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 ,~[~<=m (F~)IDOeu[2dx] 1/2
where dx is the Lebesgue measure and the binomial coefficients
are introduced for convenience. Under this norm I-Im(~) is a Hilbert space. Let p(x) be a C ~ positive function on f~. We denote by dox the measure p(x)dx and by ( , )o the scalar product: (1.2)
(f,g)p = f
J~
f(x)g(x)dax
(1) The research of the first author reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18, through the European Office of Aerospace Research (OAR) United States Air Force. (2) This paper is to be part:of the second author's Ph.D. thesis written under the direction of the first author at the Hebrew University of Jerusalem. Received March 1, 1967.
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S. AGMON AND YAKAR KANNAI
[January
We denote by L2,p(f~) the Hilbert space which is the completion of C~°(fl) (class of infinitely differentiable functions with compact support in fl) under the norm (f,f)~/2. Let A = A(x,D) =
(1.3)
]E a,(x)D ~ I~l_~m
be a linear differential operator of order m with C ~ coefficients in f~. We denote its principal part by A' = A'(x, D). We assume that A is a positive elliptic operator and that it is p-formally self-adjoint. That is we assume that
A'(x, ¢) =
E a~(x)4" > O, 4" = 4]'"" 4,7~, [~1=m for all real 4 = ( ¢ ~ , . . . , 4 , ) ~ 0 and xEf~, and that (Au,v)p=(u,Av)p for all u, v e C~°(t~).
We denote by ~ a self-adjoint realization of A in Lz,p(f~). That is, X is a selfadjoint operator in the Hilbert space Lz.p(f~) with domain of definition ~ 2 such that any u e ~,~ is a solution in the distribution sense of the differential equation:
A(x, D)u = Xu
(1.4)
By well known regularity results for weak solutions of elliptic equations (e.g. [1]) it follows from (1.4) that ~,~ = Htm°c'(f~). More generally, since ,,Ik is a realization of Ak: (1.5)
~
IOC. = Hkm (tq) for k = 1,2, ....
Assume that the self-adjoint realization X is bounded from below and let {Et} be its spectral resolution (normalized by left continuity):
It is (essentially) well known that Et is an integral operation:
Etf = f a e(t; x,y)f(y)dpy,
feL2.,(fl),
with a C ~ Carleman type kernel e(t; x,y) called the spectral function of (see G~trding [9] and section 2 of this paper). In particular e(t; x,x) is a real non-negative non-decreasing function of t. The problem:of the asymptotic behavior of spectral functions was first investigated by Carleman [7] for a class of second order elliptic operators. In more recent years the problem was studied by many authors for more general operators (e.g. [14; 15] [8; 9; 10] [5,6], [12], [4], [2]). In particular G~rding [8] proved that in the general situation discussed above, when p = 1:
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ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
3
e(t; x,x) -- c(x)t n/m= o(tnl=) as t ~ + oo, (1.6)
c(x) = (2r0 -n f
d~.
J A'(x,~)
GriMing has also shown that when A has constant coefficients then the remainder term o(t "/") in (1.6) can be replaced by the term: O(t (n- 1)i,,). For elliptic operators with variable coefficients a remainder estimate in the asymptotic formula (1.6) is known in the literature only for a class of second order operators. Avakumovic [-3] proved that for the Laplace-Beltrami operator on a compact Riemannian manifold the remainder term o(t nlz) in (1.6) can be replaced by O(t (~- 1)/2) (this result was proved explicitly only for n = 3). Recently S. Agmon proved (unpublished) that for elliptic operators of any order the following estimate for the remainder term in (1.6) holds: (1.7)
e(t; x,x) - c(x)t n/m
=
O(t(n-O)lm),
t -'* + o0,
where 0 is any number < ½in the general case and 0 any number < ~ if the principal part A' has constant coefficients. The main purpose of this paper is to improve further the last mentioned remainder estimate. We shall prove that (1.7) holds with any 0 < ½ in the general case and any 0 < 1 if A' has constant coefficients (actually we shall prove a somewhat more general result, see Theorem 3.2). In this connection we mention that a short time after the derivation of our results we were informed by L. H6rmander that he has also obtained very recently the same remainder estimates for the spectral function. His method, however, seems to be different from the method employed by us. About our method we shall say here only that it uses a procedure of estimating kernels introduced in [2] to derive a fine asymptotic expansion theorem for resolvent kernels of elliptic operators. The proof of this expansion theorem (Theorem 3.1 and the more general Theorem 6.2) takes up most of this paper. Once this result is established the remainder estimates for spectral functions follow easily with the aid of a tauberian theorem due to Malliavin 1-13]. In another paper S. Agmon will use a modified approach to derive various extensions of the results of this paper. In particular, similar remainder estimates will be proved for the asymptotic distribution formula of eigenvalues. Remainder estimates will also be proved for spectral matrix functions corresponding to a self-adjoint realization of an elliptic system of differential operators. The selfadjoint realization will not be assumed to be semi-bounded. In conclusion we wish to thank L. H6rmander for informing us about his results and for acquainting us with his recent, as yet unpublished, work 1-11] on spectral functions.
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S. AGMON AND YAKAR KANNAI
[January
2. Certain integral operators. We shall have to deal with bounded linear operators T in Lz(f~) such that range of T is contained in Hs(f~) for some m > 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
<2.2)
Ig<x,y>l <-_o<11 rll. +
HT* II.)"/mllr Jig-'/"
where to is a constant depending only on m,n and on the dimensions of the cone in the cone property of ~. Using the last theorem one can easily prove the existence of a continuous kernel in the more general situation when T is a bounded linear operator in L2,p(f~) such that the range of T and the range of T* are contained in H~C(f~), m > n. To see this let {f~j}, j = 1,2, ..., be a sequence of open bounded sets possessing the cone property, f i ~ c f ~ , f ~ j c ~ j + 1 and u j f ~ j = ~ Let Jj: L2,o(~)~L2,o(f~i) be the restriction operator restricting f e L 2 , p ( ~ ) t o ~j. Its adjoint J j*.: Lz,p(f~i) ~ Lz,p(f~) is an extension operator extending f e Lz,p(f~J) as zero in f~ - f~j. Let: Tj = JjTJ*. It is clear that T/can be considered as a bounded linear operator in L2(f~j) and that as such it verifies all the conditions of Theorem 2.1.Applying the theorem it follows that Tj is an integral operator with a continuous kernel Kj(x, y) on f~j. x f~j. It is easy to see that K~(x, y) = Kj(x, y) on f~i x £~ for any i < j . Hence the kernel K(x,y) defined by p(y)K(x,y)= Kj(x,y) in f~j x f~j for j = 1, 2,..., is a well defined kernel on f~ x f~ such that (2.3)
Tf= f
K(x,y)f(y)day
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ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
5
for all f e L2,p(f0 with compact support in f~. Finally (2.3) actually holds for all f e L2.p(f~) since K is a Carleman kernel with respect to the measure dpy (indeed by Sobolev's inclusion relations: f ~ (TJ)(x) is a bounded linear functional on L2,p(f~) for each fixed x which implies that K(x, ")~L2,p(f~)). Thus we have proved: THEOREI~ 2.1. bis. Let T be a bounded linear operator in L2,p(f])such that range T and range T* are contained in H2C'(f~) for some m > 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 -
# . - - <3O
2) d s
is a bounded operator. As before, using (1.5): (2.5)
lOC. range (R~k) = -@~,,c Hk, . (f~).
From (2.4) and (2.5) it follows that
range( ,) j=l
Hence by Theorem 2.1 bis. Et is an integral operator with a continuous (actually C °°) and bounded kernel e(t; x, y). This proves the existence of the spectral function of 2
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S. AGMON AND YAKAR KANNAI
[January
Suppose that X is positive and that m > 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
for t ~ 0, C a suitable constant independent of N. Put: (2.8)
T,, = f :
[(t -- 2)- 1 _ ON(t)]dEt.
Clearly TN is an integral operator with a kernel (2.9)
KN(x, y) = Ra(x, y) - f :
~ ( t ) de(t; x, y).
Choose any rio c c ri, rio possessing the cone property, and let Jo : L2,p(ri) ~ L2,p(rio) be the restriction operator from f~ to D 0. Its adjoint J~': L2,p(rio)~ L2,p(ri) is an extension operator. Set: TO= JoTNJ *. Then T° which is a bounded operator in L2,p(rio) can also be considered as a bounded operator in L2(rio). Considered as such it satisfies the conditions of Theorem 2.1. Some simple computations (using (2.8) and (2.7)) show that (2.10)
II T° Iio,°o = O(N-1), II : I1-,Oo= O(1), II(z°)* Ilm,flo
=
O(1) as N ~ oo.
Applying the estimate (2.2) to the kernel p(y)KN(x, y) of T ° (on rio x rio) it follows from (2.10) that (2.11)
KN(x,y) = O(N -l+":m) = o(1) as N ~ 0%
uniformly on rio x rio. From (2.11), (2.9) and (2.7) the representation formula (2.6) follows easily. In particular the absolute convergence of (2.6) for x = y follows in this way by taking 2 = - 1 and choosing ON > 0, using the fact that e(t;x,x) is a non-decreasing function of t. The absolute convergence of (2.6) for x ¢ y follows from that for x = y using the following easily established estimate for the total variation of e(t; x, y) on any finite interval a < t < b (see [4]):
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ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
7
var e(t; x,y) < [var e(t; x,x)" var e(t; y,y)] 1/2. 3. The main theorems. The key result of our paper is the following asymptotic expansion theorem for resolvent kernels of elliptic operators. THEORE~Ct3.1. Let ,~ be a positive self-adjoint operator in L2,p(f~) which is the realization of a p-formally self-adjoint (positive) elliptic differential operator: A(x,D)= ~,l~l~_ma~(x)D~ of order m > n = d i m ~. For each x~f~ define O(x) as follows: O(x)= 1 if (3.1)
~
[a,(y)-a,(x)[ =
O(ly-xl')
as y ~ x
lal =m
for all integers p. Otherwise O(x) = p/(p + 1) where p > 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, = [ I m 2[ if Re 2 > 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
as t --* + oo for any ~ > O, uniformly in x in any compact subset of f~, where O(x) is the function defined in Theorem 3.1 (1/2 __
n. Let R~(x, y) be the resolvent kernel of A. By the representation formula (2.6) we have:
Ra(x,x) = f o
(t - 2)-1de(t; x,x).
Applying Theorem 3.1, using only the first term in the asymptotic expansion (3.2), we have (3.6)
R~(x, x) - Co(X) ( - 2)~/m- a = O( 12 ]("- 1)/m- a)
as ) . ~ along the curve IIm21=12[ Re 4__> 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
J
= O(t(,-"(x)~l ~,.+~ ) uniformly in x in any compact subset of f~. Replacing t I/kby t in (3.8) we obtain (3.5) and complete the proof. The remainder of the paper is devoted to the proof of a general asymptotic formula for resolvent kernels containing Theorem 3.1 as a special case. 4. Preliminary results on fundamental solutions and related kernels. In this section we consider integral operators acting of functions on R". We denote by Hoo = H~(R") the class of functions u ~C~(R ") such that D'u ~L2(R") for Ict] >__0. By C,(R") we denote the class of functions u ~ C~(R ") such that u and all its derivatives are bounded on R n. Let s be a real number. The s-norm of u e Hoo is defined in the usual way: (4.1)
Ilull = f.. (1 + 1 12) 1 ( )12d¢
Here and in the following /~(~) stands for the Fourier transform:
a(¢) =(27r) -":~ f a , u(x)e
Let A(D) be a positive elliptic differential operator with constant coefficients of order m and with no lower order terms. It is well known that A(D) has a unique self-adjoint realization in L2(R") which we shall denote by A. The operator .~ is positive and its domain of definition is H,,(R"). Let F~ = ( . 4 - 2 ) - 1 be the resolvent of A defined for any complex 2 not contained in the non-negative axis and denote by F~ its jth power (j > 1). As before we denote by d(2) the distance of 2 from the positive axis. LV_MMA 4.1. The operator F, Sdefines a one to one linear map: H®---,H~. For any two real numbers s, t with s <- t <- s + j m the following inequality holds: (4.2)
IIF TII,z
I
IIf[l
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s. AGMON AND YAKAR KANNAI
[January
for f ~ H ~ and 14[ > 1 where T is the ellipticity constant:
(4.3)
~ = sup
(1 +
~,~
I¢1') *2
1 +A(O
For t = s the constant in (4.2) can be replaced by 1.
Proof.
By Fourier transformation:
.~(e)
(4.4)
(/~3) ( 0 = (A(0 - 4),"
which implies that Fat yields a one to-one-map: H~ -~ Hoo. From (4.4) it follows further that
I:(¢)1 ~ ,. IIr/fll, ~ -- f.. la(23-_-~leP + lelb'de
(4.5)
(1 +
=L . I/(e)12o + lel~)' IA(e)-lel.),-~ 41.,
de <= c?ll fll~
where C ~ = sup
~
(1 + l e 12)'-" la(e)- 21,,
Clearly C~ = d(2) - j for t = s. Using the estimate ]A(e) - 41 >- d(2) we have for s < t<= s + j m ,
141>=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<,-,): ,n
2+11'('-"/'d(2)-2s+~'-"/"
< "e'('-')/" 1 + A ( O - ~
-<__(3~12 I)'"-~:md(2) -'' Hence
c~ __< (3r)s[21 c'-'~:" d(2) -j for 121 > 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 <_ s + mj - l: (4.10)
[Is fll,
where T is the eUipticity constant (4.3) and C is a constant depending only on j, l, m, n and on the B, (here and in the following when we say that a constant depends on {B,} we mean that it depends on a common bound for the coefficients of {B,} and their derivatives up to a certain order N = N ( j , l, m, n)). Indeed it suffices to verify (4.10) for the extreme values t = s - l and t = s - 1 + mj; the
12
s. AGMON AND YAKAR KANNAI
[January
result for an intermediate t will then follow from the well known interpolation inequality:
IIu 11,<--(IE u 11.),,2-,,,(,2-,,,(11 u t[.) `,-,,,,`,~-,,,,
t,
Now, the estimate (4.10) for t = s - l + mj follows by considering each factor F j * of Sx as a bounded linear operator: H, o H,+mjv with norm estimated in Lemma 4.1:
(4.11)
I[ ~ :ll,÷-,v =< (3~) ;v \ d(~) ]
IIG"
Thus, if C denotes a generic constant depending only on the B~ and on j, l, m, n we have by (4.9) and (4.11): [[nk+lF]~n~...FXJ l Blf[[,_t+,,j < C I I F ~ k B k . . . F2J ' n I J,:l [ s - - l + l k + l + m j Jk
< C(3?)ik
=<
...
.
I]Bk "'" FJ ~Blf]l,-(t-tk+,)+,tj-j~ ,
=< c(3~)'llf[l,.
Similarly the estimate (4.10) for t = s - I follows considering this time each factor F~~ of S~ as a bounded linear operator: H, ~ H, with norm d(;0 -jr. From now on we assume that mj - l > 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<"+''" d(2)l
'
Ixl >= 1,
where y is the ellipticity constant and Co is a constant depending only on the B, and on j , l , m , n .
Proof. Set m ' = m j - I. By the preceding remarks S~ is a bounded linear operator in L2(R") having its range in Hr,,(R"). Similarly, range (S*) ¢ H,,,(R"). Hence, since m' > n it follows from Theorem 2.1 that S~ is an integral operator with a continuous and bounded kernel. Moreover, using (4.10) we obtain for the zero and m' norms of S~ and S* (def. (2.1)) the estimates:
1967]
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
13
ils,[10 <(37)J cl41 (141~ J, d ( - -'/" ~ , IIs~ II,. <(37) j C ~d(2)] (4.13)
IIs: 11., =< (3~)~ c \d(4)I' [1413 '
141>1,
with a constant C depending only on the By and on j, l, m, n. Applying now the inequality (2.2) to the kernel S~(x,y) of S~, using (4.13), we find:
Is~(x,y)l <= Const.(l[s~llm,+ I[sa I[°,)°"'IIs~[I~ -n'm' < r~Co 141(~+')/" d(4)J
'
which is the estimate (4.12). Finally, to show that Sz(x,y) is continuously differentiable up to the order mj - l - n - 1, consider the operator: S~'a = D~SzDa for any multi-indices a, fl with ]~ I + l a I__<mj - l - n - 1. One checks readily that S~'a is a bounded linear operator in L2(R") which satisfies the conditions of Theorem 2.1. Hence, S~ 'a is an integral operator with a continuous bounded kernel S]'a(x, y). Now from the definition of S~'a it follows that in the distribution sense:
S~'P(x,y) = ( - 1)1#1D:OPrSz(x,y), which together with the continuity of S~'P(x,y) imply the existence of the derivatives in the classical sense. This completes the proof. In addition to Sx we consider an operator valued function T~ of the form: (4.14)
Tz = SzGz
where for each complex 4 not on the non-negative axis G~ is a bounded linear operator in L2(R") such that range (G~) and range (G*) are contained in H,,(R") and such that: C
(4.15)
for 141 ~ 1, c some constant. THEOREM 4.2. Suppose that m > 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<(3w)'c I~1" = d(,~))
c
"d(~)
= c(3~)JC d(2)J+~ "
(4.17)'
11z*ll. <-- I1o~'ll.. IIs:' Iio __
= c(3~)JC d(,~)J+l Also, since for every f ~ L2(R"):
lira
II ~fll. = IIS~G~JI[. <=(3~¢ c (using (4.10) for t = s = m) we have: (4.17)"
ilz~l[ =<(3r)~c ~-~11 t~1"" G~II.__
l+t/m
Combining now (4.17), (4.17)' and (4.17)" with the inequality (2.2) applied to the kernel Tx(x, y) we arrive at the estimate (4.16). Finally the proof of the differentiability of the kernel Tx(x, y) is very much the same as the proof of the differentiability of the kernel Sx(x,y) given above. We omit the details. 5. Some properties of commutators. In this section we shall prove some results for multiple commutators of operators which wiLl be needed later on. Although the case of interest to us is that of differential operators we shall start by considering a more general situation. Let M be a linear space over a field K and let A, B: M ~ M be linear operators. Denote by S(r, t) the set of r-vectors J = (Ja, "",J,) with integral components 0 <j~ = t, i = 1, ...,r. (The elements of S(r,t) are multi-indices in R"; to avoid confusion we use here Latin and not Greek letters). Set [J] = Jl + " " + J,, S(r) = I,.J~°=1S(r, t) and denote by J u (Jr+ 1) (J ~ S(r)) the vector ( j , . . . , j , , j,+ 1) ~ S ( r + 1). Define a zero dimensional vector (belonging to S(o, t)) to be the empty vector. For the empty vector J -- 121 set: IjI = 0 and JZ U(jl) =(Jl)"
1967]
ASYMPTOTIC B E H A V I O R O F SPECTRAL F U N C T I O N S
15
We shall now define inductively multiple commutators [B, A; Y], Y non-empty, in the following way:
(5.1)
[B, A; (0)] = B
(5.2)
[B,A; (j + 1)] = [B,A; (j)]A - A[B,A; (j)]
(5.3)
[B,A; J k)(J,+ 0] = [BIB, A; J], A; (J,+l)]
(Note that [B,A;(1)] = B A - A B is the usual commutator of B and A). Let 2 e K be such that A - 2 = A - M is one-one and onto, and set F~ = (A - 2)-1.
Let r and k be positive integers. Then
THEOREM 5.1.
(F~B)" =
x
[B,A;J] e~,l+,
JeS(r,k-1)
(5.4)
r-1
+ ~, (F~B)SFa s=O
~,
[B,A; J U (k)]F Ixsl+k+'-'-t
J eS(r-s- l,k-1)
Proof. We shall proceed inductively in several steps. Consider first the case r = k = 1. Formula (5.4) reduces then to
(5.5)
FaB = BFa + Fa(BA - AB)Fx
which is immediately verified by applying A - 2 on both sides of (5.5) from the left. Suppose that (5.4) has been established already for r = 1 and some k, i.e., suppose that k-I
FxB = )2 [B,A;(j)]Fzj+~ + F~[B,A;(k)]F~
(5.6)
j=0
is true. Using (5.5) with [B,A; (k)] replacing B we find that (5.7)
Fain, A; (k)] = [B,A; (k)]e~ + Fx([B,A; (k)]A - A[B,A; (k)])Va = [B,A; (k)]Fa + F~[B,A; (k + 1)]Fx .
Inserting (5.7) in (5.6) we see that k-I
F~B= Z [B,A; (j)]F/+I+ [B,A; (k)]F~+ t+ F~[B,A; (k + I)]F~ +' j=O k
Z
[B,A; (j)]F/+'+F~[B,A; (k +
O]F? +I
j=0
Thus (5.4) is proved for r = 1. Assume now that the theorem has been proved for some r. Then
16
S. AGMON A N D YAKAR KANNAI
(FxB)"+' = FxB(FaB)"..~ FaB (5.8)
Z
[January
[B,A; J]Fa [#l+'
' +s(,,k- 1) r--I
+ X (FxB)S+IFa
Z
[ B , A ; J U ( k ) ] F Y I+k+'-'-l"
Je S(r-s-l,k-1)
s=O
According to (5.6) with B replaced by B[B,A; J] we may write the first sum in (5.8) in the form
X
[B,A; s]F m+,
d ¢ S ( r , k - 1) k-1
= ~,
]E
[B[B,A;J],A;(,j)'JFxl Jl+'+l+l
1=0 deS(r,k-1)
+
~, d ~
F~[B[B,A; J],A; (k)]Fy I+'+k"
S i r , k - 1)
According to (5.3) this is equal to
]E
[B,A; J]Fa Isl+'+l
J e S(r+ 1 , k - 1)
+
~Z
Fa[B,A; S k) (k)]Fa I'q +'+~
leS(r,k-1)
Inserting this in (5.8) we get
(FaB)"+1 =
[B,A; J ] F y I+'+1
~2 JeS(r+ I,k-I}
(5.9)
+
)2
Va[B,A;SW(k)]fx I'l+~+"
J e $ ( r , / - 1) r--1
+ )2 (FxB)~+IFx s=O
=
X
[B,A;SU(k)]Va l$l+k+'-s-I
/ e S(r-- s-- 1,k-- 1)
Z
[B,A; J]Fa lsl+'+l
S~S(r+l,k-1)
+ (E (FaB)~Fa
X
z=O
[B,A;JU(k)]F~. I'q+k+'-~
g es(r-s,k-1)
But (5.9) is (5.4) with r + 1 instead of r, so that the theorem is proved. Assume now that there exists a subring ~ of the ring of linear transformations from M to M and a function o from ~ to the real line so that the following conditions hold: (5.10) o(0) = -- oo
o(AB) < o(A) + o(B)
(5.11) (5.12)
(5.13)
o(AB
BA) < o(A) + o(B) - 1
-
o(0
o(A) is called the order of A.
= o
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNC"TIONS
19671 Lr~U_A 5.1. (5.14) Proof. (5.15)
17
Let J ~ S(r), r > O. Then o[B,A; J] < [dlo(A) + to(B) - [ J I
•
If r = 1 then (5.14) reads:
o([B,,4;(j)])
< j o(A) + o(B) - j
.
When j = 0, (5.15) follows from (5.1). Assume that (5.15) has been proved for a certain j . Using (5.2), (5.12), and the induction hypothesis, we find that
o([B,A;(j + 1)]) = o([B,A;(j)]A - A[B,A;(j)]) < o([B,A;(j)])+o(A) - 1 < j o(A) + o(B) - j + o(A) - 1 = (j + 1 ) o ( a ) + o ( e ) - ( j + 1)
which proves the theorem in the case r = 1. Assuming that the theorem has been proved for a certain r, we get from (5.3), (5.11) and (5.15) that
o([B,A ; J k3 (j,+ t)]) = o([B,A ; J],A ; (j,+ I)] ) < jr+ xo(A) + o(B[B,A ; J]) - J,+ l < jr+~o(A) + o(e) + o ( [ B , a ; J ] ) - J r + t <= L+loCa) + o(B) +
ISlo(h) + roCB)- Is[--Jr+l
= I J k) (j,+ a)[ o(a) + (r + 1) o(B) - I J L) (Jr+ 2) 1 and thus the theorem is proved for every r. LEMMA 5.1 will be applied in the sequel to the case where M is the linear space Hoo(R") and ~ is the ring of differential operators with C , coefficients. In this case we denote by o(A) the usual order of the differential operator A. We conclude this section by establishing for commutators of differential operators a result which we shall need later on. In this connection let us agree to say that a C ~° function u(x) has a zero of type p at a point x o (p > 0 an integer) if u and all its derivatives up to the order p - 1 vanish at x 0 . TrmOREM 5.2. Let [B,A; J] be a multiple commutator of two differential operators B, A (J =J(1,'",Jr)). Suppose that the coefficients of the principal part of B possess a zero of type p > IJl/r at some point x ° ~ R ". Put N j = ISl(o(A) - 1) + ro(B) (so that o([B,A,;J]) < Nz by (5.14)) and write: (5.16)
[B,A; d] =
Then each coefficient b,,j for N j - r + ( r + l ~ l - N s ) p - I J} at x °.
]E b,,s(x)D ~. I~I<-NJ
ISl/p <
<=N j has a zero of type
Proof. For the purpose of the proof it is convenient to agree that for any integer q < 0 the statement: "u(x) has at a point x o a zero of type q" is a true
18
S. AGMON AND YAKAR KANNAI
[January
statement which holds in the emptly sense. With this convention we have to prove that each coefficient b~,j has a zero of type (r + I~1- N,)p -IJI at x °. We shall prove this using a double induction on J. We shall first show that if the theorem is true for J = (j~,--.,j,) then it is also true for J ' = (Jl, " " , L - t, J, + 1). Indeed, let A = ~a,,D ~. By definition, using ( 5 . 1 ) - (5.3):
~.,
b~,.t.(x)n'=[B,h;J'] = [ B , A ; J ] A - A [ B , A ; J ]
(5.17)
=
E ~ [bp.fl~, a,O';(1)]. I#I~NJ [rl-
From (5.17) it is clear that the coefficient b~,:, is a linear combination of terms of the form:
bp..rDP'ar
(5.18)
or
arDr'bp,.t
with IBI + 1~1-1 ___I~1 where fl' is a multi-index such that f l = fl' + fl" and = f l " + V, fl" some complementary multi-index, and similarly V' is a multiindex such that ? = ?' + ?", ~ = fl + ?" for some V". Note that these restrictions imply that l~'l----1, If'l=> 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 = I ~ I + 18'1 - o(B) + 1, we find that DB'br,z has at x ° a zero of type:
(r + I~1- N , ) p - I J I -
18'1-- (r + I~1 + 18'1- 0(8) + 1 - N , ) p - l J [ - 1 8 ' [
= ( r + 1 + I = l - NjO)p_ Ijol + (p _ 1)18'1->--(r + 1 + I ~ l - N,O)p-IJol, (using also that N,o = Nj + o(B) and ]do ] = ] j ]). Next suppose that 181 -- m. In this case bp has a zero of type p at x °. Using this, the induction assumption and the estimate: lel~l~l + 18'1-oW), it follows that baDP'b~,j has at x ° a zero of type: p + (r + [ Y l - N j ) p - [ J [ -
18'[
(1 + r + I~I + Is'I - o W ) - N j ) p - IJI -18' ~_ (r + 1 + I~1- N,o)p-Isol. The above computations show that b, zo has at x o a zero of type
(r + 1 + 1o~I - N,o)p --IJOl, which is the desired result for r + 1. This completes the proof. 6. The asymptotic expansion of resolvent kernels. We shall first discuss a class of operators on R". Let A(x,D) be a positive elliptic differential operator on R n, p-formally self-adjoint and of order m > n. We assume that the coefficients of A are in C,(R'), that p ~ C,(R') and that p(x) > ~ > o, ~ some constant. We also assume that A is uniformly elliptic: A'(x, ¢) > C I ~ I "for x and ~ in R', C a positive constant. Considering A as a symmetric operator in the Hilbert space L2,p(R") with domain C~°(R~) we denote its closure by .~. It is well known: that X is a self-adjoint operator with domain of definition H,,(R~). Moreover, ~ is the unique
20
S. AGMON AND YAKAR KANNAI
[January
self-adjoint realization of A in L2.p(Rn). All these facts follow easily from the a-priori estimate: IIu Ilm< Const. ([I An 110+ l[ u Ho) which holds for u ~ Hm(R n) and from the regularity theory of weak solutions elliptic equations (e.g. [1]). In addition it follows from G~rding's inequality that Xis bounded from below. In the following we shall assume without loss of generality that ,t is positive. Consider now the resolvent operator Ra = (.4 - 2)-1. From our previous discussion it follows that Rx is an integral operator in L2,p(R*)with continuous and bounded kernel Rz(x,y). Since L2.,(R") and L2(R*) are the same function spaces on which two equivalent Hilbert norms are defined, we may consider ,,~ and Rx as operators in Lz(R,,). We shall denote by Gx the resolvent operator Rx when considered as an operator in L2(R~). It is an integral operator with a kernel: Gx(x, y) = Rx(x, y)p(y). The operator Gx: L2 ~ L2 can also be considered as an operator: L2 ~ Hm. We have the following norm estimates: (6.1)
141
IlGxllo <= cd(~)-l'
IlG;tllm
IIGx* lira < C ~
for I~1 => 1,
c a constant, where as before d(,~) denotes the distance of 2 from the positive axis. Indeed the first inequality is immediate since X is a positive operator in Lz,p(R"). To derive the second inequality write Gx in the form: Gx = G-1 Ux where Ua = ( , ~ + 1)((.~-2)-1. Clearly U~. is a bounded operator in L2 whose norm when considered as an operator: L2,p-~ L2.o does not exceed
sup - oo<,< ~o
It ÷ 11 I t - A[ --- 1 +
141 + 1
[ 2 [ for d(A---'~ <-- 3 d ( 2 )
>
1.
=
Hence:
II II- 11G-1 II-I1 IIo I d@ The last estimate in (6.1) follows of course from the second noting that G~* = pG~p- 1 We proceed now to derive the asymptotic expansion of G~(x,y). To this end we fix an arbitrary point x ° in f~ and set: Ao(D)=A'(x°,D), B(x, D) = A'(x o, D) - A(x, D). As in section 4 we denote by X o the unique selfadjoint realization of Ao in L2(Rn) and by Fa the resolvent of Ao, Fa = (¢]o - 2)- 1. Let f eL2(R n) and set u = Gff. We have: ( A o - $)u = ( A - 2)u + Bu = f +
Bu,
1967]
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
21
so that u = F~I + F~Bu,
or equivalently
Hence (6.2)
G~ = Fa + F~BGa = F:~+ F~BFx + (F~B)2G~ . . . . . 1-1
= Y., (FaB)'F~ +
(F~B)zG~
r=0
for every integer l > 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 _<_ c -
I~1"" [ l ~ l l - " ~ ' d(~)'\ ~ !
for a(,~)> 1.~1' - ' + ' , 1,~1> 1. Consider now the last member of (6.3). Since G~ verifies (6.1) it follows from Theorem 4.2 that (FxB)'Gx is an integral operator with a continuous and bounded kernel such that
(6.8)
I((FaB)'G~)(x,y)l < c - I~1"". ~
[l~l°'B)l') d-~ ',
x
lal>l. -
Suppose that o ( B ) < m. From (6.8). (6.7), (6.3) and (6.4) we obtain for H~ J the following estimate when k ~ l - 1: (6.9) for
k"
n~ (x,y) < c
d(~)=>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<0<
=
1)
'
and
Mj=M~(xO)=min{N.r,N~-l+
IJ[}p,
J~S(l).
By Theorem 5.2 the coefficients b,,,r in (6.11) vanish at x = x ° for I~1 > M j . Observe that
1
p (
Mj < ~ Nj + Nj-I (6.12)' - p + 1 p--+--I = (IJl+l)(m-1)+l
-
~ P p+l
IJI)p
+ l
1 -t- ~
IJl-(IJl
+ 1) (m-O).
We proceed now to estimate the kernel: (6.13)
([B, Ao; J]Fxl'q+tGx) (x,y) =
b,,~(x, xO) (D'try t +z a~) (x, y)
with J e S ( l , q - 1). Consider first a term in the last sum with Is] =< M j ~ ( I J I + l) ( m - 0)
24
$. AGMON AND YAKAR KANNAI
[January
(by (6.12)'). Applying Theorem 4.2 to the operator D~FxlJI+kG z it follows from (4.16) that
,
< cl'~l", m I'~1 '~""
Ib~s(D'FyI+tGz)(x'Y)] =
d(2) " d(;t) ul+t
(6.14)
=
d(2)
d(2)
=
d(2)
d(2)
for d(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
for d(2)>_-I~l '-°'m, z => 1 and x satisfying (6.15). Apply now Theorem 4.2 to the operator: (6.19)
(F~B)'F~[B, Ao; S L) (q)]F;t Ist +q+'- ' - 1G,t,
with J e S(l - s - 1, q - 1), s < l - 1. Since o ( [ B , A o ; J O (q)] _-_(I J[ + q) (m - 1) + m ( l - s),
it follows from Theorem 4.2 that (6.19) is an integral operator with a continuous and bounded kernel satisfying:
1967]
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
25
I((F~B)'Fa[B, Ao; J U (q)]Fa IJI +q+'-s- 1G,) (x, y) [ (6.20)
121":" [1211-':') I~l+~+'
]2l>1.=
For any given 6 > o we choose now q in (6.20) as the smallest integer q > 1~me. With this choice we have: (6.20)'
[((F~B)'FxrB'A°;J U(q)]FzlJl+~+t-S-lG~)(x'Y)l
< c 121°" •-'' -[121'-"m~ ' =
d(~)
~, d(2)
l
fo, d(2)___121'-':-+', 121_->1. Combining now (6.10), (6.18) and (6.20)' we find that (6.21)
12l./m (121,-0, I((&B)'G*) (x'Y)l < C d--d(-~\ d-~
for d(2)_ max{121'-°', 121'-"'+'}, 121->-1, and
)'
x verifying (6.15). Finally,
from (6.3), (6.4), (6.7)' and (6.21) we find that when o(B)= m the kernel H~ '~ for k > 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'-<°/m', 1211+.-.,-,1, 121 _->1, for any y
26
S. AGMON AND YAKAR KANNAI
[January
but x restricted to the neighborhood (6.15) of x o. Under these restrictions the 0 estimate is uniform in x,y and x o. The asymptotic representation formula of Ga(x,y) takes a particularly simple form on the diagonal of R" x R". As a matter of fact in this case (6.23) can be replaced by an asymptotic series expansion in powers of ( - 2)-1/~. To see this take in (6.23) x = y = x °. From (4.7) it follows that (6.24)
( - 2)1-"/:([Ao - A, Ao; tifF IJl+r+ l ) (Xo, Xo)
is a polynomial in ( - 2) -1/m with coefficients which are C , functions in x °. It is easy to check (using Theorem 5.2) that the constant term in this polynomial is zero. Also, F~(x°,x °) = go(x °) ( - 2)-l+t"/m) with (6.25)
go(X) = (2r0-"fR" [A'(x, 4) + 1]- td~.
These observations and Theorem 6.1 show that on the diagonal the kernel Ga has the asymptotic series expansion: (6.26)
G~(x°,x °) ,'0 (
-
2) "/m-1 ~, gj(x °) (
-
2) -i/m
j=O
with coefficients gj which are C , functions in x o (gj for j > 0 is the sum of the coefficients of ( - 2) -j/m in the polynomials (6.24) taken over all J eS(r,k), r = 1,..-, l where l is chosen large enough so that (l + 1)0 > mj). The asymptotic expansion (6.26) holds in the usual sense for ;t ~ oo in the region:
d(a)_> lal
,
141____1,
if A' has constant coefficients, and for ~ ~ oo in the region
nxl
141 _-_ 1,
in the general case. Here ~ is an arbitrary fixed positive number. The asymptotic expansion in the regions mentioned is uniform in x % R " . Recall that the standard resolvent kernel of X is the kernel
Rz(x, Y) = p(y)-lGz(x, y) (the kernel with respect to the measure dpy = p(y)dy). The asymptotic expansion which we have derived for R~(x, x) (via (6.26)) is precisely the asymptotic expansion (3.2) o f Theorem 3.1. Thus we have proved Theorem 3.1 for the class of operators X in L2,p(R"). We now extend Theorem 6.1 to the case of a self-adjoint realization of an elliptic differential operator on f~ for any open set f~ c R". Tm~OREM 6.2. Let X be a positive self-adjoint operator in L2,o(~) which is the realization of a p-formally self-adjoint (positive) elliptic operator A(x,D)
1967]
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
27
of order m > n (p and the coefficients of A belong to C®(f~)). Let Rx(x, y) be the resolvent kernel of.4. Then the conclusion of Theorem 6.1 holds for the kernel Gx(x,y) = p(y)Ra(x,y) with the modification that the statements on the uniform dependence of the 0 estimate in (6.23) hold for x, y and x ° restricted to any compact subset of fL As above Theorem 6.2 yields for Gx(x °, x °) the asymptotic series expansion (6.26) valid for 2--* oo in the region: d(2)> Ixl ~-,,-,÷', I~1 1, if a ' has constant coefficients, and in the region: d(2)~121 ~-(°/m'+', I~1----1, in the general case (the expansion being uniform in x o in every compact subset of f~). Now, the existence of such an expansion is precisely the statement of Theorem 3.1. Thus we see that Theorem 6.2 implies Theorem 3.1 as a special case. For the proof of Theorem 6.2 we shall need the following: =
L ~ 6.1. For every complex 2 which is not on the non-negative axis, let Tx be a bounded linear operator in L2(f~) such that its range and the range of its adjoint T'are contained in H~C'(f~), m > n. By Theorem 2.1 bis. Ta is an integral operator with a continuous kernel Ka(x,y). Suppose that (6.27)
[IT II
c
d(X)'
c a
constant.
Suppose moreover that there exist posilive elliptic differential operators A(x,D) and Ax(x,D) (C ° coefficients) of order m such that (6.28)
(A(x,D) - 2)Taf= 0 and (Ax(x,D) - 2)T*f = 0
for all f ~ L2(f~), Then for every integer j >=0 and every f~o c c f~ (i.e. f~o open, ~o compact and ~o c f~), the following estimate holds:
where C is a constant independent of 2. Proof. It will suffice to prove (6.29) for f~o with a smooth boundary. We first prove that for t) o c c fl and j = 0,1, 2,... :
IIT,Sllo..o <-(6.30)
c
)'11Silo.,
llT~Sll...o~ - c I~l (1~1'-"')' llSlto,o for allfE Lz(fl). Here and in the following C, C1,.-., denote constants independent of ~ orS. Indeed, write u = Tar and note that by assumption: Au = 2u. Using well known a-priori estimates for solutions of elliptic equations we have:
28
(6.31)
S. AGMON AND YAKAR KANNAI
[January
IIu II.,.o-~ c,(ll Au IIo,. + IIu Iio°) i
.
I
= cx(t21 + x)Ilu tlo.° s c~
]t/Ho,.
for [2[ ~ 1, by (6.27). (6.31) together with (6.27) yield (6.30) f o r j -- 0. We continue by induction. Suppose that (6.30) was proved for j we shall prove it for j + 1, To this end observe that our assumption that A is a positively elliptic operator implies that A can be written in the form: A = A o + B where B is an operator of order < m - 1 and A ° is a formally self-adjoint operator such that (6.32)
II° Iio,, ~ 42)-'11 ( A° - 2)0 llo,°
for all v e Hm(fl) with compact support in ta. Now, given tao choose tax with a smooth boundary such that tao'= = tax = c ta. Pick (E C~(tal) such that (--- 1 on tao. Write as before u = T~f and apply (6.32) to v = (u. We have:
(6.33)
Ilullo,°o-llcullo,.-- d(2)-'ll (AO- 2)(¢u)11o.~ z d(2)-'ll(a- 2) (¢u) Iio,. + C2d(2)-ltlull.-,,., ~_ c¢(2)-'11ulI.-,,.,,
since (A - 2)u = 0. We now apply the well known interpolation inequality: (6.34)
IIu II.-x.°, <=r ll u o,., u'' II m,.,'-"",
y a constant. Using our induction assumption, it follows from (6.30) (applied to tal) and (6.34) that
(6.35)
/12tt-1/m)J +1
IIu II.-x,°,-<-- ~c~, ~-~
Ilfllo,o.
Combining (6.33) and (6.35) we obtain the first inequality (6.30) for j + 1. To derive the second inequality we use again the interior a-priori estmates: (6.36)
IIu II-.~ =< c3(11 An Iio., + II" I1o°,) = c_.(121 + 1)11.11o,., =<2ql2111 r~fllo,.,,
121_->x.
Combining (6.36) with the first inequality (6.30) with j replaced by j + 1 (which we have just established for all tao c c ta) we obtain the second inequality (6.30) for j + 1. This completes the proof of (6.30). Let, now, ./o:L2(t'l)~ L2(f~o) be the restriction operator from ta to tao (tao as above with a smooth boundary). Its adjoint ,/* : L2(tao)~ L2(ta) is an extension operator. We define: Tx,o = JoTxJ *. It is clear that Ta,o is a bounded operator in L2(tao) which verifies the conditions of Theorem 2.1 (its kernel is Kx(x, y) restricted to f~o x tao). From (6.30) it follows that
1967]
ASYMPTOTIC BEHAVIOR OF SPECTRAL FUNCTIONS
(6.37)
il r~,ollo,no --< d(---)"
-
29
"
]"
-
Since T* is an operator with the same properties as Tx, the estimates (6.30) also hold for Tx*.Hence: (6.37)'
1[T~*oHm,ao -< C d~2D"
~)
']'
Applying now the inequality (2.2) to the kernel of Tz.o, using the norm estimates (6.37) and (6.37)' we obtain (6.29). This establishes the lemma. We conclude with the
Proof of Theorem 6.2. Let f~o c c f/. Choose a real function p°(x) ~ C~°(Rn) such that p°(x) = p(x) on t)o, p°(x) =>_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
=
llsll
=
Let f e S,~x). I f f e Q u ( - Q ) there is nothing to prove so we may assume that f ¢ Q u ( - Q ) . Define the following affine continuous functions on K: fx(k)
=
2f(k) - 1 +f(ko) 1 +f(ko) , f2(k) =
2f(k) + 1 - f ( k o ) k e K. 1 +f(ko) '
It is easy to check that hi = f l V ( - 1 ) < f 2 A 1 =hz and h2(k:o)---1. From Edwards' separation theorem [8] we infer that there exists a g~ e A(K) such that h 1 < gl < h2 and gl(ko) = 1. If
gz(k)
=
2 f ( k ) - (1
+ f(ko))g,(k ), keK,
1 -f(go) then it is dear that g2 e - Q and f = ½[(1 +f(ko))g 1 + (1 -f(ko))gz]. Since - 1 < f ( k o ) < 1 we proved that f e c o n v ( Q u ( - Q ) ) and this concludes the proof of the theorem. 3. We now pass to the space A(K, X) and its dual. The following lemma is an easy consequence of Lemma 2.4 of [12]. LEXeaA 3.1. Let K be a simplex and X a Banach space. The following subset of A(K, X) is norm dense in A(K, X):
{~,},"=,
=x,
{~,}ro~ ~A(r),
~, = X, ¢, ~_0. i=l
1967] AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS
35
TI-mOREM 3.2. Let K be a simplex, X a Banach space, k ~ OK and x* ~ OSx,. The functional y~,x,eA*(K,X) defined by (1)
y*,**(y)=x*(y(k)),
yeA(K,X)
is an extreme point of the closed unit ball of A * ( K , X ) . Conversely, to every extreme point of this ball there correspond a k ~ 8K and an x*~ OSx. related to it by (1). Proof. Denote Y = A ( K , X ) . Clearly if k E ~K and x*~ OSx. then the functional Yk,,* * given by (1) belongs to S t . . Suppose that there are y*, y*~ S t . such that
(2)
* = ½(Yt + Y*), Yk,**
Y* ~
*
Yk,x* '
From the preceding lemma we infer the existence of a partition of unity on K , {~b~}~=1 c A(K) and the existence of points {xi}~=l c X for which
YT
q ixi e Yk,x* * i=1
dPiXi i
"
1
Then there is an index i, 1 < i < n, such that y*(dpixi) v~ yk*,x.(~bixi). By Theorem 2.1 and Theorem 2.2 there is a ~ ~ A(K) which satisfies:
--II ll
= 1, y*(ffx,) ~ yZx,(~kxi).
Define two functionals x*, x*e Sx° by
x*(x) = y*(~kx), x*(x) = y*(~kx),
x ~ X.
From (2) it follows that ½(x'; + x * ) ( x ) =
= x*(4,(k)x) = x * ( x ) ,
that is, x* = ½(x~ + x2*). Hence x* = x* and in particular
y*(~kxi) = x*(x,) = x*(x,) = y*,x.(~xi). We obtained a contradiction and by this the first part of the theorem is proved. Now we Pass to show that any extreme point of Sy. can be represented as in (1). Let ~ = { y * ~ ° : k ~ O K , x * e O S x . } . First we prove that 0Sy° is included in the weak* closure of g . To see this it suffices to show that S t . is the weak* closure of convg (cf. [7, p. 80]). Let Y0*eSy, and suppose that Yo q~w* - cl (cony g). Then, by the separation theorem for compact convex sets there exist a y ~ Y and a real number a such that y~(y) > ct and y*(y) < a for every y* ~ w* - cl(conv or). In particular
x*(y(k)) = Yk.x,(Y) * < ct, k ~ K , x* e ~Sx..
36
A.J. LAZAR
[January
By the Krein-Milman theorem and Bauer's maximum principle [-2] it follows that Y H< a in contradiction with y*(y) > ~. 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 <= [x*(y(k)) - xT(y(k)) I +
+ I x'/(Y(k)) - xT(y(k,))l ---- I x*(Y(k)) - x;'(Y(k))l +
IIy
y
It is easily seen from the above inequality that Y* = Yk.x*.* Clearly y* e a S r . implies that k ~ ~K, x* e aSx.. This concludes the proof of the theorem. Now we turn to the space A(K,X) itself. The following theorem generalizes a result of Lindenstrauss [13, p. 43]. TrmOREM 3.3. Let X be a Banach space having the n.2.I.P. (n > 3) and K a simplex. Then A(K, X) has the n.2.I.P. Proof. According to [13, Lemma 4.2] it is enough to show that for any finite set {Yi}~=1 = A(K, X) and any e > 0 there exists a subspace Z c A(K, X) having the n.2.I.P, such that the distance between yt (1 < i < n) and Z is not greater than e. From 112, Lemma 2.4] we infer the existence of a partition of the unity on K, A(K), = 1, 1 < j < m and the existence of a set {xu:l
I1 ,11
(1)
y,(k) = ~ ~bj(k)xij]l<e, 1=1
fl
k e K , l<_i<_n.
It is easily seen that the subspace Z c A(K, X),
Z = { ~ , j x j : { x j } 7 = l c X }, 1=1
is isometrically isomorphic with (X (9 X ~ . . . ~ X)t~o. Hence Z has the n.ZI.P. (el. [13, Theorem 4.6]). By (1) we know that the distance of y~ from Z is at most 8 and this establishes the theorem. TrmoP~M 3.4• Let K be a simplex and X a Banach space. Assume that a) X has the n.2.LP. (n >=3); or
b) X is strictly convex.
1967]
A F F I N E FUNCTIONS ON SIMPLEXES A N D E X T R E M E OPERATORS
37
Then a function y • Y = A(K,X) is an extreme point of Sr if and only if y(k) • OSx for every k • OK. Proof. One implication is trivial. We prove only that the condition is necessary. a) Let y • OSr. Since Y has the n.2.I.P, then, according to [13, Theorem 4.7], we have l y*(y)l = 1 for y * • OSr.. Therefore, by Theorem 3.2, if k • OK and x* • OSx° then I x*(y(k)) [ = 1. Hence, if k • OK, and y(k) = ½(xt + x2), xl, x2 • Sx then for every x* • 0Sx. we have I x*(xl + 12) 1 = 21x*(Y) l = 2. It follows that x*(xl) = x*(x2) for each x*~OSx, and this together with the Krein-Milman theorem implies that xl = x2. b) We define the following map from Sx to 2s~
Z(x) = { x ' • S x : l l E x - x ' [ I =<_ 1} , x•Sx. It is obvious that x • T(x), T(x) is closed and Tis an affine map. We shall prove that it is also lower semi-continuous. We have to show that for any x • Sx, any X 0o x ' • T ( x ) and any sequence { n}n=l converging to x there are x ~ • T ( x , ) , n = 1,2,... such that tIXVl°o nsn=t converges to x'. If Ii x ll = 1 the above assertion is clear since in this case T(x)= {x}. Let IIx II < 1. We choose a sequence of numbers 2n • [0,1], 2n --* 1, such that
IIx + ~Xx'- x)II ---- 1-II x- x, 11. IIx- ~Xx'- x)II ---
1-II x-
x, II.
It is easy to check that x, + 2,(x' - x ) • T(x,) and 11x ' - ix, ÷ X,(x' - x)1 II - . 0 . This proves that T is lower semi-continuous. Let us consider the map T o y : K ~ 2 x where y•OSr. If for a certain k e O K we have y(k)60Sx, that is T o y ( k ) # {y(k)} then, according to Theorem 1.1, there is an affine continuous selection of To y, y' say, for which y ' ( k ) # y(k). Since
Y = y' + ( 22y - - y ' ) , y , • S r , 2 y - y ' • S r ,
y' # y,
we obtained the desired contradiction. REMARK. The conclusion of the previous theorem will no longer hold if the space X does not satisfy certain conditions like those imposed above. In [3] is given an example of a four-dimensional Banach space X such that not all the extreme points of the closed unit ball of C([0,1], X) admit the representation expressed by Theorem 3.4. 4. The following lemma, stated also in [12, Lemma 4.11, gives a representation for operators having the range in A(K). LEMMA 4.1. Let K be a simplex, X a Banach space and suppose that T is an operator from X into A(K). Then there exists an aj~ne and w*-continuona function x : K ~ X* such that:
38
A.J. LAZAR
[January
(1) Tx(k) = x(k)(x), x e X , k e K , (2) ]1 -- sup, ~K II Conversely, to any affine and w*-continuous function from K into X* there corresponds an operator T: X ~ A(K), given by (1) whose norm satisfies (2). T is compact if and only if • is continuous in the norm topology of X*. Combining Theorem 3.4 with the preceding lemma we obtain a characterization of the extreme compact operators whose range is the space A(K). If X, Y are Banach spaces we denote by ~ ( X , ¥) the space of compact operators from X to Y with the usual norm.
zll
x(k)II.
THEOREM 4.2. Let K be a simplex and X a Banach space whose dual has the n.2.I.P. (n >=3) or is a strictly convex space. The operator T ~ Se(X,A(K)) is an extreme point of the closed unit ball of .~e(X,A(K)) i f and only if there exists an affine and norm continuous function x:K--+Sx, such that
T(x) (k) = x(k) (x), x e X , k E K and z(k) ~ dSx. whenever k ~ dK. REMARKS. As pointed out above this characterization is not valid for any Banach space X. However, the theorem applies to a wide range of spaces which comprises all the Lp (1 < p < oo) spaces since they are strictly convex, the L1 spaces and those whose duals are L t spaces. The last categories of spaces enter here since they include spaces having the 3.2.I.P. (cf. [13, p. 44, Theorem 6.1]). For compact operators between two spaces of continuous functions on compact Hausdorff spaces the result was proved in [3]. Now we turn to the characterization of extreme positive operators from a C(S) space to an A(K) space. The extreme positive operators between two spaces of continuous functions (and even in more general situations) were characterized by A. and C. Ionescu Tulcea, Phelps [14] and Ellis [113] using methods which rely on the algebraic structure of the spaces. We found the idea of the proof of the next theorem in [3]. THEOREM 4.3. Let K be a simplex, S a compact Hausdorff metrizable space and -~q~lthe set of positive operators T from C(S) to A(K) which satisfy T1 = 1. Then the following statements are equivalent for an operator T from C(S) to
A(K): (i) T is an extreme point of ~1 ; (ii) There is a function z : K ~ C*(S) which is affine and continuous in the w*.topology of C*(S), such that
T(f)(k) = x(k)(f), f e C(S), k e K and which maps ~K into S;
1967] AFFINE FUNCTIONS ON SIMPLEXES AND EXTREME OPERATORS
39
(iii) T I = 1 and for any f, geC(S), T(fV g) is the least upper bound of Tf and Tg in A(K). Proof. (i) :~ (ii). Denote by ~¢/1(S) the set of probability Radon measures on S and define ~: ~a'l(S)~ 2 c°(s) in the following manner:
¢(~,) = {~' e c * ( s ) : 2t, > #' >=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
Suppose that there is a net {v,} c~¢t'~(S) converging to /~ for which O(v~) c~ U = ~ for every ~. From the Radon-Nikodym theorem we infer that there exists a Borel function g on S such that 0 < g < 2 and d#' = gdl~. Choose gl e C(S), 0 < gl < 2, which satisfies:
(2)
f lg-g ldt, <--(2IIY,I[)
If we define the measure v'~ on S by (3)
l
dv'~= gtdv~ then
IIf, l[ v" e ¢(v~) and
lim~f sf, dv'~ = lim~f sf, g~dv~ = f sf,g,d#.
We have
(4)
+ I f/igld#-fsfidv'~ t • From (1)-(4) we deduce that v'~ is eventually in U and this is the desired contradiction. We now define another map ~ ' = ~/t'1(S)-~2 ~lts) as follows:
m'(t~) = m ( ~ ) n ~ l ( s ) , # e ~ ( s ) . It is easy to see that ~ ' is affine and ~'(#) is a w*-closed subset of .~t(S) for every # e ..¢t'i(S) . We shall show that ¢ ' is lower semi-continuous too. Take # e ./t'l(S), #' e ¢ '(p) and suppose that {v,) c , / / l ( S ) is a net w*-converging to/~. By the lower
40
A.J. LAZAR
[January
semi-continuity of • there are measures v~' ¢ O(v,) such that the net {v'} converges to #'. Define
v;[I => 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 <=N and
a
tar 1
0--~U-\0xl'
~xN/
We will use an elementary method to define a set of functions (the R.F.) that enable the value of U ~ at an arbitrary point P, not on the initial surface 0, to be expressed as a quadrature of its Cauchy data on 0. The subject dealt with here is similar in its nature to that in [1, 2], further references can be found in [2]. 2. Auxiliary concepts and notation. To achieve simplicity we chose without loss of generality 0 to be the hyperplane N
(2.1)
0: •
x~=l
I=1
and the point P to be the origin. Some additional concepts and notation needed for our definition of the R.F. are introduced in this section. Many of our constructions are associated with subsets {ii, i2,... } of the first N positive integers. Since these constructions are in no way dependent on the
Received August 26, 1966.
53
54
RICHARD KRAFT
[January
particular properties of the integers composing the subset it is notationally convenient to chose some fixed but arbitrary subset and symbolize it by a letter. Hence H = {il, "", ik} is a fixed subset consisting of k integers ij such that 1 -< ij < N, j = 1, ..-,k; and /1 is the set theoretic compliment of H in the first N positive numbers. Also needed are two types of sets that are derived from H. The N - k sets Hp are defined by
np-{H~p},
peFI;
and for each p e / t the k + 1 sets Hpq are defined by
Ht,e = {Hp@q), qeHp where ~ and @ are set theoretic union and subtraction. From these definitions it is seen that Hpe = H. We associate certain volumes in R N with the subsets H, H e and Hpq. Thus letting H* denote any of these subsets or any subset of the first N positive integers we define the volume V[H*] by
{
N
V E H * ] - x=(xl,..-,xN) Ixj>O, Y, x./= j=l
}
~ x1_<- 1 JeH*
and the subset of 0, I[H*], by I[/-/*] =- x=(xl,...,xN
)1 xj>O, ZN j
= 1
xj= E xj= 1}. j eH*
These definitions are motivated by geometric considerations. Note that I[Hp] is the projection of V[Hp~-! along the axis x~ onto 0. Furthermore, V[Hp] is the volume contained "between" V[Hp~] and I[Hp] that is "swept out" when V[H~] is projected along x~ onto 0. These sets and ideas can be readily visualized in 3- dimensional space. It should be noted that V t = V[il,..., iN] is independent of the permutation of il, "',iN and consists of all points enclosed by 0 and the hyperplanes xt = 0, i = 1, ..., N. The definition of the R.F. employ auxiliary systems of equations that are derived from (1.1). Thus the system (2.2)
L[H] (U) = 0
is derived from (1.1) by striking from it every/th row and column where l e / 7 . We also use subsidiary systems formed from the system L[H](U)= 0. These subsidiary systems, called the progenitors of L [ H ] ( U ) = 0, are the ( N - k) systems, (2.3)
L[Hv] (U) = O,
defined for p e A . Furthermore L[H](U)g- I~, Uq - ~,z~n a~z U s, qeH.
1967]
RIEMANN FUNCTIONS FOR A SYSTEM OF HYPERBOLIC FORM
55
3. The definition of the Riemaun functions. The fundamental relation that we introduce below is basic for a certain reduction procedure that is to be used in the definition of the R.F. This relation expresses a particular quadrature of an arbitrary system in terms of quadratures of its progenitor systems and additional terms that can be evaluated from the given Cauchy data on the hyperplane 0. We derive this fundamental relation in two steps. When Green's identities for the volumes V[Hp], p E A are added together we get U~ L *[H p] (Vl')~dx ... pen ~V[H~] q~H~
p~H JVfH.] ~¢H_
= ?EpeHqeHZffv~H,,,] uqVPqdx....
(3.1)
f1[HpUqVpqdx''"
where L*[Hp] is the adjoint operator to L[Hp] and the vector valued functions Vp = (Vpq) p e/7, q e np are defined in V[Hp]. The second step in deducing the fundamental relationship is to specify the vector valued functions VP= (Vp~) as
(3.2)
i)
Vp is a solution of L*[Hp] (Vp) = 0 in V[Hp]
ii)
VP~=0 on V[Hpa] for q e Hp but q ~ p
iii) VPP= Z
Whtp on V[Hpv] = V[H]
where W~, 1 ~ H are unspecified functions defined in V[H]. Since from (1.1)
(3.3) 5//fv[//] l~'WlatpUPdxe/~. . . .
fv[H] t~WIL[H](U)Idx'"eH
we get by using (3.2) in (3.1) that (3.4)
f,,r ] IcH Z W'L[H](U),dx . . . .
fvt H.] qeH_ Z pe// qeH_ [H_]
VPqL[H p"1( U)adx . . .
U qVPadx...
which is the fundamental relation. It expresses the integral
0.5)
f Z W'L[I-I](U),dx... Jr" [HI I eH
of the system L[H] (U) = 0 in terms of exactly similar type integrals of its progenitor systems L[Hp] (U) = O, p ~ lq and certain quadratures over l[Hp] = 0 of
56
RICHARD KRAFr
functions which are determined on 0. Since the integrals on the right hand side of (3.4) are of the exact same form as that in (3.5) these integrals themselves can be re-expressed in terms of their own progenitors and further quadratures over 0. The process can be repeated until every integral like (3.5) is expressed in terms of the original progenitor, (1.1), and quadratures over subsets of 0. We now show how a typical component U l of U can be expressed at the point (0, ...,0) as a linear functional of its Cauchy data and certain additional functions (the R.F.) over 0. By Green's identity (3.6) VUi(O,...,O)= VU~(1,0,...,0) +
f
(o ..... o)
[VL[1](U)~+U~L*[1](V)I]dx,
. J ( 1 , o ..... o)
where L*[1] is the adjoint to L[1]. After chosing V(xt, 0,..., O) to satisfy (3.7)
L*[1] (11) = 0
and v(o,...,o) = 1
equation (3.6) becomes
(3.8)
ul(0, ...,o) =
vv (1, 0, ..., 0) +
f
(o ..... o)
vL[s]
axl
J ( t , o ..... o)
The integral term in (3.8) is of the form (3.5) and hence by using the fundamental relation (3.4) it can be expressed in terms of integrals of known quantities evaluated over subsets of 0 and an integral of the ultimate progenitor system (1.1)integrated over V' = V[1, ...,NI. Thus Ul(0, ...,0) = {expressions involving the V's defined in (3.2)and (3.7)} +
~, V~L(U)jdxl "'" dxN Cj=l
Because L ( U ) = 0 the last term in (3.9) vanishes and hence U~(0, .-.,0) has been expressed as a linear functional over 0 with the aid of the functions defined in (3.2) and (3.7); hence these functions are the R.F, for the system (1.1). REFERENCES
1. L. Bianchi Sulla estensione del metodo di Riemann alle equazioni lineari a derivateparzioli ~ordine superiore Noto Ill: Rend. Accad. Lincei, Ser. 5, 4 (1895), p. 133-142. 2. H. M. Stemberg and J. B. Diaz, Computation of the Riemannfunctionfor the operator ~"/~x t'" .ax, + a(x t"" xn) Mathematics of Computation, 19 (1965), pp. 562-569. NATIONAL BUREAU OF STANDARDS~
W~mNoroN D. C.
SOME REMARKS ON NUMBER THEORY. II. BY
p. ERDOS
ABSTRACT
Like the previous paper of the same title [5] this note contains di~,onnected
remarks on number theory.
1. Bellman and Shapiro in one of their papers l'l] prove among others the following result: Denote by Q(a, b) the number of squarefree integers n satisfying a ~ n < b and let A(n) be a strictly monotone function tending to infinity together with n. Then if we neglect a sequence n~ of density 0 we have
(1)
Q(n, n + A(n)) = (1 + o(1)) 6 A(n).
We will prove a more general theorem which will show that (1) remains true if the monotonicity of A(n) is no longer required. In fact we will prove TtmORBM 1. (2)
Let f ( k ) be a real valued number-theoretic function satisfying 1
lim -=- ~ f ( k ) = ~ n : ,,,~
M
(o~# 4-oo).
k:l
Assume further that to every ~l > 0 there is a g(tl) so that for every l > g(~) and every n > 0 1 l-1
(3)
-l
~, f ( n + k ) < o t + t
1.
k=O
Then to every e > 0 a n d 6 > 0 there is an h(e,6) so that for all but ex integers n < x we have for every l > h ( e , 5 ) Received January 10, 1967. 57
58
(4)
P. ERDOS
~-6
<
[January
]~ f ( n + k ) < ~ + 6 . k---O
Before we give the simple proof we make a few remarks. By the same method we could that if for every l > g(~/) and every n > 0 (3')
1 1-1 -- Z /(n+k)>~-t/ k=O
then (4) holds. It is easy to see that our Theorem implies that if for every A(n) --, oo 1
lira sup A ~ ~----00
a(.)
k~, =0 f ( n + k ) < o ~
then for almost all n _<_x (i.e. all n neglecting a sequence of density O) 1 (5)
lim
n=oo
atn) Y_, f ( n + k) = ~.
A--~ k=o
Now we prove our Theorem. The upper bound in (4) is trivial since it follows from (3) so it is enough to prove the lower bound. Let us assume that (4) does not hold, then there is an ~ > 0 and 6 > 0 so that for every t there are arbitrarily large values of x so that the number of integers n I -<_x for which there is an l~ > t satisfying 1 ll-1
(6)
- - ~, f ( n l + k ) < - o ~ - 6 li ~=o -
is greater than ex. We shall now show that for ~/< ½e6, t > g(q) (6) contradicts (3). To see this let mt be the largest integer for which
(7)
1 m,-.,-~ f ( t ) <- ct -- 6. m i - ni t=al
By our assumption and by (2)
(8)
g(~) <
m, -
n~ < o o .
Consider now the sequence of intervals (n~, rn~) (i.e. n, ~ x < mz). There clearly exists a subsequence of disjoint intervals (n~,,m~,), r = 1,2, ... so that each n~ is covered by one of the intervals (n~,,m~,), r = 1,2,.... To see this put nl, = n,, m , = ms and assume that the intervals (n~,, m 0 r < s have already been constructed. Let n~be the least n~ greater than m~,(rnz can not be one of the n's since by (6) this would contradict the maximality property of rn~,). Put ni = n~. ms = mr. and this sequence of intervals clearly has the required properties. By our assumption ~ , , ~ , 1 > sx holds for infinitely many x, hence if rnt is the' mallest ra,, _>-x we evidently have
1967]
SOME REMARKS ON NUMBER THEORY. II.
(9)
~
(m,.
-
59
n, ) > e m , .
r=l
Put m s . - n~. = ~t,. From (9) we have either 1 1 Z~ ~, > -~em,. or )22 ~, > ~ m , . where in )21, r-= 1 (rood2) and r < s, in ~2, r - 0 (rood 2), and r ~ s. Without loss of generality assume 1 )21 ~, > T e ml~.
(I0) By (2) we have mij
)2 f(k)=(l+o(1)czm,=
(II)
~,'+ Y.,"
k=nl
where in Z' -I and in )2" m~,~+t < t < ni, j+: 0 ~ j ~
s-3 2
We have from (7) (12)
)2'-<__(a-a) )2~ a,.
We evidently have by (8) flj = nl2~+, - mhj+, > mh~ - n~2j > g(~/). Thus by (3) (13)
)2"<(u+~/)
(s-3v2 ( )2 flj=(u+n) m~.j=O
~
) u, + 0 ( 1 )
1
since
(s-3/)2
)21 ~, +
)2
pj = m,, - nl = m~. + O(I).
j=O
Thus from (11), (12), (13) and (10) we have (1 + o ( 1 ) ) e m . =
)2 f(k) < ( e + ~ ) m . - ( ~
+ 6 ) I~1 e , + 0(1)
k----'nl
1 N (e + 7) m~. - ~-e6m~. + 0(1)
60
P. ERDOS
[January
an evident contradiction if t/ < ½e~. This completes the proof of our Theorem.
Corollary. Let as < a2 < "'" be any sequence of integers and let bl < b2 < "" the sequence of integers not divisible by any as. Assume that the b's have density ~. Then if U(n) ~ ~ together with n we have for almost all n and every l > U(n) lim
B(n + I ) - B(n) l
= ~
(B
(m) =
ll~aO
)
]E 1 . bl
The corollary easily follows from our Theorem. Let f(n) = 1 if n is a b and f(n) = 0 otherwise. To prove our corollary we only have to show that our f(n) satisfies (3). Denote by ~k the density of integers not divisible by any at, 1 < i < k. Evidently ~k exists and ~s --> ct2 >- "". It is known [4] that if the b's have density then (14)
lim
~tk = ~t
k---~
Let fk(n) = 1 if n ~ 0 ( m o d a ~ ) , 1 < i < k and fk(n) = 0 otherwise. Clearly fk(n) >f(n). A(n) is periodic mod[as,'",ak] thus A(n) clearly satisfies (3) with ~k replacing ~, hence finally by (14)f(n) satisfies (3). If ~ 1/ai < oo the proof of (14) is simple and direct and we do not need [4]. It is also easy to see that our Theorem applies forf(n) = a(n)/n orf(n) = ~b(n)/n. In fact it applies to every multiplicative function f(n) > 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
~
1
j=i-1
Pj
It seems to me that perhaps F ( A , x ) = f ( P t ...,p,;x) or that at least (15)
F(A; x) = (1 + o(1))f(p,, ...,p,;x).
It easily follows from the results of de Bruijn [2] that for x > xo(A) and A > 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 <~n, k > 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,,+... p
m=l
=
I-I P " / ' " < exp 2en log n.
Thus S < 48n < cln/2, which proves the lemma. LEmMA 2. Let u < n t/2. The number of integers m < n for which f,(m) = u is less than %n/u8 logn. The integers m < n for which f,(m) = u are of the form ut where (17)
t < n , t ~ 0(rood P), P < n+ U
By Brun's method the number of integers t satisfying (17) is for u < n 1/2 less than c,, u
p_ e
1 --
< ca/eulogn
which proves the Lemma. By Lemma 1 the number of a's with fe(ai) <- n i/2 is greater than c 1 n ]2. Thus we have for these a's by Lemma 2.
cln/2 < ~
can
~
1
|=1 "~i
hence i=
~'~ >2-~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
(19)
xk > I~ as _> |=1
II
l<_t<j<_k
P,2J~-
q,
where 2 = ql < "'" are the sequence of consecutive primes. From the prime number theorem we have
d) (20)
1-I q, = exp((1 + o(1))kZlogk r=l
Hence from (19) and (20) we have klogx > ( 1 + o(1)) 2k 2 l o g k or log x k < (1 + o(1)) 21oglogx " To complete the proof of (18) we now show that for every ~ > 0 there is an x0 so that if x > xo(8) we can construct integers at < "" < ak < x, log x k > (1 - ~) 2 log log x so that no two a's should be relatively prime but every three o f them are relatively prime. Put k = [ ( 1 - 8) logx /21og log x] and let q t < ' " < q t ~ be the first (k) consecutive primes. Form a symmetric matrix [ut J l of size k from these primes the diagonal elements are all 1 - s. at is the product of the primes in the i-th row each a~ is the product of k - I primes by the prime number theorem for every fixed 8 and x > 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.
