@article {6702,
	title = {Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts},
	journal = {Operators and Matrices},
	volume = {8},
	year = {2014},
	pages = {935-955},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>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 \(\alpha{\textquoteright}\) and \(\beta{\textquoteright}\) 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 \(U_\alpha f=(\alpha{\textquoteright})^{1/p}(f\circ\alpha)\) and \(U_\beta f= (\beta{\textquoteright})^{1/p}(f\circ\beta)\). We apply the theory of Mellin pseudodifferential operators with symbols of limited smoothness to study the simplest singular integral operators with two shifts \(A_{ij}=U_\alpha^i P_++U_\beta^j P_-\) on the space \(L^p(\mathbb{R}_+)\), where \(P_\pm=(I\pm S)/2\) are operators associated to the Cauchy singular integral operator \(S\), and \(i,j\in\mathbb{Z}\). We prove that all \(A_{ij}\) are Fredholm operators on \(L^p(\mathbb{R}_+)\) and have zero indices.</p>
},
	url = {http://dx.doi.org/10.7153/oam-08-52},
	author = {Karlovich, Alexei Yu. and Karlovich, Yuri I. and Lebre, Amarino B.}
}
