By G. E. Btown
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Example text
54) the inequality If(Z)I~exp{-Cl(l-IZI)_ln_e_} InJL;I1(A) is valid, and this concludes the proof. We cite some examples of obtaining fl~(I (A) and the corresponding estimates. 1. Let the set Aw, have positive JORDAN measure fl (Aw,). Then, it is evident that n-+ 00 If, in particular, Aw, consists of m curves, then fl~(! (A) > JL (Awr) JL~11 (A) = , n Let the operator A satisfy the following conditions. 1. For any IPI,1P2,1P2 (x) ~ IPI (x) satisfying a LIPSCHITZ condition with some constant C, the operator A;! Alpl exists where Alp is the FRECHET derivative of the operator A at the point IP. Moreover, for any l/I satisfying a LIPSCHITZ condition with the constant C, the function A;! Alpll/l satisfies a LIPSCHITZ condition with the constant CI depending on C. 2. If 1P2 (x) > IPI (x) the operator A;;Alpl can be represented as A;21Alpll/l = 1 J Qlp2 [x, e, 1P2 (e) -IPI (e)J l/I (~) d~ o where the function Qlp2 (x, ~, 'YJ) is non-negative and continuous for 'Y) > 0. 5. Analytic Continuation of Harmonic Functions from a Circle 39 Theorem 1. 86) 00 L o Let us estimate the function CI~1. 86). 86) the sum to be estimated is bounded. Let the sum reach its conditional maximum over the variables Ck for Ck = Ck (k = 0, ... , (0). 87) hold. Then _ e-r 2q C2 = . r 2p _r 2q ' -2 r 2p -e . 2 P. 90), it follows that ; . -2 L. Cke o 2k ::::; . lne} I . 87) hold. 91) § 5. Analytic Continuation of Harmonic Functions from a Circle 41 from which we obtain ~ C-z 2k {In e ·In Q} L...