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2004-2005 On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class.
Natasha Samko
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Real Anal. Exchange 30(2): 727-746 (2004-2005).


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.


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Natasha Samko. "On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class.." Real Anal. Exchange 30 (2) 727 - 746, 2004-2005.


Published: 2004-2005
First available in Project Euclid: 15 October 2005

zbMATH: 1197.26013
MathSciNet: MR2177430

Primary: 26A16 , 26A48 , 54C35‎

Keywords: Bary-Stechkin class , Boyd-type indices , H\"older space , indices of monotonic functions , modulus of continuity , Zygmund conditions

Rights: Copyright © 2004 Michigan State University Press

Vol.30 • No. 2 • 2004-2005
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