## 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

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 \({\Phi }:{\mathbb {R}}\rightarrow [0,\infty )\);

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 \(\text{ ess } V_{{\Phi }}(f)<\infty \).

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## References

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Nakamura, G., Hashimoto, K. On the essential bounded Riesz \(\Phi \)-variation.
*Rev Mat Complut* **30, **393–416 (2017). https://doi.org/10.1007/s13163-017-0222-9

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### Keywords

- Bounded variation
- Essential bounded variation
- Convexity
- Sobolev space

### Mathematics Subject Classification

- Primary 26A45
- Secondary 46E35