Israel Journal of Mathematics, Volume 5, Issue 1 (1967)

ON THE ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS AND RESOLVENT KERNELS OF ELLIPTIC OPERATORS BY SHMUEL AGMON(X) AND YAKAR KANNAI(2)

ABSTRACT Asymptotic formulas with remainder estimates are derived for spectral functions of general elliptic operators. The estimates are based on asymptotic expansion of resolvent kernels in the complex plane. 1. Introduction. Let t2 be an open set in real space R n with generic point x = (xl,..., xn). We denote by Hm(f~), m > 0 an integer, the subclass of functions u e L2(~) with (distribution) derivatives D~u E L2(~) for all I ~ l < m. Here and in the following ~ = (cq, . . . , ~ ) is a multi-index of length -- ~1 + . . + ~n and o • = o?...

D:n,

o~ = - i~-;,,

i = 4

-

1.

We denote by H t°c" (f~) the class of functions defined on tq and belonging locally to H m. In Hm(~ ) we introduce the norm:

(I.1)

][Ullm'fl:

[ffl ,~[~ 0. By the closed graph theorem Tis also bounded when considered as a linear transformation from Lz(O) into Hj(f~), 0 < j =< m. The norm of T when considered as an operator: L 2 ~ Hj will be denoted by:

(2.1)

II TIIj = l] zllJ,. --

sup

II zf[li.-

f ~ Lz(fl) ~

"

We state now one of the principal results of [2] (Theorem 3.1 of [2]) which will also play a basic role in this paper. In this connection recall that an open set f~ is said to possess the cone property if each point x ~ f~ is the vertex of a spherical cone of a fixed height and opening contained in fL THEOREM 2.1. Let T be a bounded linear operator in L~(~), ~ an open set in R" possessing the cone property. Suppose that the range of T and that the range of its adjoint T* are contained in Hm(~) for some m > n. Then T is an integral operator,

Tf f a K(x, y)f(y) dy,

f ~ L2(f~ ),

with a continuous and bounded kernel K(x,y) satisfying

l dimfL Then T is an integral operator of the form (2.3) with continuous Carleman kernel K(x,y). As in the introduction we consider now a self-adjoint operator ,,~ in L2,p(~ ) which is a realization of a p-formally selfadjoint elliptic differential operator A of order m. Let Rz = (A - 2)- 1 be the resolvent of X defined for every complex 2 not in the spectrum of X. Using (1.5), we have: range(R~) = range (R*) = ~

~ H~C'(f~).

Hence, if m > n, it follows from Theorem 2.1 bis. that Rz is an integral operator:

Raf = fn Ra(x, y)f(y) dpy, with a continuous Carleman kernel R~(x,y). We shall refer to Rx(x,y) as the resolvent kernel of .~. Next assume the ,~ is bounded from below but impose no restriction on m. Let {Et} be the spectral resolution of ,~. For any fixed 2 not in spectrum ,,~ and k = 1,2,..., we write: (2.4)

E t = R x k St,x,k

where St,~,~ = [

(s -

# . - - n. In this case both the resolvent kernel Rx(x,y) and the spectral function e(t; x,y) exist. The following relation holds: (2.6)

R~(x,y) =

(t - 2)-1de(t; x,y)

where the Stieltjes integral converges absolutely. Formula (2.6) is (essentially) well known (see [10], [4]). We note that a simple proof of (2.6) can be given with the aid of Theorem 2.1. Without giving complete details we shall sketch the proof. Choose a sequence (ON(t)} of step functions on t >_-0, each vanishing for t _~ tN sufficiently large and satisfying: (2.7)

[¢btc(t)--(t--2)-ll 1 is the largest integer for which (3.1) holds. Let Ra(x, y) be the resolvent kernel of A. Denote by d(2) the distance of ~ from the positive axis (d(,~) = 141 if __ 0). Then Rx(x, x) possesses an asymptotic expansion of the form:

(3.2)

R~(x,x) ~ ( - 2)"/m-1 ~

cj(x)( - 2) -jIm

J=o

valid for 2--, oo in the region: 141 _-> 1, _>__ where 8 is any given positive number, uniformly in x in every compact subset of fL That is, for any integer N >- 1 : N--1

(3.2)'

1( - ;t) 1-"/mRx(x,x) -

Y_, %(x)( - ;O-Jim I < Const. I;~ I-u/m j=O

for [h I >__1, d(Z)>__lzl where the constant in (3.2)' depends on N and e but is independent of x for x in any compact subset of fL In these formulas ( - 2) -j/m stands for the branch of the power which is positive on the negative axis while cj(x) are certain C°° functions on ~ depending only on the differential operator A. In particular:

(3.3)

Co(X) = (2n)-"p(x) -x f~,, [A'(x, ~) + 1]-1 d~.

Before we proceed with the rather long proof of Theorem 3.1 (we shall actually prove a more general result) we shall show how this theorem, when combined with tauberian theorem of Malliavin [13], yields the estimates for the remainder in the asymptotic formula for the spectral function which were mentioned in the introduction. A very simple proof of Malliavin's theorem is due to Pleijel [16-1 who also gave a slight extension of the theorem. It is the following: THEOREM (MALLIA¥IN). Let (r(t) be a non-decreasing function for t >-0 such that S~(1 + t)-~dtr(t) < + oo. Suppose that

8

S. AGMON AND YAKAR KANNAI

[January

.~o (t - 2)- ida(t) - Co( - 2¢ = o(] 2 Ip) as 2--, oo in the complex plane along the curve: IIm2[ = 125 Re2 > 0, where - l < fl < a < O, O < y < l; c o some non-negative constant. Then: (3.4)

tr(t) - sin,(a , ( a + +1)1) c°t~+1 + O(t~+ ~) + O(tp+l)

as t ~ + o o . We shall now prove the following result. THEOREM 3.2. Let .~ be a self-adjoint bounded from below operator in L2.p(f~) which is the realization of a p-formally self-adjoint elliptic differential operator A(x,D) of order m. Let e(t; x,y) be the spectral function of.4. Then:

(3.5)

e(t;x,x)- [P(X)-'(2")-"L,(x,o 1, for any e > 0 . We are now in a position to apply Malliavin's tauberian theorem to a(t) = e(t; x,x) (a non-decreasing function) with a = n / m - 1, fl = ( n - 1 ) / m - 1 and y = 1 - (O(x))/m + e. From (3.4) it follows that (3.7)

e(t; x,x) =.

sin(n, Im) tnlm n,/m Co(X) + O(t("-°(x))/m+~).

By checking the constants in Pleijel's proof of Malliavin's theorem [16] one also finds (since the O estimate in (3.6) is uniform in x in any compact subset of f~)

1967]

ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS

9

that the O estimate in (3.7) is uniform in x in any compact subset of f~. A simple computation (using (3.3)) shows that the coefficient of t*/m in (3.7) is the same as the coefficient of tn/m in (3.5). This proves the theorem for m > n. Suppose now that m < n. Choose an integer k > n/m and consider the spectral function ek(t, x,y) o f X k. Clearly, ek(t; x,y) = e(tl/k; x,y). Moreover, Xkis a selfadjoint realization of .~k, an elliptic differential operator of order k m > n. Hence it follows from the special case of the theorem just proved (noting that the function O(x) is independent of k) that (3.8)

e(tl/k; X , X ) -

d~]t "/~r"

[p(x)-l(270 -" f L

JA'tx,~)k- d(2) we have for s < t=1:

(1 + l e 12)'-" < [~,(1 +a(~))] 2('-s~/"

la(¢)- 21~, =

IA(e)- 212' ~,(1 + A(O) I z('-*):=.

-

~G)-_~ I

l

1

I A(O - 212,-2 1, and combining this estimate with (4.5) we obtain the desired inequality (4.2). Suppose now that mj > n. It follows from (4.4) that the operator F I is an integral (convolution) operator with a continuous and bounded kernel FI (x, y) = Fx(x - y, 0) given by

19671

ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS

(4.6)

Fai(x, y) = (270 - " f ~ ,

effX-y)

11

"

Moreover the kernel (4.6) has continuous bounded derivations up to the order mj - n - 1 on R" x R". In particular for y = x it follows by a straightforward computation that for ] ~] < mj - n - 1: (4.7)

(D,,~Fxl)(x, x) = (Dx'FxJ)(o, O)

/-

= ( _ ~,)(.+l-o/,.-J. (2~)-" :R/, (A(0 + 1)J d~, where ( - 2){"+l'l)m-s is the analytic branch of the power in the complex plane cut along the positive axis which is positive on the negative axis. We consider now an operator Sa of the form: (4.8)

Sa = Bk+ l(X, D) Fai~Bk(x, D) Fx j~-' ... F / ' B I ( x , D)

where the B , ( x , D ) are differential operators of orders l~ > 0 , v = 1 , . . . , k + 1, with coefficients belonging to C , ( R 0 . We set: k+l

l=

~ l,, v=l

k

j=

E j,. v=l

It is well known that a differential operator B, of the above kind defines a bounded linear transformation: H~ ~ H~-l.. for every real s. This follows from the easily established estimate: (4.9) where Cs,~ is a constant depending only on s, Iv, n and on a common bound for the coefficients of B v and their derivatives up to a certain order N = N(s, Iv, n). From the properties of the operators By and F~ j it is thus clear that Sa which is a well defined linear operator: H~o ~H~o is (after completion) a bounded linear operator: H, -* Ht for any s, t such that t < s + m j - I. By an alternate application of(4.9) and Lemma 4.1 to the factors of Sx it is easy to see that the following estimates hold for s - l < t 0 and consider Sx as a bounded linear operator: L2(R ~) ~ L2(R"). It is clear that Sa* the adjoint of Sx in L2(R"), is an operator of the same type: S*

~---

B*F~ J~ " " v.t" ~ J ~JJk *+l

where B* denotes the formal adjoint of B~. We have: TI-mOREM 4.1. Suppose that mj - l > n. Then Sx is an integral operator with a continuous bounded kernel S~(x,y) on R ~ x R ~, possessing continuous bounded derivatives up to the order m j - l - n - 1, satisfying the following estimate:

(4.12)

Is~(x,y)l =< ~Co I~l n and mj > I. Then T~ is an integral operator with a continuous bounded kernel Tz(x,y), possessing continuous bounded x-derivatives up to the order m - n - 1, such that

(4.16)

I T~(x,y)l =< ce'Co 14[('+'):m d(4)J+l for 141 >=1,

14

S. AGMON AND YAKAR KANNAI

[January

where c is the constant in (4.15) and C O is a constant having the same dependence as in (4.12). Proof. Since $4 is also a bounded linear operator: H , ~ H , it follows from (4.14) and the properties of G~ that range (T~)and range (T*)are contained in Hm(R"). Hence, since m > n, it follows from Theorem 2.1 that Tx is an integral operator with a continuous and bounded kernel T~(x, y). Now, using (4.14), (4.10) and (4.15) we have:

(4.17)

llT~llo =< [Is~llo" IlG~llo 1 and that (in the last case):

1,81--I'~1- I~'"l = I~'1 + I~"1- I~'1. To prove the theorem for J ' it will suffice to show that each of the terms (5.18) has at x ° a zero of type (r + 1~ ] - N.~.)p - IJ'l. Now by the induction assumption the coefficient bp.j has at x ° a zero of type (r + I~1 - N~)p- IJI. Hence, since IPl_>l~l+ 1-1~1__>1~1+ 1 - o ( A ) and N j , = N I + o ( A ) - I , IJ'l--IJl+ 1, we find that a term bp,:DP'a~ has at x ° a zero of type:

(r + 1 ~ 1 - N~)p-IJl _>_(r + I=1 + ~ - o(A) - N , ) p - I J I >(,. + l~,l - N , . ) p -

IJ'l .

Similarly, since I~1 = I~1 + I~'1- I~1 >--I~1 + I r ' l - o(A) and Ir'l--- 1, we arD~'ba,: has at x ° a zero of type:

find that the term

(r+ iPI-N,)p- IJ[- IW'I Nz)p-IJI-

= ( r + I~1 + [ ? ' l - o ( A ) -

IT']

= (r + I~1 + 1 - o ( A ) - N , ) p - ([JI + 1) + (p - 1)(I r' [ - 1) _~ (r + I~1 - N ~ , ) p - l J '

I.

These computations show that the theorem holds for J ' as claimed. We shall complete the proof by induction on r. Suppose first that r = 1. By the result just proved in order to establish the theorem for d = (Jl) it suffices to show that the theorem holds for J =(0). This, however, is trivial since [B, A; (0)] -- B and one checks readily that the statement of the theorem in this

19671

ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS

19

case reduces to our assumption on the coefficients of B. Hence the theorem holds or r = 1. Next assume that the theorem holds for some r. We shall show that it holds for r + 1. Using again the result established above it will suffice to prove the theorem for jo = j u (0) where J = (j~, ...,j,). Now, letting B = ]E b~D~, we have:

b~,zoO~ = [B,A; jo] = BIB, A; J]

(5.19)

= ~E ]~ bpDP(b~,~D~). From (5.19) it follows that b~,zo is a linear combination of terms of the type:

b~OP'b~,j

(5.20)

with 181 --- o(B), I~1 = 1. Considering Ao, B and F~ as linear operators: H®-~ H ~ (note that A o - 2 is one-to -one from H ~oonto itself) we apply Theorem 5.1 to (FaB) ~.After completion in L2(R* ) it follows from (5.4) and from (6.2) that l-1

Ga=F ~ + Z

Z

r=l !-1

(6.3)

+ ~, r=l

[B, Ao;J]F~ Isl+'+l

d~S(r,k-1)

r-1

~, (F~B)SF~

~,

[B, Ao; J u (k)]F~lJI +*+,-

JaS(r-s-l,k-1)

s=0

+ (F~B)1G~, where k is an arbitrary positive integer. According to Lemma 5.1 the order of the differential operator [B, A o ; J ] for J ~ S(r) is at most: [ J I (m - 1) + ro(B). If A' has constant coefficients o(B) < m - 1 and o([B, Ao; J]) < (IJI ÷ r)(m- 1). In the general! case: o([B, Ao; J]) < ([Jl + r) (m - 1) + r. Consider the right hand side of (6.3). Clearly Fx is an integral operator with a continuous and bounded kernel F~(x, y ) = F , ( x - y, 0). Using the results of section 4 and our estimate on the order of [B, A o; J] it follows that every term [B, Ao; J]Fx Isl+'+l which appears in the first sum in (6.3) is an integral operator with a continuous and bounded kernel

([8,A; S J t Y l

÷'÷ ~)(x, y).

Set: (6.4)

H t'k, (x, y) = G~(x, y) - F~(x, y) 1-t

-- ~2 r=l

~

([B, ao; J ] F y I+'+1) (x,y).

J~S(r,k-1)

Then H~'l(x,y) is the kernel of the operator given by the sum of the two last members of (6.3).

22

S. A G M O N

AND YAKAR

KANNAI

[January

Our object is to estimate H~J(x, y). To this end consider first the operator given by the one before last membcr of (6.3). It is a sum of operators:

(6.5)

(FaB)~Fx[B, Ao; J U (k)]Fa IJI +k+,- ~

with 1 < r < l - 1, 0 < s < r - 1. J e S ( r - s - 1, k - 1). According to Theorem 4.1 the operator (6.5) is an integral operator with a continuous and bounded kernel such that (since o([B, Ao; J U (k)] < (IJI + k) (m - 1) + (r - s) oW)): (6.6)

I ( ( F a B ) ' G [ K A ° ; J U ( k ) ] f y I+k+'-~)

(x,y) I

z c d~y f ~(g for t21 >- 1. Here and in the following C denotes a generic constant which is independent of 2,x,y and x o (C depends however on k and /). In particular it follows from (6.6) that when o ( B ) < m: (6.7)

I((FxB)'Fx[B, A0; J u (k)]F~I'I+~ +'- ') (x, Y)I

< t i l l ",- (1~11-,-) '+' = - a(~)

d(~)

for d(~) >-I~I'-'/', 121---I. If o(B) = m it follows from (6.6) that if k >= I + (I - 1)/em for some fixed 8 > 0, then:

(6.7)'

I((eaB)'Fa[n, Ao; J u (k)]f~ IJl+k+'-~) (x,y) I _I~1'-"', I~! >1.

121n:"/12l'-l'" ~' d(~) ~, d-~ '1

1967]

ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS

23

When o(B) = m the estimate (6.8) does not give us the information we look for. In this case, however, we shall show that (6.8) can be replaced by a better estimate if x is restricted to a small neighborhood of x ° which depends on 2. To this end apply formula (5.4) to (F~B)l and write (F~B)ZGa in the form:

(FxB)lGa = (6.10)

]E [B, A o; J].FaI'rl+tG~ J ~ sO,a- 1 )

!-1

+ ~, (F~B)'F~

Z

[B, Ao;JU(q)]Fl'q+'+t-S-'G~,

JeS(l-s- l,q-1)

s=0

where q > 1 is an integer to be fixed later on. Consider a typical term in the first sum on the right hand side of (6.10). According to Theorem 4.2 [B,Ao;J]FI~ ~1+lG,~ is an integral operator with a continuous and bounded kernel

([B, Ao; J]r2 ~1+ % ) (x, y). Write:

[B, A o ; J ] =

(6.11)

~, b~,.~(x;x°)D~,

JeS(1),

I~t~_ NJ

where Nj = IJI (m - 1) + l m (note that the b~.z are C~ functions in x and xO). By our definition of B it is clear that the coefficients of its principal part B' vanish at x o. We shall denote by p = p(x °) the largest integer > 1 such that all the coefficients of B' possess a zero of type p at x o. If no such largest integer exists, i.e. if all the coefficients of B' possess a zero of infinite order at x o, we let p = + oo. We set:

O=O(x°)=

(6.12)

P p+l

(2 [2 [1-0/m, [21 _-__1. Next, if Ms < Ns, consider a term in the sum (6.13) with Ict [ > Ms. By Theorem 5.2 the coefficient b~,s (as a function of x) has at x ° a zero of type ( [ ~ t [ - M s ) p . Restrict x to a neighborhood:

(6.15)

Ix-x°l~l;~l

-':,'"

where we set

(6.15)'

p' = p'(x o) = rain{p, Q},

Q being an arbitrary but fixed integer > 1 (independent of xO). Clearly for such x:

(6.16)

I b,,,s I =< cl,~l ('.'-''')''

.

Apply now Theorem 4.2 to the operator D~FxlSl+tG~. It follows from (4.16),(6.16) and (6.12)' that

[ b~,,,,(O"F,JJ'+'e,O(x,y)] -< q,~l("-'~')" d(~.)u, l'~l{"+''~>/" +~+i (6.17)

[./~^.[n/m(.[~[1-Olm)IS[+,

< = Ca(t)

~

d(2)

•

=< C

I,~[n/m ([~,[l-,/m)1 ~(2) ~ '~ •

for d(,~) ~ I.~1'-°/'', I'~1 > 1. Combining (6.13), (6.14) and (6.17) we conclude that

(6.18)

I([B, Ao;J]F~S'+'OD(x,Y)I l + (l - 1)~sin verifies the estimate:

(6.22)

I r,-(l l d(2)°,-),

In~*"(x'y)l < c d-d~"

max{121'-'%

for d(2) > 121'-',-+'}, 121 _>-1, and x satisfying (6.15). Summing up we have proved the following result. THEOREM 6.1. The kernel Ga(x,y) of the resolvent R x = ( ~ - 4) -1 (considered as an operator in L2(Rn)) has an asymptotic representation of the form: I

Gx(x,y)=Fa(x,Y) + ~, (6.23)

r=l

~,

( [ A o - A , Ao;J]FlSl+'+l)(x,y)

J~S(r,k)

\d(2) "

d(2)

)

with A o = A'(x°,D)(x ° a fixed point), F~=(,4 o - 2 ) -I, such that: (i) I f A' has constant coefficients then 0 = 1, k and 1 any integers with k > l - 1 > 0; the 0 estimate holds for 2-~ oo in the region

d(2)_>_121c'-'~:',

121___1,

uniformly in x, y and x °. (ii) I f A' has variable coefficients then 0 is given by (6.12), k and l any positive integers with k/l > 1 + (era) -x for any given e > 0; the 0 estimate holds for 2 ~ oo in the region: d(2)>>_max{12l'-_J > 0 on R n, and then choose a p°-formally self-adjoint uniformly elliptic operator A°(x,D) on R ~ with C , coefficients such that A ° coincides with the given elliptic operator A on fl o. (The proof that the extension A ° of A exists is standard). Let/T °be the unique self-adjoint realization of A ° in L2,po(R~). ~o is semi-bounded from below. Without loss of generality we shall assume that ~o is a positive operator as this may always be achieved by adding a large positive multiple of the identity to both operators A ° and A. Let R°(x, y) be the resolvent kernel of .~o and let G°(x, y) = p°(y)R°(x, y). We define operators S~ and S° from L2(f~o) into L2(£~o) by: Rx(x, y)f(y)dy,

S°f = rue R~(x, y)f(y)dy,

fe L2(flo).W e set:

& - s °. It is easily seen that T~verifies the conditions of Lemma 6.1. Indeed, T~is a bounded linear operator in L2(flo) with range in Hm(f~o). The same is true for its adjoint since T~*= Tx. That the estimate (6.27) holds is obvious from the relation of S~ and S ° to the resolvent operators of ,~ and Xo. For f ~ L2(fto) we have: ( A - ;~) T f f = ( A - ;~) S f f - ( A ° - 2) S ° f = f

p

f -pO

O,

since A = A° and p = pO on f~o. Similarly, ( A - ~)T~f= O. Hence, applying I.emma 6.1 to the kernel of Ta we find that for every £/~ ~ ~ f~o and every integer j ~_ 0, the following estimate holds:

30

S. AGMON AND YAKAR KANNAI

[January '

(6.38)

sup [ G~(x, y) - G°(x, y)[ < C ~ fit xllt

~

d(2) ] '

[ 2[ ~ 1.

By T h e o r e m 6.1 the kernel G°(x,,y) has the asymptotic representation (6.23). C o m b i n i n g this with (6.38) (taking j = l + 1), it follows that the a s y m p t o t i c f o r m u l a (6.23) also holds for the kernel G~(x y). This proves the theorem.

REFERENCES 1. S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathematical Studies, Princeton, N. J., 1965. 2.--, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems, Comm. Pure Appl. Math. 18 (1965), 627-663. 3. V. G. Avakumovie, Ober die Eigenfunktion auf gescMossenen Riemannschen Manningfaltigkeiten, Math. 65 (1956), 327-344. 4. G. Bergenda], Convergence and summability of eigenfunction expansions connected with elliptic differential operators, Med. Lunds Univ. Mat. Sere. 15 (1959), 1-63. 5. F. E. Browder, Le probl~me des vibrations pour un operateur aux ddrivdes partielles selfadjoint et du type elliptique d coefficients variables, C. R. Acad. Sci. Pads. 236 (1953), 2140-2142. 6. - - . , Asymptotic distribution of eigenvalues and eigenfunctions for non-local elliptic boundary value problems I, Amer. J. Math. 87 (1965), 175-195. 7. T. Carleman, Propridt~s asymptotiques des fonctions fondamentales des membranes vibrantes, C.R. du 8 ~m¢ Congr~ des Math. Scand. Stockholm 1934 (Lund 1935) pp. 3A. A.A.. 8. L. G~rding, On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand. 1 (1953), 237-255. 9. - - , Eigenfunction expansions connected with elliptic differential operators, C.R. du 12 ~me Congr~s des Math. Scand. (Lund 1953) pp. 44-55. 10.---, On the asymptotic properties of the spectral function belonging to a selfadjoint semi-bounded extension of an elliptic differential operator, Kungl. Fysiogr. Sallsk. i Lund FOrth. 24 (1954), 1-18. 11. L. HOrmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators. To appear. 12. B. M. Lcvitan, On the asymptotic behavior of the spectral function and the eigenfunction expansion of self-adjoint differential equations of the second order II, Izv. Akad. Nauk SSSR, Ser. Mat. 19 (1955), 33-58. 13. P. Malliavin, Un th~or~me taubdrian avec reste pour la transformde de Stieltjes, C.R. Acad. Sci. 255 (1962), 2351-2352. 14. A. Pleij¢l, Propridtds asymptotiques des fonctions et valeurs propres de certains probldmes de vibrations, Ark. fOr Mat., Astr. och Fys. 27A (1940), 1-100. 15. - - , Asymptotic relations for the eigenfunctions of certain boundary problems of polar type, Amer. J. Math. 70 (1948), 892-907. 16. ~ , On a theorem by P. Malliavin, Israel J. Math. 1 (1963), 166-168. THE HEBREWUNIVERSrrYOF JERUSALEM AND THB ISa~L INSTITUTEFOR BIOLOGICAL~ C H

AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS BY A . J . LAZAR* ABSTRACT

If K is a simplex and X a Banach space then A(K, X) denotes the space of affine continuous functions from K to X with the supremum norm. The extreme points of the dosed unit ball of A(K, X) are characterized, X being supposed to satisfy certain conditions. This characterization is used to investigate the extreme compact operators from a Banach space X to the space

A(K) = A(K, (-- cx),~))).

1. If S is a compact Hausdorff space then it is well known t h a t f ~ C(S)is an extreme point of the closed unit ball if and only if -- i everywhere on S. Our first aim is to extend this characterization to the more general situation of the space A(K, X ) - the space of affine continuous functions on the simplex K having values in the Banach space X with the supremum norm: It is shown that if X is strictly convex or if every three mutually intersecting closed balls of X have a point in common then f cA(K, X) is an extreme point of the closed unit ball if and only if it maps the extreme points of K into the extreme points of the closed unit ball of X (Theorem 3.4). We obtain this characterization through a similar one for the extreme points of the dosed unit ball of A*(K, X) (Theorem 3.2). In Section 2 we discuss the maximal convex subsets of the unit sphere of A(K)= A ( K , ( - ~ , oo)); their simple representation is helpful in the next section. In Section 4 we use our results to investigate the extreme compact operators from a Banach space X into A(K). This section contains also a characterization of the extreme positive operators from a C(S) space (S metrizable compact Hausdorff) to A(K): their representing functions m a p the extreme points of K into the point measures of S (Theorem 4.2). The paper ends with an example which shows that the extreme positive operators from an A(K) space to a C(S) space cannot be characterized in a similar manner. We deal only with Banach spaces over the real field. The dosed unit ball of a Banach space X is denoted by Sx. A Banach space is said to have the n.2. intersection property (n.2.I.P.) if every collection of n mutually intersecting closed

If(s)l

IlSll--sup ,,,[lf(k)ll.

* This note is part of the author's Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank them for their helpful advice and kind encouragement. Received February 28, 1967. 3I

32

A.J. LAZAR

[January

balls in X has a common point. By operators we always mean a bounded linear operator. I f K is a convex subset of a linear space then F c Kis a face of K if it is convex and satisfies: 0 < 2 < 1, kl, k2 e K, 2kx + (1 - 2)k2 e F =~ kt, k2 e F. An extreme point of K is a one point face of it. The set of the extreme points of K is denoted by OK. Let K be a compact convex subset of a locally convex linear topological space. The probability Radon measures on K are ordered by

for every convex continuous function ~b on K (Choquet's ordering). For each k e K there is a measure # on K, maximal in this ordering, which represents k, that is, f(k) = J'Kfd#for any atfine continuous function f on K (cf. [4], [15]). K is a simplex if for each of its points the representing maximal measure is unique. The reader may find all the fundamental facts about simplexes in [4] or [15]. If S is a compact Hausdorff space the probability Radon measures on S form a simplex in C*(S) when this spaceis endowed with its w*-topology. This simplex is of a special type since its extreme points form a dosed set. C(S) is isometrically isomorphic with the space of all the affine continuous functions on this simplex. We shall make no distinction between S and its canonic image in C*(S). Similarly, any simplex K can be imbedded by an affine homeomorphism into A*(K): Tk(f) = f ( k ) for k e K, f e A(K). It is convenient to consider K imbedded in this way into A*(K). A collection of non-negative functions (f~}7=1 c A(K) is called a partition of the unity on the simplex K if ~ = l fi = 1. If M is a set then we denote by 2 u the family of all its subsets. Let X, Y be topological spaces. A map T : X ~ 2 r is said to be lower semi-continuous if for any open set U c Y the set (x e X: T(x) n U 4 ~ } is open in X. Suppose that E and F are linear spaces and K is a convex subset of E. A map T : K - ~ 2 v is called affine if T(k) is a non-void convex subset of F for every k e K and 2T(k~) + (1 - 2) T(k2) c T(2kl + (1 - 2)k2) when 0 < 2 < 1, kt,k 2 eK. The following theorem on multi-valued maps defined on simplexes was proved in [12] and it is stated here for the convenience of the reader. THEOREM 1.1. (cf. [12, Theorem 3.1, Corollary 3.3]). Let E be a Frechet space and K a simplex. Suppose that T : K o 2 E is an affine lower semi-continuous map such that T(k) is closed for every k e K . Then there exists an affine continuous selection for T, that is, an affine continuous

19671 AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS

33

function f : K - - , E with f ( k ) e T ( k ) for each k e K . Moreover, if koeOK and x e T(ko) then f can be chosen such that f(ko) = x. 2. The following theorem was proved by Eilenberg [9] for the space of all continuous functions on a compact Hausdorff space. The proof given below is an adaptation of his proof to the more comprehensive class of the spaces of affine continuous functions on simplexes. THEOREM 2.1. Let K be a simplex and Q a maximal convex subset of {f A(K): IIfII = 1}. Then there exist kodaK and a sign 8 (i.e. 8 = -I- 1) such that

(1)

Q =

=.,

Ilsnl}.

Conversely, every set determined by a point ko e OK and a sign ~ as in (1) is a maximal convex subset of the boundary of Satr). Proof. Let Q be a maximal convex subset of { f e A(K): f e Q we define the following dosed faces of K:

F~ = { k s K : f ( k ) = l}, F ; =

Itf[I =

1I. For every

{ k ~ K : f ( k ) = - l}.

The first assertion of the theorem will be proved if we show that n s ~ ~2Fs"+~ ~ " or n s ~qF2 ¢ j~. Indeed, if one of these sets is non-void, say n s ~eFJ'+ # J~, then, being a dosed face of K, we can find k o e n s ~e(F}" n OK). Clearly (2)

Q = {f s A ( K ) : f ( k o ) =

Ilfll = 1}.

and the maximality of Q implies that (2) holds with equality sign between its members. Let us suppose that n s ~ Q F ] = n s ~QF] = ~ . By the compactness of K there are {f,}7=1, {g~}~ 1 = Q such that

i=1

j=l

Since Q is convex we have ( Z ~ = x f , + Z T = l g j ) / ( m + n ) e Q . Hence II ZT=lf, + Z?=,g~ll = m + n. Obviously this equality contradicts (3) so (1) holds for a certain point ko ~ OK. Now let ko e OK, 8 = __+1 and

Q

=

{feA(K):f(ko)=e'

llfll

Assume that there is a convex subset Q' c { f e A(K): Q' ~ Q. Pick f ~ Q' ~ Q and denote

F ÷ = {ker:f(k)=

=

1}.

Ilfll =

1} such that Q' = Q,

1}, F - = ( k ~ K ; f ( k ) = - 1 } ,

The extreme point ko does not belong to the closed face F = cony(F+ u F - )

34

A.J. LAZAR

[January

since k 0 CF + U F - . Indeed, Ico e F + u F - implies f e Q u ( - Q ) and if this were true then 0 = ½[f+ ( - f ) ] would belong to Q. By a theorem of Edwards [8] there exists f ' e A ( K ) such that f { F = 0 , f ' ( k o ) = e and Clearly IIf+ f ' < 2 therefore ½ ( f + f ' ) ¢ Q ' so Q' cannot be convex. This concludes the proof of the theorem. From a theorem of Lindenstrauss [13, Theorem 4.8] it follows that if Q is a maximal convex subset of {SeA(K): ]If = 1~ then So(r) = conv(Q u ( - Q ) ) . We are going to prove that the closure is superfluous here. Of course, this is wellknown for spaces of continuous functions (see [11]).

11f'll--1.

II

II

THEOREM 2.2.

(feZ(K): Ilfll =

I f K is a simplex and Q is a maximal convex subset of then So(x)= conv(Q u ( - Q ) ) .

Proof. We have to show that Sacr)c conv(Q u ( - Q ) ) . Without loss of generality we may suppose that there exists a k0 e OX such that Q

=

SeA ~. Consequently, dSr. ~ w* - cl(g)'. Let y * ~ S r . . By what we have just proved we can find two nets: {k,},~i ~ ~K, {x*}i~i~ ~Sx. such t h a t {YZ,x*,}i~l converges to y* in the w*-topology of Y*. We may assume that the first net converges to k e K and the second converges to x* ~ Sx. in the w*-topology of X* • Define Yk.x* * e St. by

I1

Yk*~(Y) = x*(y(k)),

y ~ Y.

We have

[ Yk,~*(Y) *

-

* Y~,,~,'(Y)I =0}, ~ ~ ~ l ( s ) . We shall prove that ¢ is a lower semi-continuous map when ./#1(S) and C*(S) are equipped with the w*-topology. We have to show that if # e.//~(S),#' e ¢(#) and U is any neighborhood of #' then there exists a neighborhood V of # such that ¢(v) (~ U # ~ whenever v e V. Let (1)

U= {v'eC*(S): fsf'dv'- fsffl#'] < l'fieC(S)'l 1, ,,: = .[,,:/I1,,'. II t. [2(1 -il ,,; I1>,'.+ 4]/(2- I1,," I1>, IIv: [[ < 1. Clearly v: ~ 0, 2v~ - v: _~ 0 and I[v:[[ = 1, therefore v'.'e ¢'(v~). Since lim~l[v'[] --lim~v'.(1) = #'(S) = 1, the net {C} is w*-converging to #'. We proved that for any #~J~'I(S), any #'~¢b'(#) and any net {v~}c Jt'l(S ) w*-converging to # there are measures v:~ ¢b'(v,) w*-converging to/~' i.e. ~ ' is lower semi-continuous. Let T be an extreme point of -oq'1 and x:K - C*(S) the function representing it given by Lemma 4.1. Obviously x ( K ) c J/I(S). The map ¢ ' o x : K - , 2 ~'(s) fulfills all the conditions of Theorem 1.1. If x(k) does not belong to S for a certain k ~ OK, that is ¢'(x(k)) ~ {z(k)} then there is an affine continuous selection X' of • oX whose value at k is different from x(k). The selection theorem may be used here since ~'~(S) can be imbedded into a Fr~chet space by the separability of C(S) (see, for instance, the proof of Theorem 3.5 in [12]). If T' is the operator from C(S) to ~'1(S) corresponding to X' then T' and 2 T - T' belong to -Oq'l. This is a contradiction since T is an extreme point of .£01 . The proof of (ii) :~ (i) is trivial. We turn to (ii) = (iii). If (ii) holds then T1 = 1. Pick f , g ~ C ( S ) . Obviously T ( f V g)>=Tf, Tg. Let h~A(K), h > T f , Tg. I f k ~ OK we have

r ( f V g) (k) = (Jr V 3) (z(k)) = fO~(k)) V g(x(k)) = T(f) (k) V T(g) (k) >= h(k). By the maximum principle of Bauer [2] this implies T(f V g) > h. (iii) :~ (ii). Let ;(: K -} C*(S) be the function representing the operator T given by Lemma 4.1. If f, g e C(S), k ~ OK then ( f V g) (x(k)) = ( r ( f ) V T(g)) (k) = T(f) (k) V T(g) (k) = f(z(k)) V g(z(k)).

This means that x(k) is a lattice homomorphism of C(S) into ( - oo, oo), which maps the function identically equal to 1 on S to 1. Hence, x(k) ~ S (cf. [7, p. 97]) and this completes the proof of the theorem. Rmt~K. The assumption ofmetrizability of S entered in the proof only through Theorem 1.1. Therefore, the conclusion of Theorem 4.2 is valid also if K is a

1967] AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS

41

metrizable simplex and S is homeomorphic with a w-compact subset of a Banach space (see [51 , [61 and [11). It is likely that the theorem is true without any restrictions on S or on K but we have not succeeded in proving it. The situation is entirely different if we interchange the roles of the spaces A(K) and C(S) in the previous theorem. Let A be the space of the sequences {xn}~= t converging to ½(Xl + x2) with the supremum norm. By [13, p. 78, Theorem 4.71 and [16] there is a simplex K such that A = A(K). For instance, K may be the positive face of the unit ball of Ix = A*. Let T be the identity operator from A to c - - t h e space of converging sequences. Then T is an extreme positive operator but the function from the compactification of the integers Noo to K representing it maps the unique non-isolated point of Noo to a non-extreme point of K . Still, a dense set of No~ is mapped into 0K. We are going to show that for any compact Hausdorff space S there are a simplex K and an extreme positive operator T T : A ( K ) ~ C(S) such that the representing function of T maps s ~ S into OK if and only if s is an isolated point of S. A similar fact was proved in [31 but there the domain was not a space of affine continuous functions on a simplex. EXAMPLE 4.4. Let S be a compact Hausdorff space and S' the set of nonisolated points of S. Denote by e,(s ~ S) the following function on S:

es(t) e,~co (S), e,~ll(S). x * -- (c*(s) • l~(s))~.

Obviously

~0, t#s,

11, The

t=s. dual

of

X = (C(S) ~ c o (S))l~ is

Consider the following subset of X*: m = {(s, +e,): s e S ' } LI {(s,O):seS 1 . M is bounded and w*-closed; thus K = w * - c l ( c o n v M ) is a w*-compact set whose extreme points belong to M . We shall show that K is a simplex but first we identify the extreme point of K . Clearly, if s ~ S' then (s, 0) 6 0K. If s E S - 8' then (es,0)~ X is a w*-continuous linear functional on X*. Its maximal value on K is 1 and it is attained only at (s,0), thus (s,O)~OK. Pick now s ~ S ' . The w*continuous linear functional (0, es) takes its maximal value on M at (s, e,) and its minimal value at (s, - e,). Consequently (s, __.e~) ~ ~K. We proved

OK = {(s,+e,):seS'} u { ( s , O ) : s e S - S ' } . Now we turn to prove that K is a simplex. Let #1,P2 be two probability Radon measures on K maximal in the ordering of Choquet. That is, if # is a positive Radon measure on K and fxdpd~ > Sxdpdp~for every continuous convex function then # =/1~. Assume that j'/t~kd/./1 = flc)~dl.t2 for each affine continuous function ~k. We have to show that #1 = #2.

42

A.J. LAZAR

lJanuary

We begin by showing that /q({(s,0)})= p2({(s:0)})= 0 if s;eS'. It suffices to carry on the proof only for btl. Suppose that this were not true and denote by ~+,e-,e the point measures of (s,e~), (s,-e,) and (s,0), respectively. The measure P

Pl-~+2(

=

5+ +

8-),

where 0~= #l({(s, 0)}) > 0 is non-negative and if ~bis a continuous convex function on K then fK ~b(d#) = fK q~d#t + ~[½(qS(s,e~)+ ~b(s,-e~))-qg(s,0)] > fK dPdpl. Since #1 is maximal we have Pl = #. Thus ~ = 0 and our assertion is proved. By a well-known property of maximal measures lq,lt2 are concentrated on 8 K (cf. [4], [15, p. 30]), i.e., p~(M) = p2(M) = 1. Thus it is enough to prove the equality of their restrictions to M . The set {(s, +__e,): s e S'} contains only isolated points of M; therefore, if E c {(s, +es):s ~ S'} and if a~i = p,({(s, e~)}) b~ =/4({(s, --e,)}), then

I~,(E) = X{a2 :(s,e~)sE} + X{b~ :(s, - e~)eE}, i = 1,2. Define two regular measures on the Borel sets of S by

mi(T ) = #i({(s,O):seT}), T c S, i = 1,2.

(1)

Let f e C(S), f ' e co(S). Since fK(f,f')d#l (2)

f

S

=

fK(f,f')dtl 2 we have

I S) + f rS f dml + ~,~ ~s,a,(J( ()) + ~,s,s,b~(J(s) - f'(s))

~s f dm2 + E ~s,a~(f(s) + f'(s)) + Es ~s,b~(f(s) - f'(s)). We choose J = 0, f ' + e, for s e S ' . From (2) we get (3)

as-

t

b ) = a,2 - bs2

,

seS'

•

Thus, if f e C(S), we have

Is fdm2+

+ bl)f(s)=

fdm2+ ~,~ ~ s,(a. + b~)f(s).

This together with (1) and ml({s}) = m2({s}) = 0, s e S', gives rnl = rn2; a ~1 + b s1= a)+ b2,

seS'.

2 By (3) we infer a~1 = a,,2 b sI = b s, hence lq = #2 and the proof that K is a simplex is completed.

1967] AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS

43

N o w define x : S - , K by X(s) = (s, 0) and consider the o p e r a t o r T: A ( K ) - - , C(S) given by T(g)(s)

= g(x(s))

= g(s,O),

g ~.4(g),

s ~ S.

Clearly T ~_ 0, T1 = 1. We are going to show that T i s an extreme positive operator despite the fact that X(s) is not an extreme point o f K whenever s e S ' . I f T were not an extreme positive o p e r a t o r then there would exist a non-identically null w*-continuous function ~: S--, A * ( K ) such that Z(s) _+ ~(s) e K for e a c h s ~ S. I f s ~ S - S ' then ~, (s) = 0, since X(s) is an extreme point of K. N o w let s ~ S'. Since K is a simplex and X(s) is the middle of the segment joining the extreme points (s, es) ( s , - es) we have X ( s ) + ~ ( s ) = (s,2~e~) where 12~1 < 1. Choose a net {s~} c S ' , s~--,s, s, ~ s. T h e n X(s~) + ~/(s~) --, X(s) + ~(s) and, on the other hand, (s~,2~ e~) -~ (s,0). Hence 2~ = 0 and ~b(s) = 0. We proved that ~ = 0, in other words Tis an extreme positive operator.

REFERENCES

1. D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces (to appear). 2. H. Bauer, Minimalstellen von Funktionen und Extremalpunkte, Archiv der Math. 9 (1958), 389-393. 3. R. M. Blumenthal, J. Lindenstrauss and R. R. Phelps, Extreme operators into C(K), Pacific J. of Math. 15 (1965), 747-756. 4. G. Choquet and P. A. Meyer, Existence et unicitd des representations intdgrales clans les convexes compacts quelconques, Ann. Inst. Fourier (Grenoble). 13 (1963), 139-154. 5. H. H. Corson and J. Lindenstrauss, Continuous selections with nonmetrizable range, Trans. Amer. Math. So¢. 121 (1966), 492-504. 6. H. H. Corson and J. Lindenstrauss, On weakly compact subsets of Banach spaces, Proc. Amer. Math. So¢. 17 (1966), 407-412. 7. M. M. Day, NormedLinear Spaces, Springer, Berlin (1958). 8. D. A. Edwards, Minimum-stable wedges of semi-continuous functions (to appear). 9. S. Eilenberg, Banach space methods in topology, Ann. of Math. 43 (1942), 568-579. 10. A. J. Ellis, Extreme positive operators, Oxford Quarterly J. Math. 15 (1964), 342-344. 11. R. E. Fullerton, Geometrical characterizations of certain function spaces, Proc. Inter. Symposium on Linear Spaces, Jerusalem (1961), pp. 227-236. 12. A. J. Lazar, Spaces of affine continuous functions on simplexes (to appear). 13. J. Lindenstranss, Extensions of compact operators, Memoirs Amer. Math. Soc. 48 (1964), 112 p. 14. R. R. Phelps, Extreme positive operators andhomomorphisms, Trans. Amer. Math. Soc. 108 (1963), 265-274. 15. R. R. Phelps, Lectures on Choquet's Theorem, Van Nostrand, Princeton (1965). 16. Z. Semadeni, Free compact convex sets, Bull. Acad. Polon. Sci. S6r. sci. math. astr. et phys. 13 (1964), 141-146. T ~ HEB~W UNr~REITY OF JERUSALEM

RIEMANN

FUNCTIONS

FOR A SYSTEM OF HYPERBOLIC FORM IN THREE INDEPENDENT VARIABLES BY RICHARD KRAFT AI~TRACT Functions are defined which permit the solution of a special hyperbolic system to be expressed as a quadrature of its initial data over the initial surface.

1. Introduction. In this paper Riemann functions (R.F.) are defined for systems of partial differential equations of the type

L(U) =

(1.1)

(D - A)

U=

0

where U - (U 1, ..., uN), A ~ (%(x)),

D=(DI,...,DIv),

./=3 Dt E

x -= (xl, x2, x3) 0

j = t % axj

and the direction numbers ~ = (%, a~2,~s3) are constant, distinct and oblique to the initial data surface. We assume that initial data is specified on an initial data surface, 0, and for simplicity and without loss of generality chose for 0 the hyperplane Xl = 0. We shall show that the value of U at any point P, not on x I = 0 is a quadrature of its initial data and the R.F. over a subset of the initial hyperplane. Also for purposes of simplicity and visualization the further non-restrictive hypothesis are made that P is in the upper half plane, and that the vectors ~ , i = 1, ..-, N are direction cosines and have positive projections in the xl direction i.e. the vectors ~ point in the general direction of the positive xl axis. A restrictive assumption we make is that no three vector-~t through P are coplanar; this assumption will finally be removed. Riemann functions were defined for systems similar to (1.1) in [1, 2, 3] and the Received August 26, 1966.

44

1967]

RIEMANN FUNCTIONS FOR A LINEAR SYSTEM

45

techniques used here are a synthesis of ideas introduced in those papers. Although this paper is almost completely selfcontained, a familiarity with [1, 2, 3] should be helpful. 2. Orientation and notation. The R.F. for each component of U is a set of vector valued functions. Each member of the set is a solution to the adjoint operator to (1.1), L*, defined in a domain Di. These domains are three dimensional conical subregions in the interior of the backward facing ray cone that is formed by taking the convex hull of the backward (negative xl direction) characteristics (the characteristic Ct is the line in the direction of 4) issuing from P. In order to describe more exactly these conical subregions of the backward ray cone it is convenient to introduce the concept of a wedge. A wedge is simply the planar area between two backward characteristics issuing from P. The wedge formed by the backward characteristics Cp and Cq issuing from P is denoted topa. Sometimes tOnqis used to denote only that part of t%q between P and the initial hyperplane, the exact meaning of tOpgbeing clear from the context. The wedges generated by every pair of backward characteristics issuing from P form the sides of the backward ray cone and divide it into the subcones D ~. Two points lie in the same subcone if the line segment connecting them does not intersect a wedge. Sometimes D~is also used to denote only that part of the subcone between P and X1 = 0 .

For an arbitrary component U K of U we will define in each subcone D~ a solution, W~, of L* = 0 and together these solutions will comprise the set of R.F. for the component U ~' of U. Throughout the paper capital K denotes the index of the arbitrary component of U for which R.F. are being defined. Cauchy data for the R.F. is defined on the wedges that form the boundaries of the domains D t. The motivation for the specification of this Cauchy data is explained in the next section. 3. Specification of the Cauehy data. The value of U ~ at P can be expressed in terms of quadratures of U and auxiliary functions over sections of the wedges and the initial hyperplane. Thus, by employing Green's identity

(3.1)

fcKV(DKUK-aKKUX)ds+fcUr(DKV+axKV)ds=VUI~I~ ~

In equation (3.1), Pk is the point where the characteristic Cx intersects the initial data plane and V is an, as yet, unspecified function. When the Kth equation of(1.1) is substituted into (3.1) and V is chosen to be the solution of

(3.2) and

D~V + a ~ V = 0

Vc~ = 1

46

[January

RICHARD KRAFr

then we have

V

(3.3)

~ arl U~lds = VUx I~K / P~

Green's identity is also applied to the area ~oxj, j # K. Using the coordinate system formed by taking Cx and Cj as coordinate axis we have (3.4) fo~j {ZK(DKUK - aKKU K - arjU 1) + ZJ(DjU1 -ajrU r - ajjUJ)}do~ +

+

,, {U~r(DxZx + arrZ g + ajxZj) + UI(D~ZJ + agiZ 'r + ajTZJ)}do~

~-- CKJ{~

ZKUgdsl+ ~ ZJUJdsg}

where the line integrals are taken around the boundary of o~x~; do~ = Cxflsflsx is an element of area of ogxj; dsj and dsx are elements of arc length along Cx and Cj; and CKj is the sine of the angle between Cx and Cj in the wedge cox~ at P. Furthermore,

(3.5)

~ ZrUgdsJ = fc ZgU'rdsj- f ZgUgds.t .'e',~P'~

the first line integrand on the right hand side being evaluated over Cj and the second integral over P~Pj, which is the line segment between Pk and P~ lying in the initial plane. The signs on the right hand side of (3.5) depend on the orientation of Cx with Cj. Similarly

(3.6)

~ZJUJdsx = fc, ZJUJdsx - [

ZlU~ds,r

Chosing Z g and Z J to be solutions, in coxj, of

(3.7)

DxZg + arrZ r + ajrZ 1 = 0 DjZJ + aKjZx + ajiZ~ = 0

satisfying

(3.8)

Zg= 0

onCj

Z J -- Vag.tC~]

on Cr

and then combining these equations with (3.4) gives

1967]

(39)

RIEMANN FUNCTIONS FOR A SYSTEM OF HYPERBOLIC FORM

47

f OK]

{ line integrals of Z'~ a n d U 's }

= fc~:VariUJdsx +

evaluated along P~P}

When the Kth and jth equations of (1.1) are inserted in the integrands of the left hand side of (3.9) that equation becomes (3.10)

f f ~. (Z%K,+ZJaj,)Utdm~J= fc VarjUJdsK toKj

+

{ line integrals of Z'S a n d U '~ } evaluated along PIMP}

After equation (3.10) is summed for all j but j = K and this sum substituted into (3.3) the result is (3.11)

(ZKaKI Jr ZJajl)UldO)Kj = U~p) - V(p,x) u(Kp,K) jC-K

I~K,j

j~,r{lineintegralsof +

Z '~ and U '' }

evaluated over P~P)

This formula plays a key role in the definition of R.F. and its significance will be seen after Green's identity has been written out for each region D j and these identities collected. Thus,

Dj

Sj

In (3.12) Wj are, as yet, unspecified vector valued functions defined in the domain D J; L ' i s the adjoint operator of L; Sj is the boundary DJ; •

1.-~

.

and N is the outwardly directed normal on Sj. The surface integrals in (3.12) can be further decomposed into a quadrature over the parts of Sj consisting of wedges and the parts lying in the initial plane

Sj

to

If in D j (3.14)

L* (Wj) = 0

Sjn 0

48

RICHARD KRAFT

[January

and Cauehy data for the functions Wj is assigned on the o~" so that

rr

(3.15)

rr

¢~'. J J

J~g J J

t~J,K

then it is seen by combining (3.11)-(3.15) that

[ line integrals of Z'_~ ~ and U " U~e) =V~ek)U~pk)+l~x [evaluated over

P~Pj

(3.16)

}

r surfa.~, integrals ] + I2 ~ of W~" and U" J L evaluated over S j n 0 Hence the value of U(~) is expressed in terms of its Cauchy data on 0; and the functions V, Z " and Wj'' are R.F. Motivated by these considerations, our objective is to assign the Cauchy data for the functions W j so that (3.15) holds. The terms on the left hand side of (3.15) can be separated into two classes, depending on whether the surface integration is over a wedge from the set of wedges, {A}, that form the sides of the backward ray cone or whether the integration is over a wedge from the set {B} of wedges which lie in the interior of the backward ray cone. The terms in the latter set can be paired naturally. Two terms form a pair if they are integrations over opposite sides of the same wedge. These terms arise from applying Green's identity to the two regions D r and D a that lie on opposite sides of a wedge o~pqfrom the set {B}. Thus, (3.17)

~

f f WI, Udog= ~C f f w I , Udco m

~

f{B}

I £0.

where ~rp and ~r~ are the outward normals from the regions D ~ and D ~ respectively on ~,q. Since we have when/~ is a normal to %q that = 0,

l=p,q

O.18) Equation (3.17) can be reduced to

(3.19) '0'"

,~_ ,(,,~

.{B}

l~p,q

z,,,~

1967]

RIEMANN FUNCTIONS FOR A SYSTEM OF HYPERBOLIC FORM

49

The Cauchy data for the set of vectors W j, which are defined in the regions D j, will be chosen in a such way that the right hand sides of (3.19) and (3.15) are equal. A simple way of accomplishing this is to specify the vectors Wj on the co's so that the coefficients of the functions U ~ in the integrands of those two expressions are equal. A comparison of these coefficients yields the analytical prescription for the conditions that the W i must satisfy (for understanding these conditions it is helpful to keep in mind that the subindices on COpsindicate that the wedge is formed by the characteristics Cp and Cs intersecting at P): If Wp is defined in a subcone Dp one of whose sides is ¢Op~ and o~ps~ {A} and also ~Ops~ {C} = {ujcoxj} then for all points on c%q, the function WPmust satisfy (3.20)

W l P-~ o~l • ~V = Z P a l , l -t- Z q a s t

for all I # p, q; but if c%s e {A} and cops¢ {C} then (3.21)

Wp(~t. ~r) = 0

for all I # p, q. Similarly if W p and W ~ are defined in the subcones D p and D~ that are separated by ¢Opqand ¢Opqe {B} r~ {C} then at all points on cops the vectors W p and Ws must satisfy (3.22)

WtP~', • N P + Wts~. ~ = ZPaF, + ZSas,

for all l # p, q; but if O~ps~ {B} and COps~ {C} then (3.23)

w,p , •

p + w,q,.

= 0

for all l # p, q. If (3.20--3.23) are satisfied then (3.15) holds; and if in addition the W~ satisfy (3.14) then we get equation (3.16). A system of integral equations for the functions Wj will be constructed such that the solution of the system satisfies (3.20-3.23) and (3.14).

4. Construction of the system of integral equations. We will obtain the system of integral equations by illustrating how a typical equation is derived. A single equation will be associated with each unknown, Wz~so that the number of unknowns and equations in the system are the same. Since the numbering of the subcones D ~ is arbitrary there is no loss in generality in taking i = 1. To derive the equation associated with W~1 at an arbitrary interior point P1ED I draw through P1 the characteristic Ct. This characteristic intersects successively wedges W1, ".-, W~-I from the set {B} terminating at a point P, on a wedge W~that forms part of the boundary of the backward ray cone. Let the points of intersection of Ct with these wedges be ordered according to their distance from P1 and denote them by P2, "", Ps. Similarly, in passing from P~ to P~ the characteristic C~ passes successively through the subcones D1, ... D ~- 1. if pro and Pn (where Pn is in the positive direction

50

RICHARD KRAFt

[January

from Pro) are any two points inside or on the boundary of a subcone D' then by integrating the system (3.15) there results +

'+"

P"

+.)=+:+ L

In the case of interest to us Pm and P, will belong to {PI, "",P+} and will hence be on opposite sides of the boundary of a subcone D"~ {DI, ..., DS}.The integral equation for ~lis obtained by combining (3.20-3.23) and (4.1). Thus putting Pm= P1, P,, = P2 in (4.1) it becomes

Since P2 ~ o)1 - %1,~1, we get by using (3.22) (for the sake of argument it is assumed 0)1 e {C}) in (4.2) that (4.3)

+ Z"a,,,)

W](P1) = WzZ(Pz)+ ( ~ " / q " ) - l ( Z " a , :

+ fpP2 (~ a,jW))ds where in deducing (4.3) the relation (4.

;+1)-1 = _ 1

and (3.18) have also been used. By employing (4.1) with r = 2 and P= = P2, P, = Pa in the right hand side of (4.3) it becomes (4.4)

Wt'(P1) = Wt2(P3)+(~t"

+ N~l)-l(Z'lapll+Z"ae,l)[ +

]EallW] ) ds

P2

P1

+/Pi'

(~

atjW]) ds

Since P3 e 092 the procedure in deriving (4.4) from (4.2) can be repeated and so onj The final equation for Wzl(P1) is (4.5)

Wtl(Pa)

=

(~t" J~l)-l(ZPlarll +

+ ]E m=l

Zqta

~ I + "'" + qtl)Ie2

:+' ( ~a,jW7 ) ds alP,,,

1=1

The system of equations consisting of equations like (4.5) for all Wj+, (i -- 1,...), l = 1, ...,N can be solved by the method of successive approximations in the standard manner. Since the way in which this system of integral equations was derived from (3.20-3.23) and (3.14) is reversible the solution of the system will satisfy those conditions; and as has been demonstrated this implies the functions W~are R.F. for (1.1).

1967]

R I E M A N N F U N C T I O N S F O R A SYSTEM O F HYPERBOLIC F O R M

51

The preceeding derivation of Equation (4.5) contains a source of ambiguity that was glossed over. The tacit assumption was made that D ~ was separated from D ~+~ along C~ by a unique wedge coi+1 whereas examples are readily constructed where the point at which Ct crosses between D i and D ~+~ lies on the line of intersection of two or more wedges. We will show that there is no real ambiguity in this circumstance by extending the prescription for the Cauchy data so as to include these cases. When C~ crosses between two subcones at a point P* on the line of intersection of two or more wedges then perturb the point PI so that C~ crosses between D ~and D i+ ~ at a point P** not on the line of intersection of two wedges. At the perturbed point P** the Cauchy data is unambiguously assigned by the conditions (3.20-3.23). The extension of our prescription is completed by defining the Cauchy data at P* to be limit of the Cauchy data for P** as P** approaches P*. This limit is independent of the way P** approaches P*.

In the previous discussion it was assumed that no three characteristics through P lie in the same plane. This assumption was employed explicitly in (3.18) and implicitly in asserting that the wedges divided the backward ray cone into 3dimensioned subcones. We will sketch how this restriction can be removed. A different proceedure must be adopted only when defining the R.F. for a component U r of U for which CK is coplanar with more than one other characteristic through P. We consider the representative case of a component U r of U for which the characteristics C~,..., CK through P are coplanar while none of the other characteristics CK+I, "", C~ through P lie in that plane. In order to successfully carry through the method of section 4 for defining the integral equation (4.5) in this case it is necessary to replace equation (3.11) with a more suitable expression for the value of U~p). This is because (3.18) is false in the present situation; and hence if the method of section 4 were carried through then equation (4.5) would contain a division by zero. This difficulty can be circumvented by substituting in place of (3.11) the new expression for U~p,)

(4.6)

U~I)

K--1 /line integrals of the functions / = V(P;)Ux(P'~) + j=IZ [Zit and U evaluated along PjPj+ J +

Z

Z

/=I

i=K+l

UJdto l

~Of*i + I

which is derrived using the methods of [1, 2]. In equation (4.6) the function V is an auxiliary function defined in a manner similar to the definition of the given in (3.2) while the functions Z u are constructed in a manner analogous to the construction of the W vs given in section 4. Equation (4.6) expresses the U ~ as a linear functional of the Cauchy data of the

52

RICHARD KRAFT

functions U 1, ..., U K evaluated along P~P~and surface integrals of U K+I, .-. U ~ evaluated over the wedges ~o~.~+1. These latter quadratures can be reexpressed as quadratures of the functions U over 0. The method of proceedure being entirely analogous to that used in going from (3.11) to (4.5) except (4.6) replaces (3.11). The function W i in this case are defined, of coarse, only in the non-degenerate 3-dimensional subcones. This procedure is feasible when beginning with (4.6) instead of (3.11) because Cauchy data does not need to be assigned for any component W~J of Wj on a wedge for which (~'l " N) = 0. REFERENCES 1. R. Kraft, Riemann functions for linear hyperbolic systems in two independent varibles, Dept. of Mechs. of Soils and Materials, Report No. 8; Negev Institute, Beersheva, Israel, September 1965. 2. R. Kraft, .4 direct successive approximation proceedure for calculating Riemann functions,

Dept. of Mechanics of Soils and Materials, Report No. 9, Negev Institute, Beersheva, Israel, September 1965. 3. R. Kraft, Riemann functiions for a linear system of hyperbolic form, This journal, Vol. 5, No. 1, p. 53. NATIONALBUREAUOF STANDARDS, W~N~TON D. C.

RIEMANN FUNCTIONS FOR A LINEAR SYSTEM OF HYPERBOLIC FORM BY

RICHARD K R A F T ABSTRACT

Functions arc defined which permit the solution of a special hyperbolic system to be expressed as a quadratur of its initial data over the initial surface.

1. Introduction. In this paper Riemann functions (R.F.) are defined for systems of partial differential equations of the type

(1.1)

~u)-T-£x U - AU=O

where U -- (U t, ..., uN); A = (afj), 1 < f,j g(~/). Thus by (3) (13)

)2" 1 which satisfies

Z f(p) - - < o--o 1 P

we leave the details to the reader [6]. On the other hand our Theorem does not seem to imply Theorem 4 of [7]. 2. In one of their papers Chowla and Vijayaraghavan [3] state that to every > 0 there is an A so that if as < "" < a~ < x is a sequence of integers satisfying k ~=I

1

- _->A, (at, a j) = 1 at

then the number of integers n < x not divisible by any a is < 8x. This result indeed easily follows by Brun's method i8]. The number of integers n < x, n ~ 0(rood ai), 1 < i < k is by Brun's method [8] less than cle-ax(cl is an absolute constant independent of al, "",ak). The following question seems to be of some interest: Let al < "" be of any sequence of integers satisfying ~ t l ~at < A. Denote by f ( a s , ' " ; x ) the number of integers not exceeding x not divisible by any at. Put

F(A; x) = minf(al,... ; x)

19671

SOME REMARKS ON NUMBER THEORY II.

61

where the minimum is to be taken over all sequences satisfying ~ 1 ~as < A. How large is F(A; x) and which sequence al < ... gives the minimum? Let p, be the largest prime < x and p, > p,_ ~ > . . . the sequence of primes < x. Define i by ~2

1

j=l

PJ

A o (exp z = e ~)

F(A; x) < f ( P i , ' " , P,; x) < x exp( - e'i) . I do not see how to prove (15) and in fact I cannot even show that for some fixed > 0 (e independent of A and x)

F(A, x) > ef(p, ... p,; x), in fact I have no satisfactory lower bound for F(A; x). 3. We prove by Brun's method [8] the following Theorem 2. To every Ca there is a c2 = c2(cl) so that if a 1 < ... < ak cln is any sequence of integers then

1 E1 "-d > c2 log n where in 531 the summation is extended over all the integers d which are divisors of some a~. Let 8 = ~(Cl) be sufficiently small and write

f~(m) =p ~l l m f , p < n" where fl Im means P'I m, f + l fro, dl < ' " < d, be the integers f~(a,), i = 1,-.., k. To prove our Theorem it will clearly suffice to show (16)

t=1 ~'~ > c2 l o g n .

We need two lemmas.

Lemc~x 1. Let e < el/8. Then for n > no the number S of integers m ~_ n f o r which fs(m)> n 112 is less than cln/2.

62

P. ERDOS

[January

We evidently have by the well known result

~

logp = log x + 0(1)

p<x

P

n

nSlZ < l-I f,(m) < I I p,l,+,l,,+... p2-~a l°gn

which proves (16) and hence Theorem 2. I have no reasonable estimate for cz as a function of cx. 4. Straus asked me the following question: What is the maximum number of integers al < "" < ak $ x no two of which are relatively prime but every three of them are relatively prime? The question is perhaps a bit artificial but it seems to me of some interest that a simple and fairly precise answer can be given. Put max k = f ( x ) , then 1 (18)

f(x) =

) log x + o(1) loglog x

1967]

SOME REMARKS ON NUMBER THEORY. II.

63

To prove (18) observe that i f a l < "" < ak < X satisfies for every 1 < it < i2 I~ as _> |=1

II

l xo(e) ai < qt'~) < x. (a~, ai) is the prime u~,j = uj.~ and ( a , a~, a,) is clearly always one. Let r be fixed and x large. Denote by f,(x) the largest value of k for which there is a sequence at < "" < ak < X SO that no r of them are relatively prime, but every r + 1 o f them are relatively prime. In (18) we showed

f 2(x) = ( l + o(1)) logx/log log x. By the same method we can prove

.)1,.-1 :

ogx

64

1,. ERDOS REFERENCE

1. R. Bellman and H. N. Shapiro, The distribution of squarefree integers in small intervals, Duke Journal 21 (1954), 629-637. 2. N. G. de Bruijn, On the number of positive integers >= x and free of prime factors >=y, Nederl. Akad. Wetensch. Proe. Ser. A. A. 54 (1951), 50-60. 3. S. Chowla and T. Vijayaragharan, On the largest prime divisors of numbers, J. Indian Math. Soc. 11 (1947), 31-37. 4. H. Davenport and P. Erd6s, On sequences of positive integers, J. Indian Math. Soc. 15 (1951), 19-24. 5. P. ErdSs, Some remarks on number theory, Israel J. of Math. 3 (1965), 6--12. 6. P. Erd6s, Some remarks about additive and multiplicative functions, Bull. Amer. Math. Soc., 52 (1946), 527-537, see in particular Theorem 8 p. 533. 7. P. Erd6s, Asymptotische Untersuchungen uber die Anzahl der peiler yon n, Math. Annaten, 169 (1967), 230--238. 8. See e.g.H. Halbca'stam and K. F. Roth, Sequences, Oxford and Clarendon Press, 1966.