On maps of bounded \(p\)-variation with \(p>1\)
Let \(E\) be a non-void set of the real axis and \(X\) a metric space with the metric \(d(.,.)\). Given a map \(f\in X^E\) the symbol \[ \omega(f,E)= \sup \{d(f(t),f(s)) : t,s \in E \} \] denotes the oscillation of \(f\) on \(E\). Further, let \[ {\mathcal T}(E)= \{T=\{t_i\}_{i=0}^m \subset E: m\in {\mathbb N}\cup\{0\},\;t_{i-1}\leq t_i, i=1,2,\dots,m\} \] be the set of all partitions of \(E\) by finite ordered collections of points from \(E\). Given a map \(f:E\to X\) and a partition \(T=\{t_i\}_{i=0}^m \in {\mathcal T}(E)\), the authors consider \[ V_p[f,T]=\sum_{i=1}^{m}(d(f(t_i),f(t_{i-1})))^p \] and define the \textit{total p-variation} of \(f\) on \(E\), by \[ V_p(f,E)=\sup\{ V_p[f,T] : T\in {\mathcal T}(E) \}. \] If \(V_p(f,E)< \infty, \) the map \( f \) is said to be of bounded \(p\)-variation. The above definition of \( V_p(f,E) \) was introduced by \textit{N. Wiener} in 1924 [Massachusetts J. Math. Phys. 3, 72-94 (1924; JFM 50.0203.01)]. Let us recall that a map \( f:E\to X \) is a Hölderian of exponent \(\gamma, 0<\gamma \leq 1, \) if there exists a positive number \( C \) such that \( d(f(t),f(s)) \leq C| t-s| ^{\gamma}, \forall t,s\in E.\) The least number \( C \) satisfying this inequality is called the Hölder constant of \( f \) and is denoted by \( H(f).\) Further let \( p>1.\) One of the main results is the following structural theorem : A map \(f: E\to X \) is of bounded \(p\)-variation if and only if there exist a bounded nondecreasing function \(\phi :E\to {\mathbb R} \) and a Hölderian map \( g:\phi(E)\to X \) of exponent \( \gamma =1/p \) and \( H(g)\leq 1 \) such that \( f=g\circ \phi \) on \( E\). Moreover, if \( X \) is a Banach space, the map \(g: \phi(E)\to X \) can be extended to a Hölderian map \(\overline{g} :{\mathbb R} \to X \) of the same exponent \( \gamma =1/p \) and with the Hölder constant \( H(\overline{g})\leq 3^{1- \gamma }H(g)\).
scientific article; zbMATH DE number 1172139