LECTURES ONTHE-EDGE-OF-THE-WEDGE THEOREM

9

If y € F, the Cauchy theorem (together with the fact that g vanishes at infinity, by

(iii ) shows that each y, can be replaced by any sufficiently close y

k

without chang-

ing (7). It föllows, for each t 6 Rn, that ey%tG (t) i s constant in each of the connect-

ed open sets V^ and VT

Also, it föllows from (iii' ) and the uniform continuity of g that y — g is a con-

tinuous map of V into L 1 , so that G (t) is (for each t 6 Rn) a continuous function

of ã on F. In conjuction with the preceding paragraph it now föllows that eymtG it)

i s independent of y, on F. Thus (6) is proved.

The Plancherel theorem [lO] implies that each G is in L 2 , but our hypotheses

do not imply that G is in L . To remedy this, let ø = ö be a nonnegative function

in L which vanishes outside a ball of radius r with center at the origin of Rnf and

whose integral is 1. Define

(8) h{z) = J ø(î)

6

ßae - î)Üî (z€Rn + if).

Then h = ø * g

f

where * indicates convolution. If Ç and Ø are the Fourier trans-

forms of h and ø, then Ç = Ø(ú . This has two consequences: first, (6) becomes

(9) HU) = e-*'*H0U) Uer,

r

e?) ,

second, y —• g is a continuous map of V into L , by (iii ' ) and the uniform contin-

uity of g; by the Plancherel theorem, the same is true of y — G ; since Ø 6 £ 2 , the

Schwarz inequality now implies that y —- » Ç is á continuous map of V into Ll.

In particular, the Fourier Inversion formula can be applied to h . In view of (9),

the result i s

(10) Ä(z)= f Ç0{ß)â1ae**Üß {z£Rn + iV).

We can now apply our lemma, with H' in place of ut V in place of K, and con-

clude from part (a) that y —* Ç is a continuous map of W into Ll (where, we recall,

W i s the convex hüll of F), which implies that formula (10) defines h a s a continuous

function on Rn + iW which is holomorphic in Rn + iW. From part (b) of the lemma we

conclude that the supremum of \h\ over Rn + iW is the same as that over Rn + iV.

Let us now write h in place of A, to indicate the dependence of h on ö9 hence

on r. As r —* 0, we see from (8) and the uniform continuity of g that h —* g uni-

formly on Rn + iV. The conclusion of the preceding paragraph (applied to the difference

of two functions h rather than to h) shows that \h } is a uniform Cauchy sequence on

Rn + i E The limit of this sequence furnishes the desired extehsion of g, and the

proof of the theorem is complete,

The set W in the conclusion of the theorem cannot be replaced by any larger one.