Functional Analysis and Applications Seminar 

22/04/2005, 15:00 — 16:00 — Room P3.10, Mathematics Building
Natasha Samko, Universidade do Algarve, Faro

Generalized Hölder spaces with non-equilibrated characteristics andFredholmness of singular integral operators

We consider quasi-monotonic functions of the Zygmund-Bary-Stechkin class $\Phi$ with the main emphasis on properties of the index numbers of functions in this class (of Boyd type indices). A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in $\Phi$ known in terms of these indices, are exact in a certain sense. We also single out some special family of non-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0 \lt \alpha , \beta \lt 1$ we explicitly construct examples of functions in $\Phi$ for which $\alpha$ and $\beta$ serve as the index numbers. The developed properties of functions in this class are applied to an investigation of the normal solvability of some singular integral operators in weighted spaces with prescribed modulus of continuity.