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

TL;DR

This paper investigates the Fredholm properties and index of specific weighted singular integral operators with two slowly oscillating shifts, extending previous results to more general complex parameters and shift compositions.

Contribution

It extends prior work by analyzing the Fredholmness and index of weighted singular integral operators with two shifts for complex parameters, including discontinuities of derivatives.

Findings

Operators are Fredholm under certain conditions on parameters.

Zero index is established for the operators under specified inequalities.

Results generalize previous findings for the case b3=0.

Abstract

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 $$ (S_\gamma f)(t)=\frac{1}{\pi i}\int_{\mathbb{R}_+} \left(\frac{t}{\tau}\right)^\gamma\frac{f(\tau)}{\tau-t}d\tau $$ with $\gamma\in\mathbb{C}$ satisfying $0<1/p+\Re\gamma<1$. We prove that the operator $A_{ij}$ is a Fredholm operator on $L^p(\mathbb{R}_+)$ and has zero index if \[ 0<\frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma), \quad \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)<1, \] where $\omega_{ij}(t)=\log[\alpha_i(\beta_{-j}(t))/t]$ and $\alpha_i$, $\beta_{-j}$ are iterations of $\alpha$, $\beta$. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $\gamma=0$.