Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data

Abstract

We extend the main result of {KKL11b} to the case of more general weighted singular integral operators with two shifts of the form \[ (aI-b U_\alpha )P_\gamma ^++(cI-dU_\beta )P_\gamma ^-, \] acting on the space $L^p(\mathbb{R} _+)$, $1\lt p\lt \infty $, where \[ P_\gamma ^\pm =(I\pm S_\gamma )/2 \] are operators associated with the weighted Cauchy singular integral operator $S_\gamma $, given by \[ (S_\gamma f)(t)=\frac {1}{\pi i}{\int _{\mathbb{R} _+}} \bigg (\frac {t}{\tau }\bigg )^\gamma \frac {f(\tau )}{\tau -t}\,d\tau \] with $\gamma \in \mathbb{C} $ satisfying $0\lt 1/p+\Re \gamma \lt 1$, and $U_\alpha ,U_\beta $ are the isometric shift operators given by \[ U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha ), \qquad U_\beta f=(\beta ')^{1/p}(f\circ \beta ), \] generated by diffeomorphisms $\alpha ,\beta $ of $\mathbb{R} _+$ onto itself having only two fixed points at the endpoints $0$ and $\infty $, under the assumptions that the coefficients $a,b,c,d$ and the derivatives $\alpha ',\beta '$ of the shifts are bounded and continuous on $\mathbb{R} _+$ and admit discontinuities of slowly oscillating type at $0$ and $\infty $.