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

Let $\alpha,\beta$ 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 $U_\alpha,U_\beta$ be the corresponding isometric shift operators on the space $L^p(\mathbb{R}_+)$ given by $U_\mu f=(\mu')^{1/p}(f\circ\mu)$ for $\mu\in\{\alpha,\beta\}$. We prove sufficient conditions for the right and left Fredholmness on $L^p(\mathbb{R}_+)$ of singular integral operators of the form $A_+P_\gamma^++A_-P_\gamma^-$, where $P_\gamma^\pm=(I\pm S_\gamma)/2$, $S_\gamma$ is a weighted Cauchy singular integral operator, $A_+=\sum_{k\in\mathbb{Z}}a_kU_\alpha^k$ and $A_-=\sum_{k\in\mathbb{Z}}b_kU_\beta^k$ 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 $\alpha',\beta'$ are bounded continuous functions on $\mathbb{R}_+$ which may have slowly oscillating discontinuities at $0$ and $\infty$.
Comment: Accepted for publication in the Proceedings of WOAT 2016 held in Lisbon in July of 2016, which will be published in the special volume "Operator Theory, Operator Algebras, and Matrix Theory" of OT series (Birkh\"auser)