New metric characteristics of nonrectifiable curves and their applications (Q530280)

Let \(\Gamma\) be a non-rectifiable closed Jordan curve in the complex plane dividing it into the finite domain \(D^+\) and the infinite domain \(D^-\). The author introduces the following new metric characteristics of the curve \[ {\mathfrak m}^{+}(\Gamma):= \sup\bigg\{p: \iint\limits_{D^+}\frac{dx\,dy}{\delta^{p}(z, \Gamma)}<\infty\bigg\}, \quad {\mathfrak m}^{-}(\Gamma):= \sup\bigg\{p: \iint\limits_{D^{-}\cap\{z: |z|<R\}}\frac{dx\,dy}{\delta^{p}(z, \Gamma)}<\infty\bigg\}, \] where \(\delta(z, \Gamma)\) stands for the distance between \(z=x+iy\) and \(\Gamma\), and \(R\) is sufficiently large. Then these characteristics are applied to the study of Riemann-Hilbert boundary value problems on non-rectifiable curves. The obtained criteria of solvability in terms of these characteristics improve the known results (see, for instance, [\textit{B. A. Kats}, Complex Var. Elliptic Equ. 59, No. 8, 1053--1069 (2014; Zbl 1314.30063)]).