On functions of bounded \((p, k)\)-variation (Q416306)

Summary: We introduce and study the concept of \((p, k)\)-variation \((1 < p < \infty, ~k \in \mathbb N)\) of a real function on a compact interval. In particular, we prove that a function \(u : [a, b] \rightarrow \mathbb R\) has bounded \((p, k)\)-variation if and only if \(u^{(k-1)}\) is absolutely continuous on \([a, b]\) and \(u^{(k)}\) belongs to \(L_p[a, b]\). Moreover, an explicit connection between the \((p, k)\)-variation of \(u\) and the \(L_p\)-norm of \(u^{(k)}\) is given which is parallel to the classical Riesz formula characterizing functions in the spaces \(RV_p[a, b]\) and \(A_p[a, b]\). This may also be considered as an alternative characterization of the one variable Sobolev space \(W^k_p[a, b]\).