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Revista Matemática Complutense

, Volume 30, Issue 2, pp 393–416 | Cite as

On the essential bounded Riesz \(\Phi \)-variation

  • Gen Nakamura
  • Kazuo HashimotoEmail author
Article
  • 71 Downloads

Abstract

In Nakamura and Hashimoto (Collect Math 65(3):407–416, 2014), the authors showed that for every \(f\in L^{1}_{\mathrm{loc}}({\mathbb {R}})\), the essential p-variation \(\mathrm{ess}\, V_p(f,{\mathbb {R}})\) of f is given by
$$\begin{aligned} \lim _{h\rightarrow 0}\int _{{\mathbb {R}}}\left| \frac{f(x+h)-f(x)}{h}\right| ^{p}\, dx. \end{aligned}$$
In this paper, more generally we treat the following convergence for a function \(f\in L^{1}_{\mathrm{loc}}({\mathbb {R}})\) and a convex function \({\Phi }:{\mathbb {R}}\rightarrow [0,\infty )\);
$$\begin{aligned} \lim _{h\rightarrow 0}\int _{{\mathbb {R}}}{\Phi }\left( \frac{f(t+h)-f(t)}{h}\right) \, dt, \end{aligned}$$
and we show that the limit is equivalent to an essential \({\Phi }\)-variation \(\text{ ess } V_{{\Phi }}(f)\). Moreover, we obtain a characterization of the class of functions f with \(\text{ ess } V_{{\Phi }}(f)<\infty \).

Keywords

Bounded variation Essential bounded variation Convexity Sobolev space 

Mathematics Subject Classification

Primary 26A45 Secondary 46E35 

References

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    Nakamura, G., Hashimoto, K.: On the linearity of some sets of sequences defined by \(L^p\)-functions and \(L_1\)-functions determining \(\ell _{1}\). Proc. Japan Acad. Ser. A Math Sci, 87, 77–82 (2011)Google Scholar
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    Nakamura, G., Hashimoto, K.: On the essential bounded variation of \(L^p(\mathbb{R}, X)\)-functions. Collect. Math. 65(3), 407–416 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
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    Riesz, F.: Untersuchungen über Systeme integrierbarer Funktionen. Math. Ann. 69(4), 449–497 (1910)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Matsue College of TechnologyMatsueJapan
  2. 2.Faculty of Liberal ArtsHiroshima Jogakuin UniversityHigashi-kuJapan

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