Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm

Abstract

Suppose $\Gamma$ is a Carleson Jordan curve with logarithmic whirl points, $\varrho$ is a Khvedelidze weight, $p:\Gamma \to (1,\infty )$ is a continuous function satisfying $|p(\tau )-p(t)|\le -\mathrm{const}/\log |\tau -t|$ for $|\tau -t|\le 1/2$, and $L^{p(\cdot )}(\Gamma ,\varrho )$ is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with $N\times N$ matrix piecewise continuous coefficients are Fredholm on $L_N^{p(\cdot )}(\Gamma ,\varrho )$.

16 pages