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

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 \[ (aI-bW_α)P_++(cI-dW_α)P_- \] acting on $L^p(\mR_+)$ with $1<p<\infty$, where $P_{\pm }=(I\pm S)/2$, $S$ is the Cauchy singular integral operator, and $W_{\alpha }f=f\circ \alpha$ is the shift operator, under the assumptions that the coefficients $a,b,c,d$ and the derivative ${\alpha }^{\prime }$ 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.

26 pages