LECTURES ON THE EDGE-OF-THE-WEDGE THEOREM

(b) If

Rn

b

for some fixed b and all ae = ÷ + iy with y € K, then the same inequality holds for

all y€L·

Proof. Fix € 0. Define a measure ì on Rn by Üì = |ì | cfc. Then y — e is a

continuous map of Ê into L (ì) . The compactness of X implies that there are fin-

itely many points yx, · · · , y in Ê such that to every ã Å Ê corresponds at least

one y. with ||e - e || e, the norm being that of L1^)* Since each e is in Ll(y),

there is a compact ball Â in Rn such that

J e

#

ø £ for 1 ß m,

where ä is the complement of ß. Hence

(1) J e ø 2e for every y € K.

Â' ¾

Let Õ be the set of all y € Rn for which the inequality (1) holds. Then Õ is convex.

To see this, suppose y', y " € Y, 0 ó 1, y = ( l — ó) y' + ay" . Since

Holder's inequality (with (1 — ó) and ó~é as conjugate exponents) shows that

y 6 Y. Thus (1) holds for every ã G K.

If now y and y ' are in K, (1) implies that

e - e ,

I y y

J ^+ /l«

y

-v

1

*·

Since ì(â ) « and e , —- e uniformly on ß as y ' —» y, it follows that |je

#

— e |j — 0

as y'—*y. This proves (a).

To prove (b), observe first that (a) implies that the function Uy defined on Rn + iK

by

(2) l/U) = J" Â(ß)â

é,-*þ

R"

is continuous and bounded. Let Õ be the set of all y € X for which

(3) sup | l / ( * + i y ) | b.

X

Since Ê C Õ by assumption, it is enough to show that Õ is convex. So suppose