x, x) - e(/'~)(x,

x)) = 0(1) ~(~-~)/r(log ~)t (~--~ + ~ ) . (21) B y (21) a n d L e m m a 1 we n o w g e t

v a r

e(/,i~)(x,

y) = 0(1) ~(%+~-2~)/e~(log

~)(t:~+tfl)/2

(Jl A + ;t~-b/')

w h e n ~-> + oo. H e r e (am, tl) a n d (am, t2) arc t h e E - n u m b e r s of (M, ~) a n d

(M, fi),

re- spectively, a n d a a n d /5 are a r b i t r a r y multi-indices, x a n d y belong to

S,

a n d r is sufficiently large. T h e same estimate holds for <~' ~) ~

eo ~.~ t+, y),

a n d so, t a k i n g

- ~ + ( + ' ~ ) + x +'~ - + ( + , Z ) l + y ) ) ,

,7(,u) - ~o,~,,~ ,~,p ~,~, ~+, we get b y the T a u b e r i a n t h e o r e m

(~,fl)

(eo. re(x, y) - e(/.'~l~)(x,

y)) = 0(1) ~(a,+~-2b)/2r (log 2)<t~+t~)/2 (22) when ~--> + oo. H o w e v e r , we w a n t the results for e~ = el,), a n d n o t for

erA.

F r o m t h e relation er,~ =e~l~, we i m m e d i a t e l y find t h a t (22) is valid n o t o n l y for r suf- ficiently large b u t also for r = 1. Our restriction t h a t A > 0, A 0 > 0, m a y also be removed, since b y a translation in t h e eigenvalue p a r a m e t e r ~ we m a y m a k e these t w o inequalities satisfied, a n d t h e translation does n o t change the a s y m p t o t i c for- mulas.

W e can also t a k e care of the case where A is n o t b o u n d e d f r o m below. W e h a v e the following lemma.

L e m m a 10. L e t A be an a r b i t r a r y self-adjoint extension in

L2(S)

of a 0 a n d E(2) t h e corresponding spectral resolution. T h e n for a n y ~, E(2) is given b y a kernel e~:

E(2) u(x) = f s e~(x, y) u(y) dy (u E

L2(S)), where e~ is infinitely differentiable in S • S a n d where

e(~'Z)(x, y) =

0(1) exp ( -

cN b(M)) (),--> - c~)

u n i f o r m l y on c o m p a c t subsets of S • H e r e c is a positive c o n s t a n t a n d ~,/5 a r e a r b i t r a r y multi-indices.

Proo/.

F o r a proof we refer to Nilsson [8], the Theorems 3 a n d 4, where the corre- sponding t h e o r e m is p r o v e d for a n elliptic differential o p e r a t o r

P(D).

The proof, however, works as well in our case. F o r it uses essentially three facts:

(a) To every x E S we h a v e a f u n d a m e n t a l solution

g~(x, y)

with pole x of

(P(D)

-~t), defined when ~ is large a n d negative a n d decreasing exponentially outside the pole w h e n ~-+ - ~ ,

ARKIV FOR MATEMATIK. B d 5 nr 3 5 (b) the f u n d a m e n t a l solution

g~(x, y)

a b o v e satisfies an i n e q u a l i t y

u n i f o r m l y on c o m p a c t subsets of S • S a n d for all 4. H e r e C a n d ~ are positive con- stants,

(c) we h a v e for

P(D)

an interior a priori L2-estimate of t h e t y p e (2) (this paper).

:Now (a) holds also for

M(D)

( L e m m a 7) a n d so does (c). F u r t h e r in [8] (b) is o n l y used to m a k e certain t h a t the m a p p i n g

u--> fj:g.~(.,y)u(y)dy

( K = c o m p a c t ~ S )

is continuous f r o m local L 2 to local L 2 a n d t h a t the c o n t i n u i t y is uniform with respect to 4. I n our case we h a v e t h a t M(~)~>CI~I k w h e n ~ is large, with some positive c,

C,

a n d it follows t h a t the t e m p e r a t e f u n d a m e n t a l solution of

(M(D) r -4)

is u n i f o r m l y b o u n d e d with respect to 4, if r is large enough a n d 4 large a n d negative.

This result m a y t h e n replace (b) in question of M ( D ) r, b u t via the e l e m e n t a r y con- nection between spectral functions of

M(D)

a n d

M(D) r,

with r odd, we get t h e desired result also for

M(D).

B y L e m m a 10 we m a y n o w see t h a t (22) is valid also if A is n o t b o u n d e d f r o m below. F o r let us consider

A r

with r even; t h e n A ~ is b o u n d e d f r o m below so t h a t (22) holds for e~.~. B u t

e.~=e_.~+er.~,

for 4 > 0 , a n d so b y l e m m a 10 we get (22) also for e~.

W e collect our results in the following theorem.

Theorem 2. L e t M(~) be a real hypoelliptic polynomial in

R ~, n>~2,

such t h a t M(~)--> + ~ when ]~[ - ~ . L e t S be an open subset of R n a n d let a 0 be t h e opera- t o r in

L2(S)

defined b y the differential operator

M(D),

acting on

C~(S).

Suppose t h a t A is a self-adjoint extension in

L2(S) of

%, n o t necessarily b o u n d e d from below. T h e n t h e spectral resolution E(4) of A is given b y a kernel

e~(x,y):

E(4) u(x) = / ~ e~(x, y) u(y) dy (u E L2(S)),

where ez is infinitely differentiable in S • S, a n d

e(~'~)(x,y)

= 0(1) exp (

-cl41 b(M)) (4--> - ~ )

for a n y multi-indices ~,fl. Here c > 0 , a n d

b(M)

is t h e largest positive n u m b e r b such t h a t with some c o n s t a n t C

IM(~)(~)I <~ C([M(~)] +

^{1) T M}

for all ~ a n d all real ~. (If M is elliptic a n d of degree m, we h a v e

b(M) = 1/m).

F u r t h e r , if A 0 is the unique self-adjoint extension in

L2(R n) of M(D),

defined on

C~(Rn),

a n d e0.~ its spectral function, t h e n

(e(~'~)(x, y) - e(o~ff)(x, y) )

= O(1)4(t~+tfl

eb(M))(log 4)(t~+t~)/2

N. NILSSON,

Asymptotic estimates for spectral functions

when)~--> + c~. Here ~,fi are arbitrary. The pair

(ar, tT)

is characterized b y the property t h a t

K-l,~.a~ ' (log ~)~ ~<

e(oVS)(x, x) <. K~%'

(log 2)~

for some K >0, all large positive ~ and y = ~,fl. That such numbers exist was proved in Theorem 1.

B I B L I O G R A P H Y

1. BERGENDAL, G., Convergence a n d summability of eigenfunction expansions connected w i t h ellipitic differential operators. Medd. fr. Lunds univ. mat. sem 15 (1959).

2. BESICOVITCH, A. S., Almost Periodic Functions. Cambridge, 1932.

3. GANELIUS, J., U n th6orbme taub6rien pour la transformation de Laplace. C. R. Acad. Sci.

Paris 242, 719-721 (1956).

4. GORTJAKOV, V. N., On t h e asymptotic behaviour of t h e spectral function of a class of hypoel- liptic operators. ]:)oklady Ak. Nauk, 152,3, p. 519-522 (1963).

5. GOURSAT, E., Cours d'analyse math6matique 2. Paris, 1924.

6. H(}RMANDER, L., On t h e theory of general partial differential operators. A c t a M a t h e m a t i c a 94 (1955).

7. - - On interior regularity of t h e solutions of partial differential equations. Comm. of pure and appl. m a t h . 9 (1958).

8. KELDISH, M. V., On a Tauberian theorem. Trudy Matematitjeskovo I n s t i t u t a Imeni V. A.

Steklova, vol. X X X V I I I , 77-86 (1951).

9. Sz-NAGu B., Spektraldarstellung linearer Transformationen des Hilbertschen Raumes.

Berlin, 1942.

10. NILSSON, N., Some estimates for eigenfunction expansions and spectral functions correspond- ing to elliptic differential operators. Math. Scand. 9 (1961).

l l . - - Some growth a n d ramification properties of certain integrals on algebraic manifolds.

Arkiv f f r m a t e m a t i k 5, 463 (1964).

12. ODH~'OFF, J., Operators generated b y differential problems w i t h eigenvalue p a r a m e t e r in equation and b o u n d a r y condition. Medd. fr. Lunds univ. mat. sere. 14 (1959).

13. SCHWARTZ, L., Th6orie des distributions, I - I I . Paris, 1950-1951.

540

Tryck~ den 19 januari 1965

17ppsala 1965. Ahnqvist & Wiksells Boktryel~er| AB