## Journal of Integral Equations and Applications

### Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data

#### Abstract

We extend the main result of {KKL11b} to the case of more general weighted singular integral operators with two shifts of the form $(aI-b U_\alpha )P_\gamma ^++(cI-dU_\beta )P_\gamma ^-,$ acting on the space $L^p(\mathbb{R} _+)$, $1\lt p\lt \infty$, where $P_\gamma ^\pm =(I\pm S_\gamma )/2$ are operators associated with the weighted Cauchy singular integral operator $S_\gamma$, given by $(S_\gamma f)(t)=\frac {1}{\pi i}{\int _{\mathbb{R} _+}} \bigg (\frac {t}{\tau }\bigg )^\gamma \frac {f(\tau )}{\tau -t}\,d\tau$ with $\gamma \in \mathbb{C}$ satisfying $0\lt 1/p+\Re \gamma \lt 1$, and $U_\alpha ,U_\beta$ are the isometric shift operators given by $U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha ), \qquad U_\beta f=(\beta ')^{1/p}(f\circ \beta ),$ generated by diffeomorphisms $\alpha ,\beta$ of $\mathbb{R} _+$ onto itself having only two fixed points at the endpoints $0$ and $\infty$, under the assumptions that the coefficients $a,b,c,d$ and the derivatives $\alpha ',\beta '$ of the shifts are bounded and continuous on $\mathbb{R} _+$ and admit discontinuities of slowly oscillating type at $0$ and $\infty$.

#### Article information

Source
J. Integral Equations Applications, Volume 29, Number 3 (2017), 365-399.

Dates
First available in Project Euclid: 14 August 2017

https://projecteuclid.org/euclid.jiea/1502676095

Digital Object Identifier
doi:10.1216/JIE-2017-29-3-365

Mathematical Reviews number (MathSciNet)
MR3695359

Zentralblatt MATH identifier
1376.45016

#### Citation

Karlovich, Alexei Yu.; Karlovich, Yuri I.; Lebre, Amarino B. Necessary Fredholm conditions for weighted singular integral operators with shifts and slowly oscillating data. J. Integral Equations Applications 29 (2017), no. 3, 365--399. doi:10.1216/JIE-2017-29-3-365. https://projecteuclid.org/euclid.jiea/1502676095

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