Abstract
We study 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. 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 none-equilibrated functions in $\Phi$ which oscillate in a certain way between two power functions. Given two numbers $0< \alpha\leq \beta <1$, we explicitly construct examples of functions in $\Phi$ for which $\alpha$ and $\beta$ serve as the index numbers. The investigation of properties of non-equilibrated functions in $\Phi$ was evoked by applications of these properties in problems of the normal solvability of some singular integral operators in the spaces with prescribed modulus of continuity.
Citation
Natasha Samko. "On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class.." Real Anal. Exchange 30 (2) 727 - 746, 2004-2005.
Information