On functions of bounded $(p,2)$-variation

In this paper we introduce the concept of $(p,2)$-variation which generalizes the Riesz $p$-variation. The following result is proved: A function $f:[a,b]\rightarrow\mathbb{R}$ is of bounded $(p,2)$-variation $(1 < p <\infty)$ if and only if f' is absolutely continuous on $[a,b]$ and $f''\in L_p[a,b]$. Moreover it is shown that the $(p,2)$-variation of a function $\int$ on [a,b] is given by $$V^2_p(f;[a,b])=\parallel f''\parallel^p_{L_p}[a,b]^\cdot$$