Revista Matemática Complutense

, Volume 27, Issue 1, pp 69–91 | Cite as

Functions of bounded second \(p\)-variation

Article

Abstract

The generalized functionals of Merentes type generate a scale of spaces connecting the class of functions of bounded second \(p\)-variation with the Sobolev space of functions with p-integrable second derivative. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of order \(2-1/p\). These results extend the results in Lind (Math Inequal Appl 16:2139, 2013) for functions of bounded variation but are not consequence of the last.

Keywords

Partition Periodic function Bounded (second) Modulus of \(p\)-continuity Bounded second (\({p, \alpha }\)) variation  Fractional moduli of continuity Steklov averages 

Mathematics Subject Classification (2000)

Primary 26A45 Secondary 46E35 

Notes

Acknowledgments

The authors are very grateful to Prof. Viktor Kolyada and their friend Martin Lind for encouragement and discussions, and for pointing out the reference [11]. The first named author would like to acknowledge the travel grant from SVeFUM (2012) which made possible the visit to the University of Barcelona when a part of this research was done. The second named author gratefully acknowledges the opportunity to work as a postdoc in Aalto University with Prof. J. Kinnunen where a part of this project was done. We would like to thank the referees for their remarks, which have improved the final version of the paper.

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Copyright information

© Universidad Complutense de Madrid 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceKarlstad UniversityKarlstadSweden
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland

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