On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let $\alpha ,\beta$ be orientation-preserving diffeomorphism (shifts) of ${\mathbb{R}}_{+}=(0,\mathrm{\infty })$ onto itself with the only fixed points $0$ and $\mathrm{\infty }$ and $U_{\alpha },U_{\beta }$ be the isometric shift operators on $L^p({\mathbb{R}}_{+})$ given by $U_{\alpha }f=({\alpha }^{\mathrm{\prime }}{)}^{1\mathrm{/}p}(f\circ \alpha )$, $U_{\beta }f=({\beta }^{\mathrm{\prime }}{)}^{1\mathrm{/}p}(f\circ \beta )$, and $P_2^{\pm }=(I\pm S_2)\mathrm{/}2$ where $$(S_{2}f)(t):=\frac{1}{\pi i}\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{t}{\tau }\right)}^{1\mathrm{/}2-1\mathrm{/}p}\frac{f(\tau )}{\tau -t}d\tau ,t\in {\mathbb{R}}_{+},$$ is the weighted Cauchy singular integral operator. We prove that if ${\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }}$ and $c,d$ are continuous on ${\mathbb{R}}_{+}$ and slowly oscillating at $0$ and $\mathrm{\infty }$, and $$\underset{t\to s}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\mid }c(t)\mathrm{\mid }<1,\underset{t\to s}{\mathrm{lim\hspace{0.17em}sup}}\mathrm{\mid }d(t)\mathrm{\mid }<1,s\in \{0,\mathrm{\infty }\},$$ then the operator $(I-c{U}_{\alpha })P_2^{+}+(I-d{U}_{\beta })P_2^{-}$ is Fredholm on $L^p({\mathbb{R}}_{+})$ and its index is equal to zero. Moreover, its regularizers are described.