A new inequality for the Riemann-Stieltjes integrals driven by irregular signals in Banach spaces

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 $f:[a,b]\rightarrow W$ ( $a< b$ are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy $\delta>0$ by signals whose total variation is of order $\delta^{1-p}$ as $\delta\rightarrow0+$ 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.