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.
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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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Barza, S., Silvestre, P. Functions of bounded second \(p\)-variation. Rev Mat Complut 27, 69–91 (2014). https://doi.org/10.1007/s13163-013-0136-0
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DOI: https://doi.org/10.1007/s13163-013-0136-0
Keywords
- Partition
- Periodic function
- Bounded (second)
- Modulus of \(p\)-continuity
- Bounded second (\({p, \alpha }\)) variation
- Fractional moduli of continuity
- Steklov averages