A Note Concerning Rado s Theorem

161

2 . Rado's Theorem for Holomorphic Functions

21 Theorem. Let G be a domain of C" Let EaG be closed in G Let E^ czdEnG be polar Let E^adEnG be such that H^''-^{E^) = {) Let FcC be such that the compact subsets of F are polar Let f G\E^(£ be a holomorphic function such that f is locally bounded at E^ and that Cl{f, z)czF for each ze{dE\{E^uE2))nG Then f has a unique holomorphic extension to G or f is locally constant If f is not locally constant, then E is polar

Proof Suppose / is not locally constant

Set E^={dE\{E^yjE2))nG Then / is locally bounded at E^uE^ Namely for points zeE^ this is presupposed For points zeE^ this follows from the facts that then C/(/, z)czF and FciCC Therefore for each zeE^^E^ there is a neighborhood U^ m G such that/|t7_\£ is bounded Set

E2= { dE\ [j U^)nG

zeEy <jEi,

Then F2 IS closed in G Since E'^^E^. Я^" ^{E'^)^^ By [21, Lemma 3(i), p 115] or by [19, Corollary 3 6, p 51] it is sufficient to show that / has a unique holomorphic extension to G\E'2 and (for the final assertion) that E\E'2 IS polar

Since Я^"~^(Е2) = 0. ^2 IS polar Thus E^uE2 is polar and we find by [10, Theorem 5 32, p 274] a polar G^-set

where Я^, /с =1,2, , are open, such that E^\jE2^E^2 Set G* = G\E2' ^* = E\E'2, £^=£12*^^^^*^^* ^^^ £^ = ((^£*\£^)nG* Then G* is a domain (see eg [8, pp 47-48]), E'^œG'^ is closed m G*, £* is polar, Ff C1E3 and ()£*nG*=£f uEJ Moreover, / is locally bounded at c^E'^nG'^ Namely for points z'eE"^ this follows from the fact that E'l^czE^ For points z'eE"^ this follows from the facts that then

z'edEni [j U^)nG

zeEivjEi ,

and for each zeE^^E^ f\U\E is bounded If

G * =U^^ / = i

where Я,, /= 1, 2, , are compact, then

£ * = Q (гЕ*п(С"\Я;)пЯ,)

к l=\

Since E%ciE^ and Cl{f.z)c:F whenever zeE^, each С/(У, г')Е*п((С"\Я;)пЯД /с, /= 1, 2, , IS a compact subset of F and hence by assumption polar Thus

00 F,= с/(/,£*)= и С/(/лЗ£*п(С"\Н;)пН,)

к /=1