By Morisuke Hasumi (auth.)
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Additional resources for Hardy Classes on Infinitely Connected Riemann Surfaces
Sample text
Be a m u l t i p l i c a t i v e such that suffix functions of m o d u l u s on an i n n e r is b o u n d e d . of c o m m o n R. 's we to fQ called IflI, the are the UQ = principal determined inner and f, r e s p e c t i v e l y . we w i l l then v = following (resp. m. occurs. (resp. u - v E I(R). the w+ un Thus we have observations. we w [(-m) v ( n A u),]. m, is. Since is a l w a y s and shows that = U n + I. sequence, so these inner be a n y log u functions IfQl and It f o l l o w s ( n + i) hand, Then, we have preceding u The meromorphie also It is c a l l e d wherever UQ R SP' u I = exp(Prl(lOg factor w £ i.
Then n n we h a v e (kb)R(a) = lim I ( k b ) R ( z ) d ~ n ( Z ) n -~ and t h e r e f o r e the f u n c t i o n b ÷ (kb)R(a)s = nlim f -~ is m e a s u r a b l e . Hence, 47 n÷ ~ ~ A [lim I ( k b ) ~ ( z ) d ~ n ( Z ) ] dB (b) : IA "n÷~ : I (kb)~(a)dz(b)' A as was to be proved. 3D. [] We are n o w in a p o s i t i o n to p r o v e the m a i n r e s u l t of this section. Theorem. cation boundary of R The M a r t i n in the A Let f The h a r m o n i c is g i v e n by be a n y r e a l - v a l u e d it to a c o n t i n u o u s f.
E. I(R) be the point l(z), once subspaee RE as an o p e n Ff(~(z)) to is a h o m e o m o r p h i s m R. 0 E R, w h i c h 6A). variables, R. the with extension take R of Since the Let subspace of the that +~]. with be fixed. coordinate in fact, I. includes the Let R the m a p As a c l o s e d space and, as function with R*. where the pole in The QM the b (and w i t h If b E ~, t h e n the on R. We that function b see also on R* x R now z ÷ k(b,z), k b. is a c o n t i n u o u s function variable function at can be ex- extended origin function the with func- values +~].