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

• Main Author: Rafał M Łochowski
• Format: Article
• Language: English
• Published: SpringerOpen 2018-01-01
• Series: Journal of Inequalities and Applications
• Subjects: regulated path; total variation; p-variation; truncated variation; the Riemann-Stieltjes integral; the Loéve-Young inequality
• Online Access: http://link.springer.com/article/10.1186/s13660-018-1611-4

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 ≥ 1 $p\ge1$ , we introduce the space of regulated signals f : [ a , b ] → W $f:[a,b]\rightarrow W$ ( a < b $a< b$ are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy δ > 0 $\delta>0$ by signals whose total variation is of order δ 1 − p $\delta^{1-p}$ as δ → 0 + $\delta\rightarrow0+$ 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.