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

Abstract

Let $α$ and $β$ be orientation-preserving diffeomorphisms (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$, where the derivatives $α'$ and $β'$ may have discontinuities of slowly oscillating type at $0$ and $\infty$. For $p\in(1,\infty)$, we consider the weighted shift operators $U_α$ and $U_β$ given on the Lebesgue space $L^p(\mathbb{R}_+)$ by $U_αf=(α')^{1/p}(f\circα)$ and $U_βf= (β')^{1/p}(f\circβ)$. For $i,j\in\mathbb{Z}$ we study the simplest weighted singular integral operators with two shifts $A_{ij}=U_α^i P_γ^++U_β^j P_γ^-$ on $L^p(\mathbb{R}_+)$, where $P_γ^\pm=(I\pm S_γ)/2$ are operators associated to the weighted Cauchy singular integral operator $$ (S_γf)(t)=\frac{1}{πi}\int_{\mathbb{R}_+} \left(\frac{t}τ\right)^γ\frac{f(τ)}{τ-t}dτ$$ with $γ\in\mathbb{C}$ satisfying $0<1/p+\Reγ<1$. We prove that the operator $A_{ij}$ is a Fredholm operator on $L^p(\mathbb{R}_+)$ and has zero index if \[ 0<\frac{1}{p}+\Reγ+\frac{1}{2π}\inf_{t\in\mathbb{R}_+}(ω_{ij}(t)\Imγ), \quad \frac{1}{p}+\Reγ+\frac{1}{2π}\sup_{t\in\mathbb{R}_+}(ω_{ij}(t)\Imγ)<1, \] where $ω_{ij}(t)=\log[α_i(β_{-j}(t))/t]$ and $α_i$, $β_{-j}$ are iterations of $α$, $β$. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $γ=0$.