On a generalization of a theorem of N. A. Davydov. (Q1432374)

Given \(\Gamma\), a simple closed rectifiable curve on the complex plane dividing this plane into \(\mathbb{D}^+\) and \(\mathbb{D}^- \ni \infty\); for any \(f\in C(\Gamma)\) the Cauchy-type integral of \(f\) \[ C_f(z):=\frac{1}{2\pi \imath}\int_{\Gamma}\frac{f(t)}{t-z}\,dt, \] is defined. N.A. Davydov proved in 1949, that for a closed rectifiable \(\Gamma\) and for \(f\) in the Lipschitz class, the Cauchy-type integral \(C_f\) is continuous in the closures of \(D^+\) and \(D^-\). In the paper under review, this Dadydov's theorem is generalized for some clases of nonrectifiable curves. Let \(\psi\) be a continuous increasing function on \(\mathbb R_+:=[0,\infty)\) such that \(\psi(0)=0\). The class \(R_{\psi}\) consists of all curves \(\Gamma\) such that \[ \sigma_{\psi}(\Gamma):=\sup\left\{\sum_{j=1}^n\psi(z_j-z_{j-1}| )\;| \;\forall \tau \right\} <\infty, \] the supremum is taken over all finite sequences \(\tau:=\{z_1,\cdots ,z_n\;| z_1<z_2<\cdots <z_n\}\) . If \(\Gamma \in R_\psi\), then \(\Gamma\) is called \(\psi\)-rectifiable; for \(\psi(x)=x^p\) with \(p\geq 1\) the curves are called \(p\)-rectifiable. For \(p\geq 1\), a \(p\)-rectifiable curve may be nonrectifiable. Theorem 1 states that if \(\psi\) is convex, \(\Gamma\in R_{\psi}\) and \(f\) is a Lipschitz function then \(C_f\) exists as the Stiltjes integral and is continuous in the closure of \(D^+\) and \(D^-\) provided that \[ \sum_{n=1}^{\infty} \varphi^2 \left(\frac{1}{n} \right)\leq\infty\tag{\(*\)} \] where \(\varphi\) is the function inverse to \(\psi\). For \(p\)-rectifiability (\(*\)) reduces to \(p<2\), and hence the conclusion of the theorem is valid for \(p\)-rectifiable curves with \(1\leq p<2\). Theorem 2 makes the same conclusion for a class of curves consisting of simple closed Jordan curves representable as unions of finitely many graphs (possibly rotated) of continuous real functions and for densities \(f\) in a subclass of Lipschitz functions.