On Functions of (p,α)-Bounded Variation

Abstract

In this paper we introduce the concept of $\;(p,\alpha)$-bounded variation which generalizes the Riesz p-variation. The following result is proved: a function $\; f:[a,b] \rightarrow \mathbb{R}\;$ is of $\; (p,\alpha)$-bounded variation $\;(1<p<\infty)\;$ if and only if $f$ is $\alpha$-absolutely continuous on $[a,b]$ and $f^{^{\prime}}_{\alpha} \in L_{(p,\alpha)} [a,b]$. Moreover it is shown that the $\;(p,\alpha)$-bounded variation of a function $f$ on $[a,b]$ is given by \begin{equation*} V_{(p,\alpha)} (f) = \|f^{^{\prime}}_{\alpha}\|^{p}_{L_{(p,\alpha)} [a,b]}. \end{equation*}