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Cp''P = 0, ] sign being valid because is arbitrary, we conclude all the A-periods that, of for every fixed $p,,p p in 20 the region R N R~, fA ~q~q,, (13) 'q = 0. J A similar computation shows that Ip' B. ] ' B. ] As we shall see below in IIC, an application 'P" of Th. , g'. @p, : -2~i @j B. 'P p' 3 Thus we have arrived at the following: fB. , g' We finally set ~(p,q) Since q" and all everywhere on R on +i Cauehy kernel, ~(p,q) aj at the point we claim that, q. of the differential (13) follows = B.
But t h e is p o s s i b l e details for multiplieative look obvious and are meromorphic thus omitted. NOTES Gunning dles [13] on R i e m a n n tions. The Some Chapter discussion basic facts theorem E47] contains with we r e f e r for c o h o m o l o g y Weyl in 3A is a d a p t e d and the reader a discussion a proof source See a l s o on v e c t o r 2 of Constantinescu tion together is o u r m a i n surfaces. 4D. groups [CC]. and for m u l t i p l i c a t i v e Kusunoki of harmonic to P h e l p s ' on h a r m o n i c of T h e o r e m from lattices Cornea [68] lecture note with func- [42].
Let using to the m e t r i c we h a v e u(z) for respect u E HP'(R). the a b o v e Writing result, we u = u _ -u + with u , u- E H P ( R ) and see t h a t u : J k b d~(b) A1 for some finite statement mined (b) Borel above uniquely by u. the canonical measure 2C. M ( A I) Let It is a l i n e a r £ ~ for subset the E measure shows space This for ~, ~ E M ( A I) of A I. correspondence on A I. is d e n o t e d ~ set of f i n i t e a n d has if a n d o n l y if gives a linear is o b v i o u s l y order-preserving.