Traces of analytic solutions of the heat equation

N. A ^RONSZAJN

Traces of analytic solutions of the heat equation

Astérisque, tome 2-3 (1973), p. 35-68

<http://www.numdam.org/item?id=AST_1973__2-3__35_0>

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

TRACES O F ANALYTIC SOLUTIONS O F T H E H E A T EQUATION

by N. Aronszajn.

Table of Contents:

§1. Definition of traces, their basic properties and identification with functions, distributions, and analytic functionals.

§2. Noncompatibility of GU with V(C

ⁿ

) and A

¹

.

§3. Some elementary operations on traces.

§4. Convolutable traces and multipliable traces.

§5. Convolutors and multipliers.

§6. Developments in Hermite polynomials.

§7. Application to inversion of convolutors.

§8, General operators and differential operators on the trace space QU.

§9. Final remarks and problems.

- 35 -

§1. Definition of traces, their basic properties and identification with functions, distributions, and analytic functionals.

In what follows the complete version of the talk at the Colloquium is given. Notation, terminology, and results stated in the preliminary notes (which will be refered to by P. N. ) will be used. However, we will slightly change one notation. It will be convenient to write the spaces (W(D), GV and G V

^R

as GV(D), GV, and G V

^R

. The preceding notation will be reserved for the corresponding spaces of traces which are the main topic of our talk.

Consider a space GV(D), a point t €dD and a function u(x, t) € GV(D).

If this function is regular for t = t^ the section u(x, t^) restricted to x € IR

ⁿ

will be called the trace of u at t^. Such traces will be called regular traces; they are entire functions on IR

ⁿ

of Laplacian order 2 and finite type.

By formula 2, P.N. Ch. Ill, §1 such a regular trace determines uniquely the corresponding solution u(x, t).

In the general case when u(x, t) is not regular at we will consider at first the trace of u at as an abstract object in one-one correspondence with the function u € QU(D) . The set of all these objects will be called the trace-space corresponding to D and € dD and denoted by GV(D;t^). The

>~ 0

one-one correspondence between GU{D) and G%{D; t ) allows one to transfer

~ 0 the topological vector space structure from GV(D) on GV(D;t ) making the space of traces a Montel space.

- 36 -

If f € GV(D;t°) we will denote by f(x, t) the corresponding function

^ ^ ^ ~ 1 in GV(D). W e will be mostly concerned with the space GV = GV^ = GV((C

+

), and sometimes with GV_ = GV(B_, (R)) and their corresponding trace-spaces at t = 0 which will be denoted briefly by GY = GV^ and GU

^R

respectively. For brevity's sake we will call the elements of GY "traces

¹¹

and those of G Y

^R

"R-traces".

If R < R' ^ co we will identify the trace f € G Y

R |

with the trace g € G Y

R

such that g(x, t) is a restriction of f(x, t). In this way we have clearly

(1) G ^

^{R I}

c.G*

^R

.

W e will see a little later that G Y

^D

, is dense and non-closed in №

xv. x\

Our next task is to identify the spaces G Y

^R

with topological duals of suitable proper functional spaces. To this effect we use the spaces V introduced in P.N. , Ch. Ill, §3, which we will now denote by V . W e recall that V is the class of all finite linear combinations of functions of the form D

r

E(x-£, t + T| for different values of the parameters

n 2

y

"

v

"'

Q € <C , and T € B (R) . Since these functions belong to and are all regular at t = 0 the class of the corresponding traces which we will denote by V is composed of regular traces which are functions of

R

x € IR , finite linear combinations of functions of the form D n k

^r E(X-£,T).

Every v(x) € V determines uniquely the function v(x, t) € V . By this correspondence we transfer the topology of V_ on V and using Theorem V of

R P.N., Ch. Ill, §3, we get

T H E O R E M I, The trace-space G V

^R

is identified with the topological dual of V

^R

, G V

^R

= Vj^ . For every f € the corresponding f (x, t) is given bv

(2) f(z,t) = < f, E(x-z,t) > •

- 37 -

For any u € G V

^R

and v € V

^R

the scalar product can be written in the form (3) < u,v > = § u(x,t

^!

Mx-t')dx

for sufficiently small positive t

^!

.

This theorem allows us to identify a large class of familiar objects ax

²

with traces. Since every function v(x) € V

^R

multiplied by e belongs to the class S of L. Schwartz for sufficiently small positive o> we obtain immediately:

-ax

²

T H E O R E M II. Every distribution T such that e T € S

¹

for every or > 0 is a trace. The scalar product with any v € V

^R

is given by

2 2

< T,v> = e "

a x

Tfe"* v(x)) = T(v(x))

for sufficiently small positive a (depending on v). The function T(z,t) € G Y

^R

is given by

2 2

T(z,t) = e ~

^{a x}

T ( e

^{a x}

E(x-z,t)) = T(E(x-z,t)).

Corollary III. Functions in the class $ of L. Schwartz, distributions with compact support and tempered distributions are traces.

T H E O R E M IV. (Density Theorem. )

a) The regular R-traces are dense in 6 Y

^R

;

b) The linear combinations of functions of the form E(x-£, T ) for £ € <C

ⁿ

, T € B^(R) are dense in GY^ ;

c) Polynomials are dense in »

d) The class $ of L. Schwartz is dense in GY„ . — — — — — — _H Proof, a) For R = oo with any function u (x, t) €GV the function

u (x, t) = u(x, t+c) is also in GY for e > 0 . Its trace is u = u(x, e) . It is a regular trace and obviously u u for € ^0 in the topology of GY.

- 38 -

/ i , o \

For 0 < R < 00 the isomorphism J&l _1_

¹

) (see P.N. , Ch. HI, §1 and

\ ZR ' y

§ 3).transforms GY onto GY . Using this isomorphism we prove our R

statement for all R's.

b) It suffices to show that for regular u € GY the corresponding R

solution u(z,t) is uniformly approximable on compacts in z and t by linear combinations of functions of the form E(z-£, t + T). Since

2 2

u(z,t) is regular for t € B

^R

(R) U (0) for certain p > 0, for any compact

K C

B*(R) we can find p

¹ > 0

and R

^f

> 0 so that

K C BZ D (( R ' ) C B2,D I! R 1 C B J ( R ) U

B

²

(0).

XV xv p "rxv xv p

Since the function u(z,t-p

^f

) c G Y

^{R l +}

we can apply formula (1) of Theorem ELI of P.N., Ch. Ill, §3, with u(z, t) replaced by u(z,t-p

^!), 0 by 0, t° by p^f ,

R by R' +p' . Noticing that now the circle B(R* +p', p\ 0) = B^, (R

^f fp¹),

from the formula (1) we see that u(z,t-p') can be approximated uniformly for t G K + p' and z in any compact of <C

ⁿ

by a linear combination of functions E(z-x, t-p') for finite number of values of x which proves our assertion.

c) Using Proposition 2 of P. N., Ch. Ill, §2, and Theorem IV of P. N. , Ch. Ill, §3 we prove immediately statement ch

d) In view of c)it is enough to show that if p(x) is any polynomial and Cp(x) € & with cp(x) = 1 in a neighborhood of 0 then cp(x/\)p(x) for 1 < X / oo converges in the topology of traces to p(x) but this is clear if one writes by the formula of Theorem EE

(cp(x/\)p(x))~(z,t) x J cp(x/X)p(x)E(z-

^Xj

t)dx . 1R

which for oo converges uniformly on compacts in z and t to p(z,t) = J p(x)E(z-x, t)dx.

R

ⁿ

- 39 -

T H E O R E M V. All analytic functionals on <E

ⁿ

are traces.

Proof. Since all functions v(x) € V are entire functions on TR

ⁿ

, hence also on <C

ⁿ

, for any analytic functional we can consider the scalar product < F, v >. It is immediately seen that this scalar product is continuous in v in the topology of V. Furthermore < F, v > cannot vanish identically on V since by Theorem IV b) and c) V is dense in M(<L

ⁿ

). This

identifies F with an element of QU. The corresponding F[z,t) = < F,E(z-x, t) >.

Furthermore, being identified with traces, the analytic functionals are automatically identified with R-traces.

W e can ask ourselves which functions u(x) defined on lR

n

can be identified as traces. The elementary case is the case when the convolution u(z,t) = u(x) * E(x, t) is valid for all z € <C

ⁿ

and t € clj. in the ordinary sense i. e., with Lebesgue integration. One sees immediately that for

-ax

²

this it is necessary and sufficient that u(x)e" be integrable for all -a

²

a > 0. u(x) will be an R-trace if and only if u(x)e is integrable for . 1

a >

2R •

However, we know already a large class of functions, namely the regular R-traces for which the convolution is valid not in the ordinary sense,since the integral has to be taken in general as the symbolic integral.

To include both cases we will make the following definition:

Definition. A function u(x) defined on IR

n

will be identified as a trace if and only if u(x) is locally integrable and the

u(z, t) = j; u(x) E(z-x, t)dx

exists and the uniformity conditions (see P.N., Ch. I, §5) are satisfied with

- 40 -

respect to the parameters z € <C

^nf

t € C^.. Replacing <£,\ by B

^R

(R) we obtain the definition of functions identifiable with R-traces.

It is obvious that u(z,t) is the corresponding solution of the heat equation. It is easy to prove also that in this case the scalar product

< u, v > for any v € V is given by ^ u(x)v(x)dx. The condition of existence of the symbolic integral is a very special kind of condition on the behavior of the function u(x) at co which is not in general connected with the behavior of | u(x) |.

For a function f(x) defined on IR

ⁿ

which is a trace or R-trace, the corresponding solution f (x, t) has f(x) as initial values in the usual sense as described in

T H E O R E M VI. If f(x) defined on IR

ⁿ

is_a R-trace, 0 < R ^ co, then

for every x^ € IR

ⁿ

for which lim p ~

ⁿ

\ | f(x)-f(x^) | dx = 0, hence for almost p =0 J ~

0 n

B l l ( x 0 )

all x € TR

ⁿ

we have ^

(4) f(x,°,t)— f(x°) for t^O.

Proof. By definition of the symbolic integral (in elementary case,

see P.N., Ch. I, §5) ^ ^ f(x°,t) = jjf(x)E(x°-x,t)dx = J u ^ r ^ M ^ x ^ r ) ^ e

^{4 t}

dr.

x

- 41 -

By the classical argument of Fatou-type theorems we obtain then (4).

~

Remark. The existence for a solution f(x, t) € GY„ of the limit

——————— xv f(x^) in (4) for almost all x^ € IR

ⁿ

does not assure in general that f(x)

is the trace corresponding to f(x, t) (see next section, Theorem I).

§ 2. Noncompatibility of G V with Y(<E

ⁿ

) and .

In the preceding section we established an identification of a class of entire functions with traces (the regular traces) and of a class of distributions with traces. Each time we have an identification of a subspace of one topological vector space with a subspace of another the question arises whether this identification is compatible with the topologies of the two spaces.

Let A^ and be linear subspaces of the topological vector spaces A and B respectively, and let J be the identification mapping of A^ onto B-^, i.e. a linear 1-1-mapping of A^ onto B^ . This identification is said to be compatible with the topologies of A and B if J considered as operator from A into B is closable with a closable inverse. A necessary and

sufficient condition for this is that if the nets (or sequences) {a^ } and {Ja^ } converge in their respective spaces A and B and if one of these nets

converges to zero then the other does also.

The importance of compatibility lies in two facts:

1? W e can extend the identification canonically to the closure of J which is then also an identification mapping.

2? By considering the graph of -J, G(-J), which is a closed subspace of the direct topological sum A + B we can consider the topological quotient space (A + B)/G(-J) in which the elements x € A and y € B are

- 42 -

identified with (x, 0) + G ( - J ) and ( 0 , y ) + G ( - J ) r e s p e c t i v e l y . In this way (A + B ) / G ( - J ) b e c o m e s A + B (not a d i r e c t s u m ) with A and B c o n t i n u o u s l y e m b e d d e d in A + B . T h u s , a c o m p a t i b l e identification l e a d s to a c o m m o n e n l a r g e m e n t of A and B to a t o p o l o g i c a l s p a c e A + B w h e r e the i d e n t i ­ f i c a t i o n is the natural o n e .

In m o s t c a s e s of i d e n t i f i c a t i o n the c o m p a t i b i l i t y is t r i v i a l l y t r u e . A s an i n t e r e s t i n g e x a m p l e of n o n - c o m p a b i l i t y c o n s i d e r the c l a s s 7/1 of all m e a s u r a b l e functions on IR* and the c l a s s &¹ of all d i s t r i b u t i o n s

on TR}. T h e c l a s s L>] c 7ft i s identified with a s u b s p a c e of $^f and the l o c

s e q u e n c e of functions ^NX^N (^x) » w h e r e Xⁿ i s the c h a r a c t e r i s t i c function of ( - l / 2 n , l / 2 n ) , c o n v e r g e s to z e r o in the i n t r i n s i c t o p o l o g y of % and c o n v e r g e s to the D i r a c m e a s u r e 6 ^ in . H e n c e the i d e n t i f i c a t i o n i s not c o m a p t i b l e .

T H E O R E M I. The i d e n t i f i c a t i o n s b e t w e e n GU and M<Cⁿ) and a l s o b e t w e e n GU and a r e not c o m p a t i b l e .

P r o o f . W e u s e an i d e a s u g g e s t e d to us b y F . T r e v e s . W e c o n s i d e r the function h(t) = e *0^ ^ . One p r o v e s without m u c h difficulty that f o r e v e r y 0 , with 0 < Q < TT/2, and e v e r y e > 0 t h e r e e x i s t s a function C^g ^(t) > 0 defined in the angle | A r g t | = Q and independent of p s u c h that

(1) | h^{( p )}( t ) | ^ C^{e > 0}( t ) e^P( 2 p ) ! , p = 0 , 1 . 2 F u r t h e r m o r e f o r fixed c and G

( 2 ) C^e 0(t)-*O when t — 0 in the angle | A r g t | ^ 6 . It i s i m m e d i a t e l y s e e n that the function

oo . . 2 p

( 3 ) u ( z , t ) = E h ^ t ) ^ . b e l o n g s to G.U in one s p a c e - v a r i a b l e z .

- 4 3 -

If | z | < ^ and | A r g t | ^ 0 , w e h a v e , a c c o r d i n g to (1),

| u ( z , t ) I ^ 2 C Q( t ) w h i c h tends to z e r o with t a c c o r d i n g t o ( 2 ) . I f f o l l o w s that U ( Z , T ) as e n t i r e functions o f z c o n v e r g e to z e r o in Y(<C*) for T ~ * 0 in any a n g l e . H o w e v e r , U ( Z , T ) i s a r e g u l a r t r a c e c o r r e s p o n d i n g to the s o l u t i o n u ( z , t + T) and in GY it c o n v e r g e s to the t r a c e u / 0

c o r r e s p o n d i n g to the s o l u t i o n u ( z , t ) . H e n c e , i n c o m p a t i b i l i t y .

S i n c e the functions U ( Z , T ) a r e a l s o d i s t r i b u t i o n s and they c o n v e r g e t o 0 in J^', o u r t h e o r e m i s C o m p l e t e l y p r o v e d .

R e m a r k 1. A s a c o u n t e r p a r t to the s o m e w h a t n e g a t i v e r e s u l t o f T h e o r e m I it should b e n o t i c e d that it i s e a s y t o p r o v e that the i d e n t i f i c a t i o n of the s p a c e s S ' and Y^!( < Cⁿ) , with a s u b s p a c e o f t r a c e s i s c o m p a t i b l e w i t h the usual t o p o l o g y of t h e s e s p a c e s s i n c e the i d e n t i f i c a t i o n m a p p i n g i s a c o n t i n u o u s m a p p i n g o f &, S ' and Y ' ( Cⁿ) into G Y .

§ 3 . S o m e e l e m e n t a r y o p e r a t i o n s on t r a c e s .

W e w i l l u s e the t r a n s f o r m a t i o n s d e f i n e d in P . N . C h . I l l , § 1 , t o e s t a b l i s h c e r t a i n e l e m e n t a r y o p e r a t i o n s on t r a c e s .

a ) T h e t r a n s f o r m a t i o n G ( a + T ) i s a t o p o l o g i c a l a u t o m o r p h i s m of G Y j ^ » h e n c e t r a n s f e r r e d t o G Y ^ it b e c o m e s an a u t o m o r p h i s m o f G f t ^ . It d o e s n ' t c h a n g e the v a r i a b l e t in G Y „ and c o r r e s p o n d s t o a change o f

XV

v a r i a b l e s in Cn o b t a i n e d b y an o r t h o g o n a l t r a n s f o r m a t i o n T f o l l o w e d b y a t r a n s l a t i o n a . S i n c e the t r a c e s a r e a t t a c h e d to I Rⁿ the s i g n i f i c a n c e o f this t r a n s f o r m a t i o n c a n o n l y b e g i v e n v i a the c o r r e s p o n d i n g s o l u t i o n o f the heat e q u a t i o n . H o w e v e r , i f the t r a c e i s r e g u l a r , i . e . i s an e n t i r e function u(x) = u ( x , 0 ) , the s i g n i f i c a n c e c a n b e d i r e c t l y p i c t u r e d b y

- 4 4 -

c o n s i d e r i n g the function u(a + T x ) r e s t r i c t e d to x 6 TRn .

R e m a r k 1. T h e r e i s an i n t e r e s t i n g i n t e r p r e t a t i o n of the t r a n s f o r m a t i o n G(a) f o r any c o m p l e x v e c t o r a_. S i n c e TRⁿ i s c o n s i d e r e d as a v e c t o r s p a c e it has a fixed o r i g i n 0 . The t r a n s l a t i o n with a r e a l v e c t o r a_ shifts the o r i g i n to the point a , I£ h o w e v e r , a i s a c o m p l e x v e c t o r , then I Rⁿ i s shifted to a p a r a l l e l h y p e r p l a n e in ( £ⁿ with o r i g i n at a. W e can then c o n s i d e r the o r i g i n a l t r a c e u as d e t e r m i n i n g an " a n a l y t i c " function defined on <Cⁿ with v a l u e s in G Y^D , the value f o r a = 0 b e i n g u i t s e l f and f o r any

R

c o m p l e x a € Cⁿ the value i s obtained f r o m u by the t r a n s f o r m a t i o n G ( a ) . T h i s i n t e r p r e t a t i o n i s s o m e t i m e s useful.

R e m a r k 2 . G(a) t r a n s f o r m s a r e g u l a r t r a c e into a r e g u l a r t r a c e . In g e n e r a l , h o w e v e r , e v e n if u i s a v e r y r e g u l a r function the t r a n s f o r m of u b y G(a) i s not a function f o r a_ c o m p l e x .

b) T h e t r a n s f o r m a t i o n & ( T ) i s a t o p o l o g i c a l i s o m o r p h i s m of G Y ( D ) onto G V ( D - T ) . N o t i c e that f o r 0 < T < 2 R , B ^ ( R ) - T => B2 _ ^ ^2)( R - ( T / 2 ) ) .

W e c a n then a c c e p t as t r a n s f o r m &.(T)U, f o r u(x, t) € G V , the r e s t r i c t i o n of the actual t r a n s f o r m to ^{B R}_ (^T/ 2 ) (^R- (^T/ 2 ) ) « With^tlus c o n v e n t i o n

$ ( T ) ( 0 < T < 2R) b e c o m e s a continuous i s o m o r p h i s m of GY.^ into G V ^ (T/Z) and b y t r a n s f e r <S(T) i s a continuous i s o m o r p h i s m of G V ^ into GY.^ fz) (for a < T < 2R) and of GY into GY ( f o r T > 0 ) . It i s c l e a r that f o r an R - t r a c e u ( B ( T ) U = u(x, T ) .

c) W e w i l l u s e the t r a n s f o r m a t i o n s C ( c ) only f o r r e a l c / 0 . T h e y t r a n s f o r m C Y ^ i s o m o r p h i c a l l y onto ^^yc2 •

d) T h e t r a n s f o r m a t i o n s fiff' 5 ) . . . , , . . ^x, .

\ y , o / w i l l b e u s e d e s p e c i a l l y , i n the f o l l o w i n g two c a s e s ( s e e P . N . , C h . I l l , § 1 ) .

- 45 -

1? *>( ^ ' ^X^² ) • If q>€*> the F o u r i e r t r a n s f o r m

Cp(x) = J cp(5) e "i^x^ d ? b e l o n g s to S and a d i r e c t c o m p u t a t i o n s h o w s that I Rn

nm

^ ( 2 п ) ^ е 4 s( 0 , Vi)-

Z - ( 2 т У² - x²/ 4 t ~ / x 1 \

<

0 i / 2 ^ ^

2i > o ^ b e i n g a t o p o l o g i c a l a u t o m o r p h i s m o f Q9/ and A b e i n g d e n s e in GV the m a p p i n g cp - " 9 extends by continuity to the w h o l e of GV and w e obtain thus the F o u r i e r t r a n s f o r m u~* 11 t r a n s f o r m i n g the w h o l e of G.U onto i t s e l f by the f o r m u l a

(1) Л , ^t, (2ттГ - x2/ 4 t ~ / x 1 >

4^-1

The i n v e r s e F o u r i e r t r a n s f o r m u is given by

-nm . „ / 2 0 /

( l ' ) f t ( * t ) = ( 2 n ) e A ^ . ; V J u = ^ > * • 4 F / - R e m a r k 3. F o r m u l a (1) f o r the F o u r i e r t r a n s f o r m l e a d s to an i n t e r e s t i n g (but s o m e w h a t v a g u e ) i n t e r p r e t a t i o n if w e r e s t r i c t t to b e p o s i t i v e and i n t r o d u c e the notion of t r a c e s at + 0 0 . W e c o u l d s a y that

n f o r a function u € G.U the t r a c e at 0 on the r e a l h y p e r p l a n e IR i s t r a n s f o r m e d b y the F o u r i e r t r a n s f o r m into the t r a c e of u at +00 on the p u r e l y i m a g i n a r y h y p e r p l a n e i I Rⁿ .

2 ? ^ \ 4^a ' 1 ) * If <P ^ ^ » then e^ax cp € A and a d i r e c t c o m p u t a t i o n 2

a x ~*

s h o w s that (e cp) i s given b y

(

e «

^x

V ( x , t ) = a U , ;

= f l - 4 a t ) "n/2o l - 4 « t a ( x _J \

<A *al) e * V l - 4 a t 9 T^4at >

a = BIT"" FR with R and R^! positive, ^ oo^f then ' ^ ) is

- 4 6 -

a t o p o l o g i c a l i s o m o r p h i s m of G Y ^ onto Q V ^ , S i n c e & i s d n e s e in GY_ as w e l l as G Yo l w e can extend by continuity the m u l t i p l i c a t i o n

a x² a x²

by e to the w h o l e of Q Y ^ and thus the m u l t i p l i c a t i o n by e ^x i s an i s o m o r p h i s m of G Y „ onto G V^{D l} g i v e n by

2 ^{a x}

(^{2 )} ( • « u ) ~ (^x. t )

=*(_l ; ^o)-

^{u = ( 1}

^.

^{4 a t )}

- ^ . i - 4 - t ; (^_

^f

ax?" ~ ^

R e m a r k 4. It is a c t u a l l y the f o r m u l a s for (e cp) and cp with cp € A w h i c h l e a d to the e s t a b l i s h m e n t of the w h o l e g r o u p of t r a n s f o r m a t i o n s

f Oi (3 V Q f ] X ^

\ y ' 6 / ^ ^{6 c o u} ^ c o n s i d e r m u l t i p l i c a t i o n by e with a r b i t r a r y c o m p l e x a but this w o u l d n e c e s s i t a t e the c o n s i d e r a t i o n of m o r e g e n e r a l t r a c e - s p a c e s G Y ( D ; t ^ ) .

R e m a r k 5. If a = < 0 , i . e . R < R¹ then using the i d e n t i f i c a t i o n c^x²

of R e t r a c e s with R - t r a c e s w e obtain that m u l t i p l i c a t i o n by e t r a n s f o r m s GY , into i t s e l f c o n t i n u o u s l y .

R

e ) : £ ( b ) . A g a i n w e take c p € $ , m u l t i p l y it b y e ^^x^^f and b y e l e m e n t a r y c o m p u t a t i o n obtain an e x p r e s s i o n f o r (e^^{b x}^cp) w h i c h then b y continuity, i s e x t e n d e d to all t r a c e s . W e define thus m u l t i p l i c a t i o n b y

e( b x ) ^a ^ R - t r a c e s . It i s g i v e n b y 2

( 3 ) ( e^{( b x )}u f ( x , t ) = £ ( b ) u = e^{(bx) + b} ^ ( x + 2tb, t) .

T h e m u l t i p l i c a t i o n by e ^^{b x}' i s a t o p o l o g i c a l a u t o m o r p h i s m of e a c h

au

^R

.

R e m a r k 6 . The f o r m u l a f o r (e^^{b x}^cp) i s the one w h i c h l e d to the e s t a b l i s h m e n t of the g r o u p of t r a n s f o r m a t i o n s C ( b ) .

We g i v e n o w a l i s t of f o r m u l a s c o n n e c t i n g the F o u r i e r t r a n s f o r m with other t r a n s f o r m a t i o n s i n t r o d u c e d in this s e c t i o n . Many o f t h e m a r e

s i m i l a r to the usual f o r m u l a s f o r F o u r i e r t r a n s f o r m s of f u n c t i o n s . T h e p r o o f s a r e o m i t t e d s i n c e they c o n s i s t in s i m p l e a p p l i c a t i o n of p r e c e e d i n g

- 4 7 -

f o r m u l a s given in this s e c t i o n . In all f o r m u l a s u € GU.

( 4 ) IT¹ * ( 2 n ) "ⁿC ( - l ) u ,

(5) (G(u)u)^A= e^{i ( a x )}u f o r any a £ <Cⁿ, (5») ( e^{( b x )}u f = G(ib)u f o r any b € <Cⁿ ,

2

(6) ( B ( T ) u r = e "^{T x} u f o r T > 0,

2

(6») ( e "^{T x} u r = © ( T ) u for T > 0,

(7)

(C(c)

^u

r

^;

=

L c ( l / c ) u f o r r e a l c / 0 .

I

^c

|

ⁿ

§ 4 . C o n v o l u t a b l e t r a c e s and m u l t i p l i a b l e t r a c e s .

If w e take two functions cp and

0

i n $ then cp(z,t) = ^cp(x) E ( z - x , t ) d x ,

Hj){z,

^{t) =}

J}

^ ( x ) E ( z - x , t)dx w h e r e the s y m b o l i c i n t e g r a l s a r e a c t u a l l y o r d i n a r y i n t e g r a l s . B y the c o n v o l u t i o n p r o p e r t i e s of E ( x , t) it i s then

c l e a r that (cp * </>)~(z, t' + t^{l !}) = (cp(x, t^f) * ^ ( x , t^M) ) ( z ) . H e r e the c o n v o l u t i o n on the r i g h t s i d e i s again g i v e n b y the o r d i n a r y i n t e g r a l

^

^cp(x,

t

^!

)4)(z-x,

t^{, !}) d x .

I R^N

W e c a n t r a n s f e r this p r o p e r t y to g e n e r a l t r a c e s to define c o n v o l u t a b l e t r a c e s n a m e l y :

D e f i n i t i o n . T w o t r a c e s u, v a r e c o n v o l u t a b l e i f f o r e v e r y p o s i t i v e t¹ and t" the c o n v o l u t i o n (taken with s y m b o l i c i n t e g r a l s )

u ( x , t ' ) * v ( x , t " ) ) ( x ) = y u ( x , t ' ) v ( z - x , t " ) d x e x i s t s r e l a t i v e to the t h r e e p a r a m e t e r s z , t '^f tⁿ ,

T H E O R E M I. If the t r a c e s u and v a r e c o n v o l u t a b l e then the e x p r e s s i o n J? u ( x , t ' ) v ( z - x , t^{f t}) d x = w ^ z ^ t ' y t " ) i s a function w ( z^tt^! + t") and a*(z,t) € GU*

- 4 8 -

P r o o f . By the p r o p e r t i e s of u and v to be analytic solutions of the heat equation, the function w ^ ( z , t^f, t " ) is an analytic solution of the heat

a

equation in v a r i a b l e s z and t", i . e . ^ w ^ = g^p*wl ' B y t be co mmut ativ ity of the c o n v o l u t i o n ( s e e P . N . , Ch. I, §5) w e can a l s o w r i t e

W j f z . t ' . t^{1 1}) = y v ( x , tⁿ) u ( z - x , t ' ) d x . H e n c e ^ w ^ = | p ^w^ ^# The equation

^-piwj = J p »wi *o r analytic functions w^(z,t' ,t") m e a n s that w^ depends only on the s u m t¹ + t^M, H e n c e w e can w r i t e v/^(z^$V^tt^u) - w ( z , t^f + tⁿ) and o b v i o u s l y w ( z , t ) i s an: analytic s o l u t i o n of the heat equation.

In v i e w of T h e o r e m I we can define:

Definition 2. If u and v a r e c o n v o l u t a b l e t r a c e s then the t r a c e w c o r r e s p o n d i n g to the solution w ( z , t) defined in T h e o r e m I w i l l be c a l l e d the c o n v o l u t i o n of u and v and denoted by w = u # v.

Our definition g i v e s i m m e d i a t e l y the f o l l o w i n g t h e o r e m : T H E O R E M II. If u, v a r e c o n v o l u t a b l e t r a c e s then (1) u * v = v * u

(2) F o r any c o m p l e x v e c t o r b , e ^^{b x}^ u and e ^^{b x}^ v a r e a l s o c o n v o l u t a b l e and ( e^{( b x )}u ) * ( e^{( b x )}v ) = < H u * v ) .

( 3 ) F o r any two c o m p l e x v e c t o r s a' and a " , G ( a!) u is c o n v o l u t a b l e with G(ai*)v and (G(a')u) * (G(a")v) = G(a^r + a")(u * v ) .

( 4 ) F o r p o s i t i v e T ' and T¹¹, ® ^¹) and ( B ( T " ) V a r e c o n v o l u t a b l e and

( £ ( T ' ) U ) * № ( T " ) V ) = $ ( T ' + T " ) ( U * v ) .

S i n c e f o r cp and 0 b e l o n g i n g to $ w e have

/s. A A

(cp * = Cp if)

we w i l l use the F o u r i e r t r a n s f o r m to define m u l t i p l i a b i l i t y of t r a c e s .

D e f i n i t i o n 3 . If f o r two t r a c e s u, v t h e i r i n v e r s e F o u r i e r t r a n s f o r m s

- 4 9 -

u~* and v~* a r e c o n v o l u t a b l e then w e w i l l s a y that u, v a r e m u l t i p l i a b l e and w e w i l l put

uv = (<W * '0~

¹

)

^A

.

Comparing the f o r m u l a s ( 4 ) - ( 7 ) of the p r e c e d i n g s e c t i o n with T h e o r e m EC we obtain i m m e d i a t e l y :

T H E O R E M III. If the t r a c e s u and v a r e m u l t i p l i a b l e then 1. uv = vu

o ( b ' x ) ( b^Mx )

2. e^v u and e^x ' v a r e m u l t i p l i a b l e f o r any c o m p l e x v e c t o r s b¹ j -Lii A , ( b ' x ) w ( b " x ) . (b» + b»')x

and b " and ( ev 'u)(ev ' v ) = ex ' uv.

3? G(a)u and G(a)v a r e m u l t i p l i a b l e f o r a~y c o m p l e x v e c t o r a_ and (G(a)u)(G(a)v) = G(a)uv .

2 ?

O - T^FX - T^MX

4. F o r any p o s i t i v e T ¹ and T " , e u and e v a r e m u l t i p l i a b l e

, , - T ' X² . . - T " X² . - ( T¹ + T " ) X²

and (e u) (e v) = e uv .

R e m a r k , ! . If u and v a r e functions in &t p a r t s 1?, 2? and 4 ? a r e o b v i o u s s i n c e f o r s u c h functions the m u l t i p l i c a t i o n as defined by us c o i n c i d e s with the usual m u l t i p l i c a t i o n . P a r t 3 ° i s a l s o o b v i o u s in the c a s e of a r e a l v e c t o r a. s i n c e then G(a) is the change of v a r i a b l e s x b y t r a n s l a t i o n b y the v e c t o r a. H o w e v e r , f o r m o r e g e n e r a l functions i d e n t i f i e d with t r a c e s the s t a t e m e n t s in T h e o r e m III a r e not o b v i o u s s i n c e w e d o n^Tt know if, in g e n e r a l , the m u l t i p l i c a t i o n as defined b y us c o r r e s p o n d s to the usual m u l t i p l i c a t i o n of f u n c t i o n s . M o r e p r e c i s e l y , the f o l l o w i n g p r o b l e m i s o p e n .

P r o b l e m . If u and v a r e functions identified with t r a c e s and if t h e i r p r o d u c t uv (as functions) i s a l s o i d e n t i f i e d with a t r a c e , i s it t r u e that the t w o functions a r e m u l t i p l i a b l e b y o u r Definition 3 and that t h e i r p r o d u c t b y Definition 3 i s i d e n t i c a l to t h e i r p r o d u c t as f u n c t i o n s ?

In the next s e c t i o n w e w i l l give a few i n s t a n c e s w h e r e the a n s w e r to this p r o b l e m is a f f i r m a t i v e .

- 5 0 -

§ 5 . C o n v o l u t o r s and m u l t i p l i e r s .

Definition 1. A t r a c e f w h i c h c o n v o l u t e s with e v e r y t r a c e u € GU is c a l l e d a c o n v o l u t o r . A t r a c e g w h i c h i s m u l t i p l i a b l e with e v e r y u € (3/ is c a l l e d a m u l t i p l i e r .

O b v i o u s l y the m u l t i p l i e r s a r e F o u r i e r t r a n s f o r m s o f c o n v o l u t o r s and v i c e v e r s a .

T h e o r e m I. If f i s a c o n v o l u t o r then the o p e r a t o r f # u t r a n s f o r m s c o n t i n u o u s l y GU into GU. If g i s a m u l t i p l i e r then gu is a continuous o p e r a t o r of GU into GU.

P r o o f . It i s enough to p r o v e the f i r s t statement s i n c e the s e c o n d f o l l o w s b y F o u r i e r t r a n s f o r m .

L e t f b e a c o n v o l u t o r . Denote f o r N = 1, 2 , . . . b y the c h a r a c t e r i s t i c function for B ^ O J c : <Cn. C o n s i d e r then the o p e r a t o r s

FN e ' C def ined for u ^ 0.U b y

( FNj€ u ) ( z , t ) = J ? ^ c , e + ~ ) xN( x ) u ( z - x , t ) d x .

l Rⁿ ^ ~

It is o b v i o u s that FM t r a n s f o r m s GU~*GU c o n t i n u o u s l y . F o r e fixed it i s c l e a r a l s o that

^lim 7 ( ^ , e + ^ ) x^N( x ) u ( z - x , t)dx =

^ f ( x , e ) u ( z - x , t ) d x = (f * u ) ~ ( ztt + e) = ((<B(e)f)* u f ( z . t ) .

By t r a n s f e r and b y u n i f o r m b o u n d e d n e s s t h e o r e m extended to F r e c h e t s p a c e s the c o n v o l u t i o n operator'33(e)f = l i m F^M i s a continuous o p e r a t o r GU-+GU. T h i s , h o w e v e i

N=oo i N' e

is sufficient to show the continuity of the c o n v o l u t i o n o p e r a t o r f.

C o n s i d e r now a c o n v o l u t o r f and the c o r r e s p o n d i n g F(x, t ^ ) , t ° € <E+ ,

- 5 1 -

F o r any u(x, t) € GY, c o n s i d e r

j i u(x, t') T ( - x , t ° - t ' ) d x = ( f * u f ( 0 , t ° ) for 0 < R e t ' < R e t ° . T h i s is o b v i o u s l y a continuous l i n e a r functional on GY.

0 1 n **" ^{/ v >}' P r o p o s i t i o n 1. If f i s a c o n v o l u t o r then for e v e r y t € <C£ , (03(t^u)f) € G Y ' .

P r o p o s i t i o n 2. If f o r a t r a c e i ((£(t⁰)f)~€G Y^M f o r e v e r y t°€ dj. then f is a c o n v o l u t o r .

In f a c t , b y T h e o r e m V I ' o f P . N . , Ch, III, § 3 , T f x . t⁰- ^ ) i s c o n v o l u t a b l e with any u(x, t') f o r 0 < R e t» < R e t ° and s i n c e this i s t r u e f o r e v e r y t ° € <L+, f is c o n v o l u t a b l e with e v e r y u € GY.

T H E O R E M II. In o r d e r that f be a c o n v o l u t o r it is n e c e s s a r y and sufficient that for e a c h t^€ <C+ , f ( x , t^) be a t r a c e c o r r e s p o n d i n g to a

0 ^ s o l u t i o n f ( x , t + t ) € GY ¹.

B y t r a n s f e r w e obtain i m m e d i a t e l y :

T H E O R E M 1 1 ' . The dual GY¹ of_ GY is c o m p o s e d of a l l r e g u l a r

~ 0

t r a c e s w h i c h a r e s e c t i o n s f ( x , t ) of c o n v o l u t o r s f.

Using T h e o r e m II of §4 w e obtain:

T H E O R E M III. If f i s a c o n v o l u t o r then I? e ^^{b x}^ f i s a c o n v o l u t o r f o r e v e r y b € ( Cⁿ, 2 ? C(a)f i s a c o n v o l u t o r f o r any a € Cn, 3 ? (B(T)f is a c o n v o l u t o r f o r e v e r y T > 0.

F o r m u l t i p l i e r s w e obtain ( s e e T h . I l l , §4):

T H E O R E M I I I¹. If g is a m u l t i p l i e r then 1? e ^^X^ g i s a m u l t i p l i e r f o r any b € <Cⁿ, 2 ? G(a)g i s a m u l t i p l i e r f o r any a € £n

2

3. e g i s a m u l t i p l i e r f o r any T > 0.

- 5 2 -

E x a m p l e s of c o n v o l u t o r s and m u l t i p l i e r s .

E v e r y a n a l y t i c functional is a c o n v o l u t o r ; e v e r y e n t i r e function of exponential o r d e r one is a m u l t i p l i e r . The s e c o n d part f o l l o w s f r o m the f i r s t by F o u r i e r t r a n s f o r m . On the other hand, the f i r s t part r e s u l t s f r o m the i d e n t i f i c a t i o n of an analytic functional F as a t r a c e of the s o l u t i o n

F ( z , t ) » < F ( x ) , E ( z - x ) , t >

and the s c a l a r p r o d u c t i s given by an i n t e g r a l o v e r a c o m p a c t set in x ( s e e P . N . , Ch. I, § 4 , (1)).

It f o l l o w s that all d i s t r i b u t i o n s with c o m p a c t s u p p o r t , in p a r t i c u l a r , functions in $ a r e c o n v o l u t o r s .

F o r all t h e s e c o n v o l u t o r s w e can w r i t e the c o n v o l u t i o n in m u c h s i m p l e r f o r m than in Definition 2. If F is an a n a l y t i c functional then (1) ( F * u ) ~ ( z , t ) = < F ( x ) , u ( z^rx , t ) > .

In p a r t i c u l a r , f o r a d e r i v a t i v e of the D i r a c m e a s u r e , D 6^ w h i c h is a d i s t r i b u t i o n with c o m p a c t s u p p o r t w e have

( Dk6Q* u f ( x , t ) = D ^ u ( x , t ) .

T h e F o u r i e r t r a n s f o r m of D ^ ^ i^s the m o n o m i a l i M x | ^ ; h e n c e all p o l y n o m i a l s a r e m u l t i p l i e r s .

B a o u e n d i ' s t h e o r e m ( s e e P . N . , Ch. I l l , § 3 , T h . VII) l e a d s to the f o l l o w i n g s t a t e m e n t :

T H E O R E M I V . ( B a o u e n d i ) . a) f is a c o n v o l u t o r if and only if f o r

eVe r v T > 0 t h e r e e x i s t n o n - n e g a t i v e c o n s t a n t s A , B , C and M , A > B , ' , ; T T T T T T

- 5 3 -

M J x | 4 B J x2| - A R e x2

s u c h that | f ( X , T ) 1 ^ C^T e ^T

b) g i s a m u l t i p l i e r if and only if g i s an e n t i r e function s u c h that for e a c h e > 0 t h e r e e x i s t s n o n - n e g a t i v e c o n s t a n t s A , B , C and M ,

w e e e — e

M |x| + B | x²| - A R e x²

Ae ~Be " e' with the p r o p e r t y that | g ( x ) | * Cee e

R e m a r k 1. It i s c l e a r b y v i r t u e o f this t h e o r e m that e a c h m u l t i p l i e r is a function i d e n t i f i a b l e with a t r a c e in e l e m e n t a r y c a s e ( s e e § 1 ) . If u(x) is any function identifiable as a t r a c e then b y definition o f m u l t i p l i e r s gu i s w e l l - d e f i n e d as a t r a c e but w e d o nft know if the function g(x)u(x) is a l w a y s i d e n t i f i a b l e as a t r a c e . H o w e v e r , i f w e know that g(x)u(x) i s i d e n t i f i a b l e as a t r a c e then w e w i l l p r o v e in the next t h e o r e m that the t r a c e c o r r e s p o n d i n g to the function g(x)u(x) i s a c t u a l l y g u . T h i s t h e o r e m w i l l s o l v e in a s p e c i a l c a s e the p r o b l e m stated in the p r e c e d i n g s e c t i o n .

T H E O R E M V . If g ( x ) i s a m u l t i p l i e r and the functions u(x) and g(x)u(x) a r e i d e n t i f i a b l e as t r a c e s then the t r a c e c o r r e s p o n d i n g t o g(x)u(x) i s a c t u a l l y gu.

P r o o f . C o n s i d e r Xj^ » N = 1, 2 , . . . , the c h a r a c t e r i s t i c function o f B ^ ( 0 ) c I R^{n an}d l e t cp £ $ b e n o n - n e g a t i v e with ^ cp(x)dx = 1. T h e n the function v ^ £( x ) =(uXj^) *

^(f

^-

))^

^{^ •} ^ n e l e m e n t a r y c o m p u t a t i o n s h o w s then that ( g ( x ) u^{N C}M)* = ( 2 n ) "^{n /} g * ^g . T h e function V^{N j 6}( x ) and v ^ f x ) = uXj^ a r e i d e n t i f i a b l e as t r a c e s in e l e m e n t a r y c a s e s . H e n c e it i s c l e a r that in t r a c e t o p o l o g y l i m v^{M =V} M - ^{S i nce t n e} m u l t i p l i e r g

e =0 AN, e IN

as o p e r a t o r i s continuous on GY, it f o l l o w s that l i m g v ^ ^g = g v^N, g ( x ) v^{N £}( x ) and g ( x ) v ^ ( x ) a r e functions i d e n t i f i a b l e b y t r a c e s in o r d i n a r y c a s e . H e n c e the l i m i t o f t r a c e s c o r r e s p o n d i n g to the f i r s t function, as e ~* 0, i s the t r a c e

- 5 4 -

corresponding to the second. Thus gv^ corresponds to the function gfxjv^fx). Finally, since g(x)u(x) is identifiable with a trace, that means that

S g(x)u(x)E(z-x,t)dx = (guf(z,t) exists. Hence it is equal to, in particular

uu

jj g(x)u(x)E(z-x,t)dx = J R

^{1 1}

"

¹

J g(R0)u(Re)E(a-R0,t)d0 dR 0 0 n-1

by the same procedure one sees immediately that

(gv

^N

f(z,t) = J R

^{N _ 1}

J g{R9)u(Re)E(

^Z

-R0, t)d© dR . 0 n-1

Since similar expansions Can be written for u(z,t) and v^(z,t) it is clear that the trace u corresponding to u(x) is the trace limit of . The trace corresponding to g(x)u(x) is the trace limit of gu^ . Finally by

continuity of the multiplier-operator g we have that the trace corresponding to g(x)u(x) is actually gu .

Corollary V

¹

. If g is a multiplier and u(x) is a function identifiable as a trace in elementary case then g(x)u(x) is also a trace in elementary case and is equal to the trace gu.

Remark 2. Except for functions, the largest class of distributions -ax

²

which we identified with traces are the distributions T such that e T € S

¹

for every a > 0. This identification is elementary (i.e. does not use

symbolic integrals). By procedures essentially similar but somehow simpler than the ones in the proof of Theorem V, it can be shown that if g is a multiplier the trace gT is identifiable with the distribution g(x)T.

- 55 -

R e m a r k 3. A v e r y c o n v e n i e n t m e t h o d a p p l i e d , in p a t i c u l a r , in p a r t i a l d i f f e r e n t i a l equations is the m e t h o d of l o c a l i z a t i o n b a s e d on m u l t i ­

p l i c a t i o n of functions and d i s t r i b u t i o n s by a function cp(x) € &. T h i s m e t h o d cannot be a p p l i e d in all g e n e r a l i t y when dealing with t r a c e s s i n c e functions in $ a r e not m u l t i p l i e r s . H o w e v e r , a kind of " a l m o s t1 1 l o c a l i z a t i o n m a y p e r h a p s be useful b a s e d on the fact that a function f(x) e n t i r e of e x p o n e n t i a l

2

—T x

o r d e r 1 when m u l t i p l i e d by e , T > 0 , b e c o m e s a c o n v o l u t o r (that is c h e c k e d e a s i l y by using B a o u e n d i!s c r i t e r i o n ) . It f o l l o w s by F o u r i e r t r a n s f o r m that if cp € o r m o r e g e n e r a l l y , if cp i s an a n a l y t i c functional then f o r e v e r y T > 0 , cp(x, T ) i s a m u l t i p l i e r . F o r any t r a c e u, cp(x, T ) U i s w e l l defined and without b e i n g of c o m p a c t s u p p o r t it has h e u r i s t i c a l l y a v e r y r a p i d d e c r e a s e at oo .

§6. D e v e l o p m e n t s in H e r m i t e p o l y n o m i a l s .

If u i s an a r b i t r a r y t r a c e , f o r m u l a 2 of P . N . , C h . I l l , § 3 , T h . IV s u g g e s t s the d e v e l o p m e n t :

O " , 5 , I U ^ A ' " H * ^ )

for any t > 0 . T h e A ^ ' s can be d e t e r m i n e d b y the f o r m u l a (3) of the a b o v e m e n t i o n e d t h e o r e m

(2) AR = ^ u ( 0 , t ° ) .

In o r d e r to know in w h i c h s e n s e equality in f o r m u l a (1) i s v a l i d , w e c o n s i d e r e a c h t e r m in the d e v e l o p m e n t as a t r a c e (which we m a y , s i n c e e a c h p o l y n o m i a l i s not o n l y a t r a c e but a m u l t i p l i e r ) . B y P r o p o s i t i o n 2 of P . N . , C h . I l l , § 3 , w e have .

- 5 6 -

л«°>

^{1 к | /}

Ч ( ^ ) ) ••'•^""'^(•¡ré)

S i n c e (again by the a b o v e - m e n t i o n e d T h . I V ) we have the d e v e l o p m e n t û ( z , t ) = Б S Ak( t ° - t) N/ 2H k( - )

p=0 к =p k K 4 zJt^-t '

the c o n v e r g e n c e of the s e r i e s being u n i f o r m on c o m p a c t s in z and t , the last s e r i e s c o n v e r g e s in the t o p o l o g y of GU, h e n c e the s e r i e s (1) c o n v e r g e s to u in the t o p o l o g y of GU. We obtain thus

T H E O R E M I. F o r e v e r y t > 0, the s e r i e s in (1) with c o e f f i c i e n t s given by (2) c o n v e r g e s in the t o p o l o g y of GV to u.

The i n t e r e s t of this t h e o r e m l i e s in the fact that it a l l o w s us to a p p r o x i ­ mate c a n o n i c a l l y any t r a c e u by p o l y n o m i a l s , n a m e l y the p a r t i a l sums

N

£ of the s e r i e s in (1).

p=0

§ 7 . A p p l i c a t i o n to i n v e r s i o n of c o n v o l u t o r s .

It is e a s i l y p r o v e d that the c o n v o l u t o r s f o r m a c o m m u t a t i v e a l g e b r a if the c o n v o l u t i o n i s taken as the m u l t i p l i c a t i o n - o p e r a t o r . T h i s a l g e b r a has a unit w h i c h i s the D i r a c m e a s u r e at 0, 6^ . It can be p r o v e d that in this a l g e b r a the

i n v e r s e e x i s t s o n l y in v e r y few e x c e p t i o n a l c a s e s s u c h as f = 6^ = Q{a.)6^ . H o w e v e r , w e m a y ask f o r the e x i s t e n c e of f* * € GU such that

(1) f * f*"¹ = f*"¹ * f = 6^Q .

Such f* ^ is in g e n e r a l not unique. We w i l l c a l l f* ^ a q u a s i - i n v e r s e of f. If f = L A, D 5n with A. c o n s t a n t s , f is a differential o p e r a t o r with

|k|SN k 0 k

constant c o e f f i c i e n t s and - is then a c o r r e s p o n d i n g e l e m e n t a r y s o l u t i o n . S i n c e f ~* i s in g e n e r a l not a c o n v o l u t o r w e cannot s o l v e an equation of the f o r m f * v = u by v = ^ *u; this w i l l give a s o l u t i o n if and only if

* i s c o n v o l u t a b l e with u .

- 5 7 -

B a o u e n d i p r o v e d the s o l v a b i l i t y of the equations f * v = u f o r l a r g e c l a s s e s of c o n v o l u t o r s f and t r a c e s u . What w e p r o p o s e to do h e r e i s to u s e the p r e c e d i n g d e v e l o p m e n t s to g i v e a s i m p l e c o n s t r u c t i o n of

* -1

f w h i c h s e e m s to be v a l i d in quite g e n e r a l c a s e s and m a y b e d e v e l o p p e d further to c o v e r e v e n m o r e g e n e r a l c a s e s .

C o n s i d e r a c o n v o l u t o r f and its F o u r i e r t r a n s f o r m g = f\ g i s a m u l t i p l i e r and by B a o u e n d i ' s c h a r a c t e r i z a t i o n it is an e n t i r e function

g(x) of an e x p o n e n t i a l o r d e r at m o s t 2. If w e denote by h the F o u r i e r t r a n s f o r m of the l o o k e d - f o r q u a s i - i n v e r s e of f, w e m u s t have the equality gh = 1 in the s e n s e of t r a c e s . H e u r i s t i c a l l y then h should b e the function l / g of x . T h i s function, h o w e v e r , is not in g e n e r a l a t r a c e f o r two r e a s o n s :

1? It m a y not be l o c a l l y i n t e g r a b l e b e c a u s e of the null m a n i f o l d s of g ( x ) .

2? Its b e h a v i o r at infinity m a y not be c o n s i s t e n t with our definition of functions i d e n t i f i a b l e as t r a c e s ( s e e definition at the end of §1).

H o w e v e r , t h e r e m a y e x i s t c o m p l e x v e c t o r s a € <Cn s u c h that the function —\—r- r e s t r i c t e d to x € TRn is identifiable as a t r a c e . T h i s

g(x + a)

i s the c a s e in w h i c h our c o n s t r u c t i o n w o u l d b e v a l i d . B y putting

h = l / g ( x + a) w e s e e that as a function h (x) s a t i s f i e s g(x + a)h (x) = 1.

a a a S i n c e g(x + a) is a m u l t i p l i e r and 1 i s a function i d e n t i f i a b l e with a t r a c e

(in e l e m e n t a r y c a s e ) w e get b y T h . V of § 5 that the t r a c e c o r r e s p o n d i n g to g(x + a) w h i c h i s G(a)g s a t i s f i e s ( G ( a ) g ) ha = 1. A p p l y i n g T h . I l l , 3 ° of § 4 w i t h the t r a n s f o r m a t i o n G ( - a ) , w e obtain that

( G ( - a ) G ( a ) g ) ( G ( - a ) h^a) = G ( - a ) l

i . e . g ( G ( - a ) ha) = 1 and the i n v e r s e F o u r i e r t r a n s f o r m of G ( - a ) ha g i v e s us the d e s i r e d f* .

- 5 8 -

E x a m p l e s .

I ) C o n s i d e r one s p a c e v a r i a b l e z and let f be the analytic functional e x p r e s s e d by e ' ^ ^ ^ 6 ^ . Its F o u r i e r t r a n s f o r m is g(x) = e^x. S i n c e its r e c i p r o c a l is e ^X w h i c h is identifiable with a t r a c e (in e l e m e n t a r y c a s e ) we have that ( e ~^x) ^ ~ * = e ' ^ / ^ ^ ó ^ is a q u a s i - i n v e r s e w h i c h i s a c t u a l l y an i n v e r s e (it is a c o n v o l u t o r ) .

I I ) C o n s i d e r n o w for one s p a c e v a r i a b l e f = sin( j - D ) 6Q . Now g(x) = s i n x and the function — - — i s not identifiable as a t r a c e . H o w e v e r , if w e take

sin x

any n o n - r e a l v e c t o r a € C , —:—? ¡—c is identifiable with a t r a c e , again

' s m ( x + a) ^&

in e l e m e n t a r y c a s e . H e n c e , this t r a c e t r a n s f o r m e d by G ( - a ) and b y i n v e r s e F o u r i e r t r a n s f o r m w i l l give us the d e s i r e d q u a s i - i n v e r s e . H o w e v e r , this qua s i - i n v e r s e is not an i n v e r s e s i n c e it i s not a c o n v o l u t o r and depending on the v e c t o r a w e m a y get s e v e r a l s u c h q u a s i - i n v e r s e s .

n k I I I ) C o n s i d e r now z € <C and take f = £ A, D 6~ with constant

| k | S N ^k °

c o e f f i c i e n t s A ^ , i . e . a differential o p e r a t o r with constant c o e f f i c i e n t s .

A Ik! i k

Then g(x) = f = L A, i¹ ' x • If the p o l y n o m i a l g(x) has no m u l t i p l e

|k| - N ^K

n u l l - m a n i f o l d s in <Cⁿ, t h e r e w i l l e x i s t v e c t o r s a 6 Cⁿ s u c h that on the h y p e r p l a n e a + I R^N the n u l l - m a n i f o l d s of g(x) a r e s i m p l e and of d i m e n s i o n at m o s t n - 2 . F o r s u c h a v e c t o r the function ^ ^ ^ is l o c a l l y i n t e g r a b l e for x € I R^N and its b e h a v i o r at infinity a l l o w s one t o p r o v e that it i s

identifiable with a t r a c e in e l e m e n t a r y c a s e . H e n c e again ^G(-a) g (^x^ ^a^ ) * g i v e s us the d e s i r e d q u a s i - i n v e r s e w h i c h i s an e l e m e n t a r y s o l u t i o n f o r

the d i f f e r e n t i a l o p e r a t o r .

A s i l l u s t r a t i o n let us take the c a s e of two s p a c e v a r i a b l e s with f = - A Ó Q , 2 2

g(x) = x¹ + x ^ . Then l / g ( x ) is not l o c a l l y i n t e g r a b l e a r o u n d the o r i g i n . H o w e v e r , if w e take the v e c t o r a =(ia^ ,0) with a^ f 0, a^ r e a l , then

- 5 9 -

2 2 2 g(x + a) = (xj + ia^) + x ^ . The n u l l - s p a c e of this p o l y n o m i a l on IR i s

the c o u p l e of s i n g l e points x^ ;= 0, x ^ = ± and the function l / g ( x + a) is a t r a c e . Taking (pt "^a) g (^x* +^a) ) ^ ^{w e 0}^^tai^{n a n} e l e m e n t a r y s o l u t i o n f o r the L a p l a c i a n . H o w e v e r , it w i l l be a different e l e m e n t a r y s o l u t i o n f o r a^ > 0 and f o r a^ < 0.

§ 8 . G e n e r a l o p e r a t o r s and d i f f e r e n t i a l o p e r a t o r s on the t r a c e s p a c e GY.

M o s t of the a p p l i c a t i o n s of t r a c e s to d i f f e r e n t i a l o p e r a t o r s w e r e i n v e s t i g a t e d b y M . S. B a o u e n d i and the r e a d e r w i l l find t h e m in the text of his l e c t u r e . W e w i l l l i m i t o u r s e l v e s h e r e to s o m e g e n e r a l

d e v e l o p m e n t s w h i c h w i l l not b e found in B a o u e n d i¹ s text. W e w i l l s t a r t by a g e n e r a l s e t t i n g .

L e t A b e a l i n e a r o p e r a t o r in GY defined on s o m e s u b s p a c e B o f GY. S i n c e the m a p p i n g u ~ * u ( x , t) f o r any fixed t € (C+ is the continuous i s o m o r p h i s m ©(t) w h o s e i n v e r s e can b e d e n o t e d b y <B(-t)^r f o r any r e g u l a r t r a c e v € (B(t)(B) w e can define the o p e r a t o r

(1) A^tv =(B(t) A(B(-t)v.

O b v i o u s l y

( lf) If v =(B(t)u then Atv = ( A u f ( x , t ) .

It f o l l o w s that a s o l u t i o n u € GY of the equation A u = v , v € GY i s o b t a i n a b l e by s o l v i n g the equation A^u^ = v^ f o r t € (C^. w h e r e v^ is the r e g u l a r t r a c e (B(t)v and ut i s a r e g u l a r t r a c e u^(x) s a t i s f y i n g the equation.

Ax^ut^(x) =lt ^ut^{( x )} •

H e n c e the g e n e r a l p r o b l e m of s o l v a b i l i t y of the equation Au = v in t r a c e s

r e d u c e s to a p r o b l e m w h e r e the g i v e n data and the r e q u i r e d s o l u t i o n s a r e r e g u l a r t r a c e s , h e n c e e n t i r e functions in x . In the c a s e of d i f f e r e n t i a l o p e r a t o r s

- 6 0 -

A with p o l y n o m i a l coefficients,, A^t turns out to be a l s o a d i f f e r e n t i a l o p e r a t o r with p o l y n o m i a l c o e f f i c i e n t s of s p e c i a l type and f o r the s o l u t i o n of the equation f o r u^ the C a u c h y - K o w a l e w s k i type t h e o r e m s can be a p p l i e d . T h i s w a s one of the m a i n t o o l s u s e d b y Baouendi to p r o v e that c e r t a i n standard d i f f e r e n t i a l o p e r a t o r s A for w h i c h the equation Au = v is not in g e n e r a l s o l v a b l e in d i s t r i b u t i o n s ( o r h y p e r f u n c t i o n s ) a r e s o l v a b l e in t r a c e s .

F o r s p e c i a l o p e r a t o r s A w e can g i v e At in a m o r e c o n c r e t e f o r m than (1). T o s i m p l i f y w e w i l l r e s t r i c t our o p e r a t o r s A m o r e than s t r i c t l y n e e d e d and w e w i l l a s s u m e that

(2) Au = g(f * u)

w h e r e f is a c o n v o l u t o r and g i s a m u l t i p l i e r . H e n c e , A : GY~**GY.

B e f o r e we continue w e w i l l give a p r o p o s i t i o n w h i c h b e l o n g s to the g e n e r a l t h e o r y of c o n v o l u t i o n s and f o l l o w s i m m e d i a t e l y f r o m the definition of c o n v o l u t a b i l i t y .

P r o p o s i t i o n 1. If u and v a r e c o n v o l u t a b l e then s o i s u and v ( x , t) f o r any fixed

t€<cjj_,

w h e r e v(x, t) is c o n s i d e r e d as the r e g u l a r t r a c e (B(t)v. F u r t h e r m o r e (3) ( u * v f (x,t) = ( u * <B(t)v)(x).

Using F o u r i e r t r a n s f o r m s and P r o p o s i t i o n 1 w e can w r i t e (An)*'¹ =fr^X* ( f * u ^ -¹

(f * u ) ~ ( x , t ) = (f * CB(t)u)(x) n/2 2

(f * u ) ^ -U( x , t ) = ( 2 T l )"/ ? e "X / 4 t ( ( £ • (B(l/4t)u)(x/-2ti)),

( 2 t ) ' -n/2 2

É^-1*(f* u/*

V ( x , t )

=

^ U f ^ e

^{"X / 4 t}( ( f * <B(l/4t)u)(x/2ti))]

( 2 t )ⁿ' *

T o this f o r m u l a w e a p p l y the F o u r i e r t r a n s f o r m . T o apply it c o n v e n i e n t l y we c h a n g e in the f o r m u l a the v a r i a b l e s x into y and obtain after a few

- 6 1 -

c a n c e l l a t i o n s

( A u f (x, t) = e "^{x 2}/^{4 t X}( y ) * [ e - ^ f f * <B(t)u)(2tiy))] } ( x / Z t i ) . It f o l l o w s that

(4) A^tv ( x ) = e "^X /^{4 t}^ "¹( y ) * [ e -^ty²( ( f „^v) ( 2 t i y ) ) ] } ( x / 2 t i ) .

A further s i m p l i f i c a t i o n happens when the m u l t i p l i e r g i s a p o l y n o m i a l . It is enough to c h e c k it in c a s e when g(x) = x | ^ , t = (k^ , • • • »^ⁿ)» when

^ - l = i 1^ I D " ^ . It is i m m e d i a t e l y s e e n that

2 2 i - | - < V[e- t yw(y)] = e- t y G^w(y)

w h e r e G' i s a d i f f e r e n t i a l o p e r a t o r in y with c o e f f i c i e n t s w h i c h a r e y

-Ml

I

p o l y n o m i a l s in y and t with p r i n c i p a l p a r t i ¹ ' D and if the c o e f f i c i e n t s of G' a r e d e v e l o p e d in m o n o m i a l s in y , y |m, then e a c h of t h e s e m o n o m i a l s

^ I ml

w i l l have f o r c o e f f i c i e n t a p o l y n o m i a l in t d i v i s i b l e b y t' . H e n c e w h e n w e r e p l a c e the v a r i a b l e y by x / 2 t i w e obtain an o p e r a t o r again with p o l y n o m i a l c o e f f i c i e n t s in x and t w h o s e p r i n c i p a l p a r t w i l l

\l I I

be (2t)' ' D and w e obtain the f o r m u l a

x x

(5) Atv ( x ) = G ^ ( ( f * v ) ( x ) ) , f o r Au = x | ^ ( f * u ) . If in a d d i t i o n , f = D 6^ then

x 0

(6) A v ( x ) = G ^ D ^ v ( x ) f o r Au = x f¹ u .

In c o n c l u s i o n , f o r a d i f f e r e n t i a l o p e r a t o r with p o l y n o m i a l c o e f f i c i e n t s

(7) A = E P^k( x ) D^k

|k| = m

w e c o n s i d e r the o r d e r of . H e n c e , w e can w r i t e

p

k

^{( x , =}

,,5

^p

^{k , t}

^x

^{i " -}

- 6 2 -