Cauchy projectors on non-smooth and non-rectifiable curves

Abstract

Let f (t) be defined on a closed Jordan curve Γ that divides the complex plane on two domains D + , D − , ∞ ∈ D − . Assume that it is representable as a difference f (t) = F + (t) − F − (t), t ∈ Γ, where F ± (t) are limits of a holomorphic in C \ Γ function F (z) for D ± 3 z → t ∈ Γ, F (∞) = 0. The mappings f → F ± are called Cauchy projectors. Let H ν (Γ) be the space of functions satisfying on Γ the Hölder condition with exponent ν ∈ (0,1]. It is well known that on any smooth (or piecewise-smooth) curve Γ the Cauchy projectors map H ν (Γ) onto itself for any ν ∈ (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 H ν (Γ), 0 < ν < 1 (considered as countably-normed Frechet space) onto itself.

Keywords

• Cauchy projectors
• non-smooth curves
• non-rectifiable curves