Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm
Abstract
Description
Suppose $Γ$ is a Carleson Jordan curve with logarithmic whirl points,
$\varrho$ is a Khvedelidze weight, $p:Γ\to(1,\infty)$ is a continuous function satisfying $|p(τ)-p(t)|\le -\mathrm{const}/\log|τ-t|$ for $|τ-t|\le 1/2$, and $L^{p(\cdot)}(Γ,\varrho)$ 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 $L_N^{p(\cdot)}(Γ,\varrho)$.
16 pages
16 pages