Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data

Abstract

Let $α,β$ be orientation-preserving homeomorphisms of $[0,\infty]$ onto itself, which have only two fixed points at $0$ and $\infty$, and whose restrictions to $\mathbb{R}_+=(0,\infty)$ are diffeomorphisms, and let $U_α,U_β$ be the corresponding isometric shift operators on the space $L^p(\mathbb{R}_+)$ given by $U_μf=(μ')^{1/p}(f\circμ)$ for $μ\in\{α,β\}$. We prove sufficient conditions for the right and left Fredholmness on $L^p(\mathbb{R}_+)$ of singular integral operators of the form $A_+P_γ^++A_-P_γ^-$, where $P_γ^\pm=(I\pm S_γ)/2$, $S_γ$ is a weighted Cauchy singular integral operator, $A_+=\sum_{k\in\mathbb{Z}}a_kU_α^k$ and $A_-=\sum_{k\in\mathbb{Z}}b_kU_β^k$ are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients $a_k,b_k$ for $k\in\mathbb{Z}$ and the derivatives of the shifts $α',β'$ are bounded continuous functions on $\mathbb{R}_+$ which may have slowly oscillating discontinuities at $0$ and $\infty$.