On Functions of Bounded (p,k)-Variation

Abstract

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 function u:[a,b]→R has bounded (p,k)-variation if and only if u(k-1) is absolutely continuous on [a,b] and u(k) belongs to Lp[a,b]. Moreover, an explicit connection between the (p,k)-variation of u and the Lp-norm of u(k) is given which is parallel to the classical Riesz formula characterizing functions in the spaces RVp[a,b] and Ap[a,b]. This may also be considered as an alternative characterization of the one variable Sobolev space Wpk[a,b].