Harnack Extension for Abstract Perron-Stiletjes Integral and Its Application in LSRSS Property

TL;DR

This paper extends the Harnack extension concept for the abstract Perron-Stieltjes integral, exploring its properties and applying it to analyze the LSRSS property within Banach space frameworks.

Contribution

It introduces new continuity definitions and extends the Harnack extension and Cauchy property for the abstract Perron-Stieltjes integral, linking it to the LSRSS property.

Findings

Extended the notion of Harnack extension for the integral.

Established connections between Perron-Stieltjes integral and LSRSS property.

Provided new properties of the integral in Banach spaces.

Abstract

In this work, we give some definitions of continuities with respect to a bilinear triple of Banach spaces which subsequently provide some further properties of the abstract Perron-Stieltjes integral. Moreover, we expand the notion of Harnack extension and Cauchy property for this integral and apply it for investigating a connection of the abstract Perron-Stieltjes integral and LSRSS property in advance introduced.