0 [
~
21. 1
We claim that the series 51 + 52 + L (Sjhj) j~3
converges in v(1 ) ,s (n;A). Indeed, let <1> E V(l) (~t;A) and let p least index for which supp
>
3 be the
1
Hl = 51 + 52 +
L
(Sjhj),
j=3
* we have that H1 E V(l
)(r~;A)
and 1
p
L
+ As for each k
f;
*
L
+
k=1
k=p+1
p + 1, I
converges in v( 1 ),s(n;A).
Call
2
XHI
s = 51 + 52 + L (Sjhj}. j~3
Then OS= ct>1T +
cj> 2
T+
L
00
O(Sjhj)
j~3
L ct>} = r.
[]
j=1
19.C. · The case s *0 ) (lRm+1 ;A) 19.4 Theorem Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1• Then there exists f E c(r)(1Rm+ 1;A), being of slow growth in 1Rm+ 1 too, such that Of= g in 1Rm+ 1• Moreover, if g has the growth factor 1, then f can be chosen in such a way that it has the grmtth factor 21 + 4. Proof Let g E c(r)(1Rm+ 1 ;A) be of slow growth in 1Rm+ 1 with ig(x)i 0 ~c*(1+1xl for some 1 E ~ and c* > 0. 162
f
Consider the sequence of open balls (Bk)k, 1, all of them being centred at the origin and having respective radius k~ Take a partition of unity (¢j)j~ 1 in V(Rm+ 1 ;R) subordinate to the locally finite open covering (Gj)j~ 1 m+1 o R oo of R , where G1 = B2 , Gj = tij+ 1'Bj_ 1, j ~ 2. Then g = I ¢jg in j=1 o m+1 . for a 11 j :;c 1 , the sequence of c(r)(R ;A). Not1ce that, as 0 ~ ¢j :: constants (C.) .. 1 with J J>
cJ. =
lgJ. (X) I0'
sup m+1 xElR
gJ.
¢J· g'
satisfies estimates of the type
Put for each j f . (X) J
Then Dfj
~
1,
= E*g . (X ) J
= gj and sup xElRm+1
Now, as for j ~ 3, Dfj
Js.J+ 1
= 0 in Bj_ 1• fj admits a Taylor development about
0
co
f.(x) J
=
I
s=O
P f.(x) s J
with
where C' depends upon the dimension m and R is chosen arbitrarily in ]O,j1[ (see 11.5.2). As for any E > 0 there exists C > 0 such that for all s E ~ E
C'(S+m+ 1)(1+S 2 S+l
)
~ CE (1+E)S,
we get that 163
IPs f.(x)l J 0
C ((1+dlx1)s (1+J.)l+1
5

E:
J1
Call for each k ~ 3, 1+2 hk(x) = 2 P5 fk(x). s=O m+1 Then hk E M(r) (~ ;A) and for all x E ~m+l for which (1+e:)lxl have that co
) 1 +1
S=l+3
<
k1, we
lx I )s ( Q+e:) k1
~c~ ((1+dlx1)1+3 (k1)1+"! 1_(1+dlxl
k1
Hence the series co
f1 + f2 +
2
(fkhk)
k=3 converges in c(r)ORm+ 1;A), say to f, and obviously Of= gin distributional sense. Now we claim that f is ~f slow growth in ~m+ 1 • Let p > 1, take 0 < e: < {and call [(1+2e:)p] the least natural number which is greater than or equal to (1+2e:)p. Then for any x E ~m+1 [(1+2e:)p]+2 lf(x) lo ~ I (f1+f2)(x) lo + 2 lfk(x) lo + lhk(x) lo k=3 . co
2
+
k=[(1+2e:)p]+3 Moreover there exist suitable positive constants Ci, of p such that (1)
sup l(f 1+f 2 )(x)l 0 xElRm+1 [ ( 1+2e:) p]+2
(2)
2 k=3
164
~
c1;
1, ••• ,4, independent
[(1+2c)p]+2 (3)
L
k=3
sup lhk(x) lo ~ C3(1+p)21+4;. XEBP 00
( 4)
sup xEBP
L
: : c4'f1+p)l+3 •
k=[(1+2£)p]+3
Hereby BP stands for the closed ball centred at the origin with radius p. The above inequalities thus yield that for some constant C' > 0, independent of p, sup lf(x)l 0 ~ C'(1+p) 21 +4 • xEBP This of course implies that f is of slow growth in mm+ 1 and that its growth factor is 21+4. c * m+1 19.5 Theorem LetT E s(l)(m ;A). such that DS = T.
Then there exists
s
* (mm+1 ;A), a E ~ m+1 and g, cont1nuous . Proof As T E S(l) and in mm+ 1, can be found such that T = aag (Theorem 5.4). In 19.4 there exists f E c0( )(mm+ 1 ;A) which is of slow growth r * m+1 Putting S = aaf, then clearly S E s(l)(m ;A) and DS = T.
* )(mm+1 ;A) E s(l
of slow growth virtue of Theorem such that Df = g. c
19.6 Proposition Let g E c(r)(n;A) where n is an open and relatively como (n,A) such that Df = g in n. pact subset of mm+1 Then there exists f E C(r) Proof Extend g with zero outside n. Df = (DE)*g = g in n. c 20,
Then f
= E*g E C0( r) (mm+ 1·A) and •
Primitives of monogenic functions
.
20.1 If n is an open subset of the complex plane and f E O(n), then primitives off inn may be constructed in a trivial way by using line integrals. In the case of monogenic functions the problem of primitivation is more comPlicated since such an elementary procedure is not available. Nevertheless it will be shown that for a wide class of open subsets of mm+ 1, primitives of monogenic functions do exist. 165
20.2 Let \lclRm+l be open and let i E {0,1, •.• ,m} = M. Then r2 is said to be xinormal if for each x E rt the set {x+sei : s E lR} n p is connected and has exactly one point in common with the hyperplane {X; = 0}. Furthermore r1 is called norma! with rn;pect to the origin if 1:2 is xinormal for all O,l, ••. ,m. Further we introduce the following notations: lRm+l + lRm+l lRm+l I
{X
E lRm+l
{X
E lRm+l
{X
E lRm+l
xo
>
0};
xo
<
0};
X
I 0}
0
lRm+l u lR~+l; + lR~+l = {X E lRm+l : X = 0 }. 0
Obviously JR~+l may be identified with lRm. 20.3 Lemma Let~ clRm+l be open and normal with respect to the origin. Furthermore let g E M(r)(n;A). Then there exists f E M(r)(~;A) such that
_a_ f = g in n. ClX 0
Proof
Consider the function X
h(x 0 +x) =
fo
0
g(s+x)ds.
Then clearly h is defined in~ and a~ 0 h
= g in ~. Moreover hE E(r)(~;A) and
an arbitrary solution of a~ 0 u =gin E(r)(~;A) can be written in the form u(x 0 +x) = h(x 0 +x) + ~(x) m+l where ~ E E(r) (1:2 n JR= ;A). Now we claim that~ can be chosen in such a way that h + Indeed, as in 1:2 X
X
= _a_(J ClXO
0
g(s+x)ds) +
Q
= g(O+x) + o 0 ~(xl, 166
~
E
M(r)(~;A).
J0°D g(s+x)ds + o 0 ~(x) 0
the condition D(h+~) = 0 in g is equivalent with o 0 ~(i) = g(O+;) in ~ n R=+ 1• But, as g(O+x) E E(r)(J n R:+ 1;A), in view of Theorem 19.2 there ought to exist I}! E E(r)(n n R~+1;A) such that D0 ,J!(X) = g(O,x). c Obviously, if :2J c Rr+J is open and normal ~Jith respect to the origin, g E M(r)J('1J;A) with M(r)J(::,J;A) = {f E C1 (:lJ;A) : D/ = 0 in SiJ} and i EMJ, an analogous reasoning as made in the above lemma leads to the existence of () . f E M(r)J ( C?J;A ) such that ax: f = g 1n ~lJ• 20.4
1
Theorem Let nJ cRMJ be open and normal with respect to the origin; let g E M(r)J(r2J;A) and let a E JNMJ. Then there exists f E M(r)J(c:iJ;A) such that aaf = g in nJ. 20.5
21.
MittagLeffler theorems
21.1 In 12.3.6 a classical version of the MittagLeffler theorem has been proved. In this section it will be shown that this result may be obtained as a corollary to another form of the MittagLeffler theorem. 21.2 Theorem (MittagLeffler) Let (gi)iEI be an open covering of the open set g cRm+ 1 and let gi,kEM(r)(gi n gk;A), i, k E I, satisfy the conditions gi,k
= gk,i
and gi,k + gk,l + gl,i
= 0 in ni n nk n n1 •
Then there exist functions gi E M(r)(gi;A) such that for all i,k E I gi,k = gk gi in gin nk. ~
Let (ws)sElN be a locally finite countable open covering of Q which is finer than (n.) "EI and let (cp s ) sElN be a partition of unity in V(~i;R) subor1 1 dinate to the covering (ws)sElN" Furthermore, for each s E JN, take is E I such that suppcp c g. • Then, whatever k E I may be, 1s s hk =
L
cps gi ,k E E(r) (nk ;A)
sEJJ
While for all k,
s 1
EI 167
hk hl
I
cps ( gi
sEJl
I
s'
k  g. 1) 1s'
cps g 1, k
sEJl = g 1,k
in r2k n rll"
As gl,k E M(r)(rlk n r~ 1 ;A), D(hkhl) = 0 in rlk n r~ 1 • h E E(r)(ri;A) may be defined in r2 by putting h(x) = Dhi(x),
X
Consequently a function
E Qi.
In virtue of Theorem 19.2 there exists u E E(r)(ri;A) such that Du = h in r2. Put for each i E I, g.1 = h.u and g.1, k = gkg.1 1 in Q.; 1 then g.1 EM( r )(n.:A) 1 21.3 Corollary Let r~ 1 and r~ 2 be open subsets of Rm+1 such that r~ 1 n n2 F cp. Furthermore let f E M(r)(n1 n r~2 ;A). Then there exist f 1 E M(r)(n 1 ;A) and f 2 E M(r)(r~ 2 ;A) such that f = f 1 + f 2 in n1 n n2 •
Proof Put gn n "1•"1 21.2. c Another direct application of Theorem 21.2 yields
f and apply Theorem
Theorem (Poincare). Let n cRm+ 1 be open and let I 1 and I 2 be closed mdimensional surfaces of ciass C1 contained in r2. Furthermore let f E M(r)(~ (I1 u I 2 );A). Then there exist f1 E M(r)(~L 1 ;A) and f 2 E M(r)(~I 2 ;A) such that f = f1 + f 2 in ~(I 1 U I 2 ). 21.5
Finally we get
Theorem (MittagLeffler) LetS= {a; : i = 1,2, ••• } be a subset of Rm+ 1 without any limit point; let (m(i) : i = 1,2, ••• ) be a sequence of natural numbers and let (G; : i = 1,2, ••• ) be a sequence of functions belonging to R(r)(S) such that for each i = 1,2, ••• , m(i)
I k=O 168
Then there exists a function f which is left meromorphic in ffim+ 1 and admits a; as poles and G; as corresponding principal parts. Proof. Put D0 = ffim+ 1,s; then D0 is open in ffim+ 1 • Moreover, for each i ~ 1, let D1 be an open ball centred at a. such that D. n D. = ¢ if i # j. Then 1 1 1 J (D 0 , D1, D2 , ••• ) is an open covering of ffim+ and Dj n Dk F ¢for j ~ k implies that j = 0 and that Dj n Dk = Dk'{ak} if k F 0. Put for each i c 1, g0 ,; = gi,o = G; and g0 , 0 = 0; then g0 ,i E M(r)(D 0 n Di;A) = M(r)(D;'{ai};A) and the sequence (g 0 ,i)i~o satisfies the conditions of Theorem 21.2. Hence there exists gi E M(r)(D;;A), i ~ 0, such that
Clearly g0 is the desired left meromorphic function. 22.
c
The dual of M(r)(D;A)
22.1 One way of characterizing the dual of M(r)(D;A) might run as follows. As M(r)(D;A) is a submodule of Harm(r)(D;A), using HahnBanach's theorem, M(r)(D;A) is completely described once Harm(r)(D;A) is known. But, considered as a real vector space Harm(D;A)
=
IT
Harm(D;ffi)
AEPN so that, Harm' (D;ffi) being well known (see e.g. [19]), Harm' (D;A) and hence * Harm * (r)(D;A) are obtained. This way of determining M(r)(D;A) is of course rather formal and the spirit of this section consists exactly in describing explicitly the dual of M(r)(D;A). 22.2 Let D be an open subset of ffim+ 1 and let (Kj)j~ 1 be the c~mpact exhaustion of D considered so far (see 18.10). Furthermore letT E M(r)(D;A). Then there exist C > 0 and j E ~ such that J
;;
CpK. (f) for all f E M(r)(Q;A). J
As M(r)(D;A) is a submodule of c(r)(Kj;A), endowed with the norm llfllj = sup jf(x)j 0 , f E c(r)(Kj;A), xEKj 169
we have that in view of the HahnBanach and Riesz representation theorems there exists an Avalued measure~ in Rm+l supported on Kj such that for all f
E M(r) (\i;A)
fd ~ (X )f (X).
Now let ¢ E V(0;R) with ¢(X) 1 on an open neighbourhood w¢ of Kj which is still contained in Q. Then by virtue of the representation formula of left monogenic functions on compact sets (see Theorem 17.12), for each x E Kj f(x) = JE(xt)D(f¢)(t)dt, so that, using Fubini's theorem, we get:
Jd~(x)
!E(xt)D(f¢)(t)dt
f[fdp(x)E(xt)]D(f¢)(t)dt J~*E(t)D(f¢)(t)dt.
If we put
then we have that, up to the constant (1), Fj equals the Cauchy transform m+1 'Kj;A). On the analogy of classical function ~*E of~ and so Fj E M(l)(R theory, F. is called the indicatrix of Fantappie associated with Tin m+1 J R 'Kj. We have thus proved Let T E M(r)(ro;A) be bounded by PK. and let Fj be its .... . m+1 J associated indicatrix of Fantapp1e 1n lR 'Kj. Then, if¢ E V(D;R) with ¢(t) = 1 on some open neighbourhood w¢ of Kj with w¢ c ro, we have that for all f E M(r)Ll;A) 22.3
Proposition
( 22. 1)
22.4 The converse of the foregoing proposition will now be shown. Denoting by ¢· the set of functions ¢ E V(ro;R) such that ¢(x) = 1 in some open neighbourho~d w¢ of Kj contained in ro, then, by taking u E M(l)(lRm+l, Kj;A) and¢ E ¢j fixed, a right Alinear functional Tu on M(r)(ro;A) is 170
defined by
u(t)D(f~)(t)dt
supptj;
=
J
(uD)(t).(fijJ)(t)dt
supplj_c~Kj
0.
We now claim that Tu E M(r)(~;A), i.e. that Tu is bounded. To this end we proceed as follows. As u determines a right Adistribution Tu in Rm+l,Kj by putting for each lJ! E Vr ) (Rm+l,K. ;A) ,r
J
J
u(t)ojJ(t)dt,
Rm+1,K. J
we obtain that for any U E v{r)(Rm+l ;A) which extends u and any (Rm+l ;A) of the form ljJ = D(frp) with ¢ E '~'j and f E r~(r)(ri;A)
r
J
ljJ
E V(r)
u(t)D(f¢)(t)dt
Rm+1,K. J
Nou let t: > 0 be such that K = {x E st:d(x,Kj) ~ 2E} is contained in w¢. Then, as UD E v*( )(Rm+ 1 ;A) with supp UD c K., there ought to exist ME~ and r m+1 J C > 0 such that for all ljJ E V(r)(R ;A) (see e.g. [6]).
I
;';
c
sup sup l3o.ijJ(x) lo [ai
c
sup sup [3a.f (X) I0. [a[:;;M xEK
But, as f is an Avalued harmonic function in S1, a suitable compact neighbourhood K of K, with K c w , may be found such that for some constant ll ll
"::
C* sup XEK
jf(x)io· ll
171
As
we finally obtain that I
or Tu E M{r)(n;A).
We thus have proved
22.5 1 Propositio~ Let ¢ E ~j where j E ~ is fixed. Then for any u E M(l) (~m+ 'Kj;A) the right Alinear functional Tu defined on M(r)(n;A) by
(22.2)
belongs to M{r)(n;A). 22.6 Proposition If F. E M(l)(~m+ 1 ,K.;A) is the indicatrix of Fantappie of * J m+1 J T E M(r)(n;A) then for all s E ~ , the function aSF. is the indicatrix of J Fantappie of the functional a8T defined by
By Leibniz's rule a8(f¢)
=
I y:;;B
172
(~)a 8 Yt.aY¢
=
~. L (~)aBYf.aYq,.
a13f.q,
Ys:S
yilO
Now for
y
I 0
(1)1BI!Fj(t).D[(asf).q,](t)dt <1)
Is I
for all f E M(r)(n;A). 22.7
c
Example Let a be a fixed point in n~ define T E M{r)(n;A) by
V f E M(r)(n;A).
So
The indicatrix of Fantappie associated
oa*E(x) =
E(ax).
Hence the functional T1 given by
corresponds to the indicatrix F1 (x)
(1)1 8 1
a~ E(ax)
a~ E(xa). 173
22.8 Other expressions for the formulae (22.1) and (22.2) will now be derived. * LetT E M(r)(~;A) be bounded by PK. and consider its associated indicatrix J
of Fantappie Fj. Fu.rthermore choose q., E cpj and let K be a suitable compact neighbourhood of Kj fot· which K c supp¢ c w¢. Then, applying Green's Identity, we have that for all f E M(r)(~;A)
!Fj(t)D(f¢)(t)dt (
 J
;)K
(22.3)
F .(t)datf(t). J
"J m+1 Conversely, fix j E ~and take¢ E cpj and u E M(l )(m 'Kj;A). Then from the preceding considerations it follows that T is bounded by pK , where K u n is a suitable compact neighbourhood of Kj. Hence the domain of fntegration in formula (22.2) may be restricted to supp¢'K so that, again using Green's T] Identity, for all f E M(r)(Q;A)
(22.4)
n
The formula (22.3) expresses the relationship between T and its associated indicatrix of Fantappie in a more classical form, namely by a Cauchy type integral (see also [12] for the case of holomorphic functions). 22.9 Definition Let j E ~be fixed. Then we denote by LJ(.l) the following m+1 m+1 . subset of M(l)(m 'Kj;A): u E M(l)(m 'Kj;A) belongs to ql) 1f there exists cp E ¢j such that for all f E M(r)(K;A) Ju(t).D(f¢)(t)dt = 0. Obviously L~l) is a submo~ule of M(l)(mm+ 1,K.;A) while for any u E Ltl)'Tu=O. Notice also that if u E Ltl) then for each¢~ E ¢j Ju(t).D(f¢*)(t)dt = 0 and this for all f E
M(r)(~;A).
j
The meaning of L( l) will become clear from
22.10 Proposition Let j E ~be fixed. Then u E Ltl) if and only if there exists i 0 > j such that u = 0 in mm+ 1,K .• 10
174
Proof Suppose that u E M(l)(IRm+ 1,K.;A) vanishes outside some K. (i J 1 0 Taking ~ E ~io then clearly ~ E ~j whence for any f E M(r}(~;A) 0

Ju(t).D(f~)(t)dt
> jj.
=0
j
or u E L(l)• . Conversely let u E Ltl)• f E M(r) (n;A)
Then there exists ~ E ~j such that for all
ju(t).D(f~)(t)dt =
0.
Now call i 0 > j the least index such that . 1Rm+1 ' K.• u = 0 1n
supp~ c
0
Ki
Then we claim that o
10
Indeed, by the density of M(r)(n;A) in M(r)(Ki 0 ;A) we have that, given f* E M( )(Ki ;A), there exists a sequence (hs)s~' in M( )(n;A) which converges . r o . c.rn m+ 1 r o un1formly on Ki to f*. Hence for arb1trary a E ~ and H c Ki compact, 0 0 (aa.hs)se:IN converges uniformly on H to aaf*. Consequently Ju(t}.D(f*~)(t)dt
=0
for all f* E M(r)(Ki 0 ;A). Now take a E lRm+1 'Ki 0 ; then clearly E(.a) E M(r)(Ki 0 ;A) so that Ju(t).D(E(ta}<(J(t)}dt
=
0.
Furthermore let K be a compact subset of w
i/1
0
= Ju(t).D(E(ta)~(t))dt = J(utji)(t).D(E(ta)
so that, using Green's Identity, J((utji)D)(t).E(ta)~(t)dt
=0
or J((utji)D)(t).E(ta)dt
J((utji)D)(t).E(ta)(1
~(t))dt
= 0. 175
As in Rm+ 1,K (uw)D = uD = 0 and 1$=0 in w$• the second term vanishes. As to the first term, notice that uw E E(r)(Rm+ 1;A) is an asymptotic extension of u so that, u being regular at infinity with respect to E,
o = (1 )J{(uw)D)(t).E(ta)dt ((uw)D•E)(a) (uw) (a) =
u(a).
As a E Rm+ 1,K.
has been taken arbitrarily, u = 0 in Rm+ 1,K. •
10
c
10
m+1 'Kj;A) be such that Tu = Tu • Then, taking 22.11 Now let u 1 ,u 2 E M(l)(R 1 2 $ E ~j fixed, for any f E M(r)(n;A)

Tu 2 ,f> J(u 1

u2 )(t).D(f$)(t)dt
=0 j
andsoulu2EL(l)• _ _ Conversely, if T E M(r)(n;A) is bounded by PIS and Fj E M(l)(Rm+ 1,Kj;A) is its associated indicatrix of Fantappie, then in view of the foregoing cons~derations, any element of the form F.+h, hE LJ(. l)' also respresents T. As J m+1 J L(l) is a submodule of M(l)(R 'Kj;A), the quotient left Amodule j ( m+1 j M(l) = M(l) R 'Kj;A)/L(l)
is well defined and it consists of elements m+1 j [u]j = u + L(l)' u E M(l)(R 'Kj;A). As for each j E ~.Ltl) clearly is a submodule of Lt;~ and any element of M(l)(Rm+ 1,Kj;A), when ~estricting it to Rm+ 1,Kj+ 1 belongs to M~ 1 )(Rm+ 1 , Kj+ 1;A), each [u]j E Mtl) may be considere~ as be~onging to M~i~ and this by means of the canonical injection Ij : Mtl) ~ Mt~~ where for uEM(l)(Rm+ 1, Kj ;A)
176
lj ( [u]j ) = u
+
j+1 L(l)
= [u]j+1" Consequently an increasing sequence M(l) c MCl) c ••• of left Amodules is obtained, the union of which detennines a left Amodule denoted by m+1 . M(l}(lR ....r~;A), 1.e. m+1 M( 1) (lR ....mA) = u Mil). j ;;:1
Define J: M{r)(r~;A)+ M(l)(IRm+l,Q;.A) in the following way: if T E M(r)(Q;A) is bounded by PK. and has Fj as associated indicatrix of Fantappie, then J
J(T) = [Fj]j.
It is clear that J is an algebraic isomorphism between those left Amodules. But there is more! First of all notice that for each j E ~. Lil) _is closed m+1 . 1n M(l)(IR ....Kj;A}. . Indeed, let (us)s~ be a ~equence in Ltl) converging to u in M(l)(IRm+l...._ Kj;A) and suppose that u ~ Ltl)" Then for~ E ¢j given, there ought to exist f E M(r)(Q;A) such that Ju(t)D(f~)(t)dt
I 0.
As for each s E ~ Jus(t)D(f~)(t)dt
= 0,
the integral being taken over the compact set argument yields Ju(t)D(f¢)(t)dt
supp~....w~
clRm+1 ....Kj, a classical
= 0,
. a left Frechet .. which is clearly a contradiction. So, Mrl )(IRm+1 ....K.;A) be1ng j . J Amodule, M(l) may be endowed with the system of seminorms Pj = {pK K clRm+l...._K. compact}, where for each K J
inf pK(h). hE[u]j 177
In this way (At1 1 )'Pj) bec:omes a left Frechet Amodule too. Furthermore, the . . t . j +1 canonical lnJeC 10n I j: MJ(l) + M(l) is continuous since for each K emm+ 1'Kj+ 1 compact,
<:
inf
pK(u+h)
hELt 1) pK([u]j) and this for all u E utl)' Hence the left Amodule M(l)(mm•l,~;A) may be equipped with the inductive limit topology. As such we denote the associated system of seminorms by . m+l Pind and we wr1te M(l)(m 'D;A)ind' 22.12 Our main objective now is to show that the algebraic isomorphism J already established between r4(r)(S1;A) and M(l)(mm+ 1'S"l;A) is also topological when M{r)(~;A) is provided with the strong topology. To this end we proceed as follows. As was already noted in Section 16, M(r)(~;A) is a real (FS)space. Consequently M(r)b(~;A)*becomes a sequentially complete bornological space (see [4]). Denoting by Ej the left Amodule consisting of those*T E M{r)(Q;A) which are bounded by PK.' then, if we define for each T E Ej, J
!ITIIj
=
• {f .
sup "I
Ej, IIII· is turned into a left Banach Amodule.
.]
Clearly for each j E ~. Ej*
*
is a submodule of Ej+ 1 and
* M(r)(S1;A)
= u
j~1
* Ej.
As moreover the canonical injection from E~ II II into E~ 1 II II is * J' • j J+ ' j+ 1 continuous, M(r)(~;A) may be endowed with the inductive limit topology. The corresponding system of seminorms is denoted by P~nd and we write M(r)(~;A)ind' 0
178
* Moreover the strong topology on M(r)(n;A) is weaker than the inductive limit topology just defined. In the following proposition we point out that these two topologies are in fact equivalent on M~r)(n;A). 22.13 Proposition If in each of the underlying cases M(r)(n;A) is conside~ as a real convex space, then the following topological isomorphisms hold:
Consequently
Proof Let us first recall that in view of Proposition ?..14 the function e:M~r)(n;A)~(r)(n;A), with e(T)=Te T, TEM{r)(n;A), is an isomorphism between these real vector spaces. Using ProBosition 2.10 for all TEM(r)(n;A) and fEM(r) (n.!t>
* We now claim that e is the desired topological isomorphism between M(r)(n;Nind and M(r)b(n;A). Indeed, for any B c M(r)(n;A) bounded and T E M{r)(n;A)
* But the strong topology on M(r)(n;A) being weaker than the inductive limit one, there ought to exist C > 0 and q E Pind such that p6 (T) ~ Cq(T) for all T E M{r)(n;A). Hence for any T E M{r)(n;A). p6(e(T))
~
Cle0 1 q(T) 0
Which implies that e is continuous. A classical corollary to the closed graph theorem then yields that e 1 is also continuous, whence (i) is proved. In order to prove (ii) notice that from the arguments used in (i), we 179
* already have that the same e:M(r)b(n;A) ~ M(r)b(n;A) is continuous so that only the continuity of e 1 is left to show. To this end, take an arbitrary T E M(r)(n;A). Then n
1
l
L leAl
A
For each B c M(r)(n;A) bounded, put BA
= BeA' AEPN, and call s*
Then s* is bounded in M(r)(n;A) and sup l
Obviously (i) and (ii)
imply (iii).
o
Using the above proposition we finally get
Theorem J is a topological isomorphism between M(r)b(n;A) and "'M(l )(Rm+1 ' n;A)ind" Proof From the considerations preceding Proposition 22.9 we already know that J is an algebraic isomorphism. To prove its continuity, it suffices to show that its restriction to each E~ ll !I . is continuous. To this end, * J, • J take T E Ej arbitrarily and consider J T = [Fj]j E M~l)" Then for any q E Pind 0
=c <
=
0
J
(sup inf hE[F j] j tEK
cJ sup IF/Y)I 0 yEK 0
;;; c. sup I J
yEK
;;; c}(Kj) ;;;
f
d~(x)E(yx) 10
supp~
sup IE(w)lo wEKjK
C( Kj ;K)
where (c.).~,, (p.) ·,~,are the sequences of constants and seminorms determJ J<:.~• J Jg• 1 ining the seminorm q E P. d' K is the compact subset of Rm+ ,K. defining 1n
180
J
p.
E
P.,
while C(K.;K) > 0 depends on q. Hence J is continuous. As both _J m+1 and 11( 1 )(lR 'n;A)ind are inductive limits of Frechet spaces, the closed graph theorem implies that J 1 is also continuous. o ~
J r~(r)(\?,;A)ind
22.15 On the analogy of the classical cases of holomorphic functions of one or several complex variables, we call ~1(r)(~1;A) the space of analytic functionaL~ in g. By the previous theorem this space may thus be identified with a space of right monogenic functions in 1Rm+ 1,n. 23.
The bidual of M(r)(Q;A)
m+1 23.1 In this section we shall prove that the strong dual of r1( 1 ) (lR '~2;A)ind is topologically isomorphic to M(r)(n;A), which means that M(r)(n;A) is reflexive. Let f E M(r)(n;A) and associate with it the leftAlinear functional Tf on M(l)(lRm+ 1'si;A) in the following way. Take T E M(r)(n;A) and consider its indicatrix of Fantappie [Fj]j. Then by the isomorphism J, T = J 1[Fj]j. Now put
=
=
JF.(t)D(f¢)(t)dt. J
Then we claim that Tf is bounded ~n M(l )(1Rm+ 1,n;A)ind" Indeed, take any decomposition [Fk]k of [F] E r.\ 1 ) (JRm+kn;A), choose uk E [Fk]k and set u =
d)
I
(k)
uk and Tk
J
_,
[Fk]k.
I
l
I (k)
I ( k)
I
(k)
Then [FkJk>lo
l
If for each k occurring in the above decomposition, ¢k E ¢k is taken arbitrarily but fixed~ constants Ck > 0 can be found such that
181
I
~
ck
inf ukE[Fk]k
= ck i\<[FkJkl and so inf [F]=
L L [Fk]k
Ckpk([Fk]k).
(k)
( k)
Then we first prove 23.2
Proposition
I is an algebraic isomorphism between the right Amodules ( M(r)(ll;A) and M(l) lRm+1 '\l;A ) • ~*
Proof It is clear that I is right Alinear. To show that I is injective, let f*E M(r)(ll;A) be such that I(f) = 0. Then of course
= 0.
Fix ¢ E ¢j and apply Green's Identity to a suitable compact neighbourhood K~ of Kj which is still contained in w¢. Then
I
E(ax)do f(x) = 0
ClK
X ~
whence, by Cauchy's Integral Formula, f(a) I is injective. To prove the surjectivity rv* m+1 1" E M( l) (lR 'll;A). Then we must find f E a E \l and call again j E lli the least index 182
Consequently f = 0 in 0 or of I we proceed as follows. Let M(r)(\l;A) such thatlT =lff. Take such that a E K.. Then obviously
= 0.
J
~
~1
E(a.) E M(l)(R put f(a)
j
'Kj;A) and [E{a.)]j E M(l)'
Hence it is meaningful to
=
Using straightforward arguments it may be shown that f E for any a E ~ m+1
E(r)(~;A)
and that
As lis left Alinear, Df(a) and so f E
= = 0 M(r)(~;A).
We now claim that l
= lf. Notice that in any case
= 0.
(23.1)
Furth~rmore, fix k ~ 1 and consider the restriction of TfT to M~ 1
If [u]k E M(l) and hE [u]k, by Theorem 18.20 a sequence (hs)seN may be found in M(l)(Rm+1,Kk;A) which converges to h. Recall that each hs is a right Alinear combination of functions of the form )'
aPE(xai) 3x 1 ••• ax 1 1
From (23.1)
p
it thus follows that for all s E ~.
As a final step we prove
Theorem
is a topological isomorphism between
M(r)(~;A)
* m+1 '~;A). and M(l)b(R
m+1 '~;A)ind' Then by Theorem 22.10 B* = J 1 B Let B be bounded in M(l)(R is bounded in M(r)b(n;A). But as M(r)(n;A) is bornological, B* is equicontinuous and henGe contained in the polar of a semiball in M(r)(Q;A), say B* c b~K.(r) (see [4]). Hence, for any f E M(r)(Q;A)
~
J
183
= PB('Jrf) = sup l
I
PK.
s
1
r
J
PK. (f), J
which means that I is continuous. To show that I 1 is also continuous, first notice that M(r)(~;A) considered as a real convex space, is evaluable and has representable seminorms. Hence its natural system of seminorms is equivalent with the system {n 8 :BcM(r)b(n;A) bounded} (see [4]) where for each B n 8 (f)
=sup j
Consequently, for any j E ~. there exist C > 0 and B c M(r)b(n;A) bounded such that for all f E M(r)(~;A) PK. (f)
s Cn8 (f).
J
In virtue of Proposition 22.9 (ii), B* = e 1(8) is bounded in M(r)b(Q;A) and * is bounded 1n . M(l)(~ m+1 so, using Theorem 22.10 J(B) ~;A)ind· We thus obtain that
s Cn 8 (f)
c
sup
[Fj] lJ(B*)
= Cp J ( B*) (11" f). 24. 24.1 184
IJ
Hilbert modules with reproducing kernel 2
In this section the classical HL 2  and H spaces and their corresponding
Bergman and SzegB kernels will be generalized to Euclidean space and this in the setting of monogenic function theory. 24.2 Defi~ition Let H(r) be a unitary right Hilbert Amodule consisting of Avalued functions defined on some set F. Then a function K:F x F ~A is called a PcpPoducing krPn8l of H(r) if for any fixed t E F K(.,t) E H(r)
(i)
f(t) = (K(.,t),f) for all f E H(r)·
(ii)
In this case H(r) is said to be a unitary right Hilbert Amodule with reproducing kernel. 24.3 Theorem (AronszajnBergman) Let H(r) be a unitary right Hilbert Amodule consisting of functions defined on some set F. Then H(r) possesses a reproducing kernel if and only if for any t E F there exists a constant C(t) > 0 such that lf(t) Proof
10
<
C(t) llfll for all f
E
H(r)"
If H(r) has a reproducing kernel K then for any t E F and f E H(r) f(t) = (K(.,t),f)
so that lf(t)J 0
= <
I(K(.,t),f)l 0 IIK(.,t)ll. llfll·
Hence the necessary condition is fulfilled. As to the sufficient condition, take t E F and consider the right Alinear functional Tt on H(r) defined by = f(t) for all f E H(r)• Then by assumption
185
In view of the Riesz representation theorem, there exists a unique element ht E H(r) such that f(t) = = (ht,f) for all f E H(r)• Put
= ht(x), (t,x) E F
K(x,t)
x
F.
Then obviously K is a reproducing kernel of H(r)•
c
24.4 Now assume that H(r) is a unitary right Hilbert Amodule admitting two different reproducing kernels K and K'. Then for some y E F, K(x,y) # K'(x,y) and so 0
<
IJK(.,y)K'(.,y)ll2
Te (KK' ,KK') 0
Te (K,KK')Te (K',KK') 0
0
0,
which yields a contradiction. Consequently, if H(r) admits a reproducing kernel then it is necessarily unique. 24.5 In what follows we suppose that of 1Rm+1.
~
is a relatively compact open subset
Definition Call ML 2 (r)(~;A) = M(r)(n;A) n L 2 (r)(~;A), i.e. ML 2 (r)(g;A) consists of those left monogenic functions in n which are square integrable in
n. Clearly ML 2 (r)(~;A) is a submodule of L 2 (r)(~;A) whence it may be endowed with the inner product and norm defined on L 2 (r)(~;A) (see also Section 7). It will be shown that in this way ML 2 (r)(g;A) is a right Hilbert Amodule with reproducing kernel. Theorem Let t E such that 24.6
lf(t) 10
186
:;:;
~
be fixed.
C(t) llfll for all f
Then a constant C(t)
E ML 2 (r)(~;A).
>
0 may be found
Proof Call R(t) = d(tJRm+l~) and consider the closed ball B(t,r) with 0 < r = R(t)s. By virtue of the Mean Value Theorem (Theorem 9.7)
I
f(t) = m+; r
Vm+ 1
r
m+ 1v m+l
f(u)du
B( t, r)
or f(t)
I
J
eA
A
fA(u)du.
B(t,r)
Consequently, applying the CauchySchwarz inequality
I
L(
2
fA(u)du)
A B(t,r)
s
r
m+1v
L m+l
J
fA(u)du
A B(t,r)
I
lf(u)\~
du
B(t,r)
I
Q
rm+1v Letting s
+
\f(t)
\f(u) \ 2 du 0
m+l
0+ we get 10
~
C(t) \\fl\
where
24.7
Corollary
Let H c
pH(f) =sup \f(x)\ XEH
Q
0
be compact.
Then for all f E MLz(r)(Q;A)
s CH \lf\1
187
24.8 Theorem ducing kernel.
ML 2 (r)(~;A)
is a unitary right Hilbert Amodule with repro
Proof Taking account of the Theorems 24.3 and 24.6 it clearly suffices to show that ML 2 (r)(0;A) is complete. Let therefore (f i ) i €lN be a Cauchy sequence in ML 2 ( r) ( r:; A). Then there exists f E L2 (r)(r!;A) such that (fi)iE:lN converges to fin L2 (r)( ;A). But in view of Corollary 24.7 (fi)iE:lN is also a Cauchy sequence in M(r)(1;A) and hence it converges to f* in M( r) (c_;A). Clearly f* = f a.e. in ,, so that f* E ML 2 (r)(rz;A) and (fi)iE:lN converges to f* in ML 2 (r)(r;A). o 24.9
Remark
By the proof of the above Theorem 24.8 the Hilbert module
ML 2 (r)(u;A) is closed in L2 (r)(a;A). In general Mlz(r)('2;A) is not closed in M(r)( ,;A). Indeed, if a E Sm then E(xa) E M(r)(s(0,1);A). By Runge's Theorem (Corollary 18.14) there exists a sequence of inner spherical monogenics converging to E(xa) in M£r)(B(0,1 );A). All those inner spherical monogenics belong to Mlz(r) (B(0,1 );A) but obviously E(xa) does not. Nevertheless we can state: If the sequence (fj)jE:lN of functions fj belonging to Mlz(r)('.;A) converges to f in M( ) C1;A) and possesses a subsequence (f. ) for which ( 11 f. Ill r Jk kE:lN Jk kE:lN is bounded, then f E
ML 2 (r)(~;A).
Indeed (f. ) has a subsequence which J k kE:lN converges weakly in 'z(r)('?;A) and hence pointwise, necessarily to f. So f E Lz(r)(];A). 24.10 Observe that, if the reproducing kernel of ML 2 (r)(~;A) is denoted by B(x,t), (x,t) E ~ x ~~.then for each f E Mlz(r)CI;A) and t E r,, f(t) 24.11
=
(B(.,t),f)
Theorem
=
L
ML 2 (r)(~I;A)
B(x,t) f(x)dx.
is separable.
Proof Consider a sequence of points (yi)iE:lN which is dense in' and associate with it the countable subset Q c ML 2 (r)(~;A) defined by Q
188
= {B(.,yi)' i
E lN}.
Then we claim that Q is total in orthogonal to Q, then
ML 2 (r)(r~;A).
Indeed, if hE ML 2 (r)(S"2;A) is
h(yi) = (B(.,y 1 ),h) = 0 for all i ElL Hence, by the continuity of h in
S"l,
h = 0 in r2.
o
24.12 Now let for any r > 0, 88r denote the boundary of the open (m+1)dimensional sphere 8r centred at the origin with radius r and consider the unitary right Amodule L2(r)(88r;A) of Avalued square integrable functions on 88r, where Lebesgue measure is taken on 88r. 24.13 Definition Let R > 0; then ML 2 (r)(88R;A) consists of those elements in M(r)(8R;A) such that lim r+R
<
I
I f 12
°
88 r
dS
< +
00 •
It will be shown that provided with the inner product
I
(f,g)R = 1 im r+R
<
fgdS 8Br
and norm llfiiR=lim J lfl~ dS, r+R 88 < r ML 2(r)(8BR;A) is a unitary right Hilbert Amodule with reproducing kernel. 24.14
Theorem
For each t E 8R there exists C(t)
> 0
such that for all
f E ML 2 (r)(88R;A)
~
Lett E BRand take 0 < r < R such that t E Br. Integral Formula, for any f E ML 2 (r)(88R;A) f(t) =  1wm+1
I
88
iit
r
lutlm+l
do
u
In view of Cauchy's
f(u)
189
so that, as dau f(t)
= ndS = (*)dS
= 1
r~+1
on aBr'
J
ut u f(u)dS. aBr iutlm+1
Hence, putting h(u) obtain:
(ut)uf(u) and using the CauchySchwarz inequality, we
lf(t)l~ ~
r2 w...;.2_
m+1
(JaB r
ds \ r r jut12m+2} \JaB
r
Call d = d(t,IRm+ 1,Br); then on aBr' d <: iutj "2R whence ihl~ ~ 4R 2 r 2 jfl~· Consequently
so that, by passing to the limit for r

~
<
R,
where
24.15
Corollary Let H c BR be compact.
Then for each f E ML 2(r)(aBR;A)
where CH =
4Rm+ 2 1 R2m+2 with RH = d(H,IRm+ 'BR). wm+1 H
24.16 Next observe that the space Hh(aBR) consisting of those real valued harmonic functions f in BR for which
190
is a real Hilbert space when endowed with the inner product (see [22]) (f,g)R =lim r+R
<
r
j
as
fg dS. r
As each component fA off E ML 2 (r)(aBR;A) belongs to Hh(aBR) we find that for all f,g E ML 2 (r)(aBR;A) and A E A, with f = L fAeA' g = L g8e 8 and A
B
Hence 1 im
r+R
<
J
as
r
which means that ML 2 (r)(aBR;A) is a unitary right Amodule. By analogous arguments it may be proved that for any f,g E ML 2 (r)(aBR;A) the expression (f,g)R = lim r+R
<
J
as
fgdS r
is meaningful and that for all f,g,h E ML 2(r)(aBR;A) and A E A (f,g+h)R = (f,g)R + (f,h)R' (f,gA)R = (f,g)RA and
Moreover, as ,
is a (bounded) real linear functional on A, for each eo f E ML 2(r)(aBR;A) and A E A
while
191
and in view of the inequality stated in Corollary 24.15
Hence ML 2 (r)(aBR,A) is an inner product space. 24.17
Finally we get
Theorem ML 2(r)(aBR;A) is a unitary right Hilbert Amodule with reproducing kernel. Proof Taking account of the Theorems 24.3 and 24.14 it clearly suffices to prove the completeness of ML 2 (r)(aBR;A). Let therefore (fi)i~ be a Cauchy sequence in ML 2(r)(aBR;A). Then for all s > 0, there ought to exist N(s) E~ such that llfjfki!R ~ s if j,k ~ N(d. By the Corollary 24.15 (fi)i~ is a Cauchy sequence in M(r)(BR;A) so that, by the completeness of this space, there exists f E M(r)(BR;A) t~ which (fi)ieN converges. Now take 0 < r < R. Then, as for each~ E H (aBR) (see [22])
Jasr
lfjfkl~dS ~
llfjfkiiR
~
sz
so that, by passing to the limit for j JaBr
lffkl~dS ~
s 2 for all k r
Again using the property that ( J we find that aBr lim r~R
~
~ oo ,
N(s). !ffki~dS)O
is increasing when r
t
J~s lffkl 02 dS ~ s 2 if k ~ N(s), a
r
< whence ffk E ML 2(r)(aBR;A). Consequently f E ML 2(r)(aBR;A) while from the above reasoning it follows that (fi)i~ converges in norm to f in 192
R,
Notes to Chapter 3 The results in Section 16 are similar to those of the space O(n) of holomorphic functions in nor, even more generally, to those of the space of nullsolutions of a hypoelliptic differential operator with constant coefficients (see [Ho 1]}. The original Montel Theorem concerning locally uniform bounded sequences of holomorphic functions (see [Mo]} gave rise to a special class of locally convex spaces, the socalled FrechetMontel spaces (see e.g. [K1]}. As to the spirit of Section 17, it should be noted that in the more general context of elliptic differential operators with constant coefficients, the notion of a regular solution at infinity with respect to a fundamental solution, which in fact replaces the vanishing at infinity for holomorphic functions, was introduced by Grothendieck in [Gro1] and elaborated afterwards by Chauveheid in [Ch]. The Cauchy transform of a measure in the plane having compact support provides a useful example of such a function. As is well known, this Cauchy transform plays a basic role in approximation theory (see [Z]}. The Runge type theorems occurring in Section 18 are very close to the situation in one complex variable (see [Ho2]}. The problem of compact approximation of holomorphic functions in a domain by rational functions was solved initially by Runge in 1885 (see [Ru]}. Refinements dealing with polynomial approximation were obtained soon afterwards. The latter form of the Runge Theorem  polynomial approximation  gave rise to the introduction of socalled Runge domains in the theory of several complex variables (see [Ho2]}. The problem of approximating any solution f of P(D)f = 0, P(D) being a differential operator with constant coefficients, by nullsolutions of a special type was completely solved by 14algrange in [Mal1]. Note that his results already include the BehnkeStein Theorem for holomorphic functions on a connected non compact Riemann surface (see also [BS]}. The problem of solving the equation P(D)f = g was treated in full generality by Malgrange and Ehrenpreis (see [Mal1], [Mal3], [E1], [E2], [E3]} and this, among other things, for the cases E(n) and V'(n). As to the case S'(Rn), which corresponds to the socalled division problem of any tempered distribution by a polynomial, it was first solved by H6rmander in [Ho3] While a still more general problem, namely the division of a distribution by 193
an analytic function, was treated by bojasiewicz (see [to]}. As to the special case of the operator~ we also refer to Martineau [Mar]. The methods az we have developed in Section 19 for solving the equation Df = g when g E E(r)(n;A) or g E v( 1)(n;A) are therefore standard. For the case g E S( 1)(Rm+ 1;A) we have made explicit use of function theoretic results occurring in Chapter 2. Note that Theorem 19.4, relating the growth condition of the solution to the right hand side of the equation Df = g, implies that for every bounded set Bin s*(Rm+ 1;A) there exists a bounded set a in s*(Rm+ 1 ;A) such that B c Da. Note also that the solution of the problem in the E(r)(n;A)context gives rise to the construction of primitives of monogenic functions (Section 20). The form of the MittagLeffler Theorem as presented in Section 21 has now become standard for the case of holomorphic functions (see [Ho2]}. It is a strengthened form of the classical MittagLeffler Theorem. Its analogue in several complex variables is called the additive or first Cousin problem (see [BT2], [Ho2], [Mal2]}. For an abstract version of the MittagLeffler Theorem and its applications see also [Pe]. Duality in the theory of holomorphic functions was tackled in K6the's paper [K2] in which he showed that if V is an arbitrary proper open subset of the Riemann sphere 0 and if H(V) denotes the space of locally holomorphic functions on V, then its dual may b~ identified with H(U) the space of locally holomorphic functions on u = ~V. The identification between H'(V) and H(U) is obtained via the indicatrix of Fantappie of an analytic functional in V. The importance of this result of course lies in the fact that in this way linear functionals may be identified with holomorphic functions. Almost simultaneously Grothendieck developed in [Gro2]a duality theory for vector valued holomorphic functions defined on a proper open subset of 0. In fact by both authors' general results concerning duality theory for topological vector spaces were interpreted in the case of spaces of holomorphic functions. In this contexte Silva's papers [Si1] and [Si2] and the work [D] of Dias should also be mentioned, while for the historical notion of an analytic functional in the sense of Fantappie, we refer to Pelligrino's paper [P]. Tillmann worked out a duality theory for harmonic functions in the ndimensional Euclidean space (n ~ 3) and for analytic functions on Riemann surfaces in respectively [T1] and [T2]. The latter case was treated differently by Gauthier and Rubel in [GR]. As to the case of several complex variables we refer to Lelong's work [Le] and Braun's thesis [Br], while in the infinite 194
dimensional case a duality theory was given by Dwyer in [Ow]. In [Ch] Chauveheid described completely the dual of the space of nullsolutions of an arbitrary elliptic differential equation with constant coefficients. If should be recalled that our proofs concerning the dual and bidual of M(r)(D;A) (Sections 22 and 23) rely heavily upon Runge type theorems established in Section 18, whereas in classical function theory the Runge approximation theorem appears to be a simple corollary to duality (see [RT]). The study of classes of functions which possess a reproducing kernel was put into an abstract Hilbert space setting by Aronszajn in [A]. The usefulness of the notion of a reproducing kernel in holomorphic function theory of one or several complex variables and in the solution of boundary value problems of elliptic differential equations has been beautifully exposed in a lot of works by Bergmann (see [Bel], [Be2]) and BergmannSchiffer ([BeS]). For a general introduction to the theory of reproducing kernels we refer to [Me] and [Hil]. It should be noticed that in Section 24 we have restricted ourselves to proving the existence of reproducing kernels for the spaces ML 2 (r)(D;A) ard No attention case where ~ is total subset of Hilbert modules Hile in [GH].
ML 2(r)(3BR;A). has thus been paid to the basis problem. However, in the open, bounded and connected, the existence of a countable ML 2(r)(D;A) may be shown (see [DB]). Finally note that with reproducing kernel were also studied by Gilbert and
Bibliography [A]
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337404. [Bel] S. Bergmann, Sur les fonctions orthogonales de plusieurs variables complexes avec les applications a la th~orie des fonctions anal)tiques, M~morial des Sciences Math~matiques, Fasc. CVI(GauthierVillars, Paris, 1947). [Be2] , The kernel function and conformal mapping, Survey Math: Soc. 5 (1950). [BeS] S. Bergmann and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics (Academic Press, New York, 1953). R. Braun, CauchyFantappi~ Formeln und Dualitat in der Funktion[Br] entheorie (Thesis, Gtlttingen, 1972). 195
[BS] H. Behnke and K. Stein, Entwicklung analytischer Funktionen auf Riemannschen Fl8chen, Math. Ann. 120 (1948) 430461. [BT2] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Ver8nderlichen, Zweite Auflage (SpringerVerlag, Berlin, 1970). [Ch] P. Chauveheid, Application de 1 'analyse fonctionnelle ~ la theorie des fonctions analytiques (Th~se, Li~ge, 1973). [0] C. Dias, Espa~os vectoriais topologicos e sua aplica~ao nos espa~os funcionais analiticos, Sao Paulo (1951). [DB] R. Delanghe and F. Brackx, Hypercomplex function theory and Hilbert modules with reproducing kernel, ~oa. London Math. Soa. 37 (1978) 545576. [Ow] T. Dwyer, Dualite des espaces de fonctions enti~res en dimension infinie, Ann. Inst. Fourier 26, no 4 ( 1976) [E1] L. Ehrenpreis, Solutions of some problems of division I, Am. J. Math. 76 (1954) 883903. [E2] , Solutions of some problems of division II, Am. J. Math. 78 (1956) 685715. [E3] , Solutions of some problems of division III, Am. J. Math. 82 (1960) 522588. [GH] R.P. Gilbert and G. Hile, Hilbert function modules with reproducing kernels, Nonlinear analysis 1 (1977) 135150. [GR] P.H. Gauthier and L.A. Rubel, Holomorphic functionals on open Riemann surfaces, Can. J. Math. 28 (1976) 885888. [Gro1] A. Grothendieck, Sur les espaces de solutions d'une classe generale d'equations aux derivees partielles, J. d'Analyse Math. 2 (1952/53) 243280. [Gro2] , Sur certains espaces de fonctions holomorphes, I, II, J. Reine Angew. Math. 192 (1953) 3564, 7795. [Hil] E. Hille, Introduction to the general theory of reproducing kernels, Roaky Mountain J. of Math. 2 (1972) 321368. [Ho1] L. H6rmander, Linear partial differential operators (SpringerVerlag, Berlin, 1976). [Ho2] , An introduction to complex analysis in several variables (NorthHolland, Amsterdam, 1973). [Ho3] , ·On the division of distributions by polynomials, Ark. Mat. 3 (1958) 555568. 196
[K1] G. K6the, Topologische lineare R~ume I (SpringerVerlag, Berlin, 1960). [K2] , Dualit~t in der Funktionentheorie, J. Reine Angew. Math. 191 (1953) 3049. P. Lelong, Fonctionnelles analytiques et fonctions entieres (n var[Le] iables) (Presses de l'Universite de Montreal, 1968). [to] S. tojasiewicz, Sur le probleme de division, Studia Math. 18 (1959) 87136. [Mal1] B. Malgrange, Existence et approximation des solutions des equations aux d~riv~es partielles et des ~quations de convolution, Ann. Inst. Fourier 6 (1955/56) 271355. [Mal2] Lectures on the theory of functions of several complex variables (Tata Institute of Fundamental Research, Bombay, 1958). [Mal3]     Sur la propagation de la r~gularit~ des solutions des ~quations a coefficients constants, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumanie 3 (1959) 433440. [Mar] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, In: Theory of Distributions, Proc. Intern. Summer Institute, Inst. Gulbenkian de Ciencia, Lisboa (1964) 113326. [Me] H. Meschkowski, Hilbertsche R~ume mit Kernfunktion (SpringerVerlag, Berlin, 1962). [Mo] P. Montel, Sur les suites infinies de fonctions, Ann. Sci. Ecole Norm. Sup. 24 (1907) 233334. F. Pelligrino, Die analytische Funktionale und ihre Anwendungen, [P] Mat. Tidskrift B (1949) 3162. [Pe] H.J. Petzsche, Some results of MittagLeffler type for vectorvalued functions and spaces of type A, in: Functional Analysis: Surveys and Recent Results II (NorthHolland, Amsterdam, 1980) 183204. [RT] L.A. Rubel and B.A. Taylor, Functional Analysis: proofs of some theorems in function theory, Amer. Math. Monthly 76 (1969) 483488; Corrections, Amer. Math. Monthly 77 (1970) 58. [Ru] C. Runge, Zur Theorie der eindeutigen analytischen Funktionen, Acta, Math. 6 (1885) 229244. [Si1] J.S. e Silva, As Funcqes analiticas e a analisa functional, Port. Math. 9 (1950) 1130. [Si2] Sobre a topologica des espayos funcionais analiticos, Rev. Fac. Cienc. Lisboa 2, Ser. A1 (1950) 23102. 197
[T1] H.G. Tillmann, DualitMt in der Potentialtheorie, Port. Math. 13 (1954) 5586. [T2] , DualitMt in der Funktionentheorie auf Riemannschen FlMchen, J. Reine Angew. Math. 195 (1956) 76101. [Z] L. Zalcman, Analytic capacity and rational approximation, Lecture Notes in Mathematics 50 (SpringerVerlag, Berlin, 1968).
198
4 Boundary values of monogenic functions and Fourier analysis on the sphere The present chapter is divided in two parts. In the first part we establish criteria under which a function f which is left monogenic in Q..Q, n cJRm+ 1 being an x0 normal open neighbourhood of~ cJRm, admits distributional boundary valuesxl~~+ f(x ± x0 ) in respectively v(l)s(~;A) and S(l)s(n;A) (§26 ). 0
In order to attack these boundary value problems we begin by studying distributional extension properties of harmonic functions (§25). Finally these boundary value results are applied to represent the spaces V{l)(n;A), V( 1 )(tl;A) and Ell)(JRm+ 1 ;A) by means of appropriate spaces of left monogenic functions in Q'n ( 27). In the second part it is shown how any Avalued function f which is either analytic or square integrable on the unit sphere Sm may be expanded into a series of surface spherical monogenics. In particular a decomposition of any surface spherical harmonic of order k into surface spherical monogenics is obtained, leading up to the representation of spaces of analytic functions, analytic functionals and L2functions on Sm by means of spaces of sequences of surface spherical monogenics. Moreover the Bergman and Szegtl kernels, the existence of which has been proved in Section 24, are now calculated explicitly (§§28 and 29). 25. Distributional extension properties of harmonic functions 25.1 Let v cJRm+ 1 be open and U c V be open too. The aim of this section is to establish criteria under which an Avalued function f which is harmonic in U admits an extension in some space of distributions in V. First we consider the case of the distributional extensibility to s( 1 )(1Rm+ 1 ;A). 25.2 Definition Let U cJRm+ 1 be open and let wE V(JRm+ 1 ;JR) be a radial function such that supp wc B(0,1), w ~ 0 and m£ 1 w(x)dx = 1. Furthermore JR suppose that there exists a strictly positive continuous function r in U satisfying
199
r1kT ~ d(x,aU),
(i)
lr
(ii) (iii)
x E U;
is of slow growth in U;
r is of weak slow growth in U.
The the function ¢x:U
~ V(U~).
x E U, given by
¢x(y) = (r(x)r+ 1lJi(r(x)(yx)),y E
u,
is called a harmonic testfield in U. Notice that, given lJi c V(ffim+ 1 ~). any open subset U of mm+ 1 admits a harmonic testfield. Indeed, if U = ffim+ 1 , choose r(x) = 1 in ffim+ 1 and if U~m+ 1 , take r(x) = 1 + d(x,1aU)' x E U. Notice also that if¢ is a harmonic testfield in U, then supp ¢x c B(x,~), x E u fixed, and that for any r\x, f E Harm(r)(U;A), f(x)
=J
U
(y)f(y)dy,
X
X
E U.
* 25.3 Theorem Let U cffi m+1 be open and let f E Harm(r)(U;A) n S(l)(U;A). Then f is of weak slow growth in U. Proof
Let ¢ be a harmonic testfield in U. f(x) =
fu
Then for any x E u
¢x(y)f(y)dy.
As f E s( 1 )(U;A), there exist g, continuous and of slow growth in a E ~m+ 1 such that f = aag in U. Hence we have that f(x) = (1) Ia I J aa¢ (y)g(y)dy Uy
X
so that If (x) Io ;'; Fa ( x )
J 1
B(x,rrxr)
Ig (y) Io dy
where both factors at the right hand side, namely F (x) a
200
=sup yEU
laa'" (y) 1 o Y ·'~'x
u,
and
and
are of weak slow growth in U.
Consequently f is of weak slow growth too.
o
25.4 Coro~~1 Let U cffim+1 be open and convex and let f E Harm(r)(U;A). Then the following are equivalent. (i)
(ii) (iii)
f E s(l)(U;A)
f is of weak slow growth in
u
f admits a distributional extension to
s( 1 )(ffim+ 1 ;A).
Proof ( i ) o=:., ( i i ) . See Theorem 25.3. (ii) ·.(iii). As U is convex Theorem 5.6 yields that f E hence f admits a distributional extension to SCl)(ffim+l;A) ( i i i ) = > ( i ) . Obvious. o
s( 1 )(U;A)
and
..,m+ 1 be an x0 norma 1 open ne1g · hbour hoo d of th e open set 25 . 5 Nex t 1e t Q c~ ;~ cffim and let U = ~~+ = {x E Q:x 0 > 0}. Furthermore let f E CCr)(SI+;A). Then we say that lim f(x + x0 ) exists in v( 1 )s(~;A) if for each compact X >0+ 0 0 interval K c C.', lim f(x + x0 ) exists in v( 1 )s(K;A). X >0 :0 Notice that, as for each compact interval K c ~. oK > 0 may be found for which ]Q,,
Using the 'principe du recollement' we obtain under the foregoing assumptions the existence of a unique distribution T E v( l)(~;A) such that T'G 1K = lim f ,_.x + x0 ) . It is ca 11 ed the die t1•ibu tiona l boundar>y va luc of f X >0+ 0 in v( l)s(0.;A) and we write T = J+f =lim f(x + xo). Now assume that X >0+ 0 0 = ]b,b[x ~,b E]O,+m~, and that f E C?r)(Q+;A). If for any x0 E ]O,b[ fixed, f(x + x0 ) E s( 1 )(r,;A) and lim f(x+x 0 ) X +0+ 0
201
exists in s(l)s(n;A), then this limit is called the distributional boundary Again we write a+f =lim f(x + x0 ). 0 Iff E C( r )(n  ;A), where n = {x E n:x o < O}~O~O+ then it is clear what is meant by the existence of af = lim f(x x ) in respectively v( 1 )s{n;A) xo~O+ o and s(l)s{n;A). Again af is called a dictributional boundary value off in v(l)s(n;A) or s(l)s(n;A) respectively. value off in s( 1)s(n;A).
25.6 The following theorem gives a sufficient condition under which f E c(r){n+;A) admits a distributional extension to 1 ){n;A).
v(
Theorem
Iff E c(r)(n+;A) is such that a+f =xl~+ f(x + x0 ) exists in 0
VCl)s(n;A), then f may be extended to a distribution in v( 1)(n;A). Proof Let K be a compact interval inn and let I = K X ]a,b[ be contained inn with a< 0
~o+
o
o
~
o
m
Theorem 4.10(2), there exist g E C( ){K x [O,b[;A) and a multiindex 1 E ~ o ~ r o such that in K x ]O,b[, f = alg. Now consider J = n n (K x J~,b[) and let h be defined in J by ,x 0 { g{x 0 ~;) ~
h{x)
g(O,x), x0
>
0
:;;
0. ~
Then h is continuous in J and hence T Furthermore for any ~ E V( 1 )(J;A) ~
alh E v( 1 ){J;A) is an extension of f.
Li
a
b
fo {JK ~{yo,y)f(y
+ yo)dy)dyo
rO +
(25.1)
j_oo
which implies that Tis completely determined by fin J. Let K., i = 1 ,2, 1 be compact intervals inn, g.1 E C0( r )(K 1. x ]O,b[ ;A), i1 E ~m be such that in o • Ki x ]O,b[ , f = a1 gi and let hi be defined in Ji = n n (Ki x Joo,b[) by 0
202
h.{x) = { 1
gi(x 0 ,x),x 0
>
o
+ g. ( 0 ,X), X
:~
0.
1
0
+
li Then, putting Ti =a hi' = 1,2, we get by (25.1) that T1 = T2 in J 1 n J 2 • The foregoing construction together with the 'principe du recollement' imply the existence of a global extension off to o( 1 )(~;A). c 25.7 Corollary Let~+ c~ m be open and let~ be an x0 normal open neighbourhood of~. Furthermore let h E Harm(r)(~+;A) be such that a+h = lim h(x+x ) ....0+
X
. * Then h admits an extens1on to V(l)(n;A).
* exists in V(l)s(n;A).
0
0
25.8 The following result indicates that the extensibility of a harmonic function f in ~+ to v(* 1 )(n;A) implies that f satisfies locally some growth conditions. Theorem Iff E Harm(n+;A) admits an extension to v( 1 )(ro;A), then for any compact interval K c ~ there exist ~K > 0 and k E N such that sup O
sup xEK
+ oo.
+
Proof Let K c ~be a compact interval. Choose a compact interval and o > 0 such that K' x [0,6] c n and K c K1 Furthermore, take 0 < ~K < o • If T E v( 1)(n;A) is a distributional extension off, then clearly * * B o T € v(l)(K' X [O,~];A) = s(l)(l\ 1 X ]0,6[ ;A) (see 5.5~. As v{l)(K'x]O,o[ ;A) is dense in S(!)(K' x ]O,f>[ ;A) and TIK' x ]0,6[ = fiK' x ]0,6[, we obtain that f E stn
0
sup sup O
k
+
lx 0 f(x + x0 )
10
<
+ ""·
c
Distributional boundary values and canonical extensions of monogenic functions
26.1 Let Q c:~m be open and let r1 c:~m+ 1 be an x0 normal open neighbourhood of Q. Furthermore let f E M(r)(~~;A). Then in this section conditions are stated under which f possesses distributional boundary values in v( 1 )s(Q;A) 203
*
A.
....
The case V(l)(n;A)
26.2
~~
We only formulate the boundary value problem for
Theorem
Suppose that f E M(r)(<\;A).
~+f =
(i)
lim
0
(~;A) S
*
0
( i i)
•
Then the following are equivalent
f(; + x ) exists in v*(l)
X +0+
+
f admits a distributional extension to V(l)(q;A) .... For every compact i nterva 1 K c r2 there exist ~:K > 0 and k E
(iii)
]II
such that sup
....
xEK
Proof
(i) ==>(ii).
(ii) =,>(iii).
See Theorem 25.5.
As f E M(r)(ll+;A), f is harmonic in II+ and so Theorem 25.8
may be applied. (iii)
:..:=>
(i).
By the 'principe du recollement' it suffices to prove that +
for any compact interval K c 12,
lim X
Choose K and let ~K > 0, k E If(;+ x0 )
1
0
0
]II
+0+
*
+
0
f(x + x 0 ) exists in V(l)s(K,A).
Kx
and C > 0 be such that in
]O,cK[,
~ C(1 + x~k).
Furthermore put
....
f(x + s)ds. Then'\ h 1 0
=
f.
Moreover as f E M(r)(rl+;A), Dh 1
>
=
0
f(t:K + x) E E(r)(K;A). 0
Find g 1 E E(r)(K;A) such that D0 g 1 (x) = f(cK + x) inK (see also Theorem 19.2). Call f 1 = h 1 + g 1 • Then f 1 E M(r)(l( x]O,cK[ ;A). Proceeding in the same way, assume call
204
tha~
h 1 _ 1 , g 1 _ 1 and f 1 _ 1 have already been constructed,
let g1(x) be such that D0 g1 (x) = f 1_1(EK + x) inn and put f 1 = h1 + g1 • Then 1n view of the growth condition satisfied by f, there ought to exist 10 E ~ such that f 1 is continuous in K x [O,EK[. Hence 0
0
1im
fl
x .....a+
(X+
X
0
0
= fl (X
)
°
+ 0)
0 0
exists in v(l)s(K;A).
But, as in K x ]O,EK[,
f
f 1 (x+o>. 0
* belongs to V( 1) (K;A). 0
c
26.3 Remark It should be noticed that the monogenicity is not necessary the weakest condition upon f in order to get some of the implications stated in Theorem 26.2. To this end we refer to the results proved in the previous section. 26.4 In what follows a relationship will be established between the existence of boundary values and the solution of the equation OS = T. Theorem * Assume that f E M( r )(n+ ;A) admits a distributional boundary value ''....::._ a+f in V(l)s(~;A). Then there exists a unique distributional extension f* off in v( 1 )(n;A) such that of*
= a(x 0 )
~
a+f(x). .....
Moreover the support of f* is contained in n+ u n. ~
Let K c ~ be a compact interval. Following the proof of Theorem 26.2 there exists 1 E ~. where 1 may be chosen to be even, say 10 = 2s, and 0 0 0 f 1 E C(r)(K x [O,+oo[) n n;A) n M(r)((K x ]O,+oo[) n n;A) such that 0
0
0
1
a of
x 0 10
= ( !J. ) s m
205
*
0
exists in V(l)(K~A). Next extend f 1
o
n n where
to yK in (K x lR)
0
* * 0 *" Then fK E v(l)(K xlR) n n;A),fki(K Moreover, if 10 K x [o.o] c
n
= 2s,
f;.
= (~)syK
X
. *0 ]O,:cx{) n n= f, Whlle fKi(K ]oo,Q[)nn:o. in (K x JR) n n.
and let ¢ E V(K x ]6,0[ ;lR).
* ~ = ( 1)
1 +1 0
I I
Takeo > 0 such that
Then, using Cauchy's Theorem
( ( D(  t.m ) s
Kx[O, o]
1 +1
(1) 0
D( ( llm)s
Kx[O, o]
I
1 +1
(1) o
do((1\n)s
( Kx[O, o])
n
1
<
=
«tf.
o
L<< ~,
s
+ o,
dx
¢(1 + O)>
=
;tf .¢>.
The same equality holds of course for any for any
E V((K x JR) n n;JR) and hence also
But this implies that
(K
x lR) n n
(26. 1) whence
f~ is a distributional extension of f i(K
x lR)
n n+ to
(K x JR)
n n sat
isfying (26.1) 0
Furthermore, if f 1 and f 2 are distributional extensions of fi(K x JR) n n 0 0 + to (K x lR) n nwhich satisfy DS = o(x 0 ) 0 a+f in (K x JR) n n. then f 1f 2 is 0
left monogenic in (K x lR) n n
206
0
and vanishes in (K x JR) n n + so that f 1 = f 2 •
Consequently, if f* denotes the distributional extension of f to Q such that f * I(K" x R) n Q = fK* for each compact interval K c ;t "• then in Q Of* = o(x 0 ) ® a+f and by the foregoing, f* is unique.
D
26.5 From Theorem 26.4 it follo~s that iff E M(r)(Q\n;A) is such that the distributional boundary values a f lim f(x+x 0 ) and af lim f(xx 0 ) X +0+ X +0+ 0 0 exist in V{!)s(Q;A) then, putting BVf = a+faf, a unique distributional extension f of f to Q may be found satisfying in Q
It is called the canonical extension of f. 26.6 Remark In what precedes we assumed that for f E M(r)(Q\~;A) the boundary values a±f exist separately. Now suppose that for f E M(r)(Q\Q;A), 1 im X
0
+0+
(f(x+x )  f(xx 0 )) 0
exists in v(l)s(~;A). Then we claim that this implies the existence of each of a+f and Indeed, ca 11 Q* = {x E Rm+l : x E Q and x E Q}. Then the function h(x 0 ,x) f(x+x 0 )  f(xx 0 ) is harmonic in Q+* and by assumption lim h(x 0 ,x) exists X +0+ 0 in v(l)s($1;A). Hence, in view of the Corollary 25.7, h(x 0 ,x) admits a distributional extension to V( 1)(Q;A) and so does of course Dh(x 0 ,x). But in
rt+
+ so that, using Theorem 26.2 for every compact interval K c Q, EK k E lN may be found such that sup
O
sup xEK
1
x~ a~f (x+x 0 ) o
10
<
+
>
0 and
oo.
Consequently f also satisfies estimates of the above type in the considered 207
* )(n;A). + domain whence, again by Theorem 26.2, a+f exists in V(l true for af.
The same is
* (+n;A) The case s(l)
B.
+
m
.
+
26.7 Let n c R be open and convex and let n be of the form n = ]b,b[ x n where b € ]O,+oo]. Then it follows from Theorem 25.3 that f € M(r) (n+;A) n s( 1 )(n+;A) if and only iff € M(r)(n+;A) and f is of weak slow growth in n+. Notice that the last condition upon f means that for some k € ~ and
c>6 (26.2)
(i )
if b is finite or that (i i )
if
b
(26.3)
= + oo,
26.8 Theorem Let f
€
M(r)(n+;A) n s( 1 )(n+;A).
Then a+f = xl~+ f(x+x 0 ) 0
Proof We only work out the proof forb<+ oo, the case b = +oo being similar. Call
Then a h1 = f. xo Moreover as f € M(r)(n+;A), Dh 1 = f(% + x) inn+ where, in view of the growth condition (26.2),
Hence there exists~ € s( 1 )(Rm;A) such that ~In= f(% + x). Furthermore, in virtue of Theorem 19.5, there exists
n. Consequently g1 E s(l)(n;A) n E(r)(n;A) may ~e found such that D(hl+g1)=0 inn+. Put f1 = h1+g1; then fl E M(r)(n+;A) n s(l)(n+;A). Now assume that we~have already ;onstructed h1_ 1 and g1 _1 such that fl1 = hl1 + gl1 E M(r)(n+;A) n s(l)(n+;A). Furthermore put ....
h1 (x ,x) = 0
rXo
....
J
f 1 1(x+s)ds b/2 
* (....n;A ) n E(r) (+n;A ) for wh1ch D g (x) = f _ ( b + ....x). and choose g 1 E S(l) 0 1 1 12 Then f 1 = h1 + g1 E M(r)(n+;A) n s{l)(n+;A). But for some 10 E lN, f 1 is 0 continuous in n+ u nand for some C* > 0 in n+ 0
....
Hence 1 im f 1 (x+x ) X
0
....0+
0
O
lim f(x+x) exists in s*(l) (n;A).
X ....0+ 0
0
S
26.9 Just as in the case of
o
v(1 )(n;A)
(see Theorem 26.4) we may prove
Theorem Let f E M(r)(n+;A) n s(1 )(n+;A). Then there exists a unique distributional extension f* off to srl){n;A) such that
Moreover the support of f* is contained in n+ 26.10 Now suppose that f 26.8 the boundary values
E
M(r)(n\n;A)
n
u
n.
s( 1)(n\n;A).
Then by Theorem
a±f = lim f(x+x ) X 0
....0+
0
exist in s(l)s(n;A) and, using Theorem 26.9, f admits a unique extension f* to s( 1 )(n;A), called the canonica~ extension off, for which Of* = o{x0 ) ® BVf where again BVf = a+f a f. 26.11
In Theorem 25.3 we have characterized the space M(r)(n;A)
n
* S(l)(n;A) 209
in terms of growth conditions. be proved.
If~= ~m+l the following finer result may
Theorem 8 * (~m+l ;A) we have (see Theorem 25.3) that Proof As f e: M(r) (~m+l ;A) n S(l) there exist C > 0 and k e: N such that jf(x)
10
s C(l + jxj)k.
But this implies that the Taylor series of f at the origin has to break off and so f e: P(r)" 0 27.
Representation of distributions by monogenic functions
v(
In this section we show how distributions in 1 ) (~;A), s( 1 ~ U1;A) and E( 1 )(~m;A) may be represented by monogenic functions in ~\~and R +l~m respectively, where ~and ~ satisfy the conditions of the previous section. 27.1
A. 27.2
The cases v( 1 )(~;A) and s( 1 )(~;A) Let us recall that if
fe:
M(,r )(~\~;A) has the boundary value
BVf = lim f(x+x )  lim f(xx ), xo~+ o xo~+ o then the canonical extension f* of f satisfies the equation
A converse of this result will now be formulated. 27.3 Theorem ;et T(x) e: v( 1 )(~;A) (resp. s( 1 )(~;A)) be given and let F be a solution in V(l)(~;A) (resp. s( 1 )(~;A)) of the equation
Then F is the canonicai extension of f = FI~\~"* V(l)s(~;A) (rcsp. S(l)s(~;A)).
210
e:
"* M(r) ( ~\~;A) and BVf = T in
Proof We only work out the proof for v(1 )s(~;A), the other case being treated analogously. First observe that as DF = 0 in ~\Q,f = FI~\Q indeed belongs to M(r)(~\Q;A) (see 17.4.2.). Moreover, as f has a distributional extension F to~. by means of Theorem 26.2 the boundary values lim f(x±x ) ...0+
X +
0
0
>
= af exist in+v(1 )s(S'l;A). Let f* be the canonical extension off; then Ff* = 0 in ~\~ and so supp(Ff*) c n. Now let K c ~ be a compact interval and put ~K = (KY~) n ~. Then in ~K'Ff* may be written as 0
n
Ff* =
Y.
o(h)(x 0 ) e eh(x)
k=O
*
0
where eh c V(l)(K;A). Consequently, as Of* = o(x 0 ) e BVf in
~.
we have that in
~K
o(x0 ) e (TBVf) = D(Ff*)
L
(o(h+l)(x0 ) e eh(x)+o(h)(x 0 ) e D0 eh(x))
O~h~n 0
from which it follows that en (x) Hence F = f* in ~K and as moreover o(x0 ) e (TBVf)
o(x 0 ) e
0 in K.
o0 e0 (x)
0 0
we find that T = BVf in K. Using the 'principe du recollement', we get the desired result.
Q
* + * + 27.4 Theorem (Representation Theorem) LetT E v(l)(~;A) (resp. s(l)(~;A)). Then there exists f c M(r)(Q\n;A) such that BVf = T. Furthermore if g ~ M(r)(~\~;A) is such that BVg = T, then fg is extendible to a function h
c M(r)(~;A).
~Put f = Flm~ where F is a solution in v(l)(~;A) (resp. s( 1 )(~;A)) of the equation OS= o(x) e T(x). Then Theorem 27.3 implies that BVf = T. 0 .... As to the second part, let g c M(r)(~\~;A) be such that BVg = T. Then, denoting by g* c V(l)(~;A) (resp. S( 1 )(~;A)) the canonical extension of g we get that D(Fg*) = 0 in ~ so that h=Fg*extends monogenically fg to ~. 0
211
B.
The case E(l) (lRm;A)
27.5 The representation of E{l)(lRm;A) may be given in a much more direct way than in the cases V( 1l(n;A) and S(l)(0,;A) and this by using the socalled Cauchy tmnsform of T E E( 1) (lRm;A). A
27.6 Definition given by
LetT E E{ 1 )(lRm;A).
Then the Cauchy transform T of T is
+
T(x)
~+1
I
U+X ) ' X E JRm+1~m. U+X 1m+1
+
Notice that f is well defined since for
u E lRm
and x E JRm+ 1 ~m.
u+x U+X 1 m+1 m m+1 m E(l)(lR ;A). f4oreover one easily verifies that T E f·1(r)(lR ~;A). But there is more, namely I
+ 
E
A
27.7 Theorem LetT E E{ 1 )(lRm;A): Then Tmay be extended to a left monogenic function, still denoted by T, in lRm+ 1,supp T. Furthermore lim T(x)=O X>=
and for any~ E V(l)(lRm;A)
x +0+ 0
J lR
m
~(x)(T(x+x )  T(xx 0 ))dx. 0
Finally there exists kEN such that for all E sup
1
x~supp
k " (d(x,supp T+Bm(d)) T(x)
T+Bm( d d(x,suppT + Bm(d)
<
>
10
0 <
+
oo.
1
Proof We first prove that T may be extended to a left monogenic function in Rm+ 1,supp T. By Painlev~'s Theorem (see Theorem 10.6) it is sufficient to show that T admits a continuous extension to Rm+ 1,supp T. Let y E JRm,supp T and choose E > 0 such that B(y,E) clRm+ 1,supp T. Furthermore let a E V(lRm;lR) be such that a = 1 in supp T and a = 0 in B(y,E) n lRm. Then in B(y,E) the function f given by 1
+
U+X
f(x) =  
Moreover f coincides
" o m " with T in B(y,£)~ and hence T indeed admits a continuous extension to Rm+l,supp T.
Clearly lim T(x)
0 and we claim that in
X+oo
svf = T = lim X
0
+0+
v( 1 )(Rm;A)
(T(x+x )  T(xx )). 0
To this end observe that V(l)(Rm+l;A) given by

x
lxlm+l
0
T has the distributional extension Text in
*(o(x 0 )
@
T(x)).
Furthermore, as
using Theorem 27.4 we get that
1im
=
X +0+ 0
whence for any
4> E
V( 1) (lRm ;A)
= lim X +0+ 0
T
Finally, as Tadmits the distributional extension t to v*( )(Rm+l ;A). it " ex 1 follows from Theorem 26.2 that T satisfies the stated growth conditions. c 27.8 Now we formulate a characterization for the Cauchy transform of distributions in E(1 )(Rm;A). Theorem Let K cRm be compact and let f E M(r) (Rm+1 'K;A) be such that lim f(x) = 0 and that for some k E ~ X+oo
sup l(d(x,K))k f(x)i 0 < + oo. d(x,K)<1 xtK Then f admits a boundary value BVf in v( 1)s(1Rm;A). supp BVf c K and f = BVf.
Moreover BVf E E{1 )(lRm;~. 213
Proof In view of Theorem 26.2 BVf exists in v{l)s(Rm;A) and as f E M(r)(Rm+ 1, K;A), we have that for any~ E V(l )(Rm,K;A) lim xo....O+
J Rm,K
~(x)(f(x+x 0 )

f(xx 0 ))dx
=o
whence BVfiRm,K = 0. Hence BVf E E{ 1 )(Rm;A) with supp BVf c K. By virtue of Theorem 27.7 BVf may be defined and, using Theorem 27.4, f BVf admits a left monogenic extension to Rm+ 1• But, as lim (f(x)  BVf(x)) = 0, Liouville's Theorem yields X+oo that f = BVf. c 27.9 Putting together the results of the conclude as follows: Let +Q cRm be open, neighbourhood of~ and call M(r)V*(~~;A) f E M(r)(~~;A) such that for any compact and k E N for which
Theorems 27.4, 27.7 and 27.8 we may let ::2 cRm+1 be an x0 normal open the space of all functions interval K c ~there exist cK > 0
sup 1x 0k f(x 0 ,x+) 10 < + oo, O
k
If ( x) Io ~ c( 1+ Ix I ) ( 1 +
a(x,1au) )k
m+1 Finally let M(r)E* denote the space of functions f E M(r) (R 'K;A) for some K cRm compact such that (i)
there exists k E ~for which sup l(d(x,K))kf(x) d(x,K)<1 x~K
and (ii)
lim f(x) X>oo
214
0.
10
<+co
Next consider the boundary value mappings, denoted by BV, which map the spaces M(r)V*(:z,0;A), M(r)S*(:.~;A) and M(r)E* respectively upon V( 1) (0.;A) ,S( l) (n;A) and E(ll(Rm;A). Then these mappings are left Alinear and surjective. Moreover the following algebraic isomorphisms hold. Theorem ( i)
(ii) (iii) 28.
v( l) (~;A)
~ M(r)V*(s1'n;A)/M(r) (12;A).
S(l)(n;A) ~ M(r)S*(7'~;A)/M(r)S*(~). E{l)(Rm;A) : M(r)E*.
Analytic functions and functionals on the unit sphere
28.1 In this section we treat of the following problem. Given an Amodule E of functions or functionals on the unit sphere, determine an Amodule ME consisting of monogenic functions which represents E. This leads up to the decomposition of the elements in E into a series of surface spherical monogenies satisfying certain estimates. 28.2 Lemma Let f be an Avalued harmonic function in B(0,1). f = f 1 + f 2 where Df 1 = 0 and Df 2 0 in B(0,1).
Then
Proof Let f E Harm(r)(B(0,1);A); then Df = g is le~t monogenic in B(0,1) so that, in view of Leffima 20.3, there exists hE M(r)(B(0,1);A) for which a h=g h h h xo and so 02 = g = Of. Take f 1 = 'Z and f 2 = fz. c 28.3 The proof of the following Painlevetype theorem will be omitted since it may be carried out as in 10.6. Let us just point out that when speaking of an mdimeno·ional. injec:tive C surfac:e L: in Rm+ 1 the range is meant of a C injection from an open subset U of Rm into Rm+l 00
00
Theorem (Painleve). Let n cRm+ 1 be open and let L: be an mdimensional injective C surface in n such that D'L is open. Furthermore let f E M(r)(D'L:;A) n c(r)(n;~). Then f E M(r)(n;A). 28.4 We now state a CauchyKowalewski type theorem for analytic functions on the unit sphere. Let us recall that an Avalued function is said to be analytic: on the unit sphere Sm if it is locally analytic on Sm. The set of m such functions clearly constitutes a biAmodule which is denoted by a(S ;A). 215
Theorem (CauchyKowalewski) Let f E a(r)(Sm;A). Then there exist 0 < R2 < 1 < R1 and a unique function f* which is left monogenic in B(O,R 1 ),B(O,R 2 ) such that f*ISm = f. Proof Take P E Sm and define a system of spherical coordinates in a neighbourhood n of P in Rm+ 1• Then in n, D may be written as D = war + 1 a (see r w also 8.9). Obviously g(w) = ~3 f(w) is analytic in Sm n n and as g does not depend w m m on the coordinate system chosen on S , g is well defined on the whole of S • By the CauchyKowalewski theorem for the Laplacian (see e.g. [21]} there exist 0 < R2 < 1 < R1 and a unique Avalued harmonic function h in B(O,R 1 ) ' B(O,R 2 ) such that h!Sm = f and arhiSm(w) = ~awf(w) = g(w) = ~ ~awhiSm(w). We claim that h is the desired left monogenic extension f* of f. Indeed, let P and n be as before and chooser> 0 such that B(P,r) c B(O,R 1 ),B(O,R 2 ). Then, using Lemma 28.2, h = f 1 + f 2 in B(P,r) where Df 1 = 0 and Df 2 = 0. Put 0 y = Dh in B(P,r). Then o
1 
m
o
m
y!B(P,r) n S = w(a r +wa w)h!B(P,r) n S = 0. r o
o
m
Consequently, as Dy = 0 in B(P,r) and as moreover y!B(P,r) n S = 0, we get that y = 0 in B(P,r). Since P E Sm has been taken arbitrarily we finally obtain that Dh = y = 0 in B(O,Rl),B(O,R2 ). c 28.5 By means of the foregoing theorem each f E a(r)(Sm;A) admits a unique (left) monogenic extension f* to a suitable open neighbourhood of Sm in Rm+ 1; it is called the CauchyKo~aZe~ski extension of f. Of course; if g E M(r)(B(0,1+E)'B(1E);A), 0 < E < 1, then g!Sm E a(r)(Sm;A) and the CauchyKowalewski extension g* of g!Sm coincides with g. Now call
m o _ M(r)(S ;A)= ind lim M(r)(B(0,1+E),B(0,1E);A); 1>£>0 then the previous considerations yield that a(r)(Sm;A) and M(r)(Sm;A) are isomorphic right Amodules. But, as each of the spaces M(r)(B(0,1+E)'B(0,1E);A) may be provided with its natural topology (see 16.1), the inductive limit topology can be defined on M(r)(Sm;A) (see 2.15) and hence be carried over to ~(r)(Sm;A). Analogously a(l)(sm;A) is isomorphic to M(l)(Sm;A) =lim ind M(l)(B(0,1+E)'B(0,1E);A) 1>£>0 216
from which it inherits a locally convex topology. *(m *(m . 28.6 The duals a(r) S ;A) and a(l) S ;A) are respect1vely called the spaces of right and left analytic functionals in Sm. If r\r)(lRm+ 1.....sm;A) +O denotes the right Amodule of all left monogenic functions in lRm+ 1.....sm which vanish at infinity, then in what follows it will be sketched how this space represents a( 1)(Sm;A). The details of the proof are omitted since they may be reconstructed by using arguments similar to those developed in Section 22. LetT E a( 1 )(Sm;A) and consider its Cauchy transform
1
A
T(x) =  
xw m+1>'
( 28. 1)
IXwl
then f is left monogenic in lRm+ 1.....sm and lim T(x)
0.
Furthermore
X>oo A
T(x)
Xo (PkT)(x) B( 0,1)
where for H clRm+ 1 , XH stands for the characteristic function of H. PkT and QkT are given by PkT(x) = _1_ wm+ 1
Hereby
(k~m11)
and
(28.2)
Now Tis represented by Tin the following way. Let¢ E a(l)(Sm;A); then for some 0 < s < 1,¢ admits a CauchyKowalewski extension to M(l)(i3(0,1+s)..... B(0,1s);A), still denoted by¢, such that for any 0 < n < s r
A
¢(x)doxT(x) 
J
3B(0,1+n) Conversely, if f
E.
I <
n
<
:JB(0,1n)
f1(l )(lRm+ 1.....sm;A) ~~o then Tf defined on a(l )(Sm;A) by
ZJB(0,1+n) 0
I
A
¢(x)dox T(x).
¢(x)doxf(x) 
I
¢(x)doxf(x),
()8(0,1s)
1 being suitably chosen and depending upon ¢, is a left analytic 217
· Sm an d T"' f = f • consequently M(l) * ( Sm;A ) ·= a(l) * ( Sm;A ) is iso1n morphic to M(r)(lRm+ 1.....sm;A) .... o· f·1oreover this isomorphism is continuous when 1 )(Sm;A) carries the strong m+1 m topology and ~1(r) (lR ".S ;A) .... 0 is provided with the system of seminorms in. her1ted from ~4(r) (lR m+1 .....s m;A). · 1 f unct1ona
M(
28.7 We now show how analytic functions on the unit sphere may be expanded into a series of surface spherical monogenics. Theorem Let f E a(r)(Sm;A). Then there exists a sequence (Pkf, Qkf)keN of surface spherical monogenics such that on Sm
k=O
k=O
Moreover the series
L
00
Pkf(x) +
l
Qkf(x) converges in an open neighbour
k=O k=O hood of sm to the CauchyKowalewski extension f* of f and there exist 1 :· and C ~ 0 such that for all k E ~
r :.
0
and
Conversely, if a sequence (Pk,Qk)k8N of surface spherical monogenics satisfies estimates of the above type, then 00
I Pk(v) + I Qk(r.,l! k=O k=O is analytic in sm. f(w) =
m
Proof Let f E a( )(S ;A). Then by Theorem 28.4 there exist L " r 1 m f* E M(r)(B(0,(12,:) ).....B(0,12c);A) such that f*!S =f. If 00
f*(x) =
I
k=O
>
0 and
00
Pkf*(x) +
I
Qkf*{x)
k=O
is the Laurent expansion off*, then by the Cauchy estimates (see (11.17) and 218
(12.6)) there ought to exist C , 0 such that for all k
E .N
Hence the sequence (Pkf,Qkf)k8N with Pkf = Pkf*ISm and Qkf = Qkf*ISm satisfies the stated estimates. Conversely, let (Pk,Qk)k8N be a sequence of surface spherical monogenics fulfilling estimates of the prescribed type. Put for each k EN, Pk(x) X ) X • lXI k Pk (TXT and Qk ( x ) = 1x1  ( k+m) Qk(TXf). Then 1t may be shown that 00
f*(x) =
00
I
pk(x) +
k=O
I
Qk(x)
k=O
is left monogenic in an annular neighbourhood of Sm.
I k=O iS analytiC ir. Sm.
"'
I
Pk(w) +
Hence
Qk(w)
k=O D
28.8 Now let Sk be an Avalued surface spherical harmonic of order k. clearly Sk E a(r)(Sm;A) and so, by Theorem 28.7, \(w) =
I
pl \(w) +
1=0
I
QlSk((ll) •
Then
(28.3)
1 =0
But from the sections 11.2.1 and 12.1.9 it follows that
and
Consequently, as ~+ 1 sk in (28.3) and for which Theorem
k(k+m1)Sk, all terms P1 Sk and Q1_1sk appearing f k vanish. Hence we obtain
Let Sk be an Avalued surface spherical harmonic of order k.
Then
i.e. each surface spherical harmonic of order k may be decomposed in a unique 219
way into the sum of a surface inner spherical monogenic of order k and a surface outer spherical monogenic of order (k1). Notice too that we have (see also (11.14) and (12.5)): Pk\ () w  1 (k+m1)J k+1 m Kk m+1 Wm+1 S '
I
o(ll
c,))do
I
W
Sk(w 1 )
and
these expressions give rise to the Cauchy estimates (see also (11.17) and (12.6)):
~~~m
IPk\(w) lo;;:
c(k~~1 1 )( 1 +k 2 ) ~~m
15kL,))
lo
and
(28.4)
Observe furthermore that the decomposition Sk = PkSk + Qk_ 1sk is consistent with the dimension 2nN(m+1,k) of the space of Avalued surface spherical harmonics of order k. In fact, by the CauchyKowalewski theorem for analytic functions in Rm we obtain that the dimension of the Amodule of (left) inner surface spherical monogenics of order k equals 2nM(m;k), whereas the dimension of the Amodule of (left) outer surface spherical monogenics of order (k1) is equal to 2nM(m;k1). But, as is well known (see also 11.1) N(m+1,k) = M(m,k) + M(m,k1). 28.9 Now let us recall some fundamental results, which have been obtained by several authors, concerning the decomposition of analytic functions and functionals on Sm by means of surface spherical harmonics, the functions or functionals under consideration being [valued. (i)
f E a(Sm) if and only if f(w) =
L
Sk(c,J), (Sk)kElN
k=O being a sequence of surface spherical harmonics for which C may be found such that for all k E fi
220
~
0 and 0
< r. <
(see [18]). 00
(ii)
I
f E E(Sm) if and only if f(w)
skc,J. (sk)kEJJ
k=O being a sequence of surface spherical harmonics such that for all s e constant Cs > 0 may be found with
~
a
and this for all k E ~ (see [17]). 00
(iii)
T E V'(Sm) if and only if T
k=O spherical harmonics for which C > 0 and S > 0 may be found such that for all k E
~
sup 1\(w)l ;:: C(1+k)s. wesm (see [14]). 00
(iv)
T
t
a'(Sm) if and only if T =
I
Sk' (Sk)kEl'J being a sequence of
k=O surface spherical harmonics such that for all E > 0, there exists CE > 0 with
and this for all k E ~. (see [7]). Using Theorem 28.8 and the estimates (28.4) we get: (ii ') f E E(r)(Sm;A) if and only if there exists a sequence (Pk,Qk)keN of surface spherical monogenics such that 00
I (Pk(w) + Qk(w)) k=O and satisfying the condition: for all s E ~. Cs f(w) =
ks sup ( IPk(w) 10 , IOk(w) 10 ) wesm and this for all k E ~.
;::
>
0 may be found with
Cs
221
(iii') T E v(r)(Sm;A) if and only if a sequence,.(Pk,Qk)kEI'l of surface spherical monogenics may be found such that T = L (Pk + Qk) and satisfying k=O the condition: there exist C ~ 0 and s ~ 0 such that for all kEN sup (I Pk(t>l) wESm
J
0,
IQk(r,))
J
0)
:
C( 1+k)s.
(iv') T E a(r)(Sm;A) if and only if a sequence)Pk,Qk)kEN of surface spherical monogenics may be found such that T L (Pk + Qk) and satisfying k=O the conditon: for all s > 0 there exists C 0 such that for all k E ~ f.
0>
It should be emphasized that if T E a( 1)(Sm;A) then its expansion T=
L (Pk+Qk)
k=O is in fact a direct consequence of the isomorphism between a(lj(Sm;A) and M(r) (1Rm+ 1,sm;A) _,. 0 which associates to T its Cauchy transform T given by (28.1). Hereby notice that the sequence (Pk,Qk)keN of surface spherical monogenics is determined by the expressions (28.2). Observing that Theorem 28.7 already provided a monogenic version of statement (i) we have thus obtained ~presentations of the spaces a(r)(Sm;A), E(r)(Sm;A), v(r)(Sm;A) and a{r)(Sm;A) by means of spaces of sequences (Pk,Ok\EJ.J of surface spherical monogenics satisfying appropriate estimates. Of course these representations are topological isomorphisms when the spaces under consideration are endowed with their natural locally convex topology. 28.10 To close this section we define an inner product on Sm between analytic functions. Definition
Let f,g E a(r)(Sm;A).
(f,g) m =  1S wm+1 the
222
inne~ p~oduct
Jr
Then we call
mf';w)do g(w) S ul
between f and g.
28.11
Remarks
28.11.1 Let f, g E a(r)(Sm;A) and l~t for some R1 > 1 f* and g* be the Cauch Kowalewski extensions off and g to B(O,R 1 ),B(O,R~ 1 ). Then for any open neig bourhood S of the origin having C1 boundary as c B(O,R 1 ),B(O,R 1 1 ), 0
(f,g) m =  11J)m+1 s
J
as

(28.5)
f*(u)do g*(u). u
r1oreover, if 00
f(ul) =
l:
pk f(l.&.l) +
k=O
l:
Qkf(w)
k=O
then 00
'\,
f(u.l)
l:
'\,
00
Pk f(w) +
I
k=O I
Qk f{l.&.i)
k=O 00
00
l k=Ol:
'\,
l:
Pkf(w) +
lJkT(w)
k=O
r\
f(w)~.
Consequently, as on Sm, do = wdS , dS w
(f,g) m =  1wm+1 S
w
being the elementary surface element,
J m f(w)g(w)dS • S
w
It thus follows that
,(f,f)> ~ 0 while
If f,g E a(r)(Sm;A) admit the expansions 00
f(w) =
l:
00
Pk f(w) +
k=O.
l:
Qkf(w)
k=O
and 00
g(w)
l: k=O
00
Pkg(ul) +
l:
Qkg(w),
k=O 223
then the orthogonality relations between spherical monogenics (see 11.2.4 and 11.4.5.) imply that for any k 1 1,
Moreover, as for any pair Pk and Q1 of inner and outer spherical monogenics, k, lE.tl,
lim
0
R+oo
and
we obtain, using (28.5), that for all k, 1 E .tJ
Hence (f,g) m
s
29.
L2Functions on the unit sphere the Szeg6 and Bergmann kernels
29.1 In this section it is proved that each function f E L2(r)(Sm;A) may be expanded into a series of surface spherical monogenics. Moreover, using the Cauchy transform in L2 (r)(Sm;A), an orthogonal decomposition of this space is obtained. Finally the Szeg6 and Bergman kernels corresponding to the unit ball in Rm+ 1 are calculated explicitly. 29.2 Let f,g E L2 (r)(Sm;A) {see also 7.14) and define the inne~ p~oduct between them by 1
r

(f,g) m =   Jsm ~(w)g{w)dSw' S wm+1 dS again being the elementary surface element. w
224
Furthermore put
Jsm <
J m lf{w) lo2 dS w •
s
29.3 Theorem Let f E L2(r)(S m;A). surface spherical monogenics:
Then f may be expanded into a series of
00
I
f(w)
(Pkf(w) + Qkf(w)).
k=O r~oreover
Finally, if g E L2 (r)(Sm;A) admits the expansion 00
I
g{w) =
(Pkg{w) + Qkg{w)),
k=O then (f,g) m
s
Proof
Iff E L2 (r)(Sm;A) call for each k E fi
I
1 (k+m 1\ Pkf(x)  w1 k+1 J m Kk m+ 1 w(x)do wf(w) ' m+ 1 ' S ' '
and
I
x E 1Rm+ 1 ,
Q f(x) = _1_ (k+m1\ Kk,m+1 ,x(w) do f(w) x k wm+ 1 k+1 } 5m lxlm+k w ' Furthermore define rrk, rr(k+ 1):L 2 (r)(Sm;A)
7
€
1Rm+1. o
L2(r)(Sm;A), k ~ 0, by
rrkf(w) = Pkf(w) and rr(k+ 1 )f(w) = Qkf(w),
wE sm.
225
m
Then for each s E l, ns is a bounded right Alinear operator on L2 (r)(S ;A) satisfying n~ = ns· Moreover for all f,g E L 2 (r)(Sm;A) and k ~ 0
Jsm "Pkf(w)g(,d)dS.)
(J
1 (k+m1\\ k 1 ;f(w' )do ,Kk , (<.J)}g(r,J)dS JSm Sm ''lm+ 1 + ''I ,m+ 1 •''I .,,
Jsm f(w')Pkg(w')dS w, (f,nkg) m·
s
Analogously for all f,g E L2 (r)(Sm;A) and k ~ 0 (n_(k+1 /,g) m = (f ,n_(k+1 )g) m·
s
s
Finally, whenever s # s', s,s' E Z, we have that nsns' = 0. Hence (ns:s E l) is a family of mutually orthogonal projection operators in L2 (r)(Sm;A) and so m for any f E L2 (r)(S ;A)
Consequently
But, iff is considered k=O
as an element of
a( 1)(Sm;A), co
(see also (28.2)) so that
~
i<=O
implies that
226
w
then
I
(Pkf+Qkf) converges to f in
a( 1 ) (Sm;A)
k=O
(Pkf(w)+Qkf(w))
=
f(,,i) in L2(r)(Sm;A).
This
and that, using the orthogonality relations between spherical monogenics, (f,g) m
0
s
29.4
'"\,
Remark ·   As (f,g)
fC,,),~,,
f(,,,)
r
sm
J sm
'''m+1
then for all f,g E L2 (r)(Sm;A)
'f' (,JJ) do
g (1,1) 1.1]
so that the inner product in L2 (r)(Sm;A) generalizes the one defined in a(r)(Sm;A). 29.5
Now let f E L2 (r)(Sm;A) have the expansion f(,>J)
( Pk f(uJ) + Qk f(,u))
2:
k=O and define its Cauchy tr•ancfor>m by f(x) = 1tum+1
fsm
xw do f(,.D), m+1 w IXr.DI
(see also (28.1)). ~ m+1 m ThenfEM(r)(lR . . . s;A),lim
~
f(x)
x E lRm+ 1,sm.
Oand
X>=
f(x)
I
(x m+ 1 _ (Pkf)(x)). Qkf(x)xo lR 'B(0,1) B(0,1)
k=O
Furthermore f(lxlul) E L2 (r)(Sm;A) if 1x1 I 1 and a straightforward calculation shows that lim f(w(1±c)) exists in L2 (r)(Sm;A). Consequently the operators E~O+ 'f+ and rr_, glVen by ~
n+ f(~) = lim f(w(1+r_:)) r_:+0+ 00
L
Qk f(uJ)
k=O 1
I f\ )f(w)
k=oo and 227
lim f(w(1d) £+0+
11_f(w)
co Pkf(w)
I k=O co <
I
11s)f(w)
k=O are well defined on L2(r)(Sm;A). Moreover we have that 11+ and 11_ are projection operators in L2(r)(Sm;A) satisfying the relations
and 11+ + 11_ = 1 . Hence, the following orthogonal decomposition of L2(r)(Sm;A) is obtained: Theorem L2(r)(Sm;A) = 11+L 2(r)(Sm;A) EDl. 11_L 2(r)(Sm;A). 29.6 Now we arrive at the explicit calculation of the Szegtl and Bergman kernels, the existence of which has been proved in Section 24. We first deal with the Szegtl kernel. 29.6.1 Let us recall that ML 2(r)(Sm;A) stands for the space of functions f E M(r)(~(0,1);A) such that lim r+1
Ism
Jf(rw)
1~
dSw
<
+co.
<
For convenience we modify slightly the expressions of the inner product and norm on ML 2(r)(Sm;A) in t~e following way (see also Section 24). For all f,g E ML 2(r)(S ;A) we put (f,g)
m
1,S
wm+1
1 im
r+1
<
228
I
m f(rw)g(rw)dS
S'
w
and wm+1
lim J if(rw)i 2 dS. 0 sm w
r~1
< Calling for each f E ML 2 (Sm;A) and 0
<
r
1, fr(w) = f(rw), wE Sm, we have
<
Theorem Let f E ML 2 (r)(Sm;A). Then the generalized sequence (fr)rE]0, 1[ converges in L2 (r)(Sm;A) when r ~ 1. If g(w) =lim fr(w), then
29.6.2
<
in L2 (r)(Sm;A), g(w) =
~
r~1
Pkf(w) and f =gin B(0,1).
k=O g E L2 (r)(S ;A) has the expansion g(w)
<
Conversely, if 00
00
m
L
Pkg(w)' then f(x)
k=O m 8(0,1). belongs to ML 2 (r)(S ;A) and f =  g in A
0
Proof We first prove the converse statement. 0 immediately get that for each x E 8(0,1)
Taking account of 29.5 we

00
L Pkg(x) k=O is well defined. mMoreover f ~ M(r)(B(0,1);A) and ~1~ fr E L2(r)(Sm;A). Hence f E ML 2 (r) (S ;A) and in 8(0,1), f =g. < Now let f E ML 2 (r)(Sm;A) admit in B(0,1) the Taylor series expansion about the origin: f(x) =
00
L Pkf(x).
f(x) =
k=O Then clearly fr E L2 (r)(Sm;A) may be expanded into surface spherical monogenies by (see Theorem 29.3) 00
fr(w) =
L
Pkfr(w)
with
Pkfr(w)
k=O Consequently, using Theorem 29.3 and the fact that f E ML 2 (r)(Sm;A), we get that 1 im
< +
00
r~1
<
229
or 00
L
k=O Hence, again by Theorem 29.3 the function g given by 00
L
g(w) =
Pkf(w)
k=O limfr(w) r+1
<
~elongs to L2 (r)(Sm;A) and in view of 29.5 we have that in B(0,1), f(x)
g(x).
o
29.6.3 Remarks (1) By means of the foregoing theorem we know that for each f E ML 2 (r)(Sm;A), lim1 fr(t;l) exists in L2 (r)(Sm;A). In the sequel we still r+ < m write f(w) = lim fr(w). If f,g E ML 2 (r)(S ;A) we thus have: r+1
<
(f,g)
1 ,S
m=
I < ,I 
1r.um+1

lim m f(rw)g(rw)dS r+1 Sw
m f(w)g(w)dS S w
= 
m+1
(;1
(f,g)
(29.6)
m
s
and llf!l 2 m = llfll 52m• 1 ,s
(2)
Iff E ML 2 (r)(Sm;A) and t E B(0,1) then . f(t) = 1 1 1m wm+1 r+1
< 
, I
 wm+l
I
sm
;;;i
s~ lwtlm+1
'V
(K(.,t),f) m
s
230
r~t lrwtlm+1
d a f( rw )
da f((li)
w
w
( 29. 7)
where
'\,
.
29.6.4 Theorem The Szeg6 kernel S(u,t) is given by S(u,t) = K(x,t)(u) where ~(x,t) = xt stands for the right monogenic function in x, depending on •. m+1 I
x ~ 1
the parameter t E ~(0,1). 0
.0
1
Proof FortE B(0,1) fixed, S(u,t) is left monogenic in B(O,TfT). Hence, as a function of u, S(u,t) E ML 2 (r)(Sm;A). In view of the uniqueness of thereproducing kernel it suffices to prove that for each f E ML 2(r)(Sm;A) f(t) = (S(.,t),f)
1 ,s
m·
But in view of (29.6) and (29.7) f(t)
'\,
= K(.,t),f) m
s
(S(.,t),f) m
s
= (S(.,t),f)
1 ,s
m·
c
29.7 We now end up this section with the determination of the Bergmann kernel in the case of the unit ball. 29.7.1 Let us recall that ML 2(r)(B(0,1);A) 0 iS the right Hilbert Amodule consisting of the left monogenic functions in B(0,1) which at the same time belong to L2(r)(B(0,1);A). If(,) denotes the inner product in L2(r)(B(0,1);A), then for convenience we put for all f,g E ML 2 (r)(B(0,1);A) 1 (f,g)B =(f,g). wm+1
We have
J
f(x)g(x)dx
s
J~
(J 5m fr(w)gr(w)dsw)rmdr 231
0
0
29,7.2 Now let f E ML 2 (r)(B(0,1);A). Then f E M(r)(B(0,1);A) and so it admits a Taylor series expansion about tne origin: 00
f(x)
~
I
Pkf(x).
k=O 0
We now claim that this series still converges i~ L2 (r) (B(O, 1) ;A) t~ f. Indeed, define for each k E :N the operator nk t4L 2 (r)(B(0,1);A) .... ML 2 (r)(B(0,1);A) by nkf(x)
= Pkf(x) 1 ( k+m1 \ _ wm+ 1 k+ 1 )
J
() () o ) 1 Kk,m+ 1 ,y x dayf y , x E 8(0,1 • aB(O,z)
Then clearly rrk is a right Alinear operator. Moreover it is bounded. Indeed, a straightforward calculation yields the existence of Ck > 0 such that for all X E B(0,1)
so that, taking account of Theorem 24.6,
and hence
By the uniqueness of the Taylor series we also have that rrk = rrk and, as for k # 1, (rrkf,rr 1g)Sm = 0, for all f,g E ML 2(r)(B(0,1);A), we get that (rrkf,rr1 g) 8 =
1 k+1+m( ) r rrkf,n 1g smdr
Jor
= o. 232
Finally for all f,g E ML 2 (r)(B(0,1);A) k E ~.
(nkf,g)B = (f,nkg) 8 ,
Consequent!y (rrk)keN is a se~uence of mutually orthogonal projection operators in ML 2(r)(B(0,1);A) whence k~O Pkf converges in the L2sense to f. It thus follows that 00
(f,g}B =
L
(Pkf'Pkg)B
k=O
Y J~ k=O
(29.8)
r2k+m(Pkf, Pkg) mdr S
0
0
and hence that f E M(r)(B(0,1);A) belongs to L2(r)(B(0,1);A) if and only if (29.9) 00
As we have already seen, f
E
ML 2(r)(Sm;A) if and only if
Consequently, for such f 00
L (2k+m+1)
1
k=O
I
k=O
11Pkf11 2 m S
< + oo.
11/Zk+m+l Pk fW m < + oo S
and so, taking account of (29.9),
y /2k+m+1
Pkf E ML 2(r)(B(0,1);A).
k=O H~nce it is meaningful to define the operator /m+12I: ML 2(r}(Sm;A)+ML 2(r) (B(0,1);A) by 00
~r
f =
I
12k+m+1 Pkf.
k=O Clearly this operator is injective; but it is easily seen that it is also onto. Obviously the operator ~r conserves inner products. Finally, if 233
g E M(r)(~(O,R);A) where R > 1, then its Taylor series about the origin 00
g(x) =
I
Pkg(x)
k=O converges uniformly on the compact subsets of B(O,R). Moreover, taking account Qf the Cauchy estimates satisfied by Pkg, k E ~. (see 11.17) and repeating the arguments developed above, we get that 00
,A11+12r g(x) =
I
/Zk+m+1 Pkg(x)
k=O belongs to ML 2(r)(B(O,R');A), 0 < 1 < R'
I (2k+m+1) Pkg(x) k=O still belongs to ML 2(r)(B(O,R');A). (m+12r)g(x) =
(29.10)
0
29.7.3 Theorem The Bergmann kernel B(u,t) for the unit ball is given by B(u,t) = 5 u
u  t juj2
ITur
lui
t
r+1
"' (m+12f)K(x,t)(u). Proof First observe that fortE B(0,1) fixed, B(u,t) is left monogenic in 1 0 B(O, TfT) so that B(~,t) E ML 2(r)(B(0,1);A). So we only have to show that for each f E ML 2(r)(B(0,1);A)
0
f(t) = (B(.,t),f) 8• For r
>
ltl we have that
I
X  t 1 dcrxf(x) f(t) = 1wm+ 1 aB{O,r) lxtlm+
lim _r_ m J r+1 wm+1 sm 234
<
r~t dcr f (w) lrwtlm+1 wr
m
lim _r_ J K(rx,t)(u.1)ruf CuddS r+1 lllm+1 sm r ru
<
. \ m lim r m (w K(rx,t)(w),fr(w);
s
~1
<
m(~
\
(29.11)
lim r \K(rx,t)(w), fr(w); m r+1 s
<
xt is looked upon as a right monogenic function !xt!m+1 in X and where f Cul) = f(rul)' E sm. ~ r o Since K(rx,t)(u) is left monogenic in B(O, I~[) with~> 1 we may consider its Taylor deve1opment about the origin wh1ch converges uniformly on co r : B ,R ') , 1 < RI < TfT
where again K(x,t) =
(J)
'\,
00
L
K(rx,t)(u) =
Pk,r(u)
k=O ~
where Pk,r(u) = Pk K(rx,t)(u). ~ Hence, in view of (29.10), (m+12r)K(rx,t)(u) to ML 2(r)(B(0,1);A). But, in virtue of (29.8) 0
00
L
(2k+m+1)Pk,r(u) belongs
k=O
(
m(
\
(29.13)
f(t) =lim r \(m+12r)K(rx,t)(u), f(ru>;s· r+1
'<
Now
~
~
t
rm(m+12r)K(rx,t)(u) = (m+12r)K(x, ·r)(u) belongs to M(r)(B(O, I~I);A).
Hence, choosing r > ~ we obtain that in
0~
M(r)(B(O,
~);A) ~
~
rm(m+12r)K(rx,t)(u) + (m+12r)K(x,t)(u) if r+ 1 0
so that also in L2(r)(B(0,1);A)
<
235
Furthermore, as llf(ru) 11 8 converges to llf(u) 11 8 when r<1 and as for each ¢ E V(l)d3(0,1);A) lim (fr,¢) 8 = (f,¢) 8 , r+1
<
0
we have that for all g E L2(r)(B(0,1);A) lim (f ,g)B = (f,g) 8 , r+1 r
<
0.
which implies that lim llfrf11 8 r+1 < (29.13) we get:
Hence, taking the limit for r+1 in
<
0
0
As f E ML 2(r){B(0,1);A) and t E B(0,1) have been chosen arbitrarily, we finally obtain that '\,
B(u,t) = (m+12f)K(x,t)(u). Using a general result concerning adjoint monogenic functions (see Section 13) we also have: '\,
B(u,t) = (m+12f)K(x,t)(u) 00
L
'\,
(2k+m+1)PkK(x,t)(u)
k=O 00
L (2k+m+1){QkK(x,t)(u) k=O
I k=O
(2k+m+1)
1m s{QkK(x,t))(u) 2k+m+1
00
L (1m) s(QkK(x,t))(u) k=O
236
00
L
(1m) s(
QkK(x,t))(u)
k=O
= (1m) s(K(x,t)(u) 
u
t
lulz
0u ''m+.1 •
j{ur
0
tl
1u1
29.7.4 As a final result we establish a relationship between the Bergmann and Szeg6 kernels. Lett E B(0,1) be fixed; then 0
S(u,t)
'\,
= K(x,t)(u)
where Pk,t(u) = PkS(u,t).
Furthermore
00
s(K(x,t))(u)
= L Pk,t(u) k=O
with Pk,t(u) = Pk s(K(x,t))(u).
Consequently
1m
=~
=
k +m+1 2
~
1 lul(m+1)/2
c.
Jlul r(m+1)/2 P' (r u )dr 0 k,t TuT
whence Theorem In the case of the unit ball in Rm+ 1 the Bergmann and Szeg6 kern~ls are related by S( U, t) =
1 Jlul r(m+1)/2 B(r (m+1)/2
21 u1
o
~u 1u1
,t)dr.
237
Notes to Chapter 4 The problem of representing distributions by holomorphic functions goes back to K6the's paper [K3]. He investigated under which conditions a holomorphic function in [~ 1 defines a distribution on S1 • In [T3J Tillmann continued K6the's investigations to the case of unbounded domains in[; he thus obtained· a representation of distributions with compact support in R by means of holomorphic functions in NR. In [T4] and [T5] Tillmann represented the spaces VL (RN) and S'(RN) by means of subspaces of O(([~)N). In fact for the case p
N = 1 he obtained algebraic isomorphisms between respectively the spaces S'(R) and HS' and V'(R) and H0 ,. The procedure used in the plane could also be followed to represent S'(RN) by means of a space of holomorphic functions in ([~)N, but it appeared that the cases V' (R) and V' (RN) (N > 1) had to be treated differently. It was Vogt who gave in [V1] a complete description for the representation of V' (RN). f1eanwhile r1artineau studied in [t1ar] the boundary value problems connected with V'(R) and S'(R) in a more general framework. In fact he established a relationship between the existence of a distributional boundary value off E 0([~) and the extensibility off as a distribution in [, which in its turn could be characterized by means of growth conditions upon f. Moreover he put a link between the problem of representing V'(R) (resp. S'(R)) and the solution inV'(R 2 ) (resp. S'(R 2 )) of the equation a_S = T, T being given. In this way Martineau obtained the z algebraic isomorphisms already established by Tillmann. The question of course arose whether or not these isomorphisms were also topological when the spaces H0 , and HS' are provided with a suitable topological structure and when V'(R) and S'(R) carry the strong topology. This question was first answered by Konder in [Ko] while a refined study of the topology in H0 , was worked o~t by Vogt in [V2]. As was already mentioned, some problems arose with respect to the representation of V'(RN) (N > 1), especially concerning the surjectivity of the boundary value mapping. Vogt proved this surjectivity in [V1] by using the representation and boundary value theory for vector valued distributions and holomorphic functions. The case S'(RN;E),E being a locally convex space, was studied by r4eise in [t1ei1], [Mei2] and [Mei3], while the cases S'(RN;E) and V'(RN;E) were investigated by Vogt in [V1], [V3], [V4]. The problem of the representation of distributions as boundary values of holomorphic functions 238
thus gave rise to intensive research during the last decades. As furtha· contributions to this theory we mention the papers by Luszczki  Zielesny, Carmichael and Vladimirov (see [LZ], [Ca1], [Ca2], [Ca3], [Ca4], [Ca5], [Vl1], [Vl2]). An excellent survey concerning its development up to 1977 may be found in [Mei4]. Applications of the theory of distributional boundary values of holomorphic functions to quantum field theory and network theory can be found in [BW] and [Bre]. It should be mentioned that the first investigations made in the beginning of the fifties were followed up only in the seventies by results concerning the representation of ultradistributions and this in the works of Komatsu [Kom] and K6rner [Kor]. Recently Petzsche succeeded in giving a unified approach to the relationship between generalized functions (distributions, ultradistributions, hyperfunctions) and boundary values of holomorphic functions, both in the one and several dimensional cases (see [Pe1]). Note that the concepts of hyperfunctions and microfunctions were worked out essentially by Sato, Kashiwara and Kawai (see e.g. [Sa], [SKK]). In the Sections 2527 we deal with the problem of representing distributions in Rm by monogenic functions in Rm+ 1 ~m. Although the main idea goes back to the work of Martineau, our techniques are quite different. Note that the isomorphisms we finally obtain in 27.9 are of a pure algebraic nature. Nevertheless, just as in the case of one complex variable, it may be proved that they are also topological. The nature of our approach, namely to represent distributions in Rm as boundary values of functions in (m+1)variables, should be compared with the one using several complex variables in order to distinguish its own character. In this context we also want to mention Langenbruch's papers [la1], [La2], [La3], [La4] and [la5], where distributions in Rm are represent~d as boundary values of nullsolutions of hypoelliptic differential equations; his methods again differ completely from our function theoretic ones. Finally note that ~yperfunctions may also be described in terms of monogenic functions as shown in Sommen's paper [So1]. It should be noted that hereby, just as in the case of one complex variable, the use of cohomology theory could be avoided. The theory of Fourier series has a long history and some basic questions already posed at the appearance of Fourier's work "Th~orie analytique de la chaleur" (1822) could only be answered by using Lebesgue's integration theory. The decomposition of a function f E L2 (S 1 ) , where the latter space may be 239
identified with L2 {[0,2rr]), may be found in most textbooks on real analysis (see e.g. [HS]). Also in the more dimensional case it is well known that the set of surface spherical harmonics forms a complete orthonormal system of eigenfunctions of the LaplaceBeltrami operator~*m+ 1 acting on the Hilbert space L2 (Sm) (see e.g. [Tr]). The problem of decomposing functions or functionals on Sm into a series of surface spherical harmonics, and hence the problem of generalizing the classical Fourier expansion for f E L2 (Sm), has been the subject of more recent research. As was already mentioned in Section 28, growth conditions upon the sequence (Sk)kEN of surface spherical harmonics have to be imposed in order to get series expansions of the functions or functionals under consideration. Such estimates were obtained by Seeley (see [Se1], [Se2]) for C and analytic functions, by Morimoto (see [Mor]) for distributions and by HashizumeMinemuraOkamoto (see [HMO]) for analytic functionals. Furthermore Helgason showed in [Hel] that any harmonic function f in the unit ball admits a boundary value in a'(Sm), whence it may be decomposed into a series of surface spherical harmonics satisfying certain estimates. But also the converse is true, i.e. if a sequence (Sk)kEN of surface sphe~ical harmonics is given which satisfies the required estimates, then f(x) = L lxlk Sk(w) is harmonic in the unit k=O ball. The space of analytic functionals on Sm thus appears to be the largest space of functionals which may still be represented as boundary values of harmonic functions. The representation of a( 1 )(Sm;A) by M(r)(ffim+l,Sm;Al~a as sketched in 28.6 thus yields a refinement of Helgason's result. In this context it should be mentioned that HashizumeKowataMinemuraOkamoto constructed in [HKMO] spaces of functionals on Sm which contain a'(Sm) as a proper subset. These spaces are connected with the FourierBorel transform (see also Chapter 5). Finally it should be no~ed that the representation of any f E L2 (r)(Sm;A) as the L2 boundary value of f(w(1±E)) and the derived orthogonal decomposition of L2(r)(Sm;A) seem to be inherent in the function theory under consideration. Note also that the splitting of any surface spherical harmonic of order k into a sum of an inner and outer surface spherical monogenic, being themselves eigenfunctions of the spherical CauchyRiemann operator, has applications in the theory of electron spin (see [L]). Further investigations in this context have been carried out by Sommen in [So2]. The form of the Bergmann and Szeg6 kernels obtained in Section 29 for respectively the unit ball and sphere in ffim+l again illustrates the closeness 00 
240
of monogenic function theory to the theory of holomorphic functions of one complex variable. Bibliography [Bre] H. Bremermann, Distributions, Complex Variables and Fourier Transforms (AddisonWesley, New York, 1965). [BW] E.J. Beltrami, M.R. Wohlers, Distributions and the Boundary Values of Analytic Functions (Academic Press, New York, 1966). [Ca1] R.D. Carmichael, Distributional boundary values in VL , Rend. Sem. Math. di Padova 43 (1970) 3553. p [Ca2] , Distributional boundary values in Vlp II, Rend. Sem. Math. di Padova 45 (1971) 249277. [Ca3] , Distributional boundary values of functions analytic in tubular radial domains, Indiana u. Math. J 20 (1971) 843853. [Ca4] , Representation of distributions with compact support, ManuscPipta Math. 11 (1974) 305338. [CaS] , Distributional boundary values in the dual spaces of spaces of typeS, Pacific J. Math. 56 (1975) 385422. [Hel] S. Helgason, Eigenspaces of the Laplacian: Integral representations and irreducibility, J. Functional Analysis 17 (1974) 328353. [HKMO] M. Hashizume, A. Kowata, K. Minemura, K. Okamoto, An integral representation of the ~aplacian on Euclidean space, Hiroshima Math. J. 2 (1972) 535545. [HMO] M. Hashizume, K. Minemura, K. Okamoto, Harmonic functions on Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973) 81108. [HS] E. Hewitt, K. Stromberg, Real and Abstract Analysis (Springer Verlag, Berlin, 1965). [H3] G. K6the, Die Randverteilungen analytischer Funktionen, Mach. z. 57 (1952) 1333. [Ko] P.P. Kender, Funktionentheoretische Charakterisierung der Topologie in DistributionenrMume, Math. z. 123 (1971) 241263. [Kom] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo 20 (1973) 25105. [Kor] J. K6rner, Romieusche Ultradistributionen als Randverteilungen holomorpher Funktionen (Thesis, UniversitMt Kiel, 1975).
241
[LJ
P. Lounesto, Spinor valued regular functions in hypercomplex analysis (Thesis, Helsinki University of Technology, 1979). [La1] M. Langenbruch, Randverteilungen von Null6sungen hypoelliptischer Differentialgleichungen, ManusrJl'ipta Uath. 26 (1978) 1735. [La2] PFunktionale und Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 239 (1979) 5574. [La3] , Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partiellen Differentialgleichungen, J. Reine AngeUJ. f.1ath. 311/312 (1979) 5779. [La4] , Darstellung von Distributionen endlicher Ordnung als Randwerte zu hypoelliptischen Differentialoperatoren, Math. Ann. 248 ( 1980) 11 7. [La5] , Dualraum und Topologie der (lokal) langsam wachsenden Nu116sungen hypoe 11 i pti scher Differentia 1operatoren, f.1anuacripta Math. 32 ( 1980) 2949. [LZ] Z. Luszczki, Z. Zielesny, Distributionen der RMume VL und Randverteilungen analytischer Funktionen, Colloq. Math. 8 (1961) 12~131. [Mar] A. Martineau, Distributions et valeurs au bord des fonctions holomorphes, in Theory of Distributions, Proc. Intern. Summer Inst., Inst. Gulbenkian de Ciencia, Lisboa (1974) 113326. [Mei1] R. Meise, Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen I, Math. Ann. 198 (1972) 147159. [Mei2] , Darstellung temperierter vektorwertiger Distributionen durch holomorphe Funktionen II, Math. Ann. 198 (1972) 161178. [Mei3] , Ra6me holomorpher Vektorfunktionen mit Wachstumbedingungen und topologische Tensorprodukte, Math. Ann. 199 (1972) 293312. [14ei4] , Representation of distributions and ultradi stributions by holomorphic functions, in Functional Analysis: Surveys and Recent Results, Proc. Paderborn Conference on Functional Analysis (NorthHolland, Amsterdam, 1977) 189208. [Mor] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations, to appear in a volume of the Banach Center Publication. [Pel] H.J. Petzsche, Verallgemeinerte Funktionen und Randwerte holomorpher Funktionen (Habilitation, UniversitMt D6sseldorf, 1981).
242
[Sa] M. Sa to, Theory of hyperfunctions, I and I I, J. FarJ. Sci. Univ. 1':;kyo 8 (1959/60) 139193 and 387436. [SKK] M. Sato, M. Kashiwara, T. Kawai, Hyperfunctions and pseudodifferential equations, in Lecture Notes in Mathematics 287 (SpringerVerlag, Berlin, 1973) 265529. [Se1] R.T. Seeley, Spherical harmonics, Af.J:: :.Jortthly 73, part II, no 4 (1966) 115121. [Se2]      , Eigenfunction expansions of analytic functions, Pr>oc. A/?l .. i". l·iath. Soc. 21 ( 1969) 734738. [So1] F. Sommen, Hyperfunctions with values in a Clifford algebra, to appear in ,)'·::"!lor:. :Jt.·vin. [So2] Spherical monogenic functions and analytic functionals on the unit sphere, to appear in To~:;1o Jo;B•nal of 1iath. [T1] H.G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, f.1ath. z. 59 (1953) 6183. [T2] , Distributionen als Randverteilungen analytischer Funktionen I I, l1cith. z. 76 ( 1961) 521. [T3] , Darstellung der Schwartzschen Distributionen durch analytische Funktionen, Math. z. 77 (1961) 106124. H. Triebel, HBhere Analysis (VEB Deutscher Verlag der Wissenschaften, [Tr] Berlin, 1972). [Vl1] V.S. Vladimirov, On functions holomorphic in tubular cones, Izv. Akad. Nauk :JSSR Scr>. Matlz,;m. 27 ( 1963) 75100. [V12] , On the construction of envelopes of holomorphy for regions of a special type and their applications, Amu•. Math. Soc. Tmnslation 48 (series 2) (1966) 107150. [V1] D. Vogt, Distributionen auf dem ffiN als Randverteilungen holomorpher Funktionen, J. Reine Angcw. Matlt. 261 (1973) 134145. [V2]  Randverteilungen holomorpher Funktionen und die Topologie von V', Nath. Ann. 196 (1972) 281292. [V3] , Temperierter vektorwertiger Distributionen und langsam wachsende holomorphe Funktionen, !lath. z. 132 (1973) 227237. [V4] Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen, f.Janu;;cripta Math. 17 (1975) 267290.
243
5 Transform analysis in Euclidean space
The first two sections of this last chapter are devoted to the Fourier transform and the generalized Laplace transform in higher dimension within the framework of monogenic functions and Adistributions. First we treat of the Fourier transform in S(ffim;A), V(ffim;A) and E(ffim;A) and their respective duals of Adistributions. This leads a.o. to a generalization of the GelfandShilov Zspace in one complex variable, which consists of all entire functions of the form f(z) =
JR
eitz ¢(t)dt
for some¢ E V(R). Moreover an analogue of the classical PaleyWienerSchwar~ theorem is obtained (§30). Next the generalized Laplace transform of tempered Adistributions is studied, extending the complex Laplace transform im
z
> 0
im
z
<
LT(z)
0.
It turns out that forTE s(l)(ffim;A) its generalized Laplace transform LT is a left monogenic function in ffi~+l having its Fourier transform FT as s(l )boundary value for x0 + 0±. ~1oreover the following boundary value result is generalized: let OR([~) and OR([) be the spaces of holomorphic functions in respectively [~ and [ satisfying the respective estimates k, 1 € l'J,
c
> 0
and 1 € :D'l,
then the boundary value mapping 244
c>
0;
is surjective, bounded and open, while ker BV =OR([). In Section 32 we introduce a generalized version of the FourierBorel transform, which maps monogenic functions in annular domains into monogenic functions in the complement of the origin. This transform extends the transform o = o+ + o in the complex case, where T ~
T being an analytic functional with carrier in an annular domain. the classical result
Moreover
Exp'([) being the class of Helgason's entire functionals, is generalized, 0 leading up to an integral form for the Laurent series expansion of a monogenic function in an annular domain. In the last section the Radon transform P:
O'(B(0,1)). O(B(0,1))
given by 1
P(T)(u) =
~
is generalized to higher dimensions using a monogenic version P(u,z),(u,z) E Rm+ 1 x [m of the function (1u~) 1 , (u,~) E [ x [. This results into an optimal version of the CauchyKowalewski theorem (see 14.2) concerning the monogenic extension of an Avalued analytic function in some open subset of Rm 30.
Hypercomplex Fourier transforms
30.1 In this section we first introduce the Fourier transform of rapidly decreasing Avalued C functions; the classical Fourier kernel eitx,(t,x) E m m R x R is replaced here by the function E(t,x), (t,x) E R x R introduced in 15.4. 00
~.
~~
245
Next we introduce the Fourier transform of Avalued testfunctions by means of the exponential function E(t,x), (t,x) E ~m x ~m+ 1 , introduced in 15.4, which itself is a natural generalization of eitz, (t,z) E ~ x [and which fort fixed is monogenic in the whole of ~m+ 1 • This leads to a generalization of the Gel 'fandShilov Zspace in one complex variable. Finally the generalized Fourier transform of Adistributions with compact support is studied. A.
The Fourier transform in S(~m;A) and S*(~m;A)
30.2
Let ~ E S(r)(~m;A); then we define 1 , ••• ,m
and
The function F¢ is called the
Four•ier' tmncfor•m
of ¢·
30.3 Just as in the classical theory it is obtained that Fj is a topological automorphism of S(r)(~m;A) and
Theorem
F.1 q,(t 1 , ••• ,t. 1,x.,t. 1 , ••• , t ) = 21 J JJ J+ m IT Proof
Let ¢
=
J+ooexp(t.x.e.)~(t)dt + .• _ 00
J J J
1eA~A' where ¢A E S(~m;~) for all A E PN.
J
Then for each
component ¢A the classical Fourier inversion formula holds since exp(t.x.e.) J J J
= e0 cos(t.x.) + e.sin(t.x.). J J J J J
So for each A E PN, 1
~ Ti
J+ooexp(t.x.e.)F.¢Adx. co
J J J
J
J
1
[ F.~
J
LTI
J+ooexp(t.x.e.)tAdt.Jl oo
J J J
J
1,A(t).
As F. and F~ 1 are right Alinear the above inversion formula also holds for J J 1 ¢· Furthermore Fj and Fj are continuous. o 246
30.4
F 1¢(x) 30.5
F is a topological automorphism of s(r)(Rm;A) and
Corollary
JRm E(t,;)~(t)dt. If we define for~ E s(l)(Rm;A)
Remark
= J+'~'q;(t)exp( t .x .e. )dt., oo
J J J
J
j
1 , ••• ,m
and
then of course Fj{j S( l ) (Rm; A) • 30.6
1 , ••• ,m) and Fare topological automorphisms of
Definitions
(i) LetT E s(l)(Rm;A); its Fourier transform is the tempered left Adistribution FT given by
LetS E s(r)(Rm;A); its Fourier transform is given by , ¢ E S(r)(Rm;A).
. of both S(l)s * (Rm;A) It can be proved that F is a topological automorph1sm and S(l)b(Rm;A). 30.7
Introducing the r>eflection oper>ator>u Si,
1, ••• ,m, given by
S/ ("Jt) = f (X 1 , ••• , Xi _1 , Xi , Xi+ 1 , ••• , Xm) where f stands for a function or an Adistribution in Rm, some calculation formulae for the Fourier transform can be stated: (i)
F(3t_f)("it) = xiei s 1 ••• si_ 1 Ff(x) 1
247
(i i )
(iii )
F(e.f)(;) =e.1 s 1••• S.1 1s.1+ 1••• 5m Ff(x) 1 ~
~
F(eiat_f)(x) =X; Si+ 1••• Sm Ff(x) 1
(iv) (v)
F(t;f)(x) = ei aX; s 1••• S;_ 1 Ff(x) ~
~
F(t;e;f)(x) =ax. Si+ 1••• Sm Ff(x) 1
(vi)
~~
~
~
F(f(t+a))(x)=exp(a 1x1e 1) ••• exp(a 2x2e 2s 1) ••• exp(amxmems 1••• Sm 1)Ff(x)
where
B.
The generalized Fourier transform in V(Rm;A)
30.8 Let¢ E V(r)(Rm;A); then we define
f
~
~
~
F¢(x) = Rm E(t,x)¢(t)dt,
m+1 x ER •
It is clear that F¢ is left entire; moreover it is the unique left monogenic extension of F¢Cit)
=
~
~
~
E(t,x)¢(t)dt
which belongs to S(r)(Rm;A). Hence the following inversion formula is obtained:
30.9 Call Z(r)(m;A) the module of all functions F¢ where¢ runs through V(r)(Rm;A). Obviously Z(r)(m;A) is a submodule of M(r)(Rm+ 1;A). This module Z(r)(m;A) is now characterized by means of estimates in the following theorem, the proof of which is rather straightforward. Theorem Let¢ E V(r)(Rm;A) have its support contained in B(O,R). Then for each ~s E ~ m and s > 0 a constant C 6 > 0 may be found such that for all m+1 s '" X E R
248
s
30.10 Remark In virtue of Cauchy's Representation Theorem, for any E ~m. m+1 m a E~ and c > 0 there exists c~R.a,c > 0 such that for all ~ E V(r)(R ;A)
In the following theorem such estimates are used to determine completely the space Z(r)(m;A). 30.11 Theorem Let f E M(r)(Rm+ 1;A) be such that for a certain R > 0 and for any ! E ~m. a E ~m+ 1 and c > 0, there exists c~ > 0 such that f3,c:t.E
IXB aaf(x) I X
0
<: 
c~
f3,a,c
e(R+c) IXol.
Then there exists a unique testfunction ~ E V(r)(Rm;A) with supp~ such that f(x) = F¢(x).
c
B(O,R)
Proof In view of the stated estimates the restriction off to R:+ 1 , f(X), belongs to Sir)(Rm;A). So there exists a unique testfunction ~ E s(r)(Rm;A) such that f(x) = F~(x) or~ = F 1f, or still
We now prove that the support of ~ is contained in B(O,R+c) for any s > 0. Choose s > 0 arbitrarily and take o > 0; applying Cauchy's Theorem yields
From the assumed estimates it follows that for a certain constant C > 0 ( R+c) IX I o 1+1XIm+1 o ~ m and as A is bounded for x E R we get lf(x)lo:;;:Ce
249
In an analogous way it can be shown that for any ,.,
0
I r m B(t,x)f(x)dxlo ~ C"e(R+cltl ),'.. J]R
Now iftElRm.....S(O,R+d then R+s!tl < 0 and taking limits for,~++oo in the + above inequalities leads to ~(t) = 0. So supp~ c B(O,R) and it is possible to consider its generalized Fourier transform F¢(x). The function fF¢ is left entire and it vanishes in JR~+ 1 • Hence it is identically zero in lRm+ 1 or f(x) = F¢(x). o 30.12 Now we are able to endow the module Z(r)(m;A) with a locally convex topology. Let k,s E ~and call Z(r),k,s(m,A) the submodule of Z(r)(m;A) consisting of those left entire functions f which are such that for each E ~m and a E ~m+ 1 there exists a constant C+ > 0 such that
e
s.a
+
lxB
(
a~f(x) lo ~ c!.ae k+s
1)
lxol.
Provided with the proper system of seminorms {Pj,l :(j,l) E ~ 2 }, where sup e(k+s lu.l:;:;l
1
+
Z(r),k,s(m;A) becomes a right Frechet Amodule. Z(r)(m;A) = lim ind lim proj kE:tl
sEJl
a~f(x)lo•
)lxollxB
Then we put
Z(r),k,s(m;A).
Notice that z(r)(m;A) is an inductive limit of right Frechet Amodules. 30.13 The following topological result may be proved along the same lines as in the classical case of one complex variable. Theorem The Fourier transform
is a topological isomorphism. 30.14 250
Denote by z(r)(m;A) the dual module of Z(r)(m;A), i.e. the left A
module of bounded right Alinear functionals on z( )(m;A). By means of this . * m r dual module a Four1er transform on V(r)(R ;A) may be defined. Indeed, if T E Vtr)(Rm;A) then its Fourier transform is defined as to be the element of z(r)(m;A) which acts on functions f E Z(r)(m;A) as follows:
If¢ E V( l)(ffim;A) then we define ¢F 1 (x) = J ¢(t)E(t,x)dt. Rm
It is clear that ¢F 1 is right entire. Call z(l)(m;A) the module of functions ¢F 1 where ¢ runs through V(l)(Rm;A). This module Z(l)(m;A) may be characterized in the same way as was done for z(r)(m;A). Moreover a locally convex topology may be defined on it such that Z(l)(m;A) and z~ 1 )(m;A) are topologically isomorphic to v(l)(P.m;A) and v( 1 )(ffim·A) respectively. C.
. E(l)(ffi * m;A) The generalized Fourier trans f orm 1n
30.16 LetT E E( 1 )(ffim;A); then of course T E S(l)(ffim;A) and soFT is defined and belongs to s(1 ) (Rm;A); moreover for any ¢ E s(l) (ffim;A) holds
+ +
4
+
¢(x)E(t,x)dx>
f""\J
+
+
¢(x)T(x)dx where we have put
251
So if we consider the function "' T(x)
..... = , x E Rm+ 1 ,
we have at once that .....
FT(x}.
Hence it is quite natural to define 30.17 Definition function
* (Rm;A) is the The generalized Fourier transform ofT E E(l)
... m As T is left Alinear and bounded on E(l)(R ;A) and as E(t,x) is analytic in Rm x Rm+ 1 it is easy to show that FT is left entire; it is in fact the unique '\, ..... left monogenic extension of T(x). 30.18 Theorem LetT E E{ 1 )(Rm;A) and let R > 0 be such that supp T c B(O,R). Then there exist C > 0 and k E J~ such that for all x E Rm+ 1
Proof The desired inequality follows immediately from the definition of FT. 30.19 The converse of Theorem 30.18 may be regarded as the analogue of the famous PaleyWienerSchwartz Theorem. The proof, which is rather technical and may be developed analogously to the proof of Theorem 30.11, is omitted. Theorem (PaleyWienerSchwart~) Let f E M(r)(R m+1 ;A) and let R that for some C > 0 and k E ~
0 be such
. * m+1 ;A) with supp T c B(O,R) Then there ex1sts T E E(l)(R such that f =FT. 30.20 Let k E ~and let R > 0; call ~(r),k,R the module of all left entire functions f satisfying an estimate of the form
252
c
for some C > 0. Equipped with the norm
~
(r),k,R is a right Banach Amodule.
Putting
~(r)(m;A) = li~.~nd ~(r),k,R
the following result is obtained. 30.21
Theorem The generalized Fourier transformation
* m;A) .... ~(r)(m;A) F:E(l),b(lR is a topological isomorphism. Proof Use Theorem 30.3 and 30.19 and an analogous reasoning as in the case of one complex variable. c 31.
The generalized Laplace transform in
s(1 )(1Rm;A)
31.1 First we introduce the Laplace transform ~L 1 of a testfunction ~ E S(l)(lRm;A) vanishing in a closed ball centered at the origin. As ~L 1 is right monogenic in lR~+ 1 we may investigate its S(l)boundary value for xo .... 0±.
~F
1 ....
(x)
= JRm
~(t)E(t,x)dt
....
= JlRm
~(t)[B(t,x)A(t,x)]dt
+
0
....
....
o++
....
o++
belongs to S(l)(lRm;A). Now take ~ E.S(l)(lRm;A) with ~ ~L 1 in JR~+ 1 given by
+
0 in Bm(O,R) and consider the function
253
¢L
1
JIRm (x)
+ ¢(t)B(t,;)eltixodt,
{J!Rm ¢(t)A(t,;)elt[xodt,
X
E
1Rm+1 •
1 . c 1ear th a t ¢L 1 1s . 1n . 1Rm+ h 11 prove th at ¢L 1 . r1g . ht monogen1c It 1s 1 ; we s a 1 admits ¢F as S(l )boundary value. Thereto we need a lemma on estimates of ¢L 1 , the proof of which is omitted since it is based on straightforward estimates.
e
31.3 Lemma Let¢ E S(l)(!Rm;A) vanish in Bm(O,R). Then for each E Nm, m~ > 0 such that for all a E Jl and s > 0 there exists a constant C+ B,u.' r: m+1 1R X E f
31.4
Proposition 1im ¢L X +0+
1
The boundary values
+ (x±x 0)
0
exist in S(l)s(!Rm;A) and BV¢L 1(x) = lim X +0+ 0
Proof It follows from the previous Lemma 31.3 that for each S E Jlm and a E Jlm+ 1 there exists a constant C+ > 0 such that for all x0 E]0,1]
s.a.
1 I+s X dCJ. ¢ L (X) I x o
Hence for each
254
Z E Nm
~
C+
s.a. •
and x~ 1 ), x~ 2 ) E ]0,1]
Hence
exists in S(l)s(lRm;A).
Analogously
exists in S(l)s(lRm;A) and moreover
converges to 4
0
+ +
+
¢(t)E(t,x)dt for x0
+
0+.
¢F
1
~
(x)
o
31.5 A converse of Lemma 31.3 runs as follows. m+1 Lemma Let f E ~1(ll(lR# ;A) be such that there exists R > 0 for which, given any E ~m. a E ~m 1 and 0 < E < R, a constant C > 0 may be found such S,a,E that
s
Then there exists a testfunction that f ¢L 1
E S(l)(lRm;A) with
0 in Bm(O,R) such
Proof From the given estimates and the proof of Proposition 31.4 it follows that the boundary values 1 im
fCx±x 0 )
X rO+ 0
exist in S(l )s(IRm;A). +
So put +
¢(t) = [f(x+O)  f(xO)]F; 255
then clearly~ E S(l)(Rm;A). Using Cauchy's Theorem it is easily shown that ~
~
4
0
(f(x±x 0 )F)(t) = 0 for t E Bm(O,R) which implies that
that~=
0 in Bm(O,R).
It then follows from Proposition 31.4
1 im X
0
>0+
and as f and ~L 1 both satisfy esti~ates of the above type, we obtain, using Liouville's Theorem, that f = ~L 1 • o 31.6 Now letT E S~ )(Rm;A); then pered distribution. Our aim is to of FT; this is done by introducing which is a left monogenic function for x0 + 0±.
its Fourier transform FT is again a temconstruct representing monogenic functions the generalized Laplace transform LT ofT, in R;+ 1 having FT as S(l)boundary value
0
31.7 First assume that T 0 i~ Bm(O,R). Introduce the realvalued C functions a E (t), depending onE E]O,R[ such that 00
+
+
{
aE(t) =

0, t E B(O,RE) +
m
+
++
o
1, t E R ,B(O,R
E
~).
Then the functions ++
AE (t,x) = a E (t)A(t,x) and
+
and their derivatives with respect to t, are coofunctions of slow growth in Rm x lRm. Put +
E1 (t,x) ,E
and 256
__,.
E2 (t,x) ,E: then for X fixed, both functions belong to s(l )(Rm;A). definition makes sense. 0
LetT E s(1 )(~m;A) with T alized Laplace transfo~ is given by 31.8 Definition
__,.
f
LT(x)
,
1.
X
.....
X
So the following
0 in Bm(O,R).
Then its gener
.Rm+1 E + .Rm+1 E  •
Notice that this definition is independent of E E]O,R[ and that LT is left monogenic in .R~+ 1 • r~oreover it can be proved in a straightforward manner that LT satisfies an estimate of the following type. 0
31.9 Theorem LetT E s( 1 )(Rm;A) be zero in Bm(O,R) and let E E]O,R[. there exist k, r E ~ and CE > 0 such that for x E ~~+ 1
31.10
Then
In view of the above estimate and Theorem 26.8 the boundary values LT(x±" 0 )
and lim [LT(x+x )  LT(xx )J
BVLT X
0
.....0+
0
0
exist in s( 1 )s(.Rm;A). In the following theorem the nature of those boundary values is examined; they turn out to be Fourier transforms.
* m;A) be zero in Bm(O,R). o 31.11 Theorem LetT E S(l)(.R ct> E S(l) (Rm;A)
Then for any
lim J cp(x)[LT(x+x 0 )LT(xx 0 )Jdx =
257
m
~rs>_o_:f
:,..
+
Let¢ E S(l)(JR ;A); for x0 "0 fixed the function rp(x)LT(x+x 0 ) belongs to S(l)(lRm;A) and hence, using an approximation by Riemannsums N ~
..
.. ~ ~ JtJx ¢(X N)B (t,x N)e O;(K N)>. V>
F.
V>
V>
\;=0
Observe that the sequence of Riemannsums N ~
+
~
¢(X
v.
.... ltJx N)B (t,x N)e I 0dK c
v.
v,
N), N
1 ,2' ...
( 31. 1)
\!=0
converges uniformly on the compact subsets to
:t even converges in E( 1) (lRm;A). r~oreover this sequence a~d all sequences of tderivatives are uniformly bounded with respect to N and t. Hence the sequence (31.1) converges in S(l)(lRm;A), which leads to >>
>
J~ ¢(x)LT(x+x 0 )dx =
as >
.. .... JtJx Odx.. .. J m¢(x)B .. (t,x)dx ...... JJRm ¢(x)B E (t,x)e JR E
In an analogous manner we get that
... r .. .. .. .. ! ¢(x)A (t,x):ix > JlRm ¢(x)LT(xx o )dx =
1i m X
0
.0+
and hence >>) JlRm¢(x)[LT(x+x o
1 im X .0+
>
>
 LT(xx )]dx o
0
jo.
++
jo.jo.
+
= "L
258
lRm
c
E.
+
~
=
which yields BVLT
=
FT.
c
31.12 The converse of Theorem 31.9 runs as follows • .!_heorem Let f E r~( r) (JR~+ 1 ;A) be such that for some R > 0, k, r 0 < f: ..: R, a constant CE: > 0 may be found such that in JR~+ 1
E
f'l and any
* (lRm;A ) such that Then there exists a unique tempered Adistri bution T E S(l) T is zero in Bm(O,R) and LT = f. Proof As for x0 E] 1,0[U]0,1[
f admits an s( 1 )(JRm;A)boundary value for x0 exists a unique T E s(1 )(JRm;A) such that FT
= BVf = lim X
0
+0+
+
0 (see 26.8).
Hence there
[f(x+x )  f(xx 0 )]. O
First we prove that TiBm(O,R) = 0. Take a testfunction ~ E V(l)(JRm~A) wit~ suppcp c: Bm(O,R) and take E: > 0 sufficiently small such that suppcp c: Bm(O,R2,J Then
l
= lim x +0+ IRm
~F
1
+
+
0
But as
~F
+
+
(x)[f(x+x )f(xx )]dx. o
o
1
E z(l)(m;A) there exists C > 0 such that 3 1 1 (R 7d IXol I( 1+ Ix 12 ) r+m+
and so by Cauchy's Theorem for any x0 > 0 and 6
which, letting
>
0
+ oo, leads to
a+
It can be shown analogously that
f
JIRm
~F 1 ("it)f(xx )d"it = O·for x0 o
>
0,
0
so that T = 0 in Bm(O,R). Now consider LT; we know from Theorem 31.11 that BV LT
= H = BVf,
and so the function LTf has a left entire extension in 1Rm+ 1• Furthermore as LT  f E s( l)(IRm+ 1 ;A) it is a polynomial and as both LT and f satisfy an estimate of the given form, (LTf)("it+x 0 ) + 0 if lx 0 1 + + oo. Hence LT = f. c 31.13 Now we return to the general case where no restrictions on the support ofT E s( 1)(1Rm;A) are imposed. If R > 0 and 0 < E
260
support contained in Bm(O,R ~). From 30.17 we know that FT 2 (x) is the restriction to ffi~+ 1 of the left entire function FT 2 (x); so FT 2 (X) is the s( 1 )boundary value of the function g E M(r)(ffi;+ 1 ;A) given by
and for which there exist C > 0 and l E ~ such that
s(
By Theorem 31.11 FT 1 (X) is the 1 )boundary value of the function f = LT 1 E M(r)(ffim+ 1 ;A) for which, given 0 E ]O,Rs[ there exist C0 > 0 and k,r E ~ such that
Hence FT(x) = FT 1 (x) + FT 2 (x) is the m+1 LT(x) E ~1(r) (Rf ;A) given by
s( 1 )boundary
value of the function
LT 1 (x) + ~FT(x), x E R:+ 1 LT(x) = { LT 1 (x)  ~FT(x). x E R~+ 1 and satisfying an inequality of the form j LT ( x)
I
o
~ C( 1+  1 ) ( 1 + 1 x2 lxolr
1)
k e RI Xo I • x E R~+ 1 ~
( 31.2)
for some C > 0 and k, r E ~. Conversely let the function hE t4(r)(R;+ 1 ;A) satisfy an estimate of type (31.2). Then h admits an s( l)(Rm;A)boundary value which equals FT(x) for •a unique T E s( 1 )(Rm;A). So BVh(x) = FT(x) = BVLT(x). and it is easily shown that hLT is a left entire function satisfying 261
(31.3) 31.14 TakeR> 0 to be fixed. Call M(r),R,k,s(R~+ 1 ;A) the module of left monogenic functions in R~+ 1 satisfying an estimate of type (31.2). Provided with the norm
= sup
xeR~+ 1 (1+ 1s)(1+lxl 2 )k lx 0 1 it is a right Banach Amodule. Next consider the module lim ind t~( ) R k s(R~+ 1 ;A). k • sEJl
r ' ' '
r
Each function hE M(r),R(R~+ 1 ;A) admits a boundary value BVh in S~l)(Rm;A). So we can consider the boundary value mapping * (Rm;A) BV .. M(r),R (Rim+1 ;A ) ~ S(l) As it was shown in 31.13 that for each T E s( 1)(Rm;A) there exists a function g* E t\r) ,R(R~+ 1 ;A) such that BVg* = T, this boundary value mapping BV is surjective. Next, putting ~(r),R(m;A}
=lim kind ~ (r),k,R
where ~(r),k,R is the right Amodule of left entire functions satisfying an estimate of type (31.3), already introduced in 30.20, we have ker BV = ~(r),R(m;A). This leads to 31.15 Theorem (i)
The boundary value mapping
is bounded and open. 262
(1°1°)
1 * (IRmA) s(l) ; an d M(r),R (1Rm+ I ; A)/ ~{r),R (m;A)
are topologically isomorphic. 31.16 Remark Define
Let R > 0 be fixed and letT
E
1Rm+ 1
X E
1Rm+ 1
X E
1Rm+1
LT(x), x
{
0
,
E
s( 1)(1Rm;A) vanish in Bm(O,R).
+
and 0
L_T(x)
{
LT ( X )
'
+
x
,
E
IRm+ 1•
1 d Bot h f unct1ons are 1e ft monogen1c 1n IRm+ 1 an 0
0
0
Moreover, just as LT does, they both satisfy an estimate of the form appearing in Theorem 31.9. Then by Theorem 31.12 there exist unique tempered diso( O,R ) and such that tributions P+T, P_T E S(* l) (lRm;A ) vanishing in B
and LP T = LT. Furthermore P!T = P+T,
P~T
P_T,
P+T + P T = T and P+ P T
= P P+T = 0.
Moreover it can be shown that
263
m j =1
s1 ••• s.J 1t.)T. J
s(
This decomposition ofT E 1)(Rm;A) is illustrated as follows in the cases where m = 1 and m = 2. If m = 1 then it is an easy matter to check that
notice that here p_ is the restriction operator toR+. For m = 2, let + +t = (t 1 ,t 2 ) E R2 and call e the polar angle of t. Then it is obtained that
+
P+Tt = ~ (1±cose)T(t 1 ,t 2 ) ± ~(sine)T(t 1 ,t 2 )· In complex analysis the Laplace transform can be defined as follows. e.g. f be a continuous function of slow growth in R2 and put
Let
+
p±1,±1f = fi{t E IR 2
:
+t 1 > 0, :;t 2 > 0}.
Then for (o 1 ,o2 ) E {1 ,1} 2 one can define
this function is holomorphic in transform which is defined by L 01•02
is holomorphic in
{[~) 2
f(~ 1 ,~ 2 )
{(~ 1 ,~ 2 )
if sgn
E [ 2 : sgn
Im~ 1.
Im~i
oi}.
So the Laplace
= o1.,
and can be split up into four parts:
which correspond to the Laplace transforms of the restriction of f to the 'octants'; hence one could say that this Laplace transform is of a 'cartesian nature'. In the hypercomplex approach here presented, the Laplace transform always consists of two parts Lf
= LP +f + LP· f
where P± are the above introduced 'orientation operators' in the Euclidean 264
space, which rather have a 'spherical nature'. 'natures' coincide. 32.
Only in the case m
1 both
A generalized FourierBorel transform
32.1 In this section we study a generalized version of the FourierBorel transform of certain classes of analytic functionals with support in an annular domain or the complement of an annular domain in ~m+ 1 • The transform a = a+ + a_ under consideration reduces in the complex case to
o_
T being a complex analytic functional with carrier in an annular domain. In the present monogenic approach the function~ eu/z, (u,z) E [ x [ 0 is replaced by the function H(u,y), already introduced in 15.8.3~ which for y E ~~+ 1 fixed is left and right monogenic in y E ~~+ 1 with lim H(u,y) = 0. Y>
Notice that first some modules of monogenic functions and analytic functionals in annular domains and the complement of annular domains are introduced, the images under o of which will be determined.
32.2 Let R1 and R2 be positive constants. The notation
First assume that 0
<
R2
<
Rt,
for the right Amodule of left monogenic functions in the considered annular domain, is now abbreviated to ~·1 ( r) ( R10 R2
) •
We know from Theorem 12.2.2 that iff E f(x) =
k
L IX I k=O
M(r)(R 1 ~R 2 )
then
00
Pkf(w)
+
L
k=O 265
0
the first and the second series being normally convergent in B(O,R 1 ) and Rm+1 'B(O,R Moreover those left inner and outer spherical 2 ) respectively. monogenics satisfy the following estimates (11.17) and (12.6)
for all R' E]0,R 1 [ and R" E]R 2 ,+oo[ respectively. Hence by restricting to the unit sphere Sm it is obtained that for any 0 > 0 there exists c6 > 0 such that 1Pkf(w)l 0
~
C0(1+o)
k k R1
( 32. 1)
IOkf(w) lo ~ C0(1+o)k R~. ')!
1
1
Notice that f E M(r)(R 1 ,R 2 ) if and only if t E M(l)(R 2 ,R 1 adjoint off (see 13.12) with Laurent expansion 'V
f(x) =
'V
)
where f is the
k 'V IXI(k+m) Pkf(w) + 1: lXI Qkf(w) .... k=O k=O 00
'V
00
)
where 'V
Pkf(w) = Pkf(w) ~
and 'V
Qk f(w) = QkT(w) ~.
32.3
Now assume that Rl :;:; R2.
We keep the notation
for the right Amodule of left monogenic functions in
with limit zero at infinity. Notice however that this notation is consistent with the one in 32.2 since for R1 > R2 266
in view of the Laurent series expansion of a function in M(r)(R 1 ,R 2 ). of course for f E M(r)(R 1 ,R 2 ) we have alternatively
Then
or f(x) =
~
I
1x1
k
.
0
PkfC,1) 1n B(O,R 1 )
k=O where again the left inner and outer surface spherical monogenics satisfy estimates of the form (32.1). 1 Observe that f E M(r)(R 1 ,R 2 ) if and only if ~t E M(l)(R 1 2 ,R 1 ) where now
"J
f(x) =
I
00
1x1
k ""
Qkf(w)
0
1
in B(O,R 2
)
k=O or f(x) =
Y
IXI(k+m) Pkf(w) in Rm+ 1 ,B(O,R~ 1 ).
k=O 32.4 Combining the considerations made in 32.2 and 32.3, to any f E M(r) (R 10 R2 ) a sequence (Pkf(w), Qkf((l]))kElN of left surface spherical monogenics may be associated satisfying the estimates (32.1). So call S(r)(R 1 ,R 2 ) the right Amodule of the sequences s = (Pk(w), Qk(w))kElN Pk(~)
k
k
E P(r) and Qk(w) E Q(r) are such that for each ~ > 0 there exists c6 > 0 for which (32.1) hold. Then S(r)(R 1 ,R 2 ) is a right (FS)Amodule fqr the system of seminorms P = {pj : j E ~} where
where
k 1 k IPkCu)l • Rzk (1+ ~J) 1 k IQk (ul ) 1 }. p.(s) =sup sup {Rl(1+ J.) 0 0 J k€14 tuESm
267
Furthermore the mapping from M(r)(R 1 ,R 2 ) to S(r)(R 1 ,R 2 ) which maps a function f into its associated sequence of surface spherical monogenics, is a topological isomorphism. Moreover if (Pkf(u)), 0,~f(uJ\EINES~r)(R 1 ,R 2 ) is the sequence corresponding to f E M(r)(Rl'R 2 ) , then to f E H(l)(R2, R~1) the sequence (Qkf(w), P'kf( 0 1))kEINES(l)(R;1, R~ 1 ) is associated. m+1 32.5 As for each ~2 clR open, M(r)(~;A) and r~(l)(";A) are (FS)Amodules when endowed with the topology of uniform convergence on the compact subsets of~. it is meaningful to introduce the following modules. Definition 1 i m i nd O
with
Of course those Amodules are equipped with the inductive limit topology. 1 1 Note that f E M(r)(R 1 ,R 2 ) if and only iff E r~(l )(R 2 ,R 1 ). ru

32.6 In view of the topological isomorphism between M(r)(R 1 ,R 2 ) and S(r)(R 1 ,R 2 ) (see 32.4) it is evident to introduce the following modules. Definition
Endowed with the inductive limit topology, S(r)(R 1 ,R 2 ) is topologically isomorphic to M(r)(R 1 ,R 2 ). 1 1 '\, '\, Observe that (Pk,Qk)kEINES(r)(R 1 , R2 ) if and only if {Qk,Pk\Ef'.JES(l)(R 2 ,R 1 ). 32.7 In the following proposition, the proof of which is straightforward, the module S(r)(R 1 ,R 2 ) is characterized by means of estimates. Proposition 268
A sequence of left surface spherical monogenics
belongs to S(r)(R 1 ,R 2 that
if and only if there exist 0
)
<
n
<
1 and C
>
0 such
:;: C(1n)k R~k (32.2)
~ C(1n)k R~ 32.8 It follows readily from the duality theory enunciated in the Sections 22 and 23 that M(r)(R1,R 2 ) and M(l)(R 2 ,Rl) form a dual pair; the same is true for M(l)(R1,R 2 ) and M(r)(R 2 ,Rl). By the topological isomorphisms introduced in 32.4 and 32.6 it is obtained that S(r)(R1,R 2 ) and S(l)(R 2 ,R 1) and also S(l)(R1,R 2 ) and S(r)(R 2 ,Rl) form dual pairs. Notice that those duality phenomena imply that for both M(r)(R1,R 2 ) and S(r)(R 1,R2) the inductive limit topology is equivalent to a (DFS)topology. Notice also that for hE M(r)(R1,R 2 ) and h* E M(l)(R 2 ,RI), the duality between M(r)(R1,R 2 ) and M(l)(R 2 ,RI) is expressed as follows:
( i ) if R1
>
R2 then
I
=
a(B(Rln),B(R 2 +n)) (i i )
h*(u) do u h(u);
if R1 <; R2 then
=
where twice n
J
h*(u) do u h(u), a(B(R 2 +n)' B(Rln)) > 0
is chosen sufficiently small.
32.9 The duality between M(r)(R1,R 2 ) and M(l )(R 2 ,RI) may be regarded as an 'inner product dual1ty between M(~)(R1,R 2 ) and M(r)(R11 R21 ) in the following sense. 1 1 '\, Take g E M(r)(R1,R 2 ) and f E M(r)(R1 , R2 ) ; then f E t1(l )(R 2 ,RI). Hence it is quite natural to define an inner product between f and g by putting •
1

'\,
(f,g)
269
Furthermore this duality may be expressed in terms of the associated sequence spaces. ForgE M(r)(R 1 ,R 2 ) the associated sequence (Pkg(,,J), Qkg(u.J\Ei'J is in 1 1 . . ~(r)(R.:;R 2 ~i for f E M(r)(R 1 ~ R2 ) the sequence (Pkf(,D), Qkf( 0 J))k.QJ 1s 1n S(r)(R 1 ,R 2 ), while for~ E M(l)(R 2 ,R 1 ) the associated sequence (Qkf(w), 'V PkfC<~))kEI'J is in S(l)(R 2 ,R 1 ). Now first assume that R1 < R2 ; then there exists n > 0 such that 'V f E M(l)(R 2 +n, R1 n), which leads to the Laurent expansion f(x) =
I
I
~kf(x) +
k=O
Pkf(x)
k=O
" in B(O,R 2 +n),B(O,R 1 n). Meanwhile forgE M(r)(R 1 ,R 2 ) holds
g(x) =
,,
L
Pkg(x)
in
B(O,R 1 )
Qkg(x)
i n IR
k=O or g(x) =
I
m+1 'B ( 0 , R2
) •
k=O of the estimates satisfied by the sequences (Pkg'Qkg)kEI'J and (Qkf, Pkf)kEI'J we arrive at I~ vie~
00
I k=O 00
=
I
(Pkf,Pkg) +
I
(Qkf, Qkg)
k=O co
(Pkf,Pkg) m + I (Qkf'Qkg) m k=O S k=O S
where (.,.) m is the inner product on a(Sm;A) defined in 28.10.
s
Now if
'V
R1 > R2 there exists n > 0 such that f E M(l)(R 2 +n, R1 n) where still R1 n > R2 +n; this leads to the expansions
270
I
f(x)
~kf(x) in B(O,R 2 +n)
k=O or
L
"'
oo
f(x)
Pkf(x) in mm+1 'B(O,Rln).
"'
k=O ForgE M(r)(R 1 ,R 2 ) we now have
L
g(x) =
k=O
00
Pkg(x) +
L
Qkg{x) in B(O,R 1 ),B(O,R 2 ),
k=O
and a~ain,.,in view of the estimates satisfied by the sequences (Pkg,Qkg)keN and {Qkf,Pkf)kSN we arrive at <>0
(f,g) =
L k=O co
00
(Pkf, Pkg) +
L
(Qkf,Qkg)
k=O 00
(Pkf,Pkg) m  L (Qkf, Qkg) m· s s k=O k=O
L
32.10 It should be noticed that some modules of left monogenic functions and left analytic functionals, already introduced in the foregoing sections, fit into the framework set up about M(r)(R 1 ,R 2 ) or M(r)(Rl,R 2 ). 32.10.1
M(r)(1,1) =lim ind M(r)(1+c,1c) is nothing else but M(r)(Sm;A) and 1>c>O it is isomorphic to a(r)(sm;A), the module of Avalued analytic functions on sm (see 28.5). 32.10.2 r"\r)(1,1) is the module of left monogenic functions in mm+ 1 ~m with limit zero at infinity, i.e. M(r)(mr.t+ 1,sm;A) +O" It is isomorphic to the dual of M:1)(1,1~ = r1(1)(Sm;A) = a(l)(Sm;A), i.e. the module of left analytic functionals on S (see 28.6). 32.10.3 t4(r)(oo,O) = l~m kr~~ M(r)(R 1 ,R 2 ) coincides with M(r)(m~+ 1 ;A). over we have I• z
Mote
Proposition M(r)(oo,O) is isomorphic to the right module of Avalued harmonic functions in mm+1. 271
Proof
Let h be an Avalued harmonic function in ffi 00
h(x) =
I
k=O
I
, and let
k
00
Skh(x) =
m+·l
Ix 1 \h(w)
k=O
be its expansion into spherical harmonics. Now in view of Theorem 28.8 we have the decomposition Skh(w) = Pk(w) + Qk_ 1(w) where Pk(w) and Qk(w) satisfy estimates of the form sup IPk(cu) lo :;; C (k+m1\ ( 1+k2) sup 1\h(w)lo \ k+1 ) wESm wESm and sup IOk1 (w) lo wESm
~
c(k+~2)(1+k2) sup
1\h(w) lo
wESm
But as h is harmonic in ffim+ 1, the sequence (Skh\EJJ satisfies
(see (28.4)).
So, putting e(h)(x) k=O
k=1
1• th e f unc t 1on e h ) 1s 1e ft monogen1c 1n lRm+ 0 Conversely let f E M(r)(oo,O) and put 0
h(x) =
(
I
0
0
IXIk Pkf(w) +
k=O
l
0
lxlk Qk1 f(w)
k=1 00
Then as the sequeilce (Pkf(uJ), Qkf(uJ))kEJIJ satisfies estimates of the form IPkf(w)lo:::; Cssk 272
and IQkf(w)lo ~ CEEk, · h lS · harmon1c · 1n · lRm+ 1 and clearly e(h)(x) = f ( x ) 1n . JRm+ 1• t he f unct1on 0
c
32.10.4 M(r)(O,~) =lim ind M( ) (E,!) consists of the germs of left monoE>O r E genic functions in a neighbourhood of the origin and the point infinity, while vanishing at the latter point. It is the dual of M(l)(oo,O) = M(l) (lR~+ 1 ;A) and thus represents all left analytic functionals with carrier in some annular domain.
32.11 Now we introduce some submodules of M(r)(oo,O) which consist of left monogenic functions in lR~+ 1 satisfying some growth conditions. 32.11.1 Definition Exp(r)(R1,R 2 ) is the submodule of M(r)(oo,O) consisting of those functions such that for each E > 0 there exists CE > 0 for which lf(x)l 32.11.2
o
~ C exp[{1+E)(J;l+_&_)]. ~1
E
lxl
Provided with the proper system of seminorms P = {pk
k E ~}where
Exp(r)(R1,R 2 ) becomes a right Frechet Amodule. 32.11.3 A function f belonging to M(r)(oo,O) and a fortiori belonging to Exp(r)(R1,R 2 )  has a sequence representation (Pkf(w), Qkf(w))keNES(r)(oo,O) such that for any E > 0 there exists CE > 0 for which JIPkf{w)
k
10
LIQkf(w)lo
~ CE E
~
CE Ek.
The module Exp(r)(R1,R2 ) is now characterized in terms of its sequence space representation, by means of sharper estimates than the above ones; the proof is postponed to 32.22(i). 273
Theorem A function f E M(r)(oo.O) with sequence representation (Pkf(u,), Qkf(rll))kEJ.I E S(r)(oo,O) belongs to Exp(r)(Rl'R 2 ) if and only if for any r; there exists C > 0 such that
>
0
n
( 1+n) k JIPkf ((•) l Io .:: cn k! Rlk 1 (1+!J)k k IQkf(<.:)lo <: cri k! R2. 'I
3~.11.4
We know that f E rvt1r)(".,O) if and only iff E r\ )(""•0). Now as IQkf(ul) I = IQkf(t,,) 1 and l~kf(,,,) 1 = IPkf(,,,) 1 , it is oblained that o o . 'l
Definition
Exp(r)(R 10 R2 ) =lim ind Exp(r)(R 1 +s, R2  , ) .
This mod
c·O
ule is equipped with the inductive limit topology. 32.12.2 In the following proposition Exp(r)(R 1 ,R 2 ) is characterized by means of estimates; the proof is straightforward. Proposition A function f belongs to Exp(r)(R 1 ,R 2 ) if and only iff E r4(r)("",O) and there exist C > 0 and 6 > 0 such that lf(x)
10
;:;
C
exp [(1r;)
(Jtf + Vr)].
32.12.3 This module Exp(r)(R 10 R2 ) nay also be characterized in terms of the sequence space representation. The proof of this theorem is also postponed to 32.22. Theorem A function f E M(r){oo,O) with sequence representation (Pkf(w), Qkf(w))keN E S(r)(oo,O) belongs to Exp(r)(R 1 ,R2 ) if and only if there exist C > 0 and f > 0 such that ( {IPkf w)l 0
:;;;
IQk f( w ) Io<
274
(1f)k k CIT~ R1 c ( 1o)
k!
Rk 2 •
C.
A generalized FourierBorel transform
32.13 In 15.8.3 we introduced the function H(u,y) as a generalization of ~ eu/z, (u,z) E [ x [ 0 • Recall that this function is left and right mono1 separate 1y, w1t . 1n · u E IRm+ 1 an d y E .,...m+ · h 11m · H( u,y ) = 0; moreover 1· t gen1c ~0 Y+«>
takes its values in spR {e 0 ,e 1 , ••• ,em}· Now it will be used in the definition of a generalized FourierBorel transform. 32.14 LetT E M(r)(R 1 ,R 2 ); then T may be considered as a bounded linear functional on M(l )(R 2 ,R 1 ) =lim ind M(l) (R 2 +E, R1 E). If R1 ~ R2 then E>O
clearly H(u,y) E M(l)(R 2 ,R 1 ) in both variables separately. to define in IR~+ 1 :
So it makes sense
o+(T)(y) =
J
H(u,y)dou T(u)
a[B(R2+n)~(Rln)J
=
f
aB(~2+n)
H(u,y) dou T(u)
where of course n has to be chosen sufficiently small. Clearly o+(T) is left monogenic in IR~+ 1 and lim o+(T)(y) = 0. Analogously we put in 1Rm+ 1 Y+«>
o_(T)(y)
J
H(y,u)dou T(u)
a[B(R 2+n),B(Rln)] r
J H(y,u) dou T(u), aB(R 1 n) which is a left entire function. On the other hand if R1 > R2 we introduce the functions
275
H(u,y),
1 ( 2R +R ), y E Rm+1 1 2 0
IU I
< ~
luI
>~
f 1
H+(u,y)
0
1 ( 2R 1+R 2 ), y E Rm+1 0
and u E Rm+1' IYI
0
f 1
H_(u ,y)
H( u ,y) ,
u
€
m+1 R • IY I
1
< ~(R 1 +2R 2 )
1
> ~(Rl+2R2).
Then clearly H+(u,y) E M(l)(R 2 ,R 1) with respect to the variable u, while H_(u,y) E M(l )(R 2 ,R 1) with respect toy. So we are able to define in R~+ 1 :
(
J
H+(u,y) dou T(u)
a[B(Rln),B(R2+n)]
J
H(u,y) dou T(u)
aB(R 2+n)
=I
H(u,y) dou T(u) as{R1+R2) 2
which is a left monogenic function in R~+ 1 with lim o+(T)(y) = 0. y~ ogously we put in Rm+1': o_(T)(y) =
=
f
a[B(~ln),B(R2+n)J
=I
H (y,u) dou T(u)
H(y,u) dou T(u) aB(Rln)
=
I
H(y,u)dou T(u)
aB( R1 +R2) 2
which is a left entire function. 276
Anal
Now letT E M(r)(R 1 ,R 2) =lim ind M(r)(R 1 +E,R 2c); then T can be
32.15
oO
looked upon as a bounded linear functional on M(l)(R 2 ,R 1 ) . If R1 < R2 then clearly H(u,y) E M(l )(R 2 ,R 1 ) in both variables u andy separately and so it is again meaningful to define in R~+ 1 : 0
+(T)(y) =
I
H(u,y) d0 u T(u)
a[B(R2n)'S(Rl+n}J =
J H(u,y) d0 u T(u) aBCR2n)
which is a left monogenic function in R~+l with lim 0 +(T)(y) ously we put in Rm+l Y~
0.
Analog
o_(T)(y) =
H(y~u) dau T(u)
f a[B(R2n,'B(R 1 +n)J =
 I H(y ~u) d0 u T(u) aB(R 1 +n)
which is a left entire function. On the other hand if R1 ~ R2 we define
r
H(u,y) dau T(u)
J
asu~2 n)
=
J
H(u,y) dau T(u)
aB(R1+R 2 ) 2
which is left monogenic in R~+l with lim o+(T}(y) = 0, and Y~
277
=
a_(T)(y)
r
H (y ,u) da T(u)
J a[B(R 1 +n),B(R 2 n)J
r J

u
H(y,u) dau T(u)
aB(Rl+n)
r R R H(y,u) dau T(u). a~(~) 2 32.16 In view of the considerations made in 32.14 and 32.15 we define the generalized FourierBorel transform to be the map M
{ _(r)
a
M(r) given by a(T)
a (T) + a (T) + 
where a+ and a_ are to be taken in the sense of 32.13 and 32.14. So a transforms modules of monogenic functions in annular domains or the complement of annular domains into modules of monogenic functions in the complement of the origin. But there is more as shown in the following theorem. 32.17 Theorem The following inclusions hold: (i)
a(M(r)(R 1 ,R 2 ))
c
Exp{r)(R 1 ,R 2 );
(ii)
a(M(r)(R 1 ,R 2 ))
c
Exp(r)(R 1 ,R 2 ).
Proof It is clear that (ii) follows from (i) by taking inductive limits. So letT E M(r)(R 1 ,R2 ). If R1 ::;; R2 then for every n > 0 we have Ja+(T)(y)
J0
~ I·
J aB(R 2 +n)
278
H(u,y) dau T(u) I o
~
C sup JH(u,y) Ia· n 1u:=R2+n
Now by Proposition 15.8.6 we have JH(u,y) J0 when juJ
>
;;
C(1+ ~~~~J)m ~ eJUJIJYI IYI
JYl, while for JYI
~
JuJ
JH(u,y) J0 s C' ~n IYI Hence for all y E R~+ 1 holds Ja+(T)(y) lo :;; c~ e(R2+2n)/ IYI. In an analogous way it is obtained that
Consequently for each Rm+1
E
>
0 a constant C
>
0 may be found such that in
E
0
which means that a(T) E Exp(r)(R 1 ,R 2).
The case R1
>
R2 is treated similarly.1
32.18 Actually the inclusions stated in the above Theorem 32.17 are equalities between the considered function modules. This will be proved in 32.19, but first the images of M(r)(R 1 ,R 2 ) and M(r)(R 1 ,R 2 ) under the transformation a are characterized in terms of their associated sequence spaces. _Theorem Let f E ~4(r)(oo,O) = M(r)(JR~+ 1 ;A) have (Pkf(c,l),Qkf(w))kElN as sequence representation in S(r)(oo,O). Then (i) f E a(M(r)(R 1 ,R 2 )) if and only if for every n such that
>
0 there exists en> 0
279
{ (ii) that
IPkf(w)lo
:; c11
IQkf(w) lo
:; c
( 1+n)K
k!
k Rl
( 1+~)k
k Rz;
k.
f E o(M(r)(R 1 ,R 2 )) if and only if there exist C
>
0 and 6 > 0 such
Proof Assume that R1 :;; Rz • Take T E M(r)(R 1 ,R 2 ) and put f(y) (IR~+ 1 ;A). Then
I
f(y) = aB( =
I
2
H(u,y) dau T(u) +n)
L ~!
aB(R 2 +n) k=O
I
o(T)(y)
E
M(r)
H(y,u) dou T(u)
aB(Rln) qk(u,y)dou T(u)aB( 00
L
r
t
00
I
1 TI qk(y,u) dou T(u)
n) k=O
1
IT pk T(y)
k=O wm+1 PkT(w ) and Qkf ( w) = ~ ~+1 which means that Pkf(w) = ~ QkT ( w). The statement (i) then follows from the sequence representation (PkT(w), QkT(w)) E S(r)(R 1 ,R2 ). The cases T E M(r)(R 1 ,R2 ) with R1 > R2 and T E M(r)(R 1 ,R 2 ) are treated along the same lines. c 32.19 Finally we show that a is a topological isomorphism between M(r)(R 10 R2 ) (resp. M(r)(R 1 ,R2 )) and Exp(r)(R 1 ,R2 ) (resp. Exp(r)(R 1 ,R2 )), all modules under consideration being endowed with their respective natural topologies. Theorem The following topological isomorphisms hold:
280
Proof It is clear that (ii) follows from (i) by taking induct·ive limits. Now by Theorem 32.17 it is already known that
In view of Theorem 32.18 we still have to prove the necessary condition of Theorem 32.11.3 in order to conclude that
m+1 So let therefore f E Exp(r) (R 10 R2 ) . i.e. f E t1(r) (R0 ;A) and for every E: there exists C > 0 such that E: lf(x) lo:;;; CE: exp [(1+d
>
(~ + 1~1 )].
Expending f into its Laurent series f(x) =
I
lxlk Pkf(w) +
k=O
I
lxl(k+m) Qkf(w)
k=O
where, independently of r E ]O,+oo[,
it is obtained that forE:
:; ; C'
E:
>
0 and for all r
>
r0
>
0
1 exp(1+E: ) R r • (1+k) m .. rll. 1
For all k E N such that r 0
<
~ l+E:
,
we ar~ive at
and, using Stirlings inequality _k_l_:;; kkek, at l21il< 281
0
An analogous estimate holds for !Qkf(w) lo·
c
32.20 The isomorphism a established above is now carried over by transposition to an isomorphism between the respective dual modules. Thereto we define: Definition LetT E Exp(r)(R 2 ,R 1 ) (resp. T E IXP(r)(R 2 ,R 1 )); then its generalized FourierBorel transform is the function o(T) E M(l)(R 1 ,R2 ) (resp. o(T) E M(l)(R 1 ,R2 )) such that
32.21 Theorem (i) (ii) 32.22
=
In view of Theorem 32.19 it is obtained by transposition that The following topological isomorphisms hold: cr(Exp~r),b(R 2 ,R 1 ))= ~(lf(R 1 ,R 2 );
o(Exp(r),b(R 2 ,R 1 ))
=
t4(1)(R 10 R2 ) .
Remarks
(i) The sequence characterizations of the modules Exp(r)(R 1 ,R2 ) and Exp(r)(R 1 ,R 2 ), already stated in 32.11.3 and 32.12.3, now follow immediately from Theorems 32.19 and 32.18. (ii) Observe that Exp(r)(O,oo) is the generalization of Helgason's space of entire functionals in the complex plane (see [9]). Moreover holds o(Exp(r)(O,oo)) = t4(l )(oo,O) = f4(l )(lR~+ 1 ;A). (iii) The isomorphism established in Theorem 32.21 (ii) means that, mutatis mutandis, for any f E M(r)(R 10 R2 ) (R 1 > R2 ) there exists T E Exp(l)(R2 ,R 1 ) such that o(T) = f, or
282
When x E B(O,R 1 ),B(O,R 2 ) then g(u) particular case we get
= E(ux) belongs to M(l )(R 2 ,R 1 ) and in thi:
I
E(ux)dcu f(u)
f(x)
;;[B( R1 n)'B ( R2 +n>J
while
I
o(E(ux)) (y)
J
E(ux) dou H(u,y) 
aB(R 1n)
H(x,y)
E(ux)dou H(y,u)
;)B( R2+n)
H(y,x)
+
and hence f(x)
=
+
H(y,x)>,
0

x E B(O,R 1 ),B(O,R 2 ),
yielding, when~ is an Avalued measure representing T E Exp( 1)(R2 ,R 1 ), f(x) =
I
H(x,y)dpy
+
1Rm+1
r J
o
H(y,x) dpy' x E B(O,R 1 ),B(O,R 2 ).
1Rm+1
In terms of the Laurent series expansion of f this leads to
and
L
oo
k=O
Qkf(x) =
I
H(y,x)
d~y'
x E lRm+1 'B(O,R 2
).
1Rm+1
Those two integrals might be called the "Taylor and Law•ent integrals" of f E M(r)(R 1 ,R 2 ) (R 1 > R2 ). A same reasoning may be followed in the case where R1 s R2 leading up to the same expressions for the "Taylor and Laurent integrals".
283
33.
Connections with complex analysis a generalized Radon transform
33.1 In [8] Hayman proved that every harmonic function, and hence every holomorphic function, in the open disc {r; E [ : lsi < R} admits a multiple Taylor series expansion about the origin converging absolutely in the domain {(x,y) E JIF : JxJ + IYI
<
R}.
In this section this result is generalized to the monogenic functions in the following sense : taking an analytic function f the multiple Taylor series of which converges absolutely in a domain of Rm we look for the optimal domain in Rm+ 1 where the multiple Taylor series of the monogenic extension of f converges absolutely. To this end a hypercomplex version of the Radon transform ~
P: O'(B(0,1))
0(8(0,1))
given by 1 u> P(T)(u) =
is defined. However it is necessary to leave the framework of the real Clifford algebra A used up to now in this book, and to handle the complex version of it, which will be denoted by Ac • 33.2
For u = u0 + u E Rm+ 1 , (z 1, ••• ,zm) E [m and {r; 1 •••• ,r;m) E [m we put m
m ~
z =
I j=1
e .z. "' J J
I
~
~
I
Jz J·12
j=1 m
I
~
X
j=1
m JzJ2 =
ej(xj+iy) = m
I
(xj+Yj),
j=1
r;jzj
j =1 and m
I j=1
284
ujzj
+ i.
.~
+ ly,
Next to the classical ball B( 0 , R) = {x E Rm+ 1 :
1
x1
<
R} ,
the following domains in Rm+ 1 and [m will be used constantly. (R 1 , ••• ,Rm), Rj > 0, j = 1, ••• ,m, we put B*(O,R) = (i
E
[m :
111
<
With
R=
R}; m
n (R)
{x
E
Rm+ 1 : 1x0 1 1R1 +
L j
Rj 1xj 1 < 1};
=1
where n.(R.) = {X E Rm+1 J
J
{X E Rm+1
n*(R)
33.3
+ /\(R)
+ {z E [m
'K
+ {Z E [m
If Q* f =
c
lx 0 1 + IXj I < Rj}; + 1 . lxol + lXI < R} • lzj I
[m is a domain of holomorphy then we consider functions
L eAfA
: Q*+ Ac
A
where each of the components fA : Q*+ [ is holomorphic in Q*. The Acmodule of those functions is denoted by O{Q*;Ac); it may be considered as well as a left or as a right Acmodule, and it is provided with the usual topology of uniform convergence on the compact subsets of Q*, turning it into a Frechet Acmodule. The dual modules O(l)(Q*;Ac) and O(r)(Q+;Ac) consist of all left, respectively right, Aclinear analytic functionals in Q*. For a relatively compact convex open neighbourhood Q* of the origin iA am we put lim ind O(l)((1+s)Q*;Ac). s>O
285
33.4 Just as was done in 15.7 for the real Clifford case, we consider the left and right inner spherical monogenic of order one: m
L
p(u,z) =
(u 1e0

u0 e1 )z 1
11 .... .... e 0
....
u0 z,

which for any u E Rm+1 clearly belongs to 0([ m;A c ). obtain
For its Clifford norm we
The function p(u,z)
k
.... kQ
= p(u,z)
is a left and right inner spherical monogenic of order k, taking c in sp[{e 0 ,e 1, ••• ,em} cA. Just as in 15.7.4 it is proved that Proposition
its values
If the function k
00
L
ckz:; '
ck E [ for all k E ~
k=O is holomorphic in the open disc {z:; E [ and right spherical monogenics F(u,z) =
[z:;[
<
p}, then the series of left
I
ck p(u,z)k k=O when considered as a multiple power series in the real variables u0 ,u 1 , ••. ,um, converges absolutely in the region given by [u 0
[
.... [z[
m +
L
[uj[ [zj[
< p
j=1
In the particular case where z = X E Rm this domain is the largest one in which the multiple Taylor series of F(u,z) about the origin converges absolutely.
286
Remark When m = 1 and by identifying e 1 with i, the above left and right monogenic function F(u,z) reduces to f(iuz), (u,z) E [ x [.
33.5
1Rm+ 1 of the origin is called m+1 optimal with I'e::pect to absolute c'onuel'gcn,H:; if for every u E lR 'S"2 there exists a left (or right) monogenic function f in ~ such that the multiple Taylor series expansion of f about the origin converges absolutely in Q but not in u. So by Proposition 33.4 all domains which are given by inequalities of the form 33.6
Definition An open neighbourhood
'> c:
m
>
L
luol IRI +
lujiRj
1,
<
Rj
>
0
j=1 are optimal with respect to absolute convergence. 33.7 The following example of a function of type F(u,z) is essential in the construction of the generalized Radon transform. Starting with the well known Taylor series expansion 1
k
00
T=Z = l: s ,
lsi
<
k=O the above Proposition 33.4 leads to the function P(u,z)
= 
1p(u,z)
1u 0 z m (1) 2 +u~(
l:
zj)
j =1 with Taylor series expansion about the origin 00
P(u,z)
=
z:
+ k p(u,z)
k=O which, considered as a multiple power series in u0 , ••• ,um, converges absolutely in the domain given by
Notice that P(u,z) is defined, holomorphic with respect to z and monogenic with respect to u in (1Rm+ 1 x [m)....s where the set S is determined by the equations 287
{
+
+ u~lxlz
(1)z
++
++
(1)
~+
= z
+
+ u~IYiz
++ = u~
The function P(u,z) is thus a direct generalization of 11u~
(u,~) E [
x [.
33.8 Just as was done in 15.8 we may consider series of the form
where the coefficients ck, k E ~. are now chosen to be complex, the series thus representing left and right monogenic functions in both variables u and y separately in a certain region, and taking values in sp[{e 0 ,e 1 , ••• ,em}. This yields a generalization of the function
.! f(~), ~
~
(u,d E [
x [
0,
the function f being holomorphic in an open disc. Proposition If the function with Taylor series expansion
f(~)
We have in fact
is holomorphic in the open disc
{~€[: l~l
00
f(d
=
2 k=O
then the series 00
¢(u,y)
=
2
ckqk(u,y)
k=O converges normally in the region
to a left and right monogenic function in both u and y separately. 33.9 The above two monogenic generalizations as presented in 33.4 and 33.8 are linked to each other in the following way. 288
Proposition f( rJ
Let the function 00
2
=
k=O be holomorphic in the open disc {s E [ regions:
2
00
F(u,i)
ck p(u,i)
lsi
~
p} and put in the appropriate
k
k=O and 00
q,(u,y) Then for sufficiently small r > 0 and lui< rp we have F(u,z)
¢(u,y)
(
day
P(y,i).
asc6,r) Proof Take all lzjl < R, j = 1, ••• ,m, for some R > 0; then P(y,i) is left and right monogenic in the region U given by m
+
IYo I Iz I
+
1
2 IYj I Izj I < R" j=1
Chooser> 0 small enough such that B(O,r) cU. Taking into account that ¢(u,y) is left and right monogenic in both variables in the region lui
(
aB (
6, r)
¢(u,y)
day
P(y,i)
is a function of u and z which is left and right monogenic in lui < pr. function is nothing else but
This
00
2
ck PkP(u,z)
k=O
289
00
L ck p(u,i>
k
k=O F(u,z).
c
33.10 Now we define a generalized Radon transform. Definition Let D* Then we put
c
[m be a domain of holomorphy and let T E
o( 1 ) CJ*;Ac).
PT(u) =
* ( g* ;Ac) and all >.. E Ac • for all T E O(l) Finally note that form = 1 and by the 'classical' identifications, P reduces to P( T)( u)
=
E [
X
[
1uz;,
which implies, by K6the's duality theory [12], that Pis then a topological isomorphism from O'(B(0,1)) to 0(8(0,1)). 33.11 Our aim now is to characterize P(O*(l)(n*;Ac)) in the particular cases .... 'V+ where g* is either A(R) or A(R) or still B*(O,R). We first prove .... *  .... c ). Then P(T) is left monogenic in rr(R) and its Theorem LetT E O(l)(A(R);A multiple Taylor series expansion about the origin converges absolutely in .... rr(R).
*  .... c ). Proof Take T E O(l)(A(R);A that
290
Then for each
E >
0 there exists C
E
>
0 such
+
c
m
>
for all 1> E 0(1) (~(R+2t) ;A ) , where R + [~ stands for (R 1+E•· •• ,Rm+E) E R • Now let u E n(R) be fixed and choose~> 0 such that u E n(J + 2l ); then + + c + of course P(u,z) E O(l)(AlR + 2t);A ). So for all u E n(R} the function P(u,!) belongs to O(l)(f.(R);Ac) and P(T)(u) =
is well defined. Hence P(T) is left monogenic in rr(R). + u E n(R) as above we have P(T)(u) =
z
00
z
++
I
'f
With the same fixed
p(u.1}>
k=O +k
I
l:
k=O k0 +
+ km=k
k! k I k I o · • • • m" +
ko k 1 z1
k
u
0
0
k um m
km zm >.
( 33. 1 )
Now
whence ko
l:
+ ••• +
k! km=k m + +
:; : CE:(Ju 0 J JR+E:I +
L
JujJ (Rj+d)
k
j=1
for some
c5
E:
> 0.
291
This means that the multiple Taylor series (33.1) converges absolutely in >the point u. As u was chosen arbitrarily in rr(R) the multiple Taylor series >expansion of P(T) about the origin will converge absolutely in rr(R). c 33.12 In a completely analogous way or OCl)(B*(R);Ac) respectively, then respectively, with a multiple Taylor verging absolutely in rr(R) anJ rr*(R) 33.13 In order to prove the result due to Martineau (see B*(O,R) and put respectively mapping P* : 0'(~*).. O'(Q*) p*(T)(i)
=
*
'\,
>
c
it may be proved that if TEOQ) (t,(R) ;A )
P(T) is left monogenic in rr(RJ or rr*(R) series expansion about the origin conrespectively.
converse of Theorem 33.11 we need the following >'\,>[13]). Let g* be either A(R) or A(R) or still '\,>>1 Q* equal to A(R) or A(RJ or B*(O,R ). Then the given by
1
1
>
is a topological isomorphism. If T E 0( 1 )(~*;Ac) then there exist TA E 0'(~*), A c {1 , ••• ,m}, such that T = L eATA in 0( 1 )(~*;Ac), and it is an easy matter to check that the transform~tion which maps Ton L eAP*(TA), and which is still denoted by P*, is a topological isomorphism fro~ the Frechet Acmodule O{l)(~*;Ac) onto the Frechet Acmodule O(r)(n*;Ac). 33.14 Theorem Let f be left monogenic in a neighbourhood of the origin such ..... '\,+ that its multiple Taylor series converges absolutely in either rr(R) or rr(R) or still IT'ii(R). Then there exists T E OCl)(A(R),Ac), respectively T E 0(1 ) (A(R);Ac) or T E o( 1)(B*(R);Ac), such that P( T)
= f.
Proof Let f be left monogenic such that its Taylor series expansion about the origin converges absolutely in n(R). Then f(t) = f(x)IR~+ 1 admits an absolutely converging multiple Taylor series expansion in m
rr(R)JR:+ 1
=
{tERm :
L
Rj Jxj I
< 1}.
j =1
Hence its complex extension f(!) is holomorphic in A(R) and so by the con* )(A(R);A  . . . c ) such that siderations made in 33.13 there exists T E O(l 292
p*(T) (!)
f(t)
or
)
f(!)
whence .....
1
f(u)
. . . =

in n(R) 1R:+ 1 • As the left monogenic extensions of analytic functions in open subsets of Rm+ 1 are unique (see Section 14), it follows that in n(R) .....
f(u) =
c
..... "'n(R) ..... 33.15 Proposition The domains n(R), and n*(R) are optimal with respect to absolute conve1~ence.
.....
Proof We already observed in 33.6 that a domain such as n(R), which is given by the inequality ..... m !x 0 1 IRI + l: Rj 1xj1 < 1, Rj > 0, j=1 is
with respect to absolute convergence. Next n(R) is an intersection of optimal domains with respect to absolute convergence; so it is optimal itself. Finally as n*(R) can be written as opti~al
'\,+
n*(R) =
.....
n rr(R) ..... IRI =R
it follows that rr*(R) is optimal with respect to absolute convergence too. 293
~
+ The above characterizations of P(O(l)(i).*;Ac)) for ;.2* ~qual to II(R), A(R) or B*(O,R), lead to an optimal version of the CauchyKowalewski ex tension theorem stated in 14.2. 33~16
Theorem Let f(!) be an Acvalued holomorphic function in 1\(R) (respectively A(R), B*(O,R)) and let f(;) be its Acvalued analytic restriction to Rm. Then f(;> admits a unique left monogenic extension f*(x> to rr(R> (respectively rr(R), rr*(R 1)) which is given by 2k+1 f*(x) = (1)k D[ xo A~f(;}]. (2k+1)! k=O
I
The multiple Taylor series expansion about the or1g1n of f* converges absolutely in rr(R) (respectively rr(R), rr*(R 1>> which is optimal with respect to absolute convergence. Proof The uniqueness of the CauchyKowalewski extension was already proved in Section 14. + + c . As f(r;} E O(r)(A(R)!A+), 1n view of t4artineau's result mentioned in 33.13, there exists T E 0( 1 )tA~R);Ac) such that P*(T) = f, or more explicitly 1
++> 1
294
33.17
In the particular case where m = 2 the following pictures may be drawn:
33.17.1
+
c
+
For f € O(!I(R);A ), f(x) E rv+
c
+
a({x
m
E 1R: Jx.l J
<
R., J
.
J
= 1,2};Ac) and
f* E M(r)(n(R);A ):
33.17.2
rv+
c
+
For f E 0(/\(R);A ), f(x) E +
+
a({x
E lR
m
c
f* E M(r)(n(R);A ):
295
33.17.3
For f E O(B*(O,R);Ac). f(;) E a({~ E Rm
f* E M(r)(rr*(R 1);Ac): Xo
296
Notes to Chapter 5 The Fourier transform F for functions can be extended to distributions in two ways. On the one hand, as Schwartz did (see [Sch]), a space of testfunctions vaster than V(mm), namely s(mm), is introduced such that the Fourier transform becomes a topological isomorphism on it~ which may then be carried over to the dual space S'(mm) by putting
~
297
with the space of C solutions of (1). In [Mor] also an interpretation is given of B(Sm 1) and this by using a theory of analytic functionals on m m {z = (z 1 , ••• ,zm) E 0: : I zj = 1}. j=1 . * (Sm1 ;A ) , FT (+) + Sm1 , Now 1f we put for each T E E(l) x =
_a
= ~+1, a refinement is obtained of t4orimoto's result for the case E'(Sm 1). Observe too that when m = 2, the entire functionals in 0:'{0} introduced by "' . Helgason in [Hel] represent the space B(S 1 ). The theory of analytic functionals and their FourierBorel transforms is of fundamental importance in solving existence and approximation problems for the equation P(Dz)u = h, h being holomorphic in an open convex subset ~ of O:m and P(Dz) being a differential polynomial in az , ••• ,az • The FourierBorel transform itself was 1 m introduced by Martineau in his basic work [Mar1]. To fix the ideas, let T be an analytic functional in 0: and call FT(z) =
298
F(u,O)
F(O,v)
and so also by o+T
eu/z>
a_T
...!_
u. z
1 F T(1 D) 1 z. •
=z =
T F1 (u)(O ,z).
Putting a = a+  a_, this atransform leads to a generalization of Helgason's enti1·e functionals already mentioned, and thus to a representation of B(Sm 1) in the higher dimensional case. Note that in [Mor2] Morimoto also extended the FourierBorel transform to analytic functionals having non compact carrier, while in [Mor3] analytic functionals on the Lie sphere are considered. In Section 32 a generalized version of the atransform cited above is introduced and this by using the kernel H(u,y) defined in Section 15.8.3. In this connection also the paper [So4] of Sommen should be mentioned. If T E O'(B(0,1)) and PT(z) =
0
299
Bibliography [Ca3] R.D. Carmichael, Distributional boundary values of functions analytic in tubular radial domains, Indiana Univ. Math. J. 20 (1971) 843853. L. Ehrenpreis, Analytic functions and the Fourier transform of dis[E4] tributions, I, Ann. of Math. 63 (1956) 129159; II, Trans. Amer. Math. Soc. 89 (1958) 450483. [GS] I.M. Gel 'fand and G.E. Shilov, Generalized functions, II, Spaces of fundamental and generalized functions (Academic Press, New York, 1968). [HKMO] M. Hashizume, A. Kowata, K. r4inemura, K. Okamoto, An integral representation of the Laplacian on Euclidean space, Hiroshima Math. J. 2 (1972) 535545. [Hay] W.K. Hayman, Power series expansions for harmonic functions, Bull. London Math. Soc. 2 (1970) 152158. S. Helgason, Eigenspaces of the Laplacian : Integral representations [Hel] and irreducibility, J. Functional Analysis 17 (1974) 328353. [Mar1] A. l~artineau, Sur les fonctionnelles analytiques et la transformation de FourierBorel, J. d'Anal. Math. 11 (1963) 1164. [Mar2] , Equations differentielles d'ordre infini, Bull. Soc. Math. France 95 (1967) 109154. [Mor] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations, to appear in a volume of the Banach Center Publications. [Mor1]     , A generalization of the FourierBorel transformation for the analytic functionals with nonconvex carrier, Tokyo J. Math 2 (1979) 301322. [Mor2] , Analytic functionals with noncompact carrier, Tokyo J. Math. 1 (1978) 77103. [r4or3] , Analytic functional s on the Lie sphere, Tokyo J. Math. 3 (1980) 135. [Sch] L. Schwartz, Theorie des distributions (Hermann, Paris, 1966). [Sic] J. Siciak, Holomorphic continuation of harmonic functions, Ann. Pol. Math. 29 (1974) 6773. [So3] F. Sommen, Hypercomplex Fourier and Laplace Transforms II, to appear. [So4] ,A generalized version of the FourierBorel transform, to appear in Bull. Soc. Roy. Sci de Liege. 300
[So5] F. Sommen, Some connections between Clifford analysis and complex analysis, to appear in Complex Variables : Theory and Application. [StWi] R.F. Streater and A.S. Wightman, PCT, Spin and Statistics, and All That (Benjamin, New York, 1964).
301
References
[1]
W. Ambrose,
Structure theorems for a special class of Banach algebras,
Trans. Amer. Math. Soc. 57 (1945) 364386.
[2]
F.F. Bonsall and A.W. Goldie, Algebras which represent their linear functionals, Proc. Cambridge Philos. Soc. 49 (1953) 114. [3] H.G. Garnir, Problemes aux limites pour les ~quations aux d~riv~es partielles de la physique II (Cours de troisi~me cycle de 1 'Univ. de Li~ge, 197475). [4] H.G. Garnir, M. De Wilde and J. Schmets, Analyse fonctionnelle, tome I (Birkh~user Verlag, Basel, 1968). [5] , Analyse fonctionnelle, tome II (Birkh~user Verlag, Basel, 1972). [6] ,Analyse fonctionnelle, tome III (Birkh~user Verlag, Basel, 1973). [7] M. Hashizume, K. Minemura and K. Okamoto, Harmonic functions and Hermitian hyperbolic spaces, Hiroshima Math. J. 3 (1973) 81108. [8] W.K. Hayman, Power series expansions for harmonic functions, Bull. London Math. Soc. 2(1970) 152158. [9] S. Helgason, Eigenspaces of the Laplacian: integral representations and irreducibility, J. Funct. Anal. 17 (1974) 328353. [10] L. H~rmander, Linear Partial Differential Operators {SpringerVerlag, Berlin, 1963). [11] O.D. Kellogg, Foundations of Potential Theory (SpringerVerlag, Berlin, 1929). [12] G. K~the, Dualit~t in der Funktionentheorie, J. Reine und Angew. Math. 191 (1953) 3049. [13] A. Martineau, Sur les fonctionnelles analytiques et la transformation de FourierBorel, J. d'Anal. Math. 11 (1963) 1164. [14] M. Morimoto, Analytic functionals on the sphere and their FourierBorel transformations (to appear in a volume of the Banach Center Publications). [15] C. MUller, Spherical Harmonics. Lecture Notes in Mathematics 17 {SpringerVerlag, Berlin, 1966). 302
[16] L. Schwartz, Theorie des distributions (Hermann, Paris, 1966). [17] R.T. Seeley, Spherical harmonics, A.M.;;. f.Janthly 73, II 4 (1966) 115121. [18] , Eigenfunction expansions of analytic functions, P1•oc. AmeP. Math. Soc. 21 (1969) 734738. [19] H.G. Tillmann, DualiUt in der Potentialtheorie, Po1•t. Math. 13 (1954) 5586. [20] F. Tr~ves, Linear partial differential equations with constant coefficients (Gordon and Breach, New York. 1966). [21] Basic linear partial differential equations (Academic Press, New York, 1975). [22] A. Paulik, Optimale Approximation linearer Funktionale auf analytischen und harmonischen Funktionen (Dissertation, Universit8t G6ttingen, 1975). [23] N. Vilenkin, Special functions and the theory of group representations (Translations of Math. Monographs 22, A.M.S., Providence, 1968).
303
Author index
The roman numbers refer to the Notes at the end of the respective chapters W. Ambrose 1.12 N. Aronszajn III M.F. Atiyah I H. Behnke II, III E.J. Beltrami IV S. Bergmann III F.F. Bonsall 1.12 R. Bosshard II R. Bott I F.F. Brackx III R. Braun III H. Bremermann IV R.D. Carmichael IV, V P. Charpentier II P. Chauveheid III C. Chevalley I W.K. Clifford I A. Crumeyrolle I C.A. Deavours II R. Delanghe II, III M. De Wilde 2.4, 3.2, 17.11, 22.4, I C. Dias III A. Dinghas II T. Dwyer III J.D. Edmonds, jr. L. Ehrenpreis III, v. R. Fueter 304
I, I I
H.G. Garnir 2.4, 3.2, 8.3, 17.11, 22.4, 23.3, I P.M. Gauthier III I .M. Gel' fand V J.W. Gibbs R.P. Gilbert III A.W. Goldie 1.12 B. Goldschmidt II A. Grothendieck III F. GUrsey I, II K. Habetha I I H. Haefel i II P.R. Halmos W.R. Hamilton M. Hashizume 28.9, IV, v W.K. Hayman 33. 1 ' v o. Heaviside I s. Helgason 32.22, IV, v G. Henkin II D. Hestenes I , I I E. Hewitt IV G. Hile I I , III E. Hille I I I L. H6rmander 17.4.2, III V. Iftimie K. Imaeda
II
M.A. Jafarizadeh M. Jimbo I
Yu. N. Kafiev I M. Kashiwara IV T. Kawai IV O.D. Kellogg 11.5.3 H. Komatsu IV P.P. Konder IV J. K6rner IV G. K6the 22.8, I I I , IV A. Kowata IV, V M. P. S. P.
z.
Langenbruch IV Lelong III L.ojasiewicz III Lounesto II, IV Luszczki IV
B. Malgrange III A. Martineau 33.13, III, IV, V J . C. Max we 11 I R. Meise IV H. Meschkowski III K. Minemura 28.9, IV, V T. Miwa I P. Montel III M. Morimoto 28.9, IV, V c. M!Jller 11.1.1, 11.1.3, 11.4.2 W. Nef
II
K. Okamoto 28.9, IV, V A. Paulik 24.16, 24.17 F. Pelligrino III H.J. Petzsche III, IV I.R. Porteous E. Ramirez II M. Riesz I L.A. Rubel III C. Runge III
J. Ryan
I, II
M. Sa to I, IV P.P. Saworotnow R. Schapiro I M. Schiffer III J. Schmets 2.4, 3.2, 17.11, 22.4, 23.3, I L. Schwartz 4.7, 5.3, 17.6, I, V R.T. Seeley 11.1.3, 28.9, IV G.E. Shi lov V J. Siciak V J.S. e Silva III M. Snyder I F. Sommen IV, V
v.
sou~ek
E.M. Stein II K. S~ein III R.F. Streater V K. Stromberg IV A. Sudberry II B.A. Taylor III P. Thullen II, III H.G. Tillmann 22.1, III, IV F. Tr~ves 19.1, 28.4 H. Triebel IV H.C. Tze I, II N. Vilenkin 11.1.3 V.S. Vladimirov IV D. Vogt IV G. Weiss II A.S. Wightman V E.B. Wilson I M.R. Wohlers IV Zalcman III Z. Zielesny IV L.
305
Adjoint function 13.12 algebra Clifford algebra 1.3, 33.1 Dirac algebra 1.7 H*algebra 1.12 Pauli algebra 1.7 quaternion algebra 1.7 trace algebra 1.12 analytic function 8.4 analytic functional 22.15, 28.6 AronszajnEergmann Theorem 24.3 asymptotic extension 17.2
CauchyRiemann operator 8.3 CauchySchwarz inequality 7.5 convolution 4.6
Bergmann kernel 24.1, 29.7 Bessel's inequality 7.4 boundary value 25.5, 26.5 boundary value mapping 27.9 bounded set 2.7
Functional Alinear functional 2.3 bounded Alinear functional 2.6 fundamental solution 8.8 Fourier transform 30.2, 30.6, 30.8, 30.16, 31.11, 31.13 FourierBorel transform 32.16
Canonical extension 26.5 Cauchy's Integral Formula 9.6, 9.8 Cauchy kernel 8.8 Cauchy's Theorem 9.4 Cauchy transform of an Ameasure 17.9 of a distribution with compact support 27.6 of an analytic functional 28.6 CauchyKowalewski extension 14.5, 28.5, 33.16 CauchyKowalewski product 14.11 CauchyKowalewski Theorem 14.2, 28.4 306
Distributions 4.3, 5.2, 5.6, 6.2 dual algebraic dual 2.3 dual module 2.6 dual of M(r)(~;A) 22 bidual of M(r)(Q;A) 23 Entire function 12.3.5 exponential function 15.4, 15.5
GelfandShilov Zspace
30.19
HahnBanach Theorem 2.11 harmonic harmonic function 8.4, 16.1 harmonic testfield 25.2 spherical harmonic 11.1.1 HartogsRosenthal Theorem 18.17 Helgason's entire functiorials 32.22 hermitian conjugation 1.9 Hilbert Amodule 7.2 Indicatrix of
Fantappi~
22.2
inductive limit 2.15 inner product on A 1.11 on L 2 (r)(H;A;~) 7.14 on ML 2 (r)(~;A) 24.5 on ML 2 (r)(aBR;A) 24.13 on a(r)(sm;A) 28.10 on L2 (r)(sm;A) 29.2 inversion 1.8 involution 1.10 Koebe's Theorem
10.5
Laplace operator 8.4, 8.7 Laplace transform 31.2, 31.8, 31.13 LaplaceBeltrami operator 8.7 Laurent integral 32.22 Laurent series 12.2 Legendre polynomial 11.1.2 Liouville's Theorem 12.3.11 Maximum Modulus Theorem 9.9 Mean Value Theorem 9.7 measure 3.2 meromorphic function 12.3.3 MittagLeffler's Theorem 12.3.6, 21.5 module unitary left Amodule 2.2 Hilbert Amodule 7.2 monogenic monogenic function 8.3 (surface) inner spherical monogenic 11.2.1, 16.4 (surface) outer spherical monogenic 12.1.6, 12.1.9 monogenic point 12.3.1 Montel Theorem 16.3 Morera's Theorem 10.4
Norm on A 1.11 normal normal convergence 11.3.1 x0 normal 14.3 xinormal 20.2 normal w.r.t. the origin 20.2 Optimal w.r.t. absolute convergence
33.6
pvector
1.4 Painlev~ Theorem 10.6, 28.3 PaleyWienerSchwartz Theorem 30.19 Poincar~ Theorem 21.3 primitive 20.6 projection operator 7.13, 29.5 Radon transform 33.10 reflection operator 30.7 regular at infinity 17.2, 17.3 reproducing kernel 24.2 residue 12.4.1 Residue Theorem 12.4.3 Riesz representation theorem 3.3, 7.6 Rodriguez Formula 11.1.2 Runge Approximation Theorem 18.4, 18.7, 18.9, 18.12, 18.20 Scalar part 1.11 singular point 12.3 slow growth (functions) 5.5 spherical CauchyRiemann operator 8.7 spherical transform 13.3, 13.6, 13.9 strong topology 2.7 system of seminorms 2.4, 2.5, 2.15 Szegtl kernel 24.17, 29.6 Taylor integral 32.22 Taylor series 11.3.5, 11.5 tensor product of functions and distributions 4.6 307
trace
1.12
Weak slow growth 5.5 weak topology 2.7 Weierstrass Theorem 9.11
308