On Functions of Bounded 𝑝,𝑘-VariationReport as inadecuate




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Journal of Function Spaces and ApplicationsVolume 2012 2012, Article ID 202987, 9 pages

Research Article

Universidad Central de Venezuela, Caracas, Venezuela

Universidad Nacional Abierta, Caracas, Venezuela

Received 7 April 2010; Accepted 24 November 2010

Academic Editor: Jürgen Appell

Copyright © 2012 N. Merentes et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study the concept of 𝑝,𝑘-variation 1<𝑝<∞, 𝑘∈ℕ of a real function on a compact interval. In particular, we prove that a function 𝑢∶𝑎,𝑏→ℝ has bounded 𝑝,𝑘-variation if and only if 𝑢𝑘−1 is absolutely continuous on 𝑎,𝑏 and 𝑢𝑘 belongs to 𝐿𝑝𝑎,𝑏. Moreover, an explicit connection between the 𝑝,𝑘-variation of 𝑢 and the 𝐿𝑝-norm of 𝑢𝑘 is given which is parallel to the classical Riesz formula characterizing functions in the spaces 𝑅𝑉𝑝𝑎,𝑏 and 𝐴𝑝𝑎,𝑏. This may also be considered as an alternative characterization of the one variable Sobolev space 𝑊𝑘𝑝𝑎,𝑏.





Author: N. Merentes, S. Rivas, and J. L. Sanchez

Source: https://www.hindawi.com/



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