## Banach Journal of Mathematical Analysis

### Fredholmness and index of simplest weighted singular integral operators with two slowly oscillating shifts

Alexei Yu. Karlovich

#### 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 $01/p+\Re\gamma\in (0,1)$. We prove that the operator $A_{ij}$ is a Fredholm operator on $L^p(\mathbb{R}_+)$ and has zero index if $\frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma),\,\, \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)\in (0,1),$ 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 $\gamma=0$.

#### Article information

Source
Banach J. Math. Anal., Volume 9, Number 3 (2015), 24-42.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.bjma/1419001701

Digital Object Identifier
doi:10.15352/bjma/09-3-3

Mathematical Reviews number (MathSciNet)
MR3296123

Zentralblatt MATH identifier
1335.47016

#### Citation

Karlovich , Alexei Yu. Fredholmness and index of simplest weighted singular integral operators with two slowly oscillating shifts. Banach J. Math. Anal. 9 (2015), no. 3, 24--42. doi:10.15352/bjma/09-3-3. https://projecteuclid.org/euclid.bjma/1419001701

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