Functional Analysis and Applications Seminar 

28/02/2003, 14:00 — 16:00 — Room P3.10, Mathematics Building
Stefan Samko, Universidade do Algarve, Faro

A Further Progress in the Theory of Lebesgue Spaces with VariableExponent: Singular Integral Equations and Sobolev Theorem forPotentials

The talk provides a discussion of recent results for the generalized Lebesgue spaces with variable exponent \(p(x)\) (GLSVE) including the criterion for the weighted singular operator (with a power weight) to be bounded in such spaces. This result is applied to "localize" the Gohberg-Krupnik criterion of Fredholmness of singular integral operators in such spaces on Lyapunov curves. Some abstract Banach space reformulation of the Gohberg-Krupnik scheme of investigation of Fredholmness is given, from which the result for GLSVE, in particular follows due to the boundedness criterion for the weighted singular operator. Another new result for GLSVE presented is the Sobolev theorem for potentials over the Euclidean space, in which the "new word" is a possibility to consider the variable exponent \(p(x)\) not necessarily constant at infinity. However, the "payment" for this possibility is an additional power weight fixed to infinity, which turns to be equal to \(1\) in the traditional case \(p(x)=\operatorname{const}\).