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

TL;DR

This paper establishes necessary and sufficient conditions for the Fredholmness of a class of singular integral operators with shifts and slowly oscillating data on the positive real line.

Contribution

It extends previous sufficiency results by proving that the conditions for Fredholmness are also necessary for these operators.

Findings

Necessary conditions match previously known sufficient conditions.

Fredholmness depends on the behavior of coefficients and shift derivatives at 0 and infinity.

Results apply to operators with slowly oscillating discontinuities.

Abstract

Suppose $\alpha$ is an orientation-preserving diffeomorphism (shift) of $\mR_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$. In \cite{KKLsufficiency} we found sufficient conditions for the Fredholmness of the singular integral operator with shift \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] acting on $L^p(\mR_+)$ with $1<p<\infty$, where $P_\pm=(I\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_\alpha f=f\circ\alpha$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative $\alpha'$ of the shift are bounded and continuous on $\mR_+$ and may admit discontinuities of slowly oscillating type at $0$ and $\infty$. Now we prove that those conditions are also necessary.