8 WALTER RTJDIN

y' € Õ, y " € Õ, 0 ó

0

1 , and y = ( l - aQ)y ' + aQy ". Fix x€ Rn, put ae ' = ÷ + iy ',

ae " z= ÷ + iy"

9

and define

(4) y(s) = i/((l - s)z' + sz"), s = a + ir, 0 ó 1 .

Since

ImLCl ~ s)z ' + 52 "] = (1 - a)y ' + ay" € X,

ã is well defined on this strip and is bounded and continuous; (2) shows that ã is

holomorphic in the interior of the strip. Also,

\ã(ßô)\ b and |y( l + ir)\ b

for all real r, since y' C Õ and y" € Õ. Á well-known consequence of the maximum

modulus theorem [10, Theorem 12.8] therefore implies that |y(^

0

)| 6. But yCcrJ =

U(x + i*y). Since ÷ was arbitrary, we have shown that y € Y, and the proof of the

lemma is complete.

Proof of the theorem. For ae € Rn + iV, define

gU ) = / (

Z

) e

X

p { - U + l ) (

2

2

+

. . .

+ z

2)}.

The desired conclusion about / will clearly follow if we prove it for g in place

of /.

Hypotheses (i) and (ii) hold for gf but (iii) is replaced by the better estimate

(iii') lg(* + *"y)| c exp{-*^ x2n\

in Rn + iV; here c oo depends on Á and V (note that V is a bounded set). It fol-

lows from (iii ') that g vanishes at infinity; in conjuction with (ii) this shows that g is

uniformly continuous on Rn + iV. Recall that g is defined on i!.n by

Another consequence of (iii') is that g € L l Ð L2 for every y 6 F.

The Fourier transform £ of g is given by

(5)

G

y

( i ) =

( i ) " /

s(*+iye~x"'«k

(t € Ä").

Rn

Our next objective is the relation

(6) Gy(t) =

e-'mtG0U) {t€Rn,y€V).

To prove (6), multiply (5) by e y - t and rewrite the right-hand side as an n-fold integral

(7) / • • • J ^ ( z i ' · · · ' zn] expl-i'£tkzk}dzl...dza

in which the Integration with respect to z. is over the line x, + iy., with y, fixed.