Noetherian solvability of an operator singular integral equation with a Carleman shift in fractional spaces

TL;DR

This paper establishes conditions for the Noetherian solvability and derives an index formula for a singular integral equation with a Carleman shift in fractional Besov spaces, expanding understanding beyond classical Hölder spaces.

Contribution

It introduces new solvability criteria and an index formula for singular integral equations with Carleman shifts in fractional Besov spaces, not previously addressed in classical function spaces.

Findings

Derived solvability conditions for the integral equation.

Established an index formula in fractional Besov spaces.

Extended classical results to non-Hölder continuous function spaces.

Abstract

In this paper, we obtain conditions of Noetherian solvability and the Index formula for a singular integral equation with a Cauchy kernel and a Carleman shift in Besov space, which is embedded into the space of continuous functions on a closed Lyapunov contour, but not into the class of functions satisfying H\"{o}lder condition.