Rend. Sem. Mat. Univ. Poi. Torino Voi. 49, 3 (1991)

R. Agliardi

A DIVISION T H E O R E M IN G E V R E Y CLASSES

A b s t r a c t . A division theorem is proved for Gevrey functions f(x,t) which

are divided by a polynomial in t with Gevrey coefficients with respect to x.

An estimate of the Gevrey class of the quotient and the remainder is given,

which depends on the Gevrey index of / and of the dìvisor and on the order

of the polynomial.

The purpose of this paper is to prove a division theorem for Gevrey functions f(x,t),t E M, # E M", where / (# ,£) is divided by a polynomial in t with Gevrey coeifìcients with respect to x.

Our result (see Th. 2) provides an index for the Gevrey class of the quotient and the remainder which is generally better than the one given in [3].

The proof we give here follows Nirenberg's proof to the Malgrange preparation theorem; liowever we use an adaptation of Mather's version to the extension lemma (see Th. 1).

Here a refìnement is needed to be able to deal with Gevrey classes.

Finally we recali here that in [3] the author claims that in the general case it is impossible to obtain a quotient and a remainder in the same Gevrey class as the one of the dividend and the coefficients of the polynomial by which we divide. However, if the polynomial is assumed to be hyperbolic, there is no loss of smoothness of the quotient and the remainder, as is shown in [1].

I wish to thank prof. L. Zanghirati and an anonymous referee for helpful comments on an earlier draft.

386

Notat ion

Here notation concerning Gevrey classes is recalled.

If K is a regular compact set in l n and a > 1, then S^ìA{K)^A > 0 denotes the Banach space of ali / E C°°(!n) such that

\\f\\£{,hA = sup \d°f(x)\A-Ma\-" < oc . xeK oGNn

Let V^t'A^K) be the subspace of E^,A{K) whose elements are supported in K. If Q is an open set in En the following spaces are defined:

£A(K) KCCtl A-+0

£W(Q)= lim ^W(A') €W(K)-= lim ^ ' V O

Z>(*)(fì) = lim 2 ) ^ £>£} = Imi p M ^ f t f )

X>W(fì)= i]m 2)£ } p [ f } = lhn X>W'A(J5T)

In what follows we frequently use the following Lemma concerning multi-integers, without quoting it explicitly:

LEMMA 1.

a) Va e Nn |a|! < n^ctl /a\ / l a | \

CJ ( 7 i ) b ••• ( £ ) < r 7 i : : : ? ; ) where a n 7 , G N,i = 1, . . . ,n.

|7l=?.7

387

e) is a consequence of b).

The following Lemma will be often used in what follows, usually with W = 0:

LEMMA 2. Let sup \dya f°r every compact set yeH

H C Mn and with a > 1. Let sup \d%fe(x)\ < C / A ^ ' M ! * £ = 1 , . . . ,ra, for

every compact set K C Mm. Put f(x) = (/i(#),. . . , /«,(«)) and denote òy byipW. Then

(i) sup | ^ ^ ) ( / ( ^ ) ) | < CvJ{K)(2°{n + l J ^ A / j H A J f (|a| + |/?|)!" ,

for every compact set K C M.m and with A^Ct = m.ax(A(pCf)j{ì 1).

Proof. By induction on |a|,V/0 E Nn. It is obvious for |a| = 0,V/9 G Nn. In general, Va; 6 À', it is:

i^VH/w)! = i5rej^.9 ' < a - e j

a a,

6jrW+e'//(*))l • ^ " ' ' ^ ^ W w l

By the induction hypothesis and by using the following inequality (|a'| -f 1)! < 2'a' ,l+1|a ,|!, wege.t

i^W)(/(*))i < ̂ ^ W * +...i)â G7A/}itìi A!?1

>

i < | a | - l

The terms in the summation over i are bounded above by (|a| + \/3\)]-(

< x - l

E -——-r. Hence (i) holds for any a E Nn and 3 G Nn . i=o v y

388

LEMMA 3. Let P = td + £?=i A ^ - > , A, G C. Put

* 0 sucii that

\d°id"*dP

389

Proof. From i) in Lemma 3 we deduce that 6(y, A) < l/(16d3A(ì -f

lei)1"1^) implies 8d3, and hence po(cr(y,\)/(A(l +

1_ ì_

| £ | ) 1 _ l/(16Ad3(l + \Z\)~ °) then

\dlldl2d§

> l /4A(l+ |e l ) 1" l / < T ) , Vf'GM.

Hence n) is true.

Finally, in order to prove our last claim, we observe that dyp vanishes where

•/tioWy,A.))/C4(H-|è|)1-l/

390

ii) \d?d§dldlFj(z,\)\ < C,,Aa^^^8\a\l3\1W.Yd

expi-BfS^X)-1/^-1)) if6(y,\) < D)

(where Hf, Bf, Dp,Dn are suitable positive constants).

Proof. It is a consequence of Lemma 4, where we take

F(z,\)= [e**p(y,\,t)~f(0dt

with p defìned as in Lemma 4 and y = $sm x.

Indeed if t e 1 then F(t, A) = JV^/>(0, A ,£)7(0# = /(*)• Moreover we have

\s*t4did±F(z,\)\< £ (fy-0^ i

391

As \dzF(z,\)\ = | j el^zdyp(yìX^)f(^)d^\1 where the integrai is over

|£| < l/Af\y\)^ and \y\ > 1 / (24/(1 + | ? | )1 - l / < T ) , then in the case |y| < Dp

(where £ / 0 = l/(22~1/(TAf)) and hence l^

1/*7 > l/(l1/(T(2Af\y\)^), we have:

| ^ ^ | ^ | F j ( ^ , A ) | < C / 2 a ^ + l ^ ^ l ( a ! / ? ! 7 M ! ) ^ e x p ( - / / / | i / | ^ r )

where we have put Hf = C//(21/

392

LEMMA 5. Let Pd(z,\) = zd + E Ì - I Af*d~J'. Put 6(y,\) = inf

\y — S m ^o|, witii y = ^ m 2.

Then for every compact set K\ C E2, A'2 C C*,3C > 0 such that

e)01—— xP(z,X)

for every (3fte a:, y) G A'i, A e #2» Ò(y, A) ^ 0.

Proof. Notice that |P«j(*,A)| = J J |*-*i(A)| > f j |t/-j=l,...,d j=i,...,d

Pd(*jW,*)=o />,(*,• (A),A)=o

Smzj(A)| > 6(2/, À) . Then our statement follows by induction on |a|.

LEMMA 6. For every 0 and for every ^ E N it holds:

where C\ is a suitable positive Constant depending only on ero.

Proof. The maximum of $.+ 3 y —• exp (—y~1'

393

T H E O R E M 2. Assume that f(t,x) G £ M . M ( K X ln ) . Let Pd(t,x) =

d

td + J2xj(x)td~j where xj(x) is a complex valued function in £i°j(lRn) .

(Tfere and in what follows we shall assume a > 1). Then for every compact set K C W1, I C M, we can find qjjfK(t,x) G £M.{*d} and rfj)K(x) =

(rifJtK(x)> * ' ' ' ^ / " l ^ ) ) G (£M.fa})

394

If J 'C I1 •= compact set in IR and we restrict ourselves to x E K = compact set in En or, however we consider bounded |Àj|, then {ZJ(X(X))\ j = 1 , . . . , d Pd(zj(\(x)), X(x)) = 0} is included in a compact set C. Then we take D in order to enclose C and V7.

If we take

395

à?d^dld{ il ... (idem)... < ... (idem)... J Jun{\y\>Df0 and tf(y)>min(l,JDj)}

\Jl Un{\y\>Dj0 and min(l,i?J)} e-tt'M'-'lyl-o-'dzAdz

'l7n{|y|

396

Indeed we can write as usuai that the quotient and the remainder are given by:

0(t x) - — / li^)—dz

R{t x) = J_ / f(z x)P(z,X{x))-P{t,X(x))

where OD is a suitable circle surrounding t and the zeroes of P.

It is clear that Q in analytical in t and R is a polynomial in t of degree < d- 1.

Moreover the evaluation of the Gevrey index with respect to x is trivially obtained by differentiating under the integrai sign and by employing Lemma 5.

REFERENCES

[1] M.D. BRONSTEIN, Division theorem for hyperbolic polynomials, Proc. of the Conf. on H.E.R.T., University of Padova, 1985, 33-46.

[2] L. HÒRMANDER, An Introduction to Complex Analysis in Severa] Variables, Van Nostrand Comp. Inc., Princeton, 1966.

[3] P. LÉVY-BRUHL, Un théorème de division pour les fonctions de Gevrey, C.R.A.S., Paris-Serie A., 282 (1976), 149-151.

[4] J.N. MATHER, Stability of C°° mappings: I. The division theorem, Ann. Math. 87 (1968), 89-104.

[5] J.N. MATHER, On Nirenberg's proof of Malgrange's preparation theorem, Lecture Notes 192 (1971), Springer-Verlag, 116-120.

[6] R. MEISE, B.A. TAYLOR, Whitney's extension theorem for uìtradifferentiable functions of Beurling type, Ark. Mat. (1988), 265-287.

[7] L. NIREMBERG, A proof of the Maìgrange preparation theorem, Lecture Notes 192 (1971), Springer-Verlag, 97-104.

[8] H.J. PETSCHE, D. VOGT, Almost analytic extension of uìtradifferentiable functions and the boundary values of holomorphic functions, Math. Ann. (1984), 17-35.

[9] C.T.C. WALL, Introduction to the preparation theorem, Lecture Notes 192 (1971), Springer-Verlag, 90-96.

397

Rossella AG LI ARDI

Department of Mathematics, University of Ferrara,

Via Machiavelli 35, 44100 Ferrara, Italy.

Lavoro pervenuto in redazione il 24.4.1991.