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 \(p(x)=\operatorname{const}\).