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.
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