On Functions of Bounded -Variation

We introduce and study the concept of (,)-variation (1<<∞, ∈？) of a real function on a compact interval. In particular, we prove that a function ∶[,]→？ has bounded (,)-variation if and only if (？1) is absolutely continuous on [,] and () belongs to [,]. Moreover, an explicit connection between the (,)-variation of and the -norm of () is given which is parallel to the classical Riesz formula characterizing functions in the spaces [,] and [,]. This may also be considered as an alternative characterization of the one variable Sobolev space [,].