Sufficient Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Suppose $α$ is an orientation preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. We establish sufficient conditions for the Fredholmness of the singular integral operator $(aI-bW_α)P_++(cI-dW_α)P_-$ acting on $L^p(\mR_+)$ with $1<p<\infty$, where $P_\pm=(I\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_αf=f\circα$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative $α'$ of the shift are bounded and continuous on $\mR_+$ and may admit discontinuities of slowly oscillating type at $0$ and $\infty$.

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