Singular Integral Operators on an Open Arc in Spaces with Weight

Abstract

The aim of this work is to study a singular integral operator ${\mathbf{A}=aI+bS_\Gamma}$ with the Cauchy operator S _Γ (SIO) and Hölder continuous coefficients a, b in the space ${\mathbb{H}^0_\mu(\Gamma,\rho)}$ of Hölder continuous functions with an power “Khvedelidze” weight. The underlying curve is an open arc. It is well known, that such operator is Fredholm if and only if, along with the ellipticity condition ${a^2(t)-b^2(t)\not=0,\, t\in\Gamma}$, the “Gohberg–Krupnik arc condition” is fulfilled (see Duduchava, in Dokladi Akademii Nauk SSSR 191:16–19, 1970). Based on the Poincare–Beltrami formula for a composition of singular integral operators and the celebrated Muskhelishvili formula describing singularities of Cauchy integral, the formula for a composition of weighted singular integral operators is proved. Using the obtained composition formula and the localization, the Fredholm criterion of the SIO is derived in a natural way, by looking for the regularizer of the operator ${\mathbf{A}}$ and equating to 0 non-compact operators. The approach is space-independent and this is demonstrated on similar results obtained for SIOs with continuous coefficients in the Lebesgue spaces with a “Khvedelidze” weight ${\mathbb{L}_p(\Gamma,\rho)}$, investigated earlier by Gohberg and Krupnik (Studia Mathematica 31:347–362, 1968; One Dimensional Singular Integral Operators II, Operator Theory, Advances and Applications, vol. 54, chapter IX, 1979) with a different approach.