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