@article {6704, title = {On a weighted singular integral operator with shifts and slowly oscillating data}, journal = {Complex Analysis and Operator Theory}, volume = {10}, year = {2016}, pages = {1101-1131}, abstract = {

Let \(\alpha,\beta\) be orientation-preserving diffeomorphism (shifts) of \(\mathbb{R}_+=(0,\infty)\) onto itself with the only fixed points \(0\) and \(\infty\) and \(U_\alpha,U_\beta\) be the isometric shift operators on \(L^p(\mathbb{R}_+)\) given by \(U_\alpha f=(\alpha{\textquoteright})^{1/p}(f\circ\alpha)\), \(U_\beta f=(\beta{\textquoteright})^{1/p}(f\circ\beta)\), and \(P_2^\pm=(I\pm S_2)/2\) where \[ (S_2 f)(t):=\frac{1}{\pi i}\int\limits_0^\infty \left(\frac{t}{\tau}\right)^{1/2-1/p}\frac{f(\tau)}{\tau-t}\,d\tau, \quad t\in\mathbb{R}_+, \]
is the weighted Cauchy singular integral operator. We prove that if \(\alpha{\textquoteright},\beta{\textquoteright}\) and \(c,d\) are continuous on \(\mathbb{R}_+\) and slowly oscillating at \(0\) and \(\ infty\), and \[ \limsup_{t\to s}|c(t)|\<1,\quad \limsup_{t\to s}|d(t)|\<1, \quad s\in\{0,\infty\}, \] then the operator \((I-cU_\alpha)P_2^++(I-dU_\beta)P_2^-\) is Fredholm on \(L^p(\mathbb{R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

}, url = {http://link.springer.com/article/10.1007/s11785-015-0452-0}, author = {Karlovich, Alexei Yu. and Karlovich, Yuri I. and Lebre, Amarino B.} }