Fredholmness and Index of Simplest Weighted Singular Integral Operators with Two Slowly Oscillating Shifts

Alexei Yu. Karlovich

Published 2014-05-02, updated 2014-10-01Version 2

Let $\alpha$ and $\beta$ be orientation-preserving diffeomorphisms (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points 0 and $\infty$, where the derivatives $\alpha'$ and $\beta'$ may have discontinuities of slowly oscillating type at 0 and $\infty$. For $p\in(1,\infty)$, we consider the weighted shift operators $U_\alpha$ and $U_\beta$ given on the Lebesgue space $L^p(\mathbb{R}_+)$ by $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$ and $U_\beta f= (\beta')^{1/p}(f\circ\beta)$. For $i,j\in\mathbb{Z}$ we study the simplest weighted singular integral operators with two shifts $A_{ij}=U_\alpha^i P_\gamma^++U_\beta^j P_\gamma^-$ on $L^p(\mathbb{R}_+)$, where $P_\gamma^\pm=(I\pm S_\gamma)/2$ are operators associated to the weighted Cauchy singular integral operator $$ (S_\gamma f)(t)=\frac{1}{\pi i}\int_{\mathbb{R}_+} (\frac{t}{\tau})^\gamma\frac{f(\tau)}{\tau-t}d\tau $$ with $\gamma\in\mathbb{C}$ satisfying $0<1/p+\Re\gamma<1$. We prove that all $A_{ij}$ are Fredholm operators on $L^p(\mathbb{R}_+)$ and have zero indices. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $\gamma=0$.