By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)
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Extra info for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982
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. in in D U Co (d U C (d by 1 D U Co (e) U C (d. 1 if in D U Co (E) U C, (E),' NOw, let us extend av/ay~ exists, otherwise. And now, IVul ELP(D) every compact set implies, by (5), that FeD, IVvl ELP(D).
Hyperbolic simply connected domain in ~ and let More generally, let ~:Q+6 Q be an be a Riemann mapping. 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 ), 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.
Another generalization may be obtained if we get rid of the condition Eo ,E CO. 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.