On Functions of Bounded(p,k)-Variation

摘要

We introduce and study the concept of(p,k)-variation (1<p<∞,k∈ℕ)of a real function on a compact interval. In particular, we prove that a functionu:[a,b]→ℝhas bounded(p,k)-variation if and only ifu(k-1)is absolutely continuous on[a,b]andu(k)belongs toLp[a,b]. Moreover, an explicit connection between the(p,k)-variation ofuand theLp-norm ofu(k)is given which is parallel to the classical Riesz formula characterizing functions in the spacesRVp[a,b]andAp[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev spaceWpk[a,b].