Steven G. Krantz Harold R.Parks A Primer of Real Analytic Functions
Birkhauser Verlag Base1 Boston Berlin
Authors' addresses: Steven G. Krantz Department of Mathematics Washington University St.Louis, MO 63130 USA
Harold R. Parks Department of Mathematics Oregon State University Corvallis, OR 97331-4605 USA
Library of Congress Cataloging-in-Publication Data Krantz. Steven G. (Steven Georse), 1951 A primer of real analytic functions / Steven G. Krantz, Harold R. Parks. (Basler Lehrbiicher ;vol. 4) Includes bibliograhpicd references and index. ISBN 3-7643-2768-5 (acid-free paper). -ISBN 0-8176-2768-5 (acid-free paper). 1. Analytic functions. I. Parks, Hamld R., 1949 - 11. Title. - 111. Series. QA331.K762 1992 51SY.73-dc20
Deutsche Bibliothek Cataloging-in-Publication Data
Krantz, Steven G.: A primer of real analytic functions / Steven G . Krantz ;Harold R. Parks. - Base1 ;Boston ;Berlin ; Birkhauser, 1992 (Basler Lehrbiicher. a series of advanced textbooks in mathematics : Vol. 4) ISBN 3-7643-2768-5 (Basel.. .) ISBN 0-8176-2768-5 (Boston) NE: Parks. Harold R.: GT This work is subject to copvright- All rights are reserved. whether the whole or part of the material is concerned, specifically those of translation. reprinting, re-use of illustrations. broadcasting. reproduction by photocopying machine or similar means. and storage in data banks. Under $ 5 4 of the German Cop)right Law. where copies are made for other than private use a fee is payabie to <~WrwcrtungsgeselIschaft Wort=. Munich. @ I992 Birkhauser Verlag. PO. Box 133. CH-4010 Basel. Switzerland
Printed in Germany from the authors' cameri-ready manuscript on acid-free paper ISBN 3-7643-27a-5 ISBN 0-8176-2768-5
To Frederick J. Almgren, Jr., teacher and friend
Table of Contents 1. Elementary Properties
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10
Basic Properties of Power Series ....................... Analytic Continuation ................................. Composition of Real Analytic Functions ............... Inverse Functions ..................................... Power Series in Several Variables ...................... Real Analytic Functions of Several Variables ........... Cauchy-Kowalewsky Theorem Special Case .......... The Invcrsc Function Theorem ........................ Real Analytic Subl-nanifoldsof Rn ..................... The General Cauchy-Kowalewsky Theorem ............ -
1 11 15 18 21 25 30 35 38 44
.
2 Classical Topics
Introductory Remarks ................................. The Thcorein of Pringshoim and Boas ................. Besicovitch 's Theorem ................................ Whit ney 's Exterlsion and Approximation Theoreins ............................................. 2.4 The Theorerr1 of S. Bernstein ..........................
2.0 2.1 2.2 2.3
49
50 55 59 64
.
3 Some Questions of Hard Analysis
3.1 Quasi-analytic and Gevrey Classes .................... 3.2 Puiseux Series ........................................ 3.3 Separate Real Analyticity .............................
67 80 90
.
4 Results Motivated by Partial Differential Equations
4.1 Divisiou of Distributions 1 ............................. 103 4.2 Division of Distributiorls 11 ............................ 113 4.3 The FBI Transform ................................... 123 4.4 The Paley-Wiener Thcore~n........................... 133
5. Topics in Geometry 5.1 Resolution of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 Lojaciewicz's Structure Theorem for Real Analytic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.3 T h e Embedding of Real Analytic Manifolds . . . . . . . . . . . . 158 5.4 Semianalytic and Subanalytic Sets . . . . . . . . . . . . . . . . . . . . . 165
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Chapter 1
Elementary Properties 1.1
Basic Properties of Power Series
We begin with power series on the real line R. A formal expression
with the aj 's being either real or complex const ants, is called a pouler series. It is usually convenient to take the coefficients a j to all be real; there is no loss of generality in doing so. Our first task is to determine the nature of the set on which a power series converges. Proposition 1.I. 1 Asswne that th.e power series
converges at th,e z~aZue = c. Let r = Ic - 01. Then the series converges unzfonnlg and absolutely o n co~npactsubsets of Z = {s : I;r - 01 < r } . Proof: We may take the compact subset, of I to be K = [(Y for sonlcl number O < s < r. For LC E Ji- it then holds that
-
s, (1
+S]
2
CHAPTER, 1. ELEhIENTARY PROPERTIES
the sum on the right. the first expression in absolute valnes is The bounded by some constlant C (by the convergence hyp~t~hcsis). quotient in absolute values is majorized by L = s / r < 1. The series on the right is thus dominated by 111
.
This geolnet ric series converges. By the Weierstrass hl-Test the original series converges absolut.cly and uniformly on I<.
An immediate consequence of the proposition is that the set on which the power series 3C
converges is an interval centered about t u . This interval is terrlied the interval of copwet'gersce. The series will converge absolutely and uniformly on compact subsets of the interval of convergence. The radius of the interval of convergence is defined to he half its length. Whether convergence holds at the eildpoints of the iuterval will depend on the particular series. Let us use the noratioii C to denr>t(? the open interval of convergence. While we have seen that a power series is uniforrllly convergent on cornya~t~ subilztervals of C : it is an intclresting and nontrivial fact that if the series converges at either of the endpoints, then the convergence is liniforlii up to that endpoint. This fact is a consequence of the followirlg lemlnx due to Abel (see [ABE]). Lemma 1.1.2 Lrf uo t r l , . . . be u sequence of reals, and set
If and i f th.en
1.2. BASIC PR.OPEI(TIES O F POWER SERIES Proof: One can w i t e
Hence
We also have
for j = 0 , l . . . . . and Ena 5 Ensn 5 cnA.
Adding up these inequalities and using the equality above, we obtain t'he result.
Remark: Lemma 1.1.2 implies the claim about uniform convergence as follows: We may assume that C = ( - 1 , l ) and that the series converges at x = 1. We take E j = 1 3 , uj = aj arld consider summation from j = m to j = m + n, with m large. The assertion is then immediate. The procedure exhibited in Lemma 1.1.2 and its proof is often referred to as "summation by parts." Indeed, the usual integration by parts procedure in calculus may be verified by applying summation by m parts to the Riemann sums for the integral. On the interval of convergence C , the power series defines a function f . Such a function is said to be real analytic at N. More precisely, we have
Definition 1.1.3 A function f , with domain an open set U C R and range either the real or the complex numbers, is said to be real analytic at N if the function f may be represented by a convergent power series 0x1 so~ne interval of positive radius centered at N:
The function is said to be real analytic on V C U if it is real analytic at each cu f V.
CHAPTER 1. ELEMENTARY PROPERTIES
4
Remark: It is true, but not obvious, that the function which a convergent power series defines is real analytic on the open interval of convergence. This will be shown in the next section. A consequence is that the set V in the preceding definition may as well always be chosen to be open. We need to know both the algebraic and the calculus propert,ies of a real analytic function: is it continuous? differentiable? How does one add/subtract/multiply/divide two such functions?
Proposition 1.1.4 Let
Caj(z-a)'
and C b j ( a - a ) j
be two power series with open intcrllals of convergence C1 and C2. Let f ( x ) be th,e function defined b y the first series o n C1 and g(a) th,e function deftn,ed by the second series on Cz. Th.en on th.eir common domain C = Cl n C1 it holds th.at
Proof.. Let
AN = C a j ( x - C Y ) ~ and B~ = C b j ( a -
a)j
be, respectively, the N~~ partial surns of the power series that define f and g . If CN is the N~~ partial sum of the series
then f ( x ) fg(z)
=
lim AN
N-lx
This proves (1).
+ N-cc lim
BN
1.1. BASIC PIEOPEI(,TIES OF 1'0 WER SERIES
For (2), let
We have.
Clearly,
converges to g(z) f (x) as N approaches m. It will thus suffice to show that l a o R ~+a& - (Y)RN-I - .. + a N @- Q ) ~ R ~ J coilverges to 0 as N approaches m. Now. we know that
+
is absolutely convergent so we may set
Given have
E
> 0 , we can find No so that N 2 No implies lRNl <_
E.
So we
+
~ ~ o R aNl ( z - o ) R ~ - l + . . . + a N ( z- ( Y ) ~ R ~ ~ < l a o R ~ + . . . + a ~ - ~ ~ ( xN --Na O)R N o l + I ~ N - N ~ +-~a()XN-No+l RNo-l...+alv(a - a )N Rol ~ ~ ... - +1~ N ( J -: 0 ) N < ~ U N - - N ~ + ~- (aZ) N-No+~R -
~ ~ l -
By holding No fixed and letting N approach oo we obtain the result.
6
CHAPTER 1. ELEMEN'IIAKY PRUPEH~YG'S
Next we turn to division of real analytic functions. If f and g are real analytic functions at a and if g does not vanish on an operl interval containing a, then we would like to show that f / g real analytic at 0 (it certainly is a well-defined function) and we would like to be able to calculate its power series expansion at a by formal long division. This is what the rlext result tells us:
Proposition 1.1.5 Let
C a j(z
-
n)j and
C bj ( a
- (u)j
be two power series with open inten~alsof convergence C1 and C2. Let f ( x ) be th,e function defined by the first series o n C1 and g(z) the f~inctiondefined by th,e second series on Cz. If g does not vanish on thee open i n t e n ~ a l 1c C1 n C2 centered at a , th.en theefunction
is real analytic at a. Moreover th.e power series eaparssion of h on I mtay be obtained by formal long divzsion of the series for g into the series for. f . That is, the zeroth. coeficient co of th.e series for. h is
th,e order one coeficient cl is
etc. Proof: If we can show that the power series
just indicated converges on 1 then the result' on m~iltiplicationof series in Proposition 3 yields the proposition. There is no loss of generality in ms~irningt,hat (u = 0. Further. by dilation or contraction, it is also
no loss of generality to assume that the radius of I is 1+c, some c Assume also that bl # 0. Notice that one may check inductively that, for j 2 1 ,
> 0.
Then we see that
IcjI 5 C' (IajI + Icj-11) where C = max{l l / b o!,Ibl/bo 1). It follows that
Since the radius of I exceeds 1, it follows that laj( < oo and we see that the lcj 1 are bounded. Hence the power series with coefficients cj has r d u s of convergence at least 1. In case bl = 0 then the role of bl is played by the first non-vanishing bm, m > 1. Then a new version of formula (*) is obtained and the argument proceeds as before. ¤
We conclude this section by obtaining continuity and differentiability results for real analytic functions. For this purpose, it will be convenient to introduce the Hadamard formula for the radius of convergence of a power series.
Lemma 1.1.6 For the power series
define A and p b y
A = lim sup la, I lln n300
then p is the radius of convergence of the power series about a. Proof: Observing that
lim sup lan(x - a)n~"n = AIx - ,I n-m
we see the lemma is an immediate consequence of the root test.
.
8
CHAPTEIC, 1. ELEMENTARY PROPERTIES
Corollary 1.1.7 The power series
has radzw of convergence p if and only exists a constant 0 < C = CR such that
if, for each 0 < R < p, there
From the power series
it is natural to create the derived series
using term by term differentiation.
Proposition 1.1.8 The radius of convergence of the derived series is the same as the radius of convergence of the original power series.
Proof: For notational simplicity assume that L = lim la,l* exists and By dilation or contraction, we may suppose L We observe that
> 1.
1
2 lim lnanln 1
1
= limn~lim~an~~
= L.
On the other hand, for any choice of X > 0, we have lim sup lna,
1L n-1
= lim sup (lna,
1 A)
1 A
< -
lim(lna,ln)
=
(limn;limJa,I-) L ~ .
-
n
1
1 A
1.1. BASIC PROPERTIES OF POWER SERIES
Since X > 0 was arbitrary, we have 1 lim lna,ln-l = L 9
and the result follows from the Hadamard formula.
Proposition 1.1.9 Let
be a power series with open interval of convergence C. Let f (x) be the function defined by the series on C. Then f is continuous and has continuous derivatives of all orders which are real analytic at a. Proof: On each closed subinterval of C , f is the uniform Limit of a sequence of continuous functions: the partial sums of the power series representing f . It follows that f is continuous on that closed subinterval and thus on C . Since the radius of convergence of the derived series is the same as that of the original series, it also follows that the derivatives of the partial sums converge uniformly on any closed subinterval of C to a continuous function. It then follows that f is differentiable and its derivative is the function defined by the derived series. By induction, f has continuous derivatives of all orders, each represented by the appropriate derived series.
We can now show that a real analytic function has a unique power series represent ation: Corollary 1.1.10 If the function f i s represented bg a convergent power series on an interval of positive radius centered at a,
then the coeficients of the power series are related to the dera'vatzves of the function by
10
CHAPTER 1. ELEMENTARY PROPERTIES
Proof: This follows readily by differentiating both sides of the above equation n times, as we may by the proposition, and evaluating at x = a.
Remark: If a power series converges at one of the endpoints of its interval of convergence then, by Abel's Lemma above, we see that the function defined by the power series is continuous on the closed interval including that endpoint. On the other hand, the function defined by a power series may extend continuously to an endpoint of the interval of convergence without the series converging at that endpoint. An example is the series CO
&
which converges on (-1, I), equals ,and does not converge at x = 1 even though the function extends continuously, even analytically, to (-1,oo).
&
Finally, we note that integration of power series is as well-behaved as differentiation.
Proposition 1.1.11 The power series
and the series
obtained by term by term integration have the same radius of wnvergene, and the function F defined by
on the common interval of convergence satisfies
The proof is left a s an easy exercise.
1.2. ANALYTIC CONTINUATION
1.2
Analytic Continuation
A function on an interval I is called k times wntinuously dzflerentiable if the first k derivatives of f exist on I and are continuous. We often write f E ~ ~ (to1denote ) this circumstance. If derivatives of all orders exist (and hence are automatically continuous) then we say that f is infinitely differentiable on I and write f E Cm(I). In case f is real analytic on I we write f E C W ( I ) . We will need a result regarding summation of certain series.
Lemma 1.2.1 For each non-negative znteger n and each -1 < x < 1, we have 03
C ( p n ) , ~ ~ -- - (1~ -
m=n where we use the notation (m), (n)o
n!
-
x ) ~ +''
= m(m - l ) ( m - 2 ) . . .(m - n + I ) ,
= 1.
Proof: This is proved by differentiating the geometric series
Suppose the power series
has positive radius of convergence p and thus defines a real analytic function f on (a! - p, a + p ) . If is a point with ICY- /?I < p, then we can certainly define a power series
by setting
The following proposition shows that this new power series is well behaved.
CHAPTER 1. ELEMENTARY PROPERTIES
12
Proposition 1.2.2 The power series
defiraed above has positive r a d h of convergence at least r = p- la -PI, and on the interval (p - r, /3 + r) it converges to f. Proof.. We have
< p, there is a constant C such that
We also know that, for any R
Combining these facts and using the lemma, we see that
-
=
where D = R-Ip-a( CR power series
•
m-n
C
Rn m=n
D
n!
(R-IP-
Since R
4)"'
< p was arbitrary, it follows that the
00
has radius of convergence at least T . Define the function g on the interval
(0- r ,p + 7)by setting
1.2. ANALYTIC CONTINUATION By Taylor's Theorem, we know that
where [ is a point between p and x. But similar estimates hold for f("+') as for f (") (P), so it follows that g (x)= f (x). W
(c)
The next corollary is an immediate consequence of the preceding proposition.
Corollary 1.2.3 Let 00
be a power series with open interval of convergence C. Let f (x)be the function defined by the series on C. Then f is real analytic at e v e q point of C.
Corollary 1.2.4 i f f and g are real analytic functions on an open internal U and if there is a point xo E U such that
then
f (4= g(x),
for all x E U .
Proof: We set
v = u ~ { f(j)(x) z : =g(j)(2),
for j = O , l ,
... }.
By continuity, V is closed in the relative topology of U , while by the proposition V is open. Thus, by the connectedness of U , we conclude W that U = V. The next corollary is an immediate consequence of the preceding one.
Corollary 1.2.5 i f f and g are real analytic functions on an open interval U and there is an open set W c U such that
f (4= dx),
for
~
ZE X W,
then
f
(4= g(47
for all x E U.
CHAPTER 1. ELEMENTARY PROPERTIES
14
In fact, by repeated use of the Mean Value Theorem, the hypothesis of the preceding corollary can be weakened substantially. Corollary 1.2.6 If f and g are real analytic functions o n a n open interval U and there is a sequence X I , $ 2 , . . . in U with lim xn E U such that f ( x n ) = g(xn), for n = 1 , 2 , . . .
then
f (4= s ( x ) ,
for alE x E U.
In the next definition we find it convenient to think of a function with domain a set A IR and range in IR as a collection of ordered pairs of real numbers:
c
Definition 1.2.7 Given a real analytic function f defined on an open interval U, we see from the preceding corollary that
U{ g : g is a real analytic function on an open interval V > U } is a well-defined analytic function called the analytic continuation off. Another corollary of the above Proposit ion is the following: Corollary 1.2.8 Iff E C w ( I )for some open interval I then, for each a E I , there are a n open interval J , with a E J c I , and constants C > 0 and R > 0 such that the derivatives o f f satisfy
In fact, the converse of this Corollary is also true.
Lemma 1.2.9 If f E C m ( I )for some open interval I and if, for each, a E I , there are a n open interval J , with a E J c I , and constants C > 0 and R > 0 such that the derivatives of f satisfy (*), then f E CW( I ) . Proof: The Root Test and the inequality (*) show that
1.3. COMPOSITION OF REAL ANALYTIC FUNCTIONS
15
converges at least on the interval K = ( a -R, a+R). Taylor's Theorem and the inequality (*) show that the power series
converges to f on J n K.
W
Remark: It is interesting to note that in the reference [TA1] a generalization of this result is proved in which plain differentiation (in several W variables) is replaced by a suitable elliptic differential operator. Together the previous Corollary and Lemma provide a useful characterization of real analytic functions that will be applied in many of the sections that follow:
Proposition 1.2.10 Let f E C m ( I ) for some open interval I . The function f is in fact in CW(I)if and only if, for each a E I , there are an open interval J , with a E J c It and constants C > 0 and R > 0 such that the derivatives o f f satisfy
Remark: We note that it follows from the results of this section that if a real analytic function B(x) satisfies B(0) = 0, but does not vanish identically, then it may be written in the form
B(x) = x N B(x), for some positive integer N, where ~ ( 2 is) also real analytic and B(0) # 0. Likewise, if A(x) and C(x) are real analytic and C does not vanish identically, then A (x)/C(x) may be written for some integer M, where D(x) is real analytic and D(0) # 0.
1.3
W
Composition of Real Analytic Functions
The following formula for the derivatives of a co~npositionof two functions is not very well-known. A short proof due to S. Roman can be found in [ROM].
CHAPTER I. ELEMENTARY PROPERTIES
16
Lernrna 1.3.1 (The Formula of Fa&de Bruno [FDB]) Let I be an open interval zn Iand suppose that f E Coo(I).Assume that f takes real values in an open interval J and that g E Cm( J ) . Then the derivatives of h = g 0 f are given bg
+.
where k = kl +k2 . .tk, and the sum is taken over all k l , k2, . . . ,ka for which kl + 2 k 2 + . . . ~ n l c ,= n . To apply the Formula of FaA de Bruno we will need the following combinatorial lemma which follows from a particular application of the formula. Lemma 1.3.2 For each positive integer n and positive real number
R,
+ + .. . + k,
holds, where k = kl k l , k2,. . .,kn for which Icl Proof: We take f ( t )=
h ( t ) = 9 f ( t ) = l-(R+l)f as geometric series:
and the sum is taken over all + 2k2 + ...+ n k , = n. 1 and g ( x ) = l-R(z-l). It is immediate that But all these functions are also available
1.3. COMPOSITION OF REAL ANALYTIC FUNCTIONS
17
Evaluating f and h at t = 0 and g at x = 1 , we find that f (3) (0) = j ! , g ("(f ( 0 ) ) = E ! R ~ , and h(")(0) = n!R(l R)"-', from which the lemma follows.
+
Exercise: Equating the coefficients of x" on both sides of the equation
gives an alternate proof of the preceding Lemma. We now apply the previous two lemmas together with the proposition on the rate of growth of derivatives to study compositions of real analytic functions: Proposition 1.3.3 Let I be an open interval dn R and suppose that f E C W ( I ) Assume . that f takes real values in an open interval J and that g E C W ( J ) .Then g o f E C W ( I ) . Proof: Suppose a E I and ,6 = f(a) E J . We may assume that there are constants C ,D, R, S such that, for x near enough to a and y = f (x), the inequalities
and
hold. Now the nth derivative of h = g o f is given by
C
n! f 'l'(x) k l ! k 2 ! .. .k,! g ( k ) ( y ) (
-
+ +
f(2)(x)
f '"I(4
2!
n!
)( k1
) (
where k kl k2 .. .+ k , and the sum is taken over all E l , E 2 , . for which kl + 2762 + . . . + nk, = n. So we can esti1nat.e
'k)
. . ,kn
CHAPTER 1. ELEMENTARY PROPERTIES
18
n!
= E--
T"'
with
Thus h ( x ) satisfies the standard estimates that guarantee it to be a real analytic function.
1.4 Inverse F'unctions It is natural to inquire whether the inverse of a univalent real analytic function is also real analytic. This is too much to hope for: the function f ( x ) = x3 is real analytic and univalent in a neighborhood of the origin, yet its inverse f - ' ( 2 ) = x ' / ~ is not even differentiable near 0. An additional hypothesis (non-vanishing of the first derivative) is required for the desired result to be true. These matters are best understood in the context of the Inverse Function theorem. We now turn to that topic. Again we will need an identity which follows from a specific application of the formula of Fa&de Bruno. First, we recall
Lemma 1.4.1 (Newton's Binomial Formula) For any real numbers a, and t wzth -1 < t < 1, the equation
holds, where
(7)
- a(a-l)...(a-j+l) j!
for positive integers j
1.4. INVERSE FUNCTIONS
and
Lemma 1.4.2 For each positive integer n,
+ + + kn and the sum is taken + . . . + nk;,= n.
holds, where k = k1 k2 . . . kl, k2,. . . ,k;, for which kl + 2k2
Pmof: We take f(t) = 1 that
and g(x) =
A.It is immediate
1
and, hence, that f '"+I' (t) = h(n)(t).
Also, we have
Using these series at t = 0, we find that (for j 2 1) 1
) = - 3 . ( j )- 1( -52 )
and
By the Formula of Fa%de Bruno, we have
over all
g(k)(f(~))=k!.
...+ + +
where k = k r + k 2 + km and the sum is takenover all k l , k 2 , . .. ,k, for which kl 2k2 .. . nk;, = n. Dividing this equation by n!(-2)", we obtain the lemma.
+
THEOREM 1.4.3 (Real Analytic Inverse Function Theorem) Let f E C W ( Ifor ) some open interval I C R. Ifa E I and if f(a)# 0, then Ulere 2s a neighborhood J of a and a rml analytic fvnction g defined on some open interval K containing f (a)such that g o f ( x ) = x forx E J and f o g ( x ) = x f o r a ~ l xE K. Proof: Observe that the usual inverse function theorem of advanced calculus guarantees that a Cm inverse function g for the given f exists in a neighborhood of a. Our job is to estimate the growth of the derivatives of g at points y near p = f (a). The function g satisfies the differential equation
where
is known to be real analytic in an open interval about a. We may thus choose constants C > 0 and R > 0 such that
holds for all x sufficiently near a,and from the usual inverse function theorem, for y sufficiently near to /?,g ( y ) will be such that the estimates for h ( f ) ( x )will hold when x = g ( y ) . Fix such a y and x = g ( y ) . We claim that, for positive integers j ,
holds. We prove this by induction on j . Note that the case j = 1 is immediate from
Also, note that ( - l ) j - I
1
( 2 ) is positive. Supposing that .7
(*) is valid
for j = 1,2,. ..,n, we estimate
< -
"
n!Ckl!k2!k!...k,! Rk ((f )(2.))
.:..
((-I)"-'
(i)
PC)" Rn-I
)
k-
which proves (*) for all positive integers j . Finally, it is easy to verify, from (*), that
holds, where D and S depend only on C, R, and Jg(y)1.
W
Remark: We would be remiss not to point out that one natural way to prove the real analytic inverse function theorem is to complexify and then to use the co~nplexanalytic inverse function theorem (which can be found in many standard texts -see [KRA]). However the spirit of the present non no graph is, as much as possible, to prove all results by real methods. Moreover, the techniques using the Formula of FaA W de Bruno have considerable intrinsic interest.
1.5
Power Series in Several Variables
Set Z+ = { O , 1,2, . . .). A multi-index p. is an element of (Z+)m; we will write A(m) = ( z + ) ~ , but often the value of m for a multi-index will be understood from the context. We now recall some standard multi-index not ation:
Definition 1.5.1 For p. = (PI, p2,. .
.,pm) E A(m) and x = ( x l .zz,. . . ,x,)
E Rm,
CHAPTER I . ELEMENTARY
22
ROPERT TIES
set
a'" d x ~
-
dPl a112 -
axyi a X p22
a'"axLm -
For
we write
p
5
v
if pj 5 vj for j = 1 , 2 ...,m.
Lemma 1.5.2 For integers 1 5 m and 0 5 n and a real number -1 < t < 1, we have
The proof is left as an exercise: The first conclusion is proved using the identitv
which holds for any real t and any integer j and which should be familiar from the special cases occurring in Pascal's Triangle. The second conclusion is proved using induction and the first conclusion.
A formal expression
with a E Rm and a, variables.
E
R for each p, is called a power series zn m
Definition 1.5.3 The power series
is said to wnverge at x E Rm if some rearrangement of it converges. More precisely, the series converges if there is function 4 : Z+ -+A(m) which is one-to-one and onto such that the series
converges. Remark: For a fixed power series C , a, ( x - a ) , , we denote by B the set of points x E Rna for which (a, ( lx - a l p is bounded. It is clear that if the power series converges at x then x E B. Definition 1.5.4 For x = ( x l ,x2,. .. ,x,) E Rmdefine the silhouette, s ( x ) , of x by setting
Proposition 1.5.5 (Abel's Lemma) If the power series C , a,xP converges at a point x, then it converges uniformly and absolutely on eompaet subsets of s ( x ) .
CHAPTEIt 2. ELEMENTARY PROPERTIES
24
Let K be a compact subset of s ( x ) . Choose 0 < p < 1 such that lkjl 5 plxjc,.(holds for all k E K and for j = 1, 2, ..., m. Since x E 13, we know that there is a const ant C such that 1 a, 1 lx' 1 < - C. SO we have larllkl' C o p l ~Itl .follows that
<
00
am-1
m+j m-l
j=O
9n-1
Since the upper bound is independent of k and N, the result is proved.
w
&mark: For a fixed power series
x,a,(x
- a)',
we set
The set C is called the domain of convergence of the power series. It should be clear from the proof of Abel's Lemma that Int B c C. It is, ¤ of course, trivial that C c Int B, so we have C = Int B.
Definition 1.5.6 For a set S c Rm, we define log 11 SlI by setting
The set S is said to be logarithmicallgr convex if log llSll is a convex subset of Rm.
Proposition 1.5.7 Fop. a power series vergence C is logarithmimlly convex.
C, a,xp,
the domain of con-
Proof: Fix two points y, z E C and 0 5 X 5 1. Suppose gr = ( Y I ~ Y ~., ,. ~ m and ) t = ( z 1 , t 2 , .. . , t m )NOWy E C implies y E IntB so, for some E > 0, (1gr31 E, 1y21 E , . .., lyml E ) E 23. We conclude that there is a constant C such that
+
+
+
1.6. FUNCTIONS OF SEVERAL VARIABLES
25
By the same argument, replacing c by a smaller positive number and C by a larger number if necessary, but without changing notation, we have also that /r
Note that, since y, z, and X are fixed, we can choose E'
> 0 SO that
1-X
_>~~~ll-x+d hold for j = 1, 2 , . . ., m. Then we can choose a > 0 SO that ( 1 Y i l + c ) ' _ > l ~ j l ~ + e 'and
(IHI' + c')(1zj(l-' holds for j
1.6
= 1, 2 ,
+
((zjl+c)
6')
2 lyjlXlzjl'-X + a
. . ., m. We conclude that
Real Analytic Functions of Several Variables
Definition 1.6.1 A function f , with domain an open subset U c Rm and range R, is called reul analytic if for each a, E U the function f may be represented by a convergent power series in some neighborhood of a.
Since on compact subsets of its domain of convergence, C, a power series of several variables is uniformly absolutely convergent, we conclude that a real analytic function is continuous. It is also reasonably straight-forward to modify the proofs from Section 1.1 to prove the following:
Proposition 1.6.2 Let U , V c W m be open. Iff : -U -+ R and g : V -+R are real analytic, then f + g , f . g are real analytic on U n v, and f / g is real analytic on. U n V n {x : g(x) # 0).
CXAPTER 1. ELEMENTARY PROPERTIES Let v be a multi-index. If the power series
is differentiated term by term by
6, we obtain the derived series
As in Section 1.1,we use the derived series to show that a real analytic function is differentiable:
Proposition 1.6.3 Let f be a real analytic function defined on an open subset U C Rm. Then f is continuous and has continuous, real analytic partial derivatives of all orders. Further, the indefinite integral o f f with respect to any variable is real analytic. Proof: Let f be represented near a by the power series
We can choose T > 0 such that the series converges at a + t , where t = (T,T , . . . ,T) E Rm. But then we see that there is a constant C such that la,l~I'I 5 C holds. Choose 0 < p < 1, and consider x with 1xj - ajl 5 pT for j = 1, 2 . . ., m. For the derived series we can estimate
and the last series is seen to converge by the ratio test. A similar argument can be used to show that any indefinite integral of f is represented by a convergent power series. #
1.6. FUNCTIONS OF SEVERAL VARIABLES
27
Remark: We can now relate the coefficients of the power series representing a real analytic function to the partial derivatives of the function. By evaluating the derived series at a, we find
dl~ll
-f (a)= p!a,. axp
It is interesting to verify that a function f defined by
for x in the domain of convergence C of the power series is, in fact, real analytic on C. To this end we will need the Taylor Formula for functions of m variables (see (SM1,p. 2851)
T H E O R E M 1.6.4 iff :Rm -+ R is cN+'at each point of the line segment from g to x , then there is a point 5 on this segment such that
We will also need to know that certain series converge.
Lemma 1.6.5 i f a and b are real numbers with la1 + Ibl < 1, then
Proof: For any integer n , we have
so we have
but this is just a rearrangement of the series in (1).
CHAPTER 1. ELEMENTARY PROPERTIES
28
Conclusion (2) follows easily from ( 1 ) and the fact that
Proposition 1.6.6 Let
be a power series and C its (non-empty) domain of convergence. If f : C + R is defined by
then f is real analytic. Proof: We may assume that cu = 0. Let x E C be arbitrary. For simplicity of notation, we will suppose that xj # 0 for all j. We can choose 0 < R so that (1 R ) x E C. Then there exists a constant C such that laCL 1 1 (1 + R)xlp 5 C. Set
+
and observe that bv =
C (P+v!v)vap+vxp. CL
Choose 0 < p < R. Consider We then estimate
ZJ E
Rm with lyj - xjl 5 plxjl for all j .
1.6. FUNCTIONS OF SEVERAL VARIABLES Finally we note that, for some
29
on the line segment from x to y,
So we can estimate
and observe that the last series approaches 0 as N approaches oo. W As our last result in this section, we show that the composition of real analytic functions is real analytic.
Proposition 1.6.7 If fl, f 2 , . . . , fm are real analytic in some neighborhood of a E IRk and g is real analytic in some neighborhood of ( f ~ ( af2(.)). ), 7 fm(a)), then g[fl(x), fi(x), frn(x)] is real arialytic in a neighborhood of a. Proof: We may and shall assume that a is the origin in Elk and that 0 = f l (0) = f 2 (0) = . . . = fm (0). We can choose E > 0 such that the open ball of radius E about the origin in Rm is contained in the domain of convergence of the power series representing g. Since each f j is continuous we can choose an E' > 0 such that the open ball about the origin in EXk is contained in the domain of each f j and f, maps the open ball of radius E' into the open interval of radius € 1 6 . Now, consider an arbitrary x E IRk which is in the open ball of radius E' and is also in the domain of convergence of the power series representing f, at the origin, for all j. By the result on compositions of real analytic functions of one variable, we know the function h ( t ) defined by setting
is represented by a power series about 0 with radius of convergence exceeding 1. But then by Abel's Lemma, we know that the series obtained by substituting the series for the f' into that for g is uniformly, absolutely convergent and thus can be freely rearranged to the form B arising as the Taylor series for g[fi(x),h(x),. . . ,fm(x)].
Remark: We close by remarking that the obvious analogues of Corollary 1.2.7 and Lemma 1.2.8 hold in several variables. We invite the W interested reader to formulate and prove these results.
1.7
Cauchy-Kowalewsky Theorem Special Case
The point of the Cauchy-Kowalewsky Theorem is that, for a real analytic partial (or ordinary) differential equation with real analytic initial data, a real analytic solution is guaranteed to exist. This result is arguably the most general theorem in the lore of partial differential equations. The original papers are [CAU, pp. 52-58]) and [KOW]. The technique used in the proof is called rnajorzzation: One sets up a problem which is already known to possess an analytic solution and uses the resulting convergent power series to show that the power series arising for the original proble~riis smaller and thus is convergent. We have used essentially this technique in previous proofk, for example, in the proof of the Inverse Function Theorem. Our discussion will follow that of Courant and Hilbert, [COU]. It is simplest to prove the theorem for a certain type of system of quasi-linear first order equations with initial data given along a coordinate hyperplane. Later we show how to generalize this. Let the be real analytic on some neighborhood of the origin in functions Fi,j,a R", and let the functions 4ibe real analytic on some neighborhood of the origin in Rm, where i and j range from 1 to n and k ranges fiom 1 to m. We also assume that the functions gi vanish at the origin. The Cauchy Problem is to find real analytic functions, ul, u2,. . . ,u,, defined in a neighborhood of the origin in Rm+' such that
The plan is to write
The Cauchy Problem gives us enough data to compute the coefficients aa,3. uniquely. The difficulty is in showing that the series is convergent. To see how the coefficients are determined, let the functions Fij k and bi be represented by power series as l
9
where in the first equation the multi-index cy has n components and in the second equation the multi-index y has rn components. By hypothesis, we have c", 0. Note that by differentiating the initial data we find
while this information substituted into the differential equations gives US n
m
du; ~ ( ~ 9= 0 )C ~ i , ~ , k ( b l ( x* - ) 6n(x))-(x). , a31 j=l l~=l dxk +
Evaluating at x = 0,we see that
where we have used the ad hoc notation a ( k ) for the multi-index with a k = 1 and la1 = 1. The coefficients are obtained inductively as follows: The equation
32
CHAPTER I . ELEMENTARY PROPERTIES
is differentiated once with respect to each variable yield rn + 1 equations and the system of equations
XI,.
..,xm, y
to
is differentiated once with respect to each of the variables $ 1 , . . . ,xm to yield m m+l) independent equations. These are evaluated at x = 0, y = 0 to obtain the coefficients a& with la1 = 2, the coefficients aa,1 with la1 = 1, and the coefficients ahyz Subsequent differentiation and evaluation at x = 0, y = 0 gives the complete set of coefficients for the expansion of the ui about (0,0). It will not he necessary for us to obtain the explicit formula for the various coefficients a&; instead it will suffice to note that each a& is a polynomial function of the coefficients b p y k and ck and each such polynomial has non-negat ive co&cients. We write
and we note that P:,~ really only depends on finitely many of the arguments l$"", c;. We emphasize that the key facts are that the form of PAqi? is independent of the choice of the functions Fp,q,r and q5s and the coefficients of are non-negative (in fact non-negative integers). To make use of the preceding observations, we will find another problem
for which the coefficients of the G i j , k exceed the absolute value of those for F i V j , k and the coefficients of $i exceed the absolute value of those for 4iand for which the problem is known to have real analytic solutions vi. The coefficients of vi will then exceed the absolute value of the coefficients found above, and thus the series for each U i will converge. Recall that there exist positive constants R and C such that the inequalities Ib,i-5IR IPI 5 c IC;~R~T' < -G
hold. While we might then try using
for G and $, it will be much easier to set
and
where
It is reasonable to seek solutions
The function v should solve the problem
TOsolve a first order partial differential equation of the form
one can choose functions y (a) and Pk(v) such that
CHAPTER 1. E L E M ' A R Y PROPERTIES
34
and another function w(v) and define a solution implicitly by
To solve an associated initial value problem, the function w(v) needs to be specially chosen. Applying this method to the specific problem
so that Q ~ ( v=)
nC nw 1--R
7
for k = 1, ...,m.
we may set
and see that a solution is defined by
provided
It is routine to see that,
and conclude that
which we note is real analytic at. (0.0) as w c l r l i d .
1.8. THE INVERSE FUNCTION THEOREM
We have thus proved the
THEOREM 1.7.1 (Cauchy-Kowalewsky, Special Case) If the system of partial diflerential equations
and the initial wnditions
with #i(o)
=0
are real analytlc at the origin, then there exist functions U I , ua, . . . ,U , which are real analytic at the origin and satisfy the diflerential equations and the initial wnditions.
1.8
The Inverse Function Theorem
We return to considering the Inverse Function Theorem, but for functions of more than one variable. The theorem can be obtained as a consequence of the special case of the Cauchy-Kowalewsky Theorem proved in the previous section.
THEOREM 1.8.1 (Real Analytic Inverse Function Theorem) Let F be real analytic in a neighborhood of a = ( a l , ... ,a,) and suppose DF(a) is non-singular. Then F-I is defined and real analytic in a neighborhood of F(a). The proof of the theorem is inductive; this is legitimate since we have already proved the Inverse Function Theorem for real analytic functions of one variable. The roof of the following special case contains the heart of the argument.
Proposition 1.8.2 Let n be a positive integer. Suppose the Real Analytic Inverse Function Theorem is true for functions of n real variables. If F : Rnfl -t Rn+' is real analytic near (0,. . . ,0) with F(0,. ..,0) = (0,. . .,0) and is such that DF(0,. . . ,0) is non-sir~gdar and F(Rn x ( 0 ) ) c R" x (01, then F-I is defined and r e d analytic near (0,.. . ,0).
I
30
' 1. ELEMENTAHY PROPERTIES
Pmf: We assume the Inverse Function Theorem has been proved for functions of n variables. Let the component functions of F be Fl,. . . ,F,+l. Define the function f :R" -+ Rn by setting
There is thus a real analytic function g defined near 0 E Rn such that
g ( f ( x ) )=
for x E R".
By the usual Inverse Function Theorem, F-' is defined; let us write F-' in terms of its component functions as ( u l ,. .. ,u ~ + ~ We) .know that u;(y17 ?y?l,O)= g i ( y l ? - * Y Y Y Z ) and
where AiYnis the algebraic function of the components of an (n + 1)x ( n+ 1) matrix which gives the entry of the inverse matrix in the ith row and (n 1 ) row. ~ ~Thus we see that the component functions u1, . ..,U,+I of F-' satisfy a real analytic system of partial differential equations with real analytic initial data. Further, the initial value problem is of the restrictive type dealt with in the previous section. Therefore, the functions u l , ... ,u,+l are real analytic in a neighborhood of (0,. . . ,O).
+
Now, we can do the inductive step in the proof of the full Real Analytic Inverse Function Theorem 1.8.1. Suppose the theorem is true for functions of n real variables and suppose that F : w"+' -+ R"+' is real analytic near a = ( a l ,. . . ,a,+l ) and is such that DF(a) is nonsingular. It is clearly no loss of generality to assume that a is the origin and F ( a ) is also the origin. By an orthogonal change of coordinates in the domain, we may assume that
aFYZ+l axi ( 0 ) = 0, and
for 1
< i 5 n,
Let the component functions of F be Fl,. . .,F,+1 and once again define the function f by setting
Since the matrix of partial derivatives of components of f at the origin is the matrix M given by
we see by the inductive hypothesis that there is a real analytic function g defined near 0 E Rn such that
We now define F by setting P(x) = (fi(x),
,Fn(x),Fn+~(x) - Fn+l(g(Fl(~),. .,F,(z)),o)).
Clearly we have
= 0,
for 1
< i 5 n,
So we see that det(DF(0)) = det(DF(0)) # 0. Since we also have F(W" x (0)) c Rn x {0), we may apply the proposition to obtain G which is real analytic near (0, . . . ,0) and inverts P . But then if one defines G by setting
one sees that G is real analytic and inverts F.
W
The Implicit Function Theorem is typically obtained as a corollary of the Inverse Function Theorem. Using the usual proof (see [RUD3]) we can obtain
CHAPTER I . ELEMENTARY PROPERTIES
38
THEOREM 1.8.3 meal Analytic Implicit Function Theorem) --t Rm is real analytic in a neighborhood of (xo, yo), Suppose F : for some xo E Rn and some yo E Wm. If F(xo, yo) = 0 and the m x m matrix with entries
is non-singular, then there exists a function f : Rn -+ Rm which is real analytic in a neighborhood of xo and is such that
holds in a neighborhood of xo.
Remark: Using the machinery that we have developed, it is possible t o formulate and prove a real analytic rank theorem (see [RUD3]). We shall not provide the details here.
1.9
Real Analytic Submanifolds of IRn
In the next section we shall state and prove a very general form of the Cauchy-Kowalewsky Theorem which involves real analytic submanifolds of Rn.In this section we give the basic definitions.
Definition 1.9.1 A set S c R" is called an m-dimensional real analytic submanifold if for each p E S there exists an open subset U C Rm and a real analytic function f : U -+ Rn which maps open subsets of U onto relatively open subsets of S and which is such that p E f (U)
and
rank[Df (u)] = m, Vu E U.
This definition requires a real analytic submanifold t o be locally parameterizable. Following [FED], we note that there are a number of equivalent definitions each of which is useful in certain circumstances; we record them in the next
Proposition 1.9.2 Let S be a subset of R". The following are eqdvalent: 1. S is an m -dimensional real analytic submanifold,
1.9. REAL ANALYTIC SUBMANIFOLDS OF Rn
39
c Rn, a
real analytic difleumorphism 0 : V -+ R", and an m-dimensional linear subspace L of R" such that
2. for each p E S there exist an open V with p E V
3. for each p E S there exist an open V with p E V c Rn and a real analytic function g : V Rk, with k 2 n - rn, such that -+
s 4.
V = g-l [g(p)]
rank[Dg ( v ) ]= n - m, Vv E
and
-
for each p E S there exist an open V with p E V c Rn, a convex open U c Rm, and real analytic maps # : V -4 U, 11 : U V such that
S n V = im pl
# o pl is the identity on U,
and
5. for each p E S there exist an open V with p E V orthogonal projection ll : Rn -+ Rm such that
c IRg" and
an
II(S n V ) = n(V)is convex,
II I ( S n V ) is one-to-one,
[n 1 ( S n v)]-': f ( V )
-+
Rn is real analytic,
D[II 1 ( S n ~ ) ] - ' f ( is ~ )the adjoint of f . Proof: ( 1 2 ) Let f be the function the existence of which is guaranteed by the definition. For i = 1,.. . ,rn and u E U set
af
vi (u)= -( a ) . du; Let u0 be such that f (ao)= p. Then the set of vectors {vl (u,), . . . ,v,(u,)) is linearly independent and can be enlarged to a basis for Rn by the addition of vectors %+l, . ..,v,. Define a function F : U x Rn-"" -* Rn by setting n-m
F(u,w)= f ( ~ ) + wkVm+kr k=l
E
U,
W = (~l,-.*,Wn-m)
n-m
By construction DF(u,, 0) is non-singular, and the Inverse Function Theorem may be applied to obtain ( 2 ) .
It is trivial t o see that ( 2 3 3), while (3 a 1)follows from the Implicit Function Theorem. Finally, it is easy t o see that ( 2 a 4 3 5 + 4 + 1). It is essential t o have a notion of what it means for a function defined on a real analytic submanifold to be real analytic.
Definition 1.9.3 Let S be a real analytic submanifold of Rn ; let h : S -+ R. We say that h is real analytic at p E S if, for f as in the definition of S being a real analytic submanifold, h o f is real analytic at u, where f ( u o )= p.
It is also important t o be able t o define various real analytic vector bundles over S and their real analytic sections. We want t o avoid needless abstraction, so we shall describe the vector bundles in fairly explicit terms.
Definition 1.9.4 Suppose S C Rn is a real analytic submanifold. Associated with each point p E S are two linear subspaces of Rn,the tangent space denoted by TSp and the normal space denoted by NSp. The tangent space is defined by setting
where f is as in the definition of a real analytic submanifold,
and uo is such that f (u,) = p. The normal space is the orthogonal complement of TSp in W". The disjoint union of the TSp is the tangent bundle over S, while similarly the disjoint union of the NS, is the normal bundle over S. Specifically, TS = { ( p , V ) : p E S, v E TSp),
N S = { ( p , v ):
p S, ~v E NSp).
A less well-known characterization of real analytic submanifolds is given in the next theorem. For the theorem, we must agree that a @dimensional real analytic submanifold is a set of isolated points.
1.9. REAL ANALYTIC SUBMANIFOLDS OF Rn
41
T H E O R E M 1.9.5 Suppose S is a connected subset of Rn. Then S is a real analytic subrnan2fold if and only if there exists a real analytic map retracting some open subset of R" onto S. Proofr First, let us suppose that there is an open set U and a real analytic map # : U S retracting U onto S. To determine the dimension of the submanifold, set -+
rn = sup{rank[D#(x)] : x E U } . The good points are those for which the rank of the differential is rn; set G = U n (x : rank[Dd(x)] = rn). Since the rank is the size of the largest square submatrix with nonvanishing determinant, we see that G is open, so S nG is open relative t o S. In case rn = 0, we see that # is constant on each component of G, but since also S is connected, we see that S is a singleton. We now suppose that rn 2 1. Since # o # = #, we have
so for x E G
Thus #(G) c S n G, so S n G is non-empty. For x E SnG, we have D#(x)oD#(x) = D#(x) and rank[D#(x)] = rn, so D#(x) must be the identity map on its image. Thus for a n x E S n G 1 is a root of the characteristic polynomial with multiplicity rn, and this is certainly a closed condition. Thus S n G is also closed relative t o S. Since S is connected, it follows that S = S n G. Suppose p E S = SnG. Letting {vl ,. . . ,v, } be the rn orthonormal eigenvectors of D#(p) associated with the eigenvalue 1,we see that the function f defined by
shows that S is a real analytic submanifold at p.
CHAPTER 1. ELEMENTARY PLEOf-'EK1'L~~
42
Conversely, suppose that S is a real analytic submanifold. Let p be a point of S and let f : U -+ IW" be as in the definition of a real analytic submanifold. Proceeding in a manner similar to the first part of the proof of the above proposition 2, set
af
vi(u)= -(u). aui Let u, be such that f (u,) = p. Then enlarge the set of vectors to a basis for Rn by the addition of vectors v,+l,. . . ,vn. In a neighborhood of uo , the set { vl ( u ),. .. ,V , ( a ) ,vm+l,. . .,vn) is a basis for Rn. We apply the Gram-Schmidt Orthogonalization Procedure to obtain an orthonormal basis {Bl(u),. . . ,& ( u ) }which has the additional properties that
(i)
{el( a ) ,...,Cm (u)) is an orthonormal basis for TSf(,),
(ii)
{*m+l
( u ) ,. . .,6, ( a ) )is an orthonormal basis for NSftul,
(iii) each &(u)is a real analytic function of u.
Let F : U x W-*
4
Rn be defined by
n-m
+
F(u,W ) = f ( u )
wkek,
n-m U E U ,W = ( W ~ , . . - , W ~ - E ~ )R
Of course, DF(u,, 0) is non-singular, so the Inverse Function Theorem may be applied. We conclude that the map # = f 0 lI o F-', where II is projection on the first factor, is real analytic. Note that in a sufficiently small neighborhood of p, # coincides with the "nearest point" retraction. Since there is no difficulty in extending the nearest point retraction to other points of S, we obtain the desired real analytic I retraction.
It is clear from the preceding theorem that a function is real analytic on a real analytic submanifold if and only if it extends to a real analytic function in the ambient space. The vector fields el ( u ) ,. . . ,6, ( u )satisfying (i), (ii), and (iii) in the proof of the preceding theorem are useful in defining what it means for sections of the vector bundles over S to be real analytic. The term section of the tangent bundle simply means a function o : S -+ TS such that, for each p E S, o ( p ) E T S p -
Definition 1.9.6 A real analytic section of the tangent bundle, a, is a section such that each of the functions &(u) [ao f (u)]is real analytic for i = 1,. . . ,rn. Here denotes the action of a vector field on a smooth function.
Similarly, one defines Definition 1.9.7 A real analytic section of the normal bundle, q, is a section such that each of the functions %(u) [q0 f (a)]is real analytic for i = m + 1, ...,n.
The Cauchy-Kowalewsky Theorem involves the normal symmetric algebra bundle and sections of the normal symmetric form bundle. For each p E S let a * ( N S , ) = @go Oi (NS,) denote the symmetric , m ) = @go @"(Ns,, W") denote algebra of NS,, and let O * ( N S FW the algebra of symmetric forms on N S , with coefficients in Rm. Then the normal s ymmetric algebra bundle is
and the normal symmetric form bundle with weficients in Wm is
@*(NS,W m ) = { ( p , y ) :p
E
S, y
E
@*(NS,, Bm))-
Definition 1.9.8 A real analytic section of the normal symmetric i f o m bundle with coeficients in Rm is a function a : S 4 @ ( N S R , m), with a ( p ) E a i ( ~ sWp m ),, such that the functions
are real analytic for each choice of { j l ,. . .,ji) c {m
+ 1, ...,n ).
Remark: In Chapter 5, we shall consider an abstract real analytic manifold. By that is meant a paracompact Hausdorff space with a locally Euclidean structure such that the transition functions are real analytic. It turns out that there is no true increase in generality: Every abstract real analytic manifold can be embedded, by a r e d analytic embedding, in a Euclidean space of sufficiently high dimension. HOWever, this is a deep theorem. We shall discuss it, and related results, rn in Section 5.3.
1.10 The
The General Cauchy-Kowalewsky Theorem
lctb
derivative of a k-times continuously differentiable function u : Rn -+ Rm is, at each point p E Rn,a symmetric multilinear function on lotuples of elements of Rn taking values in Rm;the space of such symmetric functions is denoted by ak(Rn ,Rm). A differential equation of order k on R" is thus an equation of the form
where
It is harder t o describe the general initial data (also, called Cauchy data) for a differential equation if the data is t o be specified on a real analytic submanifold: this is the situation that we have in the general Cauchy-Kowalewsky Theorem. We let S be a real analytic submanifold of In.Let bo : S -+ Rm. Then we can seek a solution u of the differential equation which also satisfies
But for a differential equation of order k we should also specify the derivatives up to order k - 1. To do this, for each i = 1,.. ., k - 1, let #i be a function such that, for each p E S,gi(p) is a symmetric multilinear function on ktuples of elements of N S p with values in Rm. In the terminology of Section 1.9, these are sections of the normal symmetric form bundle with coefficients in Rm. We of course assume that each #i is real analytic. To fully determine the ith derivative of u we must know not only the effect on twtors normal t o S, but also on vectors tangent t o S. Since the functions # j , for j < i, are defined and differentiable on S, they can be used t o obtain the needed information: For vl,. . . ,vr E TSp,and wl,. . . ,w, E NSp, with r + s = s', we require
The reader has probably also noticed that much of the behavior of Dk~ ( pis) similarly restricted if the initial conditions are to be satisfied. What is not determined is
1.lo. GENEELAL CAUCHY-KOWALE WSKY THEOREM
45
when wl ,..., wk E NS,. Assume that S is a d-dimensional submanifold. Then NS, is of dimension n - d. Simple combinatorial reasoning shows that the number of unordered k-tuples of basis elements from NSp must then be (k+n-d-1 n-d-1 ) for combinatorics of this sort. Thus the dimension of the space of possible functions D ~ U (on ~) k+n-d-1 k-tuples of normal vectors is m( n-d) . Accordingly, one requires = (k++l-d-1) , and one would like t o be able t o solve F = 0, analytn d-1 ically by the Inverse Function Theorem, for the undetermined normal part of Dku(p). If this is possible we say that the equation is noncharacteristic. Even after the normal part of Dku(p) is found, it is still necessary t o have the equality of mixed partial derivatives hold for derivatives of order higher than k. If this condition is satisfied, then we say that the equation is consistent.
T H E O R E M 1.10.1 (Cauchy-Kowalewsky) Suppose S c Rn is a real analytic submanifold of dimension d. Suppose $0 : S --+Rm is real analytic on S and #i is a real analytic section of the normal symmetric form bundle O'(NS, Rm), for i = 1 , . . .,k - 1. If
with
is real analytic, non-characteristic, and consistent, then there exists a function u which is real analytic in an open set U with S c U and satisfies
o k - ' ~ ( p ) 1 @k-I (NSp) = F[x, (x),Du(x), . . . ,D k u(x)] = 0,
( x , for p E S, for x E U.
1
Proof: The first step in the proof is to apply the characterization (2) from Proposition 1.9.2 t o rid ourselves of the various bundles and reduce the problem t o more concrete notation: We write 8" = IRd x p - d , so points in Rn are (xl,... zd, y l , . . .,y n - d ) , and after solving
.
46
CXlAPTm 1 . ELEMENTARY PROPERTIES
for the highest normal derivative, the differential equation becomes
The initial conditions become
To reduce t o the special case prwed earlier, additional variables are introduced: W i y a , p , where i E { I , . . .,rn) and where a and ,B are multi-indices with 1 5 la1 IPI k and I,BI k - 1 . The w 's satisfy the following eqations:
+
<
<
for i = 1 , . . . ,rn. The initial conditions for the w's follow from those on the various derivatives of u. The solution is built up inductively. Begin by setting y2 = y3 = .- - yn-d = 0 and applying the special case of the theorem to extend the functions to a neighborhood of y1 = 0.. This provides the initial data to solve in a neighborhood of y2 = 0 with y3 = . . . - ya-d = 0. After n - d applications of the special case, a real analytic functioli u is defined in a neighborhood in IWn. The hypothesis of consistency assures that the original differential equation is satisfied. W
1 .lo. GENERAL CAUCHY-KOWALEWSKY THEOREM
47
.
Remark: The Cauchy-Kowalewsky theorem has been influential in the theory of partial differential equations. Even in such modern developments as the theory of analytic wave front sets (see [SJ]) one sees some of the ideas and techniques that have been presented here.
Chapter 2
Classical Topics 2.0
Introductory Remarks
Mathematicians prior to the middle nineteenth century thought about functions very much as do calculus students today: a function is given by a f o n d a . As an extreme example, Leonhard Euler (1707-1783) addressed one of the great questions of the late eighteenth century - whether an arbitrary set of data for the wave equation (i.e. any function representing the initial position of a vibrating string) has an expansion in terms of sines and cosines - as follows: One possible initial configuration for the string on the interval [O,2?r] is
However # is not one function but two functions (reasoned Euler). Thus it could not possibly be expanded as a sum of sines and cosines (each a single function). See [LAN] for more on this matter. While from our modern perspective the argument of Euler is preposterous, it is sobering to note that in his classic text [OSG] published in 1929 Osgood felt compelled to point out that
really zs a function, and is therefore a legitimate example of a Cm but non-real analytic function. Mathematicians of the late nineteenth and
early twentieth century struggled hard to come to grips with the facts that
1. The power series of a G* function, expanded about a point a , need not converge except at a; 2. Even if the power series converges in an open neighborhood of a, it may not converge back to f . Since the nineteenth and early twentieth centuries had been devoted in part to seeing that the Fourier series of any reasonable function converges back to that function, it came as quite a shock that nothing could be further from the truth for the power series of a G" function. In fact one can use elementary considerations to see that the collection of real analytic functions on the interval (a, b) forms a set of first category in Cm(a, b). We devote this chapter to reviewing some of the results from the period 1890- 1935. Many of the results and investigations from that time were either ill-advised or have been superseded by modern insights. We shall give little space to those. (A charming treatment of some of the issues considered in those times appears in [PIE, pp. 214- 2191.) But a number of results are quite striking. and have been essentially lost to the modern mathematician. In order to give the flavor of the investigations that were made, we shall devote some detail to several of these and shall mention several others. There is no intention here to be complete. We strive rather to provide the reader some guideposts to the classical literature.
2.1
The Theorem of Pringsheim and Boas
Much of the material in this chapter draws its inspiration from the lovely article [BOA21 by Ralph Boas. Although we shall attempt to cover a much larger territory, Boas's article was our entry point to the topics discussed. The example of the non-real analytic function 4 in Section 1 has the property that it is real analytic on the right half-line. However the power series expansion of #I about a point t > 0 has radius of convergence t. Thus the radius of convergence shrinks to zero as t
moves toward the non-analytic point 0. What if a Coofunction g on an interval (a,b) has the property that the radius of convergence of the power series of g about any t E (a, 8) is at least 6 > 01 Can we hope that g is real analytic on (a, b)? A classical theorem of Alfred Pringsheim [PRI] answers the question affirmatively. Forty years after Pringsheim's proof was published, R. P. Boas, while still an undergraduate, discovered that Pringsheim's proof was fallacious. Boas then succeeded in finding a correct proof (see [BOA21 for details of this matter). Pringsheim's theorem was formulated in extremely old-fashioned language which would be inappropriate t o the present book. We state it as follows:
THEOREM 2.1.1 (Pringsheim-Boas) Let f be a Cm, real-valued function on an open interval I = (a, b). Let a j (t) = f (j)(t)/j! be the jth Taylor coeficient off at t E I. For each t E I let 1 p(t) = lim supj,, laj (t) 1 ' l j be the radius of convergence of the power series
at t. If there is a 6 > 0 such that p(t) _> 6 for all t E (a, b), then f is real analytic on I. Before proving the theorem, we consider a weaker result the proof of which illustrates the basic technique.
Proposition 2.1.2 With the same notation as in the theorem, if [c, d] c (a, b) with c < d and p(t) > 0 for each t E [c,d], then there is a non-empty open subinterval of [c, d] on which f is real analytic. Proof: Setting
for l = 1 , 2 , . . . , we note that each Fc is closed. By hypothesis we have
so by the Baire Category Theorem some F4 must contain a non-empty open subinterval of [c, 4. But then on that open subinterval we have exactly the estimate needed to show that f is real analytic.
Corollary 2.1.3 With the same notation as in the theorem, i f p ( t ) > 0 for each t E (a,b), then f is real analytic on an open dense subset of ( a ,b)The real usefulness of the lower bound on the radius of convergence is captured in the following lemma. This is a variant of a lemma used by Hoffman and Katz, VK], in their proof of the Pringsheim-Boas Theorem.
Lemma 2.1.4 With the same notation as in the theorem, rif f is real analytic on (c,d) with a < c < d < b, p(c) > 0, and, for some x E (c,d ) , p(x) > x
-c
holds, then f ( t )=
holds for all x E [c,c
C aj(C)(t
- c)'
+ p(c))-
Proof: Fix such an x E (c,d ) . Setting
+
we see that g is real analytic on ( x - p(x), x p(x)). Since f and g and all their derivatives agree at x , they must be equal on
By continuity, we also have f ( j ) ( c )= g ( j ) ( c )for j = 0,1, . . . . We know from section 1.2 that
2
00
g("'(c) 41 ( t - C )j =
+
C a ( c )( t
-
c)j
+
.
converges to g on (c - p(c), c p(c)) n ( x - p(x),x ~ ( x )=) (a,P)Since g = f on [r.. lllill{d. T + P ( x ) } ) c ( a ,P), the lemma is proved.
Remark: A similar result clearly holds for the right-hand endpoint of the interval [c, dl. The proof of the theorem will require a second application of the Baire Category Theorem.
Proof of the Theorem: Arguing by contradiction, suppose there are a and p with a < a < p < b such that (a, p) contains a point at which f is not real analytic. Let 3 denote the set of points in [a,p] at which f is not real analytic. Then B is closed and thus may be considered in its own right as a complete metric space. Set
for t = 1 , 2 , . . . . Note that each Fe is closed. By hypothesis, we have
so by the Baire Category Theorem there must be some l and some open interval I C (a,p) such that
Since we can always replace I by a smaller interval around any of the points in B n I , it will be no loss of generality to also assume that the interval I has length less than or equal to min(6, &). Fix such a value of l and such an open interval I. Consider any point x E I \ B. There is some maximal open subinterval, (c, d) , of I which contains x. It is possible that c = a or d = 0, but not both because Bn I # 0. For definiteness, let us suppose a < c E 3. Then the hypotheses of the previous lemma are satisfied, so
holds for t E [c, d ).
Now we can estimate, as in Section 1.2,
It follows that for e v e q x E I the estimate
holds, which suffices to show that f is real analytic on I. This contradicts the fact that 0 # B n I . In fact the argument presented here suffices to prove the following strictly stronger, but somewhat more technical, result:
T H E O R E M 2.1.5 Let f be a C" , real-valued function on an open internal I = (a,b). Let a j ( t ) = f ( j ) ( t ) / j !be the jthTaylor coeficierat o f f at t E I. For each t E I let 1
'('1
= lim sup,,,
laj ( t )
be the radius of convergence of the power series of g at t. If for each point t E I we have p ( t ) > 0 and lim inf,,t p(x)/lx - tl > 1 then f i s real analytic on I. Due in some measure to the influence of Hardy and Littlewood, mathematicians of the period described here did not study functions of several real variables. However it is not difficult t o see that the theorem of Pringsheim and Boas also holds in JRN. (In fact as an exercise the reader may wish to use the separate real analyticity ideas
in Section 3.3 to prove such an N-dimensional result.) As an intuitively appealing sufficient condition for real analyticity, Pringsheim and Boas's theorem is reminiscent of an important, but unfortunately rather obscure, "converse to Taylor's theorem" that we now record. We refer the reader to [KRA2] and references therein for discussion and detailed proof.
THEOREM 2.1.6 Let f be a function defi.ned on an open domain UC Suppose that them is a C > 0 such Ulat for each x E U there is a kth degree polynomial Px(h) with
w*.
for h smdl. Then f E c ~ ( uand ) the Taylor expansion to order t of f about x E U is given by Px(h). One may view Pringsheim and Boas's theorem as the order infinity analogue of this last result. The converse to Taylor's theorem has proved to be an important tool in global analysis (see [ABR]). In the next section we consider the behavior of a real analytic function at the boundary of its domain of analyticity from another point of view (that of Besicovitch). In the third section we present some work of Whitney which will both unify and supersede that which went before.
2.2
Besicovitch's Theorem
An old theorem of E. Bore1 is as follows (see [HORI, vol. 11):
THEOREM 2.2.1 Let {aj}j",, be any sequence of real or C O V I ~ Z ~ nmbers. Then there is a C" function on the internal (-1,l) such that f ( j )(0)= j ! .a j . In other words, the Taylor coefficients of a Cw function at a point may be specified at will. The next theorem, due to A. Besicovitch [BES]7 specifies a powerful extension of Borel's result:
THEOREM 2.2.2 Let { a j } g 0 and {bj}$, b sequences of real or complex numbers. There is a C" function f on the closed interval
[o, 11 such that
CHAPTER 2. CLASSICAL TOPICS
56
1 . f is real analytic o n the interval (0,l);
It is convenient, and correct, to think of the function f in the theorem as being the restriction to the interval [O,1] of a function that is Cm on the entire real line. The conclusion is not only that one may specify all derivatives of f at both endpoints of the interval, but that the function can be made analytic on the interior of the interval. By applying Besicovitch's theorem to both sides of the point 0 E W we may obtain the following strengthening of E. Borel's theorem:
Corollary 2.2.3 Let { a j } s o be any sequence of red or complex numbers. Then there is a CaOfunction on the interual (-1,l) such that f ( 3 ) (0) = j ! aj and f is reul analytic on (- 1,0) U (0,l). We shall now present the proof of Besicovitch's result. The heart of the matter is the following lemma:
Lemma 2.2.4 Let { a j ) be a given sequence of real or wmplex nambers. Then there 2s a function f that is CC on [0, oo) and real analytic on (0,m) and such that f
(3)
(0) = aj .
Proof: We may and shall assume the the series
aj
are all real. Formally define
Here the numbers Q, cl, . . . are positive numbers t o be specified. Also the numbers E O , €1, . . . will each be specified later to take one of the values -1,0,1. Fix an interval [0, A], A > 1. Notice that the jthsummand of our series does not exceed
The integral (*) equals
2.2. BESICOVITCH'S THEOREM Of course the series
converges. We conclude that the series named F(x) converges uniformly on [0,A] regardless of the choice of the c's and E'S. A straightforward imitation of the argument just presented allows one to check that the formally differentiated series F'(x) converges uniformly, and likewise for all higher order derivatives. It follows that the series F defines a Cm function on [0, oo). The simplest way to see that F is real analytic on (0,oo) is to think of x as a complex variable and verify directly that the complex derivative exists (the estimates that we just discussed make this easy). Alternatively, one may refine the estimates in the above paragraphs to majorize the jthderivative of F by an expression of the form C ~j j ! . In any event, F is plainly analytic when x > 0. It remains to see that the parameters cj, y may be selected so that the derivatives of F take the prespecified values cuj at x = 0. Differentiating F at 0 and setting the jthderivative equal to aj leads to the equations
We may rewrite these equations as
Now we reason as follows: If a. = 0, then we set eo = 0 and the choice of Q is moot; otherwise, set €0 = sgn (ao) and co = (aola2Next we choose to be -1, 0, or 1 according to whether the righthand-side of (2) is negative, zero, or positive. In case €1 = 0 the choice
58
CHAPTER 2. CLASSICAL TOPICS
of cl is again moot; otherwise equation ( 2 ) determines the value of cl from known data. We continue in this fashion, choosing the ej in succession so that the equations are consistent with the signs of known data.
Lemma 2.2.5 Let { a j ) be a given sequence of real or complex numl bers. Then there is a function f that i s Cm on [O, 1) and ~ e a analytic on (0,1 ) and such that f(j)(0) = aj , and f b ) ( l )= 0, all j. Proof: Let h(x) be a non-negative Cm function on W which is s u p ported in [O, 11,real analytic in (0,I ) , and satisfies S h(x)dx = 1. Set 5
H ( x ) = 1-
h(t)dt.
Then H is C'O on W,real analytic on (0,I), and
Choosing F according to the previous lemma so that F ( ~ ) ( o=) aj for j = 0,1,2,. .. and setting f = F H , we see that
Proof of the Theorem: Let F be a function that is real analytic on ( 0 , l ) and Cm on [O,1] and such that ~ ( j(0) ) = j!aj for every j and F b ) ( l )= 0 for all j. Likewise, by symmetry, let G be a function that is real analytic on ( 0 , l ) and C'O on [0,11 and such that G(j)(O)= 0 for every j and G ( j ) ( l )= j!bj for all j. Set f = F G. It is now obvious that f satisfies the conclusions of the theorem.
+
In the next section we shall give Hassler Whitney's dramatic generalization of these results to N dimensions.
2.3. THE TWOREMS OF WNfTNEY
2.3
Whitney9sExtension and Approximation Theorems
When compared with higher dimensions, the analysis of one real variable is relatively simple at least in part because any open set in W is the disjoint union of countably many open intervals. It was Hassler Whitney [WHI] who discovered the correct multi-variable analogue for this fact. He was able to exploit it to prove several important extension and approximation theorems. Even today Whit ney b theorems, and especially his techniques, exert a decisive influence over the directions that real analysis has taken. The key geometric result that plays the role for W N of the decomposition of open sets in W into intervals is the following:
Lemma 2.3.1 (Whitney Decomposition) Let 51 E lRN be an open set. Then there are closed cubes Qk such that
3. For each k,diam (Qk) -< dist (Qk,'51) 5 4 diam (Qk).
In what follows, when Q C W N is a cube with center xo and c > 0 we let cQ denote the set {x E W N : x0 (l/c)(x - xO)E Q). In other words, cQ is the cube with center xo and with sides parallel to those of Q and having side-length c times the side length of Q itself-
+
Lemma 2.3.2 The Whitney decomposition of an open set R E W~ an be taken so that no point of R is contained in more than 1 2 of ~ the cubes. The Whitney decomposition is generally applied in conjunction with the following:
Lemma 2.3.3 (Partition of Unity) Let R E lRN be an open set and {Qj) a Whitney decomposition for R. Then there exist C* functions 4j on W N satisfying
GHAPTER 2. CLASSICAL TOPICS 1. 0 5 4j
5 1for a l l j ;
3. # j ( x ) = 0 when x
(4/3)Qi;
4. I( 8 a / 8 x a ) $ k( x )1 < Ka - (&am ~ k ) - l " l
for any multi-index a;
5. C j # j ( x ) = 1 when x E n.
Both of these lemmas are treated in considerable detail in [STE]. See also the original paper of Whitney [WHl]. We now present an elegant application to the theory of Cm functions:
Proposition 2.3.4 Let E & RN be any closed set. Then there is a C" function f on RN such that { x E IRN : f ( x ) = 0) = E. Proof: Let
be a Whitney decomposition for the complement of E and {#j) the corresponding partition of unity. For each j let 6j denote the diameter of QjSet {Qj)
Of course the series converges absolutely and uniformly on all of R ~ . Notice that the zero set of f is precisely the complement of E. It remains to check that f is infinitely differentiable. If a is a multi-index then the series obtained by applying 8?/8xa formally t o the series defining f has jthterm that is majorized by
Now fix a point x in ' E . If u is the distance of x to E then x is contained in at most ( 1 2 ) ~of the cubes {Qjk)i2=N, and each of those cubes has diameter 4,- Moreover 6j, 5 u 1 46jk-Thus we use (*) to see that, at this x ,
As v --+ 0 we see that this last expression tends to zero. It follows from these estimates that all drri~at~ives of f exist on C Eand that they tend
2.3. THE THEOREMS OF WHITNEY
61
to zero at points of ' E tending to E. By the same token, all derivatives 0 of f on a E are zero. Of course all derivatives of f on E are zero by definition. It follows that f is a Cm function on all of W N . The principal result of Whitney's classical paper [WHl] is to show that a smooth function on a closed subset E & W* can be extended to be Cm on all of W N in such a way that the extended function is real on the complement of E. We shall formulate and discuss, but not prove, this result. It is obviously a generalization of Besicovitch's theorem presented in the last section: in that context, the role of the set E is played by just two points - the endpoints of the interval being studied. Clearly there is an obstruction to formulating Whitney 's theorem. If E is a truly arbitrary closed set, then what do we mean by a "smooth" (or C m ) function on E? One possible definition is that a function f is smooth on E if it is obtained by restricting to E a function that is smooth on all of .KtN.For some purposes such a definition is satisfactory. However, when one is proving extension theorems such a definition is inappropriate. Therefore we proceed as follows:
Definition 2.3.5 Let E
IEkN be a closed set and f a function on E. We say that f is Ck on the set E if for each x E E there are numbers f,,,, 0 5 (a15 k, such that, for each 0 5 (a( 5 k, f(x+h)=
C
p!
fz'a+'
hP
+ 'R,(x, h).
Ia+PIIk
Here 'R,(x, h) is a remainder term with the property that, if c then there is a 6 > 0 (independent of x) so that if 1hl < 6 then
> 0,
It is not difficult to see that if E is a simple set like a closed half space then the definition of Ck function just given is equivalent to any other reasonable definition. For pathological closed sets, there is no other reasonable definition of smooth function. See [KRA2], [JON] for more on these matters. Notice in passing that this definition of smooth function on a closed set is very much in the spirit of the converse of Taylor's theorem that was presented in Section 1.
CHAPTER 2. CLASSICAL TOWhitney's main theorem (see [WHl]) is the following:
THEOREM 2.3.6 (Whitney Extension Theorem) Let E be a closed subset of WN.Let f be a function on E that is Ck according to the preceding definition. Then there is a Ckfunction on all of WN such that
2.
j
jis real analptic on the complement of E.
The proof of Whitney's theorem proceeds in two steps. First, we produce a ckextension F of f to all of WN.Then an approximation procedure (similar in spirit to the Weierstrass approximation theorem) is used to replace F by functions which (a) agree with f on "most" of E, (b) are real analytic off E, and (c) approximate F closely. The desired function f is then obtained by a limiting argument. To see how Whitney's extension technique works, we let {Qj)be a Whitney decomposition for R ' E . Choose for each j an element pj E E such that dist(pi, Qj) = dist(E, Q j ) Set
=
Then we define
It turns out (we shall not prove this) that this defines a Ckfunction on all of WN that agrees with f on E. It requires some extra work to obtain an extension operator that extends an f that is Cm on E to an f that is Cm on all of WN, and we refer the interested reader to Whitney's original paper [WHI] for this matter. The necessary approximation result that allows one to arrange for the extension of f to be real analytic on the complement of E is as follows:
w, THE T m O m S OF WRlTNEY
63
proposition 2.3.7 (Whitney Approximation Theorem) EtN be a compact set. Let f be of class ckon K . If c > 0 k tK then there exists a real analytic fundion G on lRN such &at
In fact it is not difficult, given our modern perspective, to prove such a result. Let +(x) be a positive real analytic function of total mass one (the Gaussian kernel, suitably normalized, will suffice). For -N 6 > 0 set #a(x) = 6 4 ( ~ / 6 ) .We may use the Gk extension theorem above to extend f to an open set U that contains K. Let tl, be a nonnegative cutoff function that is supported in U and is identically equal to 1 on K. Define g(x) = $(x) f (x). Now set
Then straightforward arguments show that fa -+ f uniformly on K. In fact it can be shown that fa 4 f in the cktopology of K. Now, as already outlined, the approximation result can be used to make successive alterations to the ckextension theorem to arrange that the extended function is real analytic off the set E. It is interesting to note that there is no successfu definition, analogous to 2.3.5, for a real analytic function on an arbitrary closed set E. There is, however, an interesting generalization of (the spirit of) 2.3.4 due to J. Siciak [SIC3]: Let f is a Cm fundion on an open domain R. If x E R then let r(x) be the radius of convergence of the Taylor series expansion of f about x. Then we set 1. A( f ) = { E E S-2 : fis real analytic in a neighborhood of a ) ;
4. F ( f ) = {a E S : r(a) = 0) = the points of ''false convergence".
It is straightforward to check that A is open, D is a Ga, and F is an F' of the first category. The theorem is
THEOREM 2.3.8 Let R be an open domain in lRN. Let h2 = A u D u F, where A is open, D is a Gs, and F is an F, of the first category. m e n there is a Cmfunction f on R serch that A = A(f), D = ~ ( f ) , m-hd F = F ( f ) .
CHAPTER 2. CLASSICAL TOPICS
64
2.4
The Theorem of S. Bernstein
We conclude this chapter by presenting a curious and not well-known theorem of S. Bernstein that gives a sufficient, and easily checked, condition for a function to be real analytic. For convenience we work on the real line, but there are obvious analogues in several variables.
T H E O R E M 2.4.1 Let f be a C" function on an open interval I C B. If f and all its derivatives are non-negative on the entire interval I Ulen f is real analytic on I . 2
The functions ex,ex ,x,x2,etc. on the interval (0,oo)certainly satisfy the conditions of the theorem. Of course the functions sin I,cos x, log a: do not, so the utility of the result is unclear. The theorem spawned, in its day, a rash of work on the patterns of the signs of coefficients of real analytic functions. We refer the reader to [BER] and [POL] for more on these matters. Proof of the Theorem: Let a E I. Recall Taylor's theorem with remainder:
where
This result is proved by integrating the fundamental theorem of calculus
by parts a total of n - 1 times. It is convenient to use two changes of variable to rewrite R, as
In what follows we assume that b E I , x < b, then we have
a E I and that a
< x, x
E
I. If
Here we are using the fact that f ("+'I > 0 hence f (") is monotone increasing on I. The right hand side of the last inequality is nothing other than (x - a)" ( b - a)" Rn @ISince Taylor's expansion tells us that
and since all terms on the right but the last are positive, we conclude that f (b) Rn(b). Combining our inequalities gives
>
Now letting n -+ +oo yields that &(x) + 0. This shows that the Taylor expansion converges, uniformly on compact subsets of (a, b), to f. Since a < b were arbitrary in I, we conclude that f is real analytic on I. We refer the reader to the book of Boas [BOAl] for further discus sion of the phenomenon identified in Bernstein's theorem. A
Chapter 3
Some Questions of Hard Analysis 3.1
Quasi-analytic and Gevrey Classes
In the theory of functions on W N there is a great chasm between the space of GO" functions and the space of real analytic functions. If, for instance, a real analytic function vanishes on a set of positive measure then it is identically zero. [This is most easily proved by induction on dimension, beginning with the fact that in dimension 1 we have the stronger result that if the zero set has an interior accumulation point then the function is identically zero.] By contrast, any closed set is the zero set of a Cm function. In dimension 1 this is seen by noting that the complement of the closed set is the disjoint union of open internah; it is straightforward to construct a CO" function of compact support on the closure of an open interval whose support is precisely that closed interval. In several real variables the Whitney decomposition (see [STE]) serves as a substitute for the interval decomposition of an open set and allows a similar construction to be effected (see Section 2.4). Real analytic functions have (locally) convergent power series expansions; Cm functions, by contrast, generically do not. Locally s u p ported GO" functions can be patched together using a Cm partition of unity; there is no similar construct in the category of real analytic functions.
CHAPTER 3. SOME QUWTIONS OF HARD ANALYSIS Since both Cm functions and real analytic functions play an important role in the regularity theory of partial differential equations (see [HOR2]), it is desirable to have a s a l e of spaces incrementing the differences between the space CODand the space CW.(An analogue of the scale one might wish for is the scale of spaces ck, 1 < k < 00 spanning from C = Co, the continuous functions, to Cm, the infinitely differentiable functions.) Unfortunately, no such scale is known. However there are some very interesting and useful spaces that are intermediate between COD and CWand that interpolate between the two extremes in a variety of precise senses. These are the quasi-analytic classes and the Gevrey classes. We shall discuss both of these types of spaces, and their interrelationships, in the present section. Before proceeding, we note that the classes of functions defined in this section are specified in terms of rate of growth of Taylor coefficients. For an arbitrary Cm function the Taylor coefficients can be fairly unpredictable as the next theorem will show.
THEOREM 3.1-1 (E. Bore1 [HOR2]) For each multi-index a of length N let a, be a real number. Then there exists a Cm function on the unit ball B(0,l) C lRN with the proper@ that
for every multi-index a. This theorem may be proved either by adding infinitely many small bump functions, each of which carries the information about one Taylor coefficient, or by a straightforward category argument. In fact considerable investigation was made in the late nineteenth and early twentieth centuries into the pathological nature of the Taylor expansion of a Cm function. We discussed some of these ideas in Chapter 2. Hassler Whitney considered to what extent the Taylor coefficients of a CODfunction may be specified on an arbitrary set E. His result, valid in any dimension, is described in detail in [FED] or [HOR2]. See also our Section 2.4. Whitney's results are remarkable for the fact that their hypotheses are as weak as one could possibly hope for:
3.1. QUASI-ANALYTIC AND GEVREY CLASSES
69
THEOREM 3.1 -2 (The Whitney Extension Theorem) Let E be any compact subset of IllN. Let k be a positive integer and for each mdt-index a, with (a5 k, let u, be continuous functions on E. If x, y E E are unequal then we define
Also we set U,(x, x) = 0 for x E E . If each U,, 101 5 k, is a continuous function on E x E then there is a function v E Ck(WN) such that
for all x
E
E and la1 5 k.
Now we turn to our subject proper. It is convenient in this section to do analysis not on IRN nor on W1 but rather on the unit circle. Equivalently, we do analysis on the set T = W/27rZ. We are in effect working on the interval [O,2?r] but identifying the endpoints of the interval. This is useful because we shall then be able to use some elementary ideas from Fourier series. Fourier series are built up from the characters eGt, where i = and these functions are supported in a natural way on T. We use ordinary Lebesgue length or measure in doing analysis on T. (See [KAT] for a detailed consideration of analysis on T.) In what follows we let f ( j ) denote the jthordinary derivative of a function f on T.
a,
Definition 3.1.3 If 0 < a1 5 a2 5 as 5 . . . is a sequence of real numbers then we say that the sequence is logarithmically convex if {logaj) is a convex function of j, that is if whenever l! < m < n then logam
< nn -- me
log ae
+ mO Z- e- i ?log a,.
In some sense, a logarithmically convex sequence is more convex than an ordinary convex sequence. For example, the sequence {j2}is convex but not logarithmically convex. Logarithmic convexity is an important concept in analysis; it arises in the three lines theorem, in interpolation of linear operators, and in calculating domains of convergence of the power series for (real and complex) analytic functions of several variables.
70
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
Definition 3.1.4 Let M I , Mz ,... be a monotone increasing, positive, logarithmically convex sequence of real numbers. A Cmfunction f on T is said to belong to the class C ( { M j } )if there is an R > 0 such that ~ Mj R ~ . sup 1f ( j ) ( x ) < T
EXAMPLE 1 A. If Mj = j!, each j, then it is not diflcalt to see that { M i ) is increasing and logarithmically convex. The class C ( { M j ) ) consists exactly of the real analytic functions on T . B. If Mi = 1 for all j then, by Bernstein's lemma (Section 3.3), all trigonometric polynomials lie in C ( { M j ) ) -The converse is true as well. For it is a standard fact of Fourier analysis (see [KATI) that for p E Z one has 1
any 0 < rn E Z. Bat for the specified class of Mi this gives
I f lpl is large enoagh that the fiction in parentheses is less than 1 then letting m -+ oo yields that f^(p) = 0. In other words, f is a
trigonometric polpnomial. If M j = 22' then the class C ( { M j } )will contain functions that are not real analytic. Certainly the function
C.
will lie in C ( { M j ) )but it is not real analytic. In the material that follows we shall develop a method for manufacturing functions in a given C ( { M j ) ) . We begin with an important alternative definition of quasi-analytic class in terms of the L2 norm instead of the L" norm:
Definition 3.1 -5 Let Ml, M z , . .. be a monotone increasing, positive, logarithmically convex sequence of real numbers. A C"O function f on
3.1. QUASI-ANALYTIC AND GEVREY CLASSES
71
T is said to belong to the class C # ( { M j } ) if there is an R > O such that I I ~ ( ' ) (11xL)~ ( T )c - M ~~ . j
.
(**I
The two definitions of G ( { M j } ) and C # ( { M ; } ) give rise to essentially the same spaces of functions in the following sense: First, since T is a compact measure space we have that
11
f(j)llL2
5 C .sup 1 f G ) ( . T
It follows that G ( { M j ) ) C C # ( { M j } ) for any positive, monotone increasing, logarithmically convex sequence M j . For a near converse, notice that for j 2 0 and f E C m ( T )we have
by periodicity. Thus there is a point po E T such that f G+')(rn) = 0. Hence for any x E T we have
and by Holder's inequality the expression on the right is bounded by a constant times 11f (j+') llL2. Hence
In general, we cannot place an a priori bound on M j + l / M j , so the two spaces are not exactly the same. Definition 3.1.6 A C m function f on T is said to vanish to infinite order at a point p E T if f ( f ) ( p ) = 0 for all j = 0,1,2,. . . . Definition 3.1 -7 A set or class of Cmfunctions S is called quasianalgtic if whenever a function f E S vanishes to infinite order at a point p E T then f r 0.
Obviously the class of real analytic functions is a quasi-analytic class (hence the name). The main result of this section will be the Denjoy-Carleman theorem, which gives a complete characterization of quasi-analytic classes of the form C # ( { M j } ) . To this end we introduce a final piece of notation:
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
72
Definition 3.1.8 If { M j} is a positive, monotone increasing, logarithmically convex sequence of numbers then we create from it a function on {R : R > 0) by TM, (R)
= T(R)
inf Mi- R-j. j>O
Following Katznelson [KAT], we refer to r as the associated function for the sequence {Mj}.
THEOREM 3.1.9 (Denjoy-Carleman) Let {Mj) be a positive, monotone increasing, logarithmically convex sequence of real numbers. The following three statements are equivalent: (i)
C*({M~}) is a quasi-analytic class.
(ii)
slm! dr I+r
= -00.
We prove the Denjoy-Carleman theorem in three steps. Fix once and for all a positive, monotone increasing, logarithmically convex sequence {Mj) of real numbers. Step I : Proof that (b) + ( a ) . Assume property (b) and let f E c#({M~}). To test for quasi-analyticity, we take (without loss of generality) p = 0 and assume that f (j)(O) = 0 for all 3. We shall prove that f z 0. Define the Fourier-Laplace transform
where z is a complex variable unequal to zero. We integrate by parts, using our hypotheses about f to eliminate the boundary term, to obt ain
Integrating by parts j - 1 more times yields
3.1. QUASI-ANALYTIC AND GEVREY CLASSES
Restricting attention to {z E C : Re@) 2 01, we have that
hence, by Holder's inequality,
9
Letting play the role of R in the definition of the associated function T, and taking the infimum in this inequality over all j, allows us to conclude that
l11,(z)l 5 T ( % ) or, equivalently, log l+(z)l
5 log[.r(%)l.
In conclusion,
Using the fact that T(-) is a non-increasing function and that ~ ( s E ) Mo, for 0 < s 1, we can see that the statement (ii) implies
<
This estimate provides a contradiction for the following classical reason: Observe that the function $ is holomorphic on the right half plane and continuous on the closed half plane. Moreover 11, is bounded for z large by the estimate (t) and for z small by inspection. Thus r/t is in the function space Hm of G. H. Hardy. The classical inequality of Jensen for the location zeros of such a function (see [KAT, p. 1141 or [KRAl] or [HOF]) then yields that
That is the required contradiction.
74
CHAPTER 3. SOME QUES~lONSOF HARD ANALYSIS
Step 2: Proof that (a) (c). We begin this portion of the proof with an interesting construction that provides examples of Cm functions in many of the classes C# ( { M j ) ) .
Lemma 3.1.10 Let {pt)& be positive numbers that sum to a number not exceeding 1 . Define
and set
Then f f 0 is supported on [-I, 11 (mod 27r), is infinitely differentiable, and satisfies the estimates
Proof of Lemma: Notice, using Taylor's formula, that
Thus
certainly converges and therefore the infinite product defining u(k) converges. Moreover, u(k) tends to zero faster than any negative power of k (look at the denominators in the infinite product) so that the series defining f converges uniformly and absolutely. For the same reason, the series may be differentiated term by term so that f is infinitely differentiable. Finally, f has a non-trivial Fourier series hence f is not identically 0 (see [KAT]). We do the final analysis on f by examining the partial products of the coefficients u(k). By direct calculation, the sequence
3.1. QUASI-ANALWIC AND GEVREY CLASSES
consists precisely of the Fourier coefficients of the function
forl=O,1,2 ,.... Set
and define
Then the formula (see [KAT])
yields that
fN(t) = ro* rl * ' . . * rN(t).
Since the support of lies in [-pe, pel, it follows that the support of f~ lies in [pel (mod 274. Thus, since f N --+ f uniformly, pe, the support of f lies in [- 1, I] (mod 2a). Finally, we use Plancherel's theorem and the fact that (f ( j ) )^ ( k ) = ( Z I Ef(k) ) ~ (see [KAT]) to see that
We observe that
Putting together the last two displayed lines ~ i e l d sthat
76
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
This completes the proof of the lemma.
As the reader can easily see, the lemma may be applied to the situation at hand by setting that this yields
= (MI)-' and pi = M j - l / M j . Notice 1
automatically. The condition that 5 1 may be arranged by scaling, as will be noted below in the proof of ( a ) + (c). We will prove the contrapositive of the statement ( a ) + (c). Suppose that Mj/Mj+l < 00- By replacing M j by M; = ~j ~j for R small we certainly shall not change the class C# but we may arrange that C Mj/Mj+' < 112. We define
Then C p j 5 1 and
The lemma then provides us with a non-zero function f that is in C # ( { M ~ that ) ) vanishes outside [-I, 11 modulo 27r. Thus the class C # ( { M j ) ) cannot be quasi-analytic. rn
Remark 2 The construction above demonstrates that if a class c#({M,-)) is not quasi-analytic then it contains non-zero C m functions of arbitrarily small support. This is a much stronger assertion than the definztion of quasi-analytic class suggests. Step 3: Proof that ( c ) + (21). Thus far we have not used the logarithmic convexity of the sequence { M i } but nowr this property will prove to be
3.1. QUASIANALYTIC AND GEVREY CLASSG'S
I I
important. We may as well assume that the sequence { M i ) increases faster than ~j for every R > 0 otherwise the class C ( { M j ) ) is RO different from the class defined with Mj = 1 for all j and that class consists only of the trigonometric polynomials. With this assumption about the growth of the Mi,we see that the infimum in the definition of the associated function T is actually attained. Thus T ( R )= min M~R - j . 2 0
Define p1 = M;' and pj = Mj-'/Mj for j = 2,3,. . .. Then the sequence { p i } is monotone increasing; for this assertion is equivalent
ni
which is true by logarithmic convexity. Clearly M~~ - =j (p&)- . As a result, we will minimize this expression by selecting j to be the last term ( p eR)-' that is smaller than 1. In other words,
Let us define
M ( R ) = the number of elements pl such that ptR 2 e. Here e is Euler's number. Then
We conclude that, for k = 2 , 3 , . . . , we have
On the other hand notice that the number of pc between e2-k is M ( e k ) - M ( e k - I ) . Hence we have
and
78
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
Putting together the last two displayed lines we get
Summing over k = 2,3, . . . yields
But this just says that (c) =+-(b). The three implications (b) + (a),(a) + (c), ( c ) =+ (b) complete the proof of the Denjoy-Carleman theorem. A somewhat different treatment of the Denjoy-Carleman theorem - one that uses no complex analysis or Fourier analysis but is quite technical and difficult - appears in [HOR2, v. 1, p. 231. We now say a few words about another collection of spaces known as the Gewrey classes. Following [HOR2], we define these as follows. Let Lo,L1,. .. be a sequence of positive numbers with the property that, for every k, k 5 Lk _< C *Lk+ (0 Thus the sequence grows at least arithmetically and at most exponentially. We say that a function f E Cm(T) belongs to the Gevrey class G({Lj}) if there is a constant, C such that for every j it holds that
It is easy to see that, this is just a variant of the definition of the class C({Mj)). Some modern treatments of the material in the present section often formulate the Denjoy-Carleson theorem in the language of the G({Lj }) rather than the c#({Mj)) as we did in Theorem 9. A Gevrey class G ({Lj )) is quasi-analytic if and only if C 1/ L j diverges. Each Gevrey class is closed under differentiation (exercise) and is preserved under real analytic mappings. Gevrey classes are in some ways more attractive than quasi-analytic classes because they are localizable. That is because the growth rate of the derivatives of a typical cutoff function is swamped by the right hand side of the inequality ($). One might hope to prove real analytic regularity theorems for a partial differential operator L by first proving an estimate in each
3.1. QUASI-ANALYTIC AND GEVREY CLASSES
79
Gevrey class and then amalgamating all this simultaneous information. The essential tool in such an approach is the following theorem of T. Bang [BANG]: THEOREM 3.1.11 The intersection of all the non-quasi-analytic Gevrey classes consists precisely of the real analytic functions.
Curiously, the intersection of all the Gevrey classes does not give the quasi analytic functions or the real analytic functions as one might expect. Since these matters are all quite technical, we refer the interested reader to [BANG] or to [HOR2]. Just to give the interested reader the flavor of the types of questions one might ask about the classes of functions being discussed here, we briefly describe some work of Walter Rudin [RUD]. Recall that in classical analysis it is of interest to determine under what algebraic operations a class of functions is closed. Consider the operation of taking the reciprocal of a non-vanishing function f. It is easy to see that if f is Cm then so is l/ f . A slightly trickier proof shows that if f is real analytic then so is l/ f . Recall the function classes C ( { M j)) defined at the beginning of this section. When is such a function class closed under reciprocals? In order to answer this question, we need two new definitions: Definition 3.1.12 If { M i } is a positive, monotone increasing, logarit hmically convex sequence of numbers, we define
We will call { A j ) the sequence associated with the sequence { M j ) . Definition 3.1.13 Let AI ,A2, . . . be a sequence of real numbers. The sequence is said to be almost increasing if there is a number K > 0 such that b' l < s L j . As 5 K A j 9
Then we have THEOREM 3.1.14 (Rudin) Let {Mi)be an increasing, logarithmically convex sequence of positive real numbers. If the associated sequence { A j ) is almost increasing then C ( { M j } )is closed under the taking of reciprocals.
80
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
THEOREM 3.1.15 (Hormander) If a class C({Mi)) is closed under the taking of reciprocals then the associated sequence {Aj) is almost increasing. We refer the reader to [HOR2], [RUD], and [BOMl], and to references therein, for more on the lore of Gevrey and quasi-analytic classes.
3.2
Puiseux Series
A Puiseux series is a formal power series
where N is an integer and k is a positive integer. For each k, the set of such formal power series is seen to form a field. The union of all such fields is sometimes denoted by K{x), where K is the field which contains the coefficients a j . Puiseux's Theorem, in this context, is the following
THEOREM 3.2.1 (Puiseux's Theorem) If K zs of characteristic zem and algebraically closed, then K{x ) is algebraically closed. Our interest is in convergent power series over the reals, so the preceding theorem is not the one we want to prove. We describe the situation of interest to analysts: Let A(x) and B(x) be real analytic functions near 0. Their quotient A(x)/B(x) can be written as xNc(x), where C(x) is also real analytic with C(0) # 0, and N is an integer (possibly negative); this can be done as long as B does not vanish identically. The family of functions of the form xNc(x) defined near, but not necessarily at, 0 thus forms a field. We consider a polynomial equation over that field:
It is no loss of generality to assume that A. = 1. By replacing y with x- biy r , one may assume that No 5 Ni, for e' = 1,2, . .. ,n , and then one may divide through by xN0: in the equation that remains
all the coefficients are real aaalytic. Thus it will suffice to consider a polynomial equation of the form
where each Bi(x) is real analytic near 0. We will show that there is a positive integer k such that, for t near 0,
where each of Rl ,R2,...,R, is real analytic, G (t,y) is a polynomial in y whose coefficients are real analytic in and, for small non-zero real [, G(c, y) is irreducible over the reals. This decomposition of P allows us to understand the solutions of P(x, y ) = 0 near x = 0. The main tool for our investigation is an algebraic result known as Hensel's Lemma. We consider the polynomial P(x, y) as above. The simplest situation to study is that where the coefficients Bi(x) are all polynomials. First we prove a weak form of Hensel's Lemma:
c,
Lemma 3.2.2 Let P ( x ,y ) be a polynomial in y of the forna
where each Bi is a real polynomial in x . Assume that P(0, y ) factors into relatively prime real factors of degrees p and q, wzth p q = n, so
+
with go and ho real polynomials without common factors. Then P ( x ,y) factors into G ( x ,y) and H(x, y ) of the same degrees in y with coeflcients which are polynomials in x and for which G(0,y ) = go ( y ) , H(O, Y ) = b ( y ) .
Proof: We rearrange P(x, y ) by powers of x , so that
We plan on writing
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
82
The polynomials gl (y), gz(y), . .. ,g, (y) are to be of degree at most p - 1 in y, while the polynomials hl(y), h2(y), .. . ,h,(y) are to be of degree at most q - 1 in y. Multiplying together the above expressions for G and H and equating like powers of x, we see that the following equations must be satisfied:
The first equation is satisfied by hypothesis. Arguing inductively, we suppose that gl,gz, . . . ,gc- 1 and h 1, h2, .. . ,hl-l have been chosen so that the first l equations are satisfied. The equation which must be and he can be written satisfied by
with C a polynomial of degree at most n - 1. We know that, since go and ho are relatively prime, we can find ge and he of degree at most p - 1 and q - 1, respectively, which satisfy this equation. We use the weak form of Hensel's Lemma to prove the following
Lemma 3.2.3 Let x = (xl,xz,. .. ,x,) and
Suppose that po(y) = P(bl, bZ,. . . ,b,, y) factors into relatively prime real factors of degree p and q, with p q = n, so
+
mathgo(y) and ho(y) real polynomials without common factors. Then there are m i analytic functions
and Dl (x), D2(x), . . . .D, (2).
defined near x = (bl, bz7.. .,b,) such that
satisfy P(x, Y) = G(x7 Y)H(x,Y), and
G(bi7-
b,, y) = go(y),
H(bl7
-
bn, Y) = ho(Y)-
Proof: Let us write
The plan is to show that the function mapping
to the n-tuple consisting of the coefficients of yn-',
yn-27
. . .,y, 1 in
is invertible in a neighborhood of (el, c2,. .. ,cp7dl, dz, . .. ,dq). Fix a specific (x 1, 2 2 , . ..,x, ). Apply the weak version of Hense17s Lemma (above) to y n + (b1 +xit)yn-'
+. - - + (b,-l
+ ~ , - ~ t ) y +(b, +z,t)
thought of as a polynomial in t and y. We thus obtain certain polynomials K(t, y) and L(t, y) with K(0,y) = go(y), L(0,y) = ~ o ( Y )and ,
We now write
where rl, . . . ,rp7sl, . . . ,s, are polynomials in t. It is clear by considering the terms of degree less than two in t that where ~ 1 ,. .. ,en are polynomials in t. This shows that the differential is non-singular , so the result follows from the Inverse Function Theorem.
84
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
THEOREM 3.2.4 (Hensel's Lemma) Let P ( x ,y ) be a polynomial in y of the form
where each Bi i s real analytic in x . Assume that P(0,y ) factors into relatively prime real factors of degree p and q, with p + q = n, so
with go and ho real polynomials without common factors. Then P ( x ,y ) factors into G(x,y ) and H ( x , y ) of the same degrees in y with coefFcients which are real analytic in x and for which
Pmof: We let Cl ( x ),Cz( x ) , . - . ,C, ( x ) and Dl ( x ),Dz ( x ), .. . ,D, ( x ) be the functions defined in the previous lemma. Let B(x) map x to the n-tuple ( B l ( x ) ., . . ,B,(x)). Then we may set
With the aid of Hensel's Lemma, we can give a proof of the decomposition described in the beginning of this section. While it is not short, we feel that our proof is more explicit and convincing than the other proofs in the literature.
THEOREM 3.2.5 (Decomposition) Let P(x,y ) be a polynomial in y of the form
where each Bi is a real analytic function of x . Then there is a positive integer k such that P can be written in the fonn
Hem. e w h of R', R2,. . . ,R, is real analytic, G(c,y ) b a polynomial in y whose coeficients are real analytic in t , and, for small non-zero real <, G ( t ,9) is irreducible over the r e ~ l s .
Proof: We will argue by induction on the degree n of the polynomial. Obviously, the result holds if n = 1. Let us now consider n > 1 and assume the result holds for each polynomial, with real analytic coefficients, which is of degree less than n in y. Set g = y' - lnB l (x) and subsitiute in P(x, y) to obtain a new polynomial
If I?;, B$, . . . ,BA all vanish identically, then P(x, y ) = [y + 1B~(x)ln, n and P has been put into the desired form. So now assume that not all the Bi vanish identically. For each i for which B: does not vanish identically, let x P i be the smallest power of x occurring in B:. Let a be the smallest of the numbers p i / i , and write o = l / ~in, lowest terms, with l and rc positive. Set x = (xI' )6 , y' = (x11 )L yI' and substitute in P' (x, y') to obtain the new polynomial
=
11 nt?
(X
)
((Y rr )n
+ (yff)"-2 (X")-~'B~ + ...+ (x")-"'
I?:) ,
where the argument of each Bj is x". Since we have ri - i t 2 0 for all i and ri* - i* -ke = 0 for some i,, we see that
is a polynomial in y" with coefficients that are real analytic in xu. Consider the roots of P V ( O , y"). Since at least the coefficient of (yf' )n-i* does not vanish when x" = 0, we see that P"(0, y") cannot have a root of multiplicity n. We can decompose P"(0,yf'), over the reals, into two factors which have all the real roots and which have no root in common and a third factor which has all the complex roots. In other words, we write
where both f l and f 2 have degree less than n and g is irreducible over the reals. Of course, we may assume that f l , fa, and g are monk. It is possible that f or fi may be the constant polynomial. Indeed, it may seem possible that both fl and f2 are constant. That would imply that PU(O.y") has only complex roots. Because the
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
86
roots of a polynomial depend continuously on the coefficients, it would follow that PU(x",y") also has only complex roots, for all sufficiently small x" . It would then follow that
with the right-hand-side a polynomial in x" and y, which for all sufficiently small non-zero x" has only complex roots. That is, we would then be done in this case. We may assume now that fl, f2, and g are all of degree strictly less than n. By Hensel's Lemma, we can write
P"(xu,yI' ) = Fl (x" ,yl') F2 (XI' ,yI1)Go (XI',yl'), with f i (y")
= 4 (0, y"),
f2(yU)= F2(0, y"),
g(y") = G(~,gl")*
Again we argue by continuity that, since the roots of g(yl') are all complex, then for all sufficiently small non-zero real x" it holds that G(xM,y") is also irreducible over the reals. Since the degrees of Fi and F2are both less than a,we have by induction that there are positive integers kl and k2 such that 4 ( ( t ) " ,y") = (Y" - R1,,)(G)(y" - Ri.2({1))
(y" - Ri,,, (&))GI(&, y"),
We set k = K lcm{kl, k2), and let a and b be such that lcm{kl, k2) = akl = bk2. Then with & = = tb,and r" = ( 1 c m ~ k ~we ~ kfind 2~, that P has been expressed in the desired form.
ca, c2
Note that the reduciblity of P(0, y) and P1'(O, y") may differ. A simple example to illustrate this is P(x, y) = +x2, which is reducible to linear factors when x = 0. But the construction in the proof leads to P"(xU,y") = (y")2 + 1, which is irreducible over R.
3.2. PUISEUX S E M .
87
We are now in position to state a form of Puiseux's Theorem. Let us denote by P the family of functions f (c) which are defined on some open interval (0, a ) , a > 0, and can be written in the form
for some integer N , some positive integer k, and some function g which is real analytic on an interval containing (-(a)* ,( a )f ). It is clear that P forms a field under the usual arithmetic operations.
THEOREM 3.2.6 (Puiseux's Theorem) If f (0is a continuous function, defined for suficiently small positive t, for which y = f (0 satisfies a polynomial equation
with coeficients A. (t), . . . ,A, (c) i n P, then the restriction of f (t) to some interual (0, a ) , a > 0,is i n P. This theorem followrs easily from the previous results. In practice we can proceed rather directly. First, we avoid the situation of an identically vanishing discriminant: This can be done by differentiation and finding common factors. It is an extension of the usual development of the resultant of two polynomials that the corn mon factor in two polynomials can be found by using linear algebra. After such a reduction of the problem is done, we can then change variables so that we are considering a polynomial equation
with 0 = Dl < Dz < .. . < Pt = n, in which the functions Bi (<) are real analytic with Bi(0) # 0, and a**= 0, for some 2,. We suppose we have (0, yo) # 0, then the Implicit in hand a real root yo of P(o, y). If Function Theorem can be used to conclude that there is a r e d andytic function y(x) with ~ ( 0 = ) yo which satisfies P(x, y(x)) = 0. In case aP (0, yo) = 0, we perform another change of variables as follows: Recall the construction of the classical Newton Polygon ([NEW] or [COO]). Consider the set of points S = {(ai,fi) : i = 1,2, -..,t) in t.he a. &plane. There must be at least one point of S on the postive
88
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
@axis. Choose il SO that (ail,pi,) is the point of S on the positive paxis nearest to the origin. Consider the ray from (ail, Pi, ) in the negative p direction. Rotate this ray in the positive direction until it first intersects S in one or more other points. Let (ai,,pi,) be the point on this ray most distant from (ai,, Pi,). Consider then the ray from (ai2,Pi,) in the same direction as the ray from (ail, to (ai2,Pi,). Rotate that ray in the positive direction until it intersects S in one or more other points, the most distant being (ai3,Pi,). Continue in this fashion until reaching a point (a,, P,) of S which lies on the a-axis. The polygon consisting of the line segments from (aij, Pi,) to (ai,,l,Pi,+l), for j = 1,2,. ..,u - I is defined to be the Newton Polygon. Choose one of the segments which makes up the Newton Polygon. Then we can find positive integers p and a such that
a,)
where v 2 2 and Pjl < pj2 < . . . < p&,. We may, of course, suppose that p and a are relatively prime. Replace t and y by = (c)P and y = (t')Oy' and divide through by (<')pajl+oPjl to form the new polynomial P'(t', y'). One then seeks a real root of P1(O,y'). The point of Puiseux's Theorem is that after only finitely many iterations of this construction either the Implicit Function Theorem may be applied or the polynomial is irreducible over the re&.
<
One application of Puiseux's Theorem is to obtain information about the smoothness of solutions to polynomial equations with real analytic coefficients (or with coefficients in P , which is really no more general). Among the possible results that one might exhibit as typical, we have chosen the following:
T H E O R E M 3.2.7 Let I and J be open intervals. Suppose f (x) is a continuous function on I such that (i) f(x) E J for x
E I,
(ii) P(x, f (2)) = 0 for x E I , (iii) for each x E I there exists a unique y E J with P ( x , y) = 0. Here P(x; y) is a polynomial in y with coeficents which are real analytic functaons of x. If f E @', then for each xo E I , there exists a > 0 such that f E in a neighborhood of xo. C p + ' p a
Proof: Consider an arbitrary xo E I. By the decomposition theorem, we know that there are integers N and k, a positive 6, and a real analytic function g such that
for sufficiently small I . Moreover the right-hand-side of (*) always satisfies the polynomial equation. Let the powen of t occuring with non-zero coefficient in the series for g be d l < d2 < ... . Suppose k is even. We claim that every di is even. If that were not the case then, for sufficiently small t , there would be two solutions of the polynomial equation which lie in J. Thus we can remove the common factor of 2 from k and from all the d i . It follows that k may be assumed to be odd. Suppose that k is odd. For t N g ( c ' l k )to be CP?' we must have N d l / k 2 0 and either that k divides every di or, if di* is the least di not divisible by k , then N + di*/k > p + 1. In the first case, f is real analytic and, in the second case, f is C~+'Y" with the number (~=N+d~*/k-p-l.
+
No such result is true for polynomials having coefficients which are real analytic functions of, or even polynomials in, more than one variable. An example (for which we are grateful to E. Bierstone) is
There is a unique function f ( X I ,x2) such that
and f is Co.' but is not c'. The next consequence of Puiseux's theorem follows readily and illustrates the principle that a Cm submanifold of an analytic variety is in fact analytic.
T H E O R E M 3.2.8 Let P(x, y) be a polynomial in 9 with coeficients which are real analytic at xo. I f f E COo is such that P(x, f (x)) = 0, then f is real analytic at xo. There is no exact substitute for Puiseux's theorem for functions of more than two variables. On the other hand, Section 4 of [BIM3] gives
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS a version of Puiseux's theorem in several variables, and, in some sense, Hironaka's resolution of singularities theorem (Section 5.1) provides some of the same kind of information in every dimension. We also point out that M. Artin's theorem on solutions of analytic equations [ART] can in some circumstances serve as a substitute for Puiseux's theorem, in particular, in generalizing the preceding theorem to the multivariable setting. We conclude by noting that our approach in this section is quite similar to the proof of Puiseux's theorem that appears in [BIM3], but the two proofs were developed independently.
3.3
Separate Real Analyticity
It is well known that a function of several real variables that is C" in each variable separately does not necessarily enjoy any (joint) smoothness as a function of several variables. A simple counterexample is the function In general, a function that is separately Cm can be expected to be no better than measurable (see [KRA2]). By sharp contrast, a function of several complex variables that is holomorphic (in the classical one variable sense) in each variable separately is, by a deep result of Hartogs, Cm, indeed real analytic, as a function of several variables. It also turns out to be holomorphic as a function of several complex variables by any other standard definition. These matters are discussed in detail in [KRAl]. Thus it seems natural to discuss functions of several real variables that are real analytic in each variable separately. The function f (x, y) exhibited above shows that in the absence of additional hypotheses we cannot expect a separately real analytic function to be even continuous as a function of several real variables. On the other hand, it is an astonishing fact that there exist CCOfunctions (as a function of two variables) on IR2 that are separately analytic but not jointly analytic. This assertion (from [BRO]) can be proved using category-theoretic consideiations. As early as 1912 S. Bernstein [BER] showed that in the presence of some mild ambient hypothesis (such as continuity, or local boundedness) a separately real analytic function is jointly real analytic.
3.3. SEPARATE REAL ANALYTICITY On the positive side, one can also use category theory to prove that a separately real analytic function is in fact real analytic ( a , a function of several real variables) on a dense open set. Recently, Siciak [SIC2]has completely characterized the singular sets that can arise for separately real analytic functions. However, thanks to a theorem of F. Browder [BRO] and P. Lelong [LEL] (the result of Lelong is more general and both results are subsumed by the later work in [SICl]) separate real analyticity turns out to have much in common with separate complex analyticity. But some ambient, or Tauberian, hypothesis is required to obtain a full positive result. It is this matter that we shall treat in the present section. Siciak's proof of the theorem discussed here uses complex methods (just as a real analytic function of one real variable is locally the restriction to the real line of a complex analytic function, so a real analytic function of several real variables is locally the restriction to RN of a complex analytic function of several variables). Browder's earlier proof of the same result treats the real analytic functions directly: the proof consists in estimating the size of the coefficients of the Taylor expansion. This methodology is much more in the spirit of the present book than is Siciak's. And while Siciak's proof is softer than Browder's, it is considerably longer. We present the proof that appears in [BRO].
Definition 3.3.1 Let f be a function on an open subset U of RN. We say that f is separately analytic if for each j = 1,.. .,N and each collection of N - 1 real values G = (cl, ~ 2 .., . ,cj-1, C j + l , . . . ,CN) such that
is not empty the function
is real analytic as a function of one real variable.
Definition 3.3.2 A function f on an open subset U C RN is called jointly (real) analytic if it is real analytic as a function of several variables in the sense that has been discussed in this book. NOWwe state Browder's theorem. For clarity we treat functions of two real variables only. The proof transfers directly to the N-dimensional case.
92
CHAPTER 3. SOME Q
U
E
S OF ~ HARD ~ ANALYSIS
T H E O R E M 3.3.3 Let I = ( - 1 , l ) be the "unit anterval" in the reul line. Let f ( x ,y ) be a function on I x I having the property that f (., y ) E CW(I)for each fixed y E Iand f ( x , -) E Coo(I)for each fixed x E I . Suppose that there is a positive constant Co > 0 with the property that for every k = 0 , 1 , 2 , . . . we have
for every x E I , y E I and
for every x E I , y E I . Then f is a (jointly) real analytic function of two variables on I x I . Notice that the hypothesis of the theorem is not simply that f is real analytic in each variables separately but that there is some uniformity of the analyticity in the x variable when the y variable is thought of as a parameter (and vice-versa). It is instructive to note that similar results hold in the Cm category: separately Cm functions need not be smooth. But if there is some uniformity of estimates on the derivatives then joint smoothness follows. A discussion of these matters in the Cm category appears in [ K R A 2 ] . Our proof of the theorem is broken up into several lemmas, some of which have independent interest.
Lemma 3.3.4 A function satisfying the hypotheses of the theorem is (jointly) Cm on I x I . This result is of sufficient interest that we sketch two proofs.
Proof 1: By a result in [ K U R ] ,the function f is measurable since it is separately continuous. Inequality (*) shows that f and its pure derivatives are bounded. They are of course measurable since f itself is. Hence f E LC".Thus it is easy to see that the derivatives
3.3. SEPARATE REAL ANALYTICITY
93
calculated as classical derivatives of a function, coincide with the derivatives when interpreted as distributions (this is just an exercise with integration by parts and the definition of distribution derivative). Thus for any integer r 2 0 it holds that
is bounded. Standard regularity theory for elliptic partial differential operators (of which L is an example - see [BJS]) implies that any mixed partial derivative of f , in the sense of distributions, satisfies
a" an axm axnf
--
E L?~,.
The Sobolev imbedding theorem (see [STE]) then yields that, after correction on a set of measure zero, f is infinitely differentiable. But f is already infinitely differentiable in each variable separately as p r e sented. So no correction at any point is either necessary or possible. We conclude that f is a C" function. Proof 2:
As in the first proof, f is bounded and measurable. Let 4(x, y) be a Cm function of compact support in I x I that is identically equal to 1 in a neighborhood of the origin. We will prove that g E 4 - f is a Cm function. Now the hypotheses of the theorem, together with the product rule, yield that -
dk G ~ ( ~Y),
and
ak ay ~
T
Y) ( ~
~
are bounded functions on lit2 with compact support. In particular, each of these derivatives is an L2 function. Let 3(<,I)) denote the usual two variable Fourier transform of g. Then a standard result of Fourier analysis ~ i e l d s(see [KAT]) that
are L~ functions for every k . But then it is elementary to see that for any non-negative integers m and n it holds that
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
94
is L2. But this implies (again see [KAT]) that the distribution derivat ive
6" 6" -s(x9 Y)
axm axn is an L2 function. Now the Sobolev imbedding theorem yields, as before, that f is in fact C* smooth as a function of two variables. In what follows it will be convenient to use the notation
D:
to stand for
D:
to stand for
and
ak ax"
ak ay"
Lemma 3.3.5 i n order to prove the theorem it sufices to establish the existence of a constant C1 such that, for all j,k 2 0,the inequality
holds. Proofi To see that
The inequality series
(t) implies the theorem, we define
(t) implies that
this series may be majorized by the
This last is the power series expansion of the function
which converges for x,y sufficiently small. Thus fi(x,y) is a real analytic function in a small neighborhood of the origin. What is more, for every j , k , 3 ~k (D, f 0)= (oj,~ifi)(O,0).
,
For x near the origin, the functions f (x,0) and fi(x,O)are real analytic functions of x with matching derivatives at the origin. Thus
3.3. SEPARATE REAL ANALYTICITY
95
f (x, 0) = h (x, 0) for x small. Since (t) implies that D: f (x,0) is a real analytic function of x for x small, a similar argument shows that D: f (x, 0) = D,*fi(x, 0) for x sufficiently small. But this last enables us to argue, for small x, that f (2, -) and fi(x, are both analytic near 0 and have the same derivatives of all orders at y = 0. Thus f (x, y) = fi(x, y) for x, y small. So the original function f (x, y) is real analytic in a neighborhood of the origin. Of course we could apply a similar analysis to any other point of I x I. Thus the new condition (t) implies Theorem 3. That is what we wish to prove. H 0 )
We henceforth concentrate our efforts on proving (t). Define the function 0 if ( s ( > l if Is1 < 112 2 - 214 if 112 < Is1 < 1. and set C(x, 3) = P(X) P(Y) and
Crb, Y) = (C(x9 Y)) r+2
, r = 1,2, . . - -
Then Cr (x, y ) is an (r+1)-times continuously differentiable function with support in the closed unit square. Combining the spirit of the two proofs of the first lemma, we define a partial differential operator by A2, = D": D:" 1.
+
+
Using a little Fourier analysis, we can construct a solution operator for $2 as follows. For rn > 1we define em. (r,y) =
JR JR e'(xc+")(/t12m + lqlzm+ l)-'dEd~-
By the choice of m, the integral converges uniformly on W x W. If #(I, y) is a c2"function of compact support then
Notice that this last expression is the reciprocal of the symbol of e2,. If 4, $ are L1 functions and their convolution is defined to be
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
96
then (see [KAT])
(4 *
Y)Y(C> = 6(t, q) 4 ( t , q).
It follows that if v(x, y) is a
function with support in the unit
square then u(x,y) = e2,
* (d2,v)
for x E I x I .
($)
+
Now let j , k be two non-negative integers such that j k < 2m - 2. We may differentiate the expression defining e2, a total of j + k times under the integral sign to obtain
By the choice of j and k this integral converges absolutely so the Lebesgue dominated convergence theorem guarantees that the differentiation under the integral is justified. It follows fiom the last displayed equation that (D$Die2,) is continuous and bounded for j k < 2m - 2 with a bound KOindependent of j and k. Now differentiating the equation ($) under the integral sign a tot a1 of j k times, with j k still being less than 2m - 2, we have
+
+
+
Using our estimate on the derivatives of e2, we find that
(The factor of 4 comes from the area of I x I.)We will apply this last inequality to the function
where f is the function given in our theorem and (2, was constructed above. Taking (x, y) = (0,O) and recalling that t , is identically 1in a neighborhood of the origin, we obtain that
3.3. SEPARATE REAL ANALYTICITY
97
Now we must study the term on the right hand side of this inequality. Observe that
where the remainder term 72 involves derivatives of f that are of order strictly less than 2m :
(This is a standard fact about commutation of differential operators, or more generally of pseudodifferential operators. What we are saying here is that if P is an operator of order 2m and Q is an operator of order 0 then P(Qf) = Q ( Pf ) R, where R is of order less than 2m. The verification of this assertion is a simple exercise in calculus.) Of course the derivatives of C2,, which are all of order at least one, are supported only on the set where & , is not identically zero and not identically one. On that set, by design, C2m is a polynomial of degree (2m+ 2) in each variable. The following assertion is due to V. V. Markov [LOR, p. 40, ff.]:
+
Lemma 3.3.6 (Markov's lemma) Let J W be a compact interval. There b a constant M > 0, depending on J, with the following property: Let p(t) be an algebraic polynomial of degree k and let S = SUPtE~Ip(t)l. If j is a non-negative integer then sup lIlip(t)l 5 M~ . k 2 j S. tEJ
Proof Assume without loss of generality that J = [-I, 11. It is enough to prove the result for j = 1 and then apply induction. It is convenient to first prove an analogous result for trigonometric polynomials:
xg-Najeiit
There is a constant K > 0 such that if p(t) = is a trigonometric polynomial of degree N then
98
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
This trigonometric inequality of Bernstein is proved as follows: The kernel VN( t ) = 2 K 2 ~ + (1t ) - K N ( ~ ) , where 1
{
Kj(t)= 3 1
+
sin (&At sin (ft)
)
}
2
is the standard Fejer kernel of harmonic analysis (see [ K A T ] ,pp. 12-17 or [ Z Y G ] ) ,has the property that
Therefore
27r
d
VN(X - t ) d t .
It follows that
Straightforward estimates show that
completing the proof of the inequality for trigonometric polynomials. To obtain an inequality for the classical algebraic polynomial p ( t ) of degree k on the interval [- 1 , 1 ] , we apply the above result to q ( t ) = p(cos t ) . This yields
Finally, a classical lemma of Schur (see [LOR, p. 411) yields Markov's lemma. Since the proof of Schur's lemma uses ideas about Chebyshev polynomials that would take us far afield, we omit the proof.
Remark: The best known value for K in the inequality for trigonometric polynomials is K = 2. For the inequality for algebraic polynomials as stated ill the lenlnla (with .T = [-I, I]), M = 1 is best
3.3. SEPARATE REAL ANALYTICITY
99
possible. However, for our purposes, the best value for these constants is of no interest. rn We apply the lemma to differentiation of obtain that, when j > 0,
and I(~;&m)(x,Y)I
C2,
in x and in y to
< M ' j - (2m + 2)23.
[Of course these estimates may be obtained by direct computation from the explicit definition that we have given for (2,; but Markov's lemma gives a more natural way to see the estimates.] Now we estimate the error term 72. When the differential operator D m is applied to a product of functions w1w2 there results 22m terms of the form Dg w1D:w2 with coefficient 1 (note here that, for convenience, we are not gathering like terms). Thus the sum of the coefficients
in equation (***) does not exceed 22m.By the hypotheses of the Theorem and by estimates (1) and (2) we have (assuming, as we may, that the constant Co in the hypotheses of the Theorem exceeds 1) that
By similar, but simpler, reasoning one may obtain a like estimate on the term C2, A2, f. Combining these estimates, together with our formula for A2m(C, f ) and our estimate for ((D~D;f )(o, 0) 1, we find that, for 0 5 j k < 2972 - 2 we have
+
D
( 0 ,o
4~~ sup Id2m(Cmf)l 5 4Ko(sup 1721 sup 16, A2,f 1) 5 8Ko-(2-M-~1)2m-(2m+2)4"
+
<
(
z)~". K~( 2 ~ 4 ) (~2 ~" n ) ~ "1+ &
CHAPTER 3. SOME QUESTIONS OF HARD ANALYSIS
100
By Stirling's formula ([CKP] or [HEI]), we know that
for m large. Hence there exists an absolute constant L such that
hence
(2m)4m5 L2 e 4m (4m)! Also note that
As a result,
In case j + k is odd then we choose m so that j rewrite our estimate as
+ k = 2m
+
In the case that j k is even then we choose m so that j and imitate the last argument to obtain that
I(@D: (09 0)I 5 (e
I
j+k.
(j
-
3 and
+ k = 2m - 4
+ k)!
Thus for any choice of j, k we have proved the estimate (t) (introduced in Lemma 5), showing that f is real analytic in a neighborhood of the H origin. Our proof is thus complete. We remark in passing that a useful lemma of Ehrenpreiss [TAT,p. 3041 gives a method for constructing cutoff functions that behave like real analytic functions up to any prespecified finite order. By using these, one can give a quantitative version of Proof 2 of Lemma 3.3.4 and thereby present a new attack on the questions considered here. We conclude with some general remarks about the material discussed here. The paper [TAZ] gives a characterization of vector-valued
3.3. SEPARATE REAL ANALYTICITY
101
real analytic functions that may be considered an obverse of the main theorem of this section: directions in the target space are treated instead of directions in the domain. The paper [BIM3] considers functions that are real analytic along every real analytic arc. In some sense, such functions are more natural than those that are only "separately analytic." They enjoy a number of pleasant properties. [However in PMP] the authors exhibit such a function which is not even continuous!] The paper [BOM2] of Boman proves that the analogous class of functions, with "real analytic" replaced by Cm, is just the Cm functions t hernselves.
Chapter 4
Results Motivated by Partial Differential Equations 4.1
Division of Distributions I
The Cauchy-Kowalewsky Theorem is perhaps the most general result in the theory of partial differential equations. The theory needed to state and prove that theorem is entirely elementary. While the specific constant coefficient partial differential equations of mat hematical physics - Poisson's equation, the heat equation, and the wave equation - can be dealt with by specific elementary methods, the development of the general theory of linear partial differential operators with constant coefficients is tied to the more advanced and abstract theory of distributions introduced by Laurent Schwartz (see [SCHI]). One important conjecture in the theory of distributions (see [SCH2]) concerned the problem of finding a distribution S that solves the equation
for a given distribution T and a given testing function 9.(One can think of this as dividing T by 9.) This question arises from investigations, using techniques of Fourier analysis, of the solvability of a
104
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
partial differential equation
with constant coefficients. (Here P is a polynomial in several variables.) If the Fourier transform is applied to this equation it gives rise to an equation of the form (**). Solving the equation (**) for S is equivalent to solving for the Fourier transform .Ci of u. To prove that it is possible to divide the distribution T by the function it suffices to have control on the rate of vanishing of a. Lojaciewicz [LOJl] proved the requisite estimate for a real analytic O. In case @ is a polynomial, which is the situation relevant to partial differential equations, an easier proof was found by Hormander [HOR3]. While Lojaciewicz's work has broad significance for the geometry of real analytic varieties, it is much less accessible than Hormander's. In this section we will prove Hormander's weaker version of Lojaciewicz's theorem, and in the next chapter we will present a more expository treatment of Lojaciewicz's results.
THEOREM 4.1 -1 Let Q(xl, x2, ...,x,) be a real polynomial. Let K be a compact set. Suppose the zero set, N, of Q defined by
c2,
is aon-empty, and let d(& N) denote the distance from [ = (tl, . ., ) to N . Then there exist positive constants c and p such that
By compactness, to prove the theorem it will suffice to prove that there exist positive constants c and p such that
holds for 5 1. To show this is true we will need to state the problem in a very precise fashion. What is needed is an understanding of the set of pairs (x, y) such that x > 0 and y is minimal subject to the
4.1. DIVISION OF DISTR,.lB UTIONS I
following conditions:
Now, if we let C ( x ,y) be the preceding set of conditions, then we see that we need to understand the structure of
( ( 2 ,y ) : C ( x , y ) & 'v'w
[(o < w
& C ( X , W )*)Y 5
~1)-
This is a fairly complicated problem, but it involves only polynomial functions of the variables, equalities and inequalities, and logical connectives. We will see that all the variables other than x and y can be eliminated from the definition of the preceding set; what remains will be only polynomial functions of x and y, equalities and inequalities, and logical connectives. Once that is shown, the theorem will follow easily. It is of historical interest to note that Hormander based his proof of the above theorem on Seidenberg's proof of one of A. Tarski's theorems in Mathematical Logic: The decidability of the theory of real-closed fields. Since it would take us very far afield, we shall not discuss Tarski's theorem. The interested reader should see Tarski's monograph [TAR], Seidenberg's paper, or the exposition in Chapter 5 of [JAC]. Since we have before us a narrower goal, we will take a more direct route than Seidenberg's proof of Tarski's Theorem. Put simply, we need to underst and the structure of sets of the form
where P is a polynomial in y and z . First we sketch the procedure in a case with minimal complications: Suppose that P ( Y ,2 ) = a ( y ) z + b ( y ) . Clearly, if SI = { y : a ( y ) 2 b(y)2 = 01,
+
then S1 c S.
106
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
Now, let CI be any connected component of the complement of Sl. Set 492 = { Y : [ a ( y ) a + ~ ( Y ) I [ ~ ( Y +)b(y)l P = 0). Finally, let C2 be any connected component of the complement of Slu S2.Which is the same as letting C2 be a connected component of the complement of {y : [a( y) b ( y )2] [a(y ) a b(y)][a( y)P b(y)] = 0 ) . Then either Cz c S or C2 n S = 0.
+
+
+
Thus S is the union of sets defined by polynomial equations in y and connected components of complements of such sets. To make this discussion more general, we need to develop some algebraic tools concerning Common Factors of Polynomials: Let
be polynomials with real coefficients. We assume a0 # 0 and bo # 0.It is a classical fact that there is a rational integral form in the coefficients of f and g, known as the resultant, which vanishes if and only if f and g have a common factor. (Classically, this is proved over an arbitrary coefficient field, but we do not need such generality: The real numbers suffice.) In this section, we shall explore the problem of finding a common factor of f and g which is of maximal degree. Our discussion follows the pat tern of that in [VDW] used to develop the resultant. Suppose j and g have a common factor of degree r;. Let us denote that common factor by 4, so that
f = #h
and
g = 4k.
Here h is of degree m - r; and Ic is of degree n - K . Write
Since
4.1. DIVISION OF DISTRIBUTIONS I
107
we may equate coefficients on the left and right hand sides to find that
+
+
This is a set of m n - n 1homogeneous linear equations in the rn + n - n 1- (n - 1) variables co, ...,em-,, d 0,. ..,d,-,. It will simplify the notation to consider the variables in the above linear equations to be do, . .. ,dW,, -co, .. . , -cm-,. Then the matrix of coefficients for the system can be written as
+
(Note that this is not an augmented matrix: the vertical line is there only as an aid to visualizing the organization of the matrix.) The part of the matrix to the left of the vertical line has n - n 1 columns, the part of the matrix to the right of the vertical line has m - n 1 columns, and the matrix as a whole has m + n - n 1rows. We shall denote this matrix by M,. A necessary and sufficient condition for the linear system to have a non-trivial solution is that M, have rank less than m n - n 1. This, in turn, is equivalent to all the m + n - K + 1 by m n - n 1 sub-matrices of M, having determinant zero. Let us introduce the notation A(m + n + n - 1, n - 1) for the set of all increasing maps of 1,2,. . . ,n - 1 into l , 2 , . . . ,m + n + n - 1. For E A(m n n - 1,n - I), let DA denote the determinant of the square matrix obtained by deleting rows X(1), A(2), . . . X(n - 1) from M,. Finally, let RKdenote the sum of the squares of the Dx as X runs over A(m n n - 1,n 1). We have shown that if f and g have a common factor of degree n, then R, = 0. The converse is not true, because while R, = 0 does imply that the above linear system has a non-trivial solution, it might be that for all such non-trivial solutions
+
+
+ +
+ +
+ +
+ +
-
+
108
CHAPTER 4. PARTIAL DJFFERENTIAL EQUATIONS
co = do = 0;this implies that there is a common factor of degree larger than n. The next theorem is a consequence of the preceding discussion.
T H E O R E M 4.1.2 Suppose that m 5 n. Let
be real polynomials. There are real polynomials,
in the weficients ao, .. . ,a,, boy.. . ,b, such that f and g have a common factor of maximal degree n if and only if
We can also see how to find the common factor of maximal degree. Suppose the condition
holds. Since M,+l is obtained from MK by eliminating the first row, the first column, and the first column after the vertical line (that is, column n - n + 2 ), we see that the first row in M, must be dependent on the others, so in finding h and k it can be omitted. Some m n - K columns of the remaining matrix must be independent, and by thinking of the coefficients corresponding to the other columns as parameters, we can apply Cramer's Rule to solve for the coefficients corresponding to the set of independent columns. The common factor is obtained by dividing the resulting h into f or the resulting k into g . The coefficients of the common factor are rational functions of the coefficients of f and g.
+
We shall use the above theorem to investigate the Projection of Polynomially Defined Sets: Let P and Q be polynomials in x = (y,z ) = (yl, . . . ,yn-1, 2). Let C be a connected component of the complement of {x : Q(x) = 0). Let and p be real numbers. Set
4.1. DIVISION OF DISTRIBUTIONS I
109
Note that S is the orthogonal projection onto a coordinate hyperplane of the set {(y,z) :
a < 2 < p,
P(y,z) = 0,
(y,z) E C ) .
Fix an arbitrary yo E Wn-'. We are interested in the form of S near yo. Many different things can happen. First, we shall consider the simplest possible situation: Write
The theorem 2 may be used to construct the polynomials
where p is the smaller of rn and n.
We also need to consider the possible multiple roots of P. Let A be the usual discriminant of P (see [VDW]).
ASSUMPTION IV. A(yo) # 0. Lastly, we consider whether or not the top and bottom constraints are crossed.
ASSUMPTION VI. P(yo, 0) # 0. By continuity, we select an open set U with yo E U such that the inequalities in the above Assumptions hold for all y E U.Assumption IV assures us that for each y E U there are the same number of real roots to P(y, 2). These real roots can be indexed zl(y), . . . ,z,(y) so as to be continuous in y E U. Assumption 111 assures us that (y, zi (3)) stays in the same component of the complement of {x :Q(x) = 0) as varies over U. Assumptions V and VI assure that each root zi(y) stays
110
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
always in or always out of the open interval ( a ,P ) as y varies over U. Thus either U c S or U n S = 0. In the preceding, each Assumption represents the simplest eventuality that can occur. The general situation is as follows:
< <
Select an integer j with 0 j p. We shall restrict our attention to the intersection of { y : a o ( y ) 2 a l ( y ) 2 . . . a j - l ( y ) 2 = 0) with a connected component of the complement of {y : a j ( y ) = 0). On such a set the degree of P is constantly equal to m - j, and we may as well replace the original polynomial P by the following new polynomial which, to save notational complexity, is also denoted by P :
+
+ +
< <
Select an integer k with 0 k q. We shall restrict our attention to ~ 0) with a the intersection of { y : bo ( Y ) ~ b1 ( Y ) ~ . . . bk-1 ( Y ) = connected component of the complement of { y : b k ( y ) = 0). On such a set the degree of Q is constantly equal to n - k, and we may as well replace the original polynomial Q by the following new poly~iomial which, to save notational complexity, is also denoted by Q :
+ +
+
We apply theorem 2 to obtain polynomials
where v is the smaller of m
-j
and n
< <
-
k.
Select an integer 1 with 0 1 r. We further restrict our attention to the intersection of { y : ~ ~ ( ~y ~) ~( .y. . ) R~ ~ - ~= (0) ~with ) ~ a connected component of the complement of { y : R l ( y ) = 0). On such a set P and Q have a common factor of constant maximal degree 1. Suppose we write P =
+
+ +
4.1. DIVISION OF DISTRIB UTIONS I
111
polynomial in y to assure that the coefficients remain polynomial in y. As before we retain the notation P for the new polynomial. Because the transition from two real roots to two complex roots is marked by a double real root, Assumption IV is the simplest way to insure that the number of real roots remains constant. In general, to insure that the number of real roots remains constant, one needs to keep the set of multiplicities of roots constant. This requires considering all the derivatives of P. For each integer j less than the degree of P, we apply theorem 2 to P and -, a'p to produce the sequence of polynomials A i , l ( ~ )A , j,2(~),
..
Select an integer t j with 0 5 t j 5 s. We further restrict our attention to the intersection of {y : Aj,1 ( Y ) ~ Aj,z(y)2 . . . Aj,t-l ( Y ) ~= 0) with a connected component of the complement of {y : A,,,' (y) = 0). have a common factor of constant maximal On such a set P and degree tj. The sequence of integers t 1, t2, . . . determines and fixes the set of multiplicities of the roots of P, so that the number of real roots remains constant.
+
+ +
Assumption V insured that no real roots crossed the top and bottom constraints. In general, one or several roots could cross.
av-1 P CRITERION IVT. P(yo,P) = . . . -(YO,
<
P) = O,
Select integers u b and vt with 0 u b 5 u and 0 our attention to the intersection of
dVP (YO,
P) # 0.
< vt < v. We restrict
with a connected component of the complement of
Having sufficiently restricted our attention, we see that we are now on a set which either lies in S or does not intersect S.
112
C H A P T E R 4. PARTIAL DIFFERENTIAL EQUATIONS
The preceding discussion can now be turned into a theorem. We define the following Category of Sets: Let Cn consist of all subsets, A, of the unit n-cube In such that for each xo E IIn there exists an open ball U centered at xo with the property that A n U is a finite union of sets of the form { x : P ( x ) = 0 & x E C) where C is in turn a finite intersection of connected components of sets of the form u { x : Q(x) = 0).
-
In the above, P and Q are required to be real polynomials. The next theorem is our main result about this category, and it follows, as indicated, from the preceding discussion.
THEOREM 4.1.3 If A E Cn and II is an orthogonal projection onto a coordinate hyperplane, then II(A) E Theorem 1 will follow from this and Puiseux's Theorem. We recall the discussion at the start of this section, and let A be the set of pairs (x,y) such that x > 0 and
C(X,Y)&
hl
[(o < w
& C ( x , w ) ) + y 5 w],
where C(x, y) is the set of conditions
The description of the set can be rewritten as follows: The inequalities involving 5 , 2,and # can all be replaced by disjunctions involving just =, <, and > . The logical connective + can, of course, be expressed in terms of negation and disjunction (2.e. p + q is equivalent to ( l p v q). The quantifier V can be replaced by 131. Such a rewriting would make the notation very lengthy, so we shall not actually carry it out. But such a rewriting combined with repeated application of theorem 3
4.2. DIVISION OF DISTRIBUTIONS 11
113
above shows that A is, in fact, in C2. We apply the definition of C2 at the point (0,O) to see that in a neighborhood, U, of (0,O) A is a finite union of sets of the form {(x, y) : P ( x , y ) = 0 & x E C ) where C is in turn a finite intersection of connected components of sets of the form { ( x , ~:)Q(x,Y)= 0).
u
-
From Puiseux's Theorem, we know that for such a set S, either there exists 6 > 0 such that 0
<x <6
or there exist 6 > 0, c
=+
0
$zClos{y > 0 : ( x , y ) E S),
> 0, p > 0 such that
The first choice cannot hold for all of the sets making up U n A. Since there are only finitely many sets S to consider, we see there exist S > O , c > O , p > Osuch that
It follows that if d(E, N )
< 6, then IQ([) I > c . d((, N)", as desired.
In the next section, we indicate briefly how the result on division of distributions follows from the estimate on the rate of vanishing.
4.2
Division of Distributions I1
Following Hormander, we begin this section with the corollary of theorem 4.1.1 which is needed for the proof that a tempered distribution can be divided by a polynomial.
Corollary 4.2.1 Let Q(C1,& , . . . , be its zero set. T h e n either (i)
en) be a real polynomial,
and let N
N is empty and
IQ(F)I 2 c . (1 + IEl 2
-"' , for all
EIW~,
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
114 (ii)
N is non-empty and
IQ([)I
2 c . (1+ 1<12) - " '
dist((, N)"',
for all
< E RrL.
Here c, pi, and p" are positive constants. Proof: We may and shall assume Q is non-constalit. We define another polynomial q by q(q) I S I ~ ~ Q ( ~ / I S I ~ ) ~
-
where m is the degree of Q. Denote the zero set of q by Z. Since Q is non-constant, we automatically have 0 E 2. Also, we have
If the origin is an isolated zero of q (which is the same as N being compact) then choosing r > 0 so that
we can apply theorem 4.1.1 to obtain positive c and
,!L
such that
It follows that
The conclusion of the corollary now follows easily by applying theorem 1‘) . This proves the result in the case 4.1.1 again to Q and {[ : 151 5 1 where N is compact. Now suppose that the origin is not an isolated point of Z. Again we can apply theorem 4.1.1 to obtain positive c and p such that Iq(rl)l l c - d i s t ( q , Z ) " ,
for IqI
< 1.
But in this case it is non-trivial to estimate dist (7, 2);for it may well be that dist(q,Z) < Iql. We consider the possibility dist (q, Z ) < 1q1. Let q* E Z be such that dist (q, Z ) = (q - q* I. Since lg - $1 < 1q1, we have q* # 0. Associate to q the point 5 = q/lq12 and to q* the point [* = q*/1q*I2. We have [* E N. The possibility that 0 = dist(q, 2) is uninteresting, because
4.2. DIVISION OF DjrSTRIB UTIONS II
115
e*
then ( = and both sides of the inequality in (ii) are 0. So we may assume that E # e*. The triangle with vertices 0, C* is similar to the triangle with vertices 0, q*, q, so
Thus, for
e,
I[( > 1, we have
where
H(E) = inf
If H (E)
{le*I-'le e*I : E* E N) . -
< a, then there must exist E**
But then it is easy to see that
E N with
IE**l < 2151, so
Since N is non-empty, there is a constant cl such that
H(C)> min {A
dist
2' 2(l
(e,N )
+ ~ l ) ( llei) +
I
= 2(l
dist (t,N )
el)'
+ ~ l ) ( l +
Thus
. (1+ IF^)^"-^" dist([, N)', (1) for E with > 1 associated with q such that 0 < dist(q,Z) < 1ql. For 9 such that dist(q, 2)= Iql, we can use the simpler estimates
I&([)
1 2 ~2
as before to extend the applicability of (1) to all E with 2 1. Finally, the result follows easily by one further application of theorem 4.1.1 to Q and {I : 151 1). ¤
<
Now we shall apply the estimates to the division problem. We begin with some definitions.
116
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS Definition 4.2.2 Denote by S the space of infinitely differentiable (real (i) or complex valued) functions defined on all of Rn which satisfy
for all multi-indices a and P. Such functions are called rapidly decreasing or Schwartz functions. (ii) We topologize S by using the semi-norms pap, for each choice of a and p. So equipped, S is a topological vector space. (iii) A continuous linear functional on S is called a tempered distribution or Schwartz distribution. The space of all tempered distributions is denoted by S f . To assist us in making various estimates in the remainder of this section, we introduce some notation.
Definition 4.2.3 For an infinitely differentiable function, f, nonnegative integers e, m, x E Rn, and a subset B c Rn, set
In the case B = 0, we will set 1 f 1
C, m, B
equal to 0 rather than
-00.
With this notation, we can use the
as the semi-norms on S. In general, the multiplication of distributions is ill-defined (however see recent developments due to Colombeau [COL]), but it does make sense to multiply a tempered distribution by a smooth function with polynomial growth, in particular, by a polynomial: If T is a tempered distribution and P is a smooth function that satisfies
4.2. DIVISION OF DISTRIBUTIONS II for some C and some k , then we set
for each cj E S. Certainly, P4 is a rapidly decreasing function, so the right-hand-side of (4) is defined, and one checks easily that it is, in addition, a continuous functional in the topology on S. Another way of looking at the multiplication of tempered distributions by smooth functions with polynomial growth is to consider first the operation of multiplying the rapidly decreasing functions by such a function:
Lemma 4.2.4 Let P be a n infinitely diflerentiable function satisfying (3). The map M p : S -+ S , defined by
is continuous. Then the m~iltiplicationof a tempered distribution by the fiinction P is simply a composition of continuous functions. The main results of this section are contained in the following:
T H E O R E M 4.2.5 Suppose P $ 0 is a polynomial. (i)
The map M p : S -+ S, defined by
has a continuous inverse (defined only o n its image, of course). (ii) I f T is a tempered distribution, then there exists a tempered distribution S such that
(iii) If T is a tempered distribution, then there exists a tempered distribution S which solves the partial diflerential equation
118
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
The heart of the matter is (i). Since the complement of the zero set of the polynomial P is dense, it follows that the map M p is oneto-one. Thus there is an inverse map from the image of M p to S. The proof that the inverse is continuous will clearly depend on establishing estimates on the semi-norms on S. Before we sketch the proof in the general case, we will illustrate in a simple setting why one might expect the size of Pf to control the size of f . Let f : R --+ R be infinitely differentiable. Let I denote the interval [-1,1]. Let P be the polynomial P ( t ) = t. It is obvious that
and that if the maximum of f on I occurs at -1
< to < 1, then
Since the maximum on I either occurs at -1, 1, or at a to with -1 to < 1, it follows that
<
It is not to hard to show using some inductive arguments that for nonnegative integers k , m , if P ( t ) = t " then there is another non-negative integer m' and a real constant C such that
Indeed, by these means we colild obtain similar estimates for any polynomial in one variable which does not vanish identically. The argument for a polynomial in several variables is significantly more difficult, precisely because the zero set can be much more complicated. Also, the semi-norms on S require the inclusion of a polynomial factor, which will interfere with the easy argument we used. To deal with the general case, we need to define some more technical norms:
Definition 4.2.6 For an infinitely differentiable function f, nonnegative integers l , m , and a subset B c Rn, set
4.2. DIVISION OF DISTRIB UTIONS I1
is the remainder in Taylor's formula. As before, in case B = 0,we set
Let P ( f 1 , . . . ,f n ) be a polynomial which does not vanish identically. Denote by Bk the set of points at which P has a zero of order k or greater. A particular [ is in Bk if and only if
D(&)P(S)= 0 for every multi-index a with la1 < k. We have Bo = Rn and Bd+l = 0, where d is the total degree of P. Also, we have Bd+1C Bd C ... c B1 C Bo. Using our notation, to prove (i) of the lemma we need to show that, for each pair of non-negative integers t, m, there exist non-negative integers l',m' and a positive constant K such that
This is proved by an inductive argument beginning with the trivial fact that
=I
f ([)I[, m, B ~ I+ (Pf ~ )(<)Ip, , B~
For technical reasons, it is convenient to use the more complicated * norms 1.1 .
Lemma 4.2.7 For each triple of non-negative integers k , l,m, there exist non-negative integers t', m' and a positive constant K such that
Sketch of Proof: Fix k, l ,m. We argue inductively, so we may assume that the statement of the lemma holds for k . We prove the result for k + 1. The first step is to prove that there exist l',m', K so that the conclusion holds for any function f which satisfies the additional hypothesis that it vanish to order m' on Bk+1 (this extra assumption will be eliminated afterward). Once that m' is determined, we will
120
CHAPTER 4. PARTIAL DEFERENTIAL EQUATIONS
introduce an approximation to an arbitrary f , based on the Whitney Extension Theorem. We shall assume Bk+1# 0;the other case is easier. We apply the corollary to theorem 4.1.1 to the polynomial
+
for j = 1,. . . ,m 1. Notice that each Q j vanishes exactly on Bh+1 and that all the derivatives of pj of order strictly less than j k vanish on Bk, so
It follows that
where we choose the constants p', p" large enough and c small enough that ( 5 ) holds for j = 1,. . . , m + 1. Now m' is chosen to be an integer such that m'>p'+km+k+rn. Suppose f vanishes to order at least m' on Bk+1.For la1 5 m, let L, be the differential operator of order la1 with polynomial coefficients defined by
For a multi-index P, let La,' be the differential operator of order IPI defined by L,,F = ~ ' ( 1 3 , ~ ) .
+
loll
+
Set p = m' - k(lal 1)- loll. Supposing IPI = k(lal 1 ) and dist (t,Bk+l) 1, we have easily
+
>
where we simply need to choose ll and cl large enough to dominate all the polynomial coefficients in Next, we observe that
4.2. DIVISION OF DISTRIBUTIONS 11
121
if dist(E, Bk+1)< 1 while P is as before, we can find a point t* E Bk+1 with Ie-E*I = dist(<,Bk+1).Let v be the unit vector (<-<*)/I<-E*I. At [* the function La,P(Pf) vanishes to order at least p because Pf vanishes to order m'. Now we apply Taylor's Theorem to the function h(7) = (L,P (Pf)) (F* at
T
= 0, to find 0
+
72)))
< TO < IF - I*)such that
The two estimates can be combined to give
for all
E
In. Noticing that pl"l+'D" f
= C ( P f ), we have
~ 4 [ ~ ~ s t ( F , ~ k +l P l )fIl'e z , m l , ~ ( < , lfor ) ' all ( E Rn. Then we apply (5) with j = la1
+ 1 to conclude that ,for all ( E Bk,
Since p - p' 2 0, the distance from E to Bk+1can be bounded by a constant multiple of 1 IEl, so we have
+
loaf([)I
5 c6
IPf 1 ~ m)r ,B(<,1))
for all
E ~ k .
The constants l' and K are independent of ,$ and f , but f must vanish to at least order m' on Bk+1.
122
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
Similar arguments are used to obtain estimates on
We refer the reader to [HOR3]for the details. The result is an estimate
whenever f vanishes to order m' on Bk+1. We must deal with the assumption that f must vanish to order m' on Bk+1 since, obviously, this does not generally hold true. The appropriate m' has by now been fixed. Consider an arbitrary rapidly vanishing f . Using f as the source of the data and Bk+1 as the closed set, we apply the construction from the proof of the Whitney Extension Theorem to produce a function g which agrees with f on Bk+1 up to order m'. Now we come to the point where the more complicated norms are used. By careful consideration of the construction in the proof of Whitney's Extension Theorem, one obtains the estimate
where K1 and p are independent of f and Bk+1.Thus to successfully estimate the simpler norm of g, we need information on the more complicated norm of f . Also, the smoothness claimed for g is only that it possess m' continuous derivatives, but that is sufficient. We have the easy estimate
based simply on the fact that P is a polynomial of degree d. Combining the estimates (6) and (7) and taking the supremum over Rn, we obtain
where the last inequality follows by the induction hypothesis. It follows, of course, that
4.3. THE FBI TRANSFORM
123
But f - g vanishes to order m' on Bk+l,so we can apply our earlier estimates to obtain
If
*
< C7 i,rn,Bk -
.I Pf I,'
7
,',,n.
A final application of (6) and the induction hypothesis gives us
Proof of (iz) of the theorem: Let So be the linear subspace of S defined by S o = { P f f: E S ) . We define a linear functional
s : So
=+
S(pf)=T(f),
R by for f E S .
By (i) of the theorem, s is well-defined and continuous. By the HahnBanach Theorem, there is a continuous linear S : S + R, that is S E St, such that
which is the same as PS = T. Finally, recall the proof of (iii) of the theorem was sketched at the beginning of the previous section. The best local properties are not always obtainable with the tempered fundamental solutions that we have been discussing. The reader more directly interested in partial differential equations should see
[HOR2].
4.3
The FBI Transform
The rate of decay of the Fourier transform of a function f cannot be used to give sharp information about the smoothness of f . Similarly, the decay of the Fourier transform will not detect whether or not f is real analytic. The &transform (see [FJ])is a serviceable variant of the Fourier transform that will give sharp results about the smoothness of a function. For real analyticity the correct tool is the so-called FBI
124
CHAPTER 4. PARTIAL DLFFEl?BNTLAL EQUATIONS
transform. The acronym FBI stands for the names of the mathematical physicists Fourier, Bros, and Iagnolitzer. It is noteworthy that the FBI transform is a special instance of the theory of wave packets as developed by Cordoba and Fefferman (see [FO]). Wave packet theory is an alternate method for studying propagat ion of singularit ies, a phenomenon that is most often understood by using Fourier integral operators (as defined and developed by Hijrmander [HORI] and Duistermatt and Hormander [ b ~ ] ) . Define the Fourier transform of a Lebesgue integrable function f
The most fundamental facts about the Fourier transform are these:
I. If both f and j are integrable then
i(<)exists as the
11. If f is square integrable on W then limit, in the L~ topology, of the functions lim
N-rn
The function
f(~)e-~"'""dx.
f satisfies
In. If f is integrable then f^ is a bounded, continuous function and lf^(t)I5 Ilf 11~1W. Recall from the previous section that the Schwartz space of rapidly decreasing functions consists of those Coo functions on R which have the property that the function and each of its derivatives vanishes at infinity at a rate 2 faster than ( X I - * for any N. The function h(x) = e-lXl is an example of a rapidly decreasing or Schwartz function, as is any Cm function of compact support. The space S of Schwartz functions, equipped with the semi-norms
4.3. THE FBI TRANSFORM
is a Fkechet space. Its dual S' is called the space of tempered distributions or Schwartz distributions. The Fourier transform takes the space S in a univalent, surjective, continuous fashion to itself. Note in particular that the Fourier transform maps the space CO , O(W) of Coo functions with compact support into S, but it does not map C r into itself (In fact the "Heisenberg uncertainty principle" asserts that the Fourier transform of a non-trivial compact ly supported function is never compactly supported; there are quantitative versions of this assertion as well. See [FEG].)
V. We have (e-dm2)-(t) - b-1/2 .
VI. If f E S then (f ')A(<)= (-2+<)
-7rc2
/b
f(c).
VII. Iff and g are integrable then so is their convolution
and
(f * s)^(Q = f(~)3(0By applying the inverse Fourier transform to this last identity we obtain (f -g)^(t)= f * ij(t). Further details on the elementary properties of the Fourier transform may be found in [SW]. Now we define the FBI transform Ttf (x, t ) of an f E L1(W) by the formula - 2 f l c ~ c ~ - 7 r t ( s - ~ds )~ Since the Gaussian expression is bounded above by 1, it is plain that Ttf (x, F) is well-defined for any integrable f . Now we define an exponential decay condition on the FBI transform at infinity. Fix xo E R. We say that an integrable function f satisfies the condition RA(xo) if there are constants C,o,M and a neighborhood U of xo such that for all t > 0 , all > M, and all x E U it holds that
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
126
THEOREM 4.3.1 Fix xo E R. An integrable function f lytic at xo if and only if f satisfes condition RA(xo) .
i s real ana-
Note that we work in W1 for simplicity of notation, but the results of this section hold in any R" (see [Sq). Also the theorem may be proved for f a distribution if a certain amount of extra care is taken. However to avoid a number of unpleasant technicalities we shall assume that our function f is Cm . In this way we can concentrate on the main point: as we know from Chapter 1 real analyticity is in fact a condition on the growth of derivatives. So our job is to focus on that condition. The remainder of this section will be devoted to proving the theorem with this extra hypothesis in place. The exposition here is derived from that in [FO]. We will divide the argument into several lemmas and propositions. We begin by proving the easy half of the Theorem.
Lemma 4.3.2 Fzx xo E R. Let f be an integrable function that vanishes in a neighborhood of xo. Then f satisfies condition RA(xo) .
Proof: Choose 6 > 0 such that if 1s - xol < 26 then f ( s ) = 0. Then for ( x - x0 1 < 6 it holds that
with a = d2. This establishes the result. The lemma has the effect of making our work local: if f satisfies RA(xo)and if f = g in a neighborhood of xo then g satisfies condition RA(xo). In particular, if f satisfies condition RA(xo) on a neighborhood U of xo then let II, be a Cm function of compact support in U which is identically 1 in a smaller neighborhood of xo. Write f = II, f (1 - II,) f . The second term satisfies RA(xo)by the lemma hence so does the first. As a result of these observations we may assume in the sequel that f is a Cm function of compact support.
+
Proposition 4.3.3 If f is real analytic in a neighborhood ofxo then f satisfies condition RA(xo) . Proofr For simplicity take xo to be 0. As indicated, we may take f to be globally C,Oo. [Of course we shall only verify that f satisfies RA(0) in a small neighborhood of 0.1
4.3. THE FBI TRANSFORM
127
By substituting 2's for x's in the power series expansion of f about 0 we find that f is complex analytic (or holomorphic) in a neighborhood of 0 = 0 + i - 0 in the complex plane. Choose 6 > 0 such that {t iu : It 1 < 26, lvl < 6) lies in this neighborhood. Now let $ ( t ) be a Cp with support in {t E W : It1 < 26) such that 0< - $(t) 5 1for all t and $(t) = 1when It1 < 6. Then for any # 0 we may use the Cauchy Integral Theorem to move the axis of integration in the definition of Ttf (x, t) for 1x1 < 6 to the contour
+
Notice that the region in which p(s) where 8 = 6 sgn ( = 6 (/ ) (I. differs from ~ ( s = ) s lies in the region where f is holomorphic; hence Cauchy's formula applies. We see, using the new contour, that when 1x1 < 6 and ( 1 # 0 we have
We use the definition of
8 and some obvious majorizations to see that
(*) We fix
I[(
(i) If Is\
2 6 and 1x1 < 612. There are now two possibilities:
< 6 then
(ii) If Is1 > 6 then
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
128
In any event,
(*) 5 C e -n.(b2/4).t7 as desired.
Lemma 4.3.4 Let a > 0. T h e formula
X(f)
X a ( f ) = J J e 2 n i X t - 2 n a x 2 t ~ (1+ iax sgn t )f ( x )dxdt
defines a n element of S f . Proof: Our first job is to see that the integral converges. Let g E S . Now exploit Property VII above to write, for any b > 0,
Now property V of the Fourier transform enables us to write the righthand-side more explicitly as
Therefore, setting b = 2alC1,
The right-hand-side of the last equation is a function of J and, by inspection, vanishes at infinity more rapidly than Itl-Nfor any positive integer N. In particular, it is an integrable function. Therefore we may set, for a > 0 and f E S , g ( x ) = ( 1 iax s g n t ) . f ( x ) to obtain that
JJ
e2nix(e-2nax2
+ [ I ( 1 + iax sgn t )f ( x ) d x d t
is a convergent integral. Our discussion of this integral shows that its convergence only depends on finitely many of the Pa,@. Therefore X is an element of S f . rn
Lemma 4.3.5 The functional A defined in the preceding lemma is equal t o the Dirac delta mass 6.
4.3. THE FBI TRANSFORM
129
Proof: For any z # 0 we have, by the definition of the sgn function,
= ( 1 - iax)
/('
2mi(z-iax2)c
+ (1 + iax)
e* 2 a i ( x + i a x 2 ) ~ ~ ~
+ +
1 - iax I iax 2ni(x - iax2) 2ni(x i a x 2 ) - 0.
-
This shows that the distribution X is supported at the origin. Such a distribution is a sum of derivatives of the Dirac mass. We eliminate all the derivatives but the zeroth by an iterative procedure. If f is a Schwartz function that vanishes to second order at 0 (0 = f ( 0 ) = f ' ( 0 ) )then we notice that
This shows that the integral defining X converges absolutely. Thus we may apply Fubini's Theorem and reverse the order of integration in the integral defining A. Because of (*), we conclude that A ( f ) = 0. Now suppose that f is a Schwartz function that vanishes to first order at 0. Write
+
f(x) = $(x) ff(0)x ( f ( x )- $(x) - f f ( 0 ) x )
fib)+ f&)-
where $ is an even cutoff function that is identically 1near the origin. Then fi is odd and fi vanishes to second order. It follows immediately that A( f i ) = 0. But if we apply A to fl , and perform the change of variable x + -z,[ + -6 in the integral, the result is that nothing changes but a minus sign is introduced. It follows that X ( f l ) = 0. The result of our calculations is that = c - 6.It remains to determine c. [Even though the exact value of e is not important for the
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATlONS
130
result we seek, we compute it for completeness.] Fix g a C r function that is identically 1near x = 0. Let g k ( x ) = g ( x / k ) , k = 1 , 2 , . . .. Then c = X(gk) for any k. Let k + +oo to yield =
J
/'
e 2 n k ~ e - 2 n a x 2 1 ( (/ 1
+ iaxsgn c)dx@.
We use fact V and apply fact VI to conclude that
( e - ~ ~ . '15) ) - b-1/2e-ffE2/band
( z e --nbx2
)I t ) = b - 3 / 2 i ~ e
- 7 ~ ~ ~ / b
Therefore
c = J ( e -(2naiCl)x2)
I -[)dt + Jiasgn
( x e -( ~ " ~) xI2 )E7-<)4 I
Notice that the second integrand simplifies to
Thus the integrals combine to yield
Perform the change of variable p = 7r
rn
This completes the proof.
Lemma 4.3.6 Iff and g are C r functions that both satisfg, condition RA(xo) then so does f g. Proof: As usual we assume that xo = 0. By hypothesis there is a neighborhood U of 0, and positive constants M , C, o such that when X E U and I[] > M then l T t f ( x , Y ) l 5 c .e
-ot
and
ITtg(x, t t )1 5 C - e-Ot.
4.3. THE FBI TRANSFORM
131
Put q ( s ) = e- - 7 t t ( s - ~ ) ~ / 2f (s)
and
r2(s) = e-7rt(s-~)~/2 9(s).
Then, by definition of the transform Tt, we have
Also ( 4 ~ (s) 2
=e
--7tt(s-~)~
f (s)s(s)-
As a result,
On the domain of integration in A we have that 12t - 2tP1CI > 1t1.Therefore
2 2t-' ICI hence
Next we have that
Now repeated application of III and VI above shows that
This last, by inspection of the definition of r 2 , does not exceed C" (1 t ) 2 . Putting together our estimates for Ttf and Ttg yields that
+
To estimate B we notice that
132
CHAPTER 4. P A R T U L DIFFERENTIAL EQUATIONS
.
But 2t-I (
Proposition 4.3.7 Let f be a C r function. If condition RA(xo) is satisfied then f b real analytic in a neighborhood of xg Proof: We may assume that xo = 0. Since the distribution X equals 6, we may write
(Note here that we have used a translation operator to pass from a result about the Dirac mass at 0 to a Dirac mass at x.) Set r ( s ) = s f ( s ). Then r is the product of the real analytic function s , which satisfies condition R A ( 0 ) by Lemma 3, and f , which satisfies R A ( 0 ) by hypothesis. By lemma 8, r satisfies condition RA(O). Therefore there are positive constants C, M, a,and 6 such that when 1x1 < 6 and It)> M we have
JTtf ( x ,t)l I G - e-Ot
and
We now introduce the notation z = x
I T t r ( x , t ) I < C . e -at .
(**)
+ iv with x, v real.
We choose a = (4M + 4)-' and require that
1x1<6 The result is that
and
-6,~)-
a l a ivl<min( l 6 x M ' 2 x
4.4. THE PALEY- WIENER THEOREM
133
hence, using (**) and (***),
This absolute convergence and size estimate means that the integral
.
defines a holomorphic function of z on the region in x and v specified above. Obviously this holomorphic function agrees with f on the real axis. Therefore f is real analytic in a neighborhood of the origin. The FBI transform is not well known in the mathematical community. It is a powerful tool which should prove useful in many contexts.
The Paley-Wiener Theorem The FBI transform has shown us that Fourier integral operators can be used effectively to detect real analyticity. This connection is, in retrospect, not surprising because the exponential expression ex is real analytic. In fact the connections were notice rather early in the history of twentieth century analysis by R. E. A. C. Paley and N. Wiener
[PAW]. The gist of the Paley-Wiener theorem is that the Fourier transform of a compactly supported function (or, more generally, a compactly supported distribution) is an analytic function of exponential growth. The converse is true as well: every analytic function of exponential growth arises as the Fourier transform of a compactly supported function or distribution. It is also the case that the site of the support is intimately connected with the rate of growth of the function. The Paley-Wiener theorem has been influential in twentieth century analysis. It has made its mark particularly in the area of partial differential equations, where it says a great deal about the existence of solutions to linear equations and to linear systems. The work of Malgrange and Ehrenpreiss on systems with constant coefficients is treated in some detail in [HOR] and [EHR]. In fact the Paley-Wiener theory of several dimensions has an interesting geometric flavor. It is related in spirit to the Fourier analysis
134
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
of tubes over cones (see [SW]). This in turn can be used to study the edge-of-t he-wedge theorem (see [RUD21). Our purpose here is to present the central idea of the Paley-Wiener theorem without getting distracted by ancillary technical issues. Therefore we will present the result in the context in which it was first discovered: the analysis of the real line. By making this choice we can restrict any complex analysis that needs t o be done to the familiar context of the plane. We shall make a few remarks about more general versions of the theorem at the end of our discussion. As motivation for the Paley-Wiener theorem we first present an analogous theorem in the realm of the Fourier analysis of the unit circle T = R/2?rZ. Of course in practice we identify T with the interval [O, 27r] with the obvious identifications at the endpoints. Measure theory on T is defined by pulling back Lebesgue measure from [O,2?r]under this identification. If f E L1( T ) and n E Z then we set
Our theorem is
Proposition 4.4.1 Let f E C m ( T ) . Then f is real analytic on T i f and only if there are constants c, C > 0 such that
Proof: By integration by parts we see that
f ( n )=
(in)-j - f (a). (j)
[Here the exponent ( j ) denotes the jthderivative.] It is also obvious from the definition of the Fourier coefficients that
Combining these two facts with the characterization of real analytic functions given in Proposition 1.2.9 gives the result. Matters in the non-compact setting are a bit more subtle, but exhibit the same flavor. Recall that if f E L 1 ( B ) then its Fourier transform is defined to be
4.4. THE PALEY-WIENER TWEOREM
135
<
Notice that f^ E Loo(R)and 1 1 . f 1 1 ~ I l f l l L l . Recall that the Fourier inversion theorem (see, for instance, [KAT] or [SW]), says that in case . f L1 ~ then
In case f E L2, it holds that
f^ E L2 and
In this circumstance the Fourier integral must be interpreted as
f (0 =
f (t)e-""Et
lim
(because L 2 ( R ) L1( R ) ) . The Fourier inversion formula must be interpreted in a similar fashion. Notice that Fourier inversion implies Fourier uniqueness: if f(<)= g(C) almost everywhere then f = g. Finally recall that if f,g are L' functions on R then their mnvolution is the function f * g ( x ) = f (x - t)g(t)dt. An elementary change of variables and application of Fubini's theorem reveals that f * g(Q = f ^ ( -~g(0.Now we have
5,
-
T H E O R E M 4.4.2 (Paley-Wiener) Let f lowing two statem,ents are equivalent:
E
L ~ ( R )Then . the fol-
There is a function F and constants a, C > 0 such that F (i) is holomorphic zn the strip { z E C : IIm zl < a ) and
(ii)
The function eal(lf^Ces in L2(R.)
Proof: To prove (ii)
+ (i), we define
136
CHAPTER 4. PARTIAL DIFFERENTIAL EQUATIONS
Our hypothesis guarantees that j([)egc E La (as a function of the x variable) as long as y = Im z satisfies (yl < a. By Fourier inversion, FlR = f - F'urthermore, by Plancherel's thedrem,
i2
Thus (i) is proved with C = )Ifealcl 11 (R).
+
To prove (i) + (ii), define f, (x) = F(x iy) for 1y 1 < a. Observe in particular that fa = f. We shall now prove that &(<) = f^([)e-e". Let us assume for the moment that each On the one hand, F(x
+ iy) = f,(x)
1
=27r
f;, is known to lie in L'.
1
f;([)ek'(d-
On the other hand, we may define
Both F and H are holomorphic functions on the strip {x+ iy : 1y 1 < a ) . Also they agree on the real line, hence they must agree on the entire st rip. It follows from Fourier uniqueness that (6) = e-cy. Now our hypothesis, together with Plancherel's theorem, says that
&
I([)
where C is independent of 1y( < a. But then the continuity of the integral (more formally, the Lebesgue dominated convergence theorem) implies that ealclf^E L ~ This . completes the proof of the theorem in the presence of the extra hypothesis. For the general situation, we must use the standard Fourier theory r function II, with the p r o p device of the szmmmabzlity kernel. Fix a C erty that $(x) = 1 when 1x1 < 1 and @(x)= O when 1x1 > 2. Let 4 be the inverse Fourier transform of 111; so $(() = @([). For )A1 > 0 we set #A(O= A-'$(t/A). Now define
4.4.
THE P A L E Y - W N E R THEOREM
137
Then G A is clearly holomorphic in the strip { a E (C : IIm a( < a). Set gx,r( x ) = G A( x i y ) . Then gT-, (I)= &(c) - (t).Now the uniqueness argument that we presented in the first part of the proof shows that gx,&) = gAso([)e-&. Notice that &([) = 4(Y) (just use a change of . variables). Hence, when It) < 1 / A , we have &(F) = f ( [ ) e - c ~ Since A > 0 was arbitrary, we have established that &([) = f ( t ) e - ~ yfor all [. Now the proof is finished as before.
+
-
Corollary 4.4.3 Let g be an L1 function with compact support in R. If ij also has compact support then g r 0. The corollary says that a function and its Fourier transform cannot both have small support. There exist a variety of quantitative forms of this assertion as well. This circle of ideas is often referred to as the "Heisenberg uncertainty principle" and in fact is a mathematical model for the uncertainty principle of quantum mechanics. For more on this matter see [FE].
Proof of the Corollary: Let f be the inverse Fourier transform of 9. Then f satisfies condition (ii) of the Paley-Wiener theorem for any Jal > 0. Take a = 1. Then, by the theorem, f is the restriction to the real line of a function F holomorphic on { z : lIm z 1 < 1). Since f is compactly supported, the holomorphic function F vanishes on an entire half-line. Hence F r 0 and f 0. We shall: now formulate two standard variants of the Paley-Wiener theorem. The proofk involve just the same ideas, so we shall not supply those. Details may be found in [KAT].
-
THEOREM 4.4.4 (Paley-Wiener) Let f E L ~ ( B ) .Then the following two conditions are equivalent: There is a function F, holom,orphic in the upper half plane {x E C : Im a > O), and a constant C > 0 such that
(i)
and
CHAPTER 4 . PARTIAL DLFFERENTIAL EQUATIONS
138
(ii)
f([)=Oforall(
This version of the Paley-Wiener theorem can be considered to be a desymmetrized statement of the result: the function F is defined only on one side of the real line (where f lives). This explains the necessity of the convergence statement in part (i) of the theorem. Part (ii) of the theorem is in the spirit of the F. and M. Riesz theorem on the circle (or the line): a measure on the circle is the radial boundary limit of a holomorphic function on the disc if and only all of its negative FourierStieltjes coefficients are zero; in this circumstance, the measure must be absolutely continuous with resped to Lebesgue measure. Here is our final version of the Paley-Wiener theorem in dimension one:
T H E O R E M 4.4.5 (Paley-Wiener) Let F be an entire function and a > 0. The following two conditions are equivalent: (i)
FIR€L2(lR)and
There exists a function 161 > a and
(ii)
fE
such that f([)= 0 for
Obviously the third result is the adapt ation of Paley-Wiener theory to entire functions. It has perhaps the most elegant formulation of the three. The theorem is false if the function f is replaced by a measure (that is, the little "0" in part (i) must be replaced by a big "0"). For instance, cos az is the complex Fourier transform of a compactly supported measure. AS an exercise, the reader may use Paley-Wiener theory to obtain a proof of Titchmarsh's convolution theorem:
T H E O R E M 4.4.6 Let f,g be L2 functions both supported in the interval [- 1,Ol. If f a g vanishes in a neighborhood of the origin then at least one of J or g vanishes in a neighborhood of the origin. In particular, af f * g = 0 then ezther f = 0 or g 0.
4.4. THE PALEY-WIENER THEOREM
139
Both the Titschmarsh theorem and the Heisenberg uncertainty principle may be proved by real variable techniques, but the proofs are much more difficult. Now we turn to N dimensions. What is the analogue of the interval [-a, a] in a multi-dimensional Euclidean space. One answer is the unit ball, but another is the unit cube. It turns out to be most natural not to limit ourselves to these two canonical (from the point of view of Euclidean geometry) examples, but rather consider any set that could be the unit ball of some norm on ElN. Thus we restrict attention to sets K that are convex, compact, and satisfy -x E K whenever x E K. Such a set will be called a symmetric body. If K is a symmetric body then we define K* = { y E lRN : x -y 5 1 for all x E K } . [Here "-" is the standard Euclidean inner product.] The set K* is termed the polar set of K. It too is a symmetric body. The set K* is a natural construct when one views K as the unit ball of some norm. Clearly the Euclidean unit ball is canonical in this context in that it is the only symmetric body that equals its polar set. In general it holds that K** = (K*)*= K. Now if f E L1(EIN)then we define its complex Fourier transform to be
Here z = (zl,...,z,) E CN and t - z = tlrl + - - - t N z ~ . Recall that a function of several complex variables is said to be holomorphic if it is holomorphic, in the classical one variable sense, in each variable separately. A holomorphic function defined on all of CN is called entire. See [KRAl] for more on these matters. Fix a symmetric body L. If t E CN then we define
We say that an entire function F is of exponential type L if for each t > 0 there exists a constant C,> 0 such that
all t > 0. Denot,~t . 1 class ~ of all s i ~ c hfi~ilrtionsby E(L).
CHAPTER 4. PARTLAL DIFFERENTIAL EQUATIONS
140
Now we have
T H E O R E M 4.4.7 Let f E IL2(lRN) and K a symmetric body. Then following are equivalent: The function f is the restriction to RN of a junction in (i) E(K*). The function f is the Fourzer transform of a function sup(ii) ported in the symmetric body K .
The reader is referred to [SW] for a proof of the theorem and for its history.
Chapter 5
Topics in Geometry A
5.1
Resolution of Singularities
Hironaka's great paper [HIR] carries out a program of Oskar Zariski initiated in [ZAR] to resolve the singularities of an algebraic variety. The idea is best captured with the following simple example. Consider the variety V in IW2 given by
The sketch in Figure 1 shows that this variety has a double point at (-1, o)* The philosophy of resolution of singularities is to exhibit the variety as the (locally) univalent, proper image under an algebraic mapping of an algebraic manifold without singularities. In this example, the mapping t c (-2s2 + 2,-2s3 3s)
+
sends the real line algebraically and properly onto the variety. This is a particular (but certainly not the only) resolution of the singularity of the variety V. Hironaka shows in [HIR] that any algebraic variety over the reals, the complex numbers, or any field of characteristic zero may be resolved in this fashion. He shows that hot h complex analytic and real analytic varieties may be resolved as well. Unfortunately for analysts,
142
CHAPTER 5. TOPICS IN GEOMETRY
Figure 5.1: The Variety x3
+ 2y2 - 3x - 2 = 0
Hironaka's proof is presented in the language of schemes and is for all practical purposes impenetrable. Fortunately Bierstone and Milman [BIMI] have recently constructed a proof of the resolution of singularities theorem that applies to real and complex analytic varieties and to algebraic varieties over any field of characteristic zero. However there is a basic complication that is in the nature of things and will never be removed. Namely, generic analytic varieties do not have singularities that are as simple as the singularity in the variety V exhibited above. A variety is, on an open dense set, an analytic manifold of some top dimension k, with a singular locus S of dimension not, exceeding k - 1. But then S is, on a relative open dense set, an analytic manifold of dimension k - 1 with a singular locus S' of dimension not exceeding k - 2. Continuing inductively we find a stratification of the singular locus of our analytic variety all the way down to a discrete set of singular points. Any blowing up procedure must proceed inductively, starting at the dimension zero singular locus and working up to the top dimension. A second complication is that the singular locus of an analytic variety may not have normal crossings as in the variety V above. For
5.1. RESOL UTION OF SING ULA R I T E S
Figure 5.2: The Variety y2 = (x instance, the variety W
143
+ I ) ~(2. - x )
R2 given by
has the property (see Figure 2) that the point ( - 1 , O ) is an element of the singular locus and, a t that point, two branches of the curve osculate. This type of phenomenon introduces additional complexity into the blowing up procedure. A third complication that may arise is that a singular point may be a "pinch point:" the curve
has the point (0,O) as a pinch point. That is, the curve does not cross itself a t this point but instead pinches in the sense that it bends in such a fashion that it is tangent to itself (see Figure 3). Because of the considerations described in the preceding paragraphs, we shall have to content ourselves in this monograph with a treatment of resolution of singularities in a very special situation. We shall introduce enough terminology so that the theorem may be stated
CHAPTER 5. TOPICS IN GEOMETRY
144
Figure 5.3: The Variety y2 = x 3 and discussed for real analytic varieties in full generality; however the proof will only consider algebraic varieties in three dimensions with singularities that are all double and triple points with normal crossings. A brief discussion later will explain just how special this situation really is. The key to resolving singularities is the beautiful classical idea of "blowing up" a point. While formerly the sole province of algebraic geometers, this technique is now becoming a tool for analysts as well (see, for instance, [BEF]). The process of blowing up separates all the lines passing through a point P in space so that they are disjoint. A moment's thought shows that this is a prototype for what we wish to do when resolving a singularity: namely we wish to separate the tangent spaces of the different branches of our variety that pass through a multiple point P. Now we begin our formal treatment, starting with a consideration of projective space:
Definition 5.1.1 The projective space RIPN-' is defined to be the set of (one dimensional) lines through 0 in IRN. A natural way to think about RIPN-' is as the quotient of RN \ (0) by the equivalence relation (si,. . . ,sN) ( t l , .. . ,t N )if and only if there is a non-zero real number
-
5.1. RESOLUTION OF SING ULAMTBS
145
X such that (sl,...,sN) = (A - t1, ...,X t N) The equivalence class of (sl,. . .7 sN) is denoted by [sl,. ..,sw]. In order to see that RIPN-l is a manifold, we define coordinate patches
W [ ~=O{[sly ] -
3
S N ]: sio #
0))
for io = 1,. .. ,N. Then local coordinates on W[io] are given by
It is a simple matter to see that the coordinate change functions are Cm, indeed real analytic. Thus BIJ?~-' is a compact, real analytic manifold of real dimension N - 1. It is sometimes geometrically convenient to think of RIPN as the unit (N- 1)- sphere with antipodal points identified.
Definition 5.1.2 Let U be a neighborhood of the origin in EiN. The blowup of the origin is the set
8 = {(~,e)E u x RIP-' : E e}. The manifold U covers U in a natural way by the map
Clearly ?r is univalent from (U \ (0)) x RIPN-' onto U \ (0). For if ~ ( ( xl), = ?r ((x',1')) and x # 0 # z',then x = x', but l is the line through x and 0 and t' is the line through z' and 0 hence l = F. However 0 separates the lines through the origin. For if l and F are distinct lines through 0 then ((x,l) : x E l} and ((x', !)' : X' E e'} are disjoint subsets of 8.
Definition 5.1.3 The set ~ ' ( 0 )& and is usually denoted by E.
8 is called the exceptional divisor
Definition 5.1.4 Let M be a manifold of dimension N and x a point of M. Let W be a coordinate patch on M that contains x and 6 :
CHAPTER 5. TOPICS IN GEOMETRY
146
W + U C ElN a coordinate map sending z to 0. Denote by M~ the "pullback" of the covering space u consisting of the set of all ordered pairs ( w ,<) such that u E W,{ E 0, and 4 ( w ) = a(F).We call M=the (local) blowup of the point x in the manifold M. The local blowup MXis equipped with a natural projection (still called a ) down to M defined by a ( ( w ,6)) = w. The set Ez ?r- '( x ) = Ex is called the exceptional dzviso~of the blowup.
-
We have the commutative diagram:
6
where is projection onto the second factor (recall M% c W x u). We have described the local blowup of x E M in a canonical fashion; a more heuristically appealing description of this local blowup is as follows: M, EZ M \ ( x ) U
,o,
equipped with the natural projection map ?r : M~ -+ M. Of course implicit in this intuitive description are certain identifications that need to be made vis-A-vis the map 6; the pullback construction takes care of these matters automatically.
Definition 5.1.5 If V is a subvariety of M then the proper transform of V under the blow-up procedure is defined to be T-'
(V \
( I ) ) = T-'
( V )\ Ex.
It is the blowing up procedure that we will use to separate branches of an analytic variety when performing the resolution process. In order to facilitate our understanding of these matters, we now consider local
coordinates in M ~ Let . ( t l , .. . ,t N ) be local coordinates on W M. We shall focus attention on local geometry hence will deal with the manifold W rather than with M. Therefore we shall speak only of w ~ Then W* = { ( y , ! ) E W x IRlPN-' : yj & = tj), where Y = ( ~ 1 ,..., Y N ) , e = [ e l , . . . ,e,]. We let ~ x [ i o= l { ( Y , ! ) : ti, # 0)' then on w X [ i o ]we can use the following N functions as local coordinates :
%
tio '
for 3
# io,
Yio . We introduce the notation Y [io] for these functions by defining
We see that the projection n : W, on WX[io]by
+ W is given in local coordinates
Also the exceptional divisor Ex is given in local coordinates by
Next we look at the transition functions in local coordinates. In w X [ i on ] k r i l l , Eo # i17 we have
.
CHAPTER 5. TOPICS IN GEOMETRY
148
It follows that Y(il1-j= Y[il]io Y[io]-j, provided j is equal to neither io nor il. We have also
but on W, yk . ti1 = pil . 6 0 ,
We see that W~ is a real analytic manifold. Now it is time to study the operation of resolution of singularities. We will study an analytic variety V C P3 that has only ordinary singularities of orders 2 and 3. We need to define the phrase "ordinary singularity" . If P is a point of a real analytic variety V, then define the tangent cone to V at P to be the union of all tangent lines to all analytic arcs lying in K In the example (*), with which we began this section, every point but one in the variety has tangent cone that is just a line - because every point but one is a regular (or manifold) point of the variety. The exception is the point (- 1 , O ) , where the tangent cone consists of the union of the lines
y=-x+-
fi
,h
and
y=--
fix- Z'
Now a multiple point P of order m of an analytic variety of dimension N is called "ordinary" if the tangent cone at P consists of rn distinct affine spaces each of dimension N. Thus an ordinary double (that is, order 2) point on a curve in R2 will look like X in Figure 4. Generically, triple points do not occur on curves in P2. This is why we consider an algebraic variety V in P3. In this situation, elementary dimension theory arguments show that generic triple points are isolated in the variety and the set of double points form a one dimensional subvariety called the double curve. We let pl, . . . ,pt be the triple points and Cl, . . . ,C, be the irreducible components of the dolible curve.
5.1. RESOLUTION OF SINGULARITIES
149
Figure 5.4: An Ordinary Double Point Now let rl
: Y + lR3
be the blowup at the points pl, .. . ,pt. That is, we perform the blowup procedure successively at each of the points pl through pt. Let Eibe the exceptional divisor over pi. In a neighborhood of Ei the proper transform Vl of V will consist of three smooth sheets which intersect pairwise in smooth arcs. Of course Ei is a copy of RP2, and Vi intersects Eiin three lines. The double curve of Vl is the proper transform of C , and consists of three arcs arising from the pairwise intersections of the three comp* nents of Vl. Let us verify the statements in the last two paragraphs explicitly using local coordinates. Let the coordinates about a triple point p be t l , t2, t3; we may assume, after a change of coordinates, that V is given in a neighborhood U R3 of p as the zero set of the polynomial t 1 - t t 3. We then see that r; '(U) is covered in a natural way by three open sets Ul , U2,U3 where
Ui
r;'
(u)\ {the proper transform of the hyperplane {t :ti = 0 ) ) -
150
CHAPTER 5. TOPICS IN GEOMETRY
In terms of the coordinates on Ui given by
we find that
Thus we see that the intersection of the proper transform Vl of V with Ui equals precisely the proper transforms of the two coordinate hyperplanes t j = 0 and tk = 0. Note also that the double curve 6 of Vl is the union of the arcs Y[i]j = Y [ z = ] ~0 in Ui(because we chose coordinates so that V = {tl .t2 t3 = 0 ) ) . In particular, the double curve is smooth, so that the irreducible components 6, of 6 are disjoint manifolds of dimension one. Now let 7r2 : X -+
Y
be the blowup of Y along the double curve &that is, we blow up at each point of C. Our full resolution of the variety V will be given by 7r2 0 7 r l .
Let Fibe the exceptional divisor over the irreducible component ~ iV ,the proper transform of Vl, and Eithe inverse image of Eiunder 7r2. First we check that v is smooth. There is nothing to check except at the points of 7 r ; ' ( ~ ) . Let c E 6. We may choose coordinates so that, in a neighborhood U of c, we have
Vl = {(tlrt 2 ,t3): t2- t3 = 0 ) and
6 = {(tl. t 2 . t 3 ) : t 2 = t3 = 0 ) -
Now the inverse image of U under 7r2 is then covered by open sets U2 and U3 consisting respectively of the complements of the hyperplanes { t 2 = 0) and ( t = 0).
In Uzwe have coordinates
In these new coordinates we see that F = Ufi = {Y[2I2 = 0) and
Thus we see that V is the disjoint union of smooth manifolds, hence is smooth, in U2. Similarly on U3 we have coordinates
In these new coordinates we see that F = uFi = {Y [313 = 0) and
Thus we see that is the disjoint union of smooth manifolds, hence is smooth, in U3. In summary, we have found that is smooth in a neighborhood of ?rgl(c). In fact we may note that, near c, the intersection V nF equals precisely the (disjoint) union of the two sections of the bundle F -+C that correspond to the normal directions to 6 in the two branches of Vl at c. We have proved a very special case of the following theorem of Hironaka: THEOREM 5.1.6 (Hironaka) Let f1 ,.. . ,fk be real analytic finetiom on an open subset U RN,and let
V = {x E U : fj(x) = 0 for j = 1 , . . . ,k)
CHAPTER 5. TOPICS IN GEOMETRY
152
be the w m p o n d i n g variety. Then the^ is a blowup
.Ir:X+U such that the proper transform of V in X is a smooth, real analytic manifold. We close by noting that, for algebraic varieties, the restriction to varieties in R3 (or, what is more convenient in algebraic geometry, the restriction to varieties in IRP3), poses no loss of generality. For dimension-t heoretic considerations allow one to reduce the general theorem - in the case of surfaces - to two dimensional varieties in dimension three (see pp. 612-613 in [GRH]). However by restricting to singularities with normal crossings, and not considering even pinch points (much less the more complicated stratification of singularities that is typical), we have been able to present an extremely simplified sketch of Hironaka's theorem.
5.2
Lojaciewicz's Structure Theorem for Real Analytic Varieties
A complex analytic variety is defined to be the set of common zeroes, on some open domain U, of a finite collection of holomorphic functions. Complex analytic varieties are much like complex algebraic varieties: because of the completeness of the complex field, the structure theory contains no surprises and it is fairly well understood. A good reference is [GUN]. A complex analytic variety that is the zero set V of a single holomorphic function on an open set U E Cn is in fact an ( n - 1)dimensional complex analytic manifold on a dense open subset VR of V. The exceptional set Eo is closed and has complex dimension at least one (real dimension at least two) less than the dimension of K This last assertion is established by realizing Eo locally as the zero set of a certain resultant equation on a copy of Cn-I lying in Cn.See [KRAl] for details. In turn, the set & may be analyzed and a relatively dense open subset found which is a complex analytic manifold of complex dimension at most n-2 (at mast real dimension 2n-4). The exceptional Eo is closed and has complex dimension at most n - 3 (at set El most real dimension 2n - 6).
c
5.2. LOJACLEWICZ'S STRUCTURE THEOREM
153
This analysis may be continued to obtain a stratification of E into manifolds of decreasing complex dimensions. Complete details of this construction may be found in [GUN]. A briefer treatment is in [KRAI]. The situation for real analytic manifolds is somewhat more complicated, just because real analytic polynomials do not always have roots in the reals. To give an indication of the difference between the real situation and the complex situation, observe that generically the complex variety determined b y k holomorphic functions (satisfying a natural independence position that can be expressed in terms of the rank of the space spanned by their gradients) in Cn,O< k n, is of complex dimension n - E. Nothing of the sort is true for real varieties: for example, the variety in W3 determined by the real analytic function F(xl, xz, x3) = x: $1 x i is the zero dimensional set (0). Our purpose in this chapter is to give a brief description of Lojaciewicz's structure theorem for real analytic varieties and his vanishing theorem for real analytic functions. We prove little; the primary intent is to introduce these results to the non-specialist. In any event, the detailed proofs are extremely technical and far exceed the scope and purpose of this book. Lojaciewicz's comprehensive monograph [LOJ2], recently translated into English, gives a thorough treatment of his theorem together with all necessary background. It should be noted that the paper [BIM2] gives a modern treatment of many of Lojaciewicz's results, providing much more accessible proofs of the theorems.
+ +
STEP I (The Structure Theorem): We begin by introducing some terminology. A function H(xl ,. ..,xk- 1; xk) of k real variables is called a distinguished polynomial if it has the form
It is an important fact that any analytic function is locally, up to an invertible factor, a distinguished polynomial. More precisely we have
THEOREM 5.2.1 (The Weierstrass Preparation Theorem) Let f (xl, . . . ,xk) be a real analytic function in a neighborhood of the origin in a d assvme (as we may after a normalization) that f (0,. . . ,0, xk) # 0. Then f may be written in the form
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where H is a dzstinguished polynomial and U does not vanish in a neighborhood of the origin. The Weierstrass Theorem allows one to establish properties of analytic varieties by inducting on dimension. In particular, it is straightforward to prove that the collection of (germs of) real analytic functions in a neighborhood of the origin form an integral domain, and more specifically a unique factorization domain. Thus any real analytic function that vanishes at the origin admits a unique (up to order) factorization into irreducible factors. Likewise, if H is a distinguished polynomial then H admits a (unique) decomposition into irreducible distinguished polynomials. If H is a distinguished polynomial then the discriminant (see [VDW]) ( H ) ( , .. 9 x ~ - ~vanishes ) if and only if H(x1,. . . ,xk-1; xk) has a repeated irreducible factor. By using the fact that, for a non-trivial f , the discriminant cannot vanish identically, one may prove the following result:
Proposition 5.2.2 Let f be a function that is real analytic in a neighborhood of the origin and assume that f (0,. ..,0,xk) is not identically zero. Then there is a (possibly smalbr) neighborhood U of the origin and a distingu2shed polynomial Ho on U scrch that Ho has nonvanishing discriminant on U and the zero set o f f on U is identical to the zero set of Ho on U. The polynomial Ho is unique up to invertible factors. It is called the distinguished polynomial associated to f . By means of a careful analysis of the symmetric functions of the roots of a distinguished polynomial, Lojaciewicz is able to prove the following structure theorem for varieties:
THEOREM 5.2.3 (Lojaciewicz's Structure Theorem for Varieties) Let @(xl,. ..,xN) be a m l anaZytic finction in a neighborhood of the origin. W e may assume that @(O,... ,0, IN) # 0. After a rotation of the coordinates 21, . . . ,XN+, one has that there exist numbers 6-j > 0, j = 1,. ..N, and a system of distinguished polynomials
defined on Qk= {!xi Cjl < S j , 15 j 5 k ) such that the discriminant D: of does not vanish on & and the following prbpierties are satisfied:
Ht
5.2. LOJACUEWICZ'S STRUCTURE THEOREM 1. Each mot
C of H:
- .,x k ; -) on Qk
(XI,
satisfies (CI < 6t.
2. The set
Z
{X
= (XI,...I xN) : 1xj1
< bj\dj and @(x) = 0)
has a decomposition
The set VO is either empty or consists of the origin alone. Fop. 1 5 k 5 N - 1 we may write Vk as a finite, disjoint union
of k-dimensional subvarieties which have the following explicit description: (a) (Analytic Parametrization) Each :?I of N - k equations
is defined by a system
where each function X$ is each real analytic on an open Q k g Rk, subset Q:
and ~;(x,,...,xk)# O
for all (xl,. . . ,xk) E Q:, l = k + I, . . ..N.
Qk
(b) (Non-Redundancy) For any integers k , ~ , x ' either , = or Q: n Q;, = 0. In the second instance one has, for any & = k + 1 . . . . .N. either XT$ = x'r$ on Q: O r k X T $ ( X ~ ,. . . xn) # '*'$(zl, . . . ,ze)for all x E Q,. Q k I
?
CHAPTER 5. TOPICS IN GEOMETRY
156
(c) (Stmtijication) For each k the closure of Vk contains all the subsequent 6's: that is, Q n vk > Vb-l u .. .u Vo. [This property, while technical, is an important point. The lower dimensional varieties ~ 3j < , N-1, do not occur as isolated sets; they are in fact the zero sets of certain discriminants and (in a sense) form the boundaries of the components The example (*) at the beginning of V ,...,. X of this chapter illustrates this principle.] Lojaciewicz's Theorem teaches us that a real analytic variety can be stratified into submanifolds of dimensions O , 1 , . . . ,N- 1. The statement in the Theorem that the zero dimensional manifold can be (1e cally) taken to be the origin is just another way of saying that a zerodimensional manifold is a discrete set of points. Of course Lojaciewicz's Theorem is trivial when N = 1. For N = 2 it may be derived as an easy consequence of the local Puiseux series expansion. However for N 2 3 it is deep and new. Now we present the first principal application:
THEOREM 5.2.4 (The Vanishing Theorem) Let f be a nonzero real analytic function on an open set U C IRN. Assume that the zero set Z of f in U is non-empty. Define dist (x, Z) = inf { lx - z ( : z E 2).
Let E be a compact subset of U. Then there are a constant C and an integer q > 0, depending on E,such that
I f (x)l 2 C
>0
dist (x, 2)'
for every x E E. Notice that in one variable this result is trivial: by a compactness argument we may take U to be so small that it contains a single, isolated zero P of f . Then f vanishes to some finite order m at P and we may take q = m. For N > 2 matters are less obvious. However consider a special case. In case Z has the special form
5.2. LOJACUEWXCZ'S STRUCTURE THEOEGEM
then we may write f in the form
where g is real analytic and does not vanish. Since, on compact sets, g does not vanish it follows again that the desired inequality holds with q = rn. Now it is too much to hope in general that Z has the simple form of a hyperplane. However one might hope that Z is (the union of sets each of which is) a bi-Lipschitzian manifold; more particularly, we might realize Z as (the union of sets each of which is) the graph of a real analytic function that is in some Lipschitz class. [The explicit form of the Puiseux expansion suggests rather explicitly how this might come about in two dimensions.] Consider the example
For this f there is no problem verifying Lojaciewic's inequality on a compact set E that misses the origin: just perform a real analytic change coordinates and reduce to the hyperplane case. However the zero set of f has a cusp at the origin, and the simple device of a change of coordinates does not apply. Instead we notice that, near 0, z = z (f ) can be realized as the union of the sets
and = {(O' 0))-
Because each of I?: and I?; is the graph of a real analytic function in the y-variable that is Lipschitz 213, it is not difficult to see that
f (x. 3) _> C
y2 = C'
([y2]1/3)3
> C" - dist ((z,y), z ) ~ .
(In this particularly straightforward example the set I?! plays no explicit role in the analysis; however see the discussion below.) For the general case, an import ant part of Lojaciewicz's analysis involves showing that the varieties I?: are the graphs of the functions
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CHAPTER 5. TOPICS IN GEOMETRY
and that these functions are in fact Lipschitz of some positive order. In the two-dimensional example just discussed, the (implicit) role played by the zero dimensional variety I?! is to enable us to deduce that the worst points to consider are those on the x- and y-axes. Once we have this piece of information, the analysis becomes one dimensional. In higher dimensions, the exceptional set v ~U . .-.u ~ V o is more complex and one must obtain the estimate by inducting on the V s We can say no more about the matter here. X$
Our last application is the
THEOREM 5.2.5 (The Lojaciewicz Division Theorem) Let 8 be a real analgtic function on an open set U C_ IRN that vanishes identimlly on no component of U. If T 2s a distribution on U then there exists a distribution S such that
We sketched the proof of this theorem in the previous chapter, in the case when 8 is a polynomial. The Vanishing Theorem provides the critical estimate so that the same proof can be used for 8 real analytic. In fact, Lojaciewicz proves that any infinitely differentiable function 8 whose zero set satisfies the conclusions of the structure theorem for analytic varieties, and with the additional hypothesis that the functions X$ vanish only to finite order -- in a rather strong, quantitative sense that is implied by the Vanishing Theorem - also satisfies the conclusion of the Division Theorem. We refer the reader to [LOJl], [LOJ2] for further details. The thinness of the zero set of a non-constant real analytic functions can frequently be a powerful analytic tool. In [DAT] it is used to give a strikingly easy proof of the local solvability of constant coefficient partial differential operators.
5.3
The Embedding of Real Analytic Manifolds
Recall that a rnan2fold of dimension N is a paracompact Hausdorff space M that is equipped with a locally Euclidean structure in the following fashion: There is a covering U = {Uj)-gl of M by open sets
5.3. EMBEDDING OF R.EAL ANALYTIC MANIFOLDS
159
and there are homeomorphisms dj : Ui -r B, where B C W N is the unit ball. We specify additional structure on the manifold by imposing conditions on the transition from one coordinate patch U' to another. That is, the manifold is C q o r some k = 1,2,... if all of the transition functions #j
O
#il: #k
O
# ; l ( ~ )+ # j
#kl(~)
are C k . Notice that the condition that we check here is on a function (namely # j 0 # i l h) m Euclidean space to Euclidean space; therefore it makes sense a priofi to discuss smoothness of the function. When the condition holds for k = CQ then the manifold is said to be Coo or "smooth." When the maps are real analytic then the manifold is termed real analytic. In the case that N = 2n is even then we may identify W N with @" in a natural way. If the transition maps #i o #kl are holomorphic then we say that the manifold M is a complex analytic, or simply a complex, manifold. Function theory on an abstractly presented manifold (as above) can be inconvenient and tedious, for one must make constructions locally on the coordinate patches Uj and then paste them together (usually with a partition of unity). If the manifold can be realized in a natural fashion as a subset of Euclidean space then the manifold inherits the function theory of the Euclidean space - by restriction. Thus we are led to consider embeddings. In order to give a precise description of an embedding, we first must define the notion of a smooth (resp. real analytic, complex analytic) function on a manifold. If M is a smooth manifold then a function F : M -+ R is called C* or smooth if for each coordinate mapping # j : U j -+ B it holds that f o : B --+ R is C m . The definition of real analytic and complex analytic function on a real analytic or complex analytic manifold is of course analogous. Now a smooth mapping of a smooth manifold M of dimension N into a Euclidean space Rk is a function
$7'
where each f j is a smooth function from the manifold M into R. The mapping is called an embedding if it is a homeomorphism onto its image. Of particular interest and utility are proper eabeddzngs: an embedding F : M -+ Rk is called proper if for any compact K C RN
CHAPTER 5. TOPICS IN GEOMETRY
160
it holds that f-'(K) is compact in M. Another, more informal, way to think about the concept of "proper" is that if {pj) are points of M that "run out to the edge" of M then their images F(pj)P u n out to infinityY7in RN. In general, a manifold of dimension N does not embed into ElN. For example, a torus is a two dimensional manifold but will not embed into It2. A Klein bottle is a two dimensional manifold that will not embed into R3. In 1936, H. Whitney [WHIB] proved that any smooth manifold of dimension N can be smoothly, properly embedded in w ~ ~ + This '. result is sharp. In the period 1930-1960 one of the major unsolved problems in manifold theory was to properly embed a real analytic manifold into some Euclidean space. Whitney [WHIZ] was able to prove that there is a Cm embedding of such a manifold whose image in Euclzdean space is a real analytic submanifold of space; but such a result is of little use since the map does not preserve the real analytic structure of the manifold. In order to understand why the real analytic embedding of a real analytic manifold is difficult, we briefly discuss the proof in the C* case. By the very definition of manifold, one is given a local embedding: that is, the coordinate function # j is an embedding of Uj into RN.For each j let Aj be a C* function of compact support in Uj such Aj(x) = 1 on M (such a family of functions on a manifold that is called a partition of unity and is a standard construct in manifold theory--see [MUN]). Naively, one might hope that F(x)= Ej Ajbj is an embedding of M into RN.But of course this map will generally not be one-to-one. So we must pass to higher dimensions to separate the images of the different coordinate patches. This is the spirit of Whitney 's proof. The problem with emulating the preceding argument in the real analytic category is that partitions of unity do not exist. A real analytic function, either on Euclidean space or on a manifold, that is compactly supported (more generally that vanishes on an open set) must be identically zero. Thus entirely different techniques must be developed to treat embedding of real analytic manifolds. The problem comes down to constructing a large family of globally defined real analytic functions on the manifold. By the way that a manifold is defined, one only has the ability to construct functions locally (on the coordinate patches). Thus one needs a way to patch locally defined
xj
5.3. EMBEDDING OF REAL ANALYTIC MANIFOLDS
161
objects together in the real analytic category. Much in the spirit of the Stone-Weierstrass theorem, it suffices for our purposes to find globally defined real analytic functions on the manifold that separate points. There are three known ways to address the technical problem d e scribed above in the real analytic category. Each of these methods requires deep and detailed background in either sheaf theory, several complex variables, differential geometry, or partial differential equations. Limitations of space and scope make it impossible for us to present in detail any of these methods; however we shall briefly describe each of them. The first method, for compact manifolds, proceeds as follows (for details, see [ROY]): Suppose that one is given a compact real analytic manifold M that comes equipped with a real analytic Riemannian metric. Associated to this Riemannian metric is its Laplace-Beltmmi opemtor L - a second order, positive, elliptic partial differential o p erator on M that is invariant under isometries of the manifold. The eigenfunctions of the operator L are well understood: they will be real analytic (by the real analytic hypoellipticity of elliptic partial differential operators), they are countable in number, and they will separate points in a suitable way. In fact this last assertion follows from H. Weyl's theory of eigenvalues of elliptic operators on a compact manifold: the geometry of the manifold can be reconstructed from the spectral theory of a suitable elliptic operator on the manifold (see [CHA] and the more general index theory of Atiyah and Singer [PAL]). Thus, with some additional technique, the eigenfunctions of L can be patched together to manufacture an embedding of the manifold. The difficulty with the approach just discussed (certainly the simpler of the three) is finding a real analytic Riemannian metric. To construct a C" Riemannian metric on the manifold is an exercise with partitions of unity. But the construction of a real analytic metric, that is a matrix { g i f (x)}&, of functions that is positive definite for each x, begs the problem of constructing real analytic functions on a real analytic manifolds. While in some contexts the necessary functions, indeed the metric itself, are given to us from the problem being studied, in general the problem of constructing a real analytic metric is no simpler than constructing an embedding (note here that once the manifold is embedded then a Riemannian metric is automatically inherited from the ambient Euclidean space). Thus this approach, while appealing, does not completely settle the embedding problem.
CHAPTER 5. TOPICS IN GEOMETRY The partial differential equations approach to the embedding problem, which again only applies in the compact case, is due to C. Morrey [MOR]. It can be summarized as follows: One first constructs a positive, elliptic, second order partial differential operator with real analytic coefficients on the manifold M that has characteristics similar to the Laplace-Beltrami operator described in the discussion of the first method. Then the eigenfunctions of this operator become the basic tools for constructing the embedding. We shall say no more about this method. The third method, due to H. Grauert [GRA], applies to any real analytic manifold, compact or non-compact. It is not in the spirit of the present book because it reduces the embedding problem to an even deeper and more difficult problem in the complex analysis of several variables; but Grauert's is the only known technique for solving the general embedding problem. In order to avoid an extremely technical digression into the lore and machinery of several complex variables, we give but a brief description of Grauert7sideas. Let U be an open subset of IRN and let (xl, 12,. . . ,XN) be the Euclidean coordinates on U. We may think of U as a subset of CN in a natural way by means of the mapping
In this fashion we are considering the (trivial) real analytic manifold U as a submanifold of the complex manifold ir = {(xl zy,, X, ~ Y Z*, ,. X N + ~ Y N ): (21,X2, . . . 7 xN) E U ) . The manifold 0 is called a complexification of U. If 4(x) is a real analytic function on U and P E U then 4 has a power series expansion about the point P :
+
+
Of course there is an r > 0 such that the series converges absolutely and uniformly when lxj - Pj 1 5 r for j = 1, . . . , N. But then the function
is well-defined and the series converges absolutely and uniformly when Izj - Pjl 5 r. j = 1. . . . N. The frrnction $ ( r )h a holomorphic function
5.3. EMBEDDING OF REAL ANALYTIC MANIFOLDS
163
of several complex variables (that is, it is holomorphic in each variable separately - see WRAl] for a discussion of several equivalent d e f i ~ tions of holomorphic function of several complex variables). Thus the function is a cornplexification of the original real analytic function 4. We may perform this cornplexification procedure on the power series expansion of # about each point P of U. Of course, by the uniqueness of analytic continuation, two different complexifications about two different points of U must agree on their common domain. As a result of this procedure we obtain an open subset O of CN with U C O and a complex analytic function on fi such that = #. The function U is the complexification of the original analytic function U. Now if M is a real analytic manifold then, by a procedure anal* gous to that described in the preceding paragraph, each of the inverse coordinate functions 4;' may be "complexified" to functions The image of the complexified function will lie in an N-dimensional line bundle over the coordinate patch U j . We shall not provide details here, but refer the interested reader to [BRW]. That the transition functions J j o are holomorphic functions of several complex variables is a formality that follows immediately from the BruhatIWhitney construction. This procedure creates a complex manifold M that is a submanifold of an N-dimensional line bundle over the original real analytic manifold M and which has complex analytic coordinate functions. Thus M is realized in a natural fashion as a real analytic submanifold of the complex manifold M.
6
6
$1
6
&.
4~'
Grauert in fact proves an embedding theorem for (a small modification of) the complex manifold M. By restriction, this provides an embedding of the original real analytic manifold M. In order to give a description of the procedure, we need a new definition. Let U be an open subset of C". Let u be a smooth function on U. We say that u is plurisubharmonic on U if for each fixed a, b E C" such that Ua,b {C : C E @ and C - a + b E U ) # 0 it holds that the function
is subharmonic in the classical sense of function theory of one complex variable. Subharmonic functions are much more flexible objects than are holomorphic functions. For instance, they are closed under the operation of taking a maximum. They may be constructed as potentials of positive measures. Plurisubharmonic functions are like-
164
CHAPTER 5. TOPICS IN GEOMETRY
wise flexible. And just as the Riesz representation (see [TSU]) can be used to manufacture harmonic functions from subharmonic functions, so there are analogous devices in the theory of several complex variables to pass from plurisubharmonic functions to the real parts of holomorphic functions. Naturally a function u on a complex manifold W is termed plurisubharmonic if each of the compositions uo 4:' with inverse coordinate functions is plurisubharmonic. By means of an extremely ingenious argument, Grauert constructs on (a slightly shrunken version of) M a plurisubharmonic function p with the property that for every positive T)) is compact in M . real number r > 0 the set u-l({x E HP : x Such a function p is called a plurisubhamonic exhaustion function for M. Grauert proves that any complex manifold that has a plurisubharmonic exhaustion function is a Stein manifold. What is a Stein manifold? A Stein manifold W is a complex manifold that supports a great many holomorphic functions. Indeed, given any two point a, b E W there is a holomorphic function f on W such that f (a) # f (b). A s indicated in the first portion of this section, such functions are the basic tools for constructing an embedding. It is not too difficult to imitate the Whitney construction, using Grauert's separating functions, to construct an embedding of the Stein manifold M. We mention, however, that a deep theorem of R. Remmert [REM]provides even a proper embedding of M . This, by restriction, properly embeds the original real analytic manifold M and solves the embedding problem. We conclude this section by recording some results which are related, at least philosophically, to the subject proper of the present section.
<
Riemann first developed the concept of an abstract manifold with a metric structure (what we now call a Riemannian manifold) in 1868. In attempting to understand this circle of ideas, it is natural to wonder whether every such abstractly presented manifold has a realization as a metric submanifold of Euclidean space. It should be borne in mind here that the question of embedding the manifold dzfferentiably is a much simpler one and amounts, from our modern perspective, to an exercise in the concept of general position (see [HIW). However the problem of obtaining an isometric embedding is quite subtle. It was solved, using an ingenious argument, by John Nash in 1956 (see [NAS]). A nice history of the problem is given in that paper.
'
5.3. EMBEDDING OF REAL ANALYTIC MANIFOLDS
165
Our interest in the present section of the book is in real analytic manifolds. Since a real analytic manifold is a fortiori CbO,it follows from Nash's theorem that a real analytic manifold has a Coo isometric embedding. It is natural to ask whether there is a real analytic isometric embedding. In 1971 the following result was proved by Greene and Jacobowitz ([GRJ]):
T H E O R E M 5.3.1 Let M be a wmpact, real analytic manifold of dimension n. Then there is a real analytic, isometric embedding of M into
IW(3n2+11n)/2
The principal analytic tool in the proof of all the Nash-type theorems is a powerful version of the implicit function theorem. The classical inverse function theorem says, in effect, that a smooth mapping of Euclidean spaces is surjective in a neighborhood of any point where its derivative is surjective. Nash [NAS] provides an implicit function theorem for mappings of function spaces in which the classical notion of derivative is replaced by the Frechet derivative. The additional complication that must be dealt with in embedding problems is that there is a loss of derivatives that makes the most natural application of the implicit function theorem unworkable. Thus the Nash iteration scheme involves alternate applications of smoothing operators and implicit function theorem estimates. We can say no more about this rather technical material here. A nice introduction to the subject appears in [GRE].Additional work, for non-compact real analytic manifolds, appears in [GRO]. That paper also contains results about lowering the dimension of the target space in which the Riemannian manifold is embedded. The final word about embedding of Riemannian manifolds has not been heard, and there is still activity in the field. Of the twenty three problems posed by Hilbert at the International Congress of Mathematicians held in Paris in the year 1900, one of the most important and influential has been the fifth. A good working formulation of the problem is: Is every locally Euclidean group a Lie group? Let us explain what the question means, and then give the answer to the problem. A group is locally Euclidean if it has a topological structure which makes it a finite dimensional manifold, and if the group
CHAPTER 5. TOPICS IN GEOMETRY
166
operations are continuous functions on the manifold. No assumptions are made a priori about differentiability of the manifold or of the functions describing the group operations. A group is a Lie group if it is a locally Euclidean group and, in addition, it has a manifold structure that is real analytic and the group operations are real analytic functions on the manifold. THEOREM 5.3.2 ([GLE],[MZl]) Every locally Euclidean group is a Lze group.
In point of fact the techniques in the proof of this theorem, and the consequences of the theorem, have given rise to an entire subfield of geometric analysis. We refer the reader to [MZ2] and to [KOB] for a more liesurely introduction to this circle of ideas.
5.4
Semianalytic and Subanalytic Sets
Basic Definitions The theory of semianalytic and subanalytic sets is concerned with sets of points which can be described using real analytic functions. Here we will not be able to give complete proofs or even a complete exposition. We shall try to cover the highlights. The reader interested in a deeper treatment is referred to the book of Lojaciewicz [LOJ2] and to the paper of Bierstone and Milman [BIM2] and to the references cited there. Our presentation follows [BIMS] rather closely.
Definition 5.4.1 An algebraic subset of 8" is a set of the form
where P is a real polynomial. Clearly, algebraic subsets are those which can be described by polynomial equations. If we enlarge the allowable types of descriptions to include inequalities, conjunctions, disjunctions, and negations, then we have the following larger class:
Definition 5.4.2 The family of semialgebraic subsets of
IW"
is the smallest family containing the algebraic subsets of R" which is closed under finite intersection, finite union, and complement.
5.4. SEMIANALYTIC
AND SUBANALYTIC SET'S
167
There is another class of logical connectives: The quantifiers. The use of the existential quantifier corresponds t o projection. In this way we obtain what appears t o be a larger class.
Definition 5.4.3 A subset S of R" is subalgebraic if it is the projection of a semialgebraic subset of R"+" - R" x Rm for some m. Actually the term "subalgebraic set" turns out t o be redundant. That is because of the
THEOREM 5.4.4 (Tarski-Seidenberg) Every subalgebraic set is semialgebraic. In light of the preceding theorem and the logical equivalence of b' with 13-7, it is also true that no new sets will be introduced by the use of the universal quantifier. Now we consider replacing the polynomial in (*) above by a real analytic function.
Definition 5.4.5 Let U be an open subset of R". An analytic subset of U is (i) a set of the form
where F is a real analytic function on U. (ii) Let U be an open subset of Rn.The family of semianalytic subsets of U is the smallest family containing the analytic subsets of U which is closed under finite intersection, finite union, and complement. (iii) A subset S of W" is semianalytic if each point p E S has an open neighborhood U such that S I-I U is a semianalytic subset of U. (iv) A subset S of 9" is subanalytic if each point p E S has a neighborhood U such that S I-I U is the projection of a relatively compact semianalytic subset of Rn+" - Rn x Rm.
The compactness of a topological space is a property of the space and not of how it is embedded in another space. Thus it is relevant t o recall the
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CHAPTER 5. TOPICS IN GEOMETRY
Definition 5.4.6 A subset K of a topological space X is called relatively compact if Closx(K) is compact. A simple generalization is made by replacing Rn by a real analytic manifold:
Definition 5.4.7 Let M be a real analytic manifold. Let U be an open coordinate neighborhood in M. An ana(i) lytic subset of U is a set of the form
where F is a real analytic function on U. (ii) Let U be an open coordinate neighborhood in M. The family of semianalytic subsets of U is the smallest family containing the analytic subsets of U which is closed under finite intersection, finite union, and complement.
(iii) A subset S of M is semianalytic if each point p E S has an open coordinate neighborhood U such that S n U is a semianalytic subset of U. (iv) A subset S of M is subanalytic if each point p E S has a neighborhood U such that S flU is the the projection of a relatively compact semianalytic subset of M x N, where N is also a real analytic manifold For the purposes of analysis, the main results are the following:
THEOREM 5.4.8 (Uniformisation) Suppose that S is a closed subanalytic subset of the real analytzc manifold M. T h e n there exists a real analytic manifold N and a proper real analytic mapping # : N + M such that 4(N) = S. Further, N can be assumed to be of the same dimension as S. THEOREM 5.4.9 (Rectilinearisation) Suppose that S is a subanalytic subset of the real analytic manifold M of dimension m. Let K be a compact subset of M. T h e n there exist finitely m a n y real analytic functions #i: Rm + M , i = 1 , . . . ,p, such that
5.4. SEMIANALYTIC AND SUBANALYTIC SETS
169
there are compact sets Li c Rm, i = 1, . . . ,p, for which Ui 6i(Li) is a neighborhood of K in M,
(i)
for each i, q5y1(s) is a union of quadrants in Rm, where a quadrant i n Rm is a set of the form
(ii)
with
ai
E { "=", "< ",
"> ") for each i.
We shall also need the notions of "semianalytic function" and "subanalytic functions."
Definition 5.4.10 Let M and N be real analytic manifolds. Let S be a subset of M , and let f : S -+ N be a function. We say that f is semianalytic if and only if its graph is (i) semianalytic in M x N. (ii) We say that f is subanalytic if and only if its graph is subanalytic in M x N. There is also a notion of "semialgebraic function" that is defined similarly.
Definition 5.4.11 Let S be a subset of Rn.We say that f : Rn + Rm is semialgebraic if and only if its graph is semialgebraic in Rn x Rm,
Facts Concerning Semianalytic and Subanalytic Sets We state without proof some of the fundamental facts about semianalytic and subanalytic sets. The main tool used in developing these results is the Weierstrass Preparation Theorem.
THEOREM 5.4.12 Let S be a semianalytic subset of the real analytic manifold M . Then: (i)
Every connected component of S is semianalytic.
(ii)
The family of connected components of S is locally finite.
(iii)
S is locally connected.
CHAPTER 5. TOPICS IN GEOMETRY
170 (iv)
The closure and interior of S are semianalytic.
(v) Let U be a semianalgtic subset of M with U c S which is open relative t o S. Then U b locally a finite union of sets of the fom S n { x : fl(4> O,...,fk(X) > 01, where fi, . .. ,fr, are real analytic functions. (vi) If S 2s closed, then S is locally a finite unzon of sets of the fom {x : fl(.) 2 0, - 9 f k ( 4 2 01, where fi, . .. ,f k are real analytic funct2ons. The following theorem of Lojasiewicz allows us to see that, in contrast to the algebraic situation, not all subanalytic sets are semianalyt ic.
T H E O R E M 5.4.13 Let M be a real analytic manifold of dimension k. Let S be a subset of the real analytic manifold M. Necessarg and suficient for S to be semianalytic of dimension less than or equal to k is that there exist an analytic set Z of dimension less than or equal to k such that (i)
S c Z,
(ii) Clos(S) t o k - 1,
-
S is sernianalytic of dimension less than or equal
-
(iii) S ClosZ(S) is also semianalytic of dimension less than or equal to k - 1.
By the theorem, if a semianalytic subset of W" is of dimension less than n, then, in a neighborhood of each point, there must be a nontrivial analytic function which vanishes on the subset. We consider the following example of Osgood. Set S = {(x, y , z ) : 3u,v s.t. x = u, y = uv,
Z
= uveV).
Clearly, S is subanalytic; if S were semianalytic, then there would be some real analytic function f (x, y, z ) defined near (0.0, O ) , not identically zero, which vanishes on S. Assuming such a n f exists, we write
5.4. SEMIANALYTIC AND SUBANALYTIC SETS
171
where fj(x, y, I) is homogeneous of degree j . For (u,v ) near the origin in R2 we must have
so that for each j
0 = f j ( l , V , vev). Since f j is a homogeneous polynomial of degree j , we must have fi = 0, a contradiction. Thus S is subanalytic, but not semianalytic. For the semialgebraic sets, the Tarski-Seidenberg Theorem showed that projection did not lead to a larger class of sets. It follows a fortiori that the subsequent use of the complement will not lead to a larger class. For the semianalytic sets, this a fortiori argument cannot be used. In spite of this, we still have the
-
THEOREM 5.4.14 Let M be a real analytic manifold and let S be a subanalytic subset of M. Then M S is subanalytic. An important result on subanalytic functions is the following
THEOREM 5.4.15 Let M and N be real analytic manifolds, and let S be a relatively compact subanalytic subset of M. For a subanalytic function f : M + N the number of connected components of a fiber f - ( p ) is locally bounded on N.
'
Examples and Discussion It was asserted earlier that for an analyst the main results concerning semianalytic sets and subanalytics sets are the Uniformization Theorem and the Rectilinearization Theorem. In this subsection we shall illustrate this point. We start with an elementary inequality.
Definition 5.4.16 For n a positive integer and I c P set
Lemma 5.4.17 Let n be a positive integer. If t l , tz E R then
CHAPTER 5, TOPICS LN GEOMETRY
172
Proof: Set ti = %(ti), for i = 1,2. We may assume G 5 &. There are two cases depending on whether or not t1and t2 have the same sign. First we suppose
ti < o < G Set M = rna~{l<~l,&}, so M n 5 Itz
- tll. Then
we have
Next we suppose
or&
552.
In this case, we estimate
so that
Lemma 5.4.18 Let I be an open interval with 0 E I . Suppose h: I IW is real analytic and vanishes only at 0. If hc')(0) = . .. = h("-') (0) = 0 . then g : I
+B
and
h ( " ) ( ~>) 0,
defined by setting
is continuously difemntiable on I with g'(0) =
1
[f(")(o)]".
--+
Proof: The derivative of g is easily calculated away from 0, while the behavior at 0 is determined by using the power series for h.
Lemma 5.4.19 If f:R --+ R is a wntinuous subanalytic function, then f is locallp HErlder (Lipschztz) coontinuous Proof: The continuity of f is equivalent to the graph being closed, so the Uniformization Theorem is applicable. Thus there exist a one dimensional real analytic manifold M and a proper real analytic map #: M --+ R x R such that the graph of f is the image of #. Since we need to prove a local statement, we may assume M = R. Fix po E R and xo E #-' ( p o ) Let Ill and be the projection of R x R onto the first and second factors, respectively. We know that
has an isolated zero at t = 0: suppose it is a zero of order n. Let o be the sign of h(")(0). Set
By the second lemma, the Inverse Function Theorem applies to g, so g-l is defined and continuously differentiable in a neighborhood of 0. Now note that, with T, the translation rxo (2) = x 10,
+
holds
(ii)
+ 6) if n is odd, in a half-open interval [po,po + 6) if n is even and a = +1,
(iii)
in a half-open interval (po - 6, pol if n is even and a = -1.
(i)
in an open interval (po - 6, po
By the first lemma. f is Holder continuous on the interval where the above inequality holds. Since # is proper, there is either an xo E #-' (po) for which n is odd, or there are xl, x2 E #-I (po) with n even and with opposite signs for o, so that f is Holder continuous in an open interval about po.
Proposition 5.4.20 Let f :R" -+ Rn be a subanalytic function. If f is continuous, then f is locally Holder continuous. Proofr Let U be a bounded open set. Consider
A = { ( s ,t ) : 32, y E U s.t. 1x - yI2 = s2 & B = {(s,t) : s > 0 & t > o), C={(s,t):s
O ) , D=(AnB)uC, F = D -Int(D).
If
(x) - f ( y ) I 2 < t 2 } ,
Then F is the graph of a continuous subanalytic function from R to R, which by the preceding lemma is Holder continuous. The result follows from the Holder continuity of F at 0. Note that the continuity hypothesis is necessary since
is a semianalaytic function (which even has the intermediate value property), but is not continuous at x = 0. We also have the
Proposition 5.4.21 Let S be a stdunalytic subset of Rm. Then the distance function d: Rm + R defined by
is subanalytic and Holder continuous. Proof: Clearly, the distance function is continuous Holder continuity is an exercise. Set
-
indeed the
-
Then T Int(T) is the graph of the distance function and is subanalytic. In the context of the preceding proposition. we mention the following result:
THEOREM 5.4.22 (PoIya and Raby) Let S be a closed subset of Rm and let d:Rm -+ R be the distance function given b y d(x) = dist(x, S). The square of the distance function is real analytic in an open neighborhood of xo E S if and only if S is a real analytic submanifold i n an open neighborhood of xo.
Rectilinearization The proof of the Uniformization Theorem makes use of the notion of blowing-up which was discussed Section 1 in the context of Resolution of Singularities, so we will not discuss that here. But another useful consequence of blowing-up involves the following: Definition 5.4.23 Let M be a real analytic manifold and let O(M) denote the ring of real analytic functions on M. For f E O(M) we say that f is locally n o m a l crossings if each point of M has a coordinate neighborhood with coordinates xl, . . . ,x, such that
f (x) = xi' . . .xLm with each ri a non-negative integer and where g is non-vanishing in the neighborhood. Using the blowing-up technique, one can prove THEOREM 5.4.24 Let M be a real analytic manifold and let 0 f f E O(M). Then there exist a real analytic manifold N and a proper surjective r d analytic mapping #: N --+ M such that (i)
f o # is locally normal crossings o n N ,
(ii) there is an open dense subset of N on which difleomorphisrn.
# is locally a
In this section, we shall show how the Rectilinearization Theorem follows from the previous theorem and the the Uniformization Theorem. Recall the result: THEOREM 5.4.25 (Rectilinearization) Suppose that S is a subanalytic subset of the real analytic manifold M of dimension rn. Let K be a compact subset of M. Then there exist fiaitely many real analytic functions #i: Rm + M, i = 1 , . .. ,p, such that
there are compact sets Li c Rm, i = I, .. . ,p, for which U i#i (Li) is a neighborhood of K in M,
(i)
(ii) for each i, #fl ( S ) is a union of quadrants in Rm, quadrant in Rm 2s a set of the form { ( ~ 1 ? - - - , x :m x ) ~~ ~ , - -l . , x m ~ m O ) ,
with cri E {
LC-
-
I?
, "< ", "> ") for each i.
a
176
CHAPTER 5. TOPICS IN GEOMETRY
Proof of Rectilinearization: The result is local, so we may assume that M = Rm. Next, we find a neighborhood U of K such that there are closed subanalytic subsets Si,with
It is known from the previous section that the distance function to a subanalytic set is subanalytic, so defining di,j : U -+ IW by setting
we obtain a collection of continuous subanalytic functions. We shall show that there exist a real analytic manifold, N , also mdimensional, and a proper, surjective real analytic mapping 4: N + U such that each di,j is real analytic on N. Define the subanalytic mapping f :U -4El2' by f = (dl,l, . . . ,dr,2). By the Uniformizat ion Theorem applied to the graph of f , there exist a real analytic manifold N of the same dimension as the graph of f,that is m-dimensional, and a proper real analytic mapping iP: N -+ U x R" such that the image of iP is the graph of f . Let 111 and IIz denote projection of Rm x R2' onto the first and second factors, respectively. Setting 4 = Ill o iP, we see that 4 is surjective and 112o iP = (dl,, o 4,. . . ,d,,, o 4) is real analytic. Applying the above theorem, we obtain another real analytic manifold N of dimensions m and a proper surjective real analytic mapping N --+ N such that each diYjo 6 o ll, is locally normal crossings, from which the result follows.
+:
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Index Abel, N. H. 2, 23 algebraic set 166 almost increasing 79 analytic continuation 14 set 167 Artin, M. 90 associated function 72 sequence 79 Atiyah 161 Bang 79 Bernstein, S. 90 Besicovitch 55 Bierstone, E. 101, 142, 166 blowup 145, 146 Boas, R. P. 51 Boman, J. 101 Borel, E. 55 Browder, F. 91 Cauchy-Kowalewsky theorem 35 complexification 162 Denjoy-Carleman theorem 72 distance function 174 distinguished polynomial 153 domain of convergence 24 Ehrenpreiss, L. 100 exceptional divisor 145 Fourier transform 124 Gevrey classes 78 Grauert, H. 162 Greene, R. E. 165
Hadamard formula 7 Wahn-Banach theorem 123 Hartogs7stheorem 90 Hensel's lemma 84 Hilbert 165 fifth problem 165 Hironaka, H. 141, 151 Hormander, L. 80, 104 implicit function theorem 38 infinitely differentiable 11 interval of convergence 2 inverse function theorem 20, 3! Jacobowitz, H. 165 joint analyticity 91
k times continuously differentiable 11 Laplace-Beltrami operator 161 Lelong, P. 91 Lie group 166 logarithmic convexity 24, 69 Lojaciewicz, S. 170 division theorem 158 structure theorem 154 vanishing theorem 156 majorization 30 manifold 158 Markov's lemma 97 Milman, P. 101, 145, 166 Morrey, C. 162 multi-index 21 multiplication of distributions 116
INDEX Nash, J. 165 embedding theorem 165 normal bundle 40 crossings 175 space 40 symmetric algebra bundle 43 symmetric form bundle 43 ordinary singularities 148 Osgood, W. 170 Paley- Wiener theorem 135, 137, 138 partition of unity 59, 160 plurisubharmonic function 163 exhaustion function 164 Polya-Raby theorem 174 power series 1, 23 Pringsheim, A. 51 projective space 144 proper transform 146 Puiseux series 80 theorem 80, 87 quasi-analytic class 72 radius of convergence 2 rapidly decreasing functions 116, 124 real analytic a t a point 3 function 25, 40 on a set 3 section 43 section of the normal bundle 43 section of the tangent bundle 43 submanifold 38 rect iliiear izat ion theorem 168, 175 Remmert, R. 164 Rudin, W. 79
Schwartz, L. distribution 116, 125 functions 116, 124 space 124 Seidenberg, A. 105 -theorem 167 semialgebraic function 169 set 166 semianalytic function 169 set 167 separate analyticity 91 Siciak, J. 91 silhouette 23 Singer 161 Stein manifold 164 subanalytic function 169 set 167 tangent bundle 40 cone 148 space 40 Tarski, A. 105 tempered distribution 116, 125 uniformization theorem 168 Weierstrass preparation theorem 153 Whitney, H. 160 approximation theorem 63 decomposition 59 extension theorem 62, 69, 122 Zariski, 0. 141
