Riemann-Stieltjes integrals driven by irregular signals in Banach spaces and rate-independent characteristics of their irregularity

Abstract

We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any $p\ge1$ we introduce the space of regulated signals $f:[a,b] \rightarrow W$ ($a 0$ by signals whose total variation is of order $δ^{1-p}$ as $δ\rightarrow 0+$ and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.