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

By Morisuke Hasumi (auth.)

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**Sample text**

The first objective of this chapter is to give a precise d e f i n i t i o n to such functions. fact, we will define t h e m in terms of two equivalent notions: bundles and characters of the f u n d a m e n t a l group. that, on a compact b o r d e r e d Riemann surface, n o n v a n i s h i n g bounded h o l o m o r p h i c observe the order structure sections. In line We shall then show every line bundle admits The second purpose is to in the space of h a r m o n i c functions, leading to the so-called i n n e r - o u t e r f a c t o r i z a t i o n of m u l t i p l i e a t i v e analytic functions.

With kb(0) HP(0) are combine on two every there function belong to QM" k(b',a) points exists in CK of and We that set kb A = j(A). j R*. ~R n observation kb and since the m a p converging that the m a p £ Since b ~ b' is a c o m p a c t If, and into let that is HP(0) b, b' of ~ f(b'). above 4A- function A ÷ k(b,z) A such shown. kb(0) Ch. topology A, the cannot [] both of f(b) a @ R 4. harmonic (b,z) of the to be n ÷ ~, we of In fact, boundary a point HP(0). with that Since for w h i c h on the exists as boundary j: b ÷ k b definition K A with HP(0).

Of V" = ~(i"(v)), covering i'(v) Now let y" the curve = ~'(i"(v)) c. , m. Then, B(y") = {Vi(1) .... ,Vi(m)} (resp. ~'(y") : { V i , ( 1 ) , . . , V i , ( m ) } ) is a chain of V (resp. V') c o v e r i n g the curve. Then, u s i n g the convention i(m+l) = i(1) etc. as before, we have m F(e;{E~,j,}) = F(c;~'(T");IE~,j,}) = ]-m v=l E~'(v)i'(v+l) m ]-~ v:l : (@i,,(~)~i(~)i(~+l)@i,,(~+l~ I) m : T~ ~i(~)i(v+l) v=l : F(e;B(T");{~ij}) This m e a n s {~ij}. , c k' issuing for e v e r y chain. that ~2 ~ {~ij}, using is d e n o t e d in any in F0(R).