Nonlinear conditions for differentiability of functions

N O N L I N E A R C O N D I T I O N S F O R

D I F F E R E N T I A B I L I T Y O F F U N C T I O N S

By

JOHN DUNCAN, STEVEN G. KRANTZ AND HAROLD R. PARKS

0. I n t r o d u c t i o n

Let f: R--~R be a function and for 1 <=i<=k let ej: R--~R be C | function

(*) If ~j of E C ~, j = 1,2 . . . . , k, does it follow that f E Ca?

This type of problem arises frequently in nonlinear analysis. It is of partict

interest in the nontrivial situation where the Inverse Function Theorem does

immediately give the answer. In fact, the results which we are going to state can

viewed as an extension of the Inverse Function Theorem. The problem above was communicated to the second author in 1976 by M.

Taylor in the following simple form:

(**) If f2, [3 E C | does it follow that [ ~ C|

Until recently, the answer to even this ostensibly easy question was unkno~ However, in 1982 H. Joris answered (**) affirmatively (see [7]). In addition

answering (**), Joris was able to give an affirmative answer to (*) in case

(***) ~0~(t) = t',, with 0 < nj E Z, j = 1,2 . . . . , k, and g.c.d. {m, n2 . . . . . n~} = 1.

Curiously, Joris's ingenious proof has the features that (i) it is a proof I

contradiction and (if) it requires that the range of ]' be a subset of R. He asks in

paper, [7], whether this restriction on [ is necessary. (Indeed, he was unable

decide whether the analogue of (**) is true for a complex valued/.) He further asl

which other functions may be used for the ej's besides those given in (***). In the present paper, we are able to give fairly complete answers to Joris

auestions. While we are not able to give necessary and sufficient conditions on tb q,~'s to obtain an atfirmative answer to (*), we have rather broad stlfficiel

conditions and examples which indicate where the frontiers lie. We are further abl

to replace the domain of f by a smooth real manifold and the range of [ by

complex function algebra. The former follows immediately from a theorem of Ja Boman, as was noted by Joris. The latter is new and is one of the main points of thi

paper. Again, examples indicate that our results are close to best possible.

We now formulate our results as four theorems. Theorems 1, 2 and 4 should

46

JOURNAL D'ANALYSE MATHI~h~.ATIOUE. Vol. 45 1198J)

DIFFERENTIABILITY OF FUNCTIONS 47

considered the main results, while Theorem 3 suggests further avenues for research. In each of these theorems, we adhere to the following notation:

(1) M is a C | real manifold (see [6]);

(2) A is a function algebra (see [3]);

(3) ,pt,,P2,..., ~ are functions such that either (a) ~j: R---~C is C | j = 1 , 2 , . . . , k ,

or

(b) ,pj: C-~ C is complex analytic, j = 1 , 2 , . . . , k ;

(4) the (formal) Taylor series expansion for ~j, j = 1 , 2 , . . . , k, will be written

~ j ( t ) - ~,(c ) - "~. ' aj,(c )( t - c) ",''~

where 0 < ni l (c)< n~2(c)<- . . , each 0 ~ ai~(c) E C, each rj(c) is either a positive

integer or + oo. In case (3a) holds then t, c E R. In case (3b) holds then t, c E C and the power series is not merely formal - - it converges to ~i.

Notice that in case (3b) holds, we may extend each ,pj to a map from A to A by

means of the power series expansion; we also denote the extended map by ,pj.

We have examples (see Section 8) for which the only impediment to obtaining an affirmative answer to (*) is that f may be (badly) discontinuous. Thus in all our

theorems we will assume the continuity of f. This was not a difficulty in Joris's work as at least one of the functions ~j (t) = t'~ in (***) has a global continuous inverse (since at least one nj must be odd).

T h e o r e m 1. Suppose that (3b) holds, f: M ~ A is contin,~ous, and

(t) g.c.d. {n,(c), n21(c),..., nkl(c)} = 1

for each c E C. If ~j * f is C | in the strong sense for ] = 1, 2 . . . . . k, then f is C | in the strong sense.

It may be noted that it is only necessary in Theorem 1 for (t) to hold for

c E X = {im(a): a E imff)}

and it is only necessary for the r to be holomomhic in a neighborhood of X (here

" im" denotes image of a function). In case the range of f is suitably restricted, the complex analvticity of the ej 's

becomes superfluous:

T h e o r e m 2. Suppose that (3a) holds, f: M ~ R is continuous, and

(t) g.c.d. {nl,(c), n2,(c) . . . . . n~,(c)} = 1

for each c E imff). If q~j * f E C | j = 1,2 . . . . , k, then f E C |

It should be remarked that in Theorem 1, as well as in Theorems 3 and 4, a map

48 J. DUNCAN ET AL.

f: M---~ R is a[ortiori a map f : M ~ C , so the theorems apply to the real-valued

case.

Before proceeding, we should like to comment on the stringency of the

hypothesis (t). Notice that (t) is satisfied at any point c Yor which some , j satisfies ~ ( c ) # 0; for then nn(c)= 1 and the coprimality condition is trivial. In particular,

(t) is trivially satisfied by any monomial at any c # 0. If the ~0;'s are all polynomials,

then (t) is trivially satisfied at all but at most finitely many points. In Theorem I or in Theorem 2, if the ~j's are all real analytic, then (t) is trivially satisfied except

possibly on a discrete set. Simple examples show that some condition such as (t) must be imposed to obtain an affirmative answer to (*).

While Theorems 1 and 2 are fairly satisfying in their generality, they turn out to

be far from optimal in terms of the coprimality condition on the powers of (t - c).

For instance, we can prove

T h e o r e m 3. Suppose each ~oj is a holomorphic polynomial (that is, a

holomorphic function from C to C which is a polynomial). Let O(z ) = z m for some

0 < m ~ Z. Suppose there are positive integers ],, ]2, . . . , ]k such that

( i t ) g.c.d. {m, nn(0) . . . . . nl~(0), n21(0),..., n2h(0) . . . . . n~(0) . . . . . n~j~ (0)} = 1.

l f f : M ~ A is continuous, @j * f is C | in the strong sense for j = 1,2 . . . . . k, and O * f

is C | in the strong sense, then f is C | in the strong sense.

Notice that the presence of the monomial O in Theorem 3 guarantees that (t)

holds at all c ~ 0 . Of course ( t t ) is a substitute for (t) at c =0 . An interesting

consequence of Theorem 3, motivated by (**), is the

Corollary. If f : R - * R is continuous, f2 + f3 E C | and fm E C for some

0 < m E Z , t h e n f ~ C |

We show in Section 8 that the condition f2 + f~ ~ C | alone does not imply [ E C |

The condition ( t t) of Theorem 3 strictly generalizes the condition (***) of Joris

(simply let jl = j2 . . . . . j~ = 1), but is unsatisfying in its asymmetry.

Easy examples show that no condition on the nj~'s alone can be both necessary

and sufficient to obtain a result. There is a subtle interplay between the coefficients

{aj~} and the exponents {nj~} which must be better understood (some interesting commutative algebra considerations may be involved here). To this end we have

Theorem 4 which says that, for any "reasonable" set of exponents {nj~}, almost

every choice of coefficients {aj~} gives a theorem. In order to formulate Theorem 4, it is convenient to introduce the following

terminology. Let Pt(z), P2(z) . . . . . Pk (Z) be holomorphic polynomials; we will say

that these polynomials satisfy Condition 6e if whenever f : M ~ A is continuous

and such that P, * f is C | in the strong sense for j = 1 ,2 , . . . , k, then f is C | in the

strong sense.

D1FFERENTIABILITY OF FUNCTIONS 49

T h e o r e m 4.

where

Consider the holomorphic po l ynomia l s

P~(z ) = a l , z "" + a 1 2 z ' '2 + " �9 �9 + a , , a "",,

P2(z ) = a2~z ~' + a22z "~ + �9 �9 �9 + a2~z "~'~,

P~ ( z ) = a ~ z ~, + a~2z "~ + . �9 �9 + a~,~z"~,~,

O < n , < ni2 < " " < njt, for j -~ l , 2 . . . . . k.

A s s u m e there are indices a l , a2, . . . , o~j, wi th 1 < a, <- li for j = 1 ,2 , . . . , k, such that

g.c.d. {n,~ n2 . . . . . . . nk~ ~ 1.

Set d = ~ , 1,. T h e n the collection s~ o f d - tup les o[ coefficients

r n = 1 . 2 , , . . , ~

{ a l m } i . , . 2 . . . . . k C C a

such that P,, P2 . . . . . Pk satis[y condi t ion b ~ is a dense open subset o f C ~. Indeed, ~ is a

Z a r i s k i - o p e n set in C ~.

Three avenues for further research are indicated:

(i) Find necessary a n d sufficient conditions on {air}, {n/b} to obtain a theorem;

(ii) determine how many derivatives of ~ojof, j = 1 ,2 , . . . ,k . , are needed to

control K derivatives of f ; (iii) explore how general the range of [ can be. While Theorem 2 as it stands is already the natural generalization of Joris's

result, it is reasonable to formulate Theorem 2 for complex-valued or even

A-valued functions [ and ~i's which are C | but not complex analytic. Unfortu-

nately, our methods break down when the Taylor expansion for g,j involves both

z 's and ~'s. See the ends of Sections 5 and 8 f o r further remarks.

In Section 8 we also provide examples which show that A cannot be an arbitrary

Banach algebra. Also, there are non-commutative algebras A for which Theorem 1

is true and others for which it fails. Question (iii) may shed some light on the

structure of Banach algebras and conversely. To avoid needless repetition of arguments, this paper is organized so that simple

versions of the theorems are proved first and then necessary modifications which are needed to obtain the full results are given in subsequent sections. More

precisely, the contents are as follows:

Section 1 provides the proof of Theorem 1 in case [: M ~ R, (3a) holds, and the

~j's are polynomials_ Section 2 extends the result of Section 1 to [: M ~ C with (3b)

holding and the ~,'s holomorphic polynomials; it also contains a surprising proof

that the special statement (**) implies the more general statement (***). Section 3 contains a technical lemma which is needed later. Section 4 extends the results of

50 J. DUNCAN ET AL.

Sections 1 and 2 to f : M--> A. Section 5 proves Theorems 1 and 2 in their full

generality (i.e. for complex analytic or C | ~pj's). Section 6 proves Theorem 3.

Section 7 proves Theorem 4. Section 8 gives examples and further remarks.

The second author acknowledges helpful conversations with J. Anderson, C.

Fefferman and R. Jensen. He also acknowledges partial support from the National

Science Foundation. The other two authors are grateful to J. Conway for helpful

conversations. B. Mityagin drew our attention to [3] and [5].

1. The proof of T h e o r e m 1 for f : M ~ R and po lynomia l ~Pi's

Throughout this paper we will rely on the following useful theorem of Jan Boman

[1]: if f is a function on an open set U_CR N and if fo~b is C | for every C | map

~b: R--~ U, then f is C | Since the result of Boman is local, it is obvious that U may

be replaced by any smooth manifold.

As a consequence of Boman's theorem, we may always take the domain of our

function f to be a subset of R.

Throughout this section, the ~pj's will be assumed to be real polynomials. So for

each c E R we can write

�9 (c) / -

~P,(t)=~j(c)+ ~= 1 a i l ( c ) ( t - c ) " ~ , j = l ,2 . . . . . k,

for t E R, where each rj(c) is finite and each aj~(c) is real. We suppose that

(t) g.c.d. {nn(c), n21(c) . . . . , nkt(c)} = 1

for each c E R, and we suppose that f : R ~ R is continuous and satisfies ~pj o f E C |

] = 1 , 2 , . . . , k . It is our goal to prove t h a t / • C |

It is enough to prove that, for Xo ~ R fixed, f is C | in a neighborhood of xo. We

assume, since there is no loss of generality in doing so, that

Xo = f ( x o ) = c = 0 .

We therefore drop the argument " c " from the rj's, air's, and n:'s. We further

assume, without loss of generality, that

,p,(0) = ,p2(0) . . . . . ,~k (0) = 0.

We use the condition (t) in the following simply way: It follows from (t) that

there is a positive integer R with the property (1.1) below.

(1.1) If R _--- q ~ Z, then there are non-negative integers at, tr2 . . . . . ak such

that

o f l n n + o t 2 n 2 t + " �9 �9 + ~ k n , ~ t = q.

As a result, there are monomial functions .,gq : R k ~ R, for q = R, R + 1 . . . . . such

DIFFERENT1ABILITY OF FUNCTIONS 51

that

(1.2) Mq (~o,(t), ~o2(t) . . . . . ~ok (t)) = t q + (higher order terms).

Choose distinct prime integers pz, p2 which exceed R. Then there is an integer

R, > max{p,,p2} with the property (1.3) below.

(1.3) If R I < q E Z, then there are non-negative integers a, /3 such that

a p l + [3p2 = q.

Now, the polynomials in t {Mq (~pl . . . . . ~Pk)}q=a,R*~.. are clearly linearly indepen-

dent over R by (1.2). So, by taking finite linear combinations of them, we may

obtain polynomial functions ~1, 92: R ~ ---> R such that

P , ( t ) - ~; (q~l( t ) , q~2 ( t ) , . . . , ~Ok (t)) = t r' + (terms of degree _--- R1),

P2( t ) - ~2 (q , , ( t ) , ,p2(t) . . . . . ~o~ ( t ) ) = t p2 + (terms of degree > R1).

Introduce the dummy variables s~ = t p' and s2 = F 2. By (1.3) there are polynomial functions 2r 2r R e--* R such that

P~( t ) = .Ar,(s~, s2) = s~ + (higher order terms),

P2( t ) = X2( s t , s2) = se + (higher order terms).

By the Inverse Function Theorem there are neighborhoods U, U' of the origin in

R e and a C | diffeomorphism

�9 : U ~ U '

such that

(1.4) (b(N,(s, , s2), )r s2)) = (s,, sz).

It follows that

(1.5) *[~t(~p,(f) . . . . . ~o~ (/)), ~2(~o,(/),... , ~o~ if))] = (/% fP~)

when ~(~o~ff) . . . . . q~k (,f)) and ~2(q~Lf) . . . . , q~k (J)) are small. This will be true in a

neighborhood, ~, of 0, because jr(0) = 0, jr is continuous, and the ~oj's and ~t and ~2

have no constant term. Each component of the right-hand side of (1.5) is C | on ~7, since ~, ~ , ~ ,

~0t *jr, ~o2*jr,..., ~o~ *jr are C | Thus Joris's hypothesis (***) holds; so, by [7] (the

results in [7] are easily localized with cutoff functions), [ is C" in a neighborhood

of 0. []

It should be noted that the Inverse Function Theorem argument yielding (1.4)

c a n n o t be applied to ~ and ~ individually. This is a necessary feature of the

theory, since there can be no theorem unless we are dealing with more than one polynomial (see Example 1 in Section 8).

52 J. DUNCAN ET AL.

2. The proof of Theorem 1 for f : M---~C and ho lomorphic polynomial q~j's

According to B o m a n ' s T h e o r e m [1], we may assume that f : R--~ C. We also will

a ssume in this sect ion that the funct ions ~pj are ho lomorph ic polynomials . The

a rgumen t of Sect ion 1 works just as well in this case ( indeed, ~ : U---, U ' is now

b iho lomorphic , with U, U ' C C 2) and allows us to assume that there are just two ~j ' s

and that they have the form

= : , ,

2(t) =

with n~ and n2 relat ively pr ime. (Recall that Jor is ' s results (**) and (***) do not

apply to complex-va lued funct ions ~ So we must now show that if ~ l ( t ) = t n',

~ 2 ( t ) = t ~ and [ : R - , C satisfy ~ o [ E C | ~ 2 o f E C | then f E C L )

We first do the case n~ = 2, n2 = 3. The genera l case then follows by a ra ther

startl ing inductive a rgument .

For the case n~ = 2, n2 = 3, We write [ = u + iv with u and v real valued. Since

]'2,f3 E C | then [[14,1[[~ E C | By Jor is ' s t heo rem, ][12• C ~. H e n c e

(2.1) u 2 + v 2 E C |

But

(2.2) R e { f } = u 2 - v 2 E C |

hence

(2.3) u 2, v 2 ~ C |

We next show that u 5, v s E C | By Jor is ' s result for rea l -valued functions, it

follows f rom this and (2.3) that u, v E C ~, hence f E C |

Now, it is easily checked that

(2.4) 16u 5 = - 9 Re{/ '} + 25u 2 Re{/3} - 15v 2 Re{/3},

so by (2.3) and our hypotheses we have u 5 E C | Likewise, we have

(2.5) 16v 5 = - 9 Im{/5} - 25v 2 Im{/3} + 15u 2 Im{/5},

so v 5 E C | This comple tes the p roof for n~ = 2, n2 = 3,

For the full result of this section, we need the fol lowing l emma.

L e m m a 2 .6 . Let f: R---, C be continuous. Suppose there is an integer R >= 2

such that f" E C | for every integer n >= R. Then f E C |

T h e result of this sect ion is immed ia t e f rom the l e m m a since fn, E C, [ ~ E C, with

g.c.d. { n , n2} -- 1, implies that there is an R satisfying the hypothes is of the l emma.


P r o o f of t h e L e m m a . The proof is by induction on R. The case R = 2 has

been handled by the argument for n, = 2, n2 = 3 given above.

Suppose the case R = Ro has been proved and that f" E C | for n _-> Ro + 1. Let

g = fz. Notice that

g" E C | for n > S =- ~Ro12] + 1,

where ~t~ is the greatest integer less than or equal to t. Since [Ro/2~ + 1 =< Ro, the

induction hypothesis applies and g E C | A similar argument shows that h = f3 E

C | Since f , f3 E C | it follows from the argument for m = 2, n2 = 3 given above

that f ~ C | This completes the inductive step and the proof. []

Observe that the argument used to prove the lemma shows that the solution of

the rather general problem (***) of Section 0 is an immediate consequence of the

special case (**).

3. E s t i m a t e s o n t h e d e r i v a t i v e s of f

In order to prove the results about Banach algebra valued functions, we require

not simply the algebraic ideas used so far, but we need estimates on the derivatives

of f. The required estimate is contained in Lemma 3.1.

L e m m a 3.1. Let pz, p2 . . . . . pk be positive integers with

g.c.d. {pl, p2 . . . . , p~ } = 1.

Suppose / E { 0 , 1 , 2 . . . . } and 8 > 0 . Then there exist an integer p =

p(p, , . . . , p~, I, 8) > 0 and a constant K = K (pl . . . . , p~, l, 8) > 0 with the property that

if f : R---> C is continuous and is sudh that

f', E c ' ( n ) , / o r i = 1, 2 . . . . . k,

then f E C| and, for each x ~ R,

su (f ' , ) (y) (3.2) - ~ ' r f ( x ) < K 1 +l,-,~<a ~ x ' "

We now proceed to give the proof of 3.1 under the additional assumption that f is

real-valued. (The case of complex-valued f is treated at the end of the section.) This

consists of certain modifications of Joris's arguments in [7]. For simplicity, we adopt

his notational conventions. First recall that, as usual, the coprimality of the pj's

implies that there is an R > 0 such that r E c | for all n => R.

N o t a t i o n of J o r i s . If h: R ~ R is smooth, then

d" h . ( x ) ~: D , h ( x ) - dx--"-r h ( x ) ,

h7 denotes (h~)" (not D , ( h " ) ) .

54 J. DUNCAN ET AL.

Recall that for g ~ C| U an open subset of R, and for integers tr, s, n, m

satisfying m _-> s -> 0, o- => 2, n => o's, Joris defines funct ions

F(s, n, m) = F,(s, n, m)(x)

in the following recurrent fashion:

F(0, n, m ) = Dn ( g " ) if n, m _--> 0,

F(s,n,m)=,~o ~ (n) g , F ( s _ l , n _ r , m _ l ) i f m > s > 1 , n >ors.

Also, with

R=>2 , tr_->2,

m = 1 + C - g - ( R - 1) ,

r = R - 1, n -- m ( t r - 1)

(where ~- ] means " in t ege r par t " ) , Jor is sets

F = n-~ ( - 1 ) ' (m - 1 - s)t,1 F(s, n, m).

(Here , when k E l § k<,~ deno tes the p roduc t k �9 (k - 1 ) . . . (k - r + 1)). Now we

must replace L e m m a 4 of Jor is ' s p a p e r [7] by

L e m m a 4'. F is of the form

F = Y~ ~ b w,.bg ~- ~ gr G

a + b ~ m - R

with each Wo.b a polynomial in go, g~ . . . . . g,-2 and D, (gJ) for j >= R. The form of each W~.b is independent of the choice of g.

P r o o f . We apply L e m m a 2 of [7] with s < m - R ; then m - s _ - > R and

a+b<=s. []

We replace L e m m a 5 of [7] by

L e m m a 5'. F is of the form

*~bg~-lgr F = cg~-a + ~.-~"o w* " b a + b ~ i m

a < m

with each w*b a polynomial in go, gl . . . . . 8,-2 and D~(g j) for] > R. The [orm o[ each w*,~ is independent of the choice of g.

We can use Jor is ' s p roof of L e m m a 5 of [7] to p rove L e m m a 5'.


Next, we can modify the statement of Joris's Main Lemma of [7] and use Joris's proof with Lemmas 4 and 5 replaced by Lemmas 4' and 5', respectively, to obtain

the

N e w M a i n L e m m a . Let g E C'OR), R = 2, tr > 2, and

m - - 1 + ( R - ; ) .

~ n e n

(3.3) 0

a § a < m

Here each w.., is a polynomial in go, g~ . . . . . g.-2 and D,(g j) for/>= R whose form is

independent of the choice of g.

Now, to obtain estimates: First, multiplying f by a C7 cutoff function, we may suppose f E (7, (R) is real-valued and fJ E C| for ] _-> R. By Joris's theorem, we

know that f E C**OR), so we can apply the New Main Lemma:

Let t* be chosen so that

13.4)

We get

m = ..~ f.-,f.w..b(f). f o'-I a b

0

a+b~ im a ~ m

I/.-,(t*) J = sup, ,, Jf.-,(t)l.

m - I

13.5) [f,,_,(t*)]" = ~0 [[,,-,(t*)l*(w,~o(f))l,-,-,

)ecause f . ( t * ) = O (f is real-valued!). We can think of (3.5) as a polynomial

;quation satisfied by f . - ,( t*). If

W = max{/(w,.0(f)) 1,-,. l: a = 0, 1, 2 . . . . . m - 1},

t is elementary that

3.6) I/.-l(t*)[--< 1 + roW.

ince each w.~0 is a polynomial in fo, f l , . . . , [ . - 2 and D , ~ j) for ./>-R, there exist '=> 0 an integer and K ' > 0 (depending only on R and o-) such that

~-2 d* ~'+ P' P' P'

56 J. D U N C A N ET AL.

Using (3.4), (3.6) and (3.7), we obtain (3.3) by induction on tr, completing the proof

of (3.3) in the real-valued case.

Finally, the identities (2.1), (2.2), (2.3), (2.4), (2.5) and the inductive proof of

Lemma 2.6 can be combined with the real-valued case of (3.1) to obtain (3.1) in case

[ is complex-valued.

4. The proof of T h e o r e m 1 for f : A4--~ A and h o l o m o r p h i c po lynomia l ~0j's

In this section, we assume that [: M ~ A is continuous, where A is a function

algebra. We assume that the functions q~i are holomorphic polynomials. We will use

the notation of Section 1 for the Taylor expansion of ~s, but we take t, c E C and, of

course, we have a ja (c )EC. We assume that g.c.d. {nlt(c),...,nkl(c)} = 1 for each

c E C and that ~pj *f is C | in the strong sense for j = 1,2 . . . . . k. Our goal is to prove

that [ is C | in the strong sense.

Our strategy is to reduce the problem to one about differentiability in the weak

sense by using multiplicative linear functionals. A basic fact about complex function

algebras which we use is the following:

If A is a function algebra, a ~ A, II a U is the supremum norm of a,

(4.1) and r(a) is the spectral radius of a, then

Ila II = , (a ) .

Recall that in any Banach algebra

r(a) =- sup{ I y (a ) [ : ~ U Spec(A)},

where Spec(A) denotes the set of all non-zero multiplicative linear functionals on

A. For a discussion of these matters see [2; w167 17] or [3; w

Before beginning the proof, it is worth noting that

r(a)<--Ilall holds in any Banach algebra. Moreover, our arguments apply in any Banach

algebra A which satisfies

[la I [ - -< C,r(a)

for all a E A with C1 independent of a. It turns out (see [2; w that such an A has

an equivalent norm under which it is a function algebra.

Now, we turn to proving [: M --~ A is C | in the strong sense. Again, it is enough

to consider a coordinate patch on M. Thus we may and shall assume [ : R~---~ A,

where N = dim(M). Fix x0 E R N. We will prove f is C | in the strong sense in a

neighborhood of xo. Our method is to obtain uniform estimates over a neighbor-

hood of xo. We begin by assuming that N = 1. Thus, we are going to consider

[ : R--~ A and obtain estimates valid in a neighorhood of x0 E R.


Fix 3,oESpec(A). We can apply the argument of Section 1, but now with

c = yo o f(xo) and without the assumption

,p,(c) = ~2(c) . . . . . ~,~ ( c ) = 0,

to obtain polynomials P,, P2: C ~ ---> C, relatively prime integers p, and p2, and a

biholomorphism ~: U--* U', with U and U' neighborhoods of the origin in C 2,

such that

~[P,(~p,(z) - ~o,(c ) . . . . . tp, (z ) - ~p~ (c )), P2(~o,(z ) - ~o,(c ) . . . . . tpk (z ) - ~ok (C))]

(4.2) = ((Z -- Cf ' , (Z -- C f 2)

holds when I z - c I is sufficiently small, say

Iz - c l < ~ o ,

where e,, > 0. Of course, in (4.2), P,, P2, p,, p2, and �9 (and, hence, to) depend on the

choices of x,, and 3,~ Since f is continuous, by hypothesis, we can find 80 > 0 such that if x E R with

Ix -Xol< &,,

then

(4.3) I l f tx) - f(xo)l I < �89

Replacing f ( x ) by f(x,, + 2~%(arctan(x -Xo))/cr), we may assume (4.3) holds for all

x E R; this will simplify notation. Since f(x~ Spec(A)---*C (the Gelfand trans-

form of f(xo)) is continuous when the Gelfand topology is used on Spec(A ) (see [2;

w or [3; w we can find 2r a neighborhood of 3'o in Spec(A) such that if

3' E ,hco, then

I v[f(xo)]- 3,o[f(xo)ll< �89

and, consequently,

1 3 , I f ( x ) ] - 3,off(xo)]l < to.

Thus (4.2) holds with z = 3,If(x)] when x ~ R and 3, E No. If g: R--*A is any function, x ~ R , h >0 , and n E {0,1,2 . . . . }, we define

AT, g ( x ) = ~ ( - 1 ) ' ( 7 ) g (x + ( n - 21)h);

these are the familiar finite difference operators of partial differential equations

(see [8]). Notice that if g: R-~ A is C | in the strong sense and y E Spec(A), then

3,og: R---~ C is C | in the classical sense and

d I d I

(4.4) dx ' (v ~ g ) = 3, ~ g' I = 0 , 1 , 2 , . . . .

58 J. D U N C A N E T A L .

Now, suppose y E ,No and F: R--~ C is defined by setting

(4.5) F ( x ) = y * f ( x ) - 3'0 o f(xo).

Then by (4.2) applied with z = y o f and the fact that ~0 ,*y* /= y*~,,*/, i =

1, 2 . . . . . k, we have

(4.6) ~,[p, (r . r o f - ~ , ( c ) , . . . , ~,. r o f - ~ (c)),

p,(T o ~, o f - ~ , ( c ) , . . . , T. r o f - r (c))] = (P , , F'9.

Thus, we have

F',, F p~ ~ C'Qt).

Fix n ~ {t, 2 . . . . }. By the estimates (3.8) of Section 3, there is an integer p => 0 and a

constant K > 0 such that, for x E R,

2 p d~

But then by (4.5), (4.6), and the fact that elements of Spec(A) have norm 1, it follows that, with different K and p,

I I[ I) (rof)(x) ~ K a + , , ~ , ~-r~,(, ,of)(y)

By elementary calculus it follows that

l~7,(y o f ) ( x ) l ~ L, (4.7) r~s-,~<, h"

O<k<l

where L depends on xo, y0, and n, but is independent of the choice of y E ,fro. Since

Spec(A) is compact in the Gelfand topology, we can cover Spec(A) by finitely

many neighborhoods such as ./f0. Thus we see that, with a new choice of L, (4.7) holds for all y E Spec(A). By (4.1), we have

il ll SU = s u p ~.-,~'<,II h" Jl ,~s~A, 0< k < 1 ;Ix -Zol< 1

O<k<l

= sup "r~Spce(A)

O<h<l

= L .

By Theorem 6.1 in [8] (which, by inspection, holds for Banach algebra valued functions), we may conclude that

d d 2 d "-1 I , - ~ I,-Z~ I, . . . . dx.-~- f


exist in the strong sense, are continuous, and are bounded for ]x - x01 < 1, with the

bound depending only on n and L. Since n was arbitrary, the proof in case N = 1 is

complete.

For N > 1 we reduce to the one dimensional case as follows. If f : RN---> A

satisfied our hypotheses, then the proof above shows that we can obtain local

bounds on

for every k and fo r / ' = 1 . . . . . N. Then Theorem 9.1 of [8], which by inspection

holds for Banach algebra valued functions, yields that [ is C ~ as a function on R N.

5. The proofs of T h e o r e m s 1 and 2 in their full general i ty

The proof of Theorem ! in its full generality (i.e. with complex analytic ~j's) is

identical to that for holomorphic polynomial ~t 's; in particular, the proofs of (1.4)

and (1.5) go through verbatim, for the power series expansion of qb converges to r

Thus, we turn our attention to Theorem 2.

In order to prove Theorem 2, we must inspect the proof in Section 1 for the case

where ~p~, r . . . . . ek are polynomials and ascertain where the special properties of

polynomials are used. The answer is that they were only used in steps (1.4) and (1.5)

in which we found a C ~ change of cooordinates which transforms

to

. . . . . ( f ) ) , . . . . . ( f ) ) )

(f"',fPg.

To do this in the general C | case we will need a preliminary result. Set

I=RN{t: -l_-<t_ -<I}.

We endow C=(I) with the metric topology given as follows:

If ~ E C=(1), let

I I , p l l , = s u p , / = o , 1 , 2 , . . . ; t e l

for [, g E C~(I) we define the metric distance between them to be

O(f, g)___,.~ 2-,/ I I f - g , l + l l f - ~ l b ) "

Also, C| x I) will be given a similar metric topology. All functions throughout the remainder of this section will be real-valued.

60 J. DUNCAN ET AL.

L e m m a 5.1. Let p~ and p2 be relatively prime positive integers. Let X C C|

consist of all/unctions, [, such that dJf(O)/dt j = 0 unless j = ap~ + [3p2 with 0 < a,

E Z. Then X is a closed linear subspace o/C| and every f E X can be written in

the form

for some F E C| x I).

P r o o f .

f(t) = F(t", t p~)

Let [ E X. There is a formal power series G

G(x,, x2)

such that G(t p,, t p2) is the Taylor series expansion of [ about 0. By E. Borel's

theorem there is a g E C| with Taylor series G at 0. Let

h( t ) = [ ( t ) - g(t p,, tPO.

Then h has null Taylor expansion at 0. Assume without loss of generality that pl is

odd. By Glaeser's Theorem [5], there is a function k E C| such that h(t) =

k(tPO. Thus

f(t) = k(t") + g(t'., t'2),

proving the result. []

Now, we can use Lemma 5.1 to obtain (1.5) in the general case. We continue to

use the notation of the Lemma. We formulate the result as

Proposition 5.5. Let p, and p2 be relatively prime positive integers and let

R~ > max{pl,p2} be an integer such that:

If n >_- RI is an integer, then there exist non-negative integers a and fl with

n = ap~ + /3p2 .

I[ P~ and P2 are C | on R satisfying

Pl(t) ~ t p~ + (terms of degree => R~),

P2(t) ~ t p~ + (terms of degree => Rt),

then there are neighborhoods U and U' o/ the origin in R 2 and a C | diffeomorphism

�9 : U ~ U '

such that, for small It l,

P r o o f .

�9 (P,(t), P2(t)) = (t',, t'O.

Set Qi = P, I L i = 1,2. We have QI, 02 ~ X, so by Lemma 5.1 we can


find I;:,, F2 E C~(I x I) such that

Qi (t) = F~ (t p,, tP~), for t E L j = 1,2.

Define ",It: I • I---> R 2 by setting

$(x, y) = (F,(x, y), F2(x, y)).

Since R, > max{p,,p2}, we see that detJac, ,qt(0,0)~ 0. In fact, we have

d' ' (o,o), - a t ' , a 1,-o = (p,)! ax

0 ~ d " 0__F2F (0, 0), - dt,2 Qi 1,-0 = (p2)! Or

from which we obtain det Jac~ ~(0, 0) ~ 0. Applying the Inverse Function Theorem

at (0,0) and denoting ~-1 by ~, we see that

r y), F2(x, y)) = (x, y),

so when we set x = t p', y = t ~2 and take It] small enough,

a , (e , ( t ) , = ( t ' , , : 9

holds. []

Why does this proof not go through when f is complex-valued? The formulation

of Theorem 2 would then necessitate consideration of functions tpi: C ~ C which

are C | but not complex analytic. Such functions have formal power series expansions of the form

,~ aa.#z * Y. ~.

Curiously, we do not know how to formulate or prove the necessary reductions

when both z 's and f ' s are present.

In principle, the statement of Theorem 2 even makes sense for f: M---> C , (X) ,

for the functional calculus on the Banach algebra Cc (X) extends to C | functions, indeed to continuous functions. Of course, we cannot handle this situation either.

6. P r o o f of T h e o r e m 3

We begin with a lemma that illustrates the basic technique.

L e m m a 6.1. Suppose f: M--->C is continuous and set

P ( z ) = z"~ + a2z "~ + . . . + a,z m,,

Q ( Z ) = Z 'h + bez % + . . . + b,z",,

62 J. DUNCAN ET AL.

where

O < m ~ < m 2 < . . . < m , , O < n x < n 2 < . . . < n ~

are integers. I[ po[, Q o [ E C | a2r m 2 - m ~ < n z - n ~ , g.c.d.{mt, m2, n , }= l ,

then [ is C | near any point xo with f(xo) = O.

P r o o f of L e m m t t 6 .1 . I[ P~), O ~ ) E C | then we also have

[P ( f ) ] " , - [O(/ ) ]" ' = n,azf "'"'*'~--", + (higher order terms) E C ~.

Since

g.c.d. {m,, n,, re,n, + mz - m,} = g.c.d. {m,, m2, n,} = 1,

the lemma follows by a minor modification of Theorem 1. []

The assumption m2 - ml < nz - n, in Lemma 6.1 insured that no term of [Q~f)]m

could cancel the n~a2[ m,",+'~-', arising from [P(/)]",. One way to avoid this

difficulty is to assume Q is a monomial; then we can repeat over and over the basic

method of the proof of Lemma 6.1. This notion is expressed succinctly in our

inductive proof of Theorem 3.

P r o o f of T h e o r e m 3. We will write simply n~j for n~j (0). We use induction on

k

(6.2) ~ (n,,, - n,,).

Note that if E~=t ( n q , - n i l ) = 0, then we can apply Theorem 1.

Now, suppose E~.~ (n~s,-n~)>0 and that Theorem 3 is valid for any smaller

value of (6.2). We may assume, without loss of generality, n ~ , - n . > O and

a,,(0) = 1. We consider a new, larger, system of holomorphic polynomials which

will also satisfy the hypotheses of Theorem 3, but for which the value of (6.2) is

strictly smaller. For i = 1,2 . . . . . k, set

Set

r = ,p, (z).

C k + , ( z ) = [ ~ , ( z ) ] ~ - [ O ( z ) F " " ,

where a is a positive integer to be chosen later. We will write

f f ~ ( z ) = ~ t i , . z ~,', i = 1 , 2 . . . . , k + l . /73

Also, we set

[, = 1,~ =i ,

jk§ will be chosen later.

for i = 2 , 3 , . . . , k;


F o r ] = 2 , 3 , . . . , j , , let us find the coefficient, Kj = K~ (a) , of z ~="',§ in ffk.l.

Writing

J i

B= {(/31, fl2 ..... /3.~): B' EZ, fl, >0, ~ fl,=am, ~ tim. =otmntt+nt,-n.},

we have J

Kj ( a ) = (,,.....~,)~, {(/31 ' a m / 3 j ) ~ (a ' l (O))a' l"

Because n~t _-> n . + 1 f o r 1 = 2,3 . . . . . /, we conclude that

J

7 _ ~ , < = n l j - n . for ( B , . . . , r B.

Setting j-1

. . . . . o , , - - n , n /

we see that

(6.3) K , ( a ) = ~ [ a m ( a m - 1 ) . . . ( a m - ~ ' y , + l ) ( a t,t+l(0))~"'~

(,,....~_,)cc t 3tl ! J '

so Ks(a ) is a polynomial in a of degree not exceeding nii - - nil. Note that Ks(a ) is

not identically zero, because we can part i t ion the sum on the right side of (6.3) by

the size of Y.I~ yl: the largest value of E Tt that contr ibutes a non-vanishing

coefficient of a m ( m - 1 ) - . . ( am - X "Ft + 1) de termines the degree of Ks(a) , and

we observe that for E T , = I it holds that ( y l , . . . , y / _ j ) = ( 0 . . . . . 0 ,1) and the

coefficient is a t j # 0 .

Now, we choose a to be larger than any of the roots o f / ( 2 , K3 . . . . , Kj, and we

choose/'-k+~ so that

Jv~k+l.:lt+l = f f m n l l + n l h -- n i l .

Since

we see that

Also, we have

nk+l,I ~ s -~ n12 - - ni l ,

k+ l k

g.c.d. {m, h,,, h2, . . . . . h 2 h , . . . , rik ~t . , , . . . , tik*t.rk.,}

= g.c.d. {m, n]~, n2 t , . . . , n2i7,. �9 �9 nk i , . . �9 nkik, amt lH + hi2 - n i l , . . . , otmnH + n11, - nil}

= g.c.d. {m, n,, . . . . . n~i . . . . . . nkx . . . . . mjk} = 1

64 J. DUNCAN ET AL.

and if, o f is C | in the strong sense for i = 1 . . . . . k - 1 . So, by the induction

hypothesis, f is C | in the strong sense. []

7. Proof of T h e o r e m 4

In this section, we are given holomorphic polynomials

Pj ( z ) = ~ a ~ z "~., j = 1 . . . . . k. m = l

Also, we are given integers aj with 1 _-< aj _-< li, j = 1 , 2 , . . . , k, such that

g.c.d. {hi,,, n2 . . . . . . , n~,k} = 1.

It is our goal to produce three plynomials, ~1, ~2, ~3, in the variables

en -- l ,2 , . . . , I /

,

which polynomials are not identically zero, such that if ~ ({aj,, })~ 0 for i = 1,2, 3,

then Condition .90 of Section 0 holds for the polynomials P1, P2 , . . . , Pk. This will

prove Theorem 5.

The polynomial ~1 will be the resultant (see [9, w of

d,,',/z-,,-' d z i and az i

It is clear that ~1 does not vanish identically. For the desired result it is enough (by

Theorem 1) to produce non-zero polynomials ~2, ~3 such that if

@,({a,,,}) ~ 0, i = 1,2,3,

then there are polynomials ~t(xl . . . . . XE) and ~2(x~, . . . ,x~)such that

O, (z) = ~ ( P , ( z ), . . . , P~ (z )) = z p' + (higher order terms),

O 2 ( z ) = ~2(P , ( z ) , . . . , P~ (z)) = z ~ + (higher order terms),

where p,, p2 are distinct primes. This is what we shall prove.

As usual, select R so that if N > R is an integer, then there are non-negative

integers c,,cz . . . . . ck satisfying

k

For each N > R we now set

N = j~. c~n~,,.

k

A , , = A , , ( { a . . } ; z ) - ( e j ( z ) ) <,


Here the @'s are defined by the last identity, and, for any fixed N, the sum is

actually finite. Each @N., is a polynomial in {a;,}. Define

: (7.1) AN =AN({a,, ,})=det \@~., @N.R+, " '" @N.N/

NOW, AN is a polynomial in {aj,. } and AN will be non-zero on a Zariski open set if it

is non-vanishing at a single point. However, simply observe that if a~.j = 1 for

/' = 1,2 . . . . . k and all other a~ = 0, then the matrix in (7.1) has l 's on the main

diagonal and O's elsewhere. Hence, AN is not the zero polynomial.

Let R = < q f < q 2 < - " < q R be chosen with q,,q2 . . . . . q~ prime integers. Our

polynomial ~2({ai~}) will be

(7.2) ~2 = A~,.

If {at.,} is selected so that ~2({a~})~ 0, then we can find complex linear combina-

tions BI, B2 . . . . . Be of the polynomials AR, AR+z,.. . , Aq, such that

n , ( z ) = J , . , ( z ) + z ~, + Z., (z).

Here Jz., contains only terms of degree strictly less than R and J2.~ contains only terms of degree strictly greater than qR.

Now {J,a},~, is a collection of R elements in the ( R - 1)-dimensional vector

space consisting of polynomials of degree strictly less than R which vanish at 0. Therefore, there are constants y,, y2 . . . . . yR, not all zero, such that

R

v , J , , ( z ) - = 0 .

Let

(7.3) R R

O,(z ) = wB,(z) = v , z , +

where J, contains only terms of degree strictly greater than qa+~.

Finally, choose primes qn+~ < qa+2 < �9 �9 �9 < q2a all of which exceed qn. We set

and repeat the argument from lines (7.2) to (7.3) using the primes

qR§ qs+2 . . . . . q2R. We obtain the polynomial

2 R

02(z) = ,=~+1 v'z~' +. t , (z) ,

where J2(z) contains only terms of degree strictly greater than q2R.

By the remarks at the beginning of the section, the proof is complete. []

(36 J. DUNCAN ET AL.

8. E x a m p l e s a n d f u r t h e r r e m a r k s

We now collect some examples which indicate the scope of our results.

First. it should be recorded that no theo rem along the lines of T h e o r e m s I and 2

or even Joris ' s (***) is t rue for a single funct ion ~ (i.e., k = 1). Naively, this is a

trivial s t a tement : Let ~o,(t) = t '~, n -> 2 an integer, and f ( t ) = It I '/" ; then ~, o f E C ~,

but f ~ C ~. H o w e v e r there are subtlet ies involved. In [4], G laese r constructs an

example of a C ' funct ion

g: R---~ It E R: t_>0}

such that

(i) {t: g(t)=O}={O}, (ii) g is infinitely flat at 0,

(iii) g2 ~ C ~, (iv) g ~ C 2.

Glaese r ' s example can be easily modif ied so that for any integer k => 2 we can

obtain a C ~ funct ion g: R---> R such that, for n - - 1 ,

g" E C | if k divides n, but g" ~ C | if k does not divide n.

Now we use Glaese r ' s result to construct a different sort of example :

E x a m p l e 8 .1 . There is a C ~ function

h: R---~{t ~ R : t_->O}

such that

(i) {t: h(t)-- O} = {0},

(ii) h is infinitely flat at O, (iii) ~ o h ~ h 2 + h 3 E ~ C |

(iv) h ~ C 2.

Thus, even though ~p involves two relat ively pr ime powers , r o h ~ C ~ is not

sufficient to imply h is smooth . Recall that the Corol la ry to T h e o r e m 3 states that if

m is any fixed positive integer, then ~p(h)E C * and h " ~ C ~ imply h E C |

The re fo re the p h e n o m e n o n d e m o n s t r a t e d by this example is ra ther subtle.

P r o o f of E x a m p l e 8 .1 . Let g be the funct ion of Glaeser , which we may also

suppose satisfies [g( t ) l < 1 for all t E R. Define

- g'~ + g" V 4 - 3g ~ h

2

Then

o h = hZ+ h 3 = g ~ - 2 g 4 + g'-,


so ~p o h E C ~. since g2 E CL But h ~ C:, since X/4-: 3g ~ is non-vanishing and C ~

while g is not C z at 0. []

Now we turn to some examp|es to see what happens to Theorem 1 if A is not a

function algebra. Joris's remarks in [7] indicate that he was aware of Example 8.2.

E x a m p l e 8.2. Let A be a Banach algebra which contains a non-zero nilpotent

element x of order 2 (i.e., xZ =O but x#O) . Then there is an [: R - * A such that

[~ E C | and [3 E C ~, but [ is not continuous.

Proof of E x a m p l e 8.2. Set

]'x, if t is rational,

f(t) ! O, if t irrational.

Then [2 = 0 - [3, but [ is discontinuous. []

Notice that an example of an algebra A as in Example 8.6 is the 2 • 2 real

matrices equipped with the usual l~ operator norm.

R e m a r k . The essence of Example 8.2 is the non-uniqueness of solutions of

x 2= a 2, x ~= a 3 in A. There are nilpotent-free Banach algebras in which one

obtains such non-unique solutions: One example is A = it(S), where S is the

semigroup generated by indeterminates a, b with relations a 2 = b 2, a 3 = b e.

When the range of [ is a non-commutative algebra, some interesting phenomena

can occur: Let B be the algebra of matrices

(; :) equipped with the Hilbert-Schmidt norm.

Example 8.3. There is a continuous function

F: R--~ B

such that F 2 is strongly C | F ~ is strongly C | yet F is not C a.

Proof of E x a m p l e 8.3.. Let

and define

~e -'''2, if t#0, of(t)

I 0, if t=0,

t3( t )= It(,

F ( t ) = (ot~t) fl(t)~ a(t):

68 J. DUNCAN ET AL.

Then F is continuous but not C 1. Nonetheless,

, - , (t) /

and

are both strongly C |

Fqt) = / l"'(O 3a2(t)fl(t)~ 0 a'(t) ] \

[]

REFERENCES

I. J. Boman, Differentiabitity o[ a [unction and o[ its compositions with [unctions o[ one variable, Math. Scand. 20 (1967), 249-268.

2. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973. 3. A. Browder, Introduction to Function Algebras, W. A. Benjamin, New York, 1969. 4. G. Glaeser, Racine car~e d'une [onction differentiable, Ann. Inst. Fourier Grenoble 13 (1963),

203--210. 5. G. Glaeser, Fonctions compos~es differentiables, Ann. of Math. 77 (1963), 193-209. 6. M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. 7. H. Joris, Unr C'-application non-immersive qui poss~de la propri~t~ universelle des immersions,

Arch. Math. 39 (1982), 269-277. 8. S. G. Krantz, Lipsehitz spaces, smoothness of [unctions and approximation theory, Expositiones

Mathematicae 3 (1983), 193-260. 9. B. L. van der Waerden, Modem Algebra I, Ungar, New York, 1953.

UNIVERSITY OF ST/RLING STIRLING, SCOTLAND

PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PENNSYLVANIA, USA

OREGON STATE UNIVERSITY CORVALLIS, OREGON, USA

(Received May 1, 1984)

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Nonlinear conditions for differentiability of functions