Functional Analysis and Applications Seminar 

06/12/2002, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Instituto Superior Técnico, U.T.L.

A Class of Singular Integral Operators with Flip and Unbounded Coefficients on Rearrangement-Invariant Spaces

We prove Fredholm criteria for singular integral operators of the form $P+aQ+bUQ$, where $P$ and $Q$ are the Riesz projections, $U$ is the flip operator, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that a is bounded, but $b$ may be unbounded. The function $b$ belongs to a class of, in general, unbounded functions that relates to the Douglas algebra $H^{\mathrm{\infty }}+C$. This result is new even for Lebesgue spaces. It refines and generalizes some results of Kravchenko, Lebre, Litvinchuk, and Teixeira published in the Mathematische Nachrichten in 1995.