Nonlinear maps in spaces of distributions

Math. Z. 209, 101 114 (1992) Mathematische Zeitschrift

43 Springer-Verlag 1992

Nonlinear maps in spaces of distributions*

T. Gramchcv

Mathematics Institute, Bulgarian Academy of Sciences, Department of Differential Equations, G. Bonchev B1. 8, Sofia 1113, Bulgaria

Received April 5, 1990; in final form June 3, 199l

Introduction

The purpose of this article is to obtain nonlinear superposition for certain strong- ly singular distributions and examine its relation with the weak limits for semilinear hyperbolic systems and the Colombeau algebras of generalized functions.

Let us recall that according to the classical result of L. Schwartz [171 it is not possible to define product for the distributions preserving all desirable properties. However in many model of physical significance one has to investi- gate nonlinear equations with initial data distributions cf. H. Biagione [-21, J.-F. Columbeau [-3, 41.

Using the idea of considering the generalized functions as equivalence classes of smooth regularizations, E.E. Rosinger and J.-F. Colombeau developped theo- ries adapted for certain nonlinear problems. In particular the Colombeau algebra fg[R"1 contains the space of the distributions ~'(R"), is invariant under C ~ maps having polynomial growth and H6rmander ' s distributional product is associated with the multiplication in aJ[Rnl. Moreover, as shown by M. Ober- guggenberger [12, 13] the semilinear strictly hyperbolic systems in R 2 with bounded nonlinear terms admit a unique solution in N[R2], associated to the corresponding weak limit.

Here we deal with nonlinear maps growing faster than any polynomial and semilinear hyperbolic systems with the corresponding nonlinear terms. We see easily that N[R"1 is not preserved in general by such nonlinear maps despite that some subalgebras of fr [R"] are preserved, that contain elements "associated" to the distributions see I-3], 8.3.

We propose some new results on this subject. More precisely: (a) if B e is a subspace of the Sobolev space H'~oct--tD"~J, s_< hi2, such that the

Gevrey G~-microlocal singularities of its elements (it will be precised later) are contained in a fixed strictly convex cone and F(z) is an entire function in C of exponential type I/q, q>O, then F(u) is a well defined a-ultradistribution, ~r depends explicitly on s, q and the dimension n. As the proofs and the examples

* This research was started while the author was visiting University of Pisa, University of Ferrara and I.C.T.P., Trieste in the autumn of 1988

102 T. Gramchev

show, unless F(u) is a polynomial, we can not use in general H6rmander 's condition for multiplication in the space of distributions but rather its stronger version in the framework of the a-ultradistributions. We remind to the reader the well known fact that if s>n/2 the Sobolev space H~(R ") is preserved by smooth nonlinear transformations.

(b) we exhibit a class of semilinar hyperbolic systems in space dimension one with nonlinearities being sums of polynomials and entire functions, such that the weak limits of the singular Cauchy problems with initial data from B ~ exist in the space of a-ultradistributions. The index o- depends on s, the exponential type of the entire functions and the degree of the polynomials. This is a new phenomena compared with the results on weak limits in the usual weak topology for other classes of semilinear hyperbolic systems (see [5, 6, 12, 15]).

(c) motivated by b) we construct a ~r-Colombeau algebra f#~. Two principal differences with the situation in [-3, 16] occur. First, we are not able to use compactly suported Gevrey G" functions, following the regularization approach and as a consequence we construct only a a-Colombeau algebra o f " t em p e red " generalized functions. The second point is that the algebra contains the tempered (3 a-1)-ul t radis tr ibut ions (not a-ultradistributions as one expects). Nevertheless we are still in the framework of the Rosinger scheme [,16]. The differences are explained by the topology of the Gevrey G ~ spaces.

We note that all conditions, involved in our assertions, are in some sense necessary and sufficient which is illustrated by examples.

In Sect. 1 we state the principal results on nonlinear maps into ~7-ultradistribution spaces and their applications to a class of semilinear hyperbolic systems. Section 2 proves these results while Sect. 3 is devoted to the construction of a-Colombeau algebras, a > 1.

1 Statements of the main results

We remind to the reader cf. H. Komatsu [,10] that if t~> 1 the class of Gevrey functions G~(R ") consists of all C ~ (R ") functions f(x) having the following property: VK~R", 3A > 0 s.t. (s.t =such that)

(1.1) lffll~,K= ~ Al'r(~!) -" s u p l ~ f ( x ) l < or. aEZ~ x E K

One defines G{'~(R ") via (1.1) replacing 3A > 0 with VA > 0 on the line before (1.1). Put G~ (R"),= G ~ (R ") n C~ (R") and define similarly Gto "} (R"). They contain not identically equal to zero functions iff a > 1. The space G'(R ") (respectively G~'}(R")) is supplied with the inductive (respectively projective) topology given by (1.1) for K exhausting R" and A ~ 0 (respectively A 2' ~ ) [-10].

We denote by G~f'(R"), a > 1 the dual space of G~(R") with respect to the inductive topology. Its elements are called a-ultradistributions. As in [10] we introduce the G" analogue of the Schwartz class 5~(R") of the rapidly decreasing at infinity C ~ functions with all their derivatives, namely

~(R"),={f(x)eY(R"): 3A > 0 s.t. Ftfl[~ < ~ } ,

Nonlinear maps in spaces of distributions 103

where

(1.2) IIf[l~ = ~. A"l(~!)-~suplexp(Alxl ~:'~) t~if(x)[ < o~. cs x~'Rn

The dual spaces of {~}-ultradistributions G t~' (R"), ~'~1 (R") etc. are introduced in an obvious way. One checks easily that c~(R")+ {0} iff ~ > 1/2 and 5~(R ") is contained in ~(C") - the space of the entire functions in C", provided 1/2 < a < 1. We recall the well known inclusions for a > 1

(1.3) G~ (a") = :~, (R") = c ~(a"), G'"~'(R") = ~; (a") = O~o~'(R"),

as well as G~o~m G~(R"), G~f'(R") = G~o~'(R") etc. Let us introduce the notion of microlocal G" singularities.

Definition 1.1 Let ~> 1, ueG~0~ ") and let (x ~ ~~ We shall say that (x ~ ~o) does not belong to the o--wave front set WF~(u) of u iff there exists an open cone TmR" \0 , r176 a function ~sG~(R"), q~(x~ and a positive constant R s.t.

Io~(~t,)(~)l ~ R- ' exp(- Rl~.la/~), ~ T ,

where ~(v)(~)=f(r denotes the Fourier transformation, means

Rn

Obviously WF,~(u) is a conic subset of T*(R") \0 in each fibre and WF,(u) ~_ WFo(u) if a > 0. We mention also the well known fact [7] that n(WF, u) coincides with a-singsupp(u) - the Gevrey G~-singular support of u, n(x, ~)=x, (x, r and l < a < ~ , a = ~ corresponds to the C ~ case, WF~u is the usual wave front set WFu for a distribution u.

Choose and fix an open strictly convex cone F = T*(R")\0. We define for a > 1, s~R the spaces ( ( r stands for (1 + 1~12) "2)

(1.4)

with

and

B~(r ) ,={ue~' (R") . ~(~_)ELToc(R"), 3 d > 0 s.t. Ilull~,d,r < ~ }

[lull~,d,r = I[(" )-~l~(')l [IL~lr) q-liexp(d (")l/~)l~( ')l I[L~.\r)

(1.5) B~o(F):={uE (~ B~(F): supp(fi)cff}. a > l

We observe that B~(R") is supplied with the inductive topology for d '~0 in (1.4), B~o(F)is a Banach space and B~o(F)= B~(F), B](F)~ H -~+"/2 -~(R"), u > 0 while B~(F)r a > 1, sER. Here He(R ") stands for the usual Sobolev space

{u~,~'(R"): a~L]or Ilull~==ll(.)~,~(.)JIL~.)< ~}.

By the standard methods of the microlocal analysis [7] one shows that if u6 B~(F) then WF~ ~_ R" x F. The local spaces B~, loc (F) are given by {u~5~'(R"): Vtp6G~(R"), tpu~B~(F)}, a > 1. Note that 5r~(R")cB~(F), G ~ (R") = B~. lor (F) when a > 1.

104 T. Gramchev

We introduce for q >_0

(1.6) ~,ugq (Cm):= { F (z)E ~ (C'n): F(z)

-- ~ c~(c~!)-~z=,V~>0, 3M~>0, s.t. Ic~l<M~el~l, e e Z ~"} ~ X m

the space of the entire functions of exponential type 1/q. Clearly X4~o (C ~) coincides with ~f~(C '~) - the space of all ho lomorph ic functions in C ~.

Fur ther we shall say that O(x)eY(R") is a mollifier if ~ o ( x ) d x = l and ~o~(x) = ~-"~p (x e-1) converges weakly to the Dirac delta function 6(x) for ~--* 0. If y e R" we put ~p~Y (x):= ~o~ (x-- y).

Theorem 1.1 Let F(z)EJFq(C'), q>O, meN. Then one associates to F in a canoni- cal way a continuous map F

(1.7) F: (B~ (F)) '~ -* ~ ' (R")

with respect to the weak topology of ,f](R"), where

(1.8) a = l + q / ( n + s ) if s > - n ; o=oo for s = - - n

and 5~(R"). .= (~ 5P'(R"). The composition map F is determined uniquely by the

next property: if for u~(B~o(F)) ~' we set

uF,(x):=(u * q)~)(x) = l u(x -- y) qo~(y) d y,

~o (x) being a mollifier, then u~ E (C oo (R") m (Bg (F)) m,/~ (u~) = F (u~ (x)), Ve, > 0 and

(1.9) limF(u~)=F(u) in ~2(R"). ~ 0

Finally we claim that P extends continuously in a unique way to

(1.10) P: (B~(F))~" -~ G(o~)'(a ~

with ff (u) = V (u (x)) if u ~ (G ~ (R")) m and WF~ (F (u)) c_ R" x F, u ~ (B~ (F)) m. Conversely let F ( z ) ~ ( C " ) , q>O verify (1.7), (1.9), (1.10) for certain fixed

s > - n , ~> 1. Then if for some yER" the coefficients of the Taylor expansion of F at y, O~ F(y), c~Z"+ are non-negative when ]c~] >> 1 and

(1.11) lim ((~!)v-l ~F(y)) l / l ' l=oo, Vp>q,

then it follows that q >=(a- 1)(s+ n).

Example 1.1 Let u. '=o ~ - l(ffH(~)), r e Z + with H(~) being the Heaviside function. Then exp(u)eSP~'(R) iff a < l + ( r + 1) -1 and exp(u~(x))eC~176 e > 0 converges to exp(u) for e ~ 0 in 5e" (R) iff ~ < i + (r + 1)- 1.

Next we show that one can associatc nonl inear superposi t ion to some non- ho lomorph ic functions.


Theorem 1.2 Let f ~ G~o ~ (R"), s > - n, 1 < 0 < 1 + (s + n)/2 with m, n ~ Z +. Then for every y ~ R " there exists a unique continuous map ~. with the following properties:

(1.12) ~ : (B~(F))=-, J2(R"), a = l - - (O- -1 ) / ( s+n)e[1 /2 , 1)

a n d / f (p e 5~ 1 > # > 1/2 is a mollifier then p~'*fe,~(R") c ~ l _u(C") and maps continuously (Bg) ~ into 5r~'(R"), e > 0 and

lim (q~ . f ) (u) =~, (u) in the weak topology of ~9~ (R"), u ~(B~ (F))". ~ 0

Moreover j7 is represented as

(1.13) L(u) == y~ (~ ! ) - ' a.; f (y) u ~, ~eZ'?

ue(B~(r))%

where the power series above is convergent in 5F~ (R").

Example l.2 Put u , = x - l ( e x p ( i x ) - l ) . Since o~-U(~)=constZto, 1], we have ueBo~ where Zc stands for the characteristic function of the set C. Let f ( x ) = e x p ( - x zk), x + O and f ( 0 ) = 0 for a positive integer k. Then feG~l(R), 0< 1 +(2k) -~. Fix the mollifier ~o(x)=a-t:2 exp(_x2). According to Theorem 1.2 one defines correctly f O , + u ) , = ~ ( u ) e ~ ' ( R ) , c~< 1 - (2 k ) -1 and the smooth function (c~f.f)(u(x)), e > 0 converges to fy(u) when e-~ 0 in the weak topology of 5e'(R). If f (x) has a zero of infinite order at x = y clearly ~ ( u ) = 0 in Y'(R).

Now we consider the semilinear n x n strictly hyperbolic system in one space variable x with n > 1 and some fixed integer 1 < d < n

(0 t + )~j (?~) uj = O, j = 1 . . . . , d,

(1.14) (C~t+ 2k~F~)Uk +bk(u')uk=f~lu') , k = d + 1 . . . . . n,

where the constants 2j4=2 a for k+-j, u=(u ' ,u") , u'=(u~ . . . . . Uk), bk(U'), k = d + 1 . . . . . n are polynomials and for some q > 0

(1.15) fk E ovfq (Cd), k = d + 1 . . . . . n.

We note that the example in [15] of a 3 x 3 hyperbolic system exhibiting the phenomena self-spreading of singularities and existence of delta wave solutions is a particular case of (1.14).

The initial data are singular distributions

(1.16) u(0, ")= u~ +)) ~, s > - 1 , 0 > 1

Denote by r the maximal degree of the polynomials bd+t . . . . , b. and put g : = r s + r - 1 .

Theorem 1.3 Suppose that

0 < l + ( g + l ) -1, a , = l + q ( s + l ) -~, #=min(0 , a).

Then there exists a unique solution u(t, ") to the Cauchy problem (1.14), (1.16) in the space C~176 (St,'(R)) ") and if u~(t ,x)~(C~(R2)) " is the solution of (1.16)

106 T. Gramchev

with C ~176 initial data u~ ~ e>0, then the weak limit l imu'( t , -) exists in e --* O

C~176 (~'(R))") and is equal to u(t, "). Moreover, in the case bj=O, f j being a polynomial o f degree <=p, j=d+ l . . . . , n, the unique solution u(t, ") belongs to C k (R: (B~ ~ + p - 1 - k (R +))") for every positive integer k.

Remark 2.1 Concerning the last statement of the previous assertion we note that if bs-=0 and f~ is a polynomial for j = d + 1 . . . . . n we can take as initial data distributions satisfying weaker microlocal conditions for multiplication as in [I, 9, 11]. Then the Cauchy problem (1.14), (1.16) could be resolved in the space of distributions.

2 Proof of the main results

We start with Theorem 1.1. In view of the strict convexity of F we may assume without loss of generality that F \ 0 ~ {4eR": e j> 0, j = 1, . . . , n} (we can make a proper nondegenerate linear change of the variables x). So we have that Ill is equivalent with I~jl on r for j = 1, ..., n and if u~B~o(F) one can write with certain A > 0

(2.1) 10(4)1_-<A<~,) ~/~215 x (r 4 ~ r .

We will carry out the proof in the case m = l and n = 1. The method of the proof and (2.1) will imply its validity for n > l while m > l leads only to the same arguments with summations over Z'~ instead of Z+.

One needs some auxilliary results.

Lemma 2.1 Let a(~)eL~oo(R), supp(a) c [0, oe) and la(0l < A U, V4 > 0 with A > O, s > - i. Then we claim

(2.2) lak+ ( ~ ) l < z k + l r ~(k+l)+k (IV(Sq-1))k+l 4~0 , t r ( k + l ( s+ 1))'

where ak + t (4) stands for (k + 1)-times convolution a* a *. . . * a, F(z) is the Euler gamma function.

Proo f We proceed by induction. Note that supp(a *. . . * a) _ [0, oe). If (2.2) holds for k we get for ~ > 0

[ak+ll~A i lak(~--~/)l t /~dt/<ak+l (if(s+ 1)) k o = r ( ( k ) ( s+ 1))

. ~ ( ~ _ q y k + k - 1 .~d~ = A ~+ 1 ~(k+ l)+k (F( s+ 1)) k+ 1 o r ( k ( s + l ) ) "

We have changed the variable ~/= t ~ and used one of the classical Euler integral for the B-function

r(~) r(v) t " - l ( 1 - t ) ~ - l d t = F ( p + v )

0

(e.g. see [18]).


Let us recall that if K(4)~E~%r and for some fixed a>�94 we have ~c(4) exp(-el~l~t')~L~(R"), r e > 0 , then

(2.3) K(r (R").

Since f f is an automorphism of S~'(R") it is enough to define correctly the Fourier transformation ~-(F(u))(~) [10]. Using the fact that feJgq(C) and transforms the multiplication in convolution we write formally (we take into account the identity ~ (1) = (2 ~z)" 6 (x), x e R")

(2.4) (F(u))(~)-- 2r~ F(0) 6(4) = P(~):= L ck(h !)-q U(k)(~) k = l

and in view of the Stirling formula and the estimates on Ck in (1.6) and Lemma 1.1 we get for 4 > 1 (P(4) is bounded in [-0, 1])

(2.5) IP(~_)I < M~ ~ ? A k ~k~+ k - '(k !)-"(F(s + 1))k/F((k)(s + 1)) k = l

< M, L e'k Bk ~k(~+ 1)(k !)-~-,- 1 __< M~ exp(e.' 6141 (`+ ~)/(~+~+ u), k = l

where M,>0,0<e_<_ 1, e.'. m_~3 U ( q + s + l ) and the other positive constants A, B, C are independent of e. Clearly (2.5) combined with (2.3) defines correctly F(u) as an element of 5~j(R).

Let 9 e 5~ (R) be a mollifier. Then there exists C > 0 depending on 9 statisfying for 4ER, 0 < ~ < 1

tfi~(~)l = I ~b (e 4) a(~)l < C -~ e x p ( - C(e I ~ I)'/') (4) ~.

One defines F(u9 as before via (2.4). Since u ~ C ~~ c~L ~ when e > 0 clearly JV(u 9 coincides with F(u~(x)) in the weak topology of 5e,'(R). The estimate (2.6) and Lebesgue's dominated convergence theorem imply (1.9).

Let now ueB](F). As the result (1.10) is local we can assume that u is compactly supported and consequently there is an open strictly convex cone Fo, F c~ S"- 1 ~ F 0 ~ S"- ~ such that for some R > 0

(2.7) I a ( ~ ) l ~ R - l e x p ( - R l r r162

Choose Z(r supp(x)_F o c~ {r 14[ >2-1}, Z(4) = 1, 4 ~ F n {4:1412 1) and Z(t4) = ~(4) when t > 1,141->_ 1. Denote by ~(D) the corresponding G" pseudo- differential operator

Z (D) u (x),= (2 re) -" ~ exp (~x 4) Z (4) a (4) d 4.

In view of the compact support of u and (2.7) we get with the standard integration by parts in the oscillatory integral above that ( 1 - z(D))u~Se~(R").

108 T. Gramchev

So we are reduced to the case

(2.8) u = v + w , v ( x ) e ~ ( R " ) , supp(v~')eFo c~ {~: 1~1~2-~}.

We suppose again n = 1 and thus F0 = F = R +.

Lemma 2.2 Let b(x)eSP~(R) i.e. 3 A > 0 such that Hbli']< oQ. Then

(2.9) I[bk]l~A<__(llbl[~) k, k = l , 2, ...

Proof. We have for k = 1, 2 . . . .

IJbk[[~< (r! ) -~A'suplexp(kA[xl t"~) -dx (bk(x))l r = O x~R

= ~ ( (r t+. . .+rk)! ) l ~(rl!...rk!)~ 1 r l . . . . ,rk>=O

k

- sup ~ (rj!)-~Ar~lb(~J)(x)l exp(A Ixl~/~)<(llbli~) k. x611 j = l

We used the fact that k > 1, ~r > 1.

We write formally

(F (u))(~) = ~ (FI)(~) + f f (F2) (~)

where

F1 (x) = ~ (k !)- t (FIk)(w (x)) -- F (k) (0)) (v (x)) k, k = 0

F2(x)= ~ (k!) 1F~k)(O)(v(x))k=F(v(x)). k = 0

In view of Lemma 2.2 one obtains,

IJF2(v)H]< ~ (k!)-q[CkJ([[vJ[~)k<~ if [jv][]<cc k = 0

i.e. F2 (v)e 5e~(R). As supp(~, )c R + we estimate ~-(F1) for ~ > 0 similarly to (2.5)

[ ~ (F (k~(w) -- F ck)(O)) (~)l =< ~ [cr + k J ((r -I- k) !)- q (r + k) ! (r !)-1 [(~w)(r) (~)] r = l

<=M~ek(k!) l-q ~ B" er ~'(s+ l)(r!)- l-~-q <=M, ek(k!)l -q exp(e' ~l/~), r = l

where ME>0, 0 < e < 1, e '= O ( ~ I/(1 +s+q)) an B > 0 , C > 0 are independent of ~.,


The proof of Lemma 2.2 implies that if Ilvll~<~ then there exists B = B(A, v)> 0 satisfying the inequality

I~-(wk)(OI ~ B - ' exp(-- Bl~la/'~),

The last two estimates lead to

~eR, kEZ+.

I~-(F1)(~)l ~ ~ (k !)- ~ I,~-(F(k)(w)-- F(k){o)) * .~- (vk)(~)l k = O

< I ~ ( F ( w ) - F(0))(OI + M~ ~ C k s k - 1

r

exp(~'lr BIq[1/~) dtt < D exp(~:'l~l l''~) c~

with D=D(c, q, C, B)>0, e '= O(el/(t <~+~ 0<e,< 1. This chain of inequalities shows that F~ is also well defined. Let now F e ~ ( C ' ) , q > 0 verify (1.7), (1.9), (l.10) for some fixed s > - n ,

a > l and let c?~F(0)>0 if 1~1>>1. It is enough to consider the case r e = n = 1 , F = R +, y = 0 and t?~,F(0) = 0, V0=<:~< N with certain positive integer N.

Suppose that q<(c r -1 ) ( s+ l ) . Then we choose and fix a positive number q < p < ( a - 1 ) ( s + l ) and the distribution u=.~-I(~H(O)~B~o(R'-). Set 0=1 +p/(1 +s). The calculations in the proof of Lemma 2.1 yield ~(F(u))(OeLTo~(R) and

ao

,~-(ff(u))(~)= ~ 7k ~k~+k- 1H(~)>O, k = N

where

7k = F(k)(O)( k !)- 1 (F(1 + s)) k-

r ( ( k - l)(l +s))' k = N , N + I . . . .

The assumption (1. I 1) and the Stirling formula imply after direct calculations

lim (?k kkO(~+ l~)l/k =_ 00.

The limit above shows in particular that there exists C > 0 s.t.

~(F(u))(O>C ~ k-k~176162 k(~+l~, 3>1, k = N

for k>N. This yields (e.g. take k equivalent to 41/~ the following inequality with some 6 > 0

~(F(u))(O>Cexp(6~/~ 3>1.

As 1/0> 1/~ the last estimate contradicts the condition .~-(F(u))(OeS~'(R). So q > ( a - 1 ) ( s + 1) and Theorem 1.1 is proved.

110 T. Gramchev

The proof of Theorem 1.2 repeats the arguments of the first part of the previous considerations. One observes by the estimate (2.5) that the formal power series (1.13) is convergent in 5~'(R'), namely after setting c,=O~f(y)(a!) -~ we can view formally (1.13) as an affq element, q= 1 - 0 < 0 and we write formally ( r e=n= 1)

y~(u)=f(y+u)= Z (~!)-'c~ u~, ueB~o(r) �9 GtEZ +

Then as in the proof of Theorem 1.1 we estimate

],,~ (aTy (u) - f y (0)) (~)[ < C, exp (e J ~j1/o)

with ~eF, C~>0, 0<~<{1 and a given by the expression in (1.12). The rest is as in the proof of Theorem 1.1.

We point out that evidently for 1/2_< a < 1 we could not define local spaces B~.lo~, local a-singularities etc.

The proof of Theorem 1.3 is an application of the rules for nonlinear transformations established above. First we note that the j-th component of u' is written explicitly as u ~ (x - 2 i t), = 1 . . . . , d. Hence o~ ~ ~ {uj (t,')) (4) = exp (t 2~ () ~ (u ~ ({), j = 1 . . . . . d and u'(t, ")~ C"(R,: (B~-"(R+))d), m e Z + .

According to Lemma 2.1 the range of B~(R +) under maps being polynomials of degree r is contained in B*o r +"-I(R +). Having in mind the smooth dependence in t and the definition of g we get

(2.10) bk(U')(t, . )eC"(R,: B~-"(R+)), meZ+ , k = d + l . . . . . n

and therefore bk (u') (t, ') e C ~ (R,: ~'(R)). Theorem 1.1 shows that

(2.11) fk(u')(t, ")eC~ (R,: 5r (R)), k = d + 1 . . . . . n.

Each component uk, k = d + 1, ..., n solves a linear equation with coefficients distributions from C ~ (R,: ~ ' (R)) and with the right-hand side in C ~ (R,: ~'(R)). Fixing k we can take 2 k = 0 (in the characteristic coordinates for 8 t + 2k 8~). Then one may write formally

Uk(t, x ) = e x p ( - - i bk(U')(z, x )dz ) (u~

' ( j ) ) +0~ exp bk(u')(2, x ) d 2 f~(u')(z, x) dz .

Since exp (z) e ~Y~ _ e (C), V 0 < 6 < 1 we get taking into account (2.10) and (1.7)

(2.12) exp(-- i bk(u')(r, x ) )dzeC~176 �9 5r 0 < l + ( l + s - ) -1

and with a = 1 + q ( s + i ) - 1

(2.13) t )

S exp j bk(U')(2, x )d2 fR(U')(Z, x ) d r e C ~ ( R : 5v~'(R)) 0 ' , 0


The o--wave front set of (2.12) and the a-wave front set of (2.13) in the x-variables are contained in R~• So we can multiply them in C~(R: 5e/(R)) with #=min (0 , a), justifying in this way the representation for Uk, k = d + 1,. . . , n and showing the uniqueness and the existence part of the theorem. Now it is easy to deal with the weak limit of u ~ by applying the partial Fourier transformation ~ to Uk(t, X), k = d + l , ..., n and using the previous arguments and Lebesgue's dominated convergence theorem for e ~ 0.

3 The a - C o l o m b e a u a lgebras

We will follow the Colombeau approach [3] and will try to see what differences will appear in the framework of the ultradistribution spaces. For some aspects of the usual C ~ theory of extensions of spaces of distributions and applications we refer to [2-5, 8, 13, 16, 19].

Let a > 1 and define A~ = { ~o (x) E 5~ (R): S xk ~0 (X) d x = (~k O, k E Z + }, 6 k o stand- ing for the Kronecker symbol. If n > 1, x = ( x l . . . . . x,) we set

A,%={q~(x)eSa~(R"): qb(x)= qg(x,)x ... x ~o(x,), ~oeA{}.

Evidently if a > 1A~ n C~ (R") = 0 since it contains the inverse Fourier transformations of the functions r162 which are equal to 1 in a neighbour- hood of the origin.

The next assertion suggests what kind of estimates will be needed in order to define a a-analogue of the ideal ~U, which plays a crucial role in the construction of the Colombeau algebra in the C ~ category [3].

Propos i t i on 3.1 Let w~A~'~(R") and q~EA~. Then there exist two positive constants d and C such that for ~ Z ~ , x ~ R ", 0 < e < 1 the next inequality holds

(3.1) IO~(w*g)~-w)(x)l<Cl=l+l(ct!)Oexp(-B(Ixl~/~+e-~It2~-a))).

Proof." Denote for c~eZ", e > 0

f ~ , , (x).'= I O~ (w * q~, - w)(x) l = IS (Ol w (x - - e y) - ~ w (x)) q~ (y) d Y l.

In view of the Taylor formula and the assumption ~ A , ~ one may write for every positive integer N

(i ) F~,~(x)=IN ~ (/?!)-~eNSyP (1-t)N-~O~x+aW(X-tey)dt r I~1 =N

Therefore we can estimate (taking into account that w and �9 belong to 5P~(R ") i.e. they satisfy (1.2)) for c~Z~., xeR", 0<~__< 1 the quantity above as follows

(3.2) F~(x )<c l ' l+u eN ~', (Bl ) ' - ' ( (~+f l ) ! )" It~I=N

1

j ( S e x p ( - r l x - t e y l l r ~ - r l y l ~ / ~ ) d y ) d t o

__< Cl=l + ~ (~!)~ R~'~(N!)2~ - t e x p ( _ Blxl~/~)

with some positive numbers C, e, r, B, R independent of N, ~t, e.

112 T. Gramchev

One obtains from (3.2)

(el/(2,-1)M)~(N!)- l(F~w~,~(x))l/(2~ 1)~2-N C~I + 1 (~!)~exp(_b[x[~/,),

where C1 ~~C1/(2a-1) 6=a/(2er-- 1), b = B U(2(~ 1), M = ( 2 R ) - 1/(2a-1) N~Z. Sum- mation over N proves the estimate (3.1).

Now we denote by ~ the set of all maps

(3.3) f : A~ --+ GT(R")

where GT(R ") is the set of all q/(x)eG~(R ") satisfying Ve>O, ~ C > 0 s.t.

[8~@(x)l<Cl~l+'(~!)~exp(e[xll/~), xeR", ot~Z"+.

The spaces 5~j(R") are embedded in E ~ via the convolution

(3.4) [w] (q~, x),=y w (x - y) q~ (y) d y e G~ (R")

with weSe0'(R"), 45EA~, a < 0.

Definition 3.1 We shall call a map f o g . ~ a-moderate if Vq~eA,', u V6>0, 3M > 0 satisfying for x e R", 0 < e < 1 the following estimate

(3.5) I O.~(f(~,, x))l _--< m exp(6 (e- 1/(2~- 1)+ Ixl ~/q).

Let g,~ M the space of all a-moderate elements of S~.

Proposition 3.2 The space 5ej,_~(R ") is embedded in &~,M by the convolution (3.4), a > 1.

Proof Let w ~ _ I(R"). According to the structural representation theorems for the ultradistributions [-10] we can write

(3.6) w-- ~ (:r 8~,(p,(x)),

where #~(x)~L2oo(R"), c~Z~_ and for every 6>0, d > 0

(3.7) ~, 6 -I< I1/1~11~<~ ~EZ+

with

I1~ It~ =(f I#=(xl 2 e x p ( - 2dlxl '/~) dx) '/2 �9

In particular one obtains from (3.7)

(3.8) u Vd>0, 3C=C(d, 6)>0 s.t. [ d


Then we get for some positive B depending on q~

I[W](qO~ ' X)I=< Z e I~l nBM(~!)-2~r+l ~)t~(x__y)l e x p ( - 2 d l y / e l t ! ~ ) d y ~eZ~

< ~ e-I=lBt=l(~!)-z~+~ If~=ll[~exp(2d)lxl ~/~)

<=Cexp(2dlx[~/~+(R6~ t)~/~z,,-~)), x~R", d>O, O<e__<l,

where C=C~,a,~>0, R=R~,~>0. The estimate above shows the validity of (3.5) as we can take d and 6 small

enough.

Example. Let a > 1 be fixed and let

w:= ~ (ln(j+2))-J(j!)-~ O> 1. j=0

Then straightforward calculations imply that w~0 ' (R) but wr if #>0 . Choosing ~b(x)~A~ with q~(~(0)=(-1)~(j!) ", j = 0, 1 . . . . . (it is always possible by an adjustment in the Gevrey classes G ~, tT> 1 of the Borel construction of a C ~ function with given Taylor expansion) we see that [w](qs, x) satisfies (3.5) iff 0 > 3 a - 1 . That explains why 5ej,_~(R ") is embedded in g,~,M but in the case 0 < 3 a - 1 this is not true for the space 5e0 ' (R").

Definition 3.2 Put ~,~ to be the subset of g~ consisting of all f with the n,M

next property: Vq~eA, ~, V~eZ~, qd>0, 3eo, 3 M > 0 such that

(3.9) Ic3~ f ( ~ , x)l < M e x p ( - d ( I x l l / ~ + e - 1/(2a- 1)))

for x~R", 0<e=<e0. We check with the arguments used above that J'~," is an ideal in s M and

the neutrix condition holds, namely if w~gP~,_ I(R") and [w]~JV," then w=O. The Colombeau algebra ffff is defined as the quotient space o~ M/Jff~.

In view of Proposition 3.1 and Proposition 3.2 in fact we have proved:

Theorem 3.1 The a-Colombeau algebra ~ff , a > 1 is a differential algebra, module over G~(R"). Moreover ~ contains isomorphically ~ ( R ') (respectively S f~_ I(R ") as a subalgebra (respectively as a submodule over G~(R~)).

Remark 3.1 We point out that one carries out a similar construction in order to define Colombeau algebra, containing the spaces with projective topologies , ~ _ t~(R") and ~ ( R " ) . In that case we can use the structural representation theorems in [14] for elements of ~ ( R " ) .

It seems difficult to find a reasonable class of nonlinear maps, not having polynomial growth, which preserve fg.~. But we are able to show, using the estimates in the construction of ff,~ that if f ( x )~GT(R) then the superposition map

f: CJ~e(R") ~ ffff

is correctly defined, where 5P~(R ") stands for the real valued functionals from ~ ' (a") .

114 T. Gramchev

Acknowledgements. The author expresses his gratitude to Professor M. Oberguggenberger and Professor J.-F. Colombeau for the useful and stimulating communications and to Professor B. Zemian for the enlightening discussions.

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