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 \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \] 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 \[ \limsup_{t\to s}|c(t)|<1, \quad \limsup_{t\to s}|d(t)|<1, \quad s\in\{0,\infty\}, \] 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.