Bounded harmonic functions and the Dirichlet problem on the Shilov boundary of $H^\infty(W)$
Cho-ichiro Matsuoka · Kyoto journal of mathematics · 1982
e a n d le t 11 -(W ) b e th e B a n a c h algebra of bounded analytic functions on W endowed with the uniform norm .The m axim al ideal space and the C hoquet boundary o f 11"(W) will be denoted by all(W ) and aew , respectively.K will s ta n d fo r a s e t o f continuous linear forms on H°°(W ) w i t h L II = L ( 1 ) = 1 .w e w ill, of course, consider surfaces which admit nonconstant bounded analytic functions.In this situation, we can identify 9)1(W ) with a subset o f K, i .e .t h e s e t o f a ll m ultiplicative linear forms in K .It is known that K is a weak* compact convex s e t in th e dual o f H "(W ).T h e purpose o f this paper is to investigate th e order re la tio n between th e harmonic measures o n relatively c o m p a c t su b d o m a in s o f W a n d th e representing measures supported o n th e Shilov boundary S .F o r every point p of W , we can characterize a represent , )dv is a representing measure fo r z , a n d furthermore for