## Download Hardy Classes on Infinitely Connected Riemann Surfaces by Morisuke Hasumi (auth.) PDF

By Morisuke Hasumi (auth.)

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**Example text**

SP' some is p o s i t i v e . respectively, singularities, singularities of in = I(R) ± = {I} ±i. bands. Q(R), element of then Q(R) consequence in summand inner to be t h e lattice of the of o n l y and called and I(R) We may each . yl for Y' in t h e = (-u) v 0, ± u subspace subset u 2 E Y±±, sum of these For any to e v e r y u u, r e s p e c t i v e l y . - u lul =< lvl Y' and of we d e f i n e sum decomposition I(R) direct same of = {i} ± and the Proof. bound Y tices. 5B. and consists onto SP' and is a l i n e a r When Both orthogonal ~ 0 parts + + u- orthogonal and I(R) (resp.

A We m a y a s s u m e function that R* is a r e s o l u t i v e every real-valued is r e s o l u t i v e . at the p o i n t Proof. compactifieation sense t h a t on measure, kb(a)dx(b). continuous R*, w h i c h 0 ~ f ~ 1 continuous on with function on the support ([CC], is d e n o t e d R*. eompactifi- function p. on in AI, 140) A. We e x t e n d by the same l e t t e r For e v e r y n = i, 2,... we set A i = {b e Al: E9 : {a e R*: 1 ( i - ~) =< f(b) f(a) < ( i + 7)}, < i - i} U {a e R*: = n f(a) > i + i],= n E.

Such that Let us trivial Let the A = {log UT: J u 2 / u I S i. , is so t h a t be a set of l . a . m . m, uI with u in J; UI with u in J, t h e n the sreatest satisfies factor uniquely common the has on (ii) such uI divides in (i) inner u 0. of necessarily J. Set u0 It is s e e n This then I(R) say v0, u 0 = exp v 0. m, two p r o p e r t i e s . factor 0. upper I(R). m, ! common u0 inner J. A parallel functions and (i) but n o t of constant A is a n o n p o s i t i v e is an i n n e r is d e t e r m i n e d by the set on description R.