By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)

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Additional info for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982

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In the particular cases E1 n 0 = ~ or Eo n 0 = ~, 1 we assume, obviously that cap (E ,E ,D) = 0 because, in the first p 0 1 case, the function u defined by ul = 0 and uI = is admisDUEo E1 sible, while in the other case, the function u such that uI and DUE1 ul E = 0 id adnissible too and, in the both cases, IVul ID = o. o Finally let us mention a generalization by supposing that D is only open. In this case, we precise that, if there are components with u Do n E1 =~, such that the restriction such that 01 Do it is enough to consider only admissible functions n E0 = I"lI YJ, ul = 0 and if there are components D1 Do then it is enough to consider only admissible = 1.

To see x,x' ED - (Co u C,) ; and, arguing as in the preceding proposition, i t follows that lw(x' ,e) -w(x,e:) :ii 1 , 1---'---wn[eo (x')] n I ! Wn[EO (x)] n B[ X'-X,EO (x') ]nB[Eo (x)] + ! v(x+y)dIn Wn[EO (x,)]n B[X'-X,EO (x')]-B[E:15 (x)] + 1 ! v(x+y)dIn Wn[EO(X)]n B[EO(X) ]-B[x'-X,EO(X')] , + + 1 WnlEO(X') 1 B(X'-X,EO(X') v(x+y)dm I ! dIn ]-B[EO(X) 1 ! Wn(EO(X}]n B[EO(X) J-B[x'-X,EO(X') ] dm, which, on account of Radon-Nikodym's theorem, becomes arbitrarily small for Ix - x' I small enough. e.

Then z,~dl, is the Szeg8 kernel of Q. ,·) is sesqui-holomorphic on~xO. Moreover, as is well-known, the reproducing space of k ("') is the Hardy-Szego O space H (Q} (see also [2]), and hence kg ("') is independent of the particular 2 choice of the Riemann mapping ,. One may also generalize the above concept to any hyperbolic domain (or a Riemann surface) g by using a holomorphic cover map ~:6+n 21 Positive Definiteness and Holornorphy instead of a Riemann mapping. An if We omit the details. ·} e:(D; B(u:w» Cu.

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