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

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

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Correspondence to Kazuo Hashimoto.

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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