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

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.