Cauchy projectors on non-smooth and non-rectifiable curves

Abstract

© Petrozavodsk State University, 2019. Let f(t) be defined on a closed Jordan curve Γ that divides the complex plane on two domains D+, D-, 1 2 D-. Assume that it is representable as a difference f(t) = F+(t)-F-(t), t 2 Γ, where F±(t) are limits of a holomorphic in C \ Γ function F(z) for D± ∋ z → t ∈ Γ, F(∞) = 0. The mappings f ↦ F± are called Cauchy projectors. Let Hυ(Γ) be the space of functions satisfying on Γ the Hölder condition with exponent υ ∈ (0,1]: It is well known that on any smooth (or piecewise-smooth) curve Γ the Cauchy projectors map Hυ(Γ) onto itself for any υ ∈ (0, 1), but for essentially non-smooth curves this proposition is not valid. We will show that even for non-rectifiable curves the Cauchy projectors continuously map the intersection of all spaces Hυ(Γ), 0 < υ < 1 (considered as countably-normed Frechet space) onto itself.