Functional Analysis and Applications Seminar 

04/04/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
Alexei Karlovich, Universidade Nova de Lisboa

Fredholmness and index of simplest singular integral operators with two slowly oscillating shifts

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 $\alpha'$ and $\beta'$ 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 $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$ and $U_\beta f= (\beta')^{1/p}(f\circ\beta)$. 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. This is a joint work with Yuri Karlovich and Amarino Lebre.