16/09/2016, 11:50 — 12:15 — Room 6.2.38, Faculty of Sciences of the Universidade de Lisboa
Alexei Karlovich, Universidade Nova de Lisboa
Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data
Let $\alpha,\beta$ be orientation-preserving diffeomorphisms (shifts) of $\mathbb{R}_+ =(0,\infty)$ onto itself, which have only two fixed points $0,\infty$ and let $U_\alpha f=(\alpha')^{1/p}f\circ\alpha$, $U_\beta f=(\beta')^{1/p}f\circ\beta$ be the corresponding isometric shift operators on $L^p(\mathbb{R}_+)$. We prove sufficient conditions for the right and left Fredholmness on $L^p(\mathbb{R}_+)$ of singular integral operators of the form $A_+P_\gamma^++A_-P_\gamma^-$, where $P_\gamma^\pm=(I\pm S_\gamma)/2$, $S_\gamma$ is a weighted Cauchy singular integral operator, $A_+=aI-bU_\alpha$, $A_-=cI-dU_\beta$ are binomial functional operators with shifts. We assume that the coefficients $a,b,c,d$ and the derivatives of the shifts $\alpha',\beta'$ are bounded continuous functions on $\mathbb{R}_+$, which may have slowly oscillating discontinuities at $0$ and $\infty$. This is a joint work with Yuri Karlovich and Amarino Lebre.
