Functions of bounded \(k\)th \(p\)-variation and continuity modulus
For \(p>1\) and \(0\leq \alpha \leq 1/p\), let us consider the space \(V_{p,\alpha}^{(k)}\), \(k\in \mathbb{N}\) consisting of the 1-periodic functions on the real line whose \((p,\alpha)\) \(k\)-th variation is finite (see [\textit{S. Barza} and \textit{P. Silvestre}, Rev. Mat. Complut. 27, No. 1, 69--91 (2014; Zbl 1396.26015)] for a definition in terms of the supremum of certain partitions of an interval of length \(1\)). The spaces \(V_{p,\alpha}^{(k)}\) form a scale between the space of functions of bounded \(k\)-th variation and the Sobolev space \(W_p^k\). The main result of this paper consists of the study of sharp estimates of the \((p,\alpha)\) \(k\)-th variation of functions in the space in terms of the fractional modulus of smoothness of order \(k-1/p\) and also the study of limits in the scale of spaces \(V_{p,\alpha}^{(k)}\), \(k\in \mathbb{N}\). The work is an extension of some other previous papers such as [\textit{M. Lind}, Math. Inequal. Appl. 16, No. 1, 21--39 (2013; Zbl 1261.26008)] and Barza and P. Silvestre [loc. cit.].
scientific article; zbMATH DE number 6499095