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

TL;DR

This paper provides sufficient conditions for the Fredholmness of a class of singular integral operators with shifts and slowly oscillating data on the positive real line, extending understanding of their invertibility properties.

Contribution

It establishes new criteria for Fredholmness of singular integral operators with shifts and oscillating coefficients, considering discontinuities at zero and infinity.

Findings

Derived sufficient conditions for operator Fredholmness.

Analyzed operators with slowly oscillating coefficients.

Extended classical results to include shifts and discontinuities.

Abstract

Suppose $\alpha$ 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 \[ (aI-bW_\alpha)P_++(cI-dW_\alpha)P_- \] 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 $\infty$.