AN.~. (z) ~
~ : (P~xA=f (0, l~zl...
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AN.~. (z) ~
~ : (P~xA=f (0, l~zl
z)/;~
converges to the function ~(z) locally uniformly in Cz". We shall now show that there are numbers M > 0 and r ~'~ < R(%) such that ]AN,~(z)l~
Icq<:N
i=l
From this, using again the fact that the hypersurfaces of conjuate types of the function ~(z) and the function
a r e t h e same, we f i n d
that
I~l=0 f o r a n y r > 0 and a n y zEC n
I
(z) I
r I z I+
p
z
[],
provided only that the quantities ~ = CN are sufficiently close to zero. arbitrary and r < R(%), this implies the validity of the inequality [A~.x (z) [ < 3 4 exp r*
Since g > 0 is
[z ],
where M > 0 and r* < R(~). It remains to note that for the function e~zAN,~(z)all the exponents in the exponential functions have the form ~ + 8~N, where I~I < N, and hence ~N can be chosen so that ~ - ~ N ~ Thus, as N + i e~A~.~ (z) -+ e ~
~)
in the topology of the space Exp~(~)(C~n). The lemma is completely proved. 4.
Algebra of P/D Operators with Constant Analytic Symbols
LetA(~)EG(~), i.e., A(~) is an analytic function in the domain ~ c C ~ . We assign to the function A(~) a p/d operator A(D), D = (3/8z I .... ,3/8z n) acting in the space EXps(Czn)Definition 4.1.
Let u(z)CExps(Cz") and
a (z)-- ~ e ~ z (z) be some representation of it.
where
aa(~)----O~A (~)/a]
[as usual,
(4.1)
Then
a = (~l .....
~)
are multiindices
of differentiation].
Remark 4 . 1 . S i n c e t h e f u n c t i o n u ( z ) 6 E x p ~ ( C , " ) c a n be r e p r e s e n t e d i n t h e f o r m ( 4 . 1 ) i n v a r i o u s ways t h e q u e s t i o n a r i s e s o f w h e t h e r t h e o p e r a t o r A(D) i s w e l l d e f i n e d . This question i s c o n n e c t e d w i t h t h e c h a r a c t e r o f t h e domain ft. I t t u r n s o u t t h a t i f fl i s an a r b i t r a r y open s e t , t h e n t h e v a l u e s A ( D ) u ( z ) c a n , i n f a c t , be d i f f e r e n t , generally speaking, i.e., the o p e r a t o r A(D) i s m u l t i v a l u e d . On t h e o t h e r hand i f fl i s a Runge d o m a i n , t t h e n t h e v a l u e A ( D ) u ( z ) d o e s n o t d e p e n d on t h e r e p r e s e n t a t i o n ( 4 . 1 ) and h e n c e D e f i n i t i o n 3.1 i s good. The proof of this fact is carried out in Sec. 5. In view of what has been said, everywhere below (with the exception of Sec. 8) we shall assume without special mention that ~ is a Runge domain. THEOREM 4.1.
The mapping A (D):Expa (C~n)-~ ExpQ (C~n)
(*)
is defined and continuous. fWe recall that a domain is called a Runge domain if any function analytic in this domain can be locally uniformly approximated by a sequence of polynomials or, equivalently, by a sequence of linear combinations of exponentials. 2754
Proof. Since ~(z)6ExpRc~)(Czn), there exists a number r < R(~) such thatltt(z)I~
i D~P~ (~)i ~< M~ i ~ ~,'~' cxp ~ I ~ I,
( 4.2 )
where M% > 0 is a constant. From this it follows that A (D) u (z)~ ~
(O -- kI)~ [eZ~ (z)] = ~ ex~
a= (~)D ~ a (z) ----X-- eX~*x (z),
where ~z (z)6Exp;c~)(Cfl), since by inequality (4.2)
(X) t l ~ ],z~ rill exp r l z I - - M ~ exp r j z l
I*,~ (z) J 4 M~
l [we r e c a l l
t h a t r < R(X)].
T h i s means t h a t A(D)tt(z)EExp,(C~n).
The continuity of the mapping (*) can b e established by analogous and only a little more complex arguments. The theorem is proved. From the theorem proved it follows that the collection of p/d operators A(D) with symbols A(~)CCY(~2) and domain Exp~(Cz n) form an algebra (the ring operation is composition). We denote this algebra by ~(~). Further, let s be the algebra of analytic functions in the domain ~. The following conclusion is a corollary of Theorem 4. I. Conclusion~
The algebraic isomorphism
defined by the correspondence A(D) ~-+ A(~) holds. tion A-~(~) is also analytic in ~, then
Moreover,
_!_~ oA (o) = A ( D ) o ~ A(D)
=
if together with A(~) the func-
z,
where I is the identity operator. Ex_~les. i. Let u(z) = expkz and let A(r = ~. Then A(D) exp Xz = A(~) exp Xzo 2,
Let n = !, A(~)=i/~,
be analytic in a neighborhood of the point
~2----C~\L, where L is some ray issuing from the origin.
Then
where ~z.(z)s and R(I) is the distance from the point i to the ray L. The p/d operator I/D is the inverse operator to the operator of differentiation. Thus, to each function ~(z)~Exps(Cz ~) there is assigned the function D
~ (z) = ~ e ~z ~ &
m~O
( - 1)m Dm~z (Z), Lm+~
which is the unique primitive function of u(z) which also belongs to the space Exp~(Cz"). call this primitive "natural" and write
/ D
5.
We
(z) = nat 3 ~ (z) dz.
Correctness of the Definition of a P/D Operator
Let ~ be a Runge domain. We shall show that the action of a p/d operator A(D) does not depend on the representation of a function u(z)6Exps(Cz n) in the form of a sum
u (z) =
where ~(z)~Expm~) (C2) (see
~
e~ ~ (z),
(5. i )
Definition 4. i).
2755
