By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)
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Additional resources for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982
Arguing as in the preceding lemma, we obtain COROLLARY 2. --From each of the above definitions for the p-capacity, we obtain the corresponding definition for the conformal capacity if we take and suppose that D p =n is contained in a fixed ball. 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.
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. An if We omit the details. ·} e:(D; B(u:w» Cu. w)e:UXW. )u,u)Ue:H(D) U=W, for every ue:U.
E. e. in I ;S; p(x) Firstly, we observe that ~. e. -+O dS u(x+tes}-u(x) t is the directional derivative of u(x+te ):;; JpdH s , , y x + te ' s = y(Eo'X)' Clearly, + JpdH, Y where u. (x) + J pdH' . e. in since ~, p 1 ;:;;lim t+O i JpdH At 1 = p(x) is supposed to be continuous in ~ and the points of continuity are Lebesgue points implying the last part of the preceding relation, which, by (14), yields (13), as desired. Now, we remark that h = inf u(x) E1 tion ~t) u*(x) for u(x) > h, for 0:0 u(x) :0 h, ~1.