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

Abstract

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.