Metric dimensions, generalized integrations, cauchy transform, and riemann boundary-value problem on nonrectifiable arcs

Abstract

We consider a nonrectifiable Jordan arc Γ on the complex plane \( \mathbb{C} \) with endpoints a 1 and a 2. The Riemann boundary-value problem on this arc is the problem of finding a function Φ(z) holomorphic in \( \bar{\mathbb{C}} \) \ Γ satisfying the equality $$ {\varPhi^{+}}(t)=G(t){\varPhi^{-}}(t)+g(t),\,\,\,\,\,\,t\in \varGamma \backslash \left\{ {{a_1},{a_2}} \right\}, $$where Φ ± (t) are the limit values of Φ(z) at a point t ∈ Γ \ {a 1 , a 2} from the left and from the right, respectively. We introduce certain distributions with supports on nonrectifiable arc Γ that generalize the operation of weighted integration along this arc. Then we consider boundary behavior of the Cauchy transforms of these distributions, i.e., their convolutions with (2πiz)−1. As a result, we obtain a description of solutions of the Riemann boundary-value problem in terms of a new version of the metric dimension of the arc Γ, the so-called approximation dimension. It characterizes the rate of best approximation of Γ by polygonal lines.