Steven G. Krantz Harold R.Parks A Primer of Real Analytic Functions
Birkhauser Verlag Base1 Boston Berlin
Authors' ad...
532 downloads
1097 Views
15MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
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 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 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
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:
+
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
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 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 ( 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( 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"
+
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). ¤
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( 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- 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 ( 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
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.
") 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