LECTURES ON 1¹ Å EDGE-OF-THE-WEDGE THEOREM

3

To say that / extends to F (or that F is an extension of / ) means simply that

F(z) = f{z) for every ae in the domain of /*.

It is an important part of the theorem that Ù does not depend on /.

Here i s another way of stating the assumptions about /. Suppose / is holomorphic

in W U W~, associate to each ã Å Vi) ( — V) the function / defined on Å by the

formula

/

(x)

=

f(x + iy).

The continuity assumption on / amounts to saying that lim / (x) exists, as y — 0,

uniformly on compact subsets of E. Or / has continuous boundary values on the edge

Å and these are the same whether Å is approached from W or from W~.

It is desirable to weaken this assumption of uniform convergence, and in fact it

turns out that the same conclusion can be obtained if it is only assumed that lim /

exists (as ã — 0) in the sense of the theory of distributions. This can be said in

perfectly elementary language, without any reference to distributions.

Theorem B. Let E9 W , W~, and Ù be as in Theorem A. Suppose f is holomor-

phic in Ø U W~9 and

lim

J fix + iy)

ö(÷) dx

exists (as á complex number) for every infinitely differentiable function ö with com-

pact support in E. (Of course, y—» 0 within V\J(-V).) Then f has á holomorphic

extension F in Ù.

Theorem Â is the one that is usually called the edge-of-the-wedge theorem. Here

is a version of it, the "reflection theorem", in which the main hypothesis is put only

on the imaginary part of /.

Theorem C. Let E, W , and Ù be as in Theorem A. Suppose f=u + iv is holo-

morphic in W , where u and í are real, and

lim J v(x + iy) ö(÷) dx - 0

y- 0

E

for every infinitely differentiable function ö with compact support in E. (Now y —* 0

within V.) Then f has á holomorphic extension F in Ù. In W~, F satisfies F{z) =

/ù .

Of course 1 = (z,, · - · , º ) if ae = (z_, · · · , ae ) .

i n i n

These are the principal theorems of this paper. § 2 contains a few remarks about

the case ç = 1. In § 3 I consider a special case of Theorem A, in which Å = Rn. The

proof uses Fourier trans form techniques that are very similar to those used in the

Paley-Wiener theory. The possibility of this sort of proof was mentioned to me about

four years ago by E. M. Stein. It should perhaps be pointed out that the rest of the