## Journal of Integral Equations and Applications

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

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

Alexei Yu. Karlovich, Yuri I. Karlovich, and Amarino B. Lebre

#### 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

**Permanent link to this document**

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

**Subjects**

Primary: 45E05: Integral equations with kernels of Cauchy type [See also 35J15]

Secondary: 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20] 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47G10: Integral operators [See also 45P05] 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx]

**Keywords**

Orientation-preserving shift weighted Cauchy singular integral operator slowly oscillating function Fredholmness

#### 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