Functional Analysis and Applications Seminar 

21/02/2003, 14:00 — 16:00 — Room P3.10, Mathematics Building
Alexei Karlovich, Instituto Superior Técnico, U.T. Lisboa

Necessary Conditions for Fredholmness of Singular Integral Operators with PC Coefficients on Banach Function Spaces

We prove necessary conditions for Fredholmness of singular integraloperators with coefficients in the Banach algebra of piecewise continuous functions on weighted Banach function spaces. These conditions are formulated in terms of indices of a submultiplicative function associated with local properties of the space, of the curve, and of the weight. As an example, we consider the Musielak-Orlicz space \(L_{p(t)}\) (the Lebesgue space with variable exponent). In this example the above mentioned indices coincide with \(1/p(t)\) and \(p(t)/[p(t)-1]\) at each point (for nice curves and weight \(w=1\)). Our results give a natural generalization of the necessity part of the Gohberg-Krupnikcondition (for nice curves) as well as, the Boettcher - Yu. Karlovich condition (for general Carleson curves). So, we give a partial answer on the question raised by S. Samko on the roundtable on December 17, 2002.