6

WALTER RUDIN

not seem to exist in any of the Standard texts. In fact, the passage from Á to Â is not

really any easier when ç = 1 than when ç 1.

3. Tubes with a common edge

Any set of the form Rn + iV, where V is an open subset of Rn, is called an

open tube in Cn. For instance, when ç = I the region bounded by any pair of lines

parallel to the real axis is a tube.

The convex hüll of a set Ì in Rn is, by definition, the smallest eonvex subset

of Rn that contains M. If Ì is open, so is its convex hüll, as can be seen very

easily.

We now present an analogue of Theorem Á which is more special in one sense

but is more general in another. The special feature is that the edge Å is now Rn; also

(but this is les s important) a growth condition is imposed on /. The added generality

is that the sets F and — V that go into the hypothesis of Theorem Á are replaced by

almost arbitrary open sets Vl and V2- In particular, they have nothing to do with

cones.

Theorem. Suppose V. and F . are bounded connected open sets in Rn9 both of

which have the origin as á boundary point* Put V = V. U V'. Assume

(i) / is holomorphic in Rn + iV,

(ii) / is continuous on Rn + iV,

(iii) there is á constant Á oo such that

| / U + iy)\ expM(%2

+

. . .

+

*2)j

in Rn + iV.

Then f extends to á holomorphic function in Rn + iW, where*W is the convex

hüll of F. This extension is continuous on Rn + iW.

The given / is holomorphic in each of the tubes Rn + iVt and Rn + iV2 Note

that Rn + iW is the convex hüll of the union of these tubes.

The proof uses certain convexity arguments. It seems best to State these sep-

arately as a lemma. We shall use the scalar product notation

÷.*=£vf

z'ß=Óí/

for t € Rn, ÷ € Rn, ae Å Cn. Also, for y € Rn, e will be the function defined on Rn

by e (t ) = e"y't. The Symbols L , L2 denote the usual Spaces of integrable functions,

relative to Lebesgue measure on Rn.

Lemma. Suppose u € L , and y· —» ue is á continuous map of some compact set

KCRn into L1. Then (a) and (b) are true:

(a) y —* ue is á continuous map of Ê into L , where Ê is the convex hüll of K.