On functions of \((p, \alpha)\)-bounded variation (Q1029974)

The concept of bounded \(p\)-variation introduced by Riesz in the framework of real-valued functions defined on \([a,b]\) is generalized. The new concept is called \((p,\alpha)\)-bounded variation and the related space is denoted by \(BV_{(p,\alpha)}[a,b]\). The following result is established: 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 \(f'_\alpha\in L_{(p,\alpha)}[a,b]\).